Advances in Epidemiological Modeling and Control of Viruses 0323995578, 9780323995573

Table of contents :
Contents
Contributors
Preface
1 Global dynamics of a delayed reaction-diffusion viral infection in a cellular environment
1.1 Introduction
1.2 Global convergence to the homogeneous solution
1.3 Numerical simulations
1.3.1 Example 1
1.3.2 Example 2
1.4 Spatiotemporal pattern formation
1.4.1 Maximum maps
1.4.2 Bifurcations and branches of solutions
1.5 Conclusion and open problems
Acknowledgments
References
2 Hepatitis B virus transmission via epidemic model
2.1 Introduction
2.2 Model formulation
2.3 Stability analysis
2.4 Simulation and concluding remarks
References
3 Global dynamics of an HCV model with full logistic terms and the host immune system
3.1 Introduction
3.2 Previous works
3.2.1 Modelling virus-immune system interaction
3.2.2 Modelling virus infection with full logistic terms and antivirus treatment
3.3 Mathematical preliminaries
3.3.1 Linearization
3.3.2 Lyapunov functions
3.3.3 Bifurcation analysis
3.3.4 Li and Muldowney's geometric approach
3.4 Modelling virus-immune system interaction with full logistic terms in both uninfected and infected cells
3.4.1 Model construction
3.5 Analysis of the model
3.5.1 Dissipativity, basic reproduction number, and equilibria
3.5.1.1 Dissipativity
3.5.1.2 Basic reproduction number
3.5.1.3 Existence of endemic equilibrium
3.5.2 Local and global stability analysis
3.5.2.1 Local stability analysis of the disease-free equilibrium
3.5.2.2 Local stability analysis of the endemic equilibrium
3.5.2.3 Global stability analysis of the disease-free equilibrium
3.5.2.4 Uniform persistence
3.5.2.5 Global stability analysis of the endemic equilibrium
3.6 Numerical simulations
3.7 Conclusion and discussion
Acknowledgments
References
4 On a Novel SVEIRS Markov chain epidemic model with multiple discrete delays and infection rates: modeling and sensitivity analysis to determine vaccination effects
4.1 Introduction
4.2 Description of the SVEIRS epidemic and the delays in the disease dynamics
4.3 Discretization of time and decomposition of the SVEIRS population over time
4.3.1 Decomposition of the total human population over discrete time intervals
4.3.2 One-state-at-a-time decomposition of the population states over the finite delay times
4.3.3 Joint state decomposition of the population over the finite delay times
4.4 The SVEIRS stochastic process
4.4.1 The SVEIRS Markov chain
4.5 Some special SVEIRS epidemic models
4.5.1 The SVEIRS model with correlated vaccination and infection rates
4.5.2 The SVEIRS model with no correlation between vaccination and infection rates
4.6 Numerical study: some prototype SVEIRS epidemic models and sensitivity analysis to determine the effects of infection and vaccination
4.6.1 The general algorithm for the simulations
4.6.2 The prototype SVEIRS model with correlated vaccination and infection rates
4.6.2.1 Sensitivity of the SVEIRS model when p = ϕ
4.6.2.2 Sensitivity analysis of the SVEIRS model when either p or ϕ is fixed and the other parameter continuously changes
4.6.3 The prototype SVEIRS model with uncorrelated vaccination and infection rates
4.7 Conclusion
References
5 Hopf bifurcation in an SIR epidemic model with psychological effect and distributed time delay
5.1 Introduction
5.2 Model
5.3 Direction of bifurcation and stability of periodic solution
5.4 Example: a truncated exponential distribution
5.5 Numerical simulation
5.6 Discussion
Acknowledgments
5.A Matlab code for Fig. 5.8(b)
References
6 Modeling of the effects of media in the course of vaccination of rotavirus
6.1 Introduction
6.2 Epidemic modeling
6.3 Existence of equilibria of system Υ1
6.3.1 Equilibria of system Υ2
6.4 Stability of the equilibria
6.5 Optimal control problem
6.5.1 Existence of optimal control
6.5.2 Characterization of optimal control
6.6 Efficacy analysis
6.7 Numerical simulations
6.8 Discussion
References
7 Mathematical models of early stage Covid-19 transmission in Sri Lanka
7.1 Introduction
7.2 Mathematical model to estimate initial parameters
7.2.1 Analysis of the model
7.2.2 Estimation of initial parameters
7.2.3 Optimization
7.2.4 Numerical results
7.3 Mathematical models with heterogeneity of cases
7.3.1 Analysis of the model
7.3.1.1 Basic reproduction number
7.3.1.2 Stability analysis of the disease-free equilibrium
7.3.2 Introducing optimal control measures
7.3.2.1 Mathematical model with control parameters
7.3.2.2 Mathematical analysis of the model
7.3.3 Numerical results
7.3.3.1 Algorithm for the optimal control problem
7.3.4 Simulation of the COVID 19 dynamic system without control
7.3.5 Simulation of the optimal control problem
7.3.5.1 Scenario 1
7.3.5.2 Scenario 2
7.3.5.3 Scenario 3
7.4 Mathematical model with imported cases
7.4.1 Sensitivity of the control measures (NPIs)
7.4.2 Sensitivity of the control with overseas exposed cases
7.4.3 Sensitivity of the timing of implementing combined control measures
7.5 Conclusion
References
8 Global stability of a diffusive HTLV-I infection model with mitosis and CTL immune response
8.1 Introduction
8.1.1 Mathematical models
8.1.1.1 Model with latent HTLV-infected cells
8.1.1.2 Model with delay
8.1.1.3 Model with mitosis
8.1.1.4 Model with CTL immune response
8.1.1.5 Model with mitosis and CTL immune response
8.1.1.6 Model with diffusion
8.2 Model formulation
8.3 Well-posedness of solutions
8.4 Steady state analysis
8.5 Global stability analysis
8.6 Numerical simulations
8.7 Conclusion and discussion
References
9 Mathematical tools and their applications in dengue epidemic data analytics
9.1 Introduction
9.2 Fourier transformation
9.2.1 Discrete time Fourier transform
9.2.2 Fast Fourier transform
9.2.3 Dengue epidemic data analysis
9.2.4 Study areas
9.2.5 Numerical results and discussion
9.3 Wavelet analysis
9.3.1 Wavelet transform
9.3.2 Basic wavelet functions
9.3.2.1 Haar wavelet
9.3.2.2 Meyer wavelet
9.3.2.3 Morlet wavelet
9.3.3 Wavelet power spectrum
9.3.4 Wavelet coherency and phase difference
9.3.5 Statistical significance
9.3.6 Cone of influence
9.4 Wavelet analysis in epidemiology and dengue
9.4.1 Analysis of dengue incidents in urban Colombo
9.4.2 Effect of climate
9.4.3 Analysis of dengue incidents in urban Colombo
9.4.4 Effect of human mobility: a case study
9.5 Conclusion
References
10 Covid-19 pandemic model: a graph theoretical perspective
10.1 Introduction
10.2 Preliminaries
10.2.1 Graph theory terminology
10.2.2 Epidemiological terminology
10.3 A survey of mathematical models on diseases
10.3.1 Epidemic/pandemic models on Covid-19
10.3.2 Network models
10.4 SEIRD model on Covid-19
10.5 Some results
10.5.1 Effects of R1 on a network
10.6 Conclusion and recommendation
References
11 Towards nonmanifest chaos and order in biological structures by means of the multifractal paradigm
11.1 Introduction
11.2 Mathematical model
11.2.1 Short reminder on the multifractal theory of motion
11.2.2 Stationary nonlinear behaviors through Schrödinger-type ``regimes'' as ``synchronization modes''
11.2.3 Nonstationary nonlinear behaviors through Schrödinger-type ``regimes'' as ``synchronization modes''
11.2.4 Space-time ``synchronization modes'' and nonmanifest scenarios towards chaos
11.3 Conclusions
References
12 Global stability of epidemic models under discontinuous treatment strategy
12.1 Impact of discontinuous treatments on disease dynamics in an SIR epidemic model
12.1.1 Introduction
12.1.2 Model description and preliminaries
12.1.3 Equilibria and their stability
12.1.4 Global convergence in finite time
12.1.5 Conclusion and discussion
12.2 Global stability of an SIS epidemic model with discontinuous treatment strategy
12.2.1 Introduction
12.2.2 Model and preliminaries
12.2.3 Positivity and boundedness
12.2.4 Stability of equilibria
12.2.5 Simulation
12.2.6 Conclusion
12.3 Global stability of an SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment strategy
12.3.1 Introduction
12.3.2 Model and preliminaries
12.3.3 Positivity and boundedness
12.3.4 Stability of equilibrium
12.3.5 Global convergence in finite time
12.3.6 Simulations
References
Index

Citation preview

Advances in Epidemiological Modeling and Control of Viruses

Advances in Epidemiological Modeling and Control of Viruses Edited by Hemen Dutta Department of Mathematics Gauhati University Guwahati, India

Khalid Hattaf Centre Régional des Métiers de l’Education et de la Formation (CRMEF) Casablanca, Morocco

Contents Contributors

xi

Preface

xv

1. Global dynamics of a delayed reaction-diffusion viral infection in a cellular environment

1

Mohammed Nor Frioui and Tarik Mohammed Touaoula 1.1. Introduction

1

1.2. Global convergence to the homogeneous solution

3

1.3. Numerical simulations

18

1.4. Spatiotemporal pattern formation

20

1.5. Conclusion and open problems

25

Acknowledgments

26

References

26

2. Hepatitis B virus transmission via epidemic model

29

Tahir Khan, Roman Ullah, and Gul Zaman 2.1. Introduction

29

2.2. Model formulation

31

2.3. Stability analysis

33

2.4. Simulation and concluding remarks

49

References

53

3. Global dynamics of an HCV model with full logistic terms and the host immune system

55

Jazmín G. Aguilar-Basulto, Eric J. Avila-Vales, and Arturo J. Nic-May v

vi

Contents

3.1. Introduction

55

3.2. Previous works

57

3.3. Mathematical preliminaries

59

3.4. Modelling virus-immune system interaction with full logistic terms in both uninfected and infected cells

63

3.5. Analysis of the model

65

3.6. Numerical simulations

89

3.7. Conclusion and discussion

91

Acknowledgments

92

References

92

4. On a Novel SVEIRS Markov chain epidemic model with multiple discrete delays and infection rates: modeling and sensitivity analysis to determine vaccination effects

95

Divine Wanduku, Omotomilola Jegede, Chinmoy Rahul, Broderick Oluyede, and Oluwaseun Farotimi 4.1. Introduction

95

4.2. Description of the SVEIRS epidemic and the delays in the disease dynamics

98

4.3. Discretization of time and decomposition of the SVEIRS population over time

99

4.4. The SVEIRS stochastic process

110

4.5. Some special SVEIRS epidemic models

115

4.6. Numerical study: some prototype SVEIRS epidemic models and sensitivity analysis to determine the effects of infection and vaccination

122

4.7. Conclusion

141

References

142

5. Hopf bifurcation in an SIR epidemic model with psychological effect and distributed time delay 145 Toshikazu Kuniya

Contents

vii

5.1. Introduction

145

5.2. Model

149

5.3. Direction of bifurcation and stability of periodic solution

152

5.4. Example: a truncated exponential distribution

161

5.5. Numerical simulation

164

5.6. Discussion

165

Acknowledgments

166

Appendix 5.A Matlab code for Fig. 5.8(b)

166

References

167

6. Modeling of the effects of media in the course of vaccination of rotavirus 169 Amar Nath Chatterjee and Fahad Al Basir 6.1. Introduction

169

6.2. Epidemic modeling

171

6.3. Existence of equilibria of system ϒ1

172

6.4. Stability of the equilibria

174

6.5. Optimal control problem

176

6.6. Efficacy analysis

183

6.7. Numerical simulations

183

6.8. Discussion

186

References

187

7. Mathematical models of early stage Covid-19 transmission in Sri Lanka 191 Wickramaarachchillage Pieris Tharindu Mihiruwan Wickramaarachchi and Shyam Sanjeewa Nishantha Perera 7.1. Introduction

191

7.2. Mathematical model to estimate initial parameters

194

7.3. Mathematical models with heterogeneity of cases

203

viii

Contents

7.4. Mathematical model with imported cases

219

7.5. Conclusion

226

References

228

8. Global stability of a diffusive HTLV-I infection model with mitosis and CTL immune response

231

Noura H. AlShamrani and Ahmed M. Elaiw 8.1. Introduction

231

8.2. Model formulation

236

8.3. Well-posedness of solutions

237

8.4. Steady state analysis

239

8.5. Global stability analysis

240

8.6. Numerical simulations

246

8.7. Conclusion and discussion

248

References

250

9. Mathematical tools and their applications in dengue epidemic data analytics

253

Wickramaarachchillage Pieris Tharindu Mihiruwan Wickramaarachchi, Kaluhath Karunathilaka Withanage Hasitha Erandi, and Shyam Sanjeewa Nishantha Perera 9.1. Introduction

253

9.2. Fourier transformation

255

9.3. Wavelet analysis

266

9.4. Wavelet analysis in epidemiology and dengue

275

9.5. Conclusion

280

References

281

10. Covid-19 pandemic model: a graph theoretical perspective

285

Fadekemi Janet Osaye and Alex Alochukwu 10.1. Introduction

285

Contents

ix

10.2. Preliminaries

286

10.3. A survey of mathematical models on diseases

288

10.4. SEIRD model on Covid-19

295

10.5. Some results

298

10.6. Conclusion and recommendation

301

References

302

11. Towards nonmanifest chaos and order in biological structures by means of the multifractal paradigm 305 Maricel Agop and Alina Gavrilu¸t 11.1. Introduction

305

11.2. Mathematical model

306

11.3. Conclusions

319

References

321

12. Global stability of epidemic models under discontinuous treatment strategy

323

Yanjun Zhao, Wenxuan Li, and Yang Wang 12.1. Impact of discontinuous treatments on disease dynamics in an SIR epidemic model

323

12.2. Global stability of an SIS epidemic model with discontinuous treatment strategy 337 12.3. Global stability of an SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment strategy

347

References

362

Index

365

Contributors Maricel Agop Department of Physics, Gheorghe Asachi Technical University from Ia¸si, Ia¸si, Romania Jazmín G. Aguilar-Basulto Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mérida, Yucatán, Mexico Fahad Al Basir Department of Mathematics, Asansol Girls’ College, Asansol, West Bengal, India Alex Alochukwu University of Johannesburg, Johannesburg, South Africa Noura H. AlShamrani Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia Eric J. Avila-Vales Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mérida, Yucatán, Mexico Ahmed M. Elaiw Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia Department of Mathematics, Faculty of Science, Al-Azhar University, Assiut Branch, Assiut, Egypt Kaluhath Karunathilaka Withanage Hasitha Erandi Research and Development Center for Mathematical Modeling, Department of Mathematics, University of Colombo, Colombo, Sri Lanka Oluwaseun Farotimi Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, United States Mohammed Nor Frioui Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Dépt. de Mathématiques, Université Aboubekr Belkaïd, Tlemcen, Algeria Faculty of Mathematics, University of Sciences and Technology Houari Boumedienne, Algiers, Algeria xi

xii

Contributors

Alina Gavrilu¸t Faculty of Mathematics, Alexandru Ioan Cuza University from Ia¸si, Ia¸si, Romania Omotomilola Jegede Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, United States Tahir Khan Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan Department of Computing, Muscat College, Muscat, Oman Toshikazu Kuniya Graduate School of System Informatics, Kobe University, Kobe, Japan Wenxuan Li College of Mathematics, Jilin University, Changchun, China Amar Nath Chatterjee Department of Mathematics, K.L.S. College, Nawada, Magadh University, Nawada, India Arturo J. Nic-May Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mérida, Yucatán, Mexico Broderick Oluyede Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, United States Fadekemi Janet Osaye Alabama State University, Montgomery, AL, United States Shyam Sanjeewa Nishantha Perera Research and Development Center for Mathematical Modeling, Department of Mathematics, University of Colombo, Colombo, Sri Lanka Chinmoy Rahul Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, United States Tarik Mohammed Touaoula Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Dépt. de Mathématiques, Université Aboubekr Belkaïd, Tlemcen, Algeria Roman Ullah Department of Computing, Muscat College, Muscat, Oman

Contributors

xiii

Divine Wanduku Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, United States Yang Wang College of Mathematics, Jilin Normal University, Siping, China Wickramaarachchillage Pieris Tharindu Mihiruwan Wickramaarachchi Department of Mathematics, The Open University of Sri Lanka, Nawala, Nugegoda, Sri Lanka Gul Zaman Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan Yanjun Zhao International Business School, Jilin International Studies University, Changchun, China

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2023 Elsevier Inc. All rights reserved. MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. ISBN: 978-0-323-99557-3 For information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals Publisher: Mara E. Conner Acquisitions Editor: Chris Katsaropoulos Editorial Project Manager: Tom Mearns Production Project Manager: Sreejith Viswanathan Cover Designer: Mark Rogers Typeset by VTeX

Preface The book covers various aspects of epidemiological modeling, with a focus on virus control strategies. Readers can expect new and significant research information about the dynamics of various diseases under various effects and environments. The book should be a valuable resource for graduate students, researchers, and educators interested in epidemiological modeling, including data-driven models. The chapters are organized as follows. Chapter “Global dynamics of a delayed reaction-diffusion viral infection in a cellular environment” discusses a reaction-diffusion equation with a general non-linear production term and a delayed inhibition term. Some methods for proving global convergence of solutions to the positive homogeneous in space equilibrium under certain conditions are presented. When these conditions are not met and the solution becomes unstable, numerical simulations have been used to study complex nonlinear dynamics and pattern formation. Chapter “Hepatitis B virus transmission via epidemic model with treatment function” investigates an epidemiological mathematical model that demonstrates the dynamics of hepatitis B virus transmission under the influence of generalized incidence and treatment function. It is also emphasized to describe the transfer mechanism, i.e., preventive vaccination of susceptible populations; and various control mechanisms based on the size of the infective population, in which the control measure treatment can determine whether or not there is an epidemic outbreak, as well as the number of endemic equilibrium during endemic outbreaks. Numerical simulations were also performed to graphically represent the analytical findings. Chapter “Global dynamics of an HCV model with full logistic terms and the host immune system” studies a mathematical model that describes the dynamics of the hepatitis C virus (HCV) while taking four populations into account: uninfected liver cells, infected liver cells, HCV, and T cells. It establishes the existence of two equilibrium states: the uninfected state and the endemically infected state, as well as the positivity and boundedness of the system solutions. It employs a geometrical approach to investigate the global stability of positive equilibrium. It also includes some numerical simulations to justify the analytical findings. Chapter “On a Novel SVEIRS Markov chain epidemic model with multiple discrete delays and infection rates: modeling and sensitivity analysis to determine vaccination effects” investigates a novel discrete time general Markov chain SEIRS epidemic model with vaccination. The model includes finite delay times for disease incubation, natural and artificial immunity periods, and infected individuals’ infectious period. The novel platform for representing the various disease states in the population employs two discrete time measures for the current time of a person’s state, as well as how long a person has been xv

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Preface

in the current state. There are two sub-models based on whether the motivation to get vaccinated is motivated by close contact with infectious individuals or not. To determine how vaccination affects disease eradication, sensitivity analysis is performed on the two sub-models. Chapter “Hopf bifurcation in an SIR epidemic model with psychological effect and distributed time delay” aims to replicate the recurrent epidemic wave of COVID-19 by investigating an SIR epidemic model with psychological effect and distributed time delay. It obtains the index values that determine the Hopf bifurcation characteristics for general distribution functions using the theory of center manifold and Poincaré normal form. As an example, it considers the truncated exponential distribution and demonstrates how the periodic solution can emerge via the Hopf bifurcation. Chapter “Modeling of the effects of media in the course of rotavirus vaccination” investigates the influence of media cognizance and its impact at various points along the path of rotavirus vaccination. It investigates a model of the development of rotavirus diarrhoea based on time-based ordinary differential equations. It used numerical simulations to investigate the impact of vaccination as a preventive measure. Chapter “Mathematical models of early stage COVID-19 transmission in Sri Lanka” discusses several mathematical models developed to investigate the COVID-19 transmission dynamic in Sri Lanka. An SIER compartmental model is used, and the optimal initial parameters of the early stage of the outbreak are estimated by calibrating reported cases using an optimization algorithm. Scenario-based control measures are also introduced into the model at the parameter level, and their impact is evaluated using numerical simulations. Then, an optimal control model incorporating a SIER type of model is developed, taking into account the heterogeneity of cases such as asymptomatic, symptomatic with mild indications, and cases requiring intensive care treatments. All of the measures and interventions are being implemented at a significant social and economic cost; therefore, optimal control techniques have been used to identify the most appropriate strategies to reduce this cost. Chapter “Global stability of a diffusive HTLV-I infection model with mitosis and CTL immune response” proposes a spatially dependent HTLV-I infection model. The model describes the interactions between uninfected CD4+ T cells, latent HTLV infected cells, Tax-expressing HTLV-infected cells, and HTLV-specific CTLs within the host. The solutions’ well-posedness, including the existence of global solutions and boundedness, is justified. It determines the existence and stability of the model’s three steady states by calculating two threshold parameters: the basic infection reproduction number and the HTLV-specific CTL mediated immunity reproduction number. It also investigates the global stability of all steady states using suitable Lyapunov functions and the LyapunovLaSalle asymptotic stability theorem. The validity of theoretical results is justified using numerical simulations. Chapter “Mathematical tools and their applications in dengue epidemic data analytics” aims to introduce mathematical tools that are powerful in analyzing time-dependent disease transmission data in the context of external forces such as climate variability. It

Preface

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considers the Fast Fourier Transformation (FFT) method, which converts time or space domain data series to frequency domain data series by decomposing a sequence of data into components of different frequencies. The wavelet theory is then applied to noisy epidemiological time series data. The FFT analysis makes use of reported dengue and climate data. Certain dengue weekly incident data are used for the wavelet analysis. Certain temperature and rainfall data were obtained in order to investigate the patterns of dengue transmission in relation to climate. FFT and wavelet analysis are both carried out using structure computer algorithms to investigate periodic patterns of dengue incidents in relation to external variables and to obtain spectral properties of time series data. Chapter “COVID-19 pandemic model: a graph theoretical perspective” emphasises that the COVID-19 virus spread (pandemic) pattern can be analyzed using graph theory. Each vertex in the network represents an individual at any stage of infection (asymptomatic, pre-symptomatic, or symptomatic), and edges represent transmissions from person to person. It looks into the spread of COVID-19 among those who are susceptible, exposed, infected, recovered, and dead. It takes into account each individual’s neighborhood prevalence, i.e., the proportion of each individual’s contacts who are either exposed or infected, and introduces certain parameters. Furthermore, it proposes a threshold value and describes the effects of this value on pandemic spread. Chapter “Towards nonmanifest chaos and order in biological structures using the multifractal paradigm” investigates nonlinear behaviors of biological structures (viruses systems) in Schrödinger type regimes at various scale resolutions in a multifractal motion paradigm. The functionality of a hidden symmetry of SL(2R) type then implies, via a Riccati type gauge, different synchronization modes among these virus systems in the stationary case of these regimes. Furthermore, assuming that nonmanifest chaos is not present, specific patterns corresponding to virus system dynamics can be highlighted. In such a framework, using artificial intelligence methods, it is demonstrated that, based on the dynamics of certain patterns, changes in the acoustic field can constitute a method of COVID-19 detection. Chapter “Global stability of epidemic models under discontinuous treatment strategy” takes a look at the SIR, SIS, and SEIR epidemic models under discontinuous treatment. In each compact interval, the treatment rate has a finite number of jump discontinities. The basic reproductive number is demonstrated to be a sharp threshold value that completely determines the dynamics of the model using Lyapunov theory for discontinuous differential equations and other techniques on non-smooth analysis. It discusses how the disease will die out in a finite amount of time, which the corresponding model with continuous treatment cannot do. Furthermore, the numerical simulations show that increasing treatment after infective individuals reach a certain level is beneficial to disease control. We are grateful to the contributors for their cooperation throughout the book’s development process. The reviewers deserve a much appreciation for their voluntary service in making this book a success. We would like to express their gratitude to their family members, friends, and well-wishers for their ongoing support in developing such books.

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Elsevier editors and staff also deserve our appreciation for their timely assistance and support.

Hemen Dutta Guwahati, India Khalid Hattaf Casablanca, Morocco 10 August, 2022

1 Global dynamics of a delayed reaction-diffusion viral infection in a cellular environment Mohammed Nor Friouia,b and Tarik Mohammed Touaoulaa a Laboratoire d’Analyse Nonlinéaire et Mathématiques Appliquées, Dépt. de Mathématiques, Université Aboubekr Belkaïd, Tlemcen, Algeria b Faculty of Mathematics, University of Sciences and Technology Houari

Boumedienne, Algiers, Algeria

1.1 Introduction Infections are usually initiated by a few virions that are individual viral particles. They enter healthy cells and modify the genetic structure of their hosts. Then they replicate and produce a swarm of progeny viruses. During the process of viral infection, the immune response is critical in controlling the transmission of disease. The interaction of the immune cells with a target population of viruses must be considered as a dynamic process. The mathematical modeling of such a process plays an important role in understanding the population dynamics of viruses (see, e.g., the recent papers [1], [2], [3], [7] [8], [9] and the references therein). Another interesting aspect of the dynamics of the spread of a viral infection is related to the spatial structure of the system and the effects of its processes, such as random dispersal of virions. It was showed, in many works, that the spatial structure of the system may influence the dynamics of the population under consideration. Mathematical models have been extensively used to study the dynamics of viral infections, taking into account the immune cells mostly under the simplifying assumptions of spatial homogeneity, with a few models considering the spatial spread of viruses in infected hosts. When a spatial process is taken into account, reaction-diffusion equations represent an appropriate framework to describe the evolution of the dynamic of viruses and immune cells. Many reaction-diffusion equations are formulated to investigate the roles of diffusion on the transmission of disease. Motivated by the description made, in this chapter we aim to study the following reaction-diffusion equation with delay:     ut (x, t) − Duxx (x, t) = g u(x, t) − u(x, t)f u(x, t − τ ) . Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00006-5 Copyright © 2023 Elsevier Inc. All rights reserved.

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Advances in Epidemiological Modeling and Control of Viruses

The first term in the right-hand side of Eq. (1.1) describes virus diffusion, the second one represents virus production, and the last term models virus elimination by immune cells. The parameter D is the diffusion coefficient. The function f (u(x, t −τ )) stands for the number of immune cells generated after some time-delay τ ; this means that the production of immune cells depends on the concentration of viruses some time before. In our context, we consider the concentration of immune cells as a function of the virus concentration a τ time ago; that is, c(x, t) = f (u(x, t − τ )), where c(x, t) is the concentration of the immune cells at space x and at time t. Eq. (1.1) was suggested in [8], [9] for the particular case g(u) = u(1 − u) as a model of viral infection spreading in tissues, but not only; it can also be considered as a generic model of population growth with delayed inhibition (see also [7], [8], [9]). This simple equation in appearance can have solutions with very complex behavior: even when D = 0, chaos can appear as established in [6]. When D = 0, under distributed delay, Eq. (1.1) can have many very known forms [4], [5], [12], [15], [17], [19], [34]. In [5] a unimodal function f (function having exactly one maximum) is considered for the study of   t    u (t) = r(t) f u(s) ds R(t, s) − u(t) . (1.2) h(t)

The authors succeed in proving the global convergence of solution to the unique positive equilibrium of (1.2). In [34] the authors consider the equation   u (t) = −αu(t) + f1 u(t) +



τ1

  ˆ k(s)f 2 u(s) ds,

(1.3)

τ0

and they obtain a complete analysis of the global behavior of the solutions. In addition to global asymptotic stability of the unique positive equilibrium, the author in [19] obtains an exponential stability result for the equation   u (t) = −f u(t) + 



τ

  h(a)g u(t − a) da.

(1.4)

0

When D = 0, so for a delay reaction-diffusion, there is a huge interest, as this kind of equations can appear in models describing predator-prey interaction, population dynamics, and more precisely in very well-known Nicholson’s blowflies and Mackey-Glass models. Some solutions of such problems can take the form of waves connecting different equilibria. There is a vast literature devoted to delay reaction-diffusion equations (see [6], [10], [20], [21], [23], [24], [25], [27], [28], [29], [30] [31], [32], [35] and the references therein). Most of them consider models in population dynamics. Let us mention the work [30] where the

Chapter 1 • Global dynamics of a delayed reaction-diffusion

authors studied the following delay reaction-diffusion equation:   ut (x, t) − u(x, t) = −u(x, t) + f u(x, t − τ )

3

(1.5)

with Neumann boundary conditions and unimodal function f . It is proved that the convergence to the unique positive steady state of (1.5) is strongly linked to the convergence of the sequence un determined by the difference equation un+1 = f (un ). We also mention the work [33] where the authors investigated a nonlocal reaction-diffusion equation in a semiinfinite interval and obtained the global attractivity of the nontrivial equilibrium. Investigation of delay reaction-diffusion equations is often “personalized” in the sense that the methods of their analysis should be adapted to the particular form of the equation. In this work, we develop some methods to prove global convergence of solutions to the positive homogeneous in space equilibrium under certain conditions on the function f (Section 1.2). In the case where these conditions are not satisfied and this solution becomes unstable, we use numerical simulations to study complex nonlinear dynamics and pattern formation (Sections 1.3 and 1.4).

1.2 Global convergence to the homogeneous solution In this section, we study the following class of functional differential equations: ⎧ ⎨ ut (x, t) − uxx (x, t) = g(u(x, t)) − u(x, t)f (u(x, t − τ )), (x, t) ∈ (0, 1) × (0, T ), u(0, t) = u(1, t), ux (0, t) = ux (1, t), t ∈ (0, T ], ⎩ u(x, t) = φ(x, t), (x, t) ∈ [0, 1] × [−τ, 0].

(1.6)

Throughout this chapter, we will make the following assumptions: (T1) f and g are C 1 (R) with g is a bounded function on R and f (0) > 0. (T2) There exists B > 0 such that g(0) = 0, g  (0) > 0 and g(s) < 0 for all s > B. Let C = C([0, 1], R) and X = C([0, 1] × [−τ, 0], R) be equipped with the usual supremum norm ||.||. Also, let C+ = C([0, 1], R+ ) and X+ = C([0, 1] × [−τ, 0], R+ ). For any φ, ψ ∈ X, we write φ ≥ ψ if φ − ψ ∈ X+ ; φ > ψ if φ − ψ ∈ X+ \ {0}. A function v is Hölder continuous with exponent α ∈ (0, 1) on [0, 1] × [−τ, T ] if there exists a constant L > 0 such that   |v(x1 , t1 ) − v(x2 , t2 )| ≤ L |x1 − x2 |α + |t1 − t2 |α for x1 , x2 ∈ [0, 1] and t1 , t2 ∈ [−τ, T ]. We write in this case v ∈ C α ([0, 1] × [−τ, T ]). We define the ordered intervals [φ, ψ]X := {ξ ∈ X; φ ≤ ξ ≤ ψ}, and for any χ ∈ R, we write χ ∗ for the element of X satisfying χ ∗ (x, θ ) = χ for all (x, θ ) ∈ [0, 1] × [−τ, 0]. The segment ut ∈ X of a solution is defined by the relation ut (x, θ ) = u(x, t +

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θ ) for x ∈ [0, 1] and θ ∈ [−τ, 0]. The family of maps U : R+ × X+ → X+ , such that (t, φ) → ut (φ), defines a continuous semiflow on X+ [26]. The map U (t, .) is defined from X+ to X+ , which is the semiflow Ut denoted by Ut (φ) = U (t, φ). The set of equilibria of the semiflow generated by (1.6) is given by 

E = χ ∗ ∈ X+ ; χ ∈ R and g(χ) − χf (χ) = 0 . Let T (t) (t ≥ 0) be a strongly continuous semigroup of bounded linear operators on C generated by the Laplace operator  under periodic conditions. It is well known that T (t) (t ≥ 0) is an analytic, compact, and strongly positive semigroup (see example 1.13, page 26 in [26]). Define F : X → C by     F (φ)(x) = g φ(x, 0) − φ(x, 0)f φ(x, −τ ) for all x ∈ [0, 1]. (1.7) We consider the following integral equation with the given initial data: ⎧  t ⎨ u(t) = T (t)φ(., 0) + T (t − s)F (us )ds, t > 0, 0 ⎩ u0 = φ ∈ X.

(1.8)

For each φ ∈ X, u(., t) with values in C on its maximum interval [0, σφ ) is called a mild solution of (1.6) (for the existence and uniqueness of this solution, see for instance [11], [13], [14], [26]), and it is called classical if it is in C 2 with respect to x and in C 1 with respect to t. Definition 1.2.1. We call a pair of smooth functions (u, ˆ u) ˜ sub- and supersolutions of (1.6) if it satisfies the following properties: i) (u(x, ˆ t), u(x, ˜ t)) ∈ (C α ([0, 1] × [−τ, T ]))2 ∩ (C 1,2 ((0, 1) × (0, T )))2 for all (x, t) ∈ [0, 1] × [−τ, T ]. Denote by C 1,2 (0, 1) × (0, T ) the set of all functions that are once continuously differentiable in t and twice continuously differentiable in x for all (x, t) ∈ (0, 1) × (0, T ). ii) u(x, ˆ t) ≤ u(x, ˜ t) for (x, t) ∈ [0, 1] × [−τ, T ]. iii) The functions (u, ˆ u) ˜ verify the following problems: ⎧ ˜ t)) − u(x, ˜ t)f (u(x, ˆ t − τ )), (x, t) ∈ (0, 1) × (0, T ), ⎨ u˜ t (x, t) − u˜ xx (x, t) ≥ g(u(x, u(0, ˜ t) = u(1, ˜ t), u˜ x (0, t) = u˜ x (1, t), t ∈ (0, T ], ⎩ u(x, ˜ t) ≥ φ(x, t), (x, t) ∈ [0, 1] × [−τ, 0], (1.9)

Chapter 1 • Global dynamics of a delayed reaction-diffusion

5

and ⎧ ˆ t)) − u(x, ˆ t)f (u(x, ˜ t − τ )), ⎨ uˆ t (x, t) − uˆ xx (x, t) ≤ g(u(x, u(0, ˆ t) = u(1, ˆ t), uˆ x (0, t) = uˆ x (1, t), t ∈ (0, T ], ⎩ u(x, ˆ t) ≤ φ(x, t), (x, t) ∈ [0, 1] × [−τ, 0].

(x, t) ∈ (0, 1) × (0, T ),

(1.10) Lemma 1.2.2. Suppose that a smooth function w satisfies the following problem: ⎧ ⎨ wt (x, t) − wxx (x, t) + c(x, t)w ≥ 0, (x, t) ∈ (0, 1) × (0, T ), w(0, t) = w(1, t), wx (0, t) = wx (1, t), t ∈ (0, T ], ⎩ w(x, t) = φ(x, t) ≥ 0, (x, t) ∈ [0, 1] × [−τ, 0],

(1.11)

for some bounded function c(x, t). Then w ≥ 0 in [0, 1] × [0, T ]. Moreover, w(x, t) > 0 in [0, 1] × (0, T ] provided φ(., 0) is not identically null. The proof of this result is well known (see for instance [18]). Set Lc u = ut − uxx + cu and the following problems: ⎧ Lc u¯ k (x, t) = cu¯ k−1 (x, t) + g(u¯ k−1 (x, t)) − u¯ k−1 (x, t)f (uk−1 (x, t − τ )), (x, t) ∈ (0, 1) × (0, T ), ⎪ ⎪ ⎨ k u¯ (0, t) = u¯ k (1, t), u¯ kx (0, t) = u¯ kx (1, t), t ∈ (0, T ] ⎪ u¯ k (x, t) = φ(x, t), (x, t) ∈ [0, 1] × [−τ, 0], ⎪ ⎩ 0 ˆ t), u(x, ˜ t)), (x, t) ∈ [0, 1] × [−τ, T ], (u (x, t), u¯ 0 (x, t)) = (u(x, (1.12) and ⎧ Lc uk (x, t) = cuk−1 (x, t) + g(uk−1 (x, t)) − uk−1 (x, t)f (u¯ k−1 (x, t − τ )), (x, t) ∈ (0, 1) × (0, T ), ⎪ ⎪ ⎨ k u (0, t) = uk (1, t), ukx (0, t) = ukx (1, t), t ∈ (0, T ] ⎪ uk (x, t) = φ(x, t), (x, t) ∈ [0, 1] × [−τ, 0], ⎪ ⎩ 0 ˆ t), u(x, ˜ t)), (x, t) ∈ [0, 1] × [−τ, T ]. (u (x, t), u¯ 0 (x, t)) = (u(x, (1.13) Lemma 1.2.3. Suppose that f is nondecreasing and a smooth pair (u, ˆ u) ˜ exists. Then we have the following inequalities: ˜ uˆ = u0 ≤ u1 ≤ ... ≤ uk ≤ u¯ k ≤ u¯ k−1 ≤ ... ≤ u¯ 1 ≤ u¯ 0 = u. Proof. We set w = u˜ − u¯ 1 . Then w satisfies the following problem: ⎧ ⎨ L0 w(x, t) ≥ 0, (x, t) ∈ (0, 1) × (0, T ), w(0, t) = w(1, t), wx (0, t) = wx (1, t), t ∈ (0, T ], ⎩ w(x, t) ≥ 0, (x, t) ∈ [0, 1] × [−τ, 0].

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From Lemma 1.2.2, we get w ≥ 0. Similarly, we can show that uˆ ≤ u1 . Next, we set w = u¯ 1 − u1 . Then, we have ⎧ ˆ + g(u(x, ˜ t)) − g(u(x, ˆ t)) + u(x, ˆ t)f (u(x, ˜ t − τ )) − u(x, ˜ t)f (u(x, ˆ t − τ )), ⎨ Lc w(x, t) = c(u˜ − u) w(0, t) = w(1, t), wx (0, t) = wx (1, t), t ∈ (0, T ], ⎩ w(x, t) ≥ 0, (x, t) ∈ [0, 1] × [−τ, 0]. Hence, ⎧ Lc w = (c + g  (θ (x, t)))(u˜ − u) ˆ − u(x, ˜ t)(f (u(x, ˆ t − τ )) − f (u(x, ˜ t − τ )))+ ⎪ ⎪ ⎨ f (u(x, ˜ t − τ ))(u(x, ˆ t) − u(x, ˜ t)), ⎪ w(0, t) = w(1, t), wx (0, t) = wx (1, t), t ∈ (0, T ], ⎪ ⎩ w(x, t) ≥ 0, (x, t) ∈ [0, 1] × [−τ, 0], where θ is a function that lies between uˆ and u. ˜ Since f is a nondecreasing function and u˜ ≥ u, ˆ we get ⎧  ˜ t − τ )))(u˜ − u), ˆ ⎨ wt (x, t) − wxx (x, t) ≥ (c + g (θ (x, t) − f (u(x, w(0, t) = w(1, t), wx (0, t) = wx (1, t), t ∈ (0, T ], ⎩ w(x, t) ≥ 0, (x, t) ∈ [0, 1] × [−τ, 0]. Choosing c ≥ f (u(x, ˜ t − τ )) − g  (θ (x, t)) and using Lemma 1.2.2, we obtain w ≥ 0. The assertion of the lemma can now be obtained by induction. Theorem 1.2.4. Suppose that f is a nondecreasing function, and let u, ˜ uˆ be coupled superand subsolutions of problem (1.6). Then the sequences {u¯ k }, {uk } given by (1.12)–(1.13) converge monotonically to a unique solution u of (1.8) and uˆ ≤ u ≤ u˜ in ET = [0, 1] × [−τ, T ]. Proof. From Lemma 1.2.3 we conclude that the sequences {u¯ k }, {uk } converge to some limits u¯ and u, respectively, and uˆ ≤ u ≤ u¯ ≤ u. ˜ By a classical regularity theorem for parabolic equation (see for instance [16]), these limits satisfy the equations

¯ t) = g(u(x, ¯ t)) − u(x, ¯ t)f (u(x, t − τ )), L0 u(x, (1.14) ¯ t − τ )), L0 u(x, t) = g(u(x, t)) − u(x, t)f (u(x, and the boundary and initial conditions in (1.6). Hence the limits of {u¯ k }, {uk } are solutions of (1.6) if u¯ = u in D¯ T := [0, 1] × [0, T ]. We begin by choosing t ∈ [0, τ ]. Subtracting the equations in (1.14) and using the mean value theorem, we see that the function w = u¯ − u satisfies the relation

L0 w(x, t) = (g  (θ (x, t)) − f (φ(x, t − τ )))w, w(x, t) = 0, (x, t) ∈ [0, 1] × [−τ, 0], and the same boundary conditions as in (1.6). The function v(x, t) = e−ct¯ w(x, t) satisfies

Lc v(x, t) = 0, v(x, t) = 0, (x, t) ∈ [0, 1] × [−τ, 0],

Chapter 1 • Global dynamics of a delayed reaction-diffusion

7

and the same boundary conditions as in (1.6) with      c(x, t) = c¯ − g  θ (x, t) − f φ(x, t − τ ) > 0 (after a suitable choice of c). ¯ Next, we multiply the above equation by v and integrate over (0, 1). We obtain 1 d 2 dt



1

 v 2 (x, t)dx +

0

1



1

|vx (x, t)|2 dx +

0

c(x, t)|v(x, t)|2 dx = 0.

0

Thus, in view of the equality v(x, 0) = 0, we conclude that 

1

v 2 (x, t)dx = 0.

0

Therefore, by the continuity of w, we get w = 0 for all (x, t) ∈ [0, 1] × [0, τ ]. We iterate on [0, 1] × [τ, 2τ ],..., [0, 1] × [kτ, (k + 1)τ ], and we show the existence of a solution. We proceed similarly to prove the uniqueness of the solution u ∈ [u, ˆ u]. ˜ Lemma 1.2.5. Assume (T1)–(T2) hold. Then, for φ ∈ X+ , problem (1.8) admits a unique solution u. In addition, we have the following results: (i) There exists C > 0 such that u(x, t) ≤ C for all x ∈ [0, 1] and t > 0. Moreover, lim sup u(x, t) t→∞

≤ B for all x ∈ [0, 1] with B defined in (T2). (ii) ut ∈ X+ for all t ∈ [0, ∞). Furthermore, if φ(., 0) is not identically zero in [0, 1], then ut ∈ I nt (X+ ) for all t > 0, where I nt (X) is the interior of the set X. (iii) The semiflow Ut admits a compact attractor and u(x, t) is a classical solution of (1.6) for x ∈ [0, 1] and t > τ . Proof. We first prove that each solution of (1.8) is bounded. Indeed, the function u solution of (1.6) is a subsolution of the following problem:   v (t) = g(v(t)), t > 0, (1.15) φ(x, t). v(0) = max (x,t)∈[0,1]×[−τ,0]

The function v satisfies 

Lc v(t) = cv(t) + g(v(t)), t > 0, φ(x, t), v(0) = max

(1.16)

(x,t)∈[0,1]×[−τ,0]

with c chosen such that cs + g(s) is a nondecreasing function over R. Hence, by the comparison principle (see Theorem 8.1.10 in [26]), we have u(x, t) ≤ v(t) for all x ∈ [0, 1] and

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t > 0. Next, C := max{B, v(0)} (B being defined in (T2)) is a supersolution of (1.16), then v(t) ≤ C for all t ≥ 0. Therefore u(x, t) ≤ C for all x ∈ [0, 1] and t > 0. We claim now that lim sup v(t) ≤ B. Indeed, suppose that the contrary holds, then lim sup v(t) = l > B. From the t→∞

t→∞

fluctuation method (see for instance Proposition A.22 in [22]), there exists a sequence (tn ), tn → ∞ as n → ∞ such that v(tn ) → l as n → ∞ and v  (tn ) → 0. Substituting tn in (1.15) and passing to the limit, we get g(l) = 0, which is a contradiction to (T2). The claim is proved. This completes the proof of statement (i). Now, note that φ(0) + hF (φ) ∈ C+ for all φ ∈ X+ and for all small h > 0. This implies that lim dist (φ(0) + hF (φ), C+ ) = 0. Furthermore, T (t) h→0+

maps X+ into X+ . By Theorem 8.3.1 and Remark 8.3.2 in [26], we know that ut ∈ X+ . Moreover, for each T > 0, problem (1.6) may be rewritten as ⎧ ⎨ ut (x, t) − uxx (x, t) + c(x, t)u(x, t) = 0, t ∈ (x, t) ∈ (0, 1) × (0, T ), u(0, t) = u(1, t), ux (0, t) = ux (1, t), t ∈ (0, T ], ⎩ u(x, 0) = φ(x, 0), x ∈ [0, 1], with c(x, t) = g  (θ (x, t)) − f (u(x, t − τ )), where θ is a function lying between 0 and u. Since φ is nonnegative and not identically null, in view of Lemma 1.2.2, we have u(x, t) > 0 for all x ∈ [0, 1] and t > 0, (ii) is proved. Finally, statement (iii) follows from (i), (ii), and Theorem 2.2.6 in [26]. We consider the following problem:

−U  (x) = G(U (x)), x ∈ (0, 1), U (0) = U (1), U  (0) = U  (1).

(1.17)

Suppose that G satisfies the one-sided Lipschitz condition G(u1 ) − G(u2 ) ≥ −c(u1 − u2 ) for u ≤ u2 ≤ u1 ≤ u, ¯

(1.18)

where c is a nonnegative function and (u, u) ¯ is any pair of ordered lower and upper solutions of (1.17). Lemma 1.2.6. Let u(x), u(x) ¯ be ordered bounded positive sub- and supersolutions of (1.17), G(u) respectively, and suppose that (1.18) holds and that is either a decreasing or increasing u ¯ Then problem (1.17) admits a unique positive solution u ≤ u ≤ u. ¯ function for u ≤ u ≤ u. Proof. The existence of the minimal solution u1 and the maximal solution u2 of problem (1.17) such that u ≤ u1 ≤ u2 ≤ u¯ can be deduced by Theorem 3.2.2 in [16]. (The same proof can be applied when Neumann conditions are replaced by periodic ones.) Now we claim that u1 = u2 . Indeed, we have −u2 (x)u1 (x) = u2 (x)G(u1 (x))

Chapter 1 • Global dynamics of a delayed reaction-diffusion

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and −u1 (x)u2 (x) = u1 (x)G(u2 (x)). Subtracting these two equations and integrating the result over (0, 1), we get 

1 0

(u1 (x)u2 (x) − u2 (x)u1 (x))dx

 =

1

(u2 (x)G(u1 (x)) − u1 (x)G(u2 (x)))dx.

0

By integration by parts and from the periodic conditions, we obtain 

1

(u2 (x)G(u1 (x)) − u1 (x)G(u2 (x)))dx = 0;

0

so,



1

u1 (x)u2 (x)( 0

G(u1 (x)) G(u2 (x)) − )dx = 0. u1 (x) u2 (x)

It follows from u1 u2 > 0 and the monotone property of G(u1 ) G(u2 ) = u1 u2

G(u) that u

in (0, 1),

and thus u1 = u2 . Let U (x) be a solution of the stationary problem associated to (1.6); namely,

−U  (x) = g(U (x)) − U (x)f (U (x)), U (0) = U (1), U  (0) = U  (1).

x ∈ (0, 1),

(1.19)

The following result is a direct consequence of Lemma 1.2.6. Lemma 1.2.7. Suppose that f is an increasing function and Then problem (1.19) admits at most one positive solution.

g(u) is a decreasing function. u

In the subsequent part of this section, we suppose that there exists 0 < u∗ < B such that

uf (u) < g(u) for all 0 < u < u∗ uf (u) > g(u) for all u∗ < u ≤ B.

(1.20)

Corollary 1.2.8. Under the hypotheses of Lemma 1.2.7 and (1.20), problem (1.19) admits a unique positive solution. Next, we study the persistence of solutions of problem (1.6). First observe, from (T2), that there exists s ∗ ∈ (0, B] such that g(s ∗ ) = 0.

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Definition 1.2.9. We say that u solution of (1.6) is strongly persistent in X+ \ {0} if there exists ε > 0 such that lim inf u(x, t) ≥ ε t→∞

for all x ∈ [0, 1] and for all φ ∈ X+ \ {0}. Lemma 1.2.10. Suppose that f is an increasing function and either g(s) < 0 for all s > s ∗ or g(s) is a nonincreasing function. Assume also that f (s ∗ ) < g  (0), then the solution of (1.6) is s strongly persistent provided φ ∈ X+ \ {0}. g(ε) . First, suppose that ε ∗ ∗ φ ∈ [ε, s ]X . Then, (ε, s ) is a couple of sub- and supersolutions of (1.6). Indeed, it is sufficient to note that relations (1.9)–(1.10) are verified for the couple (u, ˆ u) ˜ = (ε, s ∗ ). For the general case, where without loss of generality φ(x, 0) is nonidentically zero, we conclude from Lemma 1.2.5 that u(x, t) > 0 for all (x, t) ∈ [0, 1] × (0, ∞). Then u(x, t) > ε for all (x, t) ∈ [0, 1] × [τ, 2τ ]. Hence, if we consider u in [0, 1] × [τ, 2τ ] as the initial condition, then, as before, this last implies that u(x, t) ≥ ε for all (x, t) ∈ [0, 1] × (2τ, ∞). To finish the proof, we will show that each positive solution of problem (1.6) enters the interval [0, s ∗ ]X and stays there. For φ ∈ X+ \ {0}, we introduce the following problem:   v (t) = g(v(t)) − v(t)f (0), φ(x, s). v(0) = max Proof. Since f (s ∗ ) < g  (0), there exists ε > 0 such that f (s ∗ ) ≤

(x,s)∈[0,1]×[−τ,0]

Observe that (0, v(t)) is a pair of sub- and supersolutions of problem (1.6). Therefore we get 0 ≤ u(x, t) ≤ v(t) for all (x, t) ∈ [0, 1] × (0, ∞).

(1.21)

Now, we claim that there exists T > 0 such that v(t) ≤ s ∗ for all t ≥ T . Indeed, let v be the solution of   v (t) = g(v (t)) − v (t)(f (0) + ), (1.22) max φ(x, s). v (0) = (x,s)∈[0,1]×[−τ,0]

It is easy to show that v converges to v as tends to zero (see also Theorem 5.1.1 in [18]). Now, we prove that v (t) ≤ s ∗ for all t ≥ T . First, suppose that v (t) ≥ s ∗ for all t ≥ 0. Then, g(s) is a nonincreasing function, we have from (1.22) and the fact that s   g(v (t))  v (t) = v (t) − f (0) − , v (t)   g(s ∗ ) (1.23) ≤ v (t) − f (0) − , s∗   ≤ −v (t) f (0) + .

Chapter 1 • Global dynamics of a delayed reaction-diffusion

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Obviously, the last inequation of (1.23) also holds if g(s) < 0 for all s > s ∗ . Then (1.23) implies that lim sup v (t) = 0, which is a contradiction to v (t) ≥ s ∗ for all t ≥ 0. Therefore there t→∞

exists t0 ≥ 0 such that v (t0 ) < s ∗ . Next, suppose that there exists t1 > t0 such that v  (t1 ) ≥ 0 and v (t1 ) = s ∗ . Substituting v (t1 ) in (1.22),   0 ≤ −s ∗ f (0) + ε < 0. This contradiction proves the assertion. By passing to the limit in , we can also show that v(t) ≤ s ∗ for all t ≥ T . Consequently, this with (1.21) implies that there exists T > 0 such that u(x, t) ≤ s ∗ for all t ≥ T and all x ∈ [0, 1]. The lemma is proved. The following lemma considers the case where f (s ∗ ) ≥ g  (0). In view of (1.20), f (0) ≤ ≤ s ∗ such that f (θ ) = g  (0).

g  (0), so there exists a constant 0 ≤ θ

g(s) is a nonincreasing funcs tion. Assume also that g(θ ) < θf (0). Then, for each φ ∈ X+ \ {0}, the corresponding solution of (1.6) is strongly persistent. Lemma 1.2.11. Suppose that f is an increasing function and

Proof. First, if 0 < θ ≤ u∗ , from the first inequality of (1.20) we have θf (θ ) ≤ g(θ ) < θf (0); it yields f (θ ) < f (0), which is a contradiction to the monotonicity of f . Then u∗ < θ ≤ s ∗ . Further, the hypothesis θf (0) > g(θ ) and the regularity of f and g lead to the existence of ε > 0 such that (θ − ε)f (0) > g(θ − ε).

(1.24)

Now, it is easy to show that the pair (0, θ − ε) represents sub- and supersolutions of (1.6) provided φ ∈ [0, θ − ε]X . On the other hand, since f (θ ) = g  (0), it follows that f (θ − ε) < g  (0). This leads to the existence of small α > 0 such that f (θ − ε) ≤

g(α) . α

(1.25)

Now, from (1.24) we have f (c) ≥

g(θ − ε) θ −ε

(1.26)

for all c > 0. By taking c = α and for φ ∈ [α, θ − ε]X , assertions (1.25)–(1.26) show that the pair (α, θ − ε) is sub- and supersolutions of (1.6), and so α ≤ u ≤ θ − in [0, 1] × [0, ∞). Next, for φ nonidentically zero and φ ∈ [0, θ − ε]X , Lemma 1.2.5 implies that u(x, t) > 0 for all (x, t) ∈ [0, 1] × (0, ∞). Then, u(x, t) > α for all (x, t) ∈ [0, 1] × [τ, 2τ ]. Consequently, using again assertions (1.25)–(1.26), we get u(x, t) ≥ α for all (x, t) ∈ [0, 1] × (2τ, ∞). To finish the proof, we will show that each positive solution of problem (1.6) enters the interval [0, θ − ε]X and stays there. For a nonnegative, nonidentically zero function φ, we

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introduce the following problem:   v (t) = g(v(t)) − v(t)f (0), φ(x, s). v(0) = max

(1.27)

(x,s)∈[0,1]×[−τ,0]

Observe that (0, v(t)) is a pair of sub- and supersolutions of problem (1.6). Therefore we get 0 ≤ u(x, t) ≤ v(t). Now we claim that there exists T > 0 such that v(t) ≤ θ − ε for all t ≥ T . g(s) Indeed, suppose first that v(t) ≥ θ − ε for all t ≥ 0. Then, from (1.27) and the fact that s is a nonincreasing function, we have   g(v(t)) v  (t) = v(t) − f (0) , v(t)   g(θ − ε) − f (0) . ≤ v(t) θ −ε Since g(θ − ε) < (θ − ε)f (0), it follows that lim sup v(t) ≤ 0, and we obtain a contradiction. t→∞

Therefore there exists t0 ≥ 0 such that v(t0 ) < θ − ε. Next suppose that there exists t1 > t0 such that v  (t1 ) ≥ 0 and v(t1 ) = θ − ε. Substituting v(t1 ) in (1.27), 0 ≤ g(θ − ε) − (θ − ε)f (0) < 0. This contradiction proves the assertion. Consequently, there exists T > 0 such that u(x, t) ≤ θ − ε for all t ≥ T and x ∈ [0, 1]. The lemma is proved. By using the method of separation of variables, the following lemma is easily proved. For the sake of completeness, we add the proof. Lemma 1.2.12. Green’s function of the following problem

ut (x, t) − uxx (x, t) = 0, (x, t) ∈ (0, 1) × (0, T ), u(0, t) = u(1, t), ux (0, t) = ux (1, t), t ∈ (0, T ],

(1.28)

is given by

(x, y, t) = 1 + 2

∞ 

  2 e−(2nπ) t cos 2nπ(x − y) .

n=1

Proof. We look for a solution of the form u(x, t) = X(x)T (t) for X, T to be determined. Suppose that we can find a solution of (1.28) of this form. Substituting a function u(x, t) = X(x)T (t) into the heat equation in (1.28), we arrive at X(x)T  (t) − X  (x)T (t) = 0.

Chapter 1 • Global dynamics of a delayed reaction-diffusion

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Dividing this equation by X(x)T (t), we have T  (t) X  (x) = . T (t) X(x) Since the variables x and t are independent, this last equality remains valid if and only if T  (t) X  (x) = =λ T (t) X(x) for some constant λ. In addition, for u to satisfy the periodic boundary conditions stated in (1.28), we must have

 X (x) − λX(x) = 0, x ∈ (0, 1), (1.29) X(0) = X(1), X  (0) = X  (1). Hence we have to solve the eigenvalue problem (1.29). We are going to look for the eigenvalue λ and the associated eigenfunction X by splitting into three cases: λ > 0, λ = 0, and λ < 0. Case 1: λ = γ 2 > 0. All solutions of (1.29) are in the form X(x) = αeγ x + βe−γ x . The periodic boundary conditions give     X(0) = X(1) =⇒ α 1 − eγ = β e−γ − 1 and

It follows that

    X  (0) = X  (1) =⇒ α 1 − eγ = β 1 − e−γ .     β e−γ − 1 = β 1 − e−γ ;

consequently, β = 0 and thus α = 0. Therefore there are no positive eigenvalues. Case 2: λ = 0. Then all solutions of (1.29) are given by X(x) = αx + β. From the periodic boundary condition, X(0) = X(1) =⇒ α = 0. Therefore, λ = 0 is an eigenvalue with corresponding constant eigenfunctions X(x) = β. Case 3: λ = −γ 2 < 0.

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Then all solutions are in the form X(x) = α cos(γ x) + β sin(γ x). Using the periodic boundary conditions

α(1 − cos(γ )) − β sin(γ ) = 0, α sin(γ ) + β(1 − cos(γ )) = 0. Then, if α = 0, then γ = 2nπ, n = 1, 2, .... In this case the eigenfunctions are given by Xn (x) = βn sin(2nπx), n = 1, 2, ... If β = 0, then γ = 2nπ, n = 1, 2, .... In this case the eigenfunctions are given by Xn (x) = αn cos(2nπx), n = 1, 2, ... Consequently, for each n = 1, 2, ..., eigenvalue λn admits two orthogonal eigenfunctions, (Xn1 (x), Xn2 (x)) = (αn cos(2nπx), βn sin(2nπx)), and for λ0 = 0, we have X0 (x) = α0 . Therefore Green’s function of the periodic initial value problem (1.28) has the form  1  ∞ 1  X02 Xn2 (x)Xn2 (y) −λn t Xn (x)Xn (y) ,

(x, y, t) = + e + ||X0 ||2 ||Xn1 ||2 ||Xn2 ||2 n=1

where (Xn1 (x), Xn2 (x)) = (αn cos(2nπx), βn sin(2nπx)) are orthogonal eigenfunctions correα2 β2 sponding to eigenvalues λn . It is easy to show that ||Xn1 ||2L2 (0,1) = n and ||Xn2 ||2L2 (0,1) = n , 2 2 then Xn1 (x)Xn1 (y) Xn2 (x)Xn2 (y) + ||Xn1 ||2 ||Xn2 ||2

=

cos(2nπx) cos(2nπy) + sin(2nπx) sin(2nπy),

=

cos(2nπ(x − y)).

Accordingly,

(x, y, t) = 1 + 2

∞ 

  2 e−(2nπ) t cos 2nπ(x − y) .

n=1

Remark 1.2.13. In view of the above lemma, the solution of ⎧ ⎨ ut (x, t) − uxx (x, t) + αu = f (x, t), (x, t) ∈ (0, 1) × (0, T ), u(0, t) = u(1, t), ux (0, t) = ux (1, t), t ∈ (0, T ], ⎩ u(x, 0) = u0 (x), x ∈ [0, 1], can be written as u(x, t) = e−αt



1 0



(x, y, t)u0 (y)dy + 0

t

e−αs

 0

1

(x, y, s)f (y, t − s)dyds.

Chapter 1 • Global dynamics of a delayed reaction-diffusion

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Theorem 1.2.14. Assume that φ ∈ X+ and φ(x, 0) is not identically zero. Suppose also that f g(s) is a nonincreasing function on [0, B], and either f (s ∗ ) < is an increasing function and s g  (0) or θf (0) > g(θ ) with f (θ ) = g  (0). Then the positive steady state is globally attractive if one of the following conditions is satisfied: g(s)f (s) is nonincreasing over [0, B], s g(s) + f (s) is nonincreasing over [0, B], (H2) s g(s) g(s)f (s) (H3) + f (s) + is nonincreasing over [0, B]. s s (H1)

Proof. Let u∞ (x) := lim sup u(x, t),

u∞ (x) := lim inf u(x, t), t→∞

t→∞

and u∞ := max u∞ (x),

u∞ := min u∞ (x). x∈[0,1]

x∈[0,1]

Write g(u) = g(u) + αu − αu. Since u is uniformly bounded, we can choose a constant α > 0 such that α − f (u) is positive. From Lemma 1.2.12 and Remark 1.2.13 we have u(x, t) = e

−αt



1

(x, y, t)u(y, 0)dy    t  1    g(u(y, t − s)) −αs + u(y, t − s) α − f u(y, t − s − τ ) dyds, + e

(x, y, s) u(y, t − s) u(y, t − s) 0 0 0

where u is a solution of problem (1.6). Since f is an increasing function and plies that u∞ (x) ≤

 0

From

∞

e−αs



1

(x, y, s)(u∞ (y)

0



g(u) is a nonincreasing function, Fatou’s lemma imu

1

g(u∞ (y)) + u∞ (y)(α − f (u∞ (y))))dyds. u∞ (y)

(x, y, s)dy = 1,

∀s > 0, x ∈ (0, 1),

0

we obtain g(u∞ ) . u∞ Employing the same arguments for u∞ , we get f (u∞ ) ≤

f (u∞ ) ≥

g(u∞ ) . u∞

(1.30)

(1.31)

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Next, multiplying expression (1.30) by

g(u∞ ) and combining the result with (1.31), we get u∞

g(u∞ )  ∞  g(u∞ ) f u ≥ f (u∞ ). u∞ u∞ Therefore from (H1) we have u∞ = u∞ . In the case of (H2), the proof is practically similar to g(u∞ ) in (1.30) and combine it with (1.31). (H1). Indeed, to get the required result, we add u∞ The same argument is employed if (H3) is satisfied. Thus, in all cases, u converges to a steady state. In view of (1.20), Lemma 1.2.10, and Lemma 1.2.11, each solution of problem (1.6) converges to the positive steady state. Now, suppose that f has a single maximum f (M) = max f (u) and f is increasing over 0≤u≤1

(0, M) and decreasing over (M, 1). The following lemma will allow us to apply the results of Theorem 1.2.14 to this case.

FIGURE 1.1 Some examples of functions f with g(s) = s(1 − s).

Lemma 1.2.15. Assume that φ ∈ X+ and φ(x, 0) is not identically zero. Suppose also that g(s) is a nonincreasing function on [0, B]. For each positive solution u of problem (1.6), we s have u(x, t) ≤ M for all x ∈ [0, 1] and t > t0 if one of the following conditions is satisfied: g(M) for all s ∈ [0, B]. M g(M) g(α) g(M) (K2) f (0) > and f (s) > for all s ∈ [0, B], where α is defined as f (α) = and M α M M < α < B.

(K1) f (s) >

Proof. Let us suppose that (K1) holds (see Fig. 1.1(A)). This implies that there exists ε > 0 g(M) such that f (s) ≥ + ε for all s ∈ [0, B]. From this, each solution of (1.6) satisfies the M inequality ut − uxx ≤ u(

g(u) g(M) − − ε). u M

Chapter 1 • Global dynamics of a delayed reaction-diffusion

Let v be the solution of

⎧ ⎨ v  (t) = v(t)( g(v) − g(M) − ε), v M ⎩ v(0) = φ(x, s). max

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(1.32)

(x,s)∈[0,1]×[−τ,0]

By the maximum principle, we have u(x, t) ≤ v(t) for all (x, t) ∈ [0, 1] × [0, ∞). On the other hand, we can show that there exists t0 > 0 such that v(t) ≤ M for all t ≥ t0 . In fact, suppose, g(s) by contradiction, that v(t) > M for all t > 0. From (1.32) and the fact that is a nonins creasing function, we get v  (t)

g(v) g(M) − − ε), v M −εv(t),

= v(t)( ≤

this implies that v(t) converges to zero, and we obtain a contradiction with v(t) ≥ M for all t > 0. This concludes the existence of t1 > 0 such that v(t1 ) ≤ M. Now, we prove that v(t) ≤ M for all t ≥ t1 . Otherwise, there exists t2 ≥ t1 such that v(t2 ) = M and v  (t2 ) ≥ 0. Substituting v(t2 ) in (1.32), we have 0 ≤ v  (t2 ) = ≤

g(M) g(M) − − ε), M M −εM < 0. M(

Assertion (K1) is proved. Let condition (K2) be satisfied (see Fig. 1.1(B)). We begin by proving that there exists T > 0 such that u(x, t) < α for all x ∈ [0, 1] and t > T . Indeed, from (K2), there exists ε > 0 so small g(α) that f (s) ≥ + ε. Thus, each solution of (1.6) satisfies the inequality α ut − uxx ≤ u(

g(u) g(α) − − ε). u α

Consider the following problem: ⎧ ⎪ ⎨ v  (t) = v(t)( g(v(t)) − g(α) − ε), v(t) α ⎪ max φ(x, s). ⎩ v(0) = (x,s)∈[0,1]×[−τ,0]

By the classical maximum principle, we have u(x, t) ≤ v(t) for all (x, t) ∈ [0, 1] × [0, ∞). Further, employing the same arguments as before, we can show the existence of t0 > 0 such g(M) for all s ∈ [0, α). Indeed, that v(t) < α for all t ≥ t0 . On the other hand, note that f (s) > M g(M) for s ∈ [0, M], the result is reached by combining f (0) > and f is increasing over M (0, M). If s ∈ (M, α), then the result holds again true; to see it, it suffices to combine the fact

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g(M) . Finally, employing the same arguments M as in the first part of this proof, we show that u(x, t) ≤ M for all x ∈ [0, 1] and t ≥ T .

that f is decreasing over (M, α) and f (α) =

Corollary 1.2.16. Under the assumptions of Lemma 1.2.15, the positive steady state u∗ is globally attractive. Proof. The proof is reached by combining Lemma 1.2.15 and Theorem 1.2.14.

1.3 Numerical simulations In this section, we present some numerical simulations related to the following delayed partial differential equation:   ut (x, t) = Duxx (x, t) + g u(x, t) − u(x, t)f (u(x, t − τ ). We will consider some suitable initial conditions u(x, t) = φ(x, t), (x, t) ∈ [0, L] × [−τ, 0].

1.3.1 Example 1 We propose the following equation:   ut (x, t) = Duxx (x, t) + u(x, t) 1 − u(x, t) − 2u(x, t − τ ) , the positive steady state is given by u∗ = 1/3. Let us fix the diffusion coefficient D, and let the delay τ be free to discuss with respect to its different values. We observe that there exists a critical delayed value τ ∗ > 0, where, if τ < τ ∗ , we have the convergence to the equilibrium u∗ , and if τ > τ ∗ , we get the convergence to a periodic solution. For large values of τ , we still have a periodic solutions with low frequency. (See Figs. 1.2 and 1.3.)

FIGURE 1.2 The evolution of the solution u(x, t) with respect to time t and space x for D = 1 and τ = 0 (left), τ = 1.5 (middle), and τ = 4 (right).

Now, we fix τ such that τ < τ ∗ , then we take different values of the diffusion coefficient. We will observe that there is always convergence to the homogeneous equilibrium u∗ but with a low speed for very small values of the diffusion coefficient. (See Fig. 1.4.)

Chapter 1 • Global dynamics of a delayed reaction-diffusion

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FIGURE 1.3 The evolution of the solution u(x, t) with respect to time t and space x for D = 1 and τ = 4 (left), τ = 10 (middle), and τ = 20 (right).

FIGURE 1.4 The evolution of the solution u(x, t) with respect to time t and space x for τ = 1 and D = 1 (left), D = 0.01 (middle), and D = 0.001 (right).

Next, let τ be fixed such that τ > τ ∗ , and we keep D free. We show that we can lose the convergence to a periodic solution for some small values of the diffusion coefficient D. (See Figs. 1.5, 1.6, 1.7.)

FIGURE 1.5 The evolution of the solution u(x, t) with respect to time t and space x for τ = 4 and D = 10−1 (left), D = 10−4 (middle), and D = 10−7 (right).

1.3.2 Example 2 We consider the following equation:   ut (x, t) = Duxx (x, t) + u(x, t) log

  1 − 4u(x, t − τ ) . u(x, t)

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FIGURE 1.6 The evolution of the solution u(x, t) with respect to time t and space x for τ = 4 and D = 10−7 .

FIGURE 1.7 The evolution of the solution u(x, t) with respect to time t and space x for τ = 4 and D = 10−5 , with Neumann BC (left) and periodic BC (right).

The positive steady state takes the value u∗ = 0.3005. (See Figs. 1.8 and 1.9.)

FIGURE 1.8 The evolution of the solution u(x, t) with respect to time t and space x for D = 1 and τ = 0 (left), τ = 2 (middle), and τ = 10 (right).

1.4 Spatiotemporal pattern formation We are concerned in this section with spatial temporal pattern formation induced by the equation     ut (x, t) = Duxx (x, t) + g u(x, t) − u(x, t)f u(x, t − τ ) , (x, t) ∈ (0, L) × (0, T ). (1.33)

Chapter 1 • Global dynamics of a delayed reaction-diffusion

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FIGURE 1.9 The evolution of the solution u(x, t) with respect to time t and space x for τ = 5 and D = 1 (left), D = 10−7 (right).

The functions g and f used in the numerical simulations are given by g(u) = u(1 − u) and

⎧ ⎪ F1 , 0 ≤ u ≤ u1 , ⎪ ⎪ ⎪ ⎪ ⎨ (u; u1 , u2 , F1 , F3 ), u1 ≤ u ≤ u2 , f (u) = F3 , u2 ≤ u ≤ u3 , ⎪ ⎪ ⎪ ,

(u; u 3 u4 , F3 , F2 ), u3 ≤ u ≤ u4 , ⎪ ⎪ ⎩ F2 , u ≥ u4 ,

where

(u; u1 , u2 , F1 , F3 ) = F1 + (F3 − F1 )

(u − u1 )2 (3u2 − u1 − 2u) (u2 − u1 )3

and

(u; u3 , u4 , F3 , F2 ) = F3 + (F2 − F3 )

(u − u3 )2 (3u4 − u3 − 2u) . (u4 − u3 )3

The initial condition φ is defined by ⎧ u˜ 1 , 0 ≤ x ≤ x1 ⎨ φ(x, t) =

(x; x1 , x2 , u˜ 1 , u˜ 2 ), x1 ≤ x ≤ x2 ⎩ u˜ 2 , x2 ≤ x ≤ L. Different values of parameters F1 , F2 , F3 , u˜ 1 , and u˜ 2 are considered in the numerical simulations. We will characterize solutions by the position of their maxima. We will also consider periodic boundary conditions.

1.4.1 Maximum maps For each t fixed, we define the strict maxima of the solution u(x, t) by the points xi (t) for which u(xi (t), t) > u(x, t) for all x in some neighborhood of xi (t). Such points are of finite

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or at most countable number N (t). We put all these points in the set M(u) =

 

 xi (t), t , i = 1, ..., N (t), t ≥ 0, 0 ≤ x ≤ L.

Such a set is called the maximum map. Some examples are provided in what follows. We present here some examples of maximum maps which have some similarities with the results of numerical simulations in Section 1.4. (See Figs. 1.10, 1.11, 1.12, 1.13.) We can expect that the eigenfunctions that determine the spatial perturbations of the homogeneous in space solutions are related to the functions considered in the figures that follow. (See Figs. 1.14, 1.15, 1.16.)

FIGURE 1.10 The evolution of the function v(x, t) = cos(t) · cos(αx − β) + 1 with respect to t and x.

FIGURE 1.11 The evolution of the function v(x, t) = cos(αx − ct) + 1 with respect to t and x.

FIGURE 1.12 The evolution of the function v(x, t) = cos(α|x − x0 | − ct) + 1 with respect to t and x.

Chapter 1 • Global dynamics of a delayed reaction-diffusion

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FIGURE 1.13 Maximum maps for the functions v(x, t) = cos(t) · cos(αx − β) + 1 (A), cos(αx − ct) + 1 (B), cos(α|x − x0 | − ct) + 1 (C).

FIGURE 1.14 Maximum maps for the functions v(x, t) = cos(t) · cos(αx − β − (t)) + 1 (A), cos(α|x − x0 | + (t)(ct + β)) + 1 (B), cos(t) · cos(α|x − x0 | − ct − (t)) + 1 (C).

FIGURE 1.15 Maximum maps for the function v(x, t) = cos(t) · cos(α|x − x0 | − cos(ct) + β) + 1.

In the particular case where the solution can be represented in the form u(x, t) = (t) + v(x, t),  being a periodic function with large amplitude and low frequency, the maximum map is a powerful tool for a characteristic representation of the complex dynamics induced by Eq. (1.33).

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FIGURE 1.16 Maximum maps for the function v(x, t) = cos(t) · cos(α1 x − c1 t) + cos(α2 x + c2 t) + 1.

1.4.2 Bifurcations and branches of solutions We limit ourselves in this subsection to the case of homogeneous in space periodic solutions u(x, t) = (t) that lose stability with respect to spatial perturbation. In what follows, let    ut (x, t) = Duxx (x, t) + u 1 − u − f u(x, t − τ ) , and fix time delay τ = 1.1, the length of the interval L = 0.5, and vary the value of the diffusion coefficient D. (See Fig. 1.17.)

FIGURE 1.17 Maximum maps for one-maximum (1M) symmetric mode, τ = 1.1, D=0.001 (left), D=0.00045 (middle), and D=0.0004 (right).

Consider now a solution of the form u( x, t) = u¯ 0 + φ1 (t), u¯ 0 representing the average value of u and φ1 being a changing sign periodic function. For τ = 1.1, L = 0.5, the first bifurcation occurs for D = 0.0009, and the homogeneous solution loses its stability. A “good” approximation of the bifurcating solution can be given by u(x, t) = u¯ 0 + φ1 (t) + φ2 (t) cos(bx − h)), where φ2 (t) is periodic and b takes the value b = 2π/L. As the solution is space invariant, h can take any arbitrary value. The term φ2 (t) makes the single minimum and the single maximum of the spatial perturbation alternate their positions in time. The maximum map is represented by two vertical lines.

Chapter 1 • Global dynamics of a delayed reaction-diffusion

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FIGURE 1.18 Maximum maps for one-maximum (1M) spinning mode, τ = 1.1, D=0.00031 (left), D=0.00026 (middle), and D=0.000125 (right).

It is interesting to note that solution (Fig. 1.18, left) with a constant speed loses its stability, resulting in appearance of solutions with an oscillation wave speed (Fig. 1.18, middle). (See Fig. 1.19.)

FIGURE 1.19 Maximum maps for 2M symmetric mode, τ = 1.1, D=0.000195 (left), and D=0.000175 (right).

1.5 Conclusion and open problems We have presented and studied in this chapter a general reaction-diffusion equation with delay. Under some suitable conditions, global analysis has been established, persistence, global attractiveness, and bifurcation analysis considering the delay as a bifurcation term have been provided. With a convenient choice of the data, spatio-temporal pattern formation is well illustrated, so the reader can clearly observe it. Limiting ourselves to homogeneous in space periodic solutions and when the imposed conditions are not fulfilled, we numerically observe the passage from stable to unstable behavior. We also provide an approximation of the bifurcation branch. The chapter is ended by many numerical simulations that involve in particular the maximum maps associated to the solution. Our analysis left some questions without answers, indeed:

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1. We have found some attractiveness regions for the steady state, but we still believe that these regions are not optimal and can be improved by finding some larger ones. 2. For example and only for the particular case when g(u) = u(1 − u) and the function f has a bell-shaped form as in Fig. 1.20(B), (C) or in Fig. 1.21(A), (B), and (C), the asymptotic analysis relative to the steady state is still open. More precisely, is it possible to find a region for which the steady states attract all solutions of our problem?

FIGURE 1.20 Some forms of the function f with g(s) = s(1 − s).

FIGURE 1.21 Some forms of the function f with g(s) = s(1 − s).

Acknowledgments The authors are partially supported by DGRSDT, ALGERIA, project PRFU, code C00L03UN130120200004. They are grateful to the anonymous reviewer for his\her suggestions.

References [1] M. Bachraoui, M.A. Ichou, K. Hattaf, N. Yousfi, Spatiotemporal dynamics of a fractional model for hepatit bvirus infection with cellular immunity, Math. Model. Nat. Phenom. 16 (5) (2021) 1–13. [2] M. Bachraoui, K. Hattaf, N. Yousfi, Dynamics of a fractional order HBV infection model with capsidsand CTL immune response, Commun. Math. Biol. Neurosci. 6 (2019) 1–15. [3] M. Banerjee, A. Tokarev, V. Volpert, Immuno-epidemiological model of two-stage epidemic growth, Math. Model. Nat. Phenom. 15 (27) (2020) 1–11.

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[4] L. Berezansky, E. Braverman, L. Idels, Mackey-Glass model of hematopoiesis with non-monotone feedback: stability, oscillation and control, Appl. Math. Comput. 219 (2013) 6268–6283. [5] E. Braverman, S. Zhukovskiy, Absolute and delay-dependent stability of equations with a distributed delay, Discrete Contin. Dyn. Syst. 32 (6) (2012) 2041–2061. [6] N. Bessonov, G. Bocharov, T.M. Touaoula, S. Trofimchuk, V. Volpert, Delay reaction-diffusion equation for infection dynamics, Discrete Contin. Dyn. Syst., Ser. B 24 (5) (2019) 2073–2091. [7] G.A. Bocharov, Modelling the dynamics of LCMV infection in mice: conventional and exhaustive CTL responses, J. Theor. Biol. 192 (1998) 283–308. [8] G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk, V. Volpert, Spatiotemporal dynamics of virus infection spreading in tissues, PlosOne (20 December 2016), https://doi.org/10.1371/journal.pone. 0168576. [9] G. Bocharov, A. Meyerhans, N. Bessonov, S. Trofimchuk, V. Volpert, Modelling the dynamics of virus infection and immune response in space and time, Int. J. Parallel Emerg. Distrib. Syst. (2017). [10] K. Deng, Y. Wu, On the diffusive Nicholson’s blowflies equation with distributed delay, Appl. Math. Lett. 50 (2015) 126–132. [11] W.E. Fitzgibbon, Semilinear functional differential equations in Banach space, J. Differ. Equ. 29 (1978) 1–14. [12] T. Kuniya, T.M. Touaoula, Global dynamics for a class of functional differential equations with distributed delay and non-monotone bistable nonlinearity, Math. Biosci. Eng. 17 (2020) 7332–7352. [13] R. Martin, H.L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Am. Math. Soc. 321 (1990) 1–44. [14] R. Martin, H.L. Smith, Reaction-diffusion systems with time delay: monotonicity, invariance, comparison and convergence, J. Reine Angew. Math. 413 (1991) 1–35. [15] E. Liz, M. Pinto, V. Tkachenko, S. Tromichuk, A global stability criterion for a family of delayed population models, Q. Appl. Math. 63 (2005) 56–70. [16] C.V. Pao, Nonlinear Parabolic and Elliptic Equations, Plenum Press, New York, 1992. [17] G. Rost, J. Wu, Domain-decomposition method for the global dynamics of delay differential equations with unimodal feedback, Proc. R. Soc., Math. Phys. Eng. Sci. 463 (2007) 2655–2669. [18] H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Math, Surveys Monogr, vol. 41, Amer. Math. Soc., 1995. [19] T.M. Touaoula, Global stability for a class of functional differential equations (application to Nicholson’s blowflies and Mackey-Glass models), Discrete Contin. Dyn. Syst. 38 (9) (2018) 4391–4419. [20] T.M. Touaoula, Global dynamics for a class of reaction-diffusion equations with distributed delay and Neumann condition, Commun. Pure Appl. Anal. 19 (2020) 2473–2490. [21] T.M. Touaoula, M.N. Frioui, N. Bessonov, V. Volpert, Dynamics of solutions of a reaction-diffusion equation with delayed inhibition, Discrete Contin. Dyn. Syst. 13 (9) (2020) 2425–2442. [22] H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, 2003. [23] H.R. Thieme, X-Q. Zhao, A non-local delayed and diffusive predator-prey model, Nonlinear Anal., Real World Appl. 2 (2001) 145–160. [24] S. Trofimchuk, V. Volpert, Traveling waves for a bistable reaction-diffusion equation with delay, SIAM J. Math. Anal. 50 (1) (2018) 1175–1190. [25] A.I. Volpert, V.A. Volpert, Vl.A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translation of Mathematical Monographs, vol. 140, Amer. Math. Society, Providence, 1994. [26] J. Wu, Theory and Applications of Partial Functional Differential Equations, Appl. Math. Sci., vol. 119, Springer-Verlag, New York, 1996. [27] D. Xu, X-Q. Zhao, A nonlocal reaction-diffusion population model with stage structure, Can. Appl. Math. Q. 11 (2003) 303–320. [28] T. Yi, Y. Chen, J. Wu, Global dynamics of delayed reaction-diffusion equations in unbounded domains, Z. Angew. Math. Phys. 63 (2012) 793–812. [29] T. Yi, X. Zou, Global attractivity of the diffusive Nicholson blowflies equation with Neumann boundary condition: a non-monotone case, J. Differ. Equ. 245 (2008) 3376–3388. [30] T. Yi, X. Zou, Map dynamics versus dynamics of associated delay reaction-diffusion equations with a Newmann condition, Proc. R. Soc., Math. Phys. Eng. Sci. 466 (2010) 2955–2973. [31] T. Yi, X. Zou, Global dynamics of a delay differential equation with spatial non-locality in an unbounded domain, J. Differ. Equ. 251 (2011) 2598–2611.

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[32] T. Yi, X. Zou, On Dirichlet problem for a class of delayed reaction-diffusion equations with spatial non-locality, J. Dyn. Differ. Equ. 25 (2013) 959–979. [33] T. Yi, X. Zou, Dirichlet problem for a delayed reaction-diffusion equation on a semi-infinite interval, J. Dyn. Differ. Equ. 28 (2016) 1007–1030. [34] Y. Yuan, X.Q. Zhao, Global stability for non monotone delay equations (with application to a model of blood cell production), J. Differ. Equ. 252 (2012) 2189–2209. [35] X.Q. Zhao, Global attractivity in a class of nonmonotone reaction-diffusion equations with time delay, Can. Appl. Math. Q. 17 (1) (2009) 271–281.

2 Hepatitis B virus transmission via epidemic model Tahir Khana,b , Roman Ullahb , and Gul Zamana a Department of Mathematics, University of Malakand, Khyber Pakhtunkhwa, Pakistan b Department of

Computing, Muscat College, Muscat, Oman

2.1 Introduction The liver is an important organ in every living body. This is a vital organ that fights infection and filters blood. The liver infection causes various complications, and so the functions are affected automatically whenever there is an infection. Inflammation of the liver also known as hepatitis. Hepatitis is caused mostly by bacterial infections, viruses, continuous exposure to alcohol or drugs [1]. One of the types of hepatitis, hepatitis B, is a contagious and life-threatening liver infection. The infection of hepatitis B results from the hepatitis B virus and as a result causes the liver inflammation. This infection occurs whenever the hepatitis B virus can reach the liver through the bloodstream. Once in the liver, the virus reproduces and releases large numbers of new viruses into the bloodstream [2]. Moreover, the infection of hepatitis B has two possible phases, acute and chronic ones. In acute hepatitis, the immune system is usually able to vanish the virus from the living body, and so the body recovers in a few months completely. However, in some severe cases, the infection leads to the most serious stage know as the chronic stage. Patients with this stage often have no history of acute illness. This may also cause scarring of the liver and liver failure. It also develops liver cancer [3]. Mathematical models describe the transfer mechanism of the disease transmission for infections [4,5]. One of the incidence functions symbolized by g(I ), also know as an incidence rate function, plays an important role in the transmission of the disease [6–10]. The incidence rate function provides and ensures the reasonable qualitative dynamics of the disease. Moreover, in every disease spreading, the psychological effect also plays a key role in the dynamics and control of a disease. It is very much clear that the force of infection may increase whenever the amount of I is small, i.e., the number of infected individual’s is small; on the other hand, the force of infection decreases for taking large amount of I [11]. For a large number of infected portions, the force of infection may decreases if the infected portion increases. In the absence of a large number of infected the population may reach to the minimium contacts per unit time [12,13]. Generally, the incidence function is Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00007-7 Copyright © 2023 Elsevier Inc. All rights reserved.

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defined by the following equation: g(I ) =

βI h , 1 + dI p

where β represents the probability of transmission per unit of time and p and h are some positive constants. d > 0 is the effect of an inhibitory or psychological effect. The incidence rate g(I ) was introduced for the first time by Liu et al. [14,15], where the infection force of the disease measures by βI h and the inhibition effect from the behavioral changes of 1 the susceptible individuals measures by 1+dI p when their number increases or from the crowding effect of the infected individuals [13]. This rate seems to be more reasonable than the bilinear rate βI S. Because this contains the crowding effects of the infected individuals and the behavioral changes and also prevents the unboundedness of the contact rate by selecting reasonable parameters. Especially, Ruan et al. developed a model the incidence βI 2 rate of g(I ) = 1+dI 2 and discussed the detail analysis of bifurcation (for more details, see [13]). Serio and Capasso presented another incidence rate called saturated type incidence βI in the epidemiological models, i.e., g(I ) = 1+dI , in which g(I ) reaches the level of saturation as I becomes large [12]. Xian et al. formulated a model with another type of incidence βI rate g(I ) = 1+dI 2 called the nonmonotonic incidence rate (see [11]). The treatment is another important control measure to minimize the spread of various infectious diseases such as flu, tuberculosis, measles, hepatitis B, etc. (for more details, the readers are suggested to see Feng et al. [18], Hyman and Li [19], Wu and Feng [20]). In classical epidemiological models, the ratio of treatment is supposed to be proportional to the ratio of an infected portion. But this looks unrealistic as the resources of treatment are quite large. There is a suitable capacity of treatment in every community, and so the community pays for unnecessary costs whenever they are too large; while the community is at risk if the outbreak of a disease is small enough. It is very significant to determine a reasonable capacity for the treatment of an infection because in the usual cases the allocation, as well as the change of medical measures, does not happen in a continuous form. Moreover, treatment with a constant rate is applicable if the number of the infected is very large. A treatment function given by Wang in [21] is defined by  T (I ) =

Ir

whenever 0 ≤ I ≤ Ic ,

K

whenever Ic < I.

(2.1)

The above function is called switching function in which k = Ic r, which shows that treatment is proportional to the size of the infected population if the capacity of treatment is not reached; on the other hand, it is a constant whenever the number of the infected is more than the maximal Ic . Here, K is the treatment capacity for the infection. This renders, for example, the situation where patients have to be hospitalized: the number of hospital beds is limited. This is true for the case where medicines are not sufficient [22]. Evidently, this improves the classical proportional treatment and the constant treatment in [21].

Chapter 2 • Hepatitis B virus transmission via epidemic model

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In this chapter, we modify the model reported in Khan et al. by incorporating the role of the generalized incidence rate and the effect of the treatment function as reported by many research works [16,17]. We investigate the proposed model by incorporating the above features, i.e., treatment control and generalized incidence rate, under preventive vaccination. Once we have formulated the problem, we then mainly discuss the existence and qualitative analysis of the model. The basic reproductive number will be also calculated to determine the stability conditions. Finally, all the theoretical findings are verified with the help of numerical simulations. We end our work with a brief conclusion. We organize the chapter as follows: In Section 2.2, we present the detailed model formulation and its mathematical and biological feasibility by proving the positivity and boundedness of solutions. In Section 2.3, we calculate the model equilibria and discuss qualitative analysis of the proposed model. Particularly, we find the disease-free equilibrium and calculate the basic reproductive number to obtain the endemic equilibrium of the proposed model. We then discuss the local and global dynamics of the proposed model around disease-free and endemic equilibria subject to the basic reproductive number. Finally, we perform large numerical simulations and conclude our work in Section 2.4.

2.2 Model formulation In this section, we develop the model to investigate the dynamics. Let us assume that the entire population is represented by N (t) and is divided into different groups of the compartment, the infected, and the recovered. These groups are denoted by S(t), I (t), and R(t), respectively. Moreover,  and β are the newborn and the transmission rate, the vaccination rate is ν, and μ0 denotes the natural death of the population groups. We also assume that the death that occurs from the infection is μ1 , and the recovery rate of the infected individuals is γ . Recalling Eq. (2.1) is the treatment function, where r is positive and is called the proportionality constant. This denotes the treatment capacity for the infected, and K represents the maximal treatment capacity, while Ic is taken to be a fixed value (for more details, see [22]). Thus g(I ) is directly proportional to the infected population, because under the intervention policies, the number of per unit time contacts of the infected βI population are reducing. We assume g(I ) = π(I ) for the sake of simplicity that the infection force g(I ) is a function for the infected (this plays a significant part in investigating the 1 transfer of the infection) in which π(I ) demonstrates the effect of the intervention mechanism on minimizing β as reported in [23]. A nonmonotonic infection force is defined by placing the assumption as follows: 1. If I > 0, then π(0) = 1 and

dπ(I ) dI

> 0.

2. There exists is small positive constant δ > 0 such that   d I and dI π(I ) < 0, if δ < I .

d dI



I π(I )

 > 0 whenever 0 < I < δ

The above condition describes the effects of interference strategies of the invectives δ investigating by a critical level: whenever 0 < I < δ, the incidence is increasing, while if

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I > δ, then it is decreasing. We suggest the proposed model is given in the following form: dS(t) βS(t)I (t) = N − − (ν + μ0 )S(t) + γ1 R(t), dt π(I ) dI (t) βS(t)I (t) = − (μ0 + γ + μ1 )I (t) − T (I ), dt π(I ) dR(t) = γ I (t) + νS(t) − (μ0 + γ1 )R(t) + T (I ), dt

(2.2)

with biologically feasible initial groups of population sizes S(0) = S 0 ≥ 0, I (0) = I 0 ≥ 0, R(0) = R 0 ≥ 0.

(2.3)

The parameters and the variables in the epidemic problem are characterized as follows: the natural birth and death rates are respectively denoted by  and μ0 ; the recovery rate is γ , and the rate of vaccinated proportion is symbolized by ν. Moreover, μ1 is the death proportion due to infection, r represents the capacity of treatment for infection, the maximal treatment capacity is K, and d is the psychological effect. The variables of the problem are: S(t), which demonstrates the susceptible class, I (t), the infected class, and R(t), the recovered class. It is clear from model (2.2) that the entire population is N (t), which is the sum of S(t), I (t), and R(t). This implies that N dN (t) + (S(t) + I (t) + R(t))μ0 ≤ . dt μ0 The above equation can be also rewritten as dN (t) N , + μ0 N ≤ dt μ0 which gives that N (t) approaches N μ0 as t → ∞. We give a biologically feasible region by the following set:   N 3 :0≤S +I +R≤  = (S, I, R) ∈ R+ . (2.4) μ0 Thus we immediately give the following lemma regarding the solutions of model (2.2). 3 is positively invariant subject to the proposed model (2.2) Lemma 2.1. The region  ⊂ R+ 3. with nonnegative initial groups of population in R+

Proposition 2.1. The solutions of problem (2.2)–(2.3) are symbolized by (S, A, B, R) and are positive for all t > 0. Proof. It is clear that the right-hand side functions of system (2.2) satisfy the conditions of differentiability, which implies that the unique maximal solution for any associated

Chapter 2 • Hepatitis B virus transmission via epidemic model

33

Cauchy problem exists. Let ψ = αA(t) + γ αB(t). The first equation of model (2.2) can be written as   βI (t) dS(t) = N + γ1 R(t) − + (μ0 + v) S(t). dt π(I ) The solution of the above equation implies that   S(t) = S(0) exp − (μ0 + v)t +

t

0

t

βI (x) dx π(I )



  + exp − (μ0 + v)t +

 (N + γ1 R(y)) exp −(μ0 + v)y +

0



0

 βI (x) dx 0 π(I )  βI (u) du dy (2.5) π(I ) t

for all t > 0, which implies that S(t) > 0. On the other hand, the solution of the second equation of system (2.2) is given by I (t) = I (0) exp {−(μ0 + γ + μ1 )t} + exp {−(μ0 + γ + μ1 )t} t βI (y)S(y) exp {−(μ0 + γ + μ1 )} ydy, π(I ) 0

(2.6)

which proves that I (t) ≥ 0. Similarly, it can be shown that R(t) are nonnegative values. Proposition 2.2. The solutions of problem (2.2)–(2.3) are bounded. dN Proof. Let N = S +I +R, then it implies that dN dt +μ0 N = N −μ1 I . Clearly, dt +μ0 N ≤ N gives that   N N exp −μ0 t. + N (0) − (2.7) 0 < N (t) ≤ μ0 μ0

It could be noted from the last equation that whenever t → ∞, 0 < N ≤

N μ0 .

2.3 Stability analysis We discuss the qualitative analysis of model (2.2). For this, first we find the disease-free equilibrium, which is obtained by equating the rate of change and I to zero. We get N −

βI S − (ν + μ0 )S + γ1 R = 0, π(I )

βI S − (γ + μ1 + μ0 )I − T (I ) = 0, π(I ) νS + γ I − (γ1 + μ0 )R + T (I ) = 0.

(2.8)

Replacing the subclasses S, I , and R respectively by S 0 , I 0 , and R 0 , we obtain the fixed point symbolized by E 0 = (S 0 , 0, R 0 ), called the disease-free equilibrium: 

N (γ1 + μ0 ) νN  , R0 = . (2.9) S0 = (μ0 + ν)(μ0 + γ ) + γ ν (μ0 + ν)(μ0 + γ1 ) + νγ

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We find the basic reproductive number symbolized by R0 . For this quantity, we follow the next-generation operator method. It is clear that the infected compartment of model (2.2) is only I (t). So, according to the next-generation method, we have   

I d , V = [μ0 + μ1 + γ + r]| 0 , F = βS E dI π(I ) E 0 where F is the matrix containing the nonlinear terms and V is the matrix containing the linear terms of the model. Moreover, the proposed quantity is the spectral radius of the matrix F V −1 which takes the form R0 =

βN (μ0 + γ ) . μ0 (μ0 + μ1 + γ + r)(μ0 + ν + γ )

(2.10)

We discuss the biological interpretation of the basic reproductive number (R0 ), which describes that some of the epidemic parameters of the proposed model (2.2) are directly proportional to the disease transmission, but some have negative influence. This shows that those epidemic parameters which are directly proportional to the basic reproductive number, and if the value of these parameters are increases then the disease transmission will also increases. On the other hand, if the values of those parameters that are inversely proportional to the basic reproductive number increase, then the value of the basic reproductive number will decrease. It could be noted that the parameters {γ , r} and β have, respectively, negative and positive influence on R0 , and therefore increase (decrease) in their values leads to negative and positive influence on R0 as depicted in Figs. 2.1 and 2.2. This illustrates clearly the possibility of the infection on the basis of this analysis. One may make a control mechanism for possible control by increasing (decreasing) those parameters, which will have a negative (positive) effect on R0 . We calculate the endemic equilibrium of system (2.2). For this, we solve the system at a steady state, which gives π(I )[(μ0 + γ + μ1 )I + T (I )] , βI

 1 ν[(μ1 + μ0 + γ1 )I + T (I )]π(I ) R(t) = γI + + T (I ) , μ0 γ βI βSI − (ν + μ0 )S + γ R = 0. N − π(I ) S(t) =

(2.11)

Making use of substitution of the values of S(t) and R(t) in the last equation of system (2.11), we derive the relation given by [(γ + μ0 + μ1 )μ0 + γ (μ0 + μ1 )]I μ0 + γ (μ0 + γ1 + μ1 )π(I )μ0 (ν + μ0 + γ ) − (γ + μ0 )β (μ0 + ν + γ )μ0 π(I )T (I ) T (I )μ0 − − . (γ + μ0 )βI γ + μ0

G(I ) = N  −

(2.12)

Chapter 2 • Hepatitis B virus transmission via epidemic model

35

FIGURE 2.1 The plot visualizes the variation of the basic reproductive number against the epidemic parameters v and r, and other parameter values are: N = 100,  = 2, β = 0.0079, γ = 0.0001, μ0 = 0.16, and μ1 = 0.00001.

FIGURE 2.2 The plot demonstrates the variation of the basic reproductive number against the epidemic parameters γ and β, and other parameter values are: N = 100,  = 2, μ0 = 0.16 and μ1 = 0.00001, v = 0.2, and r = 0.4.

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Putting π(0) = 1, as stated by assumption (1), and taking lim as I approaches 0, we get the following relation: lim G(I ) =

I →0

 (μ0 + γ + ν)(μ1 + μ0 + r + γ1 )μ0 μ 0 + μ 1 + γ1 R0 − . (γ + μ0 )β μ0 + μ1 + γ + r

Since it is very much clear that, for I > 0,

dπ(I ) dI

(2.13)

> 0, then

[(μ0 + γ1 + μ1 )μ0 + γ (μ1 + μ0 )]I γ + μ0 (ν + μ0 + γ )T (I )μ0 π(I ) T (I )μ0 − − , (γ + μ0 )βI γ + μ0

G(I ) < N  −

(2.14)

and by taking lim as I approaches ∞, we have G(I ) = −∞. So, from the above, the results regarding the equilibria of the proposed model are presented in the subsequent theorem. Theorem 2.1. Let assumptions (1) and (2) hold, then the proposed model (2.2) has a diseasefree equilibrium E 0 such that  E = 0



 (γ + μ0 )N  ν N (γ + μ0 ) , , 0, (ν + μ0 )(γ + μ0 ) + γ ν (μ0 + ν)(γ + μ0 ) + νγ γ + μ0

which holds for all parameter values. Moreover, if R0
1, then model (2.2) possesses a unique 1 +γ +μ0 positive solution for T (I ) = I r; however, if μμ1 +γ +r+μ0 < R0 , then model (2.2) may give one or two equilibria whenever T (I ) = K.

Proof. We consider system (2.11) and, making use of the values of S and R in the last equation of system (2.11), we have [(μ1 + μ0 + γ )μ0 + γ (ν + μ0 )]I γ + μ0 (γ + μ1 + μ0 )μ0 (μ0 + ν + γ )π(I ) − (γ + μ0 )β (γ + μ0 + ν)μ0 T (I )π(I ) T (I )μ0 − − = 0. (γ μ0 )βI γ + μ0

G(I ) = N  −

(2.15)

Using assumption (1), i.e., π(0) = 1, we get lim G(I ) =

I →0

 (γ + ν + μ)μ0 (γ + r + μ1 + μ0 ) μ0 + γ + μ1 R0 − . β(μ0 + γ ) r + μ1 + μ0 + γ

(2.16)

Chapter 2 • Hepatitis B virus transmission via epidemic model

37

Again, in case of T (I ) = rI , the above assertion may take the form given by [(r + γ + μ1 + μ0 )μ0 + γ (μ1 + μ0 )] dG(I ) =− dI γ + μ0 (ν + μ0 + γ )μ0 (μ1 + μ0 + r + γ ) dπ(I ) < 0. − (γ + μ0 )β dI Clearly,

dπ(I ) dI

(2.17)

> 0 whenever I > 0, then from assumption (2) we have [(μ0 + γ + μ1 )μ0 + (μ1 + μ0 )γ ]I γ + μ0 (μ0 + γ + ν)μ0 T (I )π(I ) μ0 T (I ) − . − (γ + μ0 )Iβ γ + μ0

G(I ) < N  −

(2.18)

Clearly, it could be noted that lim G(I ) = −∞ as I → ∞. Also, the differentiation of G(I ) with respect to I and using T (I ) = K lead to the assertion given by dG(I ) [(μ0 + γ + μ1 )μ0 + (μ1 + μ0 )γ ] =− dI γ + μ0 (μ0 + γ + μ1 )μ0 (ν + μ0 + γ ) dπ(I ) − (γ + μ0 )β dI   (μ0 + γ + ν)μ0 K d π(I ) . − (γ + μ0 )β dI I

(2.19)

μ0 +μ1 +γ μ0 +μ1 +γ +r , then model (2.2) possesses no positive solution from T (I ) = 0 +μ1 +γ I r and T (I ) = K. Also, if μμ0 +μ < R0 < 1, then model (2.2) again possesses no positive 1 +γ +r solution, while if R0 > 1, then model (2.2) possesses a unique positive solution for T (I ) = 0 +μ1 +γ I r. Moreover, if μμ0 +μ < R0 , then system (2.2) gives one or two equilibria for T (I ) = K. 1 +γ +r

Therefore, if R0
1. Theorem 2.2. Suppose that assumptions (1) and (2) hold. If R0 < 1, then Eq. (2.11) possesses no positive solution in the case of T (I ) = I r, while it has a unique positive solution whenever 1 +γ )(μ0 +ν+γ )+βK R0 > 1. Moreover, Eq. (7) possesses no positive solution whenever R0 < (μ(μ0 +μ 0 +μ1 +γ +r)(μ0 +ν+γ ) in the case of T (I ) = K; on the other hand, it has one or two positive solutions if R0 > (μ0 +μ1 +γ )(μ0 +ν+γ )+βK (μ0 +μ1 +γ +r)(μ0 +ν+γ ) . Proof. If T (I ) = I r, then Eq. (6) implies that

 (μ0 + r + μ1 + γ1 )μ0 + (μ1 + μ0 )γ I G1 (I ) = N  − γ + μ0 (μ0 + γ1 + r + μ1 )μ0 (ν + μ0 + γ )π(I ) . − β(μ0 + γ )

(2.20)

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It could be noted from assumption (1) that π(I ) is an increasing function whenever π(0) = 1 and I > 0, thus dG1 (I ) [(μ1 + μ0 + γ1 + r)μ0 + (μ1 + μ0 )γ ] =− dI γ + μ0 (μ1 + μ0 + γ1 + r)μ0 (γ + ν + μ0 ) dπ(I ) − 0, so Eq. (2.20) possesses no solution, while it has a unique positive solution whenever R0 > 1. In a similar way, Eq. (2.12) can be rewritten as in the case of T (I ) = K: [(μ0 + γ + μ1 )μ0 + γ I (μ1 + μ0 )] (μ0 + μ1 + γ )μ0 (γ + ν + μ0 )π(I ) − μ0 + γ β(γ + μ0 ) (μ0 + ν + γ )μ0 Kπ(I ) Kμ0 − − . (γ + μ0 )βI γ + μ0 (2.24)

G2 (I ) = N  −

Now, the substitution G2 (I ) = 0 leads to the following assertion: [μ0 (γ + μ0 + μ1 ) + (μ1 + μ0 )γ ]I (ν + μ0 + γ )μ0 Kπ(I ) = N − β(γ + μ0 )I γ + μ0 (μ0 + γ + μ1 )μ0 Kμ0 × (γ + ν + μ0 )π(I ) − − . (μ0 + γ )β γ + μ0 Suppose that G3 (I ) =

(ν+μ0 +γ )μ0 Kπ(I ) (γ +μ0 )βI

(2.25)

whenever G3 (I ) is positive for every I > 0, so from

d π(I ) assumption (2), there exists δ > 0 such that dI ( I ) > 0 for 0 < I < δ and δ < I . Thus lim as I → 0 and I → ∞ gives ∞. Now

[(μ0 + γ + μ1 )μ0 + (μ1 + μ0 )γ ]I γ + μ0 Kμ0 (μ0 + γ + μ1 )μ0 (ν + μ0 + γ )π(I ) − − . (γ + μ0 )β γ + μ0

d π(I ) dI ( I )

< 0 for

G3 (I ) = N  −

(2.26)

Chapter 2 • Hepatitis B virus transmission via epidemic model

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The rate of change of the above function leads to the following assertion: dG3 (I ) [(μ0 + γ + μ1 )μ0 + γ (μ1 + μ0 )] =− dI γ + μ0 (γ + μ0 + μ1 )μ0 (ν + μ0 + γ ) dπ(I ) < 0. − (γ + μ0 )β dI

(2.27)

The substitution of I = 0 leads to G3 (0) = N  −

(μ0 + γ + μ1 )μ0 (ν + μ0 + γ ) Kμ0 . − (γ + μ0 )β γ + μ0

(2.28)

Rearrangement and the use of the value of R0 gives

(μ0 + r + γ + μ1 )(μ0 + γ + ν)μ0 R0 (γ + μ0 )β  (γ + μ1 + μ0 )(ν + μ0 + γ ) + Kβ − . (μ1 + μ0 + r + γ )(ν + μ0 + γ )

G3 (0) =

(2.29)

If conditions (1) and (2) hold and R0 < 1, then Eq. (7) possesses no positive solution in the case of T (I ) = I r, but it has a unique positive solution whenever R0 > 1. Also 1 +γ )(μ0 +ν+γ )+βK Eq. (7) possesses no positive solution whenever R0 < (μ(μ0 +μ in the case 0 +μ1 +γ +r)(μ0 +ν+γ ) of T (I ) = K; on the other hand, it gives none, one, or two positive solutions whenever 1 +γ )(μ0 +ν+γ )+βK R0 > (μ(μ0 +μ depending on the value of parameter. 0 +μ1 +γ +r)(μ0 +ν+γ ) Furthermore, let π(I ) = 1 + dI p , then the following two cases arise continuously various with T (I ). Case 1: We define a function f1 (I ) as follows:

 (γ + μ1 + r + μ0 )μ0 + γ (μ1 + μ0 ) β p . f1 (I ) = dI + (μ0 + γ + r + μ1 )(μ0 + ν + γ )μ0

(2.30)

Eq. (2.30) states that the above function is increasing, and if T (I ) = rI , then Eq. (2.12) can be written as f1 (I ) = (R0 − 1).

(2.31)

Case 2: Again, a function is denoted by f2 (I ) as follows: f2 (I ) = (μ0 + γ + μ1 )μ0 (ν + μ0 + γ )dI p + (μ0 + γ + ν)μ0 dI p−1 (μ0 + γ + ν)Kμ0 . + [(μ0 + γ + μ1 )μ0 + (μ1 + μ0 )γ ]βI + I

(2.32)

Eq. (2.32) describes that the above function is nonnegative, which implies that limI →0 f2 (I ) = limI →∞ f2 (I ) = ∞. So, the global minimum of Eq. (2.32) is defined by fmin = inf f2 (I ) = min f2 (I ). I >0

I >0

(2.33)

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Now, if T (I ) = K, then Eq. (2.12) can be rewritten as

f2 (I ) = (μ0 + γ + μ1 )μ0 (ν + μ0 + γ ) R0 −

μ 0 + γ + μ1 μ0 + γ + μ1 + r

 (2.34)

− μ0 Kβ. We state the following result on the basis of the above findings. Lemma 2.2. If R0 < 1, then Eq. (2.31) has no positive roots; while if R0 > 1, then Eq. (2.31) 0 +μ1 +γ , then both Eqs. (2.31) and (2.34) have possesses a unique positive root; and if R0 < μμ0 +μ 1 +γ +r μ0 +μ1 +γ μ0 +μ1 +γ +r < R0 < 1, then Eq. (2.31)

 0 (μ0 +γ +ν) 1 +μ0 +γ has no positive roots, while if Ic < (μ0 +μ1 +γμ)μ R0 − μμ1 +μ , it may have posi0 βr 0 +r+γ

no positive roots for either T (I ) = I r or T (I ) = K. If

(μ0 +γ +μ1 )μ0 (ν+μ0 +γ ) 1 +μ0 +γ [R0 − μμ1 +μ ] and if μ0 βr 0 +γ +r μ1 +μ0 +γ fmin > (μ0 + γ + μ1 )μ0 (ν + μ0 + γ )[R0 − μ1 +μ0 +r+γ ] − βrIc μ0 , then Eq. (2.34) has no pos)μ0 (ν+μ0 +γ ) 1 +μ0 +γ itive solutions. Moreover, if Ic < (μ1 +μ0 +γβμ [R0 − μμ1 +μ ] and fmin < (μ0 + γ + 0r 0 +r+γ μ1 +μ0 +γ μ1 )μ0 (ν + μ0 + γ )[R0 − μ1 +μ0 +r+γ ] − βrIc μ0 , then Eq. (2.34) may have two positive solutions.

tive roots along with another condition: if Ic
μ0 βr μ1 + μ0 + γ + r

41

(2.38)

holds, then Eq. (2.35) possesses no positive solution, while it has a unique positive solution if

 (μ0 + γ + μ1 )μ0 (ν + μ0 + γ ) μ1 + γ + μ0 > 0 and Ic > R0 − . (2.39) βμ0 r μ1 + μ0 + r + γ Condition (2.39) holds if the following inequality is satisfied:  

1  3  1 1 bc 1 d √ 3 1 b 1 1 b 3 1 bc 1 d √ 3 b Ic > − + + − . (2.40) + − ϒ + − − ϒ − 27 a 6 a2 2 a 2a a 3 27 6 a 2 2 a Then model (2.2) from the case of T (I ) = K = rIc has a unique endemic equilibrium with 2

3

b3 bc d c b2 ϒ = − 27a + − + − , and Eq. (2.35), if 3 2a 3a 6a 2 9a 2

 (μ0 + γ + μ1 )μ0 (μ0 + γ + ν) μ1 + μ0 + γ R0 − , ≤ 0 and Ic < βμ0 r μ1 + μ0 + r + γ

(2.41)

may have two positive solutions. Proof. On the basis of Lemma (2.2), regarding model (2.2), we obtain that • • •

whenever R0 < 1, system (2.2) has no endemic equilibrium; 1 +r+μ0 )μ0 +γ (μ1 +μ0 )]β whenever R0 > 1 and f1 = [(γ(μ+μ Ic + dI p < R0 − 1, model (2.2) has no 0 +ν+γ )μ0 (μ0 +γ +μ1 +r) positive endemic state for T (I ) = I r; 0 (r+μ0 +γ +μ1 )+(μ1 +μ0 )γ ]β whenever R< 1 and f1 (I ) = [μ(μ Ic + dI p > R0 − 1, the proposed sys0 +γ +ν)μ0 (μ0 +γ +r+μ1 ) tem, as reported by Eq. (2.2), possesses at least one positive endemic state for the case T (I ) = rI .

For the simplicity of expression, we assume that p = 2 and make use of the following substitution in Eq. (2.35): • • • •

a = (μ0 + γ + ν)μ0 d(μ1 + μ0 + γ ); b = dμ0 (μ0 + γ + ν) + β[(μ0 + γ + μ1 )μ0 + (μ1 + μ0 )γ ]; 0 +μ1 +γ c = βμ0 K + (μ0 + γ + ν)μ0 (μ0 + γ + μ1 )[R0 − μμ0 +μ ]; 1 +γ +r d = (μ0 + γ + ν)μ0 K.

Moreover, by the substitution of ψ = −3bd + c2 , ϕ = −9ad + bc and ω = −3ac + b, = ϕ 2 − 4ωψ. Thus, for polynomial (2.35), we have:

 (μ0 +γ +ν)μ0 (μ0 +γ +μ1 ) μ0 +μ1 +γ R0 − μ0 +μ1 +γ +r , it implies that the left• If c is positive and K > μ0 βr hand side of Eq. (2.35) is also always positive for every I > 0; and consequently Eq. (2.35) possesses no solution.

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If > 0, then Eq. (2.35) has one real solution. If ≤ 0, then Eq.

(2.35) has three  real roots. Since b is positive and if K > (μ0 +γ +ν)μ0 (μ0 +γ +μ1 ) μ0 βr

• •

R0 −

μ0 +μ1 +γ μ0 +μ1 +γ +r

0 (μ0 +γ +μ1 ) If K > (μ0 +γ +ν)μ [R0 − μ0 βr state in the case of T (I ) = K.

, then Eq. (2.35) may give two positive solutions.

μ0 +γ +μ1 μ0 +μ1 ++r+γ

If > 0, then it implies the above, i.e., K
 with H () < 0, then system (2.2) possesses at least one endemic

state for T (I ) = K. •

(μ1 +μ0 +γ )μ0 (ν+μ0 +γ ) μ0 βr



μ1 +γ +μ0 μ1 +γ +μ0 +r



If ≤ 0, K < R0 − , and Kμ0 < −(μ0 + γ + μ1 )μ0 dI 3 −

   

[(μ0 +γ +μ1 )μ0 +(μ1 +μ0 )γ ]β μ1 +μ0 +γ μ0 βK 2 dμ0 + I + Kμ0 +μ0 (μ1 +μ0 +γ ) R0 − μ1 +μ0 +r+γ − μ0 +ν+γ I ν+μ0 +γ () < 0, then system (2.2) may have two endemic states whenever and Ic <  with H T (I ) = K.

Thus the above conditions for Eq. (2.41) and the proposed model (2.2) for T (I ) = Ic r = K possess a unique endemic state whenever the following inequalities hold:

dμ0 (μ1 + μ0 + γ )(ν + μ0 + γ )Ic2 + dμ0 (γ + ν + μ0 )  (2.42) + [μ0 (γ + μ1 + r + μ0 ) + (μ1 + μ0 )γ ]β Ic 

μ 0 + γ + μ1 < 0. − (μ0 + γ + μ1 )μ0 (μ1 + μ0 + r + γ ) R0 − μ0 + γ + μ1 + r We suppose that A = (μ0 + γ + μ1 )μ0 d(ν + μ0 + γ ), B = (μ0 + γ + ν)dμ0 + [(μ0 + γ + μ1 + r)μ0 + (μ1 + μ0 )γ ]β,

 γ + μ1 + μ0 C = −(μ0 + γ + μ1 )(μ1 + μ0 + r + γ )μ0 R0 − μ 1 + γ + r + μ0

(2.43)

and () < 0. Ic >  and H

(2.44)

Chapter 2 • Hepatitis B virus transmission via epidemic model

43

Eq. (2.35) holds whenever conditions (2.41) and (2.44) hold. Moreover, model (2.2) from T (I ) = K = rIc may possess two endemic states when the following holds:

+ μ0 + μ1 )(μ0 + γ + ν) + (μ0 + γ + ν)μ0 d  + [(μ0 + γ + μ1 + r)μ0 + (μ1 + μ0 )γ ]β Ic 

μ1 + μ0 + γ >0 − (μ0 + γ + μ1 )μ0 (r + μ0 + γ + μ1 ) R0 − μ1 + μ0 + r + γ

dμ0 Ic2 (γ

(2.45)

and Ic < . Theorem 2.4. If R0
R0 >

demic state and fmin > (μ0 + γ + ν)[R0 − T (I ) = I r, the proposed system (2.2) possesses at least one endemic state whenever R0 > 1 and f1 (Ic ) > R0 − 1 hold.

Now, we discuss the qualitative analysis of the proposed model (2.2) around the endemic state. We find the Jacobian matrix as follows:  J=

γν μ0 +γ

−

βI π(I ) − (μ0 βI π(I )

+ ν)

γ d I μ+γ (γ + T (I )) − βS dI ( π(I ) ) d I βS dI ( π(I ) − (μ0 + μ1 + γ ) − T (I )

 .

(2.46)

We assume that λ1 and λ2 represent the eigenvalues of the above matrix (J )(26), then d λ2 + λ1 = βS dI



 I γν βI + − T (I ) − π(I ) γ + μ0 π(I )

− (μ1 + 2μ0 + γ ν+),

  I (μ0 + γ + ν)μ0 d βS λ1 λ2 = − γ + μ0 dI π(I ) 

(μ0 + γ + μ1 )μ0 + (μ1 + μ0 )γ + μT (I ) βI +β γ + μ0 π(I ) [(μ0 + γ + μ1 ) + T (I )]μ0 (μ0 + γ + ν) . + μ0 + γ

(2.47)

Theorem 2.5. Suppose that conditions (1) and (2) hold, and if R0 < 1, then model (2.2) is locally stable at E 0 , while it is unstable whenever R0 > 1. Also, in the case of T (I ) = I r, model (2.2) provides the existence of a unique endemic state E ∗ . Thus whenever R0 > 1, the

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conditions of the stability analysis are derived as    I (μ0 + r + γ + μ1 )μ0 + γ (μ1 + μ0 ) + μ0 I d |I =I ∗ < min dI π(I ) (μ0 + γ + ν)μ0 S π(I ) μ0 + γ + μ1 + r 1 I , + βS S π(I )  (μ0 + r + μ1 + γ + ν)μ0 + (γ + r + μ1 + μ0 )γ . + Sβ(γ + μ0 )

(2.48)

Moreover, the conditions imposed on π(I ) are assumed to hold. Then, from the case of T (I ) = I r, model (2.2) may give two endemic states. More precisely, the endemic state E1∗ exists if   I d (μ1 + μ0 + γ + r)μ0 + (μ0 + μ1 )γ I2∗ > dI π(I ) I =I ∗ (μ0 + γ + ν)μ0 S2∗ π(I ∗ ) 2 (2.49) (γ + μ0 + r + μ1 ) . + (βS2∗ ) Similarly, E2∗ = (S2∗ , I2∗ , R2∗ ) is always a saddle point provided   I d (μ1 + γ + r + μ0 ) + (μ0 + μ1 )γ I1∗ < dI π(I ) I =I ∗ (μ0 + γ + ν)μ0 S1∗ π(I1∗ ) 1

(r + μ0 + γ + μ1 ) + . βS1∗

(2.50)

Moreover, if d dI



 I (μ1 + ν + μ0 + γ + r)μ0 + γ (γ + μ0 + r + μ1 ) < π(I ) I =I ∗ (γ + μ0 )βS1∗

1 I∗ + ∗ 1∗ , S1 π(I1 )

1

then E1∗ is locally asymptotically stable, while it is a source whenever   I (μ0 + μ1 + γ + ν + r) + γ (μ0 + γ + r + μ1 ) d > dI π(I ) I =I ∗ (γ + μ0 )βS1∗ 1 I∗ + ∗ 1∗ . S1 π(I1 )

1

(2.51)

(2.52)

Proof. The Jacobian/linearized matrix of system (2.2) is given by the following matrix:   ⎛ ⎞ Iβ d I γ − (ν + μ ) −βS − π(I 0 ) dI π(I ) ⎜ ⎟ ⎟, J =⎜ (2.53) βI ∗ ⎝ ⎠ D 0 π(I ) ν γ + T (I ) −(μ0 + γ )

Chapter 2 • Hepatitis B virus transmission via epidemic model

d where D ∗ = −(μ0 + μ1 + γ ) + βS dI

becomes

⎛

−(ν + μ0 ) J =⎝ 0 ν



 I π(I )

45

− T (I ). Then Eq. (2.53) around E 0 = (S 0 , 0, R 0 )

−βS 0 −(μ0 + γ + μ1 ) + βS 0 γ

⎞ γ ⎠, 0 −(μ0 + γ )

(2.54)

which gives λ1 = −ν − μ0 , λ2 = (μ1 + r + γ + μ0 )(R0 − 1), and λ3 = −γ − μ0 .

(2.55)

So, it is clear from Eq. (2.55) that whenever R0 is less than unity, then λ2 is negative, and we conclude that the disease-free state is asymptotically stable with local nature. However, R0 = 1 implies that λ2 = 0. Moreover, whenever R0 is greater than unity, the disease-free state becomes unstable and loses stability due to alternative signs of eigenvalues. Now, around the endemic state E ∗ the characteristic equation of J as defined by Eq. (2.53) is given by the following relation: λ3 + λ2 σ1 + λσ2 + σ3 = 0, where

(2.56)

 ∗  I d βI ∗ ∗ + (μ , + μ + ν + T (I )) + σ1 = −S β 1 0 dI π(I ∗ ) π(I ∗ )  ∗    I d I ∗β + μ σ2 = −S ∗ β + μ + ν + γ + 0 0 dI π(I ∗ ) π(I ∗ ) ∗

+ (γ + μ)(μ1 + μ0 + γ + T (I ∗ )) I∗ + (ν + μ0 )(T (I ∗ ) + μ0 ) + νγ + (T (I ∗ ) + μ0 ) , π(I ∗ )  ∗  I d + (μ0 + γ + μ1 σ3 = (μ0 + ν + γ )βS ∗ μ0 dI π(I ∗ )   βI ∗ + ν + μ + T (I ∗ )) + γ ν 1 π(I ∗ ) βI ∗ . + (T (I ∗ ) + γ )γ π(I ∗ )

(2.57)

Upon the use of the Routh-Hurwitz property, we get d βI ∗ I∗ ( ) + γ (γ + T (I ∗ )) ∗ dI π(I ) π(I ∗ )    βI ∗ + μ1 + μ0 + (μ0 + T (I ∗ ) + γ + μ1 ) π(I ∗ ) + μ1 + νγ + γ  ∗  I I ∗β ∗ ∗ d + T ν + μ1 + μ0 + νγ +ν+ (I ) − S β ∗ π(I ) dI π(I ∗ )

σ1 σ2 − σ3 = −(γ + ν + μ0 )βμ0 S ∗

(2.58)

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Advances in Epidemiological Modeling and Control of Viruses

I ∗β I ∗β ∗ + T (I ) + (ν + μ )(T (I ) + μ ) + (T (I ) + μ ) 0 0 0 π(I ∗ ) π(I ∗ )  ∗  I d . + (γ + μ0 )(μ1 + μ0 + T (I ) + γ ) − S ∗ β dI π(I ∗ )

+

Since 0 ≤ T (I ) at I = 0 and π(I ) = 1 + dI p , so   I π(I ) I − π(I ) d =− . dI π(I ) (π(I ))2

(2.59)

The simplification of the above equation leads to the assertion given by π(I ) I − π(I ) = I p d(p − 1) − 1 > 0,

(2.60)

which implies that Ip >

1 , (p − 1)

(2.61)

which leads to the conclusion that σj > 0 for j = 1, . . . , 3 and σ1 σ2 − σ3 is positive. Theorem 2.6. Let p = 2 for π(I ) = dI p + 1, then the following must be true: 1. Whenever R0 > 1, f1 (Ic ) ≥ R0 − 1 and I ∗ > √1 .

d 

μ (μ +μ +γ )(μ +ν+γ ) μ +μ +γ 1 b3 bc d ∗ 0 0 1 0 0 1 R0 − μ0 +μ1 +γ +r , and Ic < − 27a 2. > 0, I > √ , Ic < 3 + 6a 2 − 2a + μ0 βr √

1

3

ϒ



d

√ b3 bc d + − 27a ϒ 3 + 6a 2 − 2a −

1

3

2

3

b b3 bc d c b2 − 2a with ϒ = − 27a + − + − ≥ 0. 3 2a 3a 6a 2 9a 2

Remark 2.2. Result (2.6) shows R0 and Ic for finding whether there are potential multiple endemic or endemic outbreaks, or if there are two critical parameters under the conditions: R0 is greater than unity and f1 (Ic ) > R0 − 1. The disease-free state E 0 loses the stable position, and so a unique endemic state E ∗ emerges whenever T (I ) = rI , ultimately which is stable locally if I ∗ > √1 . The local dynamics results provide a base for the global properd ties of the model as stated by the subsequent result. 0 Theorem 2.7. The disease-free state of the model  that is E is stable globally whenever R0 >

1 and Ic >

(μ1 +μ0 +γ )μ0 (ν+γ +μ0 ) μ0 βr

R0 −

μ0 +μ1 +γ μ0 +μ1 +γ +r

.

Proof. It is clear from Theorem (2.6) that whenever R0 is less than unity, E 0 of system (2.2) is stable with local nature, and thus I (t) → 0 for t → ∞. Since (S, I, R) ∈ R3+ , we may write from model (2.2) that βSI dI (t) = − (μ0 + μ1 + γ )I − T (I ), dt π(I )

(2.62)

Chapter 2 • Hepatitis B virus transmission via epidemic model

47

which implies that dI (t) < I (R0 − 1)(μ0 + r + γ + μ1 ), dt

(2.63)

i.e., monotonous, and so has negative real roots. If R0 < 1 and if t → ∞, then it implies that I (t) → 0. So the comparison theorem gives that, whenever t → ∞, I (t) → 0. Hence, for small perturbation, 0 < , there exists t0 > 0 such that I (t) ≥ , ∀ t ≥ t0 . Again system (2.2) leads to dR(t) = Sν − (γ + μ0 )R + T (I ) + γ I, dt

(2.64)

dR(t) < (r + γ ) + νS − R(γ + μ0 ) for t ≥ t0 . dt

(2.65)

which implies that

In a similar way, the first equation of model (2.2) gives βSI dS(t) = N − − (μ0 + ν)S + γ R(t), dt π(I )

(2.66)

which implies that dS(t) < N − (β + μ0 + ν)S + γ R, dt

 N (μ0 +γ ) as t → ∞ and S(t) → whose solution gives that R(t) → μ0ν+γ (μ0 +ν)(μ 0 +γ )−γ ν

(2.67) N (μ0 +γ ) (μ0 +ν)(μ0 +γ )−γ ν

as t → ∞. Remark 2.3. The above result surely (2) possesses no endemic state if

gives that model  (μ1 +μ0 +γ )μ0 (γ +ν+μ0 ) μ1 +μ0 +γ R0 − μ1 +μ0 +r+γ as it holds the phenomenon of bifurR0 < 1 and Ic > μ0 βr cations (backward). Noted that the condition implies the elimination of disease outbreak and making possible that the basic reproductive number is less than unity with large maximal capacity of treatment Ic . Theorem 2.8. Let p = 2 in π(I ) = 1 + dI p , then model (2) possesses a unique endemic state E ∗ and is globally asymptotically stable: 0 +γ 1. If R0 is greater than unity, fmin < (μ0 +γ +μ1 )μ0 (ν +μ0 +γ )[R0 − μμ10+μ +r+γ +μ1 ]−βrμ0 Ic ,

I∗ >

√1 , and f1 (Ic ) > R0 d



− 1 for T (I ) = rI .

1 1

√ 3 √ 3 b3 bc d b3 bc d b 2. If > 0, I ∗ > √1 , Ic < − 27a + − + + − + − − ϒ − 2a with ϒ 3 2a 2a 6a 2 27a 3 6a 2 d 2

3

b3 bc d c b2 + − + − ≥ 0 for T (I ) = K. ϒ = − 27a 3 2a 3a 6a 2 9a 2

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Proof. To investigate the global properties of model (2.2) at E ∗ in , we apply the method of geometrical approach. If R0 is greater than unity and π(I ) = 1 + I 2 d, then

 μ0 (μ0 + γ + ν)(μ0 + γ + μ1 ) μ1 + μ0 + γ R0 − , K< βμ0 r μ1 + γ + μ0 + r along with Theorem (2.3) satisfied, so model (2.2) possesses a unique endemic state E ∗ in . This implies that model (2.2) is persistent, which implies that there is a constant  > 0 and the solution (S(t), I (t), R(t)) in  possesses lim inf |S(t), I (t), R(t)| ≥ . t→∞

Let the linearized and additive compound matrices of model (2.2) at E ∗ be respectively J2 |2| and J2 , which takes the form ⎛

−η11 ⎝ J2 = η21 η31 and

⎛ −(η11 + η22 ) [2] ∂f |2| J2 = =⎝ η32 ∂x −η31

⎞ −η13 η23 ⎠ η33

−η12 η22 η32

η23 −(η11 + η33 ) η21

(2.68)

⎞ −η13 ⎠, η12 −(η22 + η33 )

(2.69)

where   

 d I βI + βS , μ1 + r + γ + ν + 2μ0 + π(I ) dt π(I )

   I d η22 + η33 = (μ1 + r + γ + 2μ0 ) + βS , dt π(I )   βI , η23 = 0, η13 = −γ , η11 + η33 = ν + 2μ0 + γ + π(I )   I d βI , a21 = , η31 = −ν. η32 = (r + γ ), η12 = −βS dt π(I ) π(I ) η11 + η22 =

Let B be a matrix such that H = Pf P −1 + PJ [2] P −1 =



H12 H22

H11 H21

 ,

where I Iβ + 2 + Sβ H11 = −α − r − μ2 − μ1 − π(I ) I H21 = (0, 0)T , H12 = (0, 0),



 I π (I ) 1 , − π(I ) π2

(2.70)

Chapter 2 • Hepatitis B virus transmission via epidemic model  H22 =

−ν − γ −

Iβ π(I )

− 2μ0 +

I I2

−γ − r − μ1 − 2μ0 +

0

0 

49

 I π (I ) π 2 (I )

−

1 π(I )



Sβ +

.

I I2

We take a vector (f1 , f2 , f3 ) in R3 and define ||.|| by f1 , f2 , f3 = max { f1 , f2 + f3 } . Defining the Lozinski measure symbolized by ϕ(B) as (H ) ≤ sup {h2 , h1 } = sup { (H11 ) + H12 ,  (H22 ) + H21 } , implies that   I π (I ) I Iβ Sβ h1 =  (H11 ) + H12 = − α + r + 2μ0 + + μ1 − 1 + − 2, π(I ) π(I ) π(I ) I I ≤ − − (μ2 + 2μ0 + α + r) .   I  Iβ I h2 =  (H22 ) + H21 = max − γ + 2μ0 + + ν − A∗ − 2 , π(I ) I I ≤ − (r + 2μ0 + μ1 + γ ) − , I     Sβ π (I ) I where A∗ = r + 2μ0 + μ1 + γ + Iπ(I ) − 1 π(I ) − I 2 yields that 

(H ) ≤ sup {h1 , h2 } = − min {(μ2 + 2μ0 + α + r) , (r + 2μ0 + μ1 + γ )} −

I I

holds for the system as stated by Eq. (2.2) with nonnegative initial condition contained in  ⊂  in which  represents the compact absorbing set. Now we integrate (H ) along with [0, t] to obtain 1 t I (0) 1 − d. (2.71) (H )d(x) ≤ log t 0 t I (t) The application of lim as t → ∞ leads to the assertion t d ˜ = lim sup sup 1  (H )dI ≤ − < 0, 2 t→∞ x0 ∈ t 0

(2.72)

which gives that E ∗ is globally asymptotically stable in the interior of  of the proposed dynamical system (2.2).

2.4 Simulation and concluding remarks In this section, we perform the numerics of the proposed problem to explore the answers to the following issues graphically:

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FIGURE 2.3 The plot visualizes the dynamics of the compartmental populations without treatment for p = 1.

FIGURE 2.4 The plot demonstrates the dynamics of the compartmental populations without treatment for p = 2.

1. The effect of various treatment functions; 2. Various vaccination strategies; and 3. The effect on the epidemic subjected to the nonlinearity of infection force. We use the well-known Runge-Kutta method of the fourth order and assume π(I ) = 1 + dI p to respond to the first two observations. Moreover, choosing the values of the parameter from the articles as reported in [19], we will check the treatment functions, the vaccination impact, and the effect of infection forces. It has been analyzed how d and p affect the βI dynamics of the disease by taking the generalized incidence rate g(I ) = 1+dI p . Clearly, we observed that the number of infected individuals I decreases with increasing the value of d as shown in Figs. 2.3, 2.4, 2.5, 2.6, 2.7, 2.8. However, if we fix d, the level of equilibrium for I increases whenever q increases (see Fig. 2.8). Similarly, the endemic equilibria is stable if R0 > 1.

Chapter 2 • Hepatitis B virus transmission via epidemic model

51

FIGURE 2.5 The plot visualizes the dynamical behavior of the compartmental populations with treatment for p = 1.

FIGURE 2.6 The plot visualizes the dynamical behaviors of the compartmental population with treatment for p = 2.

We formulated an epidemiological model for hepatitis B virus transmission under the effect of saturated incidence rate and saturated treatment function. It is very much clear that the role of threshold quantity is very important in the persistence as well as in the extinction of infection. We found the threshold quantity (basic reproductive number) for our proposed model, and interestingly these findings suggest that whenever T (I ) = rIc and R0 < 1 hold, the model exhibits backward bifurcation. Additionally, the model possesses multiple endemic states if R0 > 1. The analytical findings and the numerics lay out a global picture of how to affect the dynamics of the disease subjected to the treatment function, the incidence rate, g(I )S, and vaccination p. Sufficient conditions are derived, and it is shown that the model is globally stable by using the well-known geometrical approach. We also explored the effect of treatment strategies. The bifurcation analysis has been performed under some conditions on the basic reproductive number (R0 ) and Ic with a fixed

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FIGURE 2.7 The graph visualizes the dynamical behavior of the infected population against the psychological effect.

FIGURE 2.8 The plot shows the relation between the psychological effect d and the infected population.

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incident rate function (π(I ) = 1 + dI 2 ). In addition, we obtained the variation of equilibria level for different values of r, K, and p with the proportionality constant, vaccination, and constant treatment function K. If the values of these parameters are increasing, the level of the infected individuals I is decreasing. In the near future we will use the optimization theory to formulate an optimal control strategy for the elimination of the disease. Moreover, we will study the associated stochastic and fractional order epidemiological models of the reported model by using the stochastic and fractional differential equations. We will also present a comparison between the above analysis and the analysis that could be investigated in the future.

References [1] CDC, Public Health Service inter-agency guidelines for screening donors of blood, plasma, organs, tissues, and semen for evidence of hepatitis B and hepatitis C, MMWR 40 (RR-4) (1991) 1–17. [2] S.J. Zhao, Z.Y. Xu, Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China, Int. J. Epidemiol. 29 (2000) 744–752. [3] J. Mann, M. Roberts, Modeling the epidemiology of hepatitis B in New Zealand, J. Theor. Biol. 269 (2011) 266–272. [4] N.H. Tuan, H. Mohammadi, S. Rezapour, A mathematical model for Covid-19 transmission by using the Caputo fractional derivative, Chaos Solitons Fractals 140 (2020) 110107. [5] S. Rezapour, H. Mohammadi, M.E. Samei, SEIR epidemic model for Covid-19 transmission by Caputo derivative of fractional order, Adv. Differ. Equ. 1 (2020) 1–9. [6] V. Capasso, Mathematical Structures of Epidemic Systems, Lecture Notes in Biomathematics, vol. 97, Springer-Verlag, Berlin, 1993. [7] S. Funk, M. Salath, V. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: a review, J. R. Soc. Interface 7 (2010) 12471256. [8] P. Manfredi, A. Donofrio, Modeling the Interplay Between Human Behavior and the Spread of Infectious Diseases, Springer-Verlag, New York, 2013. [9] F. Verelst, L. Willem, P. Beutels, Behavioural change models for infectious disease transmission: a systematic review 2010.201, J. R. Soc. Interface 13 (2016) 125. [10] Z. Wang, C. Bauch, S. Bhattacharyya, A. donofrio, P. Manfredi, M. Perc, N. Perra, M. Salathe, D. Zhao, Statistical physics of vaccination, Phys. Rep. 664 (2016) 1–113. [11] D. Xiao, S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci. 208 (2007) 419–429. [12] V. Capasso, G. Serio, A generalization of the Kermack Mckendrick deterministic epidemic model, Math. Biosci. 42 (1978) 43–61. [13] S. Ruan, W. Wang, Dynamical behavior of an epidemic model with a nonlinear incidence rate, J. Differ. Equ. 188 (2003) 135–163. [14] W. Liu, S. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol. 23 (1986) 187–204. [15] W. Liu, H. Hethcote, S. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987) 359–380. [16] T. Khan, G. Zaman, M.I. Chohan, The transmission dynamic and optimal control of acute and chronic hepatitis B, J. Biol. Dyn. 11 (1) (2017) 172–189. [17] T. Khan, G. Zaman, A.S. Alshomrani, Spreading dynamic of acute and carrier hepatitis b with nonlinear incidence, PLoS ONE 13 (4) (2018) e0191914. [18] Z. Feng, H.R. Thieme, Recurrent outbreaks of childhood diseases revisited: the impact of isolation, Math. Biosci. 128 (1995) 93. [19] J.M. Hyman, J. Li, Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. Appl. Math. 58 (1998) 912. [20] L. Wu, Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differ. Equ. 168 (2000) 150.

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[21] W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci. 201 (2006) 58–71. [22] W. Wang, S. Ruan, Bifurcation in an epidemic model with constant removal rate of the infectives, J. Math. Anal. Appl. 291 (2004) 775. [23] W. Wang, Epidemic models with nonlinear infection forces, Math. Biosci. Eng. 3 (2006) 267–279.

3 Global dynamics of an HCV model with full logistic terms and the host immune system Jazmín G. Aguilar-Basulto, Eric J. Avila-Vales, and Arturo J. Nic-May Facultad de Matemáticas, Universidad Autónoma de Yucatán, Mérida, Yucatán, Mexico

3.1 Introduction Hepatitis C is a liver disease caused by the hepatitis C virus (HCV): the virus can cause both acute and chronic hepatitis, ranging in severity from a mild illness lasting a few weeks to a serious, lifelong illness. The hepatitis C virus is a blood-borne virus: the most common modes of infection are through exposure to small quantities of blood. This may happen through injection drug use, unsafe injection practices, unsafe health care, transfusion of unscreened blood and blood products, and sexual practices that lead to exposure to blood. Globally, an estimated 71 million people have a chronic hepatitis C virus infection, and approximately 399,000 people died from hepatitis C, mostly from cirrhosis and hepatocellular carcinoma (primary liver cancer) [1]. There is currently no effective vaccine against hepatitis C; however, research in this area is ongoing. According to [2], approximately 3.5 million people in the United States (US) have chronic hepatitis C. In the European community, around 14 million people are chronically infected with HCV, which is about 20% of the global burden of disease due to the HCV infection [3]. The highest reported prevalence rates are located in Africa and Asia [4–6]. The liver is the biggest organ in the body, and it plays an important role in all metabolic processes in the body. The main job of the liver is to filter blood and fight infections, among other functions. Once the liver is infected, its functions are affected too. Hepatitis is an inflammation of the liver that can cause a range of health problems and can be fatal. There are five main strains of the hepatitis virus, referred to as types A, B, C, D, and E. While they all cause liver diseases, they differ in important ways, including the modes of transmission, the severity of the illness, geographical distribution, and prevention methods [7]. Hepatitis can be caused by an illness, some medications, heavy alcohol usage, and toxins, but viruses, which are able to replicate inside the living cells, are the most popular cause for hepatitis. Hepatocyte proliferation plays a substantial role in liver cirrhosis and could be considered as a major risk factor for liver cancer. The decrease in hepatocyte proliferation that Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00008-9 Copyright © 2023 Elsevier Inc. All rights reserved.

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occurs during liver cirrhosis is thought to enhance cancer formation in cirrhotic livers. Cirrhosis develops following long periods of chronic liver disease and is characterized by a decrease in hepatocyte proliferation, indicating an exhaustion of the regenerative capacity of the liver [8]. After exposure to HCV, a strong host immune response is launched. Thus, some patients with hepatitis C virus infection will naturally clear the virus during the early phase of infection without medical intervention. They will become better on their own after several weeks to several months. However, the response fails to eradicate the virus, leading to a chronic infection in which the immune system of the body does not naturally clear the virus. When hepatitis becomes a chronic or long-period illness, the infection may need to be treated with specific medications called antivirals. Antiviral therapy has been used to cure chronically HCV infected patients. It is currently the only available treatment because of the lack of an HCV vaccine. For several years, combinations of antivirals such as interferon-α (IFN-α) and ribavirin (RBV), pegylated-interferon (PEG-INF) and ribavirin (RVB) have been established as effective in viral clearance. In other words, they have been used as medication to reduce the levels of HCV-RNA [9]. The lymphocytes, which are white blood cells, are the most paramount factor of the adaptive immune system. They can be grouped into two important kinds of cells: Blymphocytes, called B-cells, and T-lymphocytes, called T-cells. B-cells and T-cells are the fundamental players in the host immune response. B-cells produce antibodies, which are able to recognize and bind to specific pathogen such as bacteria and viruses interaction, which makes a figure similar to a lock and a key. When a pathogen like HCV enters the human body, there are some specific B-cells receptors which can recognize this foreign intruder and are able to bind some viral proteins. However, some B-cells do not have the specific receptors to recognize this foreign antigen [9]. T-cells come into two different types of cells, T-helper cells and T-killer cells (cytotoxic T-lymphocytes (CTLs)). T-helper cells are considered an essential part in the activation of the B-cells and, in this way, in the release of antibodies. On the other hand, CTLs can recognize the infected cells and then kill them. T-helper cells are also called CD4+ T cells, and CTLs are also called CD8+ T cells. When a cell has been infected, it produces a new virus. Inside the infected cell the viral proteins are introduced on the surface in a blend with the major histocompatibility complex type I (MHC I), which is introduced basically in each cell of the human body. The T-cell receptor with specific CTL is able to recognize these introduced proteins on the infected cells surface and then it will link to the cell. After that, the CTLs will split into memory and effector cells. The job of memory cells is to remain in the host in case of a new infection occurring in the future. On the other hand, the job of effector cells is to eliminate infected cells [9]. Thus, antibodies can reduce the virus load to very low levels to activate the CTLs when they are more efficient in capturing the virus. Similarly, CTLs can reduce the virus load to very low levels to activate the antibodies when they are more efficient in killing the pathogen [9].

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Mathematical models have become an important tool to make sense of the dynamics of viral load and its infectious processes in vivo, including hepatitis B virus (HBV), hepatitis C virus (HCV), and human immunodeficiency virus (HIV). A simple model may play a significant role in the development of a better understanding of the disease and the various drug therapy strategies used against it [10]. Regarding mathematical modelling of HBV, see [11–13]. Taking these facts into account, we propose a model that describes the interaction between uninfected target cells, infected cells, free virus particles, and the immune effector mechanisms. Our system considers the virus dynamics incorporating cytotoxic T cell response. The novelty of our model includes proving global stability of a four-dimensional HCV model that incorporates full logistics terms using the geometric method. To the best of our knowledge, there are just two previous works on four-dimensional HCV models, Banerjee et al. [14] and Ble et al. [15]. We are generalizing Blé et al. article. This chapter is organized as follows. First, we mention two previous works that establish the basis for the development of our model in Sect. 3.2. Next, in Sect. 3.3, we present some important definitions and results that will be useful to us in the following sections. The new extended HCV model is designed in Sect. 3.4 and qualitatively analyzed in Sect. 3.5. Numerical simulations are reported in Sect. 3.6. Finally, we exhibit the conclusions of this work in Sect. 3.7.

3.2 Previous works Now, to establish the basis for the model we will propose and study, we present two mathematical models of HCV dynamics mentioning the principal results obtained.

3.2.1 Modelling virus-immune system interaction Avendaño et al. [16] proposed in 2002 a mathematical model for HCV that describes the dynamics of HCV and the immune system response. For this model four populations were considered: healthy liver cells, infected liver cells, virus load, and CD8+ cytotoxic T cells, which are represented by Hs , Hi , V , and T , respectively. The equations that describe the interaction between these cells and virus particles are given by the following system of ordinary differential equations: H˙ s = βs − kHs V − μs Hs , H˙ i = kHs V − δHi T − μi Hi , V˙ = pHi − μv V ,   T − μT T , T˙ = βT V 1 − Tmax where • •

βs is the per capita production rate of healthy hepatocytes; k is the per capita infection rate;

(3.1)

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• • • • • • • •

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μs is the per capita natural death rate of healthy hepatocytes; δ is the rate at which the T killer cells destroy the infected hepatocytes; μi is the per capita natural death rate of infected hepatocytes; p is the per capita production rate of viruses from the infected hepatocytes; μv is the per capita natural death rate of viruses; βT is the rate of growth of T killer cells; Tmax is the maximum T killer cell population level; μT is the per capita natural death rate of T killer cells.

The analysis of model (3.1) made by Avendaño et al. [16] reveals the existence of two equilibrium states, the uninfected state in which no virus is present and an endemically infected state in which virus and infected cells are present. They computed the basic reproduction number R0 of (3.1) that is a threshold parameter of the outcomes of viral infections. Also they showed that if R0 ≤ 1 then the only virus-free equilibrium exists and it is locally and globally asymptotically stable, i.e., the level of virus load and infected cells will monotonically decrease and ultimately be eliminated. If R0 > 1, the virus-free equilibrium is locally unstable and a unique endemic equilibrium, which is locally asymptotically stable, appears, i.e., the virus persists within host. In 2018, using the Li-Muldowney geometric approach, Blé et al. [15] proved that under some conditions the unique endemic equilibrium is globally asymptotically stable whenever the basic reproduction number is less than unity. The authors did not include logistic terms in uninfected nor infected cells. This leads us to the following mathematical model.

3.2.2 Modelling virus infection with full logistic terms and antivirus treatment In 2017, Song et al. [10] presented a mathematical analysis on global dynamics of the viral infection model proposed by Dahari et al. [17], which included full density-dependent proliferation of hepatocytes and the efficacy of treatment in blocking virion production and reduction of new infections. This model considers three populations: uninfected hepatocytes T , infected hepatocytes I , and free virus particles V , and incorporates logistic terms with distinct proliferation rates for both uninfected and infected hepatocytes. This model is given by the following system of ordinary differential equations:   T +I T˙ = s + rT T 1 − − dT T − (1 − η)βV T , Tmax   T +I (3.2) − δI, I˙ = (1 − η)βV T + rI I 1 − Tmax V˙ = (1 − εp )pI − cV , where the efficacy of treatment in blocking virion production and reducing new infections is described by the parameters εp and η, respectively, and the values are nonnegative and less than one. The parameters used in this model are given as follows:

Chapter 3 • Global dynamics of an HCV model with full logistic terms

• • • • • • • • •

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Tmax is the total (normalized) hepatocyte number; s is the maximum de novo hepatocyte influx rate; dT is the uninfected hepatocyte death rate; δ is the infected hepatocyte loss rate; c is the HCV RNA clearance rate; p is the HCV production rate per cell; β is the infection rate constant; rT is the maximum uninfected hepatocyte proliferation rate; rI is the maximum infected hepatocyte proliferation rate.

The authors calculated in [10] the basic reproduction number R0 that is a sharp threshold parameter for the outcomes of viral infections. It was proved that whenever R0 < 1, only virus-free equilibrium exists and it is locally and globally asymptotically stable, i.e., the virus is eradicated and the disease dies out. If R0 > 1, a unique endemic equilibrium appears and system (3.2) is uniformly persistent, i.e., the virus persists within host. Also they gave some conditions for the local stability of the endemic equilibrium and analyzed its global stability using Li-Muldowney’s global stability criterion [22]. By numerical simulation techniques, they found that sustained oscillations can exist and different maximum de novo hepatocyte influx rate can induce different global dynamics along with the change of overall drug effectiveness. The authors did not include the CD8+ cytotoxic T cells compartment.

3.3 Mathematical preliminaries We present in this section some definitions and results that we will use through the entire work. The systems proposed above are of the form x˙ = f (x),

(3.3)

where x ∈ Rn and f ∈ C(Rn , Rn ). The following definitions and results are taken from [18– 23].

3.3.1 Linearization Definition 3.3.1. An equilibrium point x0 of system (3.3) is called a hyperbolic equilibrium point if none of the eigenvalues of the Jacobian matrix Df (x0 ) have zero real part. The local behavior of the nonlinear system (3.3) near a hyperbolic equilibrium point x0 is qualitatively determined by the behavior of the linear system x˙ = Ax with the matrix A = Df (xo ) near the origin.

(3.4)

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Definition 3.3.2. Two autonomous systems of differential equations, such as (3.3) and (3.4), are said to be topologically equivalent in a neighborhood of the origin or to have the same qualitative structure near the origin if there is a homeomorphism H mapping an open set U containing the origin onto an open set V containing the origin that maps the trajectories of (3.3) in U onto the trajectories of (3.4) in V and preserves their orientation by time in the sense that if a trajectory is directed from x1 to x2 in U , then its image is directed from H (x1 ) to H (x2 ) in V . We can assume that the equilibrium point x0 has been translated to the origin. Theorem 3.3.3 (The Hartman-Grobman theorem [18,19]). Let E be an open subset of Rn containing the origin, let f ∈ C 1 (E), and let φt be the flow of the nonlinear system (3.3). Suppose that f (0) = 0 and that the matrix A = Df (0) has no eigenvalue with zero real part. Then there exists a homeomorphism H of an open set U containing the origin onto an open set V containing the origin such that for each x0 ∈ U there is an open interval I0 ⊂ R containing zero such that for all x0 ∈ U and t ∈ I0 H ◦ φt (x0 ) = eAt H (x0 ); i.e., H maps trajectories of (3.3) near the origin onto trajectories of (3.4) near the origin and preserves the parameterization by time. The Hartman-Grobman theorem implies that the behavior of a dynamical system near a hyperbolic equilibrium point is qualitatively similar to the behavior of its linearization near the said point. Particularly, we can determine the stability of an equilibrium point knowing the stability of the origin in the linearized system, which depends on the eigenvalues of A. To do this, we have the following theorem. Theorem 3.3.4. [18] Let x0 be a hyperbolic equilibrium point of (3.3), where f ∈ C 1 (E) for E ⊂ Rn that contains x0 , and let A = Df (x0 ). (i) If all eigenvalues of A have a negative real part, then x0 is locally asymptotically stable. (ii) If there is an eigenvalue of A with a positive real part, then x0 is unstable.

3.3.2 Lyapunov functions Definition 3.3.5. Let D = {x ∈ Rn : |x| ≤ l} for some l > 0 and V ∈ C(D, R) such that V (0) = 0. Then V is said to be: (a) positive semidefinite if V (x) ≥ 0 for x ∈ D, and negative semidefinite if V (x) ≤ 0 for x ∈ D; (b) positive definite if V (x) > 0 for x ∈ D such that x = 0, and negative definite if V (x) < 0 for x ∈ D such that x = 0; (c) indefinite if V (x) changes sign in any neighborhood of x = 0.

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Definition 3.3.6. Let D = {x ∈ Rn : |x| ≤ l} for some l > 0 and V ∈ C 1 (D, R). Then the derivative of the function V (x) along Eq. (3.3) is defined to be   n  ∂V x˙i = ∇V · x, ˙ V˙ (x) = ∂xi i=1

where

 n  ∂V i=1

∂xi

 x˙i =

∂V ∂V ∂V x˙1 + x˙2 + · · · + x˙n . ∂x1 ∂x2 ∂xn

Theorem 3.3.7 (Lyapunov’s stability theorem [20]). Let D = {x ∈ Rn : |x| ≤ l} for some l > 0 and V ∈ C 1 (D, R). (a) If V (x) is positive definite and V˙ (x) is negative semidefinite, then the zero solution of (3.3) is uniformly stable. (b) If V (x) is positive definite and V˙ (x) is negative definite, then the zero solution of (3.3) is uniformly stable and asymptotically stable. (c) If V (0) = 0 and in any neighborhood of x = 0 in D, there exists x0 such that V (x0 ) > 0 and V˙ (x) is positive definite, then the zero solution of (3.3) is unstable. Theorem 3.3.8 (LaSalle’s invariance principle [20]). Let D = {x ∈ Rn : |x| ≤ l} for some l > 0 and V ∈ C 1 (D, R). Assume that V (x) is positive definite and V˙ (x) is negative semidefinite. Moreover, if the set D0 := {x ∈ D : V˙ (x) = 0} does not contain any nontrivial orbit of Eq. (3.3), then the zero solution of Eq. (3.3) is uniformly stable and asymptotically stable.

3.3.3 Bifurcation analysis When studying disease models represented by equations that depend on many parameters, it is important to analyze the way the dynamics of the model may change when one of the parameters is subject to variations. In some cases, there exists a critical value of the parameter such that a small perturbation from this value could cause a significant change in the qualitative structure of the system. This occurs, for example, when some of the equilibria of the system switch their stability from stable to unstable or vice versa, or when a new equilibrium appears. These changes are called a bifurcation of the system, and the parameter whose variations cause these changes is known as the bifurcation parameter. Consider a general system of ODEs with a parameter φ: dx = f (x, φ), dt

f : Rn × R → Rn , and f ∈ C2 (Rn × R).

(3.5)

Without loss of generality, it is assumed that 0 is an equilibrium for system (3.5) for all values of the parameter φ; that is, f (0, φ) ≡ 0 for all φ.

(3.6)

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Theorem 3.3.9. [21] Assume that   ∂fi (0, 0) is the linearization matrix of system (3.5) around the equiA1: A = Dx f (0, 0) = ∂x j librium 0 with φ evaluated at 0. Zero is a simple eigenvalue of A, and all other eigenvalues of A have negative real parts; A2: Matrix A has a nonnegative right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let fk be the kth component of f and a= b=

n 

v k wi wj

k,i,j =1 n 

v k wi

k,i=1

∂ 2 fk (0, 0), ∂xi ∂xj

∂ 2 fk (0, 0). ∂xi ∂φ

The local dynamics of (3.5) around 0 are totally determined by a and b. i. a > 0, b > 0. When φ < 0 with |φ|  1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; when 0 < φ  1, 0 is unstable, and there exists a negative and locally asymptotically stable equilibrium; ii. a < 0, b < 0. When φ < 0 with |φ|  1, 0 is unstable; when 0 < φ  1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; iii. a > 0, b < 0. When φ < 0 with |φ|  1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0 < φ  1, 0 is stable, and a positive unstable equilibrium appears; iv. a < 0, b > 0. When φ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly, a negative unstable equilibrium becomes positive and locally asymptotically stable.

3.3.4 Li and Muldowney’s geometric approach Let D be an open subset of Rn and f ∈ C 1 (D). Consider the autonomous dynamical system in Rn : x˙ = f (x).

(3.7)

Let x(t, x0 ) be the solution of (3.7) such that x(0, x0 ) = x0 . Definition 3.3.10. The set K is absorbing in D for system (3.7) if for every compact K1 ⊂ D, x(t, K1 ) ⊂ K for sufficiently large t, where x(t, K1 ) = {x(t, x0 ) : x0 ∈ K1 }.

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Definition 3.3.11. A be a matrix of n × n. The kth additive compound matrix A[k] of A   Let   n n is the matrix × defined by k k A[k] = D(I + hA)(k) |h=0 ,

(3.8)

where D is the derivative with respect to h and X (k) is the kth multiplicative compound of X.     n n Definition 3.3.12. Let P (x) be a matrix of size × and P ∈ C 1 (D). Suppose that 2 2 P −1 exists and is continuous in K, where K is a compact absorbing set in D. We define B = Pf P −1 + P

∂f [2] −1 P , ∂x

(3.9)

where Pf is obtained replacing every entry pij of P by the directional derivative with respect the vectorial field f . Definition 3.3.13. The Lozinskii measure μ(E) with respect to the norm | · | in RN , N =   n , is defined as 2 μ(E) = lim

h→0+

|I + hE| − 1 . h

Theorem 3.3.14 (Muldowney’s theorem [22]). If K is a compact absorbing subset in intD, and there exists ν > 0 such that the Lozinskii measure   ∂f [2] −1 ≤ −ν < 0 (3.10) P μ Pf P −1 + P ∂x for all x ∈ K, then every omega limit point of system (3.7) in intD is an equilibrium in K. Theorem 3.3.15. [22] Assume that (H1) D is a simply connected open set; (H2) there is a compact absorbing set K ⊂ D; (H3) x ∗ is the only equilibrium of (3.7) in D. Then x ∗ is globally asymptotically stable in D provided that (3.10) holds.

3.4 Modelling virus-immune system interaction with full logistic terms in both uninfected and infected cells 3.4.1 Model construction We begin to develop our model by assuming logistic growth of both uninfected and infected hepatocytes. The equations of our system have been build off of models presented

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by Avedaño et al. [16] in 2001, next retaken by Blé et al. [15] in 2018, and the model presented by Song et al. [10] in 2017. Based on the framework of these two models, we have proposed a mathematical model for an HCV infection with immune response and assuming that the growth of both the uninfected hepatocyte and the infected hepatocyte follow the logistic model depending on the existing hepatocyte and the infected hepatocyte, respectively. Also, we consider distinct cell proliferation rates for uninfected and infected hepatocytes. The model for HCV dynamics is constructed as follows. Assume that at time t, Hs (t) represents the density of host healthy (uninfected) hepatocytes, Hi (t) represents the density of infected hepatocytes, V (t) represents the density of HCV, and T (t) represents the density of CTLs (cell type CD8+T) produced by the cell-mediated immune response. It is assumed that uninfected host cells (healthy hepatocytes) are generated at a rate βs , undergo natural decay at a rate μs , maximum proliferation rate of the uninfected hepatocytes (Hs ) is represented by rs , which means that Hs and Hi hepatocytes can proliferate under a blind homeostasis process. Healthy hepatocytes (Hs ) get infected by the interaction with virus at a rate k. Infected cells die at a rate μi , the maximum proliferation rate of the infected hepatocytes (Hi ) is represented by ri , and are destroyed by the CTL response at a rate δ. HCV virions are produced within infected cells at a rate p and decay at a rate μv . The number (Hs + Hi ) represents the total hepatocyte population, which can increase up to a maximum of M. Under the presence of HCV, CTLs grow proportionally to the viral load V with a saturation rate βT (1 − T /Tmax ), where βT is the rate of reproduction of T cells and Tmax is the maximum of CTLs in the body. Furthermore, these cells decay in the absence of antigenic stimulation at a rate μT . (See Fig. 3.1.) In summary, the developed system for the dynamics of HCV in the presence of distinct cell proliferation rates for both uninfected hepatocytes and infected hepatocytes and immune response is composed by the following four ordinary differential equations:   Hs + Hi H˙ s = βs − μs Hs − kHs V + rs Hs 1 − , M   Hs + Hi , H˙ i = kHs V − δHi T − μi Hi + ri Hi 1 − M (3.11) ˙ V = pHi − μv V ,   T ˙ − μT T . T = βT V 1 − Tmax For biological significance of the parameters, three assumptions are employed: (a) Due to the burden of supporting virus replication, infected cells may proliferate more slowly than uninfected cells, i.e., ri ≤ rs . (b) To have a physiologically realistic model, the healthy hepatocyte population size does not longer increase when M is reached, i.e., βs ≤ μs M. (c) Infected cells have a higher turnover rate than uninfected cells, i.e., μs ≤ μi . Model (3.11) incorporates the following new main features: i. Cell-mediated immune response for a vanishing HCV infected cell (at a rate δ).

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FIGURE 3.1 A schematic diagram that illustrates a combination of immune response and cell proliferation in an HCV model.

ii. Full logistic growth terms for both uninfected hepatocytes (Hs ) and infected hepatocytes (Hi ). iii. Distinct cell proliferation rate for both uninfected hepatocytes, which is rs , and infected hepatocytes, which is ri . Model (3.11) will now be rigorously analyzed.

3.5 Analysis of the model 3.5.1 Dissipativity, basic reproduction number, and equilibria In this section, we obtain the dissipativity, derive the basic reproduction number, and analyze the existence of equilibria for model (3.11).

3.5.1.1 Dissipativity In the absence of HCV virions and hepatocytes cells (V = 0, Hi = 0), the susceptible cell dynamics is determined by   Hs ˙ − μs Hs . (3.12) Hs = βs + rs Hs 1 − M Now, when Hs = M, we have H˙ s = βs − μs M ≤ 0 since βs ≤ μs M. On the other hand, if Hs = 0, then H˙ s = βs > 0. By the continuity of H˙ s , there exists a value for Hs such that 0 < Hs ≤ M and H˙s = 0. To find this value, we have that if H˙ s = 0, then Hs satisfies the

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Table 3.1 Model parameter interpretation in (3.11). Parameter

Description

Units

Base value

References

M

Total (normalized) hepatocyte number

cells mL−1

300

[8,10]

βs

Production rate of uninfected hepatocyte

cells mL−1 day−1

50

[8,10]

rs

Maximum uninfected hepatocyte proliferation rate

day−1

1.55

[8,10]

ri

Maximum infected hepatocyte proliferation rate

day−1

0.2

[8,10]

μs

Uninfected hepatocyte death rate

day−1

0.25

[8,10]

μi

HCV infected hepatocyte loss rate

day−1

0.5

[8,10]

μv

HCV RNA clearance rate

day−1

5

[8,10]

μT

Decay rate of number of CTLs

day−1

0.02

[8,10]

k

Infection rate constant

virions mL day−1

0.00003

[8,10]

δ

Vanishing rate of infected hepatocytes due to CTL response

day−1

0.2

[8,10]

p

HCV production rate

virions day−1

200

[8,10]

βT

CTL expansion rate due to viral antigen

day−1

0.0003

[8,10]

Tmax

Maximum CTL number

cells mL−1

150

[8,10]

quadratic equation rs 2 H + (rs − μs )Hs + βs = 0, (3.13) M s 

s βs rs − μs ± (rs − μs )2 + 4rM . Thus, when there is no viral infec−

and its solutions are

M 2rs

tion, the amount of susceptible cells will tend to a positive constant level H0 , which is given by   β M 4r s s H0 = rs − μs + (rs − μs )2 + ≤ M. (3.14) 2rs M The following theorem shows that all solutions of model (3.11) in R4+ are ultimately bounded and that solutions with positive initial value conditions are positive, which indicates that (3.11) is well posed. Theorem 3.5.1. Under the initial value (Hs (0), Hi (0), V (0), T (0)) ∈ R4+ , system (3.11) has a unique positive and bounded solution in R4+ for all t > 0. Also, all solutions ultimately enter and remain in the following bounded and positively invariant region:

= {(Hs , Hi , V , T ) ∈ R4 |Hs + Hi ≤ H0 , V ≤ VM , T ≤ TM }, where VM =

pH0 μv

and TM =

βT pH0 Tmax μv μt Tmax +βT pH0 .

Proof. The right-hand side of system (3.11) is continuous and satisfies the Lipschitz condition in R4+ . Then system (3.11) has a unique solution (Hs (t), Hi (t), V (t), T (t)) ∈ R4+ in [0, tm ) for some tm > 0.

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Now, let N (t) = Hs (t) + Hi (t). Using the two equations of (3.11) and the inequalities ri ≤ rs and μs ≤ μi , we have     N N ˙ − δHi T − μi Hi + ri Hi 1 − N = βs − μs Hs + rs Hs 1 − M M   N − δHi T ≤ βs − μs (Hs + Hi ) + (rs Hs + ri Hi ) 1 − M   N ≤ βs − μs N + rs N 1 − . M The above inequality can be written as rs N˙ ≤ − N 2 + (rs − μs )N + βs . M

(3.15)

From (3.13) and (3.14) we know that H0 is a root of the right-hand side of inequality (3.15), then by the comparison principle [24] we have lim supt→∞ N (t) ≤ H0 for all solutions in R4+ , and Hs (t) + Hi (t) ≤ H0 for all t ≥ 0 if Hs (0) + Hi (0) ≤ H0 . Then by the third equation of (3.11) we have V˙ ≤ pH0 − μv V .

(3.16)

It then follows from the comparison principle [24] that lim supt→∞ V (t) ≤ VM for all solutions in R4+ , and V (t) ≤ VM for all t ≥ 0 if V (0) ≤ VM . Then, using the fourth equation of (3.11) and the above result, we have that T˙ ≤ βT VM − μT T . It then follows from the comparison principle [24] that lim supt→∞ T (t) ≤ TM for all solutions in R4+ , and T (t) ≤ TM for all t ≥ 0 if T (0) ≤ VM . Hence Hs (t), Hi (t), V (t), and T (t) are bounded. This in turn shows that the solution exists globally, i.e., for all t ≥ 0. Consequently, the solutions (Hs (t), Hi (t), V (t), T (t)) of (3.11) are ultimately bounded in the positively invariant region . To prove that Hs (t) is positive for all t > 0, suppose that Hs (t) is not always positive. Let τ > 0 be the first time such that Hs (τ ) = 0. By the first equation of the system, we have H˙ s (τ ) = βs > 0, which implies Hs (t) < 0 for t ∈ (τ − ε, τ ), for sufficiently small ε > 0. A contradiction. Now we show that Hi (t), V (t), and T (t) are nonnegative. We first suppose that there exists t1 > 0 such that Hi (t1 ) < 0, then there is t2 < t1 such that Hi (t2 ) = 0, H˙ i (t2 ) < 0, and Hi (t) ≥ 0 for all t ≤ t2 . However, H˙ i (t2 ) = kHs (t2 )V (t2 ), which implies that V (t2 ) < 0. Thus there is t3 < t2 such that V (t3 ) = 0, V˙ (t3 ) < 0, and V (t) ≥ 0 for all t ≤ t3 , but we have that V˙ (t3 ) = pHi (t3 ) ≥ 0. A contradiction. Hence, Hi is nonnegative. Next, we suppose that V (t1 ) < 0 for some t1 > 0. Then there exists t2 < t1 such that V (t2 ) = 0 and V˙ (t2 ) < 0. But V˙ (t2 ) = pHi (t2 ) ≥ 0, which is a contradiction. Thus V is nonnegative. Finally, we suppose that T (t1 ) < 0 for some t1 > 0. Then there exists t2 < t1 such that T (t2 ) = 0 and T (t2 ) < 0.

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But we have that T˙ (t2 ) = βT V (t2 ) ≥ 0, which is a contradiction. Therefore T is nonnegative. Then we can write  

t kHs (θ )V (θ )e(δT (θ)+μi )θ dθ e−(δT (t)+μi )t Hi (t) ≥ Hi (0) + 0 −(δT (t)+μi )t

≥ Hi (0)e > 0,  

t μv θ V (t) = V (0) + pHi (θ )e dθ e−μv t ≥ V (0)e−μv t > 0, 0  

t ηθ T (t) = T (0) + βT V (θ )e dθ e−ηt ≥ T (0)e−ηt > 0, where η = − positive.



βT V (t) Tmax

0



+ μT . Hence, any solution of (3.11) with positive initial conditions is

3.5.1.2 Basic reproduction number Using the idea of the next generation matrix for a general compartmental disease transmission model in [25], we can obtain the basic reproduction number of (3.11). We know that E0 = (H0 , 0, 0, 0) is the disease-free equilibrium that exists for all positive parameter values. Based on the notation in [25], we have     +Hi kHs V + ri Hi 1 − HsM δHi T + μi Hi and V = . F= −pHi + μv V 0 Then

F=

ri −

r i Hs M

0

−

2ri Hi M

 δT + μi kHs and V = −p 0

 0 . μv

Consequently, at the point E0 the matrices F and V are given by      ri 1 − HM0 kH0 0 μi and V = F= −p μv 0 0 with V

−1

1 μv = μi μv p

Thus the next generation matrix is   ri 1− −1 F V = μi

  1 0 = μpi μi μi μv

H0 M

0



+

kpH0 μi μv



0 1 μv

kH0 μv

.

 .

0

According to [25], the basic reproduction number of (3.11) is defined by   H0 kpH0 ri R0 = ρ(F V −1 ) = 1− + , μi M μi μv

(3.17)

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69

where ρ denotes the spectral radius (dominant eigenvalue) of the matrix F V −1 .

3.5.1.3 Existence of endemic equilibrium Theorem 3.5.2. Provided R0 > 1, the model Eq. (3.11) admits a unique endemic equilibrium point E1 = (Hs∗ , Hi∗ , V ∗ , T ∗ ). Proof. We consider the following system:   ⎧ H ∗ +H ∗ ⎪ βs − μs Hs∗ − kHs∗ V ∗ + rs Hs∗ 1 − s M i = 0, ⎪ ⎪   ⎪ ⎪ H ∗ +H ∗ ⎨ kHs∗ V ∗ − δHi∗ T ∗ − μi Hi∗ + ri Hi∗ 1 − s M i = 0, ⎪ pHi∗ − μv V ∗ = 0, ⎪ ⎪ ⎪ ⎪ ⎩ βT V ∗ 1 − T ∗ − μT T ∗ = 0.

(3.18)

Tmax

Solving the third equation of (3.18), we obtain that V∗ =

pHi∗ . μv

(3.19)

Afterwards, substituting (3.19) into the fourth equation of (3.18), we have that T∗ =

pβT Tmax Hi∗ . pβT Hi∗ + μv μT Tmax

Using (3.19), (3.20), and the first equation of (3.18) leads to    βs Hs∗ μv M ∗ Hi = . − μs + rs 1 − kpM + rs μv Hs∗ M Hs∗ is determined by the positive roots of the equation   Hs∗ + Hi∗ pkHs∗ ∗ ∗ G(Hs ) = . − δT − μi + ri 1 − μv M Note that Hi∗ → ∞, T ∗ → Tmax as Hs∗ → 0+ . Then lim G(Hs∗ ) = −∞ < 0

Hs∗ →0+

and   H0 pkH0 G(H0 ) = − μi + ri 1 − μv M     ri H0 pkH0 1− + = μi −1 μi M μi μv = μi (R0 − 1) > 0.

(3.20)

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Then equation G(Hs∗ ) has a positive root. Note that ∂Hi μv M = ∂Hs kpM + rs μv

  βs rs − ∗ 2− 0, − = μv μv pkM + rs μv since ri ≤ rs and ri μv < pkM + rs μv . Then 

G

(Hs∗ ) =

∂Hi pk δpβT μv μT Tmax − − ri ∗ 2 μv (pβT Hi + μv μT Tmax ) ∂Hs



∂Hi

1 ∂H + s M M



∂Hi μv ri rs ri δpβT μv μT Tmax pk + − − μv M(kpM + rs μv ) M (pβT Hi∗ + μv μT Tmax )2 ∂Hs ri μv M + > 0. (kpM + rs μv )(Hs∗ )2 =

This implies that G(Hs∗ ) is strictly increasing in its positive roots. Let us suppose that G(Hs∗ ) has more than one positive root. Without loss of generality, we choose one, denoted ∗ ∗ by H¯s , that is nearest to Hs∗ . Because of the continuity of G(Hs∗ ), we must have G (H¯s ) ≤ 0, which results in a contradiction with the strictly increasing property of G(Hs∗ ) at all the zero points. Thus model (3.11) admits a unique endemic equilibrium.

3.5.2 Local and global stability analysis Since we are working on a mathematical model of infectious diseases, it is important to prove local and global stability of our model. Local stability of an equilibrium would imply that if there is a small perturbation of the system then, after some time, the same situation would be happening. However, global stability would imply the existence of a situation for good, regardless of any condition.

3.5.2.1 Local stability analysis of the disease-free equilibrium Theorem 3.5.3. The disease-free equilibrium E0 = (H0 , 0, 0, 0) is locally asymptotically stable if R0 < 1 and unstable if R0 > 1.

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71

Proof. The Jacobian matrix of (3.11) in any point E = (Hs , Hi , V , T ) is given by ⎡

J11 ⎢kV − ri Hi ⎢ M J (E) = ⎢ 0 ⎣

− rsMHs J22 p

0

−kHs kHs  −μv

βT 1 −

0

T Tmax



⎤

0 −δHi 0 TV − Tβmax

⎥ ⎥ ⎥, ⎦

(3.21)

− μT

where   Hs + Hi rs Hs J11 = −μs − kV + rS 1 − − , M M   Hs + Hi ri Hi − . J22 = −μi − δT + ri 1 − M M Thus at E0 we have ⎡ ⎢ ⎢ J (E0 ) = ⎢ ⎣

− Hβs0 −

r s H0 M

−kH0

0

− rsMH0   ri 1 − HM0 − μi

0 0

p 0

−μv βT

0

⎤

⎥ 0 ⎥ ⎥. 0 ⎦

kH0

(3.22)

−μT

Then the characteristic polynomial of (3.22) is 

 βs rs H0 p(λ) = − − λ (−μT − λ)p1 (λ), − H0 M

(3.23)

where    H0 − μi − λ − kpH0 p1 (λ) = (−μv − λ) ri 1 − M       μv kpH0 ri H0 ri H0 2 −1− + 1− λ − μi μv 1− = λ − μi −1 μi M μi μi M μi μv  kpH0 μv λ − μi μv (R0 − 1). − = λ2 − μi (R0 − 1) − μi μv μi Consequently, the eigenvalues of (3.22) are λ1 = − Hβs0 − rsMH0 , λ2 = −μT and the two roots of the polynomial p1 (λ). It remains to prove that the roots of this polynomial have negative real parts.   μv 0 Let a0 = 1, a1 = −μi (R0 − 1) − kpH − μi μv μi and a2 = −μi μv (R0 − 1) be the coefficients of p1 (λ). Thus we have that 1 = a 1

and

 a 2 =  1 0

 1  = a1 a2 . a2 

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Clearly, a1 > 0, a2 > 0, and 2 > 0 whenever R0 < 1. Then, by the Routh-Hurwitz criterion [26], we conclude that E0 is locally asymptotically stable if R0 < 1 and unstable otherwise. The basic reproduction number (R0 ) is the average number of infected contacts per infected individual. At a population level, a value of R0 larger than one means that a virus will continue its propagation among susceptible hosts if no environmental changes or external influences intervene. An R0 value lower than one means that the virus is doomed to extinction at the epidemiological level under those specific circumstances [27]. The above result implies that a small influx of infected individuals would not generate large outbreaks if R0 < 1, and the disease will persist (be endemic) in the population if R0 > 1. In the next result we analyze the system dynamics when R0 = 1. Theorem 3.5.4. If μi > ri , then system (3.11) exhibits a forward bifurcation when R0 = 1. Proof. The basic reproduction number can be rewritten as R0 =

ri μi

  H0 kpH0 . 1− + M μi μv

(3.24)

We will consider k as a bifurcation parameter: we calculate a critical value   H0 μv μi μv ri 1− . − pH0 pH0 M

(3.25)

  k ∗ pH0 H0 − − μi . = ri 1 − μv M

(3.26)

k∗ = Observe that

The Jacobian matrix evaluated at the disease-free equilibrium E0 with k = k ∗ is ⎡ ⎢ ⎢ J (E0 , k ) = ⎢ ⎣ ∗

− Hβs0 − 0 0 0

r s H0 M

− rsMH0 ∗

0 − k μpH v p 0

−k ∗ H0 k ∗ H0 −μv βT

0

⎤

⎥ 0 ⎥ ⎥. 0 ⎦ −μT

(3.27)

0 Consequently, the eigenvalues of (3.27) are λ1 = − Hβs0 − rsMH0 , λ2 = −μT , λ3 = − kpH μ v − μv , and λ4 = 0. Thus λ4 = 0 is a simple zero eigenvalue of the matrix J (E0 , k ∗ ) and the other eigenvalues are real and negative. Hence, when k = k ∗ (or equivalently when R0 = 1), the disease-free equilibrium E0 is a nonhyperbolic equilibrium: assumption (A1 ) of Theorem 3.3.9 is then verified.

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73

Now we denote by w = (w1 , w2 , w3 , w4 )T a right eigenvector associated with the zero eigenvalue λ4 = 0. It follows that  rs H0 rs H0 βs w1 − w2 − kH0 w3 , + H0 M M k ∗ pH0 0=− w2 + k ∗ H0 w3 , μv 0 = pw2 − μv w3 , 

0=−

0 = βT w3 − μT w4 . So that 

H0 2 (Mpk ∗ + rs μv ) μv βT w= − , 1, , 2 μT p(βs M + rs H0 ) p

T .

(3.28)

Furthermore, the left eigenvector v = (v1 , v2 , v3 , v4 ) satisfying v · w = 1 is given by  rs H0 βs v1 , + H0 M rs H0 k ∗ pH0 v1 − 0=− v2 + pv3 , M μv k ∗ pH0 rs H0 v1 − v2 + pv3 , 0=− M μv 0 = −k ∗ H0 v1 + k ∗ H0 v2 − μv v3 + βT v4 , 

0=−

0 = −μT v4 . Then the left eigenvector v turns out to be  v = 0,

 μv p pk ∗ H0 , ,0 . pk ∗ H0 + μ2v pk ∗ H0 + μ2v

(3.29)

Although Theorem 3.3.9 assumes that the right eigenvector w is nonnegative, the authors then make a remark (see Remark 1 in [21]) in which they assert that the theorem remains valid for a nonnegative equilibrium x0 , provided that wj > 0 whenever x0 (j ) = 0 (where x0 (j ) denotes the j th component of x0 ). In our case, the zero components of the equilibrium x0 = E0 are x0 (2), x0 (3), and x0 (4). Since w2 , w3 , and w4 are positive, it follows that such a theorem can be applied using the above eigenvectors. Using the notation x = [x1 , x2 , x3 , x4 ]T with x1 = Hs , x2 = Hi , x3 = V , and x4 = T , let F (x, k ∗ ) be the function defined by the right-hand side of system (3.11). Taking into ac-

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count that v1 = v4 = 0, the coefficients a and b of the normal form are given by a=

4 

v 2 wi wj

i,j =1

b=

4  i=1

where

4  ∂ 2 f2 ∂ 2 f3 (E0 , k) + v 3 wi wj (E0 , k), ∂xi ∂xj ∂xi ∂xj i,j =1

v 2 wi

∂ 2 f2 (E0 , k) + ∂xi ∂k

4  i=1

v 3 wi

∂ 2 f3 (E0 , k), ∂xi ∂k

  Hs + Hi f2 (x, k ∗ ) = k ∗ Hs V − δHi T − μi Hi + ri Hi 1 − M

(3.30)

f3 (x, k ∗ ) = pHi − μv V .

(3.31)

and

∂ 2 f3 ∂xi ∂xj

(E0 , k ∗ ) = 0, where i, j = 1, 2, 3, 4. And the nonzero second partial derivaNote that tives of f2 evaluated at (E0 , k ∗ ) are ∂ 2 f2 ∂ 2 f2 ri (E0 , k ∗ ) = (E0 , k ∗ ) = − , ∂x1 ∂x2 ∂x2 ∂x1 M ∂ 2 f2 ∂ 2 f2 (E0 , k ∗ ) = (E0 , k ∗ ) = k ∗ , ∂x1 ∂x3 ∂x3 ∂x1 ∂ 2 f2 2ri (E0 , k ∗ ) = − , ∂x2 ∂x2 M 2 ∂ f2 ∂ 2 f2 (E0 , k ∗ ) = (E0 , k ∗ ) = −δ. ∂x2 ∂x4 ∂x4 ∂x2 2

∂ f3 Note that ∂x (E0 , k ∗ ) = 0, where i = 1, 2, 3, 4. And the nonzero second partial derivatives i ∂k of f2 evaluated at (E0 , k ∗ ) are

∂ 2 f2 (E0 , k ∗ ) = H0 . ∂x3 ∂k Then b= and

pk ∗ H02 >0 pk ∗ H0 + μ2v

  r  2r i i 2 w2 − k ∗ − w2 − 2δw2 w4 a = v2 −2w1 M M     ri μv 2r i 2 = v2 −2w1 − k∗ − w2 − 2δw2 w4 pM M   (ri − μi )μv ri 2 = 2v2 −w1 − w2 − δw2 w4 . pH0 M

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Since a < 0 and b > 0, by Theorem 3.3.9, we can conclude that there exists ε > 0 such that for k ∈ (k ∗ , k ∗ + ε) there is a locally asymptotically stable endemic equilibrium near E0 . This implies that the model undergoes a forward bifurcation at R0 = 1.

3.5.2.2 Local stability analysis of the endemic equilibrium Assuming R0 > 1, the Jacobian matrix of model (3.11) at the equilibrium point E1 is given by ⎤ ⎡ 0 A11 A12 A13 ⎢A21 A22 A23 A24 ⎥ ⎥, (3.32) J (E ∗ ) = ⎢ ⎣ 0 A32 A33 0 ⎦ 0 where

0

A43

A44

  H ∗ + Hi∗ rs Hs∗ βs rs Hs∗ A11 = −μs − kV ∗ + rs 1 − s − =− ∗ − < 0, M M Hs M rs Hs∗ A12 = − < 0, M A13 = −kHs∗ < 0, A21 = kV ∗ −

ri Hi∗ , M

  ri Hi∗ H ∗ + Hi∗ kpHs∗ ri Hi∗ A22 = −μi − δT ∗ + ri 1 − s − =− < 0, − M M μv M A23 = kHs∗ > 0,

A24 = −δHi∗ < 0, A32 = p > 0, A33 = −μv < 0,   T∗ > 0, A43 = βT 1 − Tmax βT V ∗ βT V ∗ A44 = − − μT = − ∗ < 0. Tmax T The corresponding characteristic polynomial is p(x) = x 4 + c3 x 3 + c2 x 2 + c1 x + c0 , where beginequation c3 = −(A11 + A22 + A33 + A44 ) > 0, c2 = A44 A33 + A44 A22 + A44 A11 − A32 A23 + A33 A22 + A33 A11 − A21 A12 + A22 A11 , c1 = −A43 A32 A24 + A44 A32 A23 − A44 A33 A22 − A44 A33 A11 + A44 A21 A12 − A44 A22 A11 − A32 A21 A13 + A32 A23 A11 + A33 A21 A12 − A33 A22 A11 ,

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c0 = A43 A32 A24 A11 + A44 A32 A21 A13 − A44 A32 A23 A11 − A44 A33 A21 A12 + A44 A33 A22 A11 . Note that A22 A33 − A32 A23 =

μv ri Hi∗ >0 M

and      ri Hi∗ kpHs∗ ri Hi∗ rs Hs∗ βs rs Hs∗ ∗ + kV + + − H∗ M μv M M M        s  ∗ ∗ ∗ ∗ ∗ ri Hi kpHs rs Hs  ∗  rs Hs kpHs βs + + kV > 0. = + Hs∗ μv M M μv M 

A11 A22 − A21 A12 =

Then c2 > 0 and A44 A32 A23 − A44 A33 A22 > 0. Also, −A32 A21 A13 + A32 A23 A11 + A33 A21 A12 − A33 A22 A11 = (A32 A23 − A33 A22 ) A11 + (A33 A12 − A32 A13 ) A21       μv ri Hi∗ ri Hi∗ μv rs Hs∗ βs rs Hs∗ ∗ ∗ + kV − + pkHs = + Hs∗ M M M M       ∗  μv rs Hs∗ μv ri Hi∗ pkri Hi∗ Hs∗ βs ∗ = − + kV + pkH s H∗ M M M   s  ∗ ∗ ∗H ∗ ∗ μv ri Hi kprs Hi Hs H pkr βs i i s + + pk 2 Hs∗ V ∗ − = Hs∗ M M M   ∗ ∗H ∗ kp(r r H − r )H μ βs v i i s i i s = + pk 2 Hs∗ V ∗ + > 0. Hs∗ M M Then c1 > 0 and A44 A32 A21 A13 − A44 A32 A23 A11 − A44 A33 A21 A12 + A44 A33 A22 A11 > 0.

(3.33)

Then c0 > 0. Now by the Routh-Hurwitz criterion [26] it follows that all roots of x 4 +c3 x 3 +c2 x 2 +c1 x + c0 have negative real parts if and only if (c3 c2 − c1 )c1 − c32 c0 > 0. From the above analysis, we now state the following theorem. Theorem 3.5.5. The endemic equilibrium point E1 is locally asymptotically stable if R0 > 1 and (c3 c2 − c1 )c1 − c32 c0 > 0.

3.5.2.3 Global stability analysis of the disease-free equilibrium Theorem 3.5.6. For model (3.11), the disease-free equilibrium E0 = (H0 , 0, 0, 0) is globally asymptotically stable if R0 < 1.

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77

Proof. Consider the candidate Lyapunov function L = Hs − H0 − H0 ln

Hs kH0 + Hi + V. H0 μv

Letting f represent the vector field associated with (3.11), we have

 βs rs rs + rs − μs − Hs − Hi − kV + kHs V − δHi T ∇L · f =(Hs − H0 ) Hs M M   Hs + Hi kpH0 − μi Hi + Hi − kH0 V + ri Hi 1 − M μv  βs rs rs + rs − μs − Hs − Hi − δHi T − μi Hi =(Hs − H0 ) Hs M M   kpH0 Hs + Hi + + ri Hi 1 − Hi M μv    Hs + Hi βs rs rs =(Hs − H0 ) + rs − μs − Hs − Hi + ri Hi 1 − Hs M M M   kpH0 − δHi T + − μi Hi . μv Using (3.13), we obtain that rs − μs = expression of R0 in (3.17), we get

r s H0 M

−

βs H0 .

Thus, from this last inequality and the

   Hs + Hi βs βs rs H0 rs rs − + − Hs − Hi + ri Hi 1 − Hs M H0 M M M   kpH0 − δHi T + − μi Hi μv  βs rs ri ri (Hs − H0 )2 + (Hs − H0 )Hi + Hs Hi + Hi2 =− (Hs − H0 )2 − Hs H0 M rs rs   kpH0 + ri Hi + − μi Hi − δHi T μv rs ri βs 2 (Hs − H0 )2 + (Hs − H0 )Hi + (Hs − H0 )Hi (Hs − H0 ) − =− Hs H0 M rs    ri 2 ri ri H0 − μi Hi − δHi T + Hi + H0 Hi + ri Hi + μi R0 − ri + rs rs M rs ri βs (Hs − H0 )2 + (Hs − H0 )Hi + (Hs − H0 )Hi (Hs − H0 )2 − =− Hs H0 M rs  ri 2 + Hi + μi (R0 − 1)Hi − δHi T . rs

∇L · f =(Hs − H0 )

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On the other hand, (Hs + Hi − H0 )(Hs + rrsi Hi − H0 ) = (Hs − H0 )2 + (Hs − H0 )Hi + rrsi (Hs − H0 )Hi + rrsi Hi2 . Then βs rs ri (Hs − H0 )2 − (Hs + Hi − H0 )(Hs + Hi − H0 ) Hs H0 M rs + μi (R0 − 1)Hi − δHi T .

∇L · f = −

(3.34)

Using Hs + Hi − H0 ≤ 0 (see Theorem 3.5.1), ri ≤ rs , and R0 < 1, we have ∇L · f ≤ 0, i.e., the derivative of the Lyapunov function is negative semidefinite. We consider the set {(Hs , Hi , V , T ) : ∇L · f = 0}. Observe that ∇L · f = 0 only if Hs = H0 , Hi = 0, and V = 0 simultaneously, thus for LaSalle’s theorem we conclude that all solutions will approach the set S := {(Hs , Hi , V , T ) : Hs = H0 , Hi = 0 = V }. It remains to prove that all solutions in S approach E0 = (H0 , 0, 0, 0) as t → ∞. To do this, we have that the HCV dynamic in S is determined by T˙ = −μT T .

(3.35)

Consequently, all solutions in S are given by Hs (t) = H0 ,

Hi (t) = 0,

V (t) = 0,

and

T (t) = Ce−μT t .

Therefore, if t → ∞, then T → 0, and the result follows.

3.5.2.4 Uniform persistence Theorem 3.5.7. If R0 > 1, then system (3.11) is uniformly persistent, i.e., there exists ε > 0, independent of initial conditions, such that lim inf Hs (t) > ε, lim inf Hi (t) > ε, lim inf V (t) > ε, t→+∞

t→+∞

t→+∞

and lim inf T (t) > ε for all solutions of (3.11) with positive initial conditions in . t→+∞

Proof. The result follows from an application of Theorem 4.6 in [28] with X1 = int(R4+ ) and X2 = bd(R4+ ). This choice is in accordance with the conditions stated in the theorem. Now, note that by Theorem 3.5.1 there exists a compact set in which all solutions of system (3.11) initiated in R4+ ultimately enter and remain forever after. Condition (C4,2 ) of Theorem 4.6 in [28] can be verified for this set . On the other hand, we denote the omega limit set of the solution x(t, x0 ) of system (3.11) starting in x0 ∈ R4+ by ω(x0 ), so we need to determine the following set:

2 = ∪y∈Y2 ω(y),

Y2 = {x0 ∈ X2 |x(t, x0 ) ∈ X2 ,

∀t > 0}.

(3.36)

We observe that the Hs axis is an invariant set under (3.11), and it follows that all solutions starting in bd(R4+ ) but not on the Hs axis leave bd(R4+ ). This implies that Y2 = {(Hs , Hi , V , T )T ∈ bd(R4+ )|Hi = V = T = 0}. Furthermore, we see that 2 = {E0 } as all solutions initiated on Hs axis converge to E0 . Then E0 is a covering of 2 . Since E0 is a hyperbolic steady state under the assumption of the theorem, E0 is isolated and acyclic

Chapter 3 • Global dynamics of an HCV model with full logistic terms

79

(because there is no nontrivial solution in bd(R4+ ) that links E0 to itself ). Finally, we need to prove that E0 is a weak repeller for X1 to end the proof. Since the proof is similar to that of Theorem 4.1 in [10], here we only sketch the modifications that E0 is a weak repeller for X1 .    Since R0 > 1, from (3.17) we have that kpH0 > μv μi − ri 1 − HM0 . So, there exists σ > 0 sufficiently small such that  kp(H0 − σ ) > μv

  H0 μi − ri 1 − . M

(3.37)

Now, we suppose that there exists a solution (Hs (t), Hi (t), V (t), T (t)) such that (Hs (t), Hi (t), V (t), T (t)) → E0 = (H0 , 0, 0, 0). Then, for t sufficiently large, we have that H0 − σ < Hs (t) < H0 + σ, We note that 1 −

H0 M

≤1−

Hi (t) ≤ σ,

V (t) ≤ σ,

T (t) ≤ σ.

Hs +Hi M , then from the second equation

of (3.11) we have that

  H0 ˙ Hi ≥ k(H0 − σ )V − δHi T − μi Hi + ri Hi 1 − . M Taking an auxiliary system of (3.11) as  ⎧ ˙ i = k(H0 − σ )V − δHi T − μi Hi + ri Hi 1 − ⎪ H ⎪ ⎨ V˙ = pHi −  μv V ,  ⎪ ⎪ ⎩ T˙ = βT V 1 − T − μT T .

H0 M

 , (3.38)

Tmax

Clearly, (0, 0, 0) is an equilibrium point of (3.38) and the Jacobian matrix J of (3.38) at (0, 0, 0) is given by   ⎡ ⎤ k(H0 − σ ) 0 −μi + ri 1 − HM0 ⎢ ⎥ J (0) = ⎣ (3.39) p −μv 0 ⎦. 0 βT −μT The characteristic polynomial of (3.39) is       H0 H0 2 λ + μv μi − ri 1 − p2 (λ) = (−μT − λ) λ + μv + μi − ri 1 − M M −pk(H0 − σ )] . The eigenvalues of (3.39) are λ1 = −μT < 0 and        H 1 0 − pk(H0 − σ ) , λ2,3 =  ± 2 − 4 μv μi − ri 1 − 2 M

(3.40)

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Advances in Epidemiological Modeling and Control of Viruses

 where  = −μv − μi + ri 1 −

H0 M

 . By (3.37), we have that

     H0 − pk(H0 − σ ) > 0, −4 μv μi − ri 1 − M then λ2 > 0 and λ3 < 0. Thus (0, 0, 0) is unstable in this case. This is a contradiction to that (Hi (t), V (t), T (t)) → (0, 0, 0). Consequently, E0 is a weak repeller for X1 . This completes the proof. Remark 3.5.1. Since ri ≤ rs , μs ≤ μi and all components of a solution of (3.11) are nonnegative, using the first two equations of (3.11), we have     Hs + Hi Hs + Hi ˙ ˙ − δHi T − μi Hi + ri Hi 1 − Hs + Hi = βs − μs Hs + rs Hs 1 − M M   Hs + Hi − δ(Hi + Hs )T ≥ βs − μi (Hs + Hi ) + (rs Hs + ri Hi ) 1 − M   Hs + Hi ≥ βs − (μi + δTM )(Hs + Hi ) + ri (Hs + Hi ) 1 − M and   Hs + Hi H˙s = βs − μs Hs − kV Hs + rs Hs 1 − M    H0 Hs . ≤ βs − μs + kVM − rs 1 − M Thus Hs + Hi ≥ H˜ i and Hs ≥ H˜s is valid for sufficiently large time t > 0. Here, H˜s =

and M H˜ i = 2ri

βs

 μs + kVM − rs 1 −

 ri − μs − kVM +



H0 M



4βs ri (ri − μs − kVM )2 + M

 .

Remark 3.5.2. The uniform persistence of system (3.11) in the bounded set and Remark 3.5.1 is equivalent to the existence of compact K ⊂ that is absorbing for (3.11). Denote K = {(Hs , Hi , T , V ) ∈ |CH s ≤ Hs , CH i ≤ Hi , CT ≤ T , CV ≤ V }, where 0 < CH s , CH i , CV , CT .

Chapter 3 • Global dynamics of an HCV model with full logistic terms

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3.5.2.5 Global stability analysis of the endemic equilibrium Theorem 3.5.8. Assume that R0 > 1, then the unique endemic equilibrium E1 is globally asymptotically stable if the following condition holds: b = max ψ1 −

ri H˜ i βT CV (rs − ri )H˜s 2ri H˜ i , rs − μs − + δTM , − − + δTM , M TM M M

ri H˜ i βT CV 2ri H˜ i pCHi (rs − ri )CHs , rs − μs − − − − , M VM Tmax M M βT CV (rs − ri )H˜ i βs + δTM , , rs − ri − μs + μi − kCV − − − H0 Tmax M ! βT V < 0, − Tmax

ri − μi −

where

! " βs (rs − ri )Hs ψ1 = sup − |CHs ≤ Hs ≤ H0 . − Hs M

Proof. We will use the geometric approach to global stability in order to study the global stability of the infected equilibrium. We follow the approach used in [15]. We start as follows. The Jacobian matrix J associated with a general solution to system (3.11) is given in (3.21), and its second additive compound matrix J [2] is ⎡ [2] J11 ⎢ ⎢ p ⎢ ⎢ 0 ⎢ [2] J =⎢ ⎢ 0 ⎢ ⎢ 0 ⎣ 0

kHs [2]  J22

βT 1 − kV −

T

Tmax r i Hi M



−δHi 0 [2] J33

0

0

kV −

0

0

r i Hi M

kHs − rsMHs

0 0

0

− rsMHs

[2]

0

 J44  T βT 1 − Tmax

[2] J55

0

p

⎤ 0 ⎥ 0 ⎥ ⎥ −kHs ⎥ ⎥ ⎥, δHi ⎥ ⎥ kHs ⎥ ⎦ [2] J66

where ri Hs 2ri Hi 2rs Hs rs Hi − − − , M M M M 2rs Hs rs Hi [2] − , J22 = rs − μs − μv − kV − M M 2rs Hs βT V rs Hi [2] J33 − , = rs − μs − μT − kV − − Tmax M M 2ri Hi ri Hs [2] − , J44 = ri − μi − μv − δT − M M 2ri Hi βT V ri Hs [2] − , J55 = ri − μi − μT − δT − − Tmax M M [2] J11 = rs + ri − μs − μi − kV − δT −

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[2] = −μv − J66

βT V − μT . Tmax

Let Q(t) be the following matrix of 6 × 6 that is C 1 and invertible: ⎡1 ⎤ 0 0 0 0 0 Hi ⎢0 1 0 0 0 0 ⎥ ⎢ ⎥ V ⎢ ⎥ 1 0 ⎥ ⎢0 0 0 V 0 Q=⎢ ⎥, ⎢ 0 0 T1 0 0 0 ⎥ ⎢ ⎥ ⎣ 0 0 0 0 T1 0 ⎦ 0 0 0 0 0 VHTi and its inverse is given by

⎡

Hi ⎢0 ⎢ ⎢0 ⎢ Q−1 = ⎢ ⎢0 ⎢ ⎣0 0

0 V 0 0 0 0

0 0 0 T 0 0

0 0 Hi 0 0 0

0 0 0 0 T 0

0 0 0 0 0

⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎦

VT Hi

Now, let Qf be the matrix that is obtained replacing each entry Qij of Q by the derivative of Qij in the direction of the vector field f associated with (3.11), given by ⎤ ⎡ ˙ − Hi2 0 0 0 0 0 ⎥ ⎢ Hi ˙ ⎥ ⎢ 0 0 0 0 − VV2 ⎥ ⎢ 0 ⎥ ⎢ V˙ ⎥ ⎢ 0 0 0 − 0 0 2 ⎥. ⎢ V Qf = ⎢ ⎥ T˙ 0 −T2 0 0 0 ⎥ ⎢ 0 ⎥ ⎢ T˙ ⎥ ⎢ 0 0 0 0 0 −T2 ⎦ ⎣ 0

0

0

Thus we have −Qf Q−1 = diag

0

V T H˙ i −Hi (T V˙ +V T˙ ) (V T )2

0

  ˙ ˙ ˙ ˙ ˙ ˙ T˙ H˙ i Hi V V T T V . , , , , , + − Hi V V T T V T Hi

Hence the matrix B = Qf Q−1 + QJ [2] Q−1 is written as follows:

⎡ H˙ i J [2] − H i ⎢ 11 ⎢ pHi ⎢ V ⎢ ⎢ 0 ⎢ B =⎢ 0 ⎢ ⎢ ⎢ 0 ⎢ ⎣ 0

kHs V Hi [2] V˙ J11 −V r i Hi kV − M  V βT 1 − T T T max

kHs V Hi Hs − rsM [2] V˙ J44 − V

⎤

−δT

0

0

0

0

0 δT

Hs − rsM

sV − kH H

0

  V βT 1 − T T T

0 ˙ [2] J33 − TT

0

0

0

0

Hi kV − riM 0

pHi V

kHs V Hi ˙i [2] V˙ T˙ J66 +H Hi − V − T

max

˙ [2] J55 − TT

i

⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, ⎥ ⎥ ⎥ ⎥ ⎦

Chapter 3 • Global dynamics of an HCV model with full logistic terms

where

83

  Hs + Hi H˙ i kHs V , = − δT − μi + ri 1 − Hi Hi M V˙ pHi = − μv , V V   T V T˙ − μT . = βT 1− T T Tmax

We finally obtain ⎡ kHs V b11 Hi ⎢ pHi b22 ⎢ V ⎢ r i Hi ⎢ 0 kV − M  ⎢ B =⎢ 0 T V βT 1 − Tmax ⎢ Hi ⎢ ⎢ 0 0 ⎣ 0 0

kHs V Hi − rsMHs

b33 

0

βT 1 −

T Tmax



V Hi

−δHi 0 0

0 0 0

b44

− rsMHs

kV −

0

r i Hi M

0

b55 pHi V

0 0 δHi

⎤

⎥ ⎥ ⎥ ⎥ ⎥ kHs V ⎥ − Hi ⎥ ⎥ kHs V ⎥ Hi ⎦ b66

with kHs V rs Hi 2rs Hs ri Hi − − − , Hi M M M rs Hi 2rs Hs pHi − − , b22 = rs − μs − kV − M M V 2ri Hi pHi ri Hs − − , b33 = ri − μi − δT − M V M βT V T 2rs Hs V rs Hi b44 = rs − μs − βT − kV − 1− − , − T Tmax Tmax M M   2ri Hi V ri Hs βT V T − , b55 = ri − μi − βT − − δT − 1− T Tmax Tmax M M   βT V T V H˙ i pHi b66 = . − 1− − − βT Hi V Tmax T Tmax b11 = rs − μs − kV −

The second compound system of model (3.11) can be represented by the following system of differential equations: z˙ = Bz.

(3.41)

z = max{|z1 |, U1 , U2 , |z6 |},

(3.42)

We define the norm on R6

where z = (z1 , z2 , z3 , z4 , z5 , z6 ) ∈ R6 , U1 (z2 , z3 ) are defined as " if sgn(z2 ) = sgn(z3 ) |z2 | + |z3 | U1 = max{|z2 |, |z3 |} if −sgn(z2 ) = sgn(z3 ),

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Advances in Epidemiological Modeling and Control of Viruses " U2 =

if sgn(z4 ) = sgn(z5 ) |z4 | + |z5 | max{|z4 |, |z5 |} if −sgn(z4 ) = sgn(z5 ),

where the function sgn(x) is given by ⎧ ⎨ −1 if x < 0, sgn(x) = 0 if x = 0, ⎩ 1 if x > 0. To facilitate the calculations, we note that |z2 |, |z3 |, |z2 + z3 | ≤ U1 , |z4 |, |z5 |, |z4 + z5 | ≤ U2 . The Lozinskii measure can be evaluated as μ(B) = inf{c : D+ z ≤ cz} for all solutions of z˙ = Bz, where D+ is the right-hand derivative. We proceed to estimate D+ z. The full calculation involves eight separate cases, based on the definition of  ·  within each orthant. Case 1: If |z1 | ≥ max{U1 , U2 , |z6 |}, then z = |z1 |. We have that z1 z˙ 1 |z1 | z1 kHs V z1 z1 z1 + (z2 + z3 ) − δT z4 = b11 |z1 | Hi |z1 | |z1 |  2rs Hs ri Hi kHs V rs Hi − − |z1 | ≤ rs − μs − kV − − Hi M M M kHs V (|z2 | + |z3 |) + δT |z4 | + Hi  rs Hi 2rs Hs ri Hi ≤ rs − μs − kV − − − + δT |z1 | M M M ˙  Hs ri Hi βs rs Hs = − + δT z − − Hs Hs M M ˙  Hs ri (Hi + Hs ) βs (rs − ri )Hs = − + δT z. − − Hs Hs M M

D+ z =

Case 2: If |z6 | ≥ max{|z1 |, U1 , U2 }, then z = |z6 |.

Chapter 3 • Global dynamics of an HCV model with full logistic terms

Thus z6 z˙ 6 |z6 |   ˙ βT V T z6 V pHi z6 Hi pHi z5 + − 1− z6 = − − βT V |z6 | Hi V Tmax T Tmax |z6 |  ˙ Hi βT V z. ≤ − Hi T

D+ z =

Case 3: If U1 ≥ max{|z1 |, U2 , |z6 |} and sgn(z2 ) = sgn(z3 ), then z = |z2 | + |z3 |. We have that z2 z3 z˙ 2 + z˙ 3 |z2 | |z3 |   z2 z3 pHi z2 ri Hi z3 z1 + b22 z2 + kV − z2 + b33 z3 = V |z2 | |z2 | M |z3 | |z3 | rs Hs z2 z3 z3 + δT z6 − M |z2 | |z3 |  2rs Hs pHi ri Hi pHi rs Hi |z1 | + rs − μs − − − − |z2 | ≤ V M M V M  2ri Hi pHi rs Hs ri Hs − − − + ri − μi − δT − |z3 | + δT |z6 | M M V M  2ri Hi ri Hs rs Hs − − + δT z ≤ rs − μs − M M M  2ri (Hi + Hs ) (rs − ri )Hs − + δT z, ≤ rs − μs − M M

D+ z =

using ri ≤ rs , μs ≤ μi , and

r i Hs M

≤

r s Hs M .

Case 4: If U1 ≥ max{|z1 |, U2 , |z6 |}, −sgn(z2 ) = sgn(z3 ), and |z2 | > |z3 |, then z = |z2 |. We have that z2 z˙ 2 |z2 | z2 pHi z2 rs Hs z2 z1 + b22 z2 − z3 = V |z2 | |z2 | M |z2 |   2rs Hs pHi pHi rs Hi rs Hs |z1 | + rs − μs − kV − − − |z2 | + |z3 | ≤ V M M V M

D+ z =

85

86

Advances in Epidemiological Modeling and Control of Viruses  rs Hi rs Hs = rs − μs − kV − − z M M  ˙ βs Hs z, − = Hs Hs

using 0 ≤ rs − ri , 0 ≤ μi − μs , and

r i Hs M

≤

r s Hs M .

Case 5: If U1 ≥ max{|z1 |, (|z4 | + |z5 |), |z6 |}, −sgn(z2 ) = sgn(z3 ), and |z2 | < |z3 |, then z = |z3 |. We have that z3 z˙ 3 |z3 |   z3 ri Hi z3 z3 z2 + b33 z3 + δT z6 = kV − M |z3 | |z3 | |z3 |     2ri Hi pHi ri Hi ri Hs − − ≤ −kV + |z2 | + ri − μi − δT − |z3 | + δT |z6 | M M M V  ri Hi pHi ri Hs − − + δT z ≤ ri − μi − δT − M M V  ri (Hs + Hi ) pHi − z. = ri − μi − M V

D+ z =

Case 6: U2 ≥ max{|z1 |, U1 , |z6 |} and sgn(z4 ) = sgn(z5 ), then z = |z4 | + |z5 |. Thus z4 z5 z˙ 4 + z˙ 5 |z4 | |z5 |    βT V T 2rs Hs z4 V rs Hi − kV − 1− − z4 = rs − μs − βT − T Tmax Tmax M M |z4 |    T 2ri Hi z5 βT V V ri Hs 1− − z5 − δT − + ri − μi − βT − T Tmax Tmax M M |z5 |      T V ri Hi z5 z4 z5 rs Hs z4 + βT 1 − z2 + z3 + kV − z4 − z5 Tmax T |z4 | |z5 | M |z5 | M |z4 |    βT V T 2rs Hs ri Hi V rs Hi − 1− − − |z4 | ≤ rs − μs − βT − T Tmax Tmax M M M    βT V T V ri Hs 2ri Hi rs Hs − δT − 1− |z5 | − + ri − μi − βT − − T Tmax Tmax M M M   T V + βT 1 − (|z2 | + |z3 |) Tmax T

D+ z =

Chapter 3 • Global dynamics of an HCV model with full logistic terms  βT V rs Hs 2ri Hi ri Hs − ≤ rs − μs − − − (z) Tmax M M M  βT V 2ri (Hi + Hs ) (rs − ri )Hs − (z), − ≤ rs − μs − Tmax M M using that −μi ≤ −μs and ri ≤ rs . Case 7: U2 ≥ max{|z1 |, U1 , |z6 |}, −sgn(z4 ) = sgn(z5 ), and |z4 | < |z5 |, then z = |z4 |. Thus z4 z˙ 4 |z4 |    βT V T 2rs Hs z4 V rs Hi − kV − 1− − z4 = rs − μs − βT − T Tmax Tmax M M |z4 |   V z4 rs Hs z4 kHs V z4 T + βT 1 − z2 − z5 − z6 Tmax T |z4 | M |z4 | Hi |z4 |    T 2rs Hs βT V V rs Hi 1− − |z4 | − kV − = rs − μs − βT − T Tmax Tmax M M   V rs Hs kHs V T |z6 | + βT 1 − |z2 | + |z5 | + Tmax T M Hi  rs Hs kHs V βT V rs Hi z − + − ≤ rs − μs − kV − Tmax M M Hi ˙  Hi βT V (rs − ri )(Hi + Hs ) = + rs − ri − μs + μi − kV − − + δT z. Hi Tmax M

D+ z =

Case 8: U2 ≥ max{|z1 |, U1 , |z6 |}, −sgn(z4 ) = sgn(z5 ), and |z4 | < |z5 |, then z = |z4 | + |z5 |. Thus

   T 2ri Hi z5 βT V V ri Hs D+ z = ri − μi − βT 1− − z5 − δT − − T Tmax Tmax M M |z5 |    T V z5 ri Hi z5 kHs V z5 + βT 1 − z3 + kV − z4 + z6 Tmax T |z5 | M |z5 | Hi |z5 |    βT V T 2ri Hi V ri Hs − δT − 1− − |z5 | ≤ ri − μi − βT − T Tmax Tmax M M    T V ri Hi kHs V |z4 | + |z6 | + βT 1 − |z3 | + −kV + Tmax T M Hi  ri Hi kHs V βT V ri Hs ≤ ri − μi − δT − z − + − Tmax M M Hi

87

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Advances in Epidemiological Modeling and Control of Viruses

=

˙  Hi βT V z. − Hi Tmax

From all the cases presented above, we obtain the following estimate: D+ z ≤ [max{θ1 , θ2 , θ3 , θ4 , θ5 , θ6 , θ7 , θ8 }] z,

(3.43)

where H˙s βs (rs − ri )Hs ri (Hi + Hs ) − − − + δT , Hs Hs M M H˙ i βT V θ2 = − , Hi T 2ri (Hi + Hs ) (rs − ri )Hs − + δT , θ3 = rs − μs − M M H˙s βs θ4 = − , Hs Hs ri (Hs + Hi ) pHi θ5 = ri − μi − − , M V βT V 2ri (Hi + Hs ) (rs − ri )Hs − , θ6 = rs − μs − − Tmax M M H˙ i βT V (rs − ri )(Hi + Hs ) θ7 = + δT , + rs − ri − μs + μi − kV − − Hi Tmax M H˙ i βT V θ8 = − . Hi Tmax θ1 =

On the other hand, using the upper bounds of Hs , Hi , V , and T and the uniform persistence, we obtain the following:

− βTTV ≤ − βTTMCV , μv CHi H0 βs βs −H ≤ − , H0 s

i − pH V ≤−

,

−kV ≤ −kCV . Then θ1 , θ4 ≤

H˙s Hs

(rs −ri )Hs ≤ ψ1 , M (rs −ri )Hs (rs −ri )H˜s − M ≤− M , ˜ ri (Hi +Hs ) − M ≤ − riMHi , −ri ) i )H˜s − (rsM Hs ≤ − (rs −r , M

βs −H − s

δT ≤ δTM ,

H˙ i Hi + b, and θ3 , θ5 , θ6 ≤ b. H˙ i H˙s Hs + b, μ(B) ≤ Hi + b, and μ(B) ≤ b. Along each solution (Hs (t), Hi (t), (3.11) with the initial value X0 = (Hs (0), Hi (0), V (0), T (0)) ∈ K, when

+ b, θ2 , θ7 , θ8 ≤

Therefore μ(B) ≤ V (t), T (t)) of system t > t ∗ , we have 1 t

t 0

μ(B)ds ≤

1 t

0

t∗

μ(B)ds +

Hs (t) t − t∗ 1 ln +b , ∗ t Hs (t ) t

Chapter 3 • Global dynamics of an HCV model with full logistic terms

1 t 1 t



t

1 μ(B)ds ≤ t

t

1 μ(B)ds ≤ t

0

0

t∗

0

t∗

μ(B)ds +

89

Hi (t) t − t∗ 1 ln + b , t Hi (t ∗ ) t

μ(B)ds + b

0

t − t∗ . t

The boundedness of Hs and Hi implies that 1 lim sup sup t→∞ X0 ∈K t

t

μ(B)ds ≤ b < 0.

0

As a result, the global asymptotic stability of E1 follows from the Li-Muldowney global stability criterion.

3.6 Numerical simulations In this section, we present numerical simulations to illustrate the theoretical results. We use MatLab functions ode23 and ode45 to solve the initial value problem of model (3.11). At first, we simulate the model using the parameter values given in Table 3.1, thus we fixed some parameter values as follows: βs = 50, rs = 1.55, μi = 0.5, μv = 5, μT = 0.02.

μs = 0.25, M = 300, ri = 0.2, βT = 0.0003,

k = 0.00003, δ = 0.2, p = 200, Tmax = 150,

(3.44)

Simulations of the model in this situation produce stable dynamics, as is presented in Fig. 3.2. This shows that uninfected hepatocytes, infected hepatocytes, virus and T killer cells converge to the equilibrium E0 . They show that the disease-free equilibrium E0 under some condition (see Theorem 3.5.6) is asymptotically stable. Here, H0 = 285.5083, and the basic reproduction number is given by R0 = 0.7045 < 1. Next, we use the same set of parameter values as those in (3.44), but we vary the values of μT = 0.3, ri = 1, βT = 0.003, and k = 0.0003. Thus the conditions of Theorem 3.5.2 and Theorem 3.5.5 are satisfied, where c3 = 9.1466, c2 = 13.3438, c1 = 11.6451, c0 = 5.8127, and (c3 c2 − c1 )c1 − c32 c0 = 5.8127. Then system (3.11) has a unique locally asymptotically stable endemic equilibrium E1 = (Hs∗ , Hi∗ , V ∗ , T ∗ ), where Hs∗ = 202.6982, Hi∗ = 29.0912, V ∗ = 1.1636 × 103 , and T ∗ = 10.7987, as is illustrated in Fig. 3.3. Here we have that H0 = 285.5083 and R0 = 6.9488. Next, we use the same set of parameter values as those in (3.44), but we vary the values of μs = 0.9, μT = 0.3, μi = 0.9, rs = 1.01, ri = 1, βT = 0.003, and k = 0.0003. Thus H0 = 139.2934 and R0 = 2.4525, then model (3.11) admits a unique endemic equilibrium. Also the conditions of Theorem 3.5.8 are satisfied, where CV = 400, CT = 10, H˜s = 24.6248,

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FIGURE 3.2 E0 is globally asymptotically stable.

FIGURE 3.3 E1 is locally asymptotically stable.

Chapter 3 • Global dynamics of an HCV model with full logistic terms

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FIGURE 3.4 E1 is globally asymptotically stable.

H˜ i = 29.9178, TM = 40.6266, ψ1 = −0.3636, and b = max{−0.3821, −0.0295, −0.3590, −0.3587, −0.0983, −0.0090, −0.0569, −0.0080} < 0. Then system (3.11) has a unique globally asymptotically stable endemic equilibrium E1 = (Hs∗ , Hi∗ , V ∗ , T ∗ ), where Hs∗ = 33.2322, Hi∗ = 97.7886, V ∗ = 3911.5, and T ∗ = 31.0250, as is illustrated in Fig. 3.4.

3.7 Conclusion and discussion In this chapter, we have presented the mathematical analysis on global dynamics of the viral infection model (3.11) for hepatitis C, which is a modification of the model proposed by Avendaño et al. [16] and includes full density-dependent proliferation of hepatocytes and immune system response. The basic reproduction number for system (3.11) depends on the infection rate k, the intrinsic growth rate of infected hepatocytes ri , the virus production rate p, the carrying capacity of hepatocytes, and the mortality rates μi and μv . If R0 < 1, only the disease-free equilibrium E0 exists, and it is locally and globally asymptotically stable, i.e., the virus is eradicated and the disease eventually dies out (Theorems 3.5.3 and 3.5.6). On the other hand, if R0 > 1, then the system is uniformly persistent and a unique endemic equilibrium E1 appears, i.e., the virus persists within host.

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We have chosen the parameter k (the per capita infection rate) as the bifurcation parameter; we observed that there always exists a disease-free equilibrium point that is globally asymptotically stable for k < k ∗ and unstable for k > k ∗ . At k = k ∗ , the model presents forward bifurcation. (In this type of bifurcation the disease-free equilibrium point transitions from stable to unstable and the endemic equilibrium, which is stable, appears.) Further analysis shows that E1 is locally asymptotically stable when a sufficient condition is satisfied, (c3 c2 − c1 )c1 − c32 c0 > 0 (Theorem 3.5.5). Moreover, conditions for global stability of the unique endemic equilibrium E1 are obtained (Theorem 3.5.8). That means that the virus load will tend to the level at the unique endemic equilibrium in this case. To have a better understanding of the disease, more mathematical models should be formulated. When the virus of HCV enters the body and the virus production occurs in the hepatocytes, a strong immune response is effectuated, in which not only do the T killer cells intervene, but the antibody also plays an important role in fighting the infection. Thus, for future works, we contemplate to study mathematical models that include the presence of antibody and its effect on the viral load, besides considering saturation and the effect of therapy in the acute phase of infection.

Acknowledgments This work was supported by Sistema Nacional de Investigadores (15284) and Conacyt-Becas.

References [1] World Health Organization, Global Hepatitis Report, https://www.who.int/news-room/fact-sheets/ detail/hepatitis-c, 2020. (Accessed August 2020). [2] B.R. Edlin, B.J. Eckhardt, M.A. Shu, S.D. Holmberg, T. Swan, Toward a more accurate estimate of the prevalence of hepatitis C in the United States, Hepatology 62 (5) (2015) 1353–1363. [3] V. Hope, I. Eramova, D. Capurro, M. Donoghoe, Prevalence and estimation of hepatitis b and C infections in the WHO European region: a review of data focusing on the countries outside the European Union and the European free trade association, Epidemiol. Infect. 142 (2) (2014) 270–286. [4] J. Young, N. Weis, H. Hofer, W. Irving, O. Weiland, E. Giostra, J.M. Pascasio, L. Castells, M. Prieto, R. Postema, C. Lefevre, D. Evans, H.C. Bucher, J.L. Calleja, The effectiveness of daclatasvir based therapy in European patients with chronic hepatitis C and advanced liver disease, BMC Infect. Dis. 17 (1) (2017) 45. [5] K. Razali, H.H. Thein, J. Bell, M. Cooper-Stanbury, K. Dolan, G. Doré, J. George, J. Kaldor, M. Karvelas, J. Li, L. Maher, S. McGregor, M. Hellard, F. Poeder, J. Quaine, K. Stewart, H. Tyrrell, M. Weltman, O. Westcott, A. Wodak, M. Law, Modelling the hepatitis C virus epidemic in Australia, Drug Alcohol Depend. 91 (2–3) (2007) 228–235. [6] J.S. Dazley, L.D. Sriramulu, J. Slim, Decompensated HCV patients with co morbidities including HIV who are medically treated are shown to minimize decompensation related admissions and healthcare cost: a case series, J. Infect. Public Health 10 (2) (2017) 228–231. [7] World Health Organization, Health topics (hepatitis), https://www.who.int/health-topics/hepatitis# tab=tab_1. (Accessed August 2020). [8] Khondoker Nazmoon Nabi, Chandra N. Podder, Sensitivity analysis of chronic hepatitis C virus infection with immune response and cell proliferation, Int. J. Biomath. (2020) 2050017. [9] Huda Amer Hadi, A mathematical model of hepatitis C virus infection incorporating immune responses and cell proliferation, (Ph.D. Thesis), University of Texas at Arlington, 2017.

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[10] Lijuan Song, Cui Ma, Qing Li, et al., Global dynamics of a viral infection model with full logistic terms and antivirus treatments, Int. J. Biomath. 10 (1) (2017) 1750012. [11] Hajar Besbassi, Zineb el Rhoubari, Khalid Hattaf, Noura Yousfi, Dynamics of an HBV infection model with cell-to-cell transmission and CTL immune response, Rev. Afr. Rech. Inform. Math. Appl. 30 (2019) 57–69. [12] Abdessamad Tridane, Khalid Hattaf, Yafia Radouane, Fathalla Rihan, Mathematical modeling of HBV with the antiviral therapy for the immunocompromised patients, Commun. Math. Biol. Neurosci. (2016). [13] Khalid Hattaf, Kalyan Manna, Modeling the dynamics of hepatitis B virus infection in presence of capsids and immunity, in: Mathematical Modelling and Analysis of Infectious Diseases, Springer, 2020. [14] Sandip Banerjee, Ram Keval, Sunita Gakkhar, Global dynamics of hepatitis C viral infection with logistic proliferation, Int. J. Biomath. 9 (4) (2016). [15] Gamaliel Blé, Lourdes Esteva, Alejandro Peregrino, Global analysis of a mathematical model for hepatitis C considering the host inmune system, J. Math. Anal. Appl. 461 (2018) 1378–1390. [16] R. Avendaño, L. Esteva, J.A. Flores, J.L. Fuentes Allen, G. Gómez, J. López-Estrada, A mathematical model for the dynamics of hepatitis C, J. Theor. Med. 4 (2002) 109–118. [17] H. Dahari, J.E. Layden-Almer, E. Kallwitz, R.M. Ribeiro, S.J. Cotler, T.J. Layden, A.S. Perelson, A mathematical model of hepatitis C virus dynamics in patients with high baseline viral loads or advanced liver disease, Gastroenterology 136 (2009) 1402–1409. [18] Lawrence Perko, Differenctial Equations and Dynamical Systems, 3rd edition, Springer, 2001. [19] G.C. Layek, An Introduction to Dynamical Systems and Chaos, Springer, 2015. [20] Qingkai Kong, A Short Course in Ordinary Differential Equations, 2nd edition, Springer, 2010. [21] Carlos Castillo-Chavez, Baojun Song, Dynamical models of tuberculosis and their applications, Math. Biosci. Eng. 1 (2004) 361–404. [22] M.Y. Li, J.S. Muldowney, A geometric approach to the global stability problems, SIAM J. Math. Anal. 27 (1996) 1070–1083. [23] J.S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mt. J. Math. 20 (1990) 857–872. [24] G. Birkhoff, G.C. Rota, Ordinary Differential Equations, Ginn, Boston, 1982. [25] P. van den Driessche, J. Watmoungh, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002) 29–48. [26] Yazdan Bavafa-Toosi, Introduction to Linear Control Systems, Elsevier, 2017, pp. 208–214. [27] Esteban Domingo, Virus as Populations: Composition, Complexity, Quasispecies, Dynamics, and Biological Implications, 2nd edition, Academic Press, 2019. [28] H.R. Thieme, Persistence under relaxed point-dissipativity (with application to an endemic model), SIAM J. Math. Anal. 24 (1993) 407–435.

4 On a Novel SVEIRS Markov chain epidemic model with multiple discrete delays and infection rates: modeling and sensitivity analysis to determine vaccination effects Divine Wanduku, Omotomilola Jegede, Chinmoy Rahul, Broderick Oluyede, and Oluwaseun Farotimi Department of Mathematical Sciences, Georgia Southern University, Statesboro, GA, United States

4.1 Introduction Infectious disease epidemic outbreaks, such as measles, typhoid fever, yellow fever, smallpox, tuberculosis, and more recently corona virus (SARS–CoV-2), have led to high mortality rates. For example, according to the 2017 Center for Disease Control and Prevention (CDC) report [5,6], nearly 26 million cases of typhoid fever and 5 million cases of paratyphoid fever occurred annually worldwide, leading to approximately 215,000 deaths. Moreover, the most vulnerable population included children under the age of four and adults 60 years or older [26]. More recently, according to the World Health Organization (WHO) [25], globally, as of 3:05 p.m. CET, December 20, 2020, there have been 75,098,369 confirmed cases of COVID-19, including 1,680,339 deaths, reported to WHO. The discovery of new vaccines for such epidemics has significantly curtailed human deaths and mitigated the drastic negative effects of some of these epidemics on human life and on global economies. For typhoid fever caused by Salmonella typhimurium (S. typhi), there are notable vaccines available, and an example is a typhoid conjugate vaccine (TCV), which is injectable and consists of Vi polysaccharide antigen connected to tetanus toxoid protein [23,26]. More recently, new vaccines for COVID-19 have been discovered, e.g., Pfizer-BioNTech COVID-19 vaccine (cf. [8]), Moderna, (cf. [7]), Johnson and Johnson’s Janssen (cf. [7]). The use of mathematical models to investigate disease dynamics goes as far back as the SIR (Susceptible-Infected-Recovered) disease epidemic model by Kermack and McKendrick [17]. In general, models such as [17] are called compartmental models because they represent the disease dynamics in the population over various distinct disease states Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00009-0 Copyright © 2023 Elsevier Inc. All rights reserved.

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such as susceptible, vaccinated, exposed, infectious, and removed. Research on such compartmental epidemic models has advanced, and various types of general and specific disease dynamic models have been investigated. For instance, SVEIRS (SusceptibleVaccinated-Exposed-Infected-Removed-Susceptible), SVIS (Susceptible-Vaccinated-Infected-Susceptible), SVIRS (Susceptible-Vaccinated-Infected-Removed-Susceptible), SIS (Susceptible-Infected-Susceptible), SIR (Susceptible-Infected-Removed), SIRS (Susceptible-Infected-Removed-Susceptible), SEIRS (Susceptible-Exposed-Infected-RemovedSusceptible), and SEIR (Susceptible-Exposed-Infected-Removed) epidemic models (cf. [9,14,30–37,39]) are other examples of recent advances in the study of compartmental epidemic models, beyond the earlier work [17]. In general, these compartmental models can be classified as either deterministic or stochastic. Some examples of deterministic models are [9,30,35], and examples of stochastic models are [10,19,31–34,36,37,39,40]. When the population contains a large number of susceptible and infected states, where small scale random epidemiological fluctuations can be ignored, deterministic models are suitable choices for disease dynamics. However, in most epidemic dynamics, there are significant random perturbations which are better represented by stochastic models. A review of these two types of epidemic models and their applications to study important properties of diseases dynamics are given in [13,24,27] Mathematical models including vaccination play an important role in understanding and planning new vaccination strategies, and they also aid in determining long- and shortterm effects of this disease control measure on the disease dynamics (cf. [13]). In [21] the authors investigated a deterministic differential equation SVEIR model with waning artificial immunity as a result of the vaccination age of individuals in the population; and they also investigated disease invasion, as a consequence of the waning immunity of the vaccine. In [43] the authors investigated the effects of vaccination on eradication and persistence of disease in a steady state population of a stochastic differential equation SVIR model. In [38] the authors conducted statistical inference in a discrete time Markov chain SVIR epidemic model for influenza; they estimated the basic reproduction number of the disease. Other epidemic models investigating other aspects of vaccination in controlling epidemics are given in [13,14,20,42]. Stochastic epidemic models that are Markov chains can be classified, based on the index set for the stochastic process, as either discrete time Markov chain (DTMC) models or continuous time Markov chain (CTMC) models. Some examples of CTMC epidemic models are given in [2,3,16]; the examples of DTMC epidemic models are given in [10,31,40]. Most CTMC epidemic models are based on birth-and-death processes, wherein the construction of the infinitesimal generator rate matrix for the CTMC is based on the assumption that there is only one transition between disease states in a sufficiently small time interval (cf. [2]). DTMC epidemic models that are not based on birth-and-death processes do not restrict the number of transitions between the disease states in any discrete time interval, and notable examples of families of such models include: chain-binomial epidemic model and epidemic models based on branching processes (cf. [2,10,15,18,31,38,40]). Thus DTMC epidemic models approximate CTMC epidemic models of the same family. This is

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because DTMC models have more simplified transition probabilities over time, making their calculations and analysis less challenging for dynamic optimization and statistical estimation of system parameters. However, DTMC epidemic models are more suitable for small population outbreaks and also where the epidemic can be observed for a finite time [10,31,40]. A special class of DTMC epidemic models not based on birth-and-death processes is the chain-binomial epidemic model class. Some classic examples of the chain-binomial models are the Greenwood [11] and Reed-Frost models [1]. This class of DTMC epidemic models is based on counting processes, where transition probabilities for the Markov chain follow discrete probability distributions from the multinomial distribution family. Other examples of chain-binomial models include [4,10,15,18,31,38,40]. One major limitation in the representation of several DTMC chain-binomial epidemic models (cf. [15,18,40]) is the mathematical structural expression of these models. Delays in these disease dynamics are not properly represented, and as a result of this, transitions between states always occur at regular time intervals. For instance, in [18] it can be seen in the mathematical expression of the SEIR model that some newly exposed E(t) individuals at time t are expected to move to the infectious full-blown disease state I (t + t) in the next time step t + t, where t is a constant. This expression used in the epidemic model becomes an issue when the value of t does not coincide with the realistic length of time individuals remain in a given state before moving to the next state. Indeed, in most infectious diseases, there exists an expected incubation latent period T1 for a disease, expected infectious and natural immunity periods, T2 and T3 , respectively. There is also an expected artificial immunity period T4 . Designing a mathematical model structure that incorporates the delays in the disease dynamics is an important first step to a more accurate epidemic model for the disease dynamics. Also, a more realistic model would lead to more meaningful and informative results for the disease dynamics. Tuckwell and Williams [31] presented a DTMC model for an SIR epidemic model where individuals of the infectious state Yk (t) are represented based on the number present at current time t and also based on how long k ≥ 0 they have been in the infectious state. Other authors [10,38] investigated other compartmental disease models applying variant model structural extensions of the framework in [31]. Indeed, Wanduku et al. [38] conducted statistical inference for influenza epidemics with vaccination in a DTMC SVIR epidemic model, where the susceptible, vaccinated, infected, and removed states are represented based on the current time t and how long individuals have been in these states. Applying a similar but more extensive framework than [38], we design a new platform and investigate a DTMC epidemic model for general SVEIRS disease epidemic dynamics. The new model can be applied to SVEIRS disease epidemic dynamics such as typhoid fever, COVID-19, and tuberculosis. The final DTMC epidemic model incorporates all the delays above; namely, the expected incubation latent period T1 for the disease; the expected infectious and natural immunity periods T2 and T3 , respectively; and the expected artificial immunity period T4 .

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The primary objective of this study is to explore the effects of vaccination as a control measure for an epidemic such as typhoid fever, COVID-19, and tuberculosis, which can be represented by SVEIRS models. Another objective is to understand the interconnection between the following factors on the disease dynamics: high or low disease transmission rates; high or low vaccination rates; and waning artificial immunity from the vaccines. In addition, two special sub-SVEIRS DTMC models are presented based on whether (1) the drive to get vaccinated by susceptible individuals in the population is inspired by interaction with an infected person, which does not lead to infection; or (2) the drive to get vaccinated by susceptible individuals in the population is uncorrelated with interacting with other individuals in the population. We affirm our results for the positive effects of vaccination in controlling the disease by conducting a sensitivity analysis on the model. The rest of this chapter is organized as follows. In Sections 4.2 and 4.3, we describe a novel framework to compartmentalize the SVEIR population using the delays in the disease dynamics. In Section 4.4, we define a stochastic process for the SVEIRS epidemic model. Moreover, we show that the model is a Markov chain and derive a general formula for the transition probability distribution. In Section 4.5, we derive two special SVEIRS DTMC epidemic models. In Section 4.6, numerical simulation results are presented and sensitivity analysis is conducted to determine the effects of vaccination in controlling the disease.

4.2 Description of the SVEIRS epidemic and the delays in the disease dynamics We consider a fixed human population of size n in which an outbreak of the disease such as Salmonella typhi (typhoid fever) occurs. Furthermore, there is no migration during the outbreak period of the epidemic. The total human population n at any time is partitioned into four distinct classes; namely, the susceptible (S) who are vulnerable to infection, the exposed (E) who are infected but not infectious and are incubating the disease (it is assumed that the incubation period is T1 ). The exposed class becomes the infectious class (I ) after the incubation period T1 . The infectious class (I ) is assumed to be the main carrier of the infectious agent, e.g., Salmonella typhi bacteria, and the main source of infection in the population. The infectious class exhibits symptoms of the disease and is in poor health. The class of carriers of the infectious agent, e.g., Salmonella typhi bacteria, who are asymptomatic and who are in the recovered state is not considered. The infectious state (I ) is treated and assumed totally recovered from the disease after a period of infectiousness T2 . The recovered state is denoted by R. This class is assumed to exhibit temporary acquired natural immunity, which lasts for T3 time units, after which the immunity wanes and the individual returns to the susceptible state (S). Some susceptible individuals are vaccinated against the disease. The vaccine is assumed to effectively confer temporary artificial immunity against the disease for a period of T4 time units, after which the artificial immunity wanes and the individual returns to the susceptible state. The vaccinated class

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is denoted by V . A compartmental model framework illustrating the transitions between the different states of the SVEIRS epidemic is given in Fig. 4.1.

FIGURE 4.1 The compartmental framework for the SVEIRS epidemic model with different disease states of the epidemic system: S, V , E, I, R, and the transitions between states. Moreover, the delay times of individuals in each state are exhibited.

4.3 Discretization of time and decomposition of the SVEIRS population over time In this section, we define discrete time scales for the SVEIRS model and decompose the human population into the distinct states of the disease and over different time intervals.

4.3.1 Decomposition of the total human population over discrete time intervals The different disease states: susceptible (S), vaccinated (V ), exposed (E), infectious (I ), and recovered (R) individuals are observed over discrete time intervals of unit length. That is, over the time subintervals [t0 , t0 + 1), [t0 + 1, t0 + 2), [t0 + 2, t0 + 3), [t0 + 3, t0 + 4), . . ., [t0 + t, t0 + (t + 1)), . . ., where t0 ≥ 0 is any nonnegative real number and t = 0, 1, 2, 3, . . .. We assume without loss of generality that t0 = 0. Furthermore, we let z(t) be the number of people in state z at the beginning of the tth epoch [t, t + 1), or at the end of the (t − 1)th epoch [(t − 1), t), where z ∈ {S, V , E, I, R}. Another discrete measure of time, denoted by k = 0, 1, 2, . . . t, represents how long individuals in different disease states S, V , E, I , and R, have been in the given states since their initial conversion into the states. That is, zk (t) is the number of people in state z ∈ {S, V , E, I, R} at time t who have been in state z for k = 0, 1, 2, . . . , t time units. Note that z0 (t) is the number of people in state z ∈ {S, V , E, I, R} at time t who just got converted into state z.

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Example 4.3.1. For example, S(3), V (3), E(3), I (3), and R(3) are the total number of susceptible, vaccinated, exposed, infectious, and recovered persons, respectively, present in the population at the end of the third day (i.e., t = 3). And S2 (3), V2 (3), E2 (3), I2 (3), and R2 (3) are the number of susceptible, vaccinated, exposed, infectious, and recovered persons, respectively, present in the population at the end of the third day (t = 3) who have been in these different states for k = 2 days. It is easy to see from Section 4.2 and the description above that the total population n at time t is given as follows: n = S(t) + V (t) + E(t) + I (t) + R(t),

∀t = 0, 1, 2, . . . .

(4.1)

Furthermore, for any time t < min {T1 , T2 , T3 , T4 }, S(t) =

t 

Sk (t),

k=0 t 

E(t) =

Ek (t),

t 

V (t) = I (t) =

k=0

R(t) =

Vk (t),

k=0 t 

k=0 t 

Ik (t),

Rk (t).

(4.2)

k=0

Also, ˆ S(0) =  S (0) + S(0) = S0 (0) > 0, V (0) = V0 (0) ≥ 0, E(0) = E0 (0) ≥ 0,

(4.3)

I (0) = I0 (0) > 0, R(0) = R0 (0) ≥ 0, where  S (t) represents all susceptible individuals at time t who have never been infected

ˆ represents all susceptible individuals at time t who were previously vacor vaccinated; S(t) cinated or naturally immune to the disease, and they have relapsed to the susceptible state after losing immunity. Example 4.3.2. Corresponding to Example 4.3.1, at time t = 3 < min {T1 , T2 , T3 , T4 }, the expressions in (4.2) are written as S(3) = S0 (3) + S1 (3) + S2 (3) + S3 (3), V (3) = V0 (3) + V1 (3) + V2 (3) + V3 (3), E(3) = E0 (3) + E1 (3) + E2 (3) + E3 (3),

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I (3) = I0 (3) + I1 (3) + I2 (3) + I3 (3), R(3) = R0 (3) + R1 (3) + R2 (3) + R3 (3),

(4.4)

where z3 (3) = z0 (0), ∀z ∈ {S, V , E, I, R}.

4.3.2 One-state-at-a-time decomposition of the population states over the finite delay times Observe from Section 4.2 and Subsection 4.3.1 that the delays Ti , i = 1, 2, 3, 4, set restrictions on the existence of the states zk (t), ∀k = 0, 1, 2, . . . , t and z ∈ {S, V , E, I, R}. For example, since T1 is the incubation period of the disease, Ek (t) > 0, ∀k = 0, 1, 2, . . . T1 − 1, ET1 = I0 (t), and Ek (t) = 0, ∀k > T1 whenever E0 (t) > 0. In other words, individuals recently infected will remain exposed for a maximum of T1 units. Similar observations are made for other states I , R, and V over the delays T2 , T3 , and T4 , respectively. Thus, in this section, the states of the SVEIR population are characterized over all t ∈ R+ whenever the delays Ti > 0, i = 1, 2, 3, 4. Thus observe that when the incubation period is a constant T1 > 0, the following decomposition of the exposed state holds. For t ∈ [0, T1 ) ∪ [T1 , ∞], E(t) is expressed as follows:  E(t) =

E0 (t) + E1 (t) + E2 (t) + · · · + ET1 −1 (t) E0 (t) + E1 (t) + E2 (t) + · · · + Et (t)

for t ≥ T1 , for t < T1 .

(4.5)

Note that ET1 (t) = I0 (t), that is, ET1 (t) are those who have completed the incubation period of the disease T1 and are just becoming infectious at time t. Also, when the delay T2 is the constant infectious period for all infectious individuals in the population, the total number of infectious people at time t ∈ [0, T2 ) ∪ [T2 , ∞], I (t) is expressed as  I0 (t) + I1 (t) + I2 (t) + · · · + IT2 −1 (t) for t ≥ T2 , (4.6) I (t) = for t < T2 . I0 (t) + I1 (t) + I2 (t) + · · · + It (t) Note that IT2 (t) = R0 (t), and this represents the infectious people who have been infected from the full infectious period T2 and have just now been removed. When the delay T3 is the constant temporary natural immunity period in the population, the total number of removed people at time t ∈ [0, T3 ) ∪ [T3 , ∞], R(t) is expressed as  R(t) =

R0 (t) + R1 (t) + R2 (t) + · · · + RT3 −1 (t) R0 (t) + R1 (t) + R2 (t) + · · · + Rt (t)

for t ≥ T3 , for t < T3 ,

(4.7)

and when the delay T4 is the constant vaccine effective period in the population, the total number of vaccinated people at time t ∈ [0, T4 ) ∪ [T4 , ∞], V (t) is expressed as  V (t) =

V0 (t) + V1 (t) + V2 (t) + · · · + VT4 −1 (t) V0 (t) + V1 (t) + V2 (t) + · · · + Vt (t)

for t ≥ T4 , for t < T4 .

(4.8)

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Also note that RT3 (t) + VT4 (t) = S0 (t).

(4.9)

Indeed, RT3 (t) are individuals who have completed the effective natural immunity period T3 , and VT4 (t) are individuals who have completed the effective artificial immunity (via the vaccine) period T4 , respectively, and they have just now become susceptible again to the disease. Example 4.3.3. Corresponding to Example 4.3.2, we utilize the following values of the delays: T1 = 2, T2 = 4, T3 = 3, and T4 = 4 to write expressions for (4.5)–(4.9) as shown below. We assume that initially the following conditions are satisfied: S(0) =  S (0) > 0, E(0) = E0 (0) > 0, I (0) = I0 (0) > 0, R(0) = R0 (0) > 0, and V (0) = V0 (0) > 0. The expressions below show the decomposition of the population at time t = 3. E(3) = E0 (3) + E1 (3), I (3) = I0 (3) + I1 (3) + I2 (3) + I3 (3), R(3) = R0 (3) + R1 (3) + R2 (3), V (3) = V0 (3) + V1 (3) + V2 (3) + V3 (3).

(4.10)

However, note that for the given initial conditions, not all terms on the right-hand sides of (4.10) are nonzero at time t = 3. Furthermore, some of other terms on the right-hand sides are interrelated as shown below. ⎧ E2 (3) = I0 (3) ≥ 0, R0 (3) = I4 (3) = 0, ⎪ ⎪ ⎪ ⎪ ˆ ⎪ R3 (3) + V4 (3) = R3 (3) + 0 = S(3) = S0 (3) > 0, ⎪ ⎪ ⎪ ⎪ (3) = I (2) = E (2) = E (1) = E I 1 0 2 1 0 (0) > 0, ⎪ ⎪ ⎨ I2 (3) = I1 (2) = I0 (1) = E2 (1) = 0, (4.11) ⎪ I3 (3) = I2 (2) = I1 (1) = I0 (0) = E2 (0) = 0, R0 (3) = I4 (3) = 0, ⎪ ⎪ ⎪ ⎪ R1 (3) = R0 (2) = I4 (2) = 0, R2 (3) = R1 (2) = R0 (1) = I4 (1) = 0, ⎪ ⎪ ⎪ ⎪ ˆ R = S0 (3) > 0, ⎪ 3 (3) = R0 (0) > 0; R3 (3) = S(3) ⎪ ⎩ V3 (3) = V0 (0) > 0. Applying the restrictions in (4.11) to (4.10), we obtain the following simplification: E(3) = E0 (3) + E1 (3), I (3) = I0 (3) + I1 (3), R(3) = 0, V (3) = V0 (3) + V1 (3) + V2 (3) + V3 (3).

(4.12)

4.3.3 Joint state decomposition of the population over the finite delay times Note that the decomposition in Subsection 4.3.2 represents an expression of each state in the population over the time domain R+ partitioned into two disjoint time intervals by

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using the delays, i.e., R+ = [0, Ti ) ∪ [Ti , ∞), ∀i = 1, 2, 3, 4. Since all the states are interconnected, an individual in each state can only be in the state for a finite delay time before moving to the next state, the structure of the population continuously changes over time as the disease evolves. Thus it is important to consider a decomposition for the human population over the time domain R+ , taking into account all the delays Ti , ∀i = 1, 2, 3, 4. The decomposition in this section leads to a vector space representation of the states of the SVEIRS epidemic process. Indeed, since the delays Ti , ∀i = 1, 2, 3, 4, are positive real numbers, note that taking into account the magnitude of the delays Ti , ∀i = 1, 2, 3, 4, and without any particular order between the delays, there are 16 possible unique population decompositions determined by the time subintervals for t ∈ R+ . The subintervals are listed in Hypothesis 4.3.1. That is, in each subinterval, a unique decomposition of the total population in (4.3.1)–(4.2) consisting of the states z ∈ {S, V , E, I, R} is obtained. Hypothesis 4.3.1. H1: t < min{T1 , T2 , T3 , T4 }, H2: T1 ≤ t < min{T2 , T3 , T4 }, H3: T2 ≤ t < min{T1 , T3 , T4 }, H4: T3 ≤ t < min{T1 , T2 , T4 }, H5: T4 ≤ t < min{T1 , T2 , T3 }, H6: max{T1 , T2 } ≤ t < min{T3 , T4 }, H7: max{T1 , T3 } ≤ t < min{T2 , T4 }, H8: max{T1 , T4 } ≤ t < min{T2 , T3 }, H9: max{T2 , T3 } ≤ t < min{T1 , T4 }, H10: max{T2 , T4 } ≤ t < min{T1 , T3 }, H11: max{T3 , T4 } ≤ t < min{T1 , T2 }, H12: max{T1 , T2 , T3 } ≤ t < T4 , H13: max{T1 , T2 , T4 } ≤ t < T3 , H14: max{T1 , T3 , T4 } ≤ t < T2 , H15: max{T2 , T3 , T4 } ≤ t < T1 , H16: max{T1 , T2 , T3 , T4 } ≤ t. However, for every known SVEIRS disease epidemic, the average incubation period T1 , the average period of infectiousness under treatment T2 , the average artificial immunity period T4 , and the average period of natural immunity T3 conferred by the disease are known. This suggests that while all of the orderings of the delays Ti , i = 1, 2, 3, 4, in Hypothesis 4.3.1 H 1–H 16 are possible in a general sense, not all of these hypotheses are feasible for a specific SVEIRS epidemic. For example, we observe in the case of the outbreak of an SVEIRS disease epidemic, such as typhoid fever epidemic caused by the infectious agent Salmonella typhi, that only H 1, H 2, H 7, and H 16 are feasible in the disease dynamics for typhoid fever. We characterize the typhoid fever epidemic in what follows. Indeed, according to an article by T. Newman for the Medical News Today [23], typhoid fever is an infection caused by the bacterium named Salmonella typhimurium (S. typhi). The symptoms of the disease include weakness, abdominal pains, constipation, headaches, and prolonged fever [26]. The S. typhi is transmitted mainly through ingestion of food or water contaminated by fecal material from infected human beings [23,26]. Human beings are the main hosts of the disease, and animals do not transmit the disease. In human beings, the bacterium lives in the intestines and bloodstream. In fact, after being ingested, S. typhi incubates in the intestines for approximately one to three weeks [23,26,29]. After incubation, the bacteria move through the intestinal walls and into the bloodstream [23,26,29]. Furthermore, from the bloodstream, the bacteria spread into other tissues and organs. It is important to note that the immune system is limited to fight the

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infection as the bacteria live directly within the human cells, safe from the immune defense system (cf. [23,26]). Nevertheless, some studies show a possible short-term natural protective immunity against Salmonella [12,22,28,41] for those who have recovered from the disease. The treatment for typhoid fever exists in the form of antibiotics, and the most commonly used antibiotics are ciprofloxacin and ceftriaxone [23,26]. Despite the availability of treatment against the bacteria, there is occasional resistance to specific drugs, thereby impacting the choice of drugs to treat the disease. There is artificial immunity against S. typhi through vaccination. According to WHO [23,26], there are three recommended typhoid fever vaccines, namely: (1) a typhoid conjugate vaccine (TCV), which is injectable and consists of Vi polysaccharide antigen and is connected to tetanus toxoid protein. This vaccine is recommended for children at least 6 months of age and adults at most 45 years of age; (2) an unconjugated polysaccharide vaccine, which is based on the purified Vi antigen, recommended for persons at least two years old; and (3) a Ty21a vaccine for people over six years old. Moreover, the strongest of these vaccines can last over three years [23]. It is important to note that the vaccines are not completely effective in protecting against subsequent infection by S. typhi, and caution must be exercised to practice high hygiene ethics to prevent infection. Thus, from the description of the typhoid fever above, we observe that T1 is approximately 1–3 weeks; the maximum value of T4 is 3 years. It is clear that the period of natural immunity T3 is relatively shorter than T4 ; the period of infectiousness T2 without treatment or with improper treatment, where the infected person develops resistance against a particular treatment, can be significantly longer than Ti , i = 1, 3. Thus, based on these observations, the following constraints can be applied to reduce the number of hypotheses in Hypothesis 4.3.1 H 1–H 16. Assumption 4.3.1. Assume that infected individuals receive proper treatment, and as a result T1 ≤ T3 ≤ min{T2 , T4 }.

(4.13)

The constraint in (4.13) applied to Hypothesis 4.3.1 H 1–H 16 implies that the complete evolution of an SVEIRS disease, such as typhoid fever, over time t ∈ R+ = [0, ∞) from the onset of disease outbreak at time t = 0 can be characterized over the following subintervals from Hypothesis 4.3.1. Hypothesis 4.3.2. H1: t < min{T1 , T2 , T3 , T4 } and T1 ≤ T3 ≤ min{T2 , T4 }, H2: T1 ≤ t < min{T2 , T3 , T4 } and T3 ≤ min{T2 , T4 }, H7: max{T1 , T3 } ≤ t < min{T2 , T4 } and T1 ≤ T3 , H16: max{T1 , T2 , T3 , T4 } ≤ t and T1 ≤ T3 ≤ min{T2 , T4 }. In the rest of this section, it is assumed that the behavior of the SVEIRS disease dynamics is analogous to the typhoid fever dynamics characterized above, where the evolution

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of the disease dynamics satisfies Hypothesis 4.3.2. We consider various decompositions of the total SVEIRS population whenever Hypothesis 4.3.2 is satisfied. Different decompositions of the total population over the different time intervals in Hypothesis 4.3.2 are given subsequently. Definition 4.3.1. Decomposition of the SVEIRS population when H 1 in Hypothesis 4.3.1 holds, that is, when time t < min{T1 , T2 , T3 , T4 } and T1 ≤ T3 ≤ min{T2 , T4 }: The susceptible S, vaccinated V , exposed E, infectious I , and removed states R are decomposed at time t units beyond the onset of the disease outbreak, where t is also less than the minimum of the delay times in each state T1 , T2 , T3 , and T4 , and the delays are related as shown in (4.13). The states S, V , E, I , and R at time t < min{T1 , T2 , T3 , T4 }, where T1 ≤ T3 ≤ min{T2 , T4 } holds, are given in the following. Observe that since t < min{T3 , T4 }, the susceptible state consists of only the initial susceptible state that has remained susceptible until time t. There are no susceptible individuals who were previously removed or vaccinated. That is, at time t, where 0 < t < min{T3 , T4 }, the initial susceptible population reduces via infection. There are no increments in the susceptible population as a result of individuals losing their immunity and regaining the susceptible state. Hence the number of susceptible individuals who have remained susceptible from the onset of the disease outbreak is denoted by  S (t). That is, S(t) =  S (t) = St (t),

(4.14)

where St (t) ≤ S0 (0) at time 0 < t < min{T3 , T4 } and S0 (0) is defined in (4.3). The vaccination state V at time t < T4 consists of those who were initially vaccinated and those who are getting vaccinated from the susceptible class. That is, V (t) = V0 (t) + V1 (t) + V2 (t) + · · · + Vt (t),

(4.15)

where the initial vaccinated population V (0) = Vt (t) = V0 (0) ≥ 0. Also, for t < min{T1 , T2 }, it is easy to see from (4.3.2) that the states E and I are expressed generally as follows: E(t) = E0 (t) + E1 (t) + E2 (t) + · · · + Et (t)

(4.16)

I (t) = I0 (t) + I1 (t) + I2 (t) + · · · + It (t),

(4.17)

and

where from (4.3) observe that Et (t) = E0 (0) > 0 and It (t) = I0 (0) > 0. Indeed, note that since t ≤ T1 , all initially exposed individuals E(0) = E0 (0) from (4.3) are still in the exposed state, and these individuals are represented by Et (t) = E0 (0) > 0. However, the population is continuously infected over the time interval 0 < t < T1 ; this implies that all newly exposed individuals over this time satisfy Ek (t) ≥ 0, k = 0, 1, . . . t − 1. Hence, from (4.16), E(t) > 0. Observe further that, for t < T1 , all initial and newly infected individuals are still in the exposed state, and as a result there are no conversions to the infectious state during this

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time interval. Also, since t < T2 , all initially infected persons, I (0) = I0 (0) in (4.3), are still in the infectious state. Thus the only infectious individuals present at time t ≤ min(T1 , T2 ) are those who were initially infectious, given by I0 (0). That is, Ik (t) = 0, k = 0, 1, . . . t − 1, and It (t) = I0 (0) > 0. Hence (4.17) reduces to I (t) = It (t).

(4.18)

Since t < min{T2 , T3 }, the removal state R at time t consists of only those who were initially removed, because no infectious individuals have been removed yet. That is, from (4.7), the equation R(t) = R0 (t) + R1 (t) + R2 (t) + · · · + Rt (t)

(4.19)

is such that Rk (t) = 0,

k = 0, 1, . . . , t − 1;

Rt (t) = R0 (0) = R(0) ≥ 0.

(4.20)

It follows that Eq. (4.19) reduces to R(t) = Rt (t).

(4.21)

Therefore, from (4.14)–(4.20) and (4.21), the total population of size n present at time t < min{T1 , T2 , T3 , T4 }, where (4.13) holds, is given by n = S(t) + V (t) + E(t) + I (t) + R(t) = St (t) +

t  k=0

Vk (t) +

t 

Ek (t) + It (t) + Rt (t).

(4.22)

k=0

Furthermore, the vector B(t) = (S(t), V0 (t), V1 (t), . . . , Vt (t), E0 (t), . . . , Et (t), I0 (t) . . . , It (t))T

(4.23)

is sufficient to describe the disease dynamics in population whenever H 1 in Hypothesis 4.3.2 holds. We characterize the SVEIRS population whenever Hypothesis 4.3.2 H 2 holds. Definition 4.3.2. Decomposition of the SVEIRS population when H 2 in Hypothesis 4.3.2 holds, that is, when T1 ≤ t < min{T2 , T3 , T4 } and T3 ≤ min{T2 , T4 }: Without repeating the arguments for (4.14) and (4.15) in Definition 4.3.1, observe that when t < min{T3 , T4 }, S(t) = S(t) = St (t) 

(4.24)

V (t) = V0 (t) + V1 (t) + V2 (t) + · · · + Vt (t),

(4.25)

and

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where the initial vaccinated population V (0) = Vt (t) = V0 (0) ≥ 0. Now, since T1 ≤ t, the exposed state E(t) at time t is given by (4.5). That is, E(t) = E0 (t) + E1 (t) + E2 (t) + · · · + ET1 −1 (t).

(4.26)

Also, since T1 ≤ t < T2 , it is easy to see that the infectious state I consists of those who were initially infectious and those who have converted from the exposed state. That is, I (t) = I0 (t) + I1 (t) + I2 (t) + · · · + It (t),

(4.27)

where I (0) = It (t) = I0 (0) > 0. Indeed, note that since t ≥ T1 , the infectious state consists of all previously exposed individuals E, present beyond the incubation period T1 of the disease. That is, I0 (t) = ET1 (t),

I1 (t) = ET1 +1 (t), . . . , I(t−T1 ) (t) = ET1 +(t−T1 ) (t).

(4.28)

Therefore, from (4.27)–(4.28), it follows that I (t) can be written equivalently as follows: I (t) = ET1 (t) + ET1 +1 (t) + · · · + ET1 +(t−T1 ) (t).

(4.29)

From the arguments in (4.19)–(4.21) of Definition 4.3.1, observe that when t < T3 , the removal state R(t) is given by R(t) = Rt (t),

(4.30)

where R(0) = Rt (t) = R0 (0) ≥ 0. Moreover, from (4.24)–(4.30) and (4.14), the total population of size n at time T1 ≤ t < min{T2 , T3 , T4 }, where T3 ≤ min{T2 , T4 } holds, is given by n = St (t) +

t  k=0

Vk (t) +

T 1 −1 k=0

Ek (t) +

t 

Ik (t) + Rt (t).

(4.31)

k=0

Furthermore, the vector B(t) = (St (t), V0 (t), V1 (t), . . . , Vt (t), E0 (t), . . . , ET1 −1 (t), I0 (t), . . . It (t))T

(4.32)

is sufficient to describe the SVEIRS disease dynamics population whenever H 2 holds. Definition 4.3.3. Decomposition of the SVEIRS population when t lies in max{T1 , T3 } ≤ t < min{T2 , T4 } and T1 ≤ T3 ≤ min{T2 , T4 } that is, when Hypothesis 4.3.2 H 7 holds: Define by  S (t) all those who have remained susceptible from the onset of the disease

ˆ all those who have newly converted into the susceptible outbreak until time t and by S(t) state at time t via losing natural or artificial immunity. Observe that since t ≥ max{T1 , T3 }, the susceptible population S(t) present at time t consists of both those who have remained

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susceptible from the onset of the disease outbreak and those who have lost their natural immunity. That is, S(t) can be decomposed into two categories as follows: ˆ S(t) =  S (t) + S(t),

(4.33)

S (t) = St (t) 

(4.34)

where

represents all the susceptible individuals who have remained in the susceptible state for t time units and ˆ = S0 (t) + S1 (t) + S2 (t) + · · · + St−T 3 (t) S(t)

(4.35)

represents all those who have returned to the susceptible state after losing their natural immunity lasting for T3 time units. It is easy to see that at time t ≥ T3 , S0 (t) = RT3 (t), S1 (t) = RT3 +1 (t), S2 (t) = RT3 +2 (t), · · · , St−T3 (t) = RT3 +t−T3 (t),

(4.36)

where St−T3 (t) = RT3 +t−T3 (t) = Rt (t) = R(0) ≥ 0 are those who at initial time were in the removal state R and at time t ≥ T3 they were in the susceptible state for t − T3 time units ˆ can be expressed alternatively as follows: after losing their natural immunity. Thus S(t) t−T t−T  3 3 S (t) = Sk (t) = RT3 +k (t). k=0

(4.37)

k=T3

Applying the arguments in (4.25)–(4.30) in Definition 4.3.2, observe that since T1 ≤ t < min{T2 , T4 }, where T1 ≤ T3 ≤ min{T2 , T4 }, other states are expressed as follows: V (t) = V0 (t) + V1 (t) + V2 (t) + · · · + Vt (t), E(t) = E0 (t) + E1 (t) + E2 (t) + · · · + ET1 −1 (t), I (t) = I0 (t) + I1 (t) + I2 (t) + · · · + It (t), R(t) = R0 (t) + R1 (t) + R2 (t) + · · · + RT3 −1 (t).

(4.38)

Note that since t satisfies T3 ≤ t ≤ T2 , it implies that there are no conversions from the infectious state to the removal state at time t ≤ T2 . Also, since t ≥ T3 , it implies that all the initially removed individuals R(0) have already lost their natural immunity, and they are currently in the susceptible state at time t ≥ 0. This observation suggests that from (4.38) Rk (t) = 0,

∀k = 0, 1, . . . , T3 − 1.

(4.39)

It follows that the equation for the removal state in (4.38) reduces to R(t) = R0 (t) + R1 (t) + R2 (t) + · · · + RT3 −1 (t) = 0,

(4.40)

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and the equation for the susceptible state in (4.36) and (4.35) reduces to ˆ = S0 (t) + S1 (t) + S2 (t) + · · · + St−T 3 (t) S(t) = 0 + St−T 3 (t) = St−T 3 (t).

(4.41)

Hence, from (4.33)–(4.41), the total population of size n at time t satisfying max{T1 , T3 } ≤ t < min{T2 , T4 }, given by n = St (t) +

t−T 3 k=0

Sk (t) +

t 

Vk (t) +

k=0

T 1 −1

Ek (t) +

k=0

t 

Ik (t) +

k=0

T 3 −1

Rk (t),

(4.42)

k=0

reduces to n = St (t) + St−T3 (t) +

t  k=0

Vk (t) +

T 1 −1

Ek (t) +

k=0

t 

Ik (t) + 0.

(4.43)

k=0

In addition, the vector B(t) = (S0 (t), S1 (t), . . . , St−T3 (t), St (t), V0 (t), V1 (t), . . . , Vt (t), E0 (t), . . . , ET1 −1 (t), I0 (t), I1 (t), . . . , It (t))

(4.44)

is sufficient to describe the disease dynamics in the population at time t, which satisfies max{T1 , T3 } ≤ t < min{T2 , T4 } and T1 ≤ T3 ≤ min{T2 , T4 }. Definition 4.3.4. Decomposition of the SVEIRS population at time t ≥ max{T1 , T2 , T3 , T4 }, that is, when Hypothesis 4.3.2 H 16 holds: Applying the arguments in (4.33)–(4.37) in Definitions 4.3.1–4.3.3, the following expression is obtained for the susceptible state: ˆ S(t) =  S (t) + S(t),

(4.45)

S (t) = St (t), 

(4.46)

where

ˆ is all those who have newly converted into the susceptible state at time t via losing and S(t) natural or artificial immunity. Since from Assumption (4.13), T3 ≤ T4 , it is easy to see that ˆ = S0 (t) + S1 (t) + · · · + St−T3 (t) + St−T3 +1 (t) + St−T3 +2 (t) + · · · + St−T4 (t), S(t)

(4.47)

where S0 (t) = RT3 (t) + VT4 (t),

S1 (t) = RT3 +1 (t) + VT4 +1 (t), . . . , Sk (t) = RT3 +k (t) + VT4 +k (t), . . . ,

St−T4 (t) = RT3 +(t−T4 ) (t) + VT4 +t−T4 (t) ≡ RT3 +(t−T4 ) (t) + Vt (t) = RT3 +(t−T4 ) (t) + V0 (0),

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St−T4 +1 (t) = RT3 +(t−T4 )+1 (t), St−T4 +2 (t) = RT3 +(t−T4 )+2 (t), . . . , St−T3 (t) = St−T4 +(T4 −T3 ) (t) = RT3 +(t−T4 )+(T4 −T3 ) (t) ≡ Rt (t) = R0 (0). (4.48) For other states V , E, I , R, since t ≥ max{T1 , T2 , T3 , T4 }, it follows similarly from Definitions 4.3.1–4.3.3 that V (t) = V0 (t) + V1 (t) + V2 (t) + · · · + VT4 −1 (t), E(t) = E0 (t) + E1 (t) + E2 (t) + · · · + ET1 −1 (t), I (t) = I0 (t) + I1 (t) + I2 (t) + · · · + IT2 −1 (t), R(t) = R0 (t) + R1 (t) + R2 (t) + · · · + RT3 −1 (t).

(4.49)

Thus the population of size n at time t ≥ max{T1 , T2 , T3 , T4 } is given by n = St (t) +

t−T 3 k=0

Sk (t) +

T 4 −1 k=0

Vk (t) +

T 1 −1

Ek (t) +

k=0

T 2 −1 k=0

Ik (t) +

T 3 −1

Rk (t).

(4.50)

k=0

In addition, the vector B(t) = (S0 (t), S1 (t), . . . , St−T4 (t), . . . , St−T3 (t), St (t), V0 (t), V1 (t), . . . , VT4 −1 (t), E0 (t), . . . , ET1 −1 (t), I0 (t), I1 (t), . . . , IT2 −1 (t))

(4.51)

is sufficient to describe the disease dynamics in the population at time t ≥ max{T1 , T2 , T3 , T4 }. Remark 4.3.1. Since the SVEIR population continuously evolves over time with the spread of the disease, it is clear that the population descriptions in Definitions 4.3.1–4.3.3 are valid only for a finite amount of time. The description in Definition 4.3.4 represents the most valid representation of the population whenever the disease is fully established in the population, beyond the initial population when the outbreak occurred. Thus the vector in (4.51) will be used to define a Markov chain for the SVEIRS disease dynamics. Example 4.3.4. Corresponding to Example 4.3.2, we utilize the following values of the delays T1 = 2, T2 = 4, T3 = 3, and T4 = 4, to write an expression for the vector B(6) in (4.51) at time t = 6: B(6) = S0 (6), S1 (6), S2 (6), S3 (6), S6 (6), V0 (6), V1 (6), V2 (6), V3 (6),

E0 (6), E1 (6), I0 (6), I1 (6), I2 (6), I3 (6) . (4.52)

4.4 The SVEIRS stochastic process Let (, F, P) be a complete probability space and Ft be a filtration (that is, sub σ -algebra Ft that satisfies the following: given t1 ≤ t2 ⇒ Ft1 ⊂ Ft2 ; E ∈ Ftk , ∃k, and P (E) = 0 ⇒ E ∈ Ft0 .

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Define a random vector measurable function B : Z+ ×  −→ Z+ (T1 +T2 +T4 )+(t−T3 +1) ,

(4.53)

where, for any t ≥ max{T1 , T2 , T3 , T4 } and ω ∈ , B(t, ω) = (S0 (t, ω), S1 (t, ω), . . . , St−T3 (t, ω), St (t, ω), V0 (t, ω), V1 (t, ω), . . . , VT4 −1 (t, ω), E0 (t, ω), . . . , ET1 −1 (t, ω), I0 (t, ω), I1 (t, ω), . . . IT2 −1 (t, ω)).

(4.54)

Moreover, the σ -algebra Ft , ∀t ≥ max{T1 , T2 , T3 , T4 } is generated by the random vectors B(0), B(1), . . ., B(t). The collection of random vectors {B(t, ω) : t ∈ Z+ , ω ∈ }

(4.55)

defines a random process for the SVEIRS epidemic dynamics. We introduce the following notations. Note that, for any t ∈ Z+ , the vector B(t) ∈ Z+ (T1 +T2 +T4 )+(t−T3 +1) in (4.54) is a vector of random variables. Let t ∈ Z+ and wkt ∈ Z+ be a nonnegative integer constant, where k = 0, 1, 2, . . . , t − T3 , t; xit ∈ Z+ be a nonnegative integer constant, where i = 0, 1, 2, . . . , t − T4 ; yjt ∈ Z+ be a nonnegative integer constant, where j = 0, 1, 2, . . . , T1 − 1; zlt ∈ Z+ be a nonnegative integer constant, where l = 0, 1, 2, . . . , T2 − 1. Also, let b(t) ∈ Z+ (T1 +T2 +T4 )+(t−T3 +1) be a vector of the constant nonnegative integer values wkt , xit , yjt , and zlt , where b(t)

= (w0t , w1t , . . . , w(t−T3 )t , wtt , x0t , x1t , x2t , . . . , x(T4 −1)t , y0t , y1t , . . . , y(T1 −1)t , z0t , z1t , . . . , z(T2 −1)t ) ∈ Z+ (T1 +T2 +T4 )+(t−T3 +1) .

(4.56)

Moreover, for each t ≥ max{T1 , T2 , T3 , T4 }, the vector b(t) in (4.56) is considered an observed value of the vector of random variables B(t) in (4.54) if and only if  B(t) = b(t), and P(B(t) = b(t)) = 1, (4.57) b(t)

where the sum in (4.57) is a multiple sum of the probability mass function over all possible values b(t) of the random variable B(t) at time t. That is, Sk (t) = wkt , where k = 0, 1, 2, . . . , t − T3 , t; Vi (t) = xit , where i = 0, 1, 2, . . . , t − T4 ; Ej (t) = yjt , where j = 0, 1, 2, . . . , T1 − 1; Il (t) = zlt , where l = 0, 1, 2, . . . , T2 − 1.

(4.58)

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Example 4.4.1. Corresponding to Example 4.3.2, we utilize the following values of the delays T1 = 2, T2 = 4, T3 = 3, and T4 = 4, to write an expression for the vector b(6) corresponding to (4.56) at time t = 6:

b(6) = w0t , w1t , w2t , w3t , w6t , x0t , x1t , x2t , x3t , y0t , y1t , z0t , z1t , z3t ∈ Z+ 14 . (4.59)

4.4.1 The SVEIRS Markov chain Recall that (4.45) in Definition 4.3.4 asserts that, for t ≥ max{T1 , T2 , T3 , T4 }, the susceptible state given by S(t)

ˆ + S (t) = S(t)  = S0 (t) + S1 (t) + S2 (t) + · · · + St−T3 (t) + St (t),

(4.60)

ˆ = S0 (t) + S1 (t) + S2 (t) + · · · + St−T3 (t) and S (t) = St (t), changes at any given where S(t)  time step t by the following events: vaccination, infection, and loss of natural and artificial immunity. Therefore, the susceptible population at the next time step t + 1, that is, S(t + 1), is completely determined by the random variables S(t), E0 (t + 1), V0 (t + 1), and S0 (t + 1). Indeed, since in the epoch [t, t + 1) only S(t) is infected or vaccinated, hence the expression S(t) − V0 (t + 1) − E0 (t + 1)

(4.61)

represents all the susceptible individuals who have remained susceptible at time t + 1. The random variable S0 (t + 1) represents all the new susceptible individuals who were previously in the R or V state, and they have lost their natural and artificial immunity during the epoch [t, t + 1). Thus, from (4.60)–(4.61), it is easy to see that S(t + 1) = S(t) − V0 (t + 1) − E0 (t + 1) + S0 (t + 1).

(4.62)

But using (4.60), S(t + 1) is written as S(t + 1) = S0 (t + 1) + S1 (t + 1) + · · · + St+1−T3 (t + 1) + St+1 (t + 1).

(4.63)

Therefore, from (4.62)–(4.63), note that S1 (t + 1) + S2 (t + 1) + · · · + St+1−T3 (t + 1) + St+1 (t + 1) = S(t) − V0 (t + 1) − E0 (t + 1). (4.64) In other words, at time t + 1, S(t) − V0 (t + 1) − E0 (t + 1) represents all the susceptible individuals who have been in the susceptible state for at least one time unit. This interpretation will be useful to reduce the dimension of the vector B(t) in (4.57), whenever necessary. Also, this observation indicates that S0 (t + 1) is independent of S(t) − V0 (t + 1) − E0 (t + 1), and hence independent of Sk (t + 1), k = 1, 2, . . . , t + 1 − T3 , and k = (t + 1). We consider the following vector simplifications. From (4.58), denote

(t) = (S1 (t), S2 (t), . . . , St−T3 (t), St (t)) ∈ Z+ t−T3 +1 , W

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w t = (w1t , w2t , . . . w(t−T3 )t , wtt ) ∈ Z+ t−T3 +1 ,

= (V1 (t), V2 (t), . . . , VT4 −1 (t), ) ∈ Z+ T4 −1 , X(t) x t = (x1t , x2t , . . . x(T4 −1)t ) ∈ Z+ T4 −1 , Y (t) = (E1 (t), E2 (t), . . . , ET1 −1 (t), ) ∈ Z+ T1 −1 , y t = (y1t , y2t , . . . y(T1 −1)t ) ∈ Z+ T1 −1 ,

= (I1 (t), I2 (t), . . . , IT2 −1 (t), ) ∈ Z+ T2 −1 , Z(t) z t = (z1t , z2t , . . . z(T2 −1)t ) ∈ Z+ T2 −1 .

(4.65)

Using (4.45) and (4.65), we also denote S(t) = E(t) =

wt = yt =

t−T 3

wkt + w(t)t ,

V (t) = xt =

T 4 −1

k=0

k=0

T 1 −1

T 2 −1

k=0

Ek (t),

and I (t) = zt =

Vk (t),

Ik (t).

(4.66)

k=0

Example 4.4.2. Corresponding to Example 4.3.2, we utilize the following values of the delays T1 = 2, T2 = 4, T3 = 3, and T4 = 4 to write the following expressions in (4.67) using the vectors in (4.65) at time t = 6:

(6) = (S1 (6), S2 (6), S3 (6), S6 (6)) ∈ Z+ 4 , W w 6 = (w16 , w26 , w36 , w66 ) ∈ Z+ 4 ,

X(6) = (V1 (6), V2 (6), V3 (6)) ∈ Z+ 3 , x 6 = (x16 , x26 , x36 ) ∈ Z+ 3 , Y (6) = E1 (6) ∈ Z+ , y 6 = y16 ∈ Z+ ,

Z(6) = (I1 (6), I2 (6), I3 (6), ) ∈ Z+ 3 , z 6 = (z16 , z26 , z36 ) ∈ Z+ 3 .

(4.67)

(t + 1) contains only the Remark 4.4.1. Observe from (4.64) and (4.65) that the vector W individuals in the susceptible state who have remained susceptible at the end of the epoch [t, t + 1). Since during the epoch [t, t + 1) a susceptible person can either become infected E0 (t + 1), vaccinated V0 (t + 1) or remain susceptible, it is easy to see that the events that

(t + 1) occurs are equivalent events. We denote S(t) − V0 (t + 1) − E0 (t + 1) occurs and W these equivalent events by not (V0 (t + 1) ∨ E0 (t + 1)). That is, the events

(ω, t + 1)} ≡ {ω ∈  : S(ω, t) − V0 (ω, t + 1) − E0 (ω, t + 1)} {ω ∈  : W ≡ {ω ∈  : not (V0 (ω, t + 1) ∨ E0 (ω, t + 1))}.

(4.68)

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It follows from (4.54)–(4.56) and (4.65) that, for t ≥ max{T1 , T2 , T3 , T4 },

(t), V0 (t), X(t),

B(t) = (S0 (t), W E0 (t), Y (t), Z0 (t), Z(t)) ∈ Z+ (t−T3 +2)+T4 +T1 +T2

(4.69)

and b(t) = (w0t , w

t , x0t , x t , y0t , y t , z0t , z t ) ∈ Z+ (t−T3 +2)+T1 +T2 +T4

(4.70)

and B(t) = b(t) if and only if

(t) = w t , V0 (t) = x0t , X(t)

= x t , S0 (t) = w0t , W

= z t . E0 (t) = y0t , Y (t) = y t , I0 (t) = z0t , Z(t)

(4.71)

The following result shows that the random process {B(t), t ≥ 0} defined in (4.55) and (4.69) is a Markov chain. Theorem 4.4.1. Suppose t ≥ max{T1 , T2 , T3 , T4 }, the random process {B(t), t ≥ 0} defined in (4.55) and (4.69) and that satisfies (4.62)–(4.64) is a Markov chain, where

(t), V0 (t), X(t),

B(t) = (S0 (t), W E0 (t), Y (t), I0 (t), Z(t)) ∈ Z+ (T1 +T2 +T4 )+(t−T3 +2) .

(4.72)

That is, given b(t) ∈ Z+ T1 +T2 +T4 +2+(t−T3 ) defined in (4.56)–(4.58) and also in (4.65)–(4.71), which satisfies (4.62)–(4.64), P(B(t + 1) = b(t + 1) | Ft ) = P((B + 1) = b(t + 1) | B(t) = b(t)).

(4.73)

Moreover, P(B(t + 1) = b(t + 1) | B(t) = b(t)) =

(t + 1) = w t , V0 (t + 1) = x0(t+1) , E0 (t + 1) = y0(t+1) | B(t) = b(t)). P(W

(4.74)

In other words, the distribution of the state of the stochastic process {B(t) : t ≥ 0} at time t + 1, given any past observations of the process are completely determined by the conditional distribution of the number of people just vaccinated V0 (t + 1) and the number of people just exposed E0 (t + 1). Proof. Note that the continuity of the filtration is used in this proof. That is, the filtration Ft satisfies the following: given t1 ≤ t2 ⇒ Ft1 ⊂ Ft2 ; E ∈ Ftk , ∃k. It is easy to see from (4.56)–(4.58) and (4.70)–(4.73) that

(t + 1) = w P (B(t + 1) = b(t + 1) | Ft ) = P S0 (t + 1) = w0 (t + 1), W

(t+1) ,

+ 1) = x (t+1) , V0 (t + 1) = x0(t+1) , X(t E0 (t + 1) = y0(t+1) , Y (t + 1) = y (t+1) ,

+ 1) = z (t+1) | I0 (t + 1) = z0(t+1) , Z(t

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(t) = w

= xt ,

t , V0 (t) = x0t , X(t) S0 (t) = w0t , W

= z t , Ft−1 . E0 (t) = y0t , Y (t) = yt , I0 (t) = z0t , Z(t) (4.75)

The following simplifications of (4.75) are considered. It is shown below that all coordinates in the vector B(t + 1) are already given in the vector B(t) by translation, except for the random variables E0 (t + 1) and V0 (t + 1).

+ 1) = (V1 (t), V2 (t), . . . , VT4 −1 (t)) = (V0 (t), V1 (t), . . . , Indeed, observe from (4.65) that X(t VT4 −2 (t)) is already given in B(t). Similarly, Y (t + 1) = (E1 (t + 1), E2 (t + 1), . . . , ET1 −1 (t + 1)) =

+ 1) are al(E0 (t), E1 (t), . . . , ET1 −2 (t)) is already given in the vector B(t); I0 (t + 1) and Z(t ready given in the vector B(t); S0 (t + 1) = VT4 (t + 1) + RT3 (t + 1) = VT4 −1 (t) + RT3 −1 (t) is already given in B(t). Also, at the end of the epoch [t, t + 1), the difference in the suscep (t + 1) which is a tible state after infection and vaccination has occurred is the vector W

(t) that is already given in B(t). Thus, since many terms in translation from the vector W the argument of the probability function/measure in (4.75) are already given in the conditions for the probability measure, Eq. (4.75) reduces to P (B(t + 1) = b(t + 1) | Ft )

(t + 1) = w =P W

(t+1) , V0 (t + 1) = x0(t+1) , E0 (t + 1) = y0(t+1) | B(t) = b(t) ,

(4.76)

and the result in Theorem 4.4.1 follows immediately. Remark 4.4.2. It is easy to see from Remark 4.4.1, Theorem 4.4.1, and (4.65) that an equivalent expression for (4.74) is P (B(t + 1) = b(t + 1) | B(t) = b(t)) = P (not (V0 (t + 1) ∨ E0 (t + 1)) ≡ S(t) − V0 (t + 1) − E0 (t + 1) =

t−T 3

wkt + w(t)t − x0(t+1) − y0(t+1) ,

k=0



V0 (t + 1) = x0(t+1) , E0 (t + 1) = y0(t+1) | B(t) = b(t) ,

(4.77)

where S(t) is defined in (4.66).

4.5 Some special SVEIRS epidemic models As remarked in Theorem 4.4.1, the transition probabilities of the Markov chain {B(t) : t ≥ 0} are completely defined by the joint conditional distributions of those who are just infected E0 (t + 1) and those who are just vaccinated V0 (t + 1). There are several different ways to specify the discrete probability distribution (4.77) to completely define the Markov chain {B(t), t ≥ 0}. Two discrete probability models are considered in this section.

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4.5.1 The SVEIRS model with correlated vaccination and infection rates The following assumptions are made to describe an SVEIRS epidemic model, where the drive to get vaccinated correlates with the infection rate in the population. To make this section very applicable, the example of an SVEIRS epidemic, typhoid fever [23], is used wherever necessary. Assumption 4.5.1. – It is assumed that the infectious agent (e.g., typhoid bacteria) is highly contagious and severe, the population is well sensitized about the disease, and there are vaccines readily available for those who want to be vaccinated. Furthermore, because of the high prevalence rate of the disease, every susceptible individual in the population at the beginning of the epoch [t, t + 1) is either infected, vaccinated or remains susceptible. Moreover, the probability that the ith susceptible individual in the epoch [t, t + 1) escapes infection and avoids vaccination during that time t is denoted by PSi (t), where i = 1, 2, 3, . . ., S(t) = wt ; and wt is defined in (4.66). Also, the probabilities that the ith susceptible gets infected and vaccinated, respectively, during the epoch [t, t + 1) are denoted by PEi (t) and PVi (t), ∀i = 1, 2, . . . , wt . In addition, it assumed that PEi (t) + PVi (t) + PSi (t) = 1, ∀t ≥ 0, ∀i = 1, 2, . . . , wt .

(4.78)

– Using the ideas in [1,31], we further assume p to be the probability of becoming infected from one interaction with an infectious individual of type I (t), where all interactions are independent. Hence, at time t, the probability (cf. [15]) that the ith susceptible person becomes exposed after interacting with j infectious individuals, denoted by Pji (t) ≡ Pji , and is given by Pji (t) ≡ Pji = 1 − (1 − p)j , i ∈ ∀{1, 2, 3, . . . , wt }, ∀j ∈ {1, 2, 3, . . . , zt },

(4.79)

where I (t) = zt is defined in (4.66), and Pji (t) ≡ Pji signifies that the probability in (4.79 ) is a constant in any interval [t, t + 1). – Let Ni (t) be the number of people (of any state S, V , E, I , R) the ith susceptible individual contacts at time t. It is easy to see that the collection {Ni (t), t = 0, 1, 2, . . . } is a stochastic process (cf. [40]). However, for simplicity, we assume that Ni (t) is a constant N > 0 at any time t, that is, Ni (t) = N , ∀t ≥ 0. In other words, the ith susceptible individual meets a coonstant number N of individuals every time. Moreover, N is sufficiently large such that the conditional binomial approximation for the number of infectious people a susceptible individual meets, given the N total constants is possible. In the next Subsection 4.5.2, we characterize the random process {Ni (t), t = 0, 1, 2, . . . } as a Poisson process. – It is assumed that there is homogeneous mixing in the population, and as a result the probability that the ith susceptible person at time t meets an infectious person I (t) during the epoch [t, t + 1) is given by α i (t) =

I (t) , n−1

(4.80)

Chapter 4 • SVEIRS Markov chain epidemic model with multiple discrete delays

117

where n = S(t) + V (t) + E(t) + R(t) is the total population size in (4.50); i = 1, 2, 3, . . . , wt , and S(t) = wt . We define the following random variables. Given that the ith susceptible person contacts Ni (t) = N people at time t, let Yi (t) be the number of infectious individuals the ith susceptible person meets during that time, where Yi (t) = 0, 1, 2, . . . , N . Clearly, assuming independent contacts in the population, it is easy to see from (4.80) that the conditional distribution Y i |Ni (t) ∼ binomial(Ni (t), α i (t)), i.e.,   Ni (t) i (α i (t))j (1 − α i (t))Ni (t)−j , (4.81) P (Y (t) = j |Ni (t)) = j where j = 1, 2, 3, . . . , Ni (t). – Given that the ith susceptible person meets Y i (t) = j infectious people at time t, we let Z i (t) be a categorical random variable indicating the event that the ith susceptible person gets infected in the epoch [t, t + 1). It is easy to see from (4.79) that P (Z i (t) = 1|Y i (t) = j, Ni (t) = N ) = 1 − (1 − p)j ≡ pji .

(4.82)

It is also easy to see that PEi (t), the probability that the ith susceptible person gets infected in the epoch [t, t + 1), is given by PEi (t) = P (Z i (t) = 1|Ni (t) = N ) =

N 

P (Z i (t) = 1, Y i (t) = j |Ni (t) = N ),

(4.83)

j =1

which reduces to PEi (t) = 1 − (1 − pα i (t))N pI (t) N =1− 1− . N −1

(4.84)

– In this scenario, the event of vaccination occurs as follows. It is assumed that susceptible individuals gain the motivation to seek vaccination after the realization of eluding infection from interacting with infected people. That is, it is assumed that interactions with infectious people which do not lead to infection give rise to the urgent desire for the susceptible person to get vaccinated. Indeed, during an initial outbreak of an epidemic, it is very common for people to minimize the need for vaccination because of unguarded trust for family, friends, etc. However, the realization of the disease in those one is close to would motivate the need for protection against the disease. Furthermore, it is assumed that the disease has the incubation period T1 , and the vaccine is sufficiently strong to reverse only new infections that occur before the first time unit [t, t + 1) of the incubation class. That is, only vaccination of a susceptible individual who is recently infected can be reversed, where it is assumed that there is a minimal time lapse after infection until the recently infected person can be classified as E0 . In addition, vaccination in any other classes will not result in any change of status of the individuals in the classes.

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Given that the ith susceptible person meets Ni (t) = N people during the epoch [t, t + 1) and Yi (t) = j who are infectious, we let φ ∈ (0, 1) be the probability that the ith susceptible person gets vaccinated, given that the individual eludes infection during the epoch [t, t + 1). If we further let X i (t) indicate the event that the ith susceptible individual gets vaccinated in the epoch [t, t + 1), then based on assumptions (1) − (6) above, PVi (t), the probability that the ith susceptible person gets vaccinated, given the individual eludes infection from the Ni (t) contacts, is defined as follows: pI (t) N PVi (t) = φ(1 − pα i (t))N = φ 1 − . N −1

(4.85)

Applying Assumption 4.5.1, an explicit expression for (4.77) is obtained in the following. Theorem 4.5.1. Let the conditions of Theorem 4.4.1 be satisfied, and let the probability that the ith susceptible individual gets vaccinated at time t, denoted by PVi (t), be as defined in (4.85). Also, let the probability that the ith susceptible individual gets infected at time t, denoted by PEi (t), be as defined in (4.84). It follows that PVi (t) + PEi (t) + PSi (t) = 1 and the transition probabilities of the Markov chain {B(t), t ≥ 0}, whenever t ≥ max{T1 , T2 , T3 , T4 } is a trinomial distribution, are given by P (B(t + 1) = b(t + 1) | B(t) = b(t))  = P not (V0 (t + 1) ∨ E0 (t + 1)) ≡ S(t) − V0 (t + 1) − E0 (t + 1) =

t−T 3

 wkt + w(t)t − x0(t+1) − y0(t+1) , V0 (t + 1) = x0(t+1) , E0 (t + 1) = y0(t+1) | B(t) = b(t)

k=0

 =

wt





y0 wt − x0(t+1) (t+1) x (PVi (t)) 0(t+1) × PEi (t) y0(t+1)

x0(t+1)

wt −x0 −y0(t+1) (t+1) × PSi (t) ,

x0(t+1) = 0, 1, 2, . . . , wt ; y0(t+1) = 0, 1, 2, . . . , wt − x0(t+1) , where not (V0 (t + 1) ∨ E0 (t + 1)) = E0 (t + 1) = y0(t+1) .

t−T3 k=0

(4.86)

wkt + w(t)t − x0(t+1) − y0(t+1) , V0 (t + 1) = x0(t+1) , and

Moreover, recall that not (V0 (t + 1) ∨ E0 (t + 1)) represents the number of people susceptible at time t, who remain susceptible at time t + 1. Thus the conditional marginal distributions of V0 (t + 1), E0 (t + 1) and not (V0 (t + 1) ∨ E0 (t + 1)) are given as follows:  P (V0 (t + 1) = x0(t+1) |B(t) = b(t)) =

wt x0(t+1)



x0(t+1)

(PVi (t))

wt −x0(t+1)

(PEi (t) + PSi (t))

,

(4.87)

Chapter 4 • SVEIRS Markov chain epidemic model with multiple discrete delays

119

where x0(t+1) = 0, 1, 2, . . . , wt .   P E0 (t + 1) = y0(t+1) |B(t) = b(t) =



wt y0(t+1)



y0(t+1)

(PEi (t))

wt −y0(t+1)

(PVi (t) + PSi (t))

,

(4.88)

where y0(t+1) = 0, 1, 2, . . . , wt . P (not (V0 (t + 1) ∨ E0 (t + 1))) = wt − x0(t+1) − y0(t+1) |B(t) = b(t))   wt w −x −y x +y (PSi (t)) t 0(t+1) 0(t+1) (PVi (t) + PEi (t)) 0(t+1) 0(t+1) . = wt − x0(t+1) − y0(t+1)

(4.89)

Proof. Using (4.61), it is easy to see that since each susceptible i = 1, 2, 3, . . . , wt present at time t + 1 is either vaccinated, infected or remains susceptible, it follows that P (V0 (t + 1) = x0(t+1) , E0 (t + 1) = y0(t+1) |B(t) = b(t)) = P (V0 (t + 1) = x0(t+1) , E0 (t + 1) = y0(t+1) , not (V0 (t + 1) ∨ E0 (t + 1)) = wt − x0(t+1) − y0(t+1) |B(t) = b(t)).

(4.90)

We can now easily express Eq. (4.90) as follows: P (V0 (t + 1) = x0(t+1) , E0 (t + 1) = y0(t+1) , not (V0 (t + 1) ∨ E0 (t + 1)) = wt − x0(t+1) − y0(t+1) |B(t) = b(t))     wt − x0(t+1) wt − x0(t+1) − y0(t+1) wt x y × (PVi (t)) 0(t+1) × (PEi (t)) 0(t+1) = x0(t+1) y0(t+1) wt − x0(t+1) − y0(t+1) wt −x0(t+1) −y0(t+1)

× (PSi (t))

.

(4.91)

Observe that (4.91) reduces to (4.86).

4.5.2 The SVEIRS model with no correlation between vaccination and infection rates Recall that Subsection 4.5.1 presents the transition probabilities for the Markov chain B(t), t ≥ 0, for the special scenario of the SVEIRS epidemic, where vaccination occurs whenever an individual eludes infection. In this subsection we present a special SVEIRS epidemic model without this dependence of vaccination on infection. Note that typhoid fever [23] is used as an example of the SVEIRS epidemic, wherever necessary. We let PEi (t) be the probability that in the next time step t the ith susceptible individual becomes infected, and let PVi (t) be the probability that in the next time step t the ith susceptible individual becomes vaccinated. Also, let PSi (t) be the probability that in the next time step t the ith susceptible person remains susceptible, where PVi (t) + PEi (t) + PSi (t) = 1. We use a different multinomial distribution to characterize the transition probabilities for the SVEIRS stochastic process {B(t), t ≥ 0} defined in Theorem 4.4.1.

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The multinomial distribution, where vaccination and infection are uncorrelated, is based on the following assumptions. Note that the assumptions are modifications of Assumptions 4.5.1 (1)–(6) made for the special SVEIRS model in Subsection 4.5.1. Continuing with the assumptions in Subsection 4.5.2, we make the following additional assumptions. Assumption 4.5.2. – Similar to Assumptions 4.5.1, let Zti be an indicator random variable for the event that the ith susceptible person becomes infected at time t. Also, let Yti be the discrete random variable denoting the number of infectious people the ith susceptible individual interacts with at time t. – Recall Assumptions 4.5.1 (1). It is assumed that the population is also well educated and sensitized about the disease, and there are available vaccines for those seeking to get vaccinated, and some people in the population practice natural preventive techniques against the disease. Hence, because of available vaccination and natural control measures, every susceptible individual in the population in the next time step becomes either infected, vaccinated or remains susceptible to the disease. – Using ideas in [29], and similar to Assumptions 4.5.1, Ni (t) = N is the random number of people the ith susceptible individual contacts at time t, where N = 0, 1, 2, 3, . . . . Moreover, the random process {Ni (t), t ≥ 0} is a homogeneous Poisson process with arrival rate λ. – Similar to (4.79), at time t, given that an individual can contact Ni (t) = N people independently, the probability that the ith susceptible individual becomes infected after contacting j infectious individuals is denoted by Pji (t) ≡ pji and is given by P (Zti = 1|Yti = j, Ni (t) = N ) ≡ pji (t) = 1 − (1 − p)j ,

(4.92)

where i = 1, 2, . . . , xt , j = 0, 1, 2, . . . N, and S(t) = xt in (4.67). – It is assumed that there is homogeneous mixing in the population at any time, and as a result the probability that the ith susceptible individual meets an infectious individual at (t) is given by (4.80), αti = NI (t) −1 . Thus, assuming independent interactions, the probability that the ith susceptible individual meets exactly j infectious individuals given Ni (t) = N contacts is given by   N P (Yti = j |Ni (t) = N ) = (α i (t))j (1 − α i (t))N −j j   I (t) N −j N I (t) j 1− = . (4.93) N −1 N −1 j Using assumptions in (7)–(11) above, we find PEi , the probability that the ith susceptible person becomes exposed, in the following: PEi (t) = P (Zti = 1) =

N ∞   N =0 j =0

P (Zti = 1|Yti = j, Ni (t + 1) − Ni (t) = N )

Chapter 4 • SVEIRS Markov chain epidemic model with multiple discrete delays

× P (Yti = j |Ni (t + 1) − Ni (t) = N ) × P (Ni (t + 1) − Ni (t) = N ).

121

(4.94)

Observe that for {Ni (t), t ≥ 0} a Poisson process with rate λ, P (Ni (t + 1) − Ni (t) = N ) =

λN exp−λ . N!

(4.95)

Thus, substituting Eqs. (4.92), (4.93), and (4.95) into Eq. (4.94), we obtain PEi (t) = 1 − e−pαt λ . i

(4.96)

– The following assumptions are made to derive PVi (t), which is the probability that in the next time step t the ith susceptible individual gets vaccinated. It is assumed that people get vaccinated independently at a constant rate μv per unit time. Hence the number of people who get vaccinated over a unit time interval follows the Poisson distribution. Thus the time Tv until the ith susceptible individual gets vaccinated in any interval [t, t + 1) follows the exponential distribution with mean μ1v . Therefore it is easy to see that the probability PVi (t) = 1 − exp (−μv ).

(4.97)

Similar to Theorem 4.5.1, we now use the trinomial distribution with probabilities (4.96) and (4.97) to characterize the transition probabilities of the SVEIRS stochastic process {B(t), t = 1, 2, 3, . . . } whenever t ≥ max{T1 , T2 , T3 , T4 }. The following theorem applies exactly the same technique as Theorem 4.5.1. Thus the result is given without proof. Theorem 4.5.2. Let the conditions of Theorem 4.4.1 be satisfied, and for each i = 1, 2, 3, . . . , xt , let the probability that the ith susceptible individual gets vaccinated, denoted by PVi (t), be defined in (4.97), and let the probability that the ith susceptible individual gets infected at time t, denoted by PEi (t), be as defined in (4.96), where PVi (t) + PEi (t) + PSi (t) = 1. It follows that, for any t ≥ max(T1 , T2 , T3 , T4 ), the transition probabilities for the stochastic process {B(t), t = 1, 2, 3, . . . } are given as follows: P (B(t + 1) = b(t + 1) | B(t) = b(t)) = P (not (V0 (t + 1) ∨ E0 (t + 1)) ≡ S(t) − V0 (t + 1) − E0 (t + 1) =

t−T 3 k=0

wkt + w(t)t − x0(t+1) − y0(t+1) ,  | B(t) = b(t)

V0 (t + 1) = x0(t+1) , E0 (t + 1) = y0(t+1)   

y0 wt − x0(t+1) wt (t+1) x (PVi (t)) 0(t+1) × PEi (t) = x0(t+1) y0(t+1)

wt −x0 −y0(t+1) (t+1) × PSi (t) ,

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x0(t+1) = 0, 1, 2, . . . , wt ; y0(t+1) = 0, 1, 2, . . . , wt − x0(t+1) ,

(4.98)

 3 where not (V0 (t + 1) ∨ E0 (t + 1)) = t−T k=0 wkt + w(t)t − x0(t+1) − y0(t+1) , V0 (t + 1) = x0(t+1) , and E0 (t + 1) = y0(t+1) . In addition, recall that not (V0 (t + 1) ∨ E0 (t + 1) is the random number of the susceptible people at time t who remain susceptible at time t + 1. Therefore, the conditional marginal distributions of V0 (t + 1), E0 (t + 1), and not (V0 (t + 1) ∨ E0 (t + 1) are given as follows:   wt x w −x P (V0 (t + 1) = x0(t+1) |B(t) = b(t)) = (PVi (t)) 0(t+1) (PEi (t) + PSi (t)) t 0(t+1) , (4.99) x0(t+1) where x0(t+1) = 0, 1, 2, . . . , wt .  P (E0 (t + 1) = y0(t+1) |B(t) = b(t)) =

wt

y0(t+1)

 y w −y (PEi (t)) 0(t+1) (PVi (t) + PSi (t)) t 0(t+1) ,

(4.100)

where y0(t+1) = 0, 1, 2, . . . , wt . P (not (V0 (t + 1) ∨ E0 (t + 1)) = wt − x0(t+1) − y0(t+1) |B(t) = b(t))   wt x w −x −y = (PVi (t)) 0(t+1) × (PSi (t)) t 0(t+1) 0(t+1) wt − x0(t+1) − y0(t+1) x0(t+1) +y0(t+1)

× (PVi (t) + PEi (t))

.

(4.101)

Proof. See Theorem 4.5.1.

4.6 Numerical study: some prototype SVEIRS epidemic models and sensitivity analysis to determine the effects of infection and vaccination In this section, we present and study some prototype SVEIRS models, and we also utilize inferential statistical techniques to determine the epidemiological outcomes of various disease scenarios under the influence of different levels of infection and vaccination rates. Note that the numerical simulations in this section are completely theoretical, and the primary objective of this section is to numerically explore the impacts of infection and vaccination in this study. In the following, we characterize the SVEIR population utilized to study all scenarios of the SVEIRS epidemic. In each scenario, the total population n, will be set to 3300 people. The starting number of infectious individuals at t = 0, I0 (0) is 100. The incubation period of the disease T1 is two time units (days, weeks, months, etc.). The infectious period, that is, the time until full recovery from the disease after the initial infection T2 , is three time units. The temporary immunity period in which the natural immunity wears off and an individual relapses to the susceptible class T3 is four time units. The artificial immunity

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or vaccine effective period for which the vaccine wears off and the vaccinated individual returns to the susceptible class T4 is 4. The starting amount of susceptible individuals ˆ S(0) = S(0) +S(0) is 3000. The starting number of vaccinated individuals at t = 0, V0 (0), is  150. The starting number of exposed individuals at t = 0, E0 (0), is 50. The starting number of individuals just becoming susceptible at t = 0, S0 (0), is 0. The starting number of recovered individuals at t = 0, R0 (0), is 0. Also, the number of people the ith susceptible individual meets or interacts with per unit time in the population Ni ≡ N is set to be 10 (see Assumption 4.5.1, #5). We will take a look at the behavior of the states of the process {B(t), t ≥ 0} over 30 discrete time steps or longer. Recall Remark 4.3.1 and Section 4.4, where the SVEIRS stochastic process B(t) was defined based on [H16]: max{T1 , T2 , T3 , T4 } ≤ t from Hypothesis 4.3.1.

4.6.1 The general algorithm for the simulations In each scenario, the initial conditions at t = 0 are listed as follows. S(0) = w0 = 3000. This represents the initial number of people who are at risk of the disease. V (0) = V0 (0) = x00 = 150. This represents the initial number of vaccinated individuals at the beginning of the disease outbreak. All other states of the vaccinated class, e.g., V1 (0), V2 (0), V3 (0), are zero. E(0) = E0 (0) = y00 = 50. This represents the initial number of exposed individuals who have the disease incubating in them but are not infectious at the beginning of the disease outbreak. I (0) = I0 (0) = z00 = 100. This represents the initial number of infectious individuals who can pass the infection to other susceptible individuals at the beginning of the disease outbreak. Other states of the class, such as I1 (0), I2 (0), are all zeros. R(0) = R0 (0) = r00 = 0. This represents the initial number of recovered individuals at the beginning of the disease outbreak. S0 (0) = w00 = 0. This represents the initial number of individuals who are just becoming susceptible at the beginning of the disease outbreak. PVi (t) is the probability of becoming vaccinated at any instant t (i.e., [t, t + 1)). This is calculated as pI (t) Ni (t) , (4.102) PVi (t) = PVi = φ 1 − n−1 as earlier remarked in (4.85). φ is the probability that the susceptible individual gets vaccinated after escaping infection. A value of φ will be selected for each scenario presented. PEi (t) is the probability of becoming infected at any time t (i.e., in [t; t + 1)). pI (t) Ni (t) PE ≡ PEi (t) = 1 − 1 − , n−1

(4.103)

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as earlier remarked in (4.84). p is the probability that the susceptible individual becomes infected after an interaction with an infectious individual at any time t = 0, 1, 2, 3, . . . . We will be using different values for p as we look at different scenarios for our model. We implement the following steps to generate sample paths and histograms for the different states of the SVEIRS model in the next time step. – In Section 4.6.2 we specify p and φ for all the scenarios based on the equality of p and φ at the beginning of the algorithm. Also, Eqs. (4.102) and (4.103) are calculated. For each individual in the susceptible class (note, at t = 0 the susceptible state is S(0) = ˆ = 3000), the binomial marginal conditional distributions in (4.87) and (4.88) S(0) +S(0)  are used to generate observations for V0 and E0 , while (4.60) and (4.62) are used to find S0 and S(t) in the next time step. – Note that we specify p and φ for the different scenarios at the initial step and utilize these values for all subsequent time steps. Also note that without violation of Assumption 4.5.1, PE and PV are continuously calculated at each time step based on the number of infectious individuals in the current population. Each individual in the vaccinated class, the exposed class, or the infectious class is shifted 1 time step ahead (i.e., if an individual was in V0 (0) = x00 , that person is moved to V1 (1) = x11 ). At the time step in which an individual in the exposed, infectious, natural immunity, and artificial immunity states exceeds T1 , T2 , T3 , or T4 , respectively, that individual is moved to a new state exhibited in Fig. 4.1. For example, for the exposed class, an individual who has been in that state for more than T1 time units at time t is placed into the infectious class I (t + 1) = x0t+1 . Also, for the infectious class, an individual who has been in the state for more than T2 time units at time t is placed into the recovered class R(t + 1) = r0t+1 . For the recovered class, an individual who has been in the state for more than T3 time units at time t is placed into the susceptible class S(t + 1) = w0t+1 . Also, for the vaccinated class, an individual who has been in that state for more than T4 units at time t is placed into the susceptible class S(t + 1) = w0t+1 . – Repeat Steps 1 and 2 until all observations for the states of the process {B(t), t ≥ 0} have been produced over 30 time steps. In Section 4.6.3, based on Assumption 4.5.2, we use Eqs. (4.96) and (4.97) instead of Eqs. (4.102) and (4.103) and do the same steps stated above to generate histograms and estimates. The rest of the explanation is stated in Section 4.6.3.

4.6.2 The prototype SVEIRS model with correlated vaccination and infection rates 4.6.2.1 Sensitivity of the SVEIRS model when p = φ In the following, we present sample paths for the SVEIRS model defined in Theorem 4.5.1, where the probability of becoming infected after one interaction with an infectious indi-

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vidual p and the probability that the susceptible individual gets vaccinated after escaping infection φ are assumed to be the same value of 0.15 (i.e., p = φ = 0.15). We utilize the initial conditions and algorithm from the steps in Subsection 4.6.1 to generate sample paths and distribution for the different susceptible states S, S0 , exposed states E0 , E1 , infectious states I0 , I1 , I2 , recovered states R0 , R1 , R2 , R3 , and vaccination rates V0 , V1 , V2 , V3 . The sample paths lead to samples of reasonable sizes generated from the SVEIRS population at anytime t ≥ 1, which are used to compute histograms, and confidence intervals to predict the state of the population at time t. The histograms showing the estimated distribution of the susceptible, vaccinated, exposed, infectious, and removed populations at time t = 25 are given in Figs. 4.2, 4.3, and 4.4. Note that 1000 sample paths for each state of the process {B(t), t ≥ 0} were used to construct the histogram.

FIGURE 4.2 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the susceptible, exposed, and infectious populations of the process at time t = 25 whenever the probabilities of getting infected and vaccinated are the same, i.e., p = φ = 0.15.

One can infer from the graphical representation in Fig. 4.2 that the shape of the distribution for the susceptible population is almost symmetric with the measures of center (mode, mean, and median) between 825 and 900. Furthermore, the variability is low, and the support of the random variable S(25) stretches across from 800 and 950. The graphical representation for S(25) suggests that at the 25th time step there is greater chance for values of S(25) in the range (825, 870) to occur than anywhere else. This suggests that at the 25th time step the number of people who are no longer in the initial susceptible

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state lies in the range (2130, 2175). Note that this description does not include the S0 class. The 95% confidence interval asserts that the true mean value of the susceptible state at the 25th time step (i.e., the expected value E[S(25)]) lies somewhere between 852 and 857. The confidence interval is narrow and affirms the accuracy of the estimate. Observing the exposed population in Fig. 4.2, we can infer from the graphical representation that the shape of the distribution for the number of people just exposed E0 is more symmetric than skewed with the mode, mean, and median between 210 and 250. Furthermore, the variability is low, and the support of the random variable E0 (25) stretches across from 180 and above 280. Observe that a similar description for S(25) above can be obtained for E1 (25). This is because of translations over the incubation period T1 = 2. Furthermore, it is easy to see that at the 25th time step approximately between (210, 250) number of people are just infected and exposed E0, while between (210, 250) number of people have been exposed for one time unit (E1 ). The graphical representation suggests that at the 25th time step there is a greater chance for values of E0 (25) in the range (220, 260) to occur than anywhere else. This suggests that at the 25th time step the number of initial exposed individuals is high. In addition, the graph suggests that at the 25th time step there is also a bigger chance for values of E1 (25) in the range (220, 260) to occur than anywhere else. This observation also suggests that at the 25th time step the number of exposed individuals on the second day is approximately high. Furthermore, from the 95% confidence interval we can say that the true mean value E[E0 (25)]) of the just exposed person E0 at the 25th time step lies approximately somewhere between 235 and 238, whereas the true mean value E[E1 (25)]) of the exposed person for one time unit (E1 ) at the 25th time step lies approximately somewhere between 234 and 237. In both cases, the confidence interval is narrow and affirms the accuracy of the estimate. Also observing the infectious population in Fig. 4.2, we can make conclusions from the graphical representation that the shape of the distribution for the just become infectious population I0 is more symmetric but slightly right skewed with the mode between 220 and 230, the mean and median are approximately higher than 230. For the distribution of I1 (25), it is slightly skewed to the right with the mode approximately between 220 and 240, the mean and median between 230 and 250. Furthermore, observing I2 (25), the distribution is slightly symmetric with the mode and mean between 220 and 240, and the median is approximately 230. The graphical representation suggests that at the 25th time step there is a greater chance for values of I0 (25) in the range (220, 260) to occur than anywhere else. This suggests that at the 25th time step the number of just become infectious individuals is high. The graph suggests that at the 25th time step there is a bigger chance for values of I1 (25) and I2 (25) in the range (220, 260) and (220, 250), respectively, to occur than anywhere else. That is, at the 25th time step, the number of infectious individuals on the second day and third day is approximately high. Furthermore, from the 95% confidence intervals, we can say that the true mean value of the just infected person (I0 ), E[I0 (25)], the infectious person for one time unit (I1 ), E[I1 (25)], and the infectious person for two time units (I2 ),

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E[I2 (25)] at the 25th time step lies approximately somewhere between (231, 233), (231, 233), and (226, 228), respectively. In all cases the confidence interval is narrow and affirms the accuracy of the estimate.

FIGURE 4.3 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the susceptible, vaccinated, and just becoming susceptible populations of the process at time t = 25 whenever the probabilities of getting infected and vaccinated are the same, i.e., p = φ = 0.15.

The histograms in Fig. 4.3 suggest that at the 25th time step the distribution of the number of people just vaccinated V0 is closely symmetric with low variability, and the measures of center (mean, median, and mode) lie in the range (80, 100). This range is approximately the same for all other subclasses V1 , V2 , V3 by translation over the artificial immunity period T 4 = 4. Moreover, from the 95% confidence interval we can say that the true mean value of the subclasses V0 , V1 , V2 , and V3 at the 25th time step i.e., E[V0 (25)], E[V1 (25)], E[V2 (25)],

and E[V3 (25)] lies approximately somewhere between (94, 96), (95, 98), (95, 98), and (95, 97), respectively. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. The distribution of the number of people just susceptible S0 (25) in Fig. 4.3 is also more symmetric and the center (mean, median, and mode) lies in (310, 340). The 95% confidence interval asserts that the true mean value of just susceptible person (S0 ) at the 25th time step E[S0 (25)] lies approximately somewhere between 326 and 329. The confidence interval is narrow and affirms the accuracy of the estimate.

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FIGURE 4.4 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the susceptible, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probabilities of getting infected and vaccinated are the same, i.e., p = φ = 0.15.

Furthermore, it follows from Fig. 4.4 that the distribution of the removed classes R0 , R1 , R2 is more symmetric, and their centers lie approximately in (200, 240). From the 95% confidence interval we can say that the true mean value of the subclasses R0 , R1 , R2 , and R3 at the 25th time step (i.e., E[R0 (25)], E[R1 (25)], E[R2 (25)], and E[R3 (25)]) lies approximately somewhere between (223, 226), (220, 223), (219, 222), and (230, 233), respectively. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. The decision about the aggressiveness of the disease, based on the sample statistics obtained at the 25th time step in Figs. 4.2, 4.3, and 4.4, follows easily by comparing the average number in the susceptible state at time t = 25 to the fixed total population n = 3300. Indeed, if on average approximately (825, 870) people who have never been vaccinated or infected are in the susceptible state and approximately (310, 340) of those who have been previously either vaccinated or infected are in the susceptible state, then on average a total of (1135, 1210) people are in the susceptible state at the 25th time step. Thus approximately on average between (2090, 2165) people are either exposed, infectious, or removed. That is, approximately (2090, 2165) people are ever infected at the 25th time step. Hence the disease is very aggressive in the population.

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4.6.2.2 Sensitivity analysis of the SVEIRS model when either p or φ is fixed and the other parameter continuously changes We utilize Figs. 4.5–4.10 to compare the effects of continuously changing the values of φ ∈ {0.01, 0.1, 0.5} for a fixed value of p = 0.15 on the dynamics of the disease, and the effects of continuously changing the values of p ∈ {0.01, 0.1, 0.5} for a fixed value of φ = 0.15. Note that Figs. 4.5, 4.6, 4.7, 4.8, 4.9, 4.10 depict the estimated distributions of the random variables S(25), E0 (25), V0 (25), S0 (25), I0 (25), and R0 (25).

FIGURE 4.5 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probabilities of getting infected and vaccinated are not the same, p = 0.15 and φ = 0.01.

To minimize space, we shall focus on the centers of distributions of the random variables S(25), E0 (25), V0 (25), S0 (25), I0 (25), and R0 (25) characterizing the epidemic at the 25th time step. We utilize the measures of center (mean, median, and mode) to compare the changing effects of p, φ ∈ {0.01, 0.1, 0.5}. For a fixed value of p = 0.15 and φ continuously changing from φ = 0.01 to φ = 0.5, it is easy to see that the number of susceptible people most likely present at time t = 25 on the average (mean, mode, median) lies approximately between (680, 740) whenever φ = 0.01 and changes to (820, 880) whenever φ = 0.1, and further changes to (740, 780) whenever φ = 0.5. From the 95% confidence interval we can say that the true mean value of the susceptible people at time t = 25, E[S(25)], lies approximately somewhere between (713, 717) whenever φ = 0.01, changes to (848, 852) whenever φ = 0.1, and further changes

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to (760, 765) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. The above observation suggests that as the vaccination rate continuously increases, more susceptible people get vaccinated, and as a result they are saved from infection, which explains why the interval for S(25) continuously decreases relative to the initial susceptible population S(0) = 3000.

FIGURE 4.6 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probabilities of getting infected and vaccinated are slightly different, that is, p = 0.15 and φ = 0.1.

Similarly, at time t = 25, the average for E0 (25) lies approximately between (270, 300) whenever φ = 0.01 and changes to (240, 270) whenever φ = 0.1, and further changes to (100, 130) whenever φ = 0.5. The above observation suggests that as the vaccination rate increases, the number of people just infected or exposed to the disease decreases. From the 95% confidence interval we can say that the true mean value of E0 at time t = 25, E[E0 (25)], lies approximately somewhere between (283, 286) whenever φ = 0.01, changes to (254, 257) whenever φ = 0.1, and further changes to (118, 120) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. Similarly, at time t = 25, the average for V0 (25) lies approximately between (3, 6) whenever φ = 0.01 and changes to (55, 75) whenever φ = 0.1, and further changes to (360, 390) whenever φ = 0.5. The above observation suggests that as the vaccination rate increases, the number of people just vaccinated increases. From the 95% confidence interval we

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can say that the true mean value of V0 at time t = 25, E[V0 (25)], lies approximately somewhere between (5, 6) whenever φ = 0.01, changes to (62, 64) whenever φ = 0.1, and further changes to (375, 378) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate.

FIGURE 4.7 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probabilities of getting infected and vaccinated are slightly different, that is, p = 0.15 and φ = 0.5.

Similarly, at time t = 25, the average for S0 (25) lies approximately between (210, 240) whenever φ = 0.01, changes to (280, 320) whenever φ = 0.1, and further changes to (450, 480) whenever φ = 0.5. The above observation suggests that as the vaccination rate increases, the number of people just becoming susceptible increases. From the 95% confidence interval we can say that the true mean value of S0 at time t = 25, E[S0 (25)], lies approximately somewhere between (223, 226) whenever φ = 0.01, changes to (298, 301) whenever φ = 0.1, and further changes to (469, 473) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. Similarly, at time t = 25, the average for I0 (25) lies approximately between (310, 340) whenever φ = 0.01 and changes to (240, 270) whenever φ = 0.1, and further changes to (110, 140) whenever φ = 0.5. The above observation suggests that as the vaccination rate increases, the number of people just infected to the disease decreases. This decrease corresponds to the decrease in the E0 (25) above, since by translation E0 becomes I0 over the incubation period T1 = 2 time units. From the 95% confidence interval we can say that

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the true mean value of I0 at time t = 25, E[I0 (25)], lies approximately somewhere between (321, 324) whenever φ = 0.01, changes to (252, 255) whenever φ = 0.1, and further changes to (125, 127) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate.

FIGURE 4.8 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probability of getting infected is lesser than the probability of getting vaccinated, that is, φ = 0.15 and p = 0.01.

Lastly, at time t = 25, the average for R0 (25) lies approximately between (280, 310) whenever φ = 0.01, and changes to (220, 250) whenever φ = 0.1, and further changes to (100, 115) whenever φ = 0.5. From the 95% confidence interval we can say that the true mean value of R0 at time t = 25, E[R0 (25)], lies approximately somewhere between (292, 295) whenever φ = 0.01, changes to (238, 241) whenever φ = 0.1, and further changes to (106, 108) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. The above observation suggests that as the vaccination rate increases, the number of people just recovered from the disease decreases. This decrease corresponds to the decrease in the E0 (25) and I0 (25) above. This is because E0 translates into I0 and I0 translates to R0 over the incubation and infectious periods T1 = 2, T2 = 3 time units, respectively. For a fixed value of φ = 0.15 and p continuously changing from p = 0.01 to p = 0.5, it is easy to see that the number of susceptible people most likely present at time t = 25, the average from the centers (mean, mode, and median), lies approximately between (2040, 2100) whenever p = 0.01 and changes to (1140, 1200) whenever p = 0, 1, and further changes to

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FIGURE 4.9 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probability of getting infected is lesser than the probability of getting vaccinated, that is, φ = 0.15 and p = 0.1.

(630, 650) whenever p = 0, 5. The above observation suggests that as the infection rate increases, susceptible people get infected, and as a result S(25) decreases. From the 95% confidence interval we can say that the true mean value of S at time t = 25, E[S(25)], lies approximately somewhere between (2059, 2063) whenever φ = 0.01, changes to (1172, 1177) whenever φ = 0.1, and further changes to (631, 635) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. Similarly, at time t = 25, the average for E0 (25) lies approximately between (0, 0.1) whenever p = 0.01 and changes to (150, 180) whenever p = 0.1, and further changes to (320, 360) whenever φ = 0.5. The above observation suggests that as the infection rate increases, the number of people just exposed or infected to the disease increases. From the 95% confidence interval we can say that the true mean value of E0 at time t = 25, E[E0 (25)], lies approximately somewhere between (0, 0) whenever φ = 0.01, changes to (167, 170) whenever φ = 0.1, and this further changes to (333, 336) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. Similarly, at time t = 25, the average for V0 (25) lies approximately in the range (290, 320) whenever p = 0, 01 and changes to (140, 170) whenever p = 0.1, and further changes to (20, 35) whenever φ = 0.5. The above observation suggests that as the infection rate increases, the number of people just exposed or infected to the disease increases. From the

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FIGURE 4.10 After running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probability of getting infected is lesser than the probability of getting vaccinated, that is, φ = 0.15 and p = 0.5.

95% confidence interval we can say that the true mean value of V0 at time t = 25, E[V0 (25)], lies approximately somewhere between (308, 312) whenever φ = 0.01, changes to (149, 152) whenever φ = 0.1, and further changes to (28, 29) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. Similarly, when φ fixed at 0.15 and p ∈ {0.01, 0.1, 0.5} at time t = 25, the average for I0 (25) lies approximately in the range (0, 0.2) whenever p = 0.01 and changes to (150, 180) whenever p = 0.1, and further changes to (230, 260) whenever p = 0.5. The above observation suggests that as the infection rate increases, the number of just infected individuals increases. This increase corresponds to the increase in E0 (25) above, since by translation E0 (25) becomes I0 (25) over the incubation period T1 = 2 time units. From the 95% confidence interval we can say that the true mean value of I0 at time t = 25, E[I0 (25)] = 0, whenever φ = 0.01, changes to (167, 170) whenever φ = 0.1 and further changes to (247, 250) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. Similarly, at time t = 25, the average for R0 (25) lies approximately in the range between (0, 0.2) whenever p = 0.01 and changes to (150, 180) whenever p = 0.1, and further changes to (160, 200) whenever p = 0.5. The above observation suggests that as the infection rate increases, the number of just recovered people from the disease increases. This increase

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corresponds to the decrease in the E0 (25) and I0 (25) above at the 25th time step. This is because E0 translates into I0 and I0 translates to R0 over the incubation and infectious periods T1 = 2, T2 = 3 time units, respectively. From the 95% confidence interval we can say that the true mean value of R0 at time t = 25, E[R0 (25)] = 0, whenever φ = 0.01, changes to (168, 170) whenever φ = 0.1, and further changes to (179, 182) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. Similarly, at time t = 25, the average for S0 (25) lies approximately between (290, 320) whenever p = 0.01 and changes to (300, 340) whenever p = 0.1, and further changes to (470, 510) whenever p = 0.5. The above observation suggests that as the infection rate increases, the number of people just becoming susceptible increases due to translation occurring from R0 (25) and V0 (25) above at the 25th time step. This occurs over the natural and artificial immunity periods T3 = 4, T 4 = 4 time units, respectively. From the 95% confidence interval we can say that the true mean value of S0 at time t = 25, E[S0 (25)], lies approximately somewhere between (308, 311) whenever φ = 0.01, changes to (319, 322) whenever φ = 0.1, and further changes to (486, 489) whenever φ = 0.5. In all cases the confidence interval is narrow and affirms the accuracy of the estimate. Remark 4.6.1. Discussion of the sensitivity analysis results of the SVEIRS model with correlated vaccination and infection rates. Overall, the following conclusions can be made from the results presented in the previous subsections for the sensitivity analysis of the SVEIRS model defined in Theorem 4.5.1 for the three cases, namely: when p, the probability of becoming infected after one interaction with an infectious individual, and φ, the probability that the susceptible individual gets vaccinated after escaping infection, are (1) assumed to be constant (i.e., p = φ = 0.15); (2) p = 0.15 is fixed, while φ continuously changes values in φ ∈ {0.01, 0.1, 0.5}; (3) φ = 0.15 is fixed, while p continuously changes values in p ∈ {0.01, 0.1, 0.5}. In the case where vaccination and infection rates are kept constant, it turns out that the disease becomes aggressive, as more susceptible individuals in the population are predicted to get infected by the disease. This observation suggests that keeping vaccination and infection rates constant would not lead to any reduction in the disease spread. In the case when either p or φ is fixed and φ or p is continuously changing, it can be seen that when vaccination rates are increased in the population, more susceptible people become vaccinated, and the number of people just infected decreases. Due to the decrease in disease that occurs from the people just exposed, by translation, the infected and removal classes decrease. On the other hand, when vaccination is fixed, i.e., φ is fixed and p continuously changes, more susceptible people get infected, and the overall susceptible population decreases. Also, by translation occurring in the population, it can be seen that the number of people just exposed, infected, and just recovered would increase. These observations suggest that controlling the SVEIRS epidemic requires encouraging more vaccination as the disease prevalence rises. While no vaccination is clearly not good for disease control, maintaining vaccination rates at the same level while infection rates are rising would also not lead to disease eradication.

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4.6.3 The prototype SVEIRS model with uncorrelated vaccination and infection rates In this section, we generate histograms for the state of the process {B(t), t ≥ 0} in Theorem 4.5.2. We utilize Figs. 4.11–4.14 to compare the effects of the values of p representing the probability of passing infection from one interaction in (4.96) and μv representing the average number of people getting vaccinated per unit time in (4.97), where the values of p and μv are either occurring in the absence of one another or occurring together at a low rate or high rate. We observe the effects of changing the values of the probability of getting vaccinated in (4.97) using μv ∈ {0, 0.05, 0.07} on the dynamics of the disease and the effects of changing the values of the probability of getting infected in (4.96) using p ∈ {0, 0.05, 0.15}, where it is assumed in each scenario that λ, representing the average number of people the susceptible person meets per unit time (i.e., [t, t + 1)), is 10. Note that all figures in this section are produced using the algorithm described in Subsection 4.6.1. Moreover, we utilize the initial population values also characterized in Subsection 4.6.1. Figs. 4.11–4.14 depict the estimated distributions of the random variables S(25), E0 (25), V0 (25), I0 (25), R0 (25), and S0 (25).

FIGURE 4.11 For the exponential time until vaccination, after running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probability of getting infected is high and the probability of getting vaccinated is zero, that is, V0 (25) = 0, and the probabilities are p = 0.15 and μv = 0.

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For the model {B(t), t ≥ 0} in Theorem 4.5.2, considering the absence of vaccination and high infection rates, that is, for a value of μv = 0 and p = 0.15, in Fig. 4.11, it can be observed that the number of susceptible people present at time t = 25, S(25) has the measures of center (mean, median, and mode) lying approximately between (670, 750), the distribution appears to be approximately symmetric. The 95% confidence interval asserts that the true mean value of the susceptible person (S) at 25th time step E[S(25)] lies approximately somewhere between 704 and 709. The confidence interval is narrow and affirms the accuracy of the estimate. The number of just exposed people at time t = 25, E0 (25) lies approximately between (260, 310) and the distribution also appears to be symmetric. The 95% confidence interval asserts that the true mean value of (E0 ) at the 25th time step E[E0 (25)] lies approximately somewhere between 284 and 287. The confidence interval is narrow and affirms the accuracy of the estimate. The number of just vaccinated people at time t = 25, V0 (25) is zero; this is because there is an absence of vaccination in the population at that time. The average of the number of just infected people at time t = 25, I0 (25) lies approximately between (310, 350), and the distribution appears nearly symmetric. The 95% confidence interval asserts that the true mean value of (I0 ) at the 25th time step E[I0 (25)] lies approximately somewhere between 331 and 334. The confidence interval is narrow and affirms the accuracy of the estimate. The average of the number of just recovered people at time t = 25, R0 (25) lies approximately between (280, 320), and the distribution appears to be symmetric. The 95% confidence interval asserts that the true mean value of (R0 ) at the 25th time step E[R0 (25)] lies approximately somewhere between 298 and 301. The confidence interval is narrow and affirms the accuracy of the estimate. Finally, the average of the number of people just becoming susceptible present at time t = 25, S0 (25) lies approximately between (190, 230), and the distribution appears nearly symmetric. The 95% confidence interval asserts that the true mean value of just susceptible person (S0 ) at the 25th time step E[S0 (25)] lies approximately somewhere between 210 and 213. The confidence interval is narrow and affirms the accuracy of the estimate. From the different distributions, it can be observed that in the absence of vaccination, the number of susceptible, recovered, and just becoming susceptible people decreases; there are more exposed and infectious people; there are no vaccinated people in the population. Considering the absence of infection and high vaccination in the population for the model {B(t), t ≥ 0} in Theorem 4.5.2 for values of μv = 0.7 and p = 0, in Fig. 4.12, it is easy to see that the number of susceptible people present at time t = 25 on the average (mean, median, and mode) lies approximately between (1050, 1120), the distribution appears to be symmetric. The 95% confidence interval asserts that the true mean value of (S) at the 25th time step E[S(25)] lies approximately somewhere between 1076 and 1080. The confidence interval is narrow and affirms the accuracy of the estimate. The average of the number of just vaccinated people at time t = 25, V0 (25) lies in the range (560, 620), and the distribution appears to be symmetric. The 95% confidence interval asserts that the true mean value of (V0 ) at the 25th time step E[V0 (25)] lies approximately somewhere between 582 and 586. The confidence interval is narrow and affirms the accuracy of the estimate. The average number of just becoming susceptible people present at time t = 25, S0 (25) lies ap-

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FIGURE 4.12 For the exponential time until vaccination, after running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probability of getting infected is zero and the probability of getting vaccinated is high, here E0 (25), I0 (25), R0 (25) = 0, where p = 0 and μv = 0.7.

proximately between (480, 520), and the distribution appears nearly symmetric. The 95% confidence interval asserts that the true mean value of (S0 ) at the 25th time step E[S0 (25)] lies approximately somewhere between 500 and 503. The confidence interval is narrow and affirms the accuracy of the estimate. Finally, the average number of just exposed people at time t = 25, E0 (25) = 0. E0 (25) translates into just infected I0 (25), and just infected I0 (25) translates into R0 (25), it is easy to see that I0 (25) = R0 (25) = 0. They are all zeros because the probability of getting infected p is zero. From the distributions described above, it can be observed that in the absence of infection, the number of just exposed E0 (25) people, the number of just infected I0 (25) people, and, by translation, the number of just recovered people R0 (25) becomes zero, that is, E0 (25) = I0 (25) = R0 (25) = 0. The number of just vaccinated V0 (25), susceptible S0 (25), and just becoming susceptible S0 (25) people increases due to vaccination. For the model {B(t), t ≥ 0} in Theorem 4.5.2, considering high infection and high vaccination rates in the population for values of μv = 0.7 and p = 0.15, in Fig. 4.13, on the average (mean, median, and mode) it is easy to see that the number of susceptible people present at time t = 25 lies approximately between (680, 750), the distribution appears to be nearly symmetric. The 95% confidence interval asserts that the true mean value of (S) at the 25th time step E[S(25)] lies approximately somewhere between 708 and 713.

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FIGURE 4.13 For the exponential time until vaccination, after running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probability of getting infected is high and the probability of getting vaccinated is high, p = 0.15 and μv = 0.7.

The confidence interval is narrow and affirms the accuracy of the estimate. The average of the number of just vaccinated people at time t = 25, V0 (25) lies in the range (400, 450), and the distribution appears nearly symmetric. The 95% confidence interval asserts that the true mean value of (V0 ) at the 25th time step E[V0 (25)] lies approximately somewhere between 424 and 427. The confidence interval is narrow and affirms the accuracy of the estimate. The number of just exposed people at time t = 25, E0 (25) lies approximately between (80, 110), and the distribution also appears to be symmetric. The 95% confidence interval asserts that the true mean value of (E0 ) at the 25th time step E[E0 (25)] lies approximately somewhere between 94 and 96. The confidence interval is narrow and affirms the accuracy of the estimate. The average of the number of just infected people at time t = 25, I0 (25) lies approximately between (80, 120), and the distribution appears symmetric. The 95% confidence interval asserts that the true mean value of (I0 ) at the 25th time step E[I0 (25)] lies approximately somewhere between 101 and 103. The confidence interval is narrow and affirms the accuracy of the estimate. The average of the number of just recovered people at time t = 25, R0 (25) lies in the range between (70, 100), and the distribution appears to be symmetric. The 95% confidence interval asserts that the true mean value of (R0 ) at the 25th time step E[R0 (25)] lies approximately somewhere between 85 and 88. The confidence

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interval is narrow and affirms the accuracy of the estimate. The average number of just becoming susceptible people present at time t = 25, S0 (25) lies approximately between (500, 550), and the distribution appears symmetric. The 95% confidence interval asserts that the true mean value of (S0 ) at the 25th time step E[S0 (25)] lies approximately somewhere between 531 and 535. The confidence interval is narrow and affirms the accuracy of the estimate. From the distributions described above, it can be observed that when the rate of vaccination and infection are both high, the number of just exposed E0 (25) people, the number of just infected I0 (25) people, and, by translation, the number of just recovered people R0 (25) are relatively high. The number of just vaccinated V0 (25), susceptible S0 (25), and just becoming susceptible S0 (25) people also increases. That is, at the 25th time step, as the vaccination rate increases, there are more vaccinated people, and as the infection rate also increases, there are more infected people in the population, respectively.

FIGURE 4.14 For the exponential time until vaccination, after running the simulation 1, 000 times, the histogram shows the behavior of the distribution for the initial susceptible, vaccinated, exposed, infectious, recovered, and just becoming susceptible populations of the process at time t = 25 whenever the probability of getting infected is low and the probability of getting vaccinated is also low, p = 0.05 and μv = 0.1.

For the model {B(t), t ≥ 0} in Theorem 4.5.2, considering low infection and vaccination rates in the population for values of μv = 0.1 and p = 0.05, in Fig. 4.14, on the average (mean, median, and mode) it is easy to see that the number of susceptible people present at time t = 25 lies approximately between (2100, 2200), the distribution appears to be symmetric. The 95% confidence interval asserts that the true mean value of (S) at the 25th time step E[S(25)] lies approximately somewhere between 2159 and 2165. The confidence inter-

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val is narrow and affirms the accuracy of the estimate. The average of the number of just vaccinated people at time t = 25, V0 (25) lies in the range (190; 220), and the distribution appears symmetric. The 95% confidence interval asserts that the true mean value of (V0 ) at the 25th time step E[V0 (25)] lies approximately somewhere between 203 and 206. The confidence interval is narrow and affirms the accuracy of the estimate. The number of just exposed people at time t = 25, E0 (25) lies approximately between (25, 45), and the distribution also appears to be nearly symmetric. The 95% confidence interval asserts that the true mean value of (E0 ) at the 25th time step E[E0 (25)] lies approximately somewhere between 33 and 35. The confidence interval is narrow and affirms the accuracy of the estimate. The average of the number of just infected people at time t = 25, I0 (25) lies approximately between (25, 40), and the distribution appears to be symmetric. The 95% confidence interval asserts that the true mean value of (I0 ) at the 25th time step E[I0 (25)] lies approximately somewhere between 34 and 36. The confidence interval is narrow and affirms the accuracy of the estimate. The average of the number of just recovered people at time t = 25, R0 (25) lies in the range between (25, 40), and the distribution appears nearly symmetric. The 95% confidence interval asserts that the true mean value of (R0 ) at the 25th time step E[R0 (25)] lies approximately somewhere between 35 and 37. The confidence interval is narrow and affirms the accuracy of the estimate. The average number of just becoming susceptible people present at time t = 25, S0 (25) lies approximately between (230, 260), and the distribution appears symmetric. The 95% confidence interval asserts that the true mean value of (S0 ) at the 25th time step E[S0 (25)] lies approximately somewhere between 240 and 243. The confidence interval is narrow and affirms the accuracy of the estimate. From the distributions described above, it can be observed that when the rates of vaccination and infection are both low, the number of susceptible people is high. The number of just exposed E0 (25) people, the number of just infected I0 (25) people, and, by translation, the number of just recovered people R0 (25) are relatively low. The number of just vaccinated V0 (25) and just becoming susceptible S0 (25) people also decreases. That is, at the 25th time step, as the vaccination rate decreases, there are fewer people becoming just vaccinated, and as the infection rate decreases, fewer people become just infected, respectively. Overall, a similar conclusion in Remark 4.6.1 is made for the results in Subsection 4.6.3. That is, vaccination is highly necessary for disease eradication.

4.7 Conclusion This chapter presents two chain-binomial SVEIRS epidemic models for disease outbreaks such as typhoid fever. Unlike most traditional Markov chain epidemic models, where individuals in any state (susceptible, vaccinated, exposed, infectious, and removed) transit to the next state in the next time step, the presented discrete time SVEIRS Markov chain models consider discrete delays in the disease dynamics, namely: the incubation period T1 , the infectious period T2 , the natural immunity period T3 , and the artificial immunity period via vaccination T4 . That is, two discrete time measures are used for the state of an individual

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in the disease dynamics, namely: the epoch of the current state (susceptible, vaccinated, exposed, infectious, and removed) of an individual and how long the individual has been in the current state. The obtained discrete time Markov chain models exhibit multinomial transition probabilities. Using a theoretical example and statistical inferential techniques, sensitivity analysis is conducted to determine the effects of vaccination and infection on the disease dynamics.

References [1] H. Abbey, On the statistical measure of infectiousness, Hum. Biol. 24 (1952) 201–233. [2] L. Allen, An introduction to stochastic epidemic models, in: Mathematical Epidemiology, in: Lecture Notes in Mathematics, vol. 1945, Springer, 2008. [3] N. Bailey, The Mathematical Theory of Infectious Diseases, Griffin & Co., 1975. [4] N. Becker, A general chain binomial model for infectious diseases, Biometrics 37 (1981) 251–258. [5] Center for Disease Control and Prevention (CDC), Typhoid Fever and travels, https://wwwnc.cdc. gov/travel/diseases/typhoid. [6] Center for Disease Control and Prevention (CDC), Typhoid Fever symptoms and Treatments, https:// www.cdc.gov/typhoid-fever/index.html. [7] Center for Disease Control and Prevention (CDC), Different Covid-19 vaccines, https://www.cdc.gov/ coronavirus/2019-ncov/vaccines/different-vaccines, September 2021. [8] Center for Disease Control and Prevention (CDC), CDC, vaccines and immunization, Pfizer-BioNTech Covid-19 vaccine, https://www.cdc.gov/vaccines/covid-19/info-by-product/pfizer/index.html, December 2020. [9] F. Etbaigha, A. Williams, Z. Poljak, An SEIR model of influenza A virus infection and reinfection within a farrow-to-finish swine farm, PLoS ONE 13 (9) (2008) e0202493. [10] M. Ferrante, E. Ferraris, C. Rovira, On a stochastic epidemic SEIHR model and its diffusion approximation, Test 25 (2016) 482–502. [11] M. Greenwood, On the statistical measure of infectiousness, J. Hyg. Camb. 31 (1931) 336–351. [12] A. Griffin, S. McSorley, Development of protective immunity to Salmonella, a mucosal pathogen with a systemic agenda, Mucosal Immunol. 4 (2011) 371–382. [13] H. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000) 599–653. [14] P. Narayan, K. Mathur, Dynamics of an SVEIRS epidemic model with vaccination and saturated incidence rate, Int. J. Appl. Comput. Math. 4 (118) (2015), https://doi.org/10.1007/s40819-018-0548-0. [15] J. Keeling, M. Ross, An examination of the Reed-Frost theory of epidemics, Hum. Biol. 24 (1952) 201–234. [16] J. Keeling, M. Ross, On methods for studying stochastic disease dynamics, J. R. Soc. Interface 5 (19) (2008) 171–181. [17] W.O. Kermack, A.G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A 115 (1927) 700–721. [18] P. Lekone, B. Finkenstadt, An examination of the Reed-Frost theory of epidemics, Biometrics 62 (2006) 1170–1177. [19] M. Li, et al., Global dynamics of a SEIR model with varying total population size, Math. Biosci. 160 (1999) 191–213. [20] X. Liu, Y. Takeuchi, S. Iwami, SVIR epidemic models with vaccination strategies, J. Theor. Biol. 253 (2008) 1–11. [21] R. Massoukou, S. Noutchie, R. Guiem, Global dynamics of an SVEIR model with age-dependent vaccination, infection, and latency, Abstr. Appl. Anal. 62 (2018) 8479638. [22] P. Mastroeni, N. Menager, Development of acquired immunity to Salmonella, J. Med. Microbiol. 52 (2003) 453–459. [23] Medical News Today, What you need to know about Typhoid, https://www.medicalnewstoday.com/ articles/156859.php.

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[24] J. Metz, O. Diekmann, J. Heesterbeek, On the definition and the computation of the basic reproduction ratio Ro in models for infectious diseases in heterogeneous populations, J. Math. Biol. 28 (1990) 365–382. [25] World Health Organization, WHO coronavirus disease (Covid-19) dashboard, https://covid19.who. int/, December 20 2020. [26] World Health Organization, WHO immunization, vaccines and biology, https://www.who.int/ immunization/diseases/typhoid/en/, 20 December 2020. [27] M. Robert, R.M. Anderson, Infectious Diseases Ofhumans: Dynamics and Control, Oxford University Press, 1992. [28] R. Ravindran, S. McSorley, Tracking the dynamics of T-cell activation in response to Salmonella infection, Ann. Med. Health Sci. Res. 114 (2005) 450–458. [29] A. Sharma, et al., Typhoid intestinal perforation: 24 perforations in one patient, Ann. Med. Health Sci. Res. 26 (2013) 41–43. [30] G. Tilahun, O. Makinde, D. Malonza, Modelling and optimal control of typhoid fever disease with cost-effective strategies, in: Computational and Mathematical Methods in Medicine 2017, 2017, p. 2324518. [31] H. Tuckwell, R. Williams, Some properties of a simple stochastic epidemic model of SIR type, Math. Biosci. 208 (2007) 76–97. [32] Wanduku Divine, Complete global analysis of a two-scale network SIRS epidemic dynamic model with distributed delay and random perturbations, Appl. Math. Comput. 294 (2017) 49–76. [33] Wanduku Divine, Modeling highly random dynamical infectious systems, in: Applied Mathematical Analysis: Theory, Methods, and Applications, Springer, 2020, pp. 509–578. [34] Wanduku Divine, The stochastic extinction and stability conditions for nonlinear malaria epidemics, Math. Biosci. Eng. 16 (5) (2019) 3771–3806. [35] Wanduku Divine, Threshold conditions for a family of epidemic dynamic models for malaria with distributed delays in a non-random environment, Int. J. Biomath. 11 (06) (2018) 1850085. [36] Wanduku Divine, G.S. Ladde, Fundamental properties of a two-scale network stochastic human epidemic dynamic model, Neural Parallel Sci. Comput. 19 (3) (2011) 229. [37] Wanduku Divine, G.S. Ladde, Global properties of a two-scale network stochastic delayed human epidemic dynamic model, Nonlinear Anal., Real World Appl. 13 (2) (2012) 794–816. [38] Divine Wanduku, et al., Modeling the stochastic dynamics of influenza epidemics with vaccination control, and the maximum likelihood estimation of model parameters, in: Mathematical Modelling in Health, Social and Applied Sciences, 1st edition, Springer, 2020, pp. 23–72. [39] P. Witbooi, G. Muller, G. Schalkwyk, William Budd and typhoid fever, in: Computational and Mathematical Methods in Medicine, 2015, p. 271654. [40] R. Yaesoubi, T. Cohen, A novel framework for developing dynamic health policies, Eur. J. Oper. Res. 215 (2011) 679–687. [41] S. Zaki, Re-infection of typhoid fever and typhoid vaccine (comment on “An imported enteric fever caused by a quinolone-resistant Salmonella typhi”), Ann. Saudi Med. 31 (2011) 203–204. [42] Y. Zhao, D. Jiang, Global stability of an SVIR model with age of vaccination, Ann. Saudi Med. 226 (2014) 528540. [43] Y. Zhao, D. Jiang, The behavior of an SVIR epidemic model with stochastic perturbation, Abstr. Appl. Anal. 2014 (2014) 742730.

5 Hopf bifurcation in an SIR epidemic model with psychological effect and distributed time delay Toshikazu Kuniya Graduate School of System Informatics, Kobe University, Kobe, Japan

5.1 Introduction The coronavirus disease 2019 (COVID-19) has caused enormous damage to many countries around the world. In COVID-19, recurrent epidemic waves have been observed in many countries (see Fig. 5.1 for the case of Japan). The purpose of this study is to investigate the essential mechanism of such recurrent epidemic waves from the viewpoint of mathematical modeling. In the classical SIR epidemic model, which was proposed by Kermack and McKendrick [16], the total population is divided into three classes called susceptible S, infective I , and recovered R. A typical SIR model with birth and death processes (see, e.g., [14, Section 5.5.2]) is formulated as the following system of ordinary differential equations: ⎧  S (t) = b − βS(t)I (t) − μS(t), ⎪ ⎪ ⎨ I  (t) = βS(t)I (t) − (μ + γ )I (t), ⎪ ⎪ ⎩  R (t) = γ I (t) − μR(t),

(5.1)

where S(t), I (t), and R(t) denote the susceptible, infective, and recovered populations at time t, respectively. b, β, μ, and γ are positive constants representing the birth, infection, death, and recovery rates, respectively. It is easy to see that system (5.1) has two kinds of equilibria: the disease-free equilibrium E 0 : (S, I, R) = (S 0 , 0, 0) and the endemic equilibrium E ∗ : (S, I, R) = (S ∗ , I ∗ , R ∗ ), where S0 =

b , μ

S∗ =

μ+γ , β

I∗ =

μ (R0 − 1), β

R∗ =

γ ∗ I . μ

Here, R0 is the so-called basic reproduction number [6], which implies the expected number of secondary cases produced by an infective individual in the fully susceptible populaAdvances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00010-7 Copyright © 2023 Elsevier Inc. All rights reserved.

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FIGURE 5.1 The daily number of newly reported cases of COVID-19 in Japan from January 14, 2020 to July 31, 2021 [28].

tion. For system (5.1), R0 is given by R0 =

βb . μ(μ + γ )

The classical threshold theorem is as follows: if R0 ≤ 1, then the disease-free equilibrium E 0 is globally asymptotically stable, whereas if R0 > 1, then the endemic equilibrium is globally asymptotically stable [14, Proposition 5.8]. This implies that system (5.1) excludes any nontrivial periodic solutions. That is, recurrent epidemic waves cannot be replicated by model (5.1). In many countries, nonpharmaceutical interventions such as lockdown were taken against COVID-19. The daily number of newly reported cases tends to decrease during the period of intervention [8]. However, the rebound of the epidemic often occurs after the lifting of the intervention (see Fig. 5.2 for the case of Japan). We can conjecture that people’s behavior change induced by the intervention is one of the causes of recurrent epidemic waves. Nonlinear incidence rate is known as one of the key factors that cause the periodicity in epidemic models [13, Section 4]. In [5] Capasso and Serio generalized the bilinear incidence rate βSI in the SIR model to G(I )S to consider the psychological effect. That is, G is bounded and not necessarily monotone increasing because people may reduce the number of contacts when the epidemic size is large (see Fig. 5.3). Their idea originated from the study of cholera spread in Bari, 1973 [4]. In [23] Ruan and Wang studied the case of G(I ) = βI p /(1 + αI q ) with p = q = 2 in an SIR model and showed that their system admits the periodic solution. In many previous studies, p > 1 was necessary for the existence of the periodic solution (see, for instance, [1,18–20,25]). However, the epidemiological justification of p > 1 seems not to be easy (see [20, Section 6] for a discussion on this perspective). In this paper we focus on the case of p = 1, which could be more epidemiologically comprehensible.

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FIGURE 5.2 The daily number of newly reported cases of COVID-19 in Japan from January 14 to July 31, 2020 [28]. In Japan, the first state of emergency for COVID-19 was declared on April 7, 2020 (red (dark gray in print version) dot line), and it was lifted on May 25, 2020 (green (light gray in print version) dot line) [22].

FIGURE 5.3 Example of the nonlinear incidence rate G(I ) = βI p /(1 + αI q ).

Time delay is also one of the key factors that cause the periodicity in epidemic models [13, Section 3]. There are many studies on epidemic models with discrete time delays in which the periodic solution arises through the Hopf bifurcation regarding the time delay as the bifurcation parameter (see, for instance, [3,15,17,24,26,27,29]). For the global stability result in an SIR epidemic model with discrete time delay, see, e.g., [12]. In [30], to consider the effects of media coverage, Zhao and Zhao studied a nonlinear incidence rate with discrete time delay τ : G(I (t)) = (β1 − β2 I (t − τ )/(α + I (t − τ )))I (t). They showed that the periodic solution arises through the Hopf bifurcation by changing the bifurcation parameter τ . Here, we may have to note that the discrete time delay τ in the nonlinear incidence rate implies that people’s behavior at the current time t is affected only by the infective population at time t − τ . To construct a more realistic model, we may have to incorporate the distributed time delay (see [2,7,11,21]), by which we can consider the effect of the history of the infective population on people’s behavior at the current time.

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FIGURE 5.4 Truncated exponential distribution (5.2) that represents the effect of the past infective population on the current people’s behavior.

From these perspectives, in this study we propose the following generalized nonlinear incidence rate with psychological effect and distributed time delay: 1+α

∞ 0

βS(t)I (t) , f (σ, τ )I (t − σ )dσ

where α ≥ 0 denotes the sensitivity of people’s behavior change, τ is a bifurcation parameter, and f (σ, τ ) is a distribution function that represents the effect of the infective population at σ time ago on people’s behavior at the current time. In this study, we assume that f is nonnegative, and ∞ (A1) 0 f (σ, τ )dσ = 1 for any τ ≥ 0; ∞ (A2) There exists r > 0 such that 0 |f (σ, τ )erσ |2 dσ < ∞ for any τ ≥ 0. (A1) implies that f (·, τ ) is a distribution on R+ for fixed τ . (A2) is assumed as in [10], which is needed in the analysis in Section 5.3. The following truncated exponential distribution is one example that satisfies (A1)–(A2):  0, σ < τ, f (σ, τ ) = k > 0. (5.2) −k(σ −τ ) , σ ≥ τ, ke We can regard τ in (5.2) as the incubation period, that is, the presymptomatic infective individuals cannot be detected, and they do not affect people’s behavior before time τ has passed since the infection. After the onset, the effect of the past infective population on the current people’s behavior decays in an exponential way (see Fig. 5.4). In this study, for general f satisfying (A1)–(A2), we derive the characteristic values that determine the property of the Hopf bifurcation: the direction (supercritical or subcritical), stability, and period of the bifurcated periodic solution. As an example, we consider the truncated exponential distribution (5.2) and show that the stable periodic solution, which could represent recurrent epidemic waves, can arise through the Hopf bifurcation. The organization of this chapter is as follows. In Section 5.2 we formulate our model and show a standard threshold result on the basic reproduction number R0 . In Section 5.3, by

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using the theory of center manifold and Poincaré normal form [10], we obtain the characteristic values that determine the direction, stability, and period of the bifurcated periodic solution. In Section 5.4 we show that the Hopf bifurcation can occur for the case of the truncated exponential distribution (5.2). In Section 5.5 we verify the validity of our theoretical results by numerical simulation. More precisely, we give a numerical example, in which a stable periodic solution arises through the Hopf bifurcation. Finally, we discuss our results in Section 5.6.

5.2 Model In this study, we consider the following SIR epidemic model with psychological effect and distributed time delay: ⎧ βS(t)I (t) ⎪ ⎪ ∞ S  (t) = b − − μS(t), ⎪ ⎪ 1 + α 0 f (σ, τ )I (t − σ )dσ ⎪ ⎪ ⎨ βS(t)I (t) ∞ − (μ + γ )I (t), I  (t) = ⎪ ⎪ ⎪ 1 + α 0 f (σ, τ )I (t − σ )dσ ⎪ ⎪ ⎪ ⎩ S(ρ) = ϕ1 (ρ), I (ρ) = ϕ2 (ρ),

t > 0, t > 0,

(5.3)

ρ ∈ (−∞, 0],

∞ where b, β, μ, γ > 0 and α ≥ 0. As explained above, the denominator 1 + α 0 f (σ, τ )I (t − σ )dσ of the infection term represents the psychological effect, and the time delay effect ∞ appears distributively in the integral 0 f (σ, τ )I (t − σ )dσ . Note that we omit the equation of recovered population R because it does not affect system (5.3). For r > 0 in (A2), we define the following set:   Cr := φ ∈ C((−∞, 0], R2 ) | φ(0)2 +

∞

e−rσ φ(−σ )2 dσ < ∞ ,

0

where C((−∞, 0], R2 ) denotes the Banach space of continuous functions on (−∞, 0] to R2 with the supremum norm, and  ·  denotes the Euclidean norm in R2 . Denote the initial condition by ϕ := (ϕ1 , ϕ2 )T , where T denotes the transpose operation. In what follows, we assume that ϕ ∈ Cr,+ , where Cr,+ := Cr ∩ {φ : (−∞, 0] → R2+ } denotes the positive cone of Cr . For any t ≥ 0 and u : (−∞, t] → R2 , we define ut : (−∞, 0] → R2 by ut (ρ) = u(t + ρ), ρ ∈ (−∞, 0]. For simplicity, for any constant vector ν ∈ R2 , we write ν ∈ Cr regarding it as a constant function on (−∞, 0]. For any t ≥ 0, we define Ut ∈ Cr,+ by Ut (ρ) = (S(t + ρ), I (t + ρ))T , ρ ∈ (−∞, 0]. System (5.3) can then be rewritten as the following system of delay differential equations: Ut = F (Ut ),

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where F : Cr → R2 is defined by ⎛

βφ1 (0)φ2 (0)  ⎜ b − 1 + α ∞ f (σ, τ )φ2 (−σ )dσ − μφ1 (0) ⎜ 0 F (φ) := ⎜ βφ1 (0)φ2 (0) ⎝ ∞ − (μ + γ )φ2 (0) 1 + α 0 f (σ, τ )φ2 (−σ )dσ

⎞ ⎟ ⎟ ⎟, ⎠

 φ=

φ1 φ2

 ∈ Cr .

Let us define the following set in Cr,+ :      b φ1  := φ = ∈ Cr,+  0 ≤ φ1 (ρ) + φ2 (ρ) ≤ for all ρ ∈ (−∞, 0] . φ2 μ We can easily check that F is Lipschitz in , and hence the local solution Ut uniquely exists. Moreover, we can easily prove the following proposition, which implies that the solution is global. Proposition 5.2.1.  is positively invariant, that is, Ut ∈  for all t > 0 if ϕ ∈ . Proof. Suppose that there exists t1 > 0 such that S(t) ≥ 0 for all t < t1 and S(t1 ) = 0. We then have that S  (t1 ) = b > 0, which implies that S is positive as long as it exists. I is also positive as long as it exists since it has the following exponential form: I (t) = I (0) exp

  t 0

  βS(u) ∞ − (μ + γ ) du 1 + α 0 f (σ, τ )I (u − σ )dσ

and I (0) = ϕ2 (0) ≥ 0. Moreover, we have [S(t) + I (t)] = b − μS(t) − (μ + γ )I (t) ≤ b − μ [S(t) + I (t)] , which implies that S(t) + I (t) ≤

  b b + S(0) + I (0) − e−μt . μ μ

Since S(0) + I (0) = ϕ1 (0) + ϕ2 (0) ≤ b/μ, we have S(t) + I (t) ≤ b/μ. This completes the proof.

In what follows, unless otherwise specified, we restrict our attention to the solution in . It is easy to see that E 0 : (S 0 , 0)T = (b/μ, 0)T ∈  is the disease-free equilibrium of system (5.3). By linearizing the second equation of (5.3) around E 0 , we have I  (t) = βS 0 I (t) − (μ + γ )I (t) = (μ + γ )



 βS 0 − 1 I (t), μ+γ

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which implies that the basic reproduction number R0 for system (5.3) is given by R0 =

βb βS 0 = . μ+γ μ(μ + γ )

This is similar to the one defined for the SIR model (5.1) without psychological effect and time delay. By assumption (A1), the endemic equilibrium E ∗ : (S, I )T = (S ∗ , I ∗ )T for system (5.3) satisfies b−

βS ∗ I ∗ − μS ∗ = 0, 1 + αI ∗

βS ∗ I ∗ − (μ + γ )I ∗ = 0. 1 + αI ∗

(5.4)

From these equations, we obtain S∗ =

(μ + γ )(1 + αI ∗ ) , β

I∗ =

μ (R0 − 1). αμ + β

(5.5)

The following proposition is a standard threshold result of R0 . Proposition 5.2.2. (i) If R0 ≤ 1, then the disease-free equilibrium E 0 : (S, I )T = (S 0 , 0)T = (b/μ, 0)T ∈  of system (5.3) is globally asymptotically stable in . (ii) If R0 > 1, then the system (5.3) has the unique positive endemic equilibrium E ∗ : (S, I )T = (S ∗ , I ∗ )T ∈ 0 , where 0 := {φ = (φ1 , φ2 )T ∈  | φ2 (0) > 0}. In particular, if α = 0, then E ∗ is globally asymptotically stable in 0 . Proof. (i) We now define the following Lyapunov functional on Cr to R: V1 (Ut ) := S(t) − S 0 − S 0 ln

S(t) + I (t), S0

t > 0.

Note that S(t) > 0 for all t > 0, provided ϕ ∈  as b > 0. Differentiating V1 , we have   S0 V1 (Ut ) = 1 − S  (t) + I  (t) S(t)    S0 βS(t)I (t) ∞ = 1− − μS(t) b− S(t) 1 + α 0 f (σ, τ )I (t − σ )dσ βS(t)I (t) ∞ − (μ + γ )I (t) 1 + α 0 f (σ, τ )I (t − σ )dσ 2 μ  βS 0 I (t) ∞ S(t) − S 0 + =− − (μ + γ )I (t) S(t) 1 + α 0 f (σ, τ )I (t − σ )dσ 2 μ  S(t) − S 0 + βS 0 I (t) − (μ + γ )I (t) ≤− S(t) 2 μ  S(t) − S 0 + (μ + γ )(R0 − 1)I (t) ≤ 0. ≤− S(t) +

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By the invariance principle (see, e.g., [9, Theorem 3.1 in Chapter 5]), we then see that E 0 is globally asymptotically stable. (ii) It is obvious from (5.5) that if R0 > 1, then I ∗ > 0, and hence E ∗ uniquely exists in 0  . If α = 0, then system (5.3) can be regarded as a standard SIR epidemic model without delay. Thus the assertion follows from a well-known Lyapunov functional approach (see, e.g., [14, Section 5.5.2]). This completes the proof.

5.3 Direction of bifurcation and stability of periodic solution In this section, we assume that R0 > 1, and thus, by Proposition 5.2.2 (ii), the unique positive endemic equilibrium E ∗ : (S ∗ , I ∗ )T exists. We regard τ as a bifurcation parameter and study the direction of bifurcation and the stability of the bifurcating periodic solution when the Hopf bifurcation occurs at some τ = τc . Let X := S − S ∗ and Y := I − I ∗ be the perturbation from the endemic equilibrium E ∗ . Substituting S = X + S ∗ and I = Y + I ∗ into the first equation in (5.3), we have, for all t > 0, X  (t) = b −

β[X(t) + S ∗ ][Y (t) + I ∗ ] ∞ − μ[X(t) + S ∗ ]. 1 + αI ∗ + α 0 f (σ, τ )Y (t − σ )dσ

(5.6)

Under the assumption that |Y | is sufficiently small, we have, for all t > 0,

1 + αI ∗ + α

∞ 0

1 1 = f (σ, τ )Y (t − σ )dσ 1 + αI ∗ 1 +

α

∞

1

0 f (σ,τ )Y (t−σ )dσ 1+αI ∗  ∞ ∞  α f (σ, τ )Y (t 1 = − 0 ∗ 1 + αI 1 + αI ∗ n=0

− σ )dσ

n .

We then have from (5.4) and (5.6) that, for all t > 0, β[I ∗ X(t) + S ∗ Y (t)] βS ∗ I ∗ α − μX(t) + · X (t) = − 1 + αI ∗ 1 + αI ∗ 

∞

f (σ, τ )Y (t − σ )dσ 1 + αI ∗  n ∞ ∞ α 0 f (σ, τ )Y (t − σ )dσ βX(t)Y (t) β[X(t)Y (t) + I ∗ X(t) + S ∗ Y (t)]  − − − 1 + αI ∗ 1 + αI ∗ 1 + αI ∗ n=1  n ∞ ∞ α 0 f (σ, τ )Y (t − σ )dσ βS ∗ I ∗  − − 1 + αI ∗ 1 + αI ∗ n=2  ∞ β˜ ˜ = −(β + μ)X(t) − (μ + γ )Y (t) + α(μ ˜ +γ) f (σ, τ )Y (t − σ )dσ − ∗ M(X(t), Y (t)), I 0 (5.7) 0

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where α˜ := αI ∗ /(1 + αI ∗ ), β˜ := βI ∗ /(1 + αI ∗ ), and M(X(t), Y (t)) 

n  ∞ ∞    α˜ ∗ ∗ − ∗ := X(t)Y (t) + X(t)Y (t) + I X(t) + S Y (t) f (σ, τ )Y (t − σ )dσ I 0 n=1 n   ∞ ∞   α˜ ∗ ∗ − ∗ +S I f (σ, τ )Y (t − σ )dσ . I 0 n=2

In a similar manner, we obtain, for all t > 0,  ∞ β˜ ˜ − α(μ ˜ +γ) f (σ, τ )Y (t − σ )dσ + ∗ M(X(t), Y (t)). Y  (t) = βX(t) I 0

(5.8)

Let u := (X, Y )T . We then have from (5.7) and (5.8) that ut = A(τ )ut + R(τ )ut ,

(5.9)

where, for each τ ≥ 0, A(τ ) and R(τ ) are operators on Cr1 := Cr ∩ C 1 ((−∞, 0], R2 ) defined by ⎧ dφ(ρ) ⎪ ⎪ , ρ ∈ (−∞, 0), ⎪ ⎪ ⎪ dρ ⎪ ⎪ ⎪ ⎪  ⎨  −(β˜ + μ) −(μ + γ ) A(τ )φ(ρ) := φ(0) ⎪ β˜ 0 ⎪ ⎪ ⎪ ρ = 0,   ⎪  ∞ ⎪ ⎪ 0 1 ⎪ ⎪ ˜ +γ) f (σ, τ ) φ(−σ )dσ, ⎩ +α(μ 0 −1 0 ⎧ ρ ∈ (−∞, 0), ⎪ ⎨ 0,   R(τ )φ(ρ) := −1 (5.10) ⎪ , ρ = 0, ⎩ M(φ) 1  β˜ M(φ) := ∗ φ1 (0)φ2 (0) I n  ∞ ∞   α˜ ∗ ∗ − ∗ f (σ, τ )φ2 (−σ )dσ +[φ1 (0)φ2 (0) + I φ1 (0) + S φ2 (0)] I 0 n=1  n    ∞ ∞   α˜ φ1 ∗ ∗ − ∗ ∈ Cr1 . +S I f (σ, τ )φ2 (−σ )dσ , φ= φ2 I 0 n=2

Note that ut = A(τ )ut is the linearized system around the endemic equilibrium E ∗ . In this section we make the   following assumptions on the eigenvalue λ ∈ σ (τ ) := {λ ∈ C | det λE − A(τ )(eλρ E) = 0}, where E denotes the identity matrix and the domain of A(τ ) is extended to the set of functions valued in C2 :

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There exists a pair of simple eigenvalues λ, λ ∈ σ (τ ) such that λ = λ(τ ) = η(τ ) + iω(τ ),

•

where η and ω are real and η(τc ) = 0, ωc := ω(τc ) > 0 and η (τc ) = 0 for some τc > 0. All other elements of σ (τc ) have negative real parts.

That is, we assume that the Hopf bifurcation occurs at τ = τc . In what follows, using the theory of center manifold and Poincaré normal form [10], we will obtain characteristic values for the direction (supercritical or subcritical) of the bifurcation and the stability and period of the bifurcating periodic solution. For simplicity, we write A = A(τc ), R = R(τc ), and f = f (·, τc ). We define the adjoint operator A∗ for A and the inner product < ·, · > by ⎧ dψ ⎪ ⎪ − , ρ ∈ (0, ∞), ⎪ ⎪ dρ ⎪ ⎪ ⎪   ⎪ ⎨ −(β˜ + μ) −(μ + γ ) ∗ ψ(0) A ψ(ρ) = β˜ 0 ⎪ ⎪ ⎪ ρ = 0, ⎪    ∞ ⎪ ⎪ 0 1 ⎪ ⎪ f (σ )ψ(σ ) ˜ +γ) dσ, ⎩ +α(μ 0 −1 0 



∞

˜ +γ) < ψ, φ >= ψ(0)φ(0) + α(μ 0

 φ=

φ1 φ2

σ

f (σ ) 0

 ψ(σ − ρ)

0 1 0 −1

 φ(−ρ)dρdσ,

 ∈ C 1 ((−∞, 0], C2 ),

ψ = (ψ1 , ψ2 ) ∈ C 1 ([0, ∞), C2 ).

Let q = q(ρ) = (1, q1 )T eiωc ρ and q ∗ = q ∗ (ρ) = Q(1, q1∗ )eiωc ρ be the eigenvectors of operators A and A∗ corresponding to eigenvalues iωc and −iωc , respectively. By a simple calculation, we then have β˜ + μ − iωc , β˜ 1 ∞ Q= , ∗ 1 + q1 q1 + α(μ ˜ + γ ) 0 σ eiωc σ f (σ )dσ q1 (1 − q1∗ ) q1 = −

μ + iωc , μ + γ + iωc

q1∗ =

where Q is chosen so that < q ∗ , q >= 1. Note that < q ∗ , q >= 0 as iωc is a simple eigenvalue of A. As in [10, Chapter 4], we define variables z and w as follows: z = z(t) =< q ∗ , ut >,

w = w(t, ρ) = ut (ρ) − z(t)q(ρ) − z(t)q(ρ).

(5.11)

Note that w = ut − 2Re (zq) is real and < q ∗ , w >= 0. By applying the theory in [10], we obtain the following system on a center manifold whose local coordinates are z and z in the direction of q ∗ and q ∗ , respectively: z = iωc z + g(z, z),

w  = Aw + H (z, z, ρ),

(5.12)

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where z2 z2 z2 z + g11 zz + g02 + g21 + ··· , 2 2 2 z2 H (z, z, ρ) = H20 (ρ) + H11 (ρ)zz + · · · . 2

g(z, z) = g20

From the theory of reduction of the two-dimensional system of z to Poincaré normal form (see [10, Chapter 1]), we obtain the following proposition. Proposition 5.3.1. Let us define the following characteristic values:   g21 i |g02 |2 Re c1 g20 g11 − 2|g11 |2 − + , μ2 = −  , c1 = 2ωc 3 2 η (τc ) Im c1 + μ2 ω (τc ) β2 = 2Re c1 , T2 = − . ωc Then the following assertions hold: (i) If μ2 > 0, then the Hopf bifurcation at τ = τc is supercritical. If μ2 < 0, then it is subcritical. (ii) If β2 < 0, then the periodic solution bifurcating at τ = τc is stable. If β2 > 0, then it is unstable. (iii) The period of the periodic solution bifurcating at τ = τc is given by  2π  τ − τc 1 + T2  2 + O( 4 ) , where  2 = + O((τ − τc )2 ). ωc μ2 That is, if T2 > 0 (T2 < 0), then the period of the bifurcated periodic solution increases (decreases) as τ departs from τc along the direction of the bifurcation in the neighborhood of τc . A typical example of Hopf bifurcations in connection with Proposition 5.3.1 is shown in Fig. 5.5. To apply Proposition 5.3.1, we have to calculate g20 , g11 , g02 , and g21 . From (5.9) and (5.11) we have z = < q ∗ , Aut + Rut >= iωc z+ < q ∗ , Rut >   −1 =iωc z + q ∗ (0)M(ut ) 1 =iωc z + Q(−1 + q1∗ )M(ut )

! =iωc z + κ m20 z2 + m11 zz + m02 z2 + m21 z2 z + · · · ,

(5.13)

˜ ∗ and mj k (j, k ∈ N, j + k ≥ 2) denotes the coefficient of zj zk in where κ := Q(−1 + q1∗ )β/I ∗ ˜ (I /β)M(ut ). Comparing (5.12) and (5.13), we obtain g20 = 2κm20 , g11 = κm11 , g02 = 2κm02 , g21 = 2κm21 .

(5.14)

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FIGURE 5.5 Example of Hopf bifurcations in connection with Proposition 5.3.1.

Thus we are in a position to calculate m20 , m11 , m02 , and m21 . From (5.10) we have m20 z2 + m11 zz + m02 z2 + m21 z2 z + · · ·  ∞  ∞ S∗ ˜ t,1 (0) f (σ )ut,2 (−σ )dσ − α˜ ∗ ut,2 (0) f (σ )ut,2 (−σ )dσ = ut,1 (0)ut,2 (0) − αu I 0 0    ∞ 2 ∞ S∗ α˜ + α˜ 2 ∗ f (σ )ut,2 (−σ )dσ − ∗ ut,1 (0)ut,2 (0) f (σ )ut,2 (−σ )dσ I I 0 0  ∞ 2  ∞ 2 ∗ α˜ 2 2 S + ∗ ut,1 (0) f (σ )ut,2 (−σ )dσ + α˜ ut,2 (0) f (σ )ut,2 (−σ )dσ I (I ∗ )2 0 0  ∞ 3 S∗ − α˜ 3 ∗ 2 f (σ )ut,2 (−σ )dσ , (5.15) (I ) 0 where ut = (ut,1 , ut,2 )T . Note that ut = w + qz + qz and w can be written as w = w(z, z, ρ) = w20 (ρ)

z2 + w11 (ρ)zz + · · · 2

on the center manifold. Thus we have w20,1 (0) 2 z + w11,1 (0)zz + z + z + · · · , 2 w20,2 (0) 2 ut,2 (0) = z + w11,2 (0)zz + q1 z + q1 z + · · · , 2  ∞ F20 2 z + F11 zz + q1 F10 z + q1 F01 z + · · · , f (σ )ut,2 (−σ )dσ = 2 0

ut,1 (0) =

(5.16)

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where wj k = (wj k,1 , wj k,2 )T (j, k ∈ N, j + k ≥ 2) and 



∞

F20 =

f (σ )w20,2 (−σ )dσ, 

0

F10 =

∞

F11 =

∞

f (σ )w11,2 (−σ )dσ, 0

f (σ )e−iωc σ dσ,

F01 = F10 .

0

Thus we have  w20,1 (0) w20,2 (0) 2 z z ut,1 (0)ut,2 (0) = q1 + w11,1 (0)q1 + w11,2 (0) + 2 2 

+ q1 z2 + (q1 + q1 )zz + q1 z2 + · · · ,    ∞ w20,1 (0) F20 2 z z f (σ )ut,2 (−σ )dσ = q1 F01 + w11,1 (0)q1 F10 + F11 + ut,1 (0) 2 2 0 + q1 F10 z2 + (q1 F10 + q1 F01 )zz + q1 F01 z2 + · · ·   ∞ F20 2 w20,2 (0) z z f (σ )ut,2 (−σ )dσ = q1 F01 + w11,2 (0)q1 F10 + q1 F11 + q1 ut,2 (0) 2 2 0 



+ q12 F10 z2 + q1 q1 (F10 + F01 )zz + q1 2 F01 z2 + · · · ,

2

∞

= (q1 F20 F01 + 2q1 F11 F10 ) z2 z

f (σ )ut,2 (−σ )dσ 0



∞

ut,1 (0)ut,2 (0) 

2 2 2 2 + q12 F10 z + 2q1 q1 F10 F01 zz + q1 2 F01 z + ··· ,

f (σ )ut,2 (−σ )dσ = [q1 q1 F01 + q1 (q1 + q1 )F10 ] z2 z + · · · ,

0 ∞

ut,1 (0)

! 2 z2 z + · · · , = 2q1 q1 F10 F01 + q12 F10

2

! 2 z2 z + · · · , = 2q12 q1 F10 F01 + q12 q1 F10

f (σ )ut,2 (−σ )dσ 0

 ut,2 (0) 

2

∞

f (σ )ut,2 (−σ )dσ 0

∞

3

f (σ )ut,2 (−σ )dσ 0

! 2 2 = 2q12 q1 F10 F01 + q12 q1 F10 F01 z2 z + · · · .

Comparing coefficients of z2 , zz, and z2 in (5.15), we obtain S∗ S∗ 2 2 q1 F10 + α˜ 2 ∗ q12 F10 , ∗ I I S∗ S∗ m11 = q1 + q1 − α(q ˜ 1 F10 + q1 F01 ) − α˜ ∗ q1 q1 (F10 + F01 ) + 2α˜ 2 ∗ q1 q1 F10 F01 , I I ∗ S∗ 2 2S 2 2 m02 = q1 − αq ˜ 1 F01 − α˜ ∗ q1 F01 + α˜ ∗ q1 F01 = m20 . I I

m20 = q1 − αq ˜ 1 F10 − α˜

(5.17)

As F10 and F01 do not depend on either w20 or w11 , we can calculate these values. Thus g20 , g11 , and g02 can be obtained by (5.14). Moreover, comparing coefficients of z2 z in (5.15), we

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obtain m21 =

w20,1 (0) w20,2 (0) q1 + w11,1 (0)q1 + w11,2 (0) + 2 2  w20,1 (0) F20 − α˜ q1 F01 + w11,1 (0)q1 F10 + F11 + 2 2   S ∗ w20,2 (0) F20 − α˜ ∗ q1 F01 + w11,2 (0)q1 F10 + q1 F11 + q1 I 2 2 ∗ S α ˜ + α˜ 2 ∗ (q1 F20 F01 + 2q1 F11 F10 ) − ∗ [q1 q1 F01 + q1 (q1 + q1 )F10 ] I I ! ! ∗ α˜ 2 2 2 2 S 2 2 2 2q + ∗ 2q1 q1 F10 F01 + q1 F10 + α˜ q F F + q q F 1 10 01 1 1 1 10 I (I ∗ )2 ! ∗ S 2 2 − α˜ 3 ∗ 2 2q12 q1 F10 F01 + q12 q1 F10 F01 . (I )

(5.18)

To compute this m21 , we have to derive w20 and w11 as F20 and F11 depend on them. To this end, we use the method as stated in [10, Chapter 4]. By (5.12) and (5.16), we have w  =w20 zz + w11 (z z + zz ) + · · · = iωc w20 z2 + 0 · zz + · · · Aw20 + H20 2 z + (Aw11 + H11 )zz + · · · . =Aw + H (z, z, ρ) = 2

(5.19) (5.20)

Comparing the coefficients in (5.19) and (5.20), we have Aw20 = 2iωc w20 − H20 ,

Aw11 = −H11 .

(5.21)

On the other hand, by (5.11) we have w  =ut − qz − qz =Aut + Rut − q [iωc z + g(z, z)] − q [−iωc z + g(z, z)] =Aw + Rut − qg(z, z) − q g(z, z).

(5.22)

Comparing (5.12) and (5.22), we have H (z, z, ρ) = R(ut )(ρ) − q(ρ)g(z, z) − q(ρ)g(z, z),

ρ ∈ (−∞, 0].

For ρ < 0, we have z2 + H11 (ρ)zz + · · · 2     z2 z2 = −q(ρ) g20 + g11 zz + · · · − q(ρ) g02 + g11 zz + · · · . 2 2

H20 (ρ)

(5.23)

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Comparing the coefficients in this equation, we obtain, for ρ < 0, H20 (ρ) = −q(ρ)g20 − q(ρ)g02 ,

H11 (ρ) = −q(ρ)g11 − q(ρ)g11 .

(5.24)

Substituting (5.24) into (5.21), we obtain, for ρ < 0, Aw20 (ρ) = 2iωc w20 (ρ) + q(ρ)g20 + q(ρ)g02 ,

Aw11 (ρ) = q(ρ)g11 + q(ρ)g11 .

By solving these equations, we obtain 

ρ

w20 (ρ) = w20 (0)e2iωc ρ +

e2iωc (ρ−σ ) [q(σ )g20 + q(σ )g02 ]dσ

0

g02 g20 = W20 e − q(ρ) − q(ρ), iωc 3iωc  ρ w11 (ρ) = w11 (0) + [q(σ )g11 + q(σ )g11 ] dσ 2iωc ρ

(5.25)

0

g11 g11 = W11 + q(ρ) − q(ρ), iωc iωc where W20 and W11 are constant vectors. To determine W20 and W11 , we consider the following relation derived by (5.23) for ρ = 0: z2 + H11 (0)zz + · · · 2       z2 z2 −1 = M(ut ) − q(0) g20 + g11 zz + · · · − q(0) g02 + g11 zz + · · · 1 2 2     ! 2 ˜ z β −1 2 = ∗ m20 z + m11 zz + · · · − q(0) g20 + g11 zz + · · · 1 I 2   2 z − q(0) g02 + g11 zz + · · · . 2

H20 (0)

Comparing the coefficients in this equation, we obtain   ˜ 20 −1 2βm H20 (0) = − g20 q(0) − g02 q(0), 1 I∗   ˜ 11 −1 βm H11 (0) = ∗ − g11 q(0) − g11 q(0). 1 I Substituting (5.25) and (5.26) into (5.21), we obtain, for ρ = 0,   g02 g02 g20 A(W20 e2iωc ρ )(0) − g20 q(0) + q(0) = 2iωc W20 − q(0) − q(0) 3 iωc 3iωc   ˜ 20 −1 2βm + g20 q(0) + g02 q(0), − 1 I∗

(5.26)

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˜ 11 βm AW11 + g11 q(0) + g11 q(0) = − ∗ I



−1 1

 + g11 q(0) + g11 q(0),

and thus we have   ∞   ˜ 20 −1 ˜ + γ ) 0 f (σ )e−2iωc σ dσ 2iωc + β˜ + μ μ + γ − α(μ 2βm , W20 = ∞ 1 I∗ ˜ + γ ) 0 f (σ )e−2iωc σ dσ −β˜ 2iωc + α(μ     ˜ 11 −1 β˜ + μ μ + γ − α(μ ˜ +γ) βm . W11 = ∗ 1 I −β˜ α(μ ˜ +γ) Consequently, we obtain  −1  ∞  ˜ 20 2iωc + β˜ + μ μ + γ − α(μ ˜ + γ ) 0 f (σ )e−2iωc σ dσ 2βm −1 W20 = ,  ∞ 1 I∗ ˜ + γ ) 0 f (σ )e−2iωc σ dσ −β˜ 2iωc + α(μ  −1   ˜ 11 β˜ + μ μ + γ − α(μ ˜ +γ) βm −1 . W11 = ∗ 1 I −β˜ α(μ ˜ +γ) (5.27) By (5.25) and (5.27), we can calculate w20 and w11 . As stated above, we can then calculate F20 and F11 , and thus m21 and g21 . In conclusion, from (5.14), (5.17), and (5.18), g20 , g11 , g02 , and g21 can be calculated as follows:   ∗ S∗ 2 2S 2 2 ˜ 1 F10 − α˜ ∗ q1 F10 + α˜ ∗ q1 F10 , g20 =2κ q1 − αq I I   ∗ S∗ 2S g11 =κ q1 + q1 − α(q ˜ 1 F10 + q1 F01 ) − α˜ ∗ q1 q1 (F10 + F01 ) + 2α˜ ∗ q1 q1 F10 F01 , I I   ∗ S S∗ 2 2 g02 =2κ q1 − αq ˜ 1 F01 − α˜ ∗ q1 F01 + α˜ 2 ∗ q1 2 F01 = m20 , I I  w20,1 (0) w20,2 (0) g21 =2κ q1 + w11,1 (0)q1 + w11,2 (0) + 2 2   w20,1 (0) F20 − α˜ q1 F01 + w11,1 (0)q1 F10 + F11 + 2 2   F20 S ∗ w20,2 (0) − α˜ ∗ q1 F01 + w11,2 (0)q1 F10 + q1 F11 + q1 I 2 2 ∗ S α ˜ + α˜ 2 ∗ (q1 F20 F01 + 2q1 F11 F10 ) − ∗ [q1 q1 F01 + q1 (q1 + q1 )F10 ] I I ! ! S∗ α˜ 2 2 2 + ∗ 2q1 q1 F10 F01 + q12 F10 + α˜ 2 ∗ 2 2q12 q1 F10 F01 + q12 q1 F10 I (I )  ! ∗ 3 S 2 2 2 2 2q1 q1 F10 F01 + q1 q1 F10 F01 . − α˜ (I ∗ )2

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Computing these g20 , g11 , g02 , and g21 , we can apply Proposition 5.3.1 to check the direction of the Hopf bifurcation at τ = τc and the stability and period of the bifurcated periodic solution.

5.4 Example: a truncated exponential distribution In this section, we show that the Hopf bifurcation can occur for the case of the truncated exponential distribution (5.2). Suppose that R0 > 1 and the endemic equilibrium E ∗ : (S ∗ , I ∗ )T exists. In this case, the characteristic equation at E ∗ is calculated as   ∞  λ + β˜ + μ μ + γ − α(μ ˜ + γ )kekτ τ e−(λ+k)σ dσ   0 =  ∞   −β˜ λ + α(μ ˜ + γ )kekτ τ e−(λ+k)σ dσ    λ + β˜ + μ μ + γ − α(μ ˜ + γ )ke−λτ /(λ + k)   = ,  −β˜ λ + α(μ ˜ + γ )ke−λτ /(λ + k)  which is equivalent to λ3 + a2 λ2 + a1 λ + a0 + (b1 λ + b0 )e−λτ = 0,

(5.28)

where a2 = β˜ + μ + k,

˜ + γ + k) + μk, a1 = β(μ

˜ + γ ), b1 = αk(μ

b0 = αμk(μ ˜ + γ ).

˜ a0 = βk(μ + γ ),

The following lemma immediately holds. Lemma 5.4.1. Suppose that f is given by (5.2) and R0 > 1. If τ = 0, then the positive endemic equilibrium E ∗ : (S ∗ , I ∗ )T of system (5.3) is locally asymptotically stable. Proof. For τ = 0, the characteristic Eq. (5.28) is given by λ3 + a2 λ2 + (a1 + b1 )λ + a0 + b0 = 0. It is obvious that each coefficient is positive. Moreover, we have ˜ + γ ) + μb1 − (a0 + b0 ) = 0. a2 (a1 + b1 ) − (a0 + b0 ) > k β(μ Hence, the assertion follows from the Routh-Hurwitz criterion. This completes the proof. To study the existence of the purely imaginary roots λ = ±iωc (ωc > 0), we substitute λ = iωc into the characteristic Eq. (5.28). We then have −iωc3 − a2 ωc2 + ia1 ωc + a0 + (ib1 ωc + b0 )e−iωc τ = 0.

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FIGURE 5.6 Example of cases (i) and (ii) of Lemma 5.4.2. Note that h (xc ) > 0 holds in both cases.

By considering the real and imaginary parts of this equation, we have a2 ωc2 − a0 = b1 ωc sin(ωc τ ) + b0 cos(ωc τ ), ωc3 − a1 ωc = b1 ωc cos(ωc τ ) − b0 sin(ωc τ ).

(5.29)

By adding the squares of each equation in (5.29), we obtain ωc6 + (a22 − 2a1 )ωc4 + (a12 − 2a2 a0 − b12 )ωc2 + a02 − b02 = 0. Hence, the existence of the purely imaginary roots ±iωc is equivalent to the existence of the positive root x = xc = ωc2 of the following cubic equation: h(x) := x 3 + p2 x 2 + p1 x + p0 = 0, where p2 = a22 − 2a1 ,

p1 = a12 − 2a2 a0 − b12 ,

p0 = a02 − b02 .

By an elementary observation on the cubic equation, we immediately obtain the following lemma. Lemma 5.4.2. Suppose that f is given by (5.2) and R0 > 1. The characteristic Eq. (5.28) has a conjugate pair of purely imaginary roots λ = ±iωc (ωc > 0) if either one of the following conditions holds: (i) p0 < 0. √ (ii) p0 ≥ 0,  := p22 − 3p1 > 0, x0 := (−p2 + )/3 > 0, and h(x0 ) < 0. A typical example of cases (i) and (ii) of Lemma 5.4.2 is shown in Fig. 5.6. If the assump-

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tions in Lemma 5.4.2 hold, then we can calculate τc such that λ(τc ) = ±iω(τc ) as follows: ⎧ arccos χ 1 ⎪ , χ2 > 0, ⎪ ⎨ ωc τc = 2π − arccos χ1 ⎪ ⎪ ⎩ , χ2 ≤ 0, ωc where χ1 :=

b1 ωc4 + (a2 b0 − a1 b1 )ωc2 − a0 b0 (a2 b1 − b0 )ωc3 + (a1 b0 − a0 b1 )ωc , χ := . 2 b12 ωc2 + b02 b12 ωc2 + b02

We next check the transversality condition at τ = τc , that is, Re λ (τc ) = 0. By differentiating (5.28) for τ , we have   3λ2 + 2a2 λ + a1 + b1 e−λτ − τ (b1 λ + b0 )e−λτ λ − λ(b1 λ + b0 )e−λτ = 0. We then have



 3λ2 + 2a2 λ + a1 τ  b1 − + λ − 1 = 0. λ(b1 λ + b0 )e−λτ λ(b1 λ + b0 ) λ

Note that it follows from (5.28) that (b1 λ + b0 )e−λτ = −(λ3 + a2 λ2 + a1 λ + a0 ). We then have   3λ2 + 2a2 λ + a1 1 1 b1 =− − +τ . λ λ λ3 + a2 λ2 + a1 λ + a0 b1 λ + b0 For τ = τc , we have   −3ωc2 + i2a2 ωc + a1 1 b1 1 =− − + τc λ (τc ) iωc −iωc3 − a2 ωc2 + ia1 ωc + a0 ib1 ωc + b0 "  #  −3ωc2 + a1 + i2a2 ωc −a2 ωc2 + a0 + i(ωc3 − a1 ωc ) b1 b0 − ib12 ωc i = − 2 + τc . ωc (−ωc3 + a1 ωc )2 + (−a2 ωc2 + a0 )2 b1 ωc2 + b02 We then have     b12 ωc 1 (3ωc2 − a1 )(ωc3 − a1 ωc ) + 2a2 ωc (a2 ωc2 − a0 ) 1 = − 2 Re  λ (τc ) ωc ωc6 + (a22 − 2a1 )ωc4 + (a12 − 2a2 a0 )ωc2 + a02 b1 ωc2 + b02 =

3ωc4 + 2(a22 − 2a1 )ωc2 + a22 − 2a2 a0 − b12 b12 ωc2 + b02

=

h (ωc2 ) . b12 ωc2 + b02

% $ % $ Since sgn Re λ = sgn Re (1/λ ) , Re λ (τc ) = 0 is equivalent to h (ωc2 ) = 0. We then obtain the following theorem. Theorem 5.4.3. Suppose that f is given by (5.2) and R0 > 1. If either (i) or (ii) in Lemma 5.4.2 holds, then there exists τc > 0 such that the positive endemic equilibrium

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FIGURE 5.7 Function h(x) for parameters (5.30).

E ∗ : (S ∗ , I ∗ )T of system (5.3) is locally asymptotically stable for τ ∈ [0, τc ), and it is destabilized through the Hopf bifurcation at τ = τc . Proof. It is obvious from the form of the square function h that if either (i) or (ii) in Lemma 5.4.2 holds, then there exists a positive root xc > 0 such that h (xc ) > 0 (see also Fig. 5.6). Since xc corresponds to ωc2 , we have h (ωc2 ) > 0, and thus Re λ (τc ) > 0. By Lemma 5.4.1, E ∗ is locally asymptotically stable for τ = 0, and thus, by the continuity, all eigenvalues lie in the complex left half plane for τ ∈ [0, τc ), and only the purely imaginary roots ±iωc cross the imaginary axis from left to right at τ = τc . This completes the proof.

5.5 Numerical simulation In this section, we check the validity of our theoretical results by numerical simulation. We use MATLAB R2019b for each simulation. To compute the numerical solution of system (5.3), we use the standard Euler method (see Appendix). As in the previous section, we consider the truncated exponential distribution (5.2) and fix the following values: b = μ = 0.01,

γ = 1,

R0 = 2.5,

β = R0

μ(μ + γ ) , b

k = 1,

α = 10.

(5.30)

Note that these values are chosen for technical reasons. In this case, we obtain p0 ≈ 1.897 × 10−4 > 0, x0 ≈ 0.015 > 0,

 ≈ 1.036 > 0,

h(x0 ) ≈ −3.5 × 10−5 < 0,

and thus condition (ii) in Lemma 5.4.2 holds. As shown in Fig. 5.7, there exists positive √ xc > 0 such that h(xc ) = 0 and h (xc ) > 0. For such xc , we obtain ωc = xc ≈ 0.1443. For this

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FIGURE 5.8 Time variation of the infective population I (t) in model (5.3) for distribution (5.2), parameters (5.30), and initial condition (5.31) (τc ≈ 12.54).

ωc , we obtain χ1 ≈ −0.237 and χ2 ≈ 0.972, and hence τc = (arccos χ1 )/ωc ≈ 12.54. Moreover, we can calculate the index values in Proposition 5.3.1 as follows: μ2 ≈ 2.996 × 103 > 0,

β2 ≈ −16.33 < 0,

T2 ≈ 185.

Hence we see from Proposition 5.3.1 that the bifurcation at τ = τc is supercritical, the bifurcated periodic solution is stable, and its period near τc is approximate to (2π/ωc )(1 + T2  2 ) ≈ 46.11. For the initial condition ϕ1 (ρ) = 1 − ϕ2 (ρ),

ϕ2 (ρ) = 0.1,

ρ ∈ (−∞, 0],

(5.31)

we plot the time variation of the infective population I (t) for τ < τc and τ > τc in Fig. 5.8. From Fig. 5.8 we can observe that the endemic equilibrium (I ∗ ≈ 5.714 × 10−3 ) is destabilized at τ = τc and a stable periodic solution is bifurcated. Fig. 5.9 shows the amplitude of the bifurcated periodic solution of the infective population I versus the bifurcation parameter τ . From Fig. 5.9 we can confirm that the bifurcation at τ = τc is supercritical, as we suggested.

5.6 Discussion In this chapter, to investigate the essential mechanism of recurrent epidemic waves, we have formulated the SIR epidemic model with psychological effect and distributed time delay. We have obtained the characteristic values that determine the direction of the Hopf bifurcation and the stability and period of the bifurcated periodic solution. For the case of the truncated exponential distribution, we have shown that the endemic equilibrium can be destabilized and a periodic solution can arise through the Hopf bifurcation. By numer-

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FIGURE 5.9 Amplitude of the bifurcated periodic solution of the infective population I in model (5.3) for distribution (5.2), parameters (5.30), and initial condition (5.31) (τc ≈ 12.54).

ical simulation, we have given an example, in which a stable periodic solution bifurcates supercritically from the endemic equilibrium. In our modeling we have focused on people’s behavior change and disregarded other factors such as seasonality, waning of immunity, and mutation of virus. Although these factors are of course important to be studied, our results suggest that people’s behavior change can be the cause of recurrent epidemic waves. In particular, for the case of the truncated exponential distribution, we have shown that the endemic equilibrium is stable, and thus there is no nontrivial periodic solution for R0 > 1 and α = τ = 0. This implies that the sensitivity of people’s behavior change and the incubation period of the disease are essential in the occurrence of recurrent epidemic waves. In this study, we have not discussed the validity of the truncated exponential distribution. The analysis for other distributions, such as the uniform and Gaussian distributions, has been left as future work.

Acknowledgments The author would like to thank the editors and anonymous reviewers for their helpful and constructive comments to the previous version of the manuscript. This work was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI [grant number 19K14594] and the Japan Agency for Medical Research and Development (AMED) [grant number JP20fk0108535].

Appendix 5.A Matlab code for Fig. 5.8(b) 1. 2. 3. 4.

b=0.01; mu=b; gam=1; R0=2.5;

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5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28.

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bet=R0*mu*(mu+gam)/b; k=1; alp=10; tau=13.5; dt=0.01;te=2000;nt=te/dt; Te=50;nT=Te/dt; ntau=tau/dt; for t=1:1:nT I(t)=0.1; S(t)=1-I(t); if t < ntau f(t)=0; else f(t)=k*exp(k*tau)*exp(-k*t*dt); end end for t=nT:1:nt for u=1:1:nT intf(u)=f(u)*I(t-u+1); end Intf=sum(intf )*dt; S(t+1)=S(t)+dt*(b-bet*S(t)*I(t)/(1+alp*Intf )-mu*S(t)); I(t+1)=I(t)+dt*(bet*S(t)*I(t)/(1+alp*Intf )-(mu+gam)*I(t)); end

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[10] B.D. Hassard, N.D. Kazarinoff, Y-H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [11] K. Hattaf, Global stability and Hopf bifurcation of a generalized viral infection model with multidelays and humoral immunity, Physica A 545 (2020) 123689. [12] K. Hattaf, A.A. Lashari, Y. Louartassi, N. Yousfi, A delayed SIR epidemic model with general incidence rate, Electron. J. Qual. Theory Differ. Equ. 3 (2013) 1–9. [13] H.W. Hethcote, S.A. Levin, Periodicity in epidemiological models, in: S.A. Levin, T.G. Hallam, L.J. Gross (Eds.), Applied Mathematical Ecology, Springer, 1989, pp. 193–211. [14] H. Inaba, Age-Structured Population Dynamics in Demography and Epidemiology, Springer, 2017. [15] Z. Jiang, J. Wei, Stability and bifurcation analysis in a delayed SIR model, Chaos Solitons Fractals 35 (2008) 609–619. [16] W.O. Kermack, A.G. McKendrick, Contributions to the mathematical theory of epidemics – I, Bull. Math. Biol. 53 (1991) 33–55 (reprinted from the Proc. Roy. Soc. 115A (1927) 700–721). [17] A. Kumar Nilam, Mathematical analysis of a delayed epidemic model with nonlinear incidence and treatment rates, J. Eng. Math. 115 (2019) 1–20. [18] G. Li, W. Wang, Bifurcation analysis of an epidemic model with nonlinear incidence, Appl. Math. Comput. 214 (2009) 411–423. [19] W.M. Liu, H.W. Hethcote, S.A. Levin, Dynamical behaviour of epidemiological models with nonlinear incidence rates, J. Math. Biol. 25 (1987) 359–380. [20] W.M. Liu, S.A. Levin, Y. Iwasa, Influence of nonlinear incidence rates upon the behaviour of SIRS epidemiological models, J. Math. Biol. 23 (1986) 187–204. [21] C.C. McCluskey, Complete global stability for an SIR epidemic model with delay – distributed or discrete, Nonlinear Anal., Real World Appl. 11 (2010) 55–59. [22] Prime Minister of Japan and His Cabinet, Ongoing topics, Available from https://japan.kantei.go.jp/ ongoingtopics/index.html. (Accessed 1 September 2021). [23] S. Ruan, W. Wang, Dynamical behaviour of an epidemic model with a nonlinear incidence rate, J. Differ. Equ. 188 (2003) 135–163. [24] A. Sirijampa, S. Chinviriyasit, W. Chinviriyasit, Hopf bifurcation analysis of a delayed SEIR epidemic model with infectious force in latent and infected period, Adv. Differ. Equ. 2018 (2018) 348. [25] Y. Tang, W. Li, Global dynamics of an epidemic model with an unspecified degree, Comput. Math. Appl. 53 (2007) 1704–1717. [26] W. Wang, L. Chen, Stability and Hopf bifurcation analysis of an epidemic model by using the method of multiple scales, Math. Probl. Eng. 2016 (2016) 2034136. [27] L. Wang, X. Wu, Stability and Hopf bifurcation for an SEIR epidemic model with delay, Adv. Theory Nonlinear Anal. Appl. 2 (2018) 113–127. [28] World Health Organization, WHO coronavirus disease (Covid-19) dashbord, Available from https:// covid19.who.int/. (Accessed 1 September 2021). [29] F.F. Zhang, Z. Jin, G.Q. Sun, Bifurcation analysis of a delayed epidemic model, Appl. Math. Comput. 216 (2010) 753–767. [30] H. Zhao, M. Zhao, Global Hopf bifurcation analysis of an susceptible-infective-removed epidemic model incorporating media coverage with time delay, J. Biol. Syst. 11 (2017) 8–24.

6 Modeling of the effects of media in the course of vaccination of rotavirus Amar Nath Chatterjeea and Fahad Al Basirb a Department of Mathematics, K.L.S. College, Nawada, Magadh University, Nawada, India b Department of

Mathematics, Asansol Girls’ College, Asansol, West Bengal, India

6.1 Introduction Infections caused by rotavirus are widespread for children and infants less than five years old. They cause severe gastroenteritis infection [1]. An older person caring for younger children has an increased risk of infection as well. Rotavirus infection leads to dehydration, which is very life-threatening for babies and young children [2]. The World Health Organization (WHO) estimates that 215,000 children die each year from rotavirus infection worldwide [3,4]. This virus spreads quickly through hands and mouth contact, close person-to-person contact, and contaminated environment-to-person [5,6]. The incubation period of rotavirus disease is nearly two days. A contaminated environment is the main cause for the spread of endemic infections. Due to nonhygienic sanitary conditions, the infection rate is very high among infants. Rotavirus infections are unkind. Nonpharmaceutical interventions, such as washing hands and proper sanitization, are significant to control the disease. However, vaccination is an important way to protect children against rotavirus disease. Two main rotavirus vaccines are currently recognized: Rota Teq (RV5) and Rotarix (RV1). Epidemic modeling plays a pivotal role in studying the dynamics of diseases. We can predict the infection process and the expected outbreak through epidemic modeling. Many researchers have done the dynamical model of rotavirus through mathematical modeling, for example, [5,7–12] among others. Katherine et al. [13] verified the effect of the rota vaccine in the human population. Shim et al. [8] explored the role of breastfeeding on rotavirus control incorporating the seasonal effect. Kribs et al. [14] presented both deterministic and stochastic models for the transmission of rotavirus and compared the efficacy for possible control measures to reduce the infection. Roldaoa et al. [19] studied the production of rotavirus virus-like particles (VLP). In controlling a new infection, media plays a major role [15,16]. However, the specific way of media impact epidemics is yet to be understood. Mathematical modeling can play a Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00011-9 Copyright © 2023 Elsevier Inc. All rights reserved.

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vital role in understanding the potential role of media impact in the case of vaccination and controlling the disease transmission. Several mathematical modeling articles talk about this problem [17]. Despite the success of media awareness of vaccination, long-term control is very much problematic. In this aspect, optimal control theory plays an essential role in investigating and controlling the disease. There are different types of control approaches adopted by different groups of mathematicians. Generally, the researchers use the control functions to study the vaccination, awareness for reducing infections, treatment, etc., for humans [27]. Also, some mathematicians studied the control-theoretic approach of drug dose to control the disease in human hosts caused by different pathogens [28–31]. Here our main goal for using the optimal control functions is to minimize new infections in the host population and the cost for the execution of all the control. From the literature review it is clear that researchers have given less attention to study both new infection and reinfection after vaccination using a mathematical model. In this chapter, we formulate a new infection rotavirus vaccinated model with two control inputs which acts to reduce the new infection (before vaccination) and infection after vaccination. Here we deal with the SVI (Susceptible-Vaccinated-Infected) model’s qualitative and optimal control analysis because of the effort as the time-dependent function. Initially, we study the effective role of media awareness regarding the rotavirus and the importance of vaccination analytically and show the effectiveness as it can play a significant role towards controlling the disease progression. Thereafter we derive the optimal control input on the basis of cost-effectiveness. β0 In this chapter, we propose an incidence rate β(I, p)I , where β(I, p) = 1+pI 2 , β0 is mea1 sured as infection force of the disease, and 1+pI 2 represents the inhibitory effect due to media impact. This is very important because it has been proved that media awareness effectively reduces the infection rate even when the number of infections is very large [18]. Here the function β(I, p)I is nonmonotone. That is, the function is increasing when the infection level is small and again decreasing when the infected population is huge (see Fig. 6.1). It can be interpreted that due to the media effect among a large number of infective individuals, the infection force may decrease as the number of infected individuals increases. Notice that when p = 0, the nonmonotone incidence rate β(I, p) becomes the bilinear incidence rate β0 . Continuous mathematical modeling for rotavirus diarrhea is proposed, analyzed, and simulated in this chapter. The model with media-induced vaccination is proposed in Section 6.2. Various dynamics of the system, for example, the existence of equilibria, stability analysis, are studied analytically in Section 6.3 and Section 6.4, respectively. The optimal control and efficacy analysis are studied in Sections 6.5 and 6.6. Numerical results of this model are discussed in Section 6.7. The implication of the results is discussed in Section 6.8 with final conclusion to end the chapter.

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FIGURE 6.1 Nonmonotone incidence function β(I, p), where p = 0.5.

6.2 Epidemic modeling In this rotavirus infection model, four populations are considered; namely, the susceptible population S(t), the vaccinated population V (t), the infected population I (t), and the recovered population R(t). The model we propose is as follows: dS dt dV dt dI dt dR dt

= (1 − ρ) − β(I, p)SI − (ν + μ)S, = ρ + νS − β(I, p)V I − μV , = β(I, p)SI + β(I, p)V I − (τ + κ + μ)I,

(6.1)

= κI − μR.

We assume that the population mix is homogeneous to formulate the system of ordinary differential equations. In model (6.1),  is the birth rate of S(t) by an adult and β0 is the 1 rate of contact. Here we introduce the media impact 1+pI 2 where p denotes the strength of media effect. Ic is the threshold value that decides whether media awareness is needed or not. Here we also introduce ϕ(I ) = I − Ic to show the media factor. ρ is the recruitment rate of vaccination individual. Due to vaccination the contact rate is reduced at a rate , and ν is the vaccine coverage rate. Here μ is the natural death rate and κ is the recovery rate. τ is the death rate due to rotavirus. β0 Here β(I, p) = 1+pI 2 , p represents the media effects with  p=

0, ϕ(I ) ≤ 0; 1, ϕ(I ) > 0.

(6.2)

When p = 0, the transmission rate β = β0 is constant, and when p = 1, the transmission β0 rate is β = 1+pI 2.

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The phase space splits into two parts P1 = {(S, V , I, R) ∈ R4+ ; ϕ ≤ 0},

(6.3)

P2 = {(S, V , I, R) ∈ R4+ ; ϕ

(6.4)

> 0}.

In P1 region there is no media effect and transmission rate in β = β0 , while in region P2 β0 the transmission rate reduces to 1+pI 2. Let the vector X = (S, V , I, R)T and GP1 (X) = [(1 − ρ) − β0 SI − (ν + μ)S, ρ + νS − β0 V I − μV , β0 SI, + β0 V I − (τ + κ + μ)I, κI − μR]T β0 β0 GP2 (X) = [(1 − ρ) − SI − (ν + μ)S, ρ + νS −  V I − μV , 1 + pI 2 1 + pI 2 β0 β0 SI +  V I − (τ + κ + μ)I, κI − μR]T . 2 1 + pI 1 + pI 2 The above equations can be rewritten as the following switching system: X =



GP1 (X ), GP2 (X ),

X ∈ P1 ; X ∈ P2 .

(6.5)

Here we assume that the region P1 is the subsystem ϒ1 and the region P2 is the subsystem ϒ2 , and these two subsystems are in isolation first.

6.3 Existence of equilibria of system ϒ1 The subsystem ϒ1 has one disease-free equilibrium E¯ 1 = (S¯1 , V¯1 , 0, 0), where (1 − ρ) ¯ (ρμ + ν) , V1 = . S¯1 = ν +μ μ(ν + μ) We also study the stability of the system in this section. The basic reproduction number R0 of the system is R0 =

β0 {μ(1 − ρ) + (ρμ + ν)} , μ(μ + ν)(τ + κ + μ)

(6.6)

and we can verify when R0 > 1, the subsystem ϒ1 has a unique endemic equilibrium E1∗ = (S1∗ , V1∗ , I1∗ , R1∗ ), where (1 − ρ) , β0 I1∗ + (μ + ν) {ρ(β0 I1∗ + μ) + ν} V1∗ = , (β0 I1∗ + μ + ν)(β0 I1∗ + μ) S1∗ =

Chapter 6 • Modeling of the effects of Rota vaccination

R1∗ =

173

κI1∗ , μ

(6.7)

a0 I1∗ 2 + a1 I1∗ + a2 = 0,

(6.8)

and I1∗ is derived from

where a0 = β02 (τ + κ + μ), a1 = (τ + κ + μ){(μ + ν) + μ} − β02 , a2 = μ(τ + κ + μ)(μ + ν)(1 − R0 ).

(6.9)

Here a0 is positive always; thus the following theorem is derived for the existence of E ∗ . Theorem 6.3.1. Let us define  = a12 − 4a0 a2

(6.10)

as the discriminant of Eq. (6.8). Then there will be a unique feasible equilibrium if  > 0 and a2 < 0. On the other hand, if a2 > 0, a1 < 0, and  > 0, then there exist exactly two distinct feasible endemic equilibria.

6.3.1 Equilibria of system ϒ2 The disease-free equilibria of the subsystems ϒ1 and ϒ2 are the same. Thus we study the existence of the endemic equilibrium of ϒ2 . Equating GP2 (X ) with zero, the endemic equilibrium must satisfy β0 SI − (ν + μ)S = 0, 1 + pI 2 β0 ρ + νS −  V I − μV = 0, 1 + pI 2 β0 β0 SI +  V I − (τ + κ + μ)I = 0, 1 + pI 2 1 + pI 2 (1 − ρ) −

κI − μR = 0.

(6.11)

For the positive equilibrium point, we have (1 − ρ)(1 + pI 2 ) , β0 I + (ν + μ)(1 + pI 2 ) (ρ + νS)(1 + pI 2 ) V= , β0 I + μ(1 + pI 2 ) κI R= , μ S=

(6.12)

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and I can be derived from the equation a0 I 4 + a1 I 3 + a2 I 2 + a3 I + a4 = 0,

(6.13)

where a0 = μ(μ + ν)(μ + κ + τ ) > 0, a1 = β0 (μ + κ + τ ){μ + (μ + ν)}p > 0, a2 = β0 (μ + κ + τ ) + pμ(μ + ν)(μ + κ + τ )(2 − R0 ), a3 = β0 [(μ + κ + τ ){μ + (ν + μ)} − ],

(6.14)

a4 = μ(μ + ν)(μ + κ + τ )(1 − R0 ) < 0. Now a2 > 0 if R0 < 2. Also, if R0 > 2 and the impact of media satisfies the relation 0 < β0 , then we have a2 > 0. p < p  where p = p  = μ(μ+ν)(R 0 −2)

Again, a3 > 0 if  < c , where c = (μ+κ+τ ){μ+(μ+ν)} .   Hence, if p ≤ p or R0 < 2, Eq. (6.13) has a unique endemic equilibrium. But if p > p  of R0 > 2, then Eq. (6.13) has more than one endemic equilibrium point. Thus we have the following theorem. Theorem 6.3.2. When 1 < R0 < 2 and 0 < p ≤ p  , the subsystem ϒ2 has a unique equilibrium E ∗ = (S ∗ , V ∗ , I ∗ , R ∗ ).

6.4 Stability of the equilibria In this section we verify the local and global stability of the equilibria of the subsystems ϒ1 and ϒ2 . Theorem 6.4.1. The disease-free equilibrium E¯ 1 is locally asymptotically stable if R0 < 1 and unstable when R0 > 1. Proof. The characteristic equation at E¯ 1 for the subsystem ϒ1 is of the form (ξ + μ + ν)(ξ + μ)[ξ + β0 (S¯ +  V¯ ) − (τ + κ + μ)] = 0,

(6.15)

where ξ is the eigenvalue. We get the eigenvalues as follows: ξ1 = −(μ + ν), ξ2 = −μ, ξ3 = −β0 (S¯ +  V¯ ) + (τ + κ + μ).

(6.16)

Thus, if R0 < 1, all eigenvalues are negative. Hence the subsystem ϒ1 at E¯ 1 attains its local asymptotic stability and is unstable if R0 > 1. The proofs of the disease-free equilibria of the subsystems ϒ1 and ϒ2 are same.

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Theorem 6.4.2. If R0 < 1, then the system E¯ 1 is globally asymptotically stable. Proof. To prove the global stability of the subsystem ϒ1 at E¯ 1 , we assume Lyapunov’s function L(t) = I (t).

(6.17)

Now differentiating L with respect to t, we get dL β0 β0 = SI +  V I − (τ + κ + μ)I 2 dt 1 + pI 1 + pI 2 I = [R0 − 1] . (τ + κ + μ)(1 + pI 2 )

(6.18)

If R0 < 1, then dL dt < 0. According to LaSalle’s invariance principle [20], the subsystem ϒ1 at E¯ 1 is globally asymptotically stable in . We can similarly prove the global asymptotic stability for the subsystem ϒ2 at the disease-free state. Now we consider the stability of the endemic equilibrium E1 ∗ for the subsystem ϒ1 and for the endemic equilibrium E ∗ of the subsystem ϒ2 . Using the Routh-Hurwitz stability criterion, we can easily get the theorem. Theorem 6.4.3. For the subsystem ϒ1 , the endemic equilibrium E1∗ is locally asymptotically stable if R0 > 1. Also, the endemic equilibrium E ∗ for the subsystem ϒ2 is locally asymptotically stable if R0 > 1. Now, the endemic equilibrium E1 ∗ for the subsystem ϒ1 , we get Rc = 1 + aIc2 (a0 Ic + a1 ) for which 1 < R0 < Rc , which leads to the existence of E1∗ that belongs to I1∗ . Here Rc is the point at which two positive endemic equilibria collide. If R0 > Rc , the endemic equilibrium E1∗ belonging to ϒ1 leads to the endemic equilibrium E ∗ of ϒ2 . In the case of occurrence of two positive endemic equilibria when R0 < 1, the system attains a backward bifurcation. Then we have the following results. Theorem 6.4.4. The endemic equilibrium E1∗ is globally asymptotically stable when 1 < R0 < Rc , and when R0 > Rc , the endemic equilibrium E ∗ is globally asymptotically stable. Proof. Let us assume the Dulac function as follows: D(S, V , I, R) =

1 . SV I R

We assume g1 = (1 − ρ) − β(I, p)SI − (ν + μ)S, g2 = ρ + νS − 1 β(I, p)V I − μV , g3 = β(I, p)SI + 1 β(I, p)V I − (τ + κ + μ)I,

(6.19)

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g4 = κI − μR,

(6.20)

for the subsystem ϒ2 . Then we can write  ∂ 1  g1 β0 I + (ν + μ) , (Dg1 ) = − + ∂S SV I R S 1 + pI 2  ∂ β0 1  g2 (Dg2 ) = − + I + μ , ∂V SV I R V 1 + pI 2 ∂ β0 1  g3 β0 (Dg3 ) = − + (pI 2 − 1)S +  (pI 2 − 1)V ∂I SV I R I 1 + pI 2 1 + pI 2  + (τ + κ + μ) , ∂ 1 g4 (Dg4 ) = − . ∂R SV I R R

(6.21)

Therefore we have ∂ ∂ ∂ ∂ (Dg1 ) + (Dg2 ) + (Dg3 ) + (Dg4 ) < 0. ∂S ∂V ∂I ∂I

(6.22)

Hence, for the subsystem ϒ1 , we can exclude the existence of limit cycles. Also, the same method can be applicable for the subsystem ϒ2 . The endemic equilibrium E1∗ belongs to the subsystem ϒ1 and all model trajectories converge to E1∗ when 1 < R0 < Rc . Also, if the curves start from ϒ2 and converge to E ∗ and satisfy the condition 1 < R0 < Rc , then the system must belong to ϒ1 . This results in that the system has a unique equilibrium and there exists no limit cycle in the subsystem ϒ1 , and the system starting from ϒ2 will move to the subsystem ϒ1 and converge to E1∗ . Thus the endemic equilibrium E1∗ is globally asymptomatically stable when 1 < R0 < Rc . It is also observed that when R0 > Rc , the endemic equilibrium E ∗ belongs to the subsystem ϒ2 . Hence curves starting from ϒ1 move to the subsystem ϒ2 and the system converges to the endemic equilibrium E ∗ in the subsystem ϒ2 . Thus the endemic equilibrium E ∗ is globally asymptotically stable when R0 > Rc .

6.5 Optimal control problem In this section our main aim is to minimize the cost, to minimize the infected population, and to maximize the vaccinated and susceptible populations. Here the control functions are u1 (t) and u2 (t) with values normalized to be between 0 and 1, where ui (t) = 1, i = 1, 2, represents totally effective therapy and ui (t) = 0, i = 1, 2, represents no treatment. We choose our control class as follows: Model (6.1) is modified with time-dependent control u1 (t) and u2 (t) as the control which reduces the spread of rotavirus. Model (6.1) then becomes: dS = (1 − ρ) − (1 − u1 (t))β(I, p)SI − (ν + μ)S, dt

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FIGURE 6.2 I ∗ is plotted versus R0 , and the disease-free state and the endemic state are identified. Parameter values are as in Table 6.2 except β0 ∈ (0.001, 0.1).

dV = ρ + νS − (1 − u2 (t))β(I, p)V I − μV , dt dI = (1 − u1 (t))β(I, p)SI + (1 − u2 (t))β(I, p)V I − (τ + κ + μ)I, dt dR = κI − μR. dt

(6.23)

Here the controls u1 (t), u2 (t) correspond to finding the new cases and the coefficients (1 − u1 (t)), (1 − u2 (t)) represent the efficacy to prevent the failure of case finding and treatment through vaccination. The determination to find the cases of rotavirus infection is integrated by adding a control term. This control enables us to reduce new cases from latent individuals. The new case tracking effort is incorporated by adding a control term that may lower the treatment failure rate. Also, the second control reduces the new cases from the vaccinated population and also reduces the reinfection process. The objective function J (u(t)) for the control problem is defined as  tf   1 J (u(t)) = A1 I (t) + A2 u21 (t) + A3 u22 (t) dt. (6.24) 2 ti J (u(t)), objective functional, includes the terms I (the infected population) which should be minimized. Here ti is the initial time to start the control and tf is the final time and A1 , A2 , A3 are the balancing cost factor. We assume u = (u1 , u2 ). The costs of funds required for control implementation depend on the case tracking method and controlling reinfection after vaccination. Hence we seek to find an optimal control (u∗1 (t), u∗2 (t)) such that J (u∗1 (t), u∗2 (t)) = min J (u1 (t), u2 (t)), u∈U

(6.25)

where U = {(u1 , u2 ) : [ti , tf ] → [a, b], (u1 , u2 ) is Lebesgue measurable} and a, b are positive constants.

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Here U is the control set. We know that the fundamental background of an optimal control problem is the existence condition along with features of the optimal control problem. To prove the basic framework of an optimal control, we have to prove the existence of the optimal control and characterize the optimal control with the help of an optimality system [26]. For the optimal control model (6.23), we prove the existence of an optimal control problem using the approach by Fleming and Rishel [23] and then characterizing it for optimality.

6.5.1 Existence of optimal control To prove the existence condition of the optimal control problem (6.23), we use the results of Fleiming and Rishel [23] and Lukes [22]. We have to prove the following conditions: i. ii. iii. iv. v.

The control and state variables related to the control problem are nonempty. The control set U is closed and convex. The state system is bounded by a linear function in the state and control variables. The integrand of the objective functional L(S, V , I, R) is convex on U. The ingrained of the objective function is bounded below by k1 [(u1 (t))2 + (u2 (t))2 ]σ/2 k2 , where k1 > 0, k2 > 0, and σ > 1.

Here the state variables for model (6.23) are S, V , I and the control variables are u1 (t), u2 (t). Proof of Theorem 6.5.1 is as follows. Theorem 6.5.1. The optimal control variables u∗ (t) = (u∗1 (t), u∗2 (t)) ∈ U such that J (u∗1 (t), u∗2 (t)) = min{(u1 (t), u2 (t))|uj (t) ∈ U for j = 1, 2}.

(6.26)

Proof. We can prove that the control variables are nonempty by using the result of [24] (see Theorem 7.1.1). Let S˙ = ψ1 (t, S, V , I, R), V˙ = ψ2 (t, S, V , I, R), I˙ = ψ3 (t, S, V , I, R), R˙ = ψ4 (t, S, V , I, R), where ψ1 , ψ2 , ψ3 , ψ4 are the right-hand side of system (6.23). Also, let uj (t) = Mi f or i = 1, 2 be some constants, and since all parameters are constants, S, V , I are all continuous. Then 1 ∂ψ1 ∂ψ1 ∂ψ1 ∂ψ2 ∂ψ2 ∂ψ2 ∂ψ2 ψ1 , ψ2 , ψ3 are also continuous. The partial derivatives ∂ψ ∂S , ∂V , ∂I , ∂R , ∂S , ∂V , ∂I , ∂R , ∂ψ3 ∂ψ3 ∂ψ3 ∂ψ3 ∂ψ4 ∂ψ4 ∂ψ4 ∂ψ4 ∂S , ∂V , ∂I , ∂R , ∂S , ∂V , ∂I , ∂R are all continuous. Hence the control solution is unique and the corresponding state variables are nonempty. The controls u1 , u2 and state variables S, V , I , R of system (6.24) are nonnegative values. Hence the necessary convexity of our objective functional stated in terms of u1 , u2 is satisfied. The set of admissible Lebesgue measurable control variables (u1 (t), u2 (t)) ∈ U is also convex and closed by the definition. To verify this argument, we rewrite system (6.23) in the form [25] X  = AX + F(X ), where X = [S, V , I, R]T ,

Chapter 6 • Modeling of the effects of Rota vaccination ⎛ ⎜ F(X ) = ⎜ ⎝ ⎛ and

⎜ A=⎜ ⎝

179

⎞ (1 − ρ) − (1 − u1 (t))β(I, p)SI ⎟ ρ − (1 − u2 (t))β(I, p)V I ⎟ (1 − u1 (t))β(I, p)SI + (1 − u2 (t))β(I, p)V I ⎠ κI ⎞ −(ν + μ) 0 0 0 ν −μ 0 0 ⎟ ⎟. 0 0 −(τ + κ + μ) 0 ⎠ 0

0

−μ

0

X  represents the derivative of X with respect to time t. Since system (6.23) is nonlinear with bounded coefficient, we set (X ) = AX + F(X ).

(6.27)

F(X ) satisfies |F(X1 ) − F(X2 )| ≤ C1 (|S1 (t) − S2 (t)|) + C2 (|V1 (t) − V2 (t)|) + C3 (|I1 (t) − I2 (t)|) + C4 (|R1 (t) − R2 (t)|) ≤ C((|S1 (t) − S2 (t)|) + (|V1 (t) − V2 (t)|) + (|I1 (t) − I2 (t)|) + (|R1 (t) − R2 (t)|)),

(6.28)

where the positive constant C = max{Cr , f or r = 1, 2, 3, 4} is independent of the state vari ables. Also, we have (X1 ) − (X2 ) ≤ C|X1 − X2 |, where C = 4r=1 Cr + H < ∞. Thus the function  satisfies the uniform Lipschitz continuity. Hence, according to the definition of control variables and nonnegative initial conditions, we can say that a solution of system (6.23) exists [25]. The integrand in the objective functional (6.24) is as follows: L(t, x, u) = A1 I +

 1 A2 u21 (t) + A3 u22 (t) , 2

(6.29)

which is convex in the control set U. We have to verify the condition that there exist a constant σ > 1 and positive numbers k1 , k2 > 0 such that  1 L(t, x, u) = A1 I + A2 u21 (t) + A3 u22 (t) 2  1 2 ≥ A1 u1 (t) + A2 u22 (t) 2   σ/2 1 − k2 . ⇒ A1 u21 (t) + A2 u22 (t) ≥ k1 u21 (t) + u22 (t) 2 Let k1 = inf{ A22 , A23 } and k2 = 2 supt∈[ti ,tf ] (I ) and σ = 2, then it follows that σ

L(I, u1 , u2 ) ≥ k1 (|u1 |2 + |u2 |2 ) 2 − k2 . Hence we can conclude that the optimal control pair exists.

(6.30)

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6.5.2 Characterization of optimal control Optimal control u∗1 (t) and u∗2 (t) is determined by the help of Pontryagin’s maximum principle (PMP) [26]. With the help of PMP, we convert systems (6.23) and (6.24) into a problem of minimizing pointwise Hamiltonian function H with respect to (u1 (t), u2 (t)). To construct the Hamiltonian, we use the adjoint variables along with the state variables and combining the results with the objective functional. The Hamiltonian is defined as H = A1 I +

4   1 A2 u21 (t) + A3 u22 (t) + ξ r Rr . 2

(6.31)

r=1

Here ξ1 , ξ2 , ξ3 , ξ4 are the adjoint variables associated with the state equations in (6.23). Ri , i = 1, 2, 3, 4, is the right-hand side of the differential equations of the ith state variable in system (6.23). The Hamiltonian function for (6.23) is given below:  1 A2 u21 (t) + A3 u22 (t) 2 + ξ1 {(1 − ρ) − (1 − u1 (t))β(I, p)SI − (ν + μ)S}

H = A1 I +

+ ξ2 {ρ + νS − (1 − u2 (t))β(I, p)V I − μV } + ξ3 {(1 − u1 (t))β(I, p)SI + (1 − u2 (t))β(I, p)V I − (τ + κ + μ)I } + ξ4 {κI − μR}.

(6.32)

We get the optimality equations by taking the partial derivative of the Hamiltonian i function H with respect to the control variables (u1 (t), u2 (t)), respectively, and dξ dt can be dξi ∂H obtained from dt = − ∂x , where x represents the state variable. Theorem 6.5.2. The optimal control set is (u∗1 (t), u∗2 (t)) and its corresponding state solutions are S ∗ , V ∗ , I ∗ , R ∗ for which J (u1 (t), u2 (t)) is minimized, and therefore there exist adjoint functions ξ1 , ξ2 , ξ3 , ξ4 such that dξ1 = (ξ1 − ξ3 )(1 − u1 (t))β(I, p)I + ξ1 (ν + μ) − ξ2 ν, dt dξ2 = (ξ2 − ξ3 )(1 − u2 (t))β(I, p)I + ξ2 μ, dt dξ3 1 − pI 2 = −A1 + (ξ1 − ξ3 )(1 − u1 (t)) β(I, p) dt 1 + pI 2 1 − pI 2 + (ξ1 − ξ2 )(1 − u2 (t)) β(I, p) + ξ3 (τ + k + μ) − ξ4 κ, 1 + pI 2 dξ4 = ξ4 μ, dt

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181

with the transversality conditions ξ1 (tf ) = 0, ξ2 (tf ) = 0, ξ3 (tf ) = 0, ξ4 (tf ) = 0, and the control variables (u∗1 (t), u∗2 (t)) satisfy the following optimality conditions:   (ξ − ξ )β(I, p)SI   3 1 u∗1 (t) = min max 0, ,1 , A2   (ξ − ξ )βI, pV I   3 2 u∗2 (t) = min max 0, ,1 . A3

(6.33) (6.34)

Proof. The differential equations for the adjoints are standard results from Pontryagin’s maximum principle (PMP) [26]. Given the Hamiltonian function in (6.32), the adjoint equations can be easily computed by dξ1 ∂H dξ2 ∂H dξ3 ∂H dξ4 ∂H =− , =− , =− , =− . dt ∂S dt ∂V dt ∂I dt ∂R

(6.35)

Therefore the adjoint system evaluated at optimal controls u1 (t), u2 (t) with the corresponding model state variables S(t), V (t), I (t), R(t) is given by dξ1 = (ξ1 − ξ3 )(1 − u1 (t))β(I, p)I + ξ1 (ν + μ) − ξ2 ν, dt dξ2 = (ξ2 − ξ3 )(1 − u2 (t))β(I, p)I + ξ2 μ, dt dξ3 1 − pI 2 = −A1 + (ξ1 − ξ3 )(1 − u1 (t)) β(I, p) dt 1 + pI 2 1 − pI 2 + (ξ1 − ξ2 )(1 − u2 (t)) β(I, p) + ξ3 (τ + κ + μ) − ξ4 κ, 1 + pI 2 dξ4 = ξ4 μ. dt

(6.36) (6.37) (6.38) (6.39) (6.40)

The transversality conditions are ξ1 (tf ) = 0, ξ2 (tf ) = 0, ξ3 (tf ) = 0, ξ4 (tf ) = 0. Now Pontryagin’s maximum principle [26] states that the unconstrained optimal control u∗ (t) satisfies ∂H = 0. ∂u∗

(6.41)

Here we find ∂∂uHi , i = 1, 2, and solve for u∗1 , u∗2 by setting our partial derivative of H equal to zero. Thus we have ∂H = A2 u1 (t) + (ξ1 − ξ3 )β(I, p)SI = 0, ∂u1 (t) ∂H = A3 u2 (t) + (ξ1 − ξ2 )β(I, p)V I = 0. ∂u2 (t) Hence we obtain u1 (t) =

(ξ3 − ξ1 )β(I ∗ , p)S ∗ I ∗ , A2

(6.42)

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(ξ1 − ξ2 )β(I, p)V I . A3  ∂2H  ∂u1 u2   = A2 A3 > 0. ∂2H   2 ∂u

u2 (t) =    2 Clearly, ∂ H2 = A2 > 0,  ∂u1 

∂2H ∂u21 ∂2H ∂u2 u1

2

Therefore the optimal problem is minimum at controls u1 , u2 , where (ξ3 − ξ1 )β(I, p)SI , A2 u∗2 (t) = A3 u2 (t) + (ξ1 − ξ2 )β(I, p)V I. u∗1 (t) =

(6.43) (6.44)

According to the boundedness of the standard control, we conclude for the control u1 : u∗1 (t) =

⎧ ⎪ ⎨ 0, ⎪ ⎩

(ξ3 −ξ1 )β(I,p)SI , A2

1,

(ξ3 −ξ1 )β(I,p)SI ≤ 0; A2 (ξ3 −ξ1 )β(I,p)SI 0< < 1; A2 (ξ3 −ξ1 )β(I,p)SI ≥ 1. A2

(6.45)

Hence the compact form of u∗1 (t) is   (ξ − ξ )β(I, p)SI   3 1 u∗1 (t) = max min 1, ,0 . A2

(6.46)

In a similar way, we get a compact form of u∗2 (t) in the form of   (ξ − ξ )β(I, p)V I   1 2 u∗2 (t) = max min 1, ,0 . A3

(6.47)

Utilizing Eqs. (6.46), (6.47) and taking the state system along with the adjoint system and the transversality conditions, we have the following optimal system: dS dt dV dt dI dt dR dt dξ1 dt dξ2 dt dξ3 dt

= (1 − ρ) − (1 − u1 (t))β(I, p)SI − (ν + μ)S, = ρ + νS − (1 − u2 (t))β(I, p)V I − μV , = (1 − u1 (t))β(I, p)SI + (1 − u2 (t))β(I, p)V I − (τ + κ + μ)I, = κI − μR. = (ξ1 − ξ3 )(1 − u1 (t))β(I, p)I + ξ1 (ν + μ) − ξ2 ν, = (ξ2 − ξ3 )(1 − u2 (t))β(I, p)I + ξ2 μ, = −A1 + (ξ1 − ξ3 )(1 − u1 (t))

1 − pI 2 β(I, p) 1 + pI 2

Chapter 6 • Modeling of the effects of Rota vaccination

+ (ξ1 − ξ2 )(1 − u2 (t))

183

1 − pI 2 β(I, p) + ξ3 (τ + k + μ) 1 + pI 2

dξ4 = ξ4 R. dt ξi (tf ) = 0, i = 1, 2, 3, 4.

6.6 Efficacy analysis The best control strategy is considered on the basis of efficacy strategy. In this chapter we consider two control variables. Among this control u1 and u2 , we consider three strategies STG-I, STG-II, STG-III, where STG-I is the strategy where u2 = 0, STG-II is the strategy where u1 = 0, and STG-III is the strategy where u1 = 0, u2 = 0. The efficacy index E is used to find the best strategy where  c  E = 1 − S × 100%.  Here the area under the infected population is denoted as  c when the control is used, and the infected total population curve in absence of control input is denoted as  S . The cumulative number of the infected population during the time interval [0, T ] is defined by [34] 

T

=

I (t)dt. 0

The values of  c and the efficacy index for three strategies are given in Table 6.1. We can choose the best strategy with the help of the efficiency index [32,33]. The values of Ac and the efficiency index for three strategies are given in Table 6.1. From Table 6.1, we can conclude that Strategy B is effective in comparison to Strategy A. But Strategy C (which is the combination of both controls) is the best strategy among these three strategies. Table 6.1 Efficiency index for system (6.23). Strategy

  = 0T I (t)dt

c E = 1 − S × 100%

No Strategy

29.84

0%

A

8.4092

71.84%

B

8.3271

72.09%

C

8.3212

72.12 %



6.7 Numerical simulations In this section we verify our analytical findings of the two subsystems ϒ1 and ϒ2 using the set of parameters as in Table 6.2 with an aim to study the dynamics of the systems.

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Table 6.2 Variables and biologically relevant parameter values used for numerical simulations of system (6.1). Parameters

Biological meaning

Assigned value (unit) day−1



Recruitment rate of human

0.5 ∼ 1.8

ρ

Recruitment rate of vaccination individual

0.008

β0

Effective contact rate

0.01 ∼ 0.0005

ν

Vaccine converge rate

0.002

μ

Natural death rate

0.018

κ

Recovery rate

0.02

τ

Death rate of children due to virus

0.0045



Effect of vaccination

0.0005

In Fig. 6.2, we have plotted the infected population with respect to the basic reproduction number R0 . Forward bifurcation is seen at R0 and the disease-free state is stable for R0 < 1. The endemic state is present for R0 > 1, verifying Theorem 6.3.1. It is also clear from Fig. 6.3 that when R0 < 1 the system moves to the disease-free state irrespective of the value of p. Next we consider R0 > Rc by taking R0 = 8.71 and Rc = 6, 82. We also consider two sets of initial values which belong to ϒ1 (I¯ < Ic ) and ϒ2 (I¯ > Ic ). According to Fig. 6.4, the trajectories approach towards E1∗ no matter whether I¯ belongs to ϒ1 or ϒ2 . Thus the endemic equilibrium E1∗ of the subsystem ϒ1 is globally asymptotically stable when 1 < R0 < Rc . Also, if we increase the media index p from 0.08 to 10, the system switches from the endemic state to the disease-free state. According to Fig. 6.5, the trajectories approach towards E ∗ when R0 > Rc . Also, the system remains in the endemic state in spite of the increasing value of the media index p from 0.08 to 10. Thus the system E ∗ attains its global asymptotic stability when R0 > Rc .

FIGURE 6.3 Trajectories of model variables for different values of p = 0.08, 1, 10 when R0 < 1 and Rc > R0 . The model parameters are as in Table 6.2.

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FIGURE 6.4 Trajectories of model variables for different values of p = 0.08, 1, 10 when 1 < R0 < Rc . The model parameters are as in Table 6.2.

FIGURE 6.5 Trajectories of model variables for different values of p = 0.08, 1, 10 when 1 < Rc < R0 . The model parameters are as in Table 6.2.

In the following figures, we consider the impact of media to control the disease and an effective role of vaccination by drawing the trajectories S(t), V (t), I (t) with respect to p as shown in Figs. 6.3, 6.4, and 6.5. It is also observed that as p increases, the number of infected people reduces and opposite results occur for the susceptible and vaccinated populations. This result is very common, because media coverage influences people’s awareness regarding vaccination to control and prevent themselves from the rotavirus disease through sanitation, cleanliness, vaccination, etc. So, it is clear that the media index plays a pivotal role in inhibiting and controlling the disease. Our next aim is to find out the optimal strategy to minimize the cost of the control. Here we have considered the controls as time-dependent functions. Numerically, we have investigated the optimal control (6.23) and (6.24) as a boundary value problem, and here we choose ti = 0 and tf = 200. We assume that the treatment will stop after 200 days. Fig. 6.6 is a two-point boundary value

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problem. In Fig. 6.6 it is clearly observed that the new infection control input reaches its maximum value within 5–10 days and attains its maximum value at the end of the control schedule. Regarding the reinfection control (for the vaccinated population), the control input increases and reaches its maximum value (0.2) after 100 days, and after that it gradually reduces to zero at the end of the treatment. We have also observed that the optimal control makes a positive effect on a sharp rise of the susceptible human population and vaccinated humans. Also, the infected population decreases gradually. Thus the control in presence of awareness plays a pivotal role in controlling the infection of rotavirus.

FIGURE 6.6 Model simulation of (6.23)–(6.24), showing the cases with and without control. (a) Susceptible humans, (b) Vaccinated humans, (c) Infected human without control and with time-dependent control. (d–e) Optimal control plot with different cost weights. The model parameters are as in Table 6.2.

6.8 Discussion The World Health Organization (WHO) endorses that the rotavirus vaccine will be introduced in every country’s national immunization program. To date, 96 countries around the world have introduced the rotavirus vaccine in their national immunization programs. The Government of India (GoI) has decided to introduce the rotavirus vaccine in the Universal

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Immunization Programme (UIP) in a phased manner like a pulse vaccination campaign [21]. In this chapter, a mathematical model has been proposed and an insight into the media’s impact on vaccination campaigns has been analyzed. Here we consider that public mass media impact does not work when the infected population is generally small or the disease is in the mutant stage. Incorporating the piecewise continuous transmission rate to represent media impact has its effect only when the amount of the infected population exceeds the threshold level. From the analytical findings, we have observed that when R0 < 1, the disease-free equilibrium is globally asymptotically stable. Also, the endemic equilibrium attains its global stability when R0 > 1 and 0 < p < p  . But when R0 > Rc , the system moves towards E ∗ of the subsystem ϒ2 . Numerical simulation confirms the stability of the equilibrium and the impact of media on the system trajectories. It is observed that increasing the value of the media impact reduces the amount of the infected population, and thus the disease can be controlled. We have simulated the system behavior of the SVI model in the absence and presence of the control variable, respectively (see Fig. 6.6). Given the control-theoretical study, we have found that a combination of both controls is more effective for reducing the new infected human population and leading to the awareness program’s success. Also, Strategy C (i.e., a combination of both controls) reduces the probability of a new infection from the vaccinated population. Thus we recommend the optimal control method as a standard nonpharmaceutical policy for rotavirus infection. We have dealt with three strategies of optimal control in Section 6.6. We have compared three strategies. We have found that Strategy C has a much higher efficiency index (see Table 6.1) than both Strategy A and Strategy B. The main target of Strategy A is to prevent new infections of the susceptible human population, while strategy B focuses on controlling new infections from the vaccinated population. Both strategies are adopted in the light of awareness input. However, Strategy C performs more effectively than the other two as it concentrates on both of these two points. Hence we can conclude that the optimal control with Strategy C in this model is the most beneficial way of controlling rotavirus infection cost-effectively. This study successfully describes the effect of awareness towards controlling the new infection of the vaccinated population. The results from our work can further guide in developing a cost-effective awareness programming for controlling rotavirus infection.

References [1] C.D.C. Rotavirus, Manual for the Surveillance of Vaccine-Preventable Disease, 5th edition, 2011. [2] L. Roberts, Rotavirus vaccines’ second chance, Science 305 (2004) 1890–1893. [3] L.J. White, J. Buttery, B. Cooper, D.J. Nokes, G. Medley, Rotavirus within day care centres in Oxfordshire, UK: characterization of partial immunity, J. R. Soc. Interface 5 (2008) 1481–1490. [4] Rotavirus Fact Sheet, Department of Health and Human Services, Centers for Disease Control and Prevention, 2004.

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[5] CDC, Vaccine and Preventable Disease, Rotavirus vaccine, Online available http://www.cdc.gov/ vaccines/hcp/acip-recs/vacc-specific/rotavirus.html, 2013. [6] WHO, Introduction of rotavirus vaccines into national immunization programmes, Geneva 5 (2009) 200–209. [7] P.A. Offit, H.F. Clark, The rotavirus vaccine, Curr. Opin. Pediatr. 11 (1998) 9–13. [8] E. Shim, H.T. Banks, C. Castillo-Chavez, Seasonality of rotavirus infection with its vaccination, J. Infect. Dis. 101 (2001) 62–92. [9] D.O. Matson, S.S. Long, L.K. Pickering, C.G. Prober, Principles and Practice of Pediatric Infectious Diseases on Rotavirus, New York, NY, Churchill Livingstone, 2003, pp. 1105–1109. [10] CDC, Prevention of rotavirus gastroenteritis among infants and children: recommendation of the advisory committee on immunization practices (ACIP), MMWR Recomm. Rep. 58 (2009) 1–25. [11] E.T. Van, M. Soriano-Gabarr, S. Debrus, C.E. Newbern, J. Gray, A mathematical model of the indirect effects of rotavirus vaccination, Epidemiol. Infect. 138 (2010) 884–897. [12] J.E. Tate, A.H. Burton, C. Boschi-Pinto, A.D. Steele, J. Duque, U.D. Parashar, Estimate of worldwide rotavirus-associated mortality in children younger than 5 years before the introduction of universal rotavirus vaccination programmes: a systematic review and meta-analysis, Lancet Infect. Dis. 12 (2012) 136–141. [13] E.A. Katherine, S. Eunha, E. Virginia, P.A.P. Galvani, Impact of rotavirus vaccination on epidemiological dynamics in England and Wales, Vaccine 30 (2012) 552–564. [14] M. Christopher, Z. Kribs, J. Jean-Franois, V. Philippe, C. Sandrine, Modeling nosocomial transmission of rotavirus in PediatricWards, Bull. Math. Biol. 73 (2011) 1413–1442. [15] M.C.S. De Jesus, V.S. Santos, L.M. Storti-Melo, C.D.F. De Souza, Í.D.D.C. Barreto, M.V.C. Paes, P.A.S. Lima, A.K. Bohland, E.N. Berezin, R.L.D. Machado, L.E. Cuevas, Impact of a twelve-year rotavirus vaccine program on acute diarrhea mortality and hospitalization in Brazil: 2006–2018, Expert Rev. Vaccines 19 (2020) 585–593. [16] S.P. Marbaniang, Women care and practices in the management of childhood diarrhea in Northeast India, in: Child Care in Practice, 2020, pp. 1–13. [17] F. Al Basir, S. Ray, E. Venturino, Role of media coverage and delay in controlling infectious diseases: a mathematical model, Appl. Math. Comput. 337 (2018) 372–385. [18] Dongmei Xiao, Shigui Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci. 208 (2007) 419–429. [19] A. Roldaoa, H.L.A. Vieiraa, M.J.T. Carrondoa, P.M. Alvesa, R. Oliveira, Rotavirus-like particle production: simulation of protein production and particle assembly, in: Proceedings of the 16th European Symposium on Computer Aided Process Engineering and 9th International Symposium on Process Systems Engineering, vol. 21, 2006, pp. 1673–1678. [20] D.Y. Rokityanski, Sufficient conditions for local and global stability of time-varying sets, IFAC Proc. 34 (2001) 447–452. [21] A.S. Bhadoria, S. Mishra, M. Singh, S. Kishore, National immunization programme–mission indradhanush programme: newer approaches and interventions, Indian J. Pediatr. (2019) 1–6. [22] D.L. Lukes, Differential equations: classical to controlled, 1982. [23] W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimalcontrol, Springer Science & Business Media, 2012. [24] W.E. Boyce, R.C. DiPrima, Elementary Differential Equations Andboundary Value Problems, John Wiley & Sons, 2012. [25] G. Birkhoff, G.C. Rota, Ordinary Differential Equations, Ginn, 1962. [26] L.S. Pontryagin, Mathematical Theory of Optimal Processes, Routledge, 2018. [27] Jayanta Mondal, Piu Samui, Amar Nath Chatterjee, Optimal control strategies of non-pharmaceutical and pharmaceutical interventions for Covid-19 control, J. Interdiscip. Math. (2020) 1–29. [28] P.K. Roy, S. Chowdhury, A.N. Chatterjee, J. Chattopadhyay, R. Norman, A mathematical model on CTL mediated control of HIV infection in a long-term drug therapy, J. Biol. Syst. 21 (2013) 1350019. [29] A.N. Chatterjee, P.K. Roy, Anti-viral drug treatment along with immune activator IL-2: a control-based mathematical approach for HIV infection, Int. J. Control 85 (2012) 220–237. [30] P.K. Roy, A.N. Chatterjee, Effect of HAART on CTL mediated immune cells: an optimal control theoretic approach, in: Electrical Engineering and Applied Computing, Springer, Dordrecht, 2011, pp. 595–607.

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[31] P.K. Roy, A.K. Roy, E.N. Khailov, F. Al Basir, E.V. Grigorieva, A model of the optimal immunotherapy of psoriasis by introducing IL 10 and IL 22 inhibitor, J. Biol. Syst. 28 (2020) 609–639. [32] H.M. Yang, C.P. Ferreira, Assessing the effects of vector control on Dengue transmission, Appl. Math. Comput. 198 (2008) 401–413. [33] H. Abboubakar, J.C. Kamgang, L.N. Nkamba, D. Tieudjo, Bifurcation thresholds and optimal control in transmission dynamics of arboviral diseases, J. Math. Biol. 76 (2018) 379–427. [34] S.A. Carvalho, O.S. Stella, C.C. Iraziet, Mathematical modeling of Dengue epidemic: control methods and vaccination strategies, Theory Biosci. 138 (2019) 223–239.

7 Mathematical models of early stage Covid-19 transmission in Sri Lanka COVID-19 models in Sri Lanka Wickramaarachchillage Pieris Tharindu Mihiruwan Wickramaarachchia and Shyam Sanjeewa Nishantha Pererab a Department of Mathematics, The Open University of Sri Lanka, Nawala, Nugegoda, Sri Lanka b Research and Development Center for Mathematical Modeling, Department of Mathematics, University of Colombo, Colombo, Sri Lanka

7.1 Introduction The COVID-19 global epidemic has been significantly damaging the human well-being, life style of people, and the global economy. It is clear that the entire world is moving into a dangerous phase of this epidemic at the moment since it started in early 2020. Several vaccines to fight against the virus have been successful by December, 2020; however, it is predicted that the mass production of doses to cover the entire world population and global distribution of such vaccines would be challenging. In the absence of a widely available preventive vaccine, the governments across the world implement, monitor, and manage various public health and social distancing measures to control the spread of this extremely contagious disease. Even in the presence of these control measures, the number of cases and deaths due to COVID-19 has mounted in millions causing economies in the world to disrupt moving forward, while industries such as airlines, shipping, and hospitality have been seriously struggling to recover for a foreseeable future. Mathematical models in epidemiology have a century-long history as many researchers developed a range of mathematical models to explain the dynamic of numerous communicable diseases spread across the world population over the last century. These models were extensively used to investigate the dynamic of infectious diseases, to predict the outbreaks for the future, and to assess the efficacy of control measures, public response, and treatments. However, with the introduction of computer technology, computing algorithms have been developed to simulate model-based outcomes, and subsequently the results are used in epidemiological decision-making, that is, to find optimal treatment plans, to determine best vaccination schedules, and to develop budgets optimizing disease preventive investment versus better health outcomes in vulnerable communities. These mathematical models can be either discrete or continuous in time. The widely used class of mathematical models has been SIR (Susceptible-Infected-Recovered) modAdvances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00012-0 Copyright © 2023 Elsevier Inc. All rights reserved.

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els. In these models, it is assumed that populations are homogeneous. In addition, these compartmental models are easy to solve using computer algorithms. In contrast, agentbased models are more complex in their nature as they assume the demographic heterogeneity of the individuals in populations. It is argued that these mathematical models do not produce perfectly accurate predictions of the outbreaks; however, researchers believe the information revealed by the models is extremely important to explore the qualitative features of epidemic progressions, and also they are flexible in handling the evaluation of various treatment and intervention measures. Since the start of the COVID-19 global outbreak in early 2020, researchers have rushed to adopt and develop mathematical models to understand and forecast the dynamic of the transmission basically implementing SIR models and their extended versions, so-called SEIR (Susceptible-Exposed-Infected-Recovered), with a handful of information and learning they had regarding the dynamic of the novel virus [2,18,33,34]. Initially, model parameters had been hypothetically fed into the model. But later, with more available data due to evolving outbreak, these parameters were estimated using clinical and epidemiological data [12,13,35]. In Sri Lanka, the first COVID-19 case was found on January 26, 2020, and it was successfully treated and recovered. The first local patient was found on March 11, 2020 followed by the government taking strong decisions to control the transmission of the disease over the community, including shutting down all the places of public gatherings such as schools, universities, and nonessential services, imposing travel ban to high risk countries, introducing mandatory quarantine for all arrivals to the country, shutting down the airport, and finally imposing island wide curfew [6]. The time line of early stage COVID-19-related events and responses by the government is illustrated in Fig. 7.1. Due to the timely public health and social distancing measures introduced, the country did not experience a dramatic rise in cases during the first 40 days of the outbreak; however, a sudden rise took place as a result of a large cluster in the naval base. Then again, after 75 days, another jump was reported due to repatriated Sri Lankan immigrant workers mainly from the Middle East countries. Total corona virus infections, active cases, deaths, and recoveries in Sri Lanka recorded for the first 82 days of the epidemic are presented in Fig. 7.2. A few of the high risk areas and villages have been locked down restricting any type of mobility. Even though the public health sector including the military forces is acting effectively, one of the major challenges to combat the virus in Sri Lanka has been a significant rise in the asymptomatic infections, i.e., those that are not showing any COVID-19 symptoms but are carriers of the virus in the population [7]. Since COVID-19 is a new disease that emerged in the world, lack of data is available related to the dynamic of the virus, so that they fit to the existing mathematical models to predict the outbreak. On the other hand, these models may be used to demonstrate the possible different scenarios of the disease transmission with respect to social distancing and public health intervention measures introduced by authorities [8]. This study mainly focuses on a retrospective analysis of the initial (first 82 days) government COVID-19 response that was driven majorly by combinations of NPIs. This chapter focuses on demon-

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FIGURE 7.1 COVID-19 events (red (mid gray in print version)) and control measures (blue (dark gray in print version)) in Sri Lanka since the date when the first reported case was identified [6] during the early period of the outbreak.

FIGURE 7.2 Cumulative cases (a), active cases (b), deaths (c), and recovered cases (d) of COVID-19 in Sri Lanka during the first 82 days of the epidemic [3].

strating how various mathematical models can be adopted and extended to investigate the early stage dynamic of the COVID-19 transmission in Sri Lanka. We basically study three approaches. The first approach focuses on studying a compartmental SEIR-type mathematical model to predict the infections and to reveal useful qualitative information about the dynamic of COVID-19 transmission in Sri Lanka. Reported cases are used to estimate the doubling time of cases in the community, and hence the initial transmission prob-

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ability is approximated assuming exponential growth behavior of cases. This measure is also applied to model the time-dependent transmission probability during the first 82 days of the epidemic. Next, an optimization problem is established to estimate parameter values for the transmission probability β, initial sizes of the exposed E0 and infected I0 populations, matching between total reported COVID-19 cases and cumulative infections resulted in numerical simulation of the mathematical model. In the second mathematical model, we study optimal control of the disease implemented at the parameter level introducing three control variables. A seven-dimensional compartmental model is developed assuming the heterogeneity of infections and the comorbidity. Lastly, the third model is used to show the sensitivity of the transmission probability with respect to a hypothetical scenario based on NPIs such as public health control measures, border closures, and the timing of implementation.

7.2 Mathematical model to estimate initial parameters In this study, we adopt an SEIR (Susceptible-Exposed-Infected-Recovered) compartmental approach to model the transmission of COVID-19 in Sri Lanka. It is assumed that all the clinically identified positive patients for COVID-19 virus are homogeneous with no impact of demographic factors such as age, gender, and history of chronic diseases on the disease progression. The natural birth and death process was considered to be negligible, and those who recovered were assumed to have developed complete immunity against the virus. In this simple model, the susceptible individual may become exposed to the novel corona virus at a probability of β, and this is also defined as the transmission rate of the virus. Let σ denote the rate at which exposed individuals become infectious. As a result of treatments or due to immunity, the patient can recover at a rate of γ . It is noted that some patients whose condition gets worse could end up with deaths at a rate of μ. It should be noted that the effective contact rates between the susceptible and the infected populations depend on the number of individuals in the entire population (N ). The transmission rate SI SI d(S/N ) = −β 2 , and here 2 is the fraction of those between S and I is assumed to be dt N N contacts between an infectious and a susceptible individual which result in the susceptible person becoming infected. Based on these assumptions and dynamics, the transmission of COVID-19 can be represented as the following schematic diagram (Fig. 7.3). The SEIR model of COVID-19 transmission can now be established as a system of nonlinear differential equations given by dS dt dE dt dI dt dR dt

β SI N

(7.1)

β SI − σ E N

(7.2)

=σ E − γ I − μI

(7.3)

=γ I,

(7.4)

=− =

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FIGURE 7.3 Schematic diagram.

where N is the total size of the population. The initial conditions for the model are S(0) = S0 = N − E0 − I0 , E(0) = E0 , I (0) = I0 , and R(0) = R0 . We let the set of solutions to the system of nonlinear differential equations, denoted by , be as follows:  = {(S, E, I, R) ∈ R4+ : S + E + I + R ≤ N, S, E, I, R ≥ 0}.

(7.5)

Initially, the entire population is susceptible to COVID-19, thus we let N ≈ S, and this gives β N SI ≈ βI . Now the model with infection groups can be written as dE =βI − σ E dt dI =σ E − γ I − μI dt or in a matrix form



E I



 =

−σ σ

β −(γ + μ)

   E . . I

At the early stage of the epidemic, there is a nonlinear relationship between the total number of cases I (t) and the force of infection λ, which is given as follows [36,37]: I (t) ∝ I (0)e(λt) ,

(7.6)

where I (0) denotes the initial size of the epidemic. Then the doubling time (D) of the number of infections can be obtained as ln 2 . (7.7) D= λ

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The doubling time is a useful measure in a public health point of view that explains the number of days it takes to double the number of infections. This also reveals critical information such as the level of risk the population is exposed to and the rate of transmission. The initial growth rate of the epidemic is determined by the largest eigenvalue λ of the linear system, and this is given by 1 − (γ + σ + μ) + λ= 2



 (γ + μ − σ )2 + 4βσ .

(7.8)

Combining Eqs. (7.7) and (7.8) yields the following relationship for the transmission probability β of the virus:  β=

ln 4 D

2 + (γ + μ + σ ) − (γ + μ − σ )2 4σ

.

(7.9)

7.2.1 Analysis of the model Basic reproduction number R0 stands for the number of secondary infections that can be produced by a single infected patient on average [17,38]. It is critical to distinguish new infections in the dynamic of the population to compute R0 . In general, we let x = (x1 , . . . , xn )T , xi ≥ 0, be the number of individuals in each population class. For simplicity, we arrange the compartments in such a way that first m stands for the infected individuals. We also define the set X0 = {x ≥|xi = 0, i = 1, . . . , m}. Let Fi (x) be the rate of arrival of new infections in compartment i, Vi+ (x) be the rate of transfer of individuals into compartment i in various other routes, and Vi− (x) be the rate of transfer of individuals out of compartment i. The functions Fi (x), Vi+ (x), and Vi− (x) are assumed to be continuous and at a minimum of twice differentiable on x. Now, in general terms, the system of differential equations can be represented in the form x˙i = fi (x)

= Fi (x) − Vi (x), i = 1, . . . , n,

(7.10)

where Vi (x) = Vi− (x) − Vi+ (x) and the above functions must meet assumptions A(1)–A(5) listed below. A(1) Since each function represents a directed transfer of individuals in the population, they are all nonnegative. That is, if x ≥ 0, then Fi (x), Vi+ (x), Vi− (x) ≥ 0 for i = 1, . . . , n. A(2) If a compartment is empty, then there can be no transfer of individuals out of the compartment by death, migration, infection, or any other means. That is, if xi = 0, then Vi− (x) = 0. A(3) The incidence of infection for uninfected compartments is zero. That is, Fi (x) = 0 if i > m. A(4) If the population is free of disease, then the population will remain free of disease. Thus, if x ∈ X0 , then Fi (x) = 0 and Vi+ (x) = 0 for i, . . . , m.

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A(5) If the population is held closed to the disease-free equilibrium (DFE), then the population will get back to the DFE as ruled by the linearized system x˙ = Df (x0 )(x − x0 ), where Df (x0 ) =

(7.11)

 ∂f  i

assessed at the DFE x0 . This can be written as if F(x) = 0 then ∂xi all eigenvalues of Df (x0 ) have negative real parts.

Now we present the computation of the basic reproductive number R0 for the COVID19 transmission model. For this purpose, we now define the new vector of only infected variables X = (E, I ) containing the classes which are responsible for transmitting the virus in the population. Now, we establish the following two-dimensional system of differential equations involving E and I : dE β = SI − σ E dt N dI =σ E − (γ + μ)I. dt

(7.12) (7.13)

The next generation matrix method is applied for the computation of R0 [15,16]. Accordingly, necessary matrices F and V are obtained as follows:  F= and

 V=

0 0

σ −σ

β NS

 (7.14)

0

0 (γ + μ)

 (7.15)

.

The basic reproduction number is defined as the spectral radius ρ of the matrix F V −1 [15]. Thus we obtain the following formula for R0 together with the assumption S ≈ N initially: R0 ≈

β . γ +μ

(7.16)

Substituting the expression for β in Eq. (7.9), we get  R0 ≈

ln 4 D

2 + (γ + μ + σ ) − (γ + μ − σ )2 4σ (γ + μ)

.

(7.17)

The change in R0 with respect to varying doubling time D is given in Fig. 7.4, and it clearly indicates that R0 reduces as the doubling time increases implying a reduced level of exposed risk to the disease.

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FIGURE 7.4 The change in R0 with respect to doubling time D.

7.2.2 Estimation of initial parameters In the model for COVID-19 in Eqs. (7.1)–(7.4), the transmission probability β is assumed to be fixed. However, in the realistic case this transmission probability depends on time as the governments introduce various control measures to minimize the transmission of virus with respect to time. This results in an SEIR model with a time-dependent transmission rate as follows: dS dt dE dt dI dt dR dt where

β(t) =

β(t) SI N β(t) = SI − σ E N =−

(7.18) (7.19)

=σ E − γ I − μI

(7.20)

=γ I,

(7.21)

⎧ ⎪ ⎪ ⎪β1 , if t < 14; ⎪ ⎪ ⎪ ⎪β2 , if 14 ≤ t < 28; ⎨ β3 , if 28 ≤ t < 42; ⎪ ⎪ ⎪ ⎪ β , if 42 ≤ t < 56; ⎪ ⎪ 4 ⎪ ⎩β , if t ≥ 56. 5

(7.22)

It should be noted that if βi = β for i = 1, 2, 3, 4, 5, then the model reduces to the system in Eqs. (7.1)–(7.4).

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Next, we attempt to estimate the parameter β and the initial exposed and infected populations E0 and I0 using the reported COVID-19 cases data in Sri Lanka. We define the following least-squares functional for the matching between simulated output from system (7.1)–(7.4) and the reported data for the period of eighty days since the first local COVID-19 case was identified during the second week of March, 2020 [39,40].

J (u) =

T

0

 2 1 1 1 (I (t) + R(t)) − Y (t) + C1 β 2 + C2 E02 + C3 I02 dt, 2 2 2

(7.23)

where Y (t) for t ∈ [0, T ] is the time series of clinically identified COVID-19 commutative cases in Sri Lanka, and that should be matched with the sum of simulated results for the infected I (t) and the recovered R(t) for t ∈ [0, T ]. Here u = (β, E0 , I0 ) the unknown parameters should be estimated. The regularization term 12 C1 β 2 + 12 C2 E02 + 12 C3 I02 is included to the cost functional to ensure the convexity, where Ck for k = 1, 2, 3 are constants and they are responsible to relatively balance the contribution from least-squares error. We find the optimal control measures u∗ = (β ∗ , E0∗ , I0∗ ) such that J (β ∗ , E0∗ , I0∗ ) = min J (β, E0 , I0 ) U

(7.24)

with β ∗ ∈ (0, 1), E0∗ ≥ 0, and I0∗ ≥ 0.

7.2.3 Optimization Now we discuss the method of obtaining the solution to problem (7.23). For this, it is necessary to define the Lagrangian and Hamiltonian for the optimal control problem (7.23). Thus the Lagrangian L is stated as  2 1 1 1 L(I, R, Y, u) = I (t) + R(t) − Y (t) + C1 β 2 + C2 E02 + C3 I02 , 2 2 2

(7.25)

and for the Hamiltonian, we let X = (S, E, I, R) and λ = (λ1 , λ2 , λ3 , λ4 ), and we write   β  β     SI −σ E +λ3 σ E −γ I −μI +λ4 γ I , (7.26) H (X, u, λ) = L(I, R, Y, u)+λ1 − SI +λ1 N N where λj , j ∈ {1, 2, 3, 4} are the adjoint variables. Next derivation is the necessary conditions as described in detail in [22,23] for the Hamiltonian H for the equation given in (7.26). Theorem 7.1. Given an optimal control u∗ = (β ∗ , E0∗ , I0∗ ) and a solution X∗ = (S ∗ , E ∗ , I ∗ , R ∗ ) with respect to system (7.1)–(7.4), there exist adjoint variables λj , j ∈ {1, 2, 3, 4} satisfying dλ1 Iβ(λ1 − λ2 ) = dt N dλ2 =σ (λ2 − λ3 ) dt dλ3 Sβ =2(Y − R − I ) − λ4 γ + (λ1 − λ2 ) + λ3 (γ + μ) dt N

(7.27) (7.28) (7.29)

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dλ4 =2(Y − R − I ) dt

(7.30)

with the transversality conditions λj (tf ) = 0, j ∈ {1, 2, 3, 4}.

(7.31)

Proof. System (7.1)–(7.4) is obtained by taking the derivative dX ∂H (t, u∗ , λ1 , λ2 , λ3 , λ4 ) = , dt ∂λ and the adjoint system is obtained taking dλ −∂H (t, t, u∗ , λ1 , λ2 , λ3 , λ4 ) = . dt ∂X ∗

7.2.4 Numerical results In this section we obtain the numerical solutions for the system given in Eqs. (7.18)–(7.21) considering time-varying transmission rate β as established in (7.22). The differential system is numerically solved using ODE45 solver in MATLAB package that is based on a variable step Runge-Kutta (RK4) method to solve differential equations numerically. The classical Runge-Kutta method is applied to the initial value problem in the form dy = f (t, y), y(t0 ) = y0 . dt Here y is an unknown function (scalar or vector) of time t, which we would like to approximate. At the initial time t0 the corresponding y value is y0 . The function f and the initial conditions t0 , y0 are given. Now we let the step size h such that h > 0 and define 1 yn+1 = yn + h(k1 + 2k2 + 2k3 + k4 ) 6 tn+1 = tn + h for n = 0, 1, 2, 3, . . ., using k1 = f (tn , yn )  h k1  k2 = f tn + , y n + h 2 2  h k2  k3 = f tn + , y n + h 2 2 k4 = f (tn + h, yn hk3 ). Here yn+1 is the RK4 approximation of y(tn+1 ), and the next value (yn+1 ) is determined by the present value (yn ) plus the weighted average of four increments, where each increment

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is the product of the size of the interval h and an estimated slope specified by function f on the right-hand side of the differential equation. Further, • • • •

k1 is the slope at the beginning of the interval, using y; k2 is the slope at the midpoint of the interval, using y and k1 ; k3 is again the slope at the midpoint, but now using y and k2 ; k4 is the slope at the end of the interval, using y and k3 .

The time-dependent form of β is computed considering daily COVID-19 data in Sri Lanka for the first 80 days from the date the first local case was reported. The following algorithm is applied for the computation. STEP 1 Select the COVID-19 total daily cases for the given period. STEP 2 Fit the exponential curve considering I (t) = ae(λt) , a ∈ R+ with 95% confidence interval. STEP 3 Obtain the force of infection λ, and then compute doubling time using Eq. (7.7). STEP 4 Repeat the process choosing different time intervals as indicated in (7.22). STEP 5 Compute the true value of β and 95% confidence interval for each time period using Eq. (7.9). The computed true values of β and the 95% confidence interval for each time period is presented in Table 7.1. In the context of local transmission of COVID-19 in Sri Lanka, the spread was triggered due to a large number of imported cases to the country mainly from the European Union countries and from South Korea. Thus, for the simulation, we take E(0) = 600 and I (0) = 50. We also let S(0) = N − E(0) − I (0) and R(0) = 0. Table 7.1 The estimated values for true β and its 95% confidence interval. In Eq. (7.9), the rest of the parameter values are γ = 1/25, μ = 0.00001, and σ = 1/3 [6,9,12]. Time period

True value of β

t < 14

0.0343

95% confidence interval [0.0257, 0.0439]

14 ≤ t < 28

0.0142

[0.0124, 0.0162]

28 ≤ t < 42

0.0106

[0.0099, 0.0114]

42 ≤ t < 56

0.0126

[0.0121, 0.0132]

t ≥ 56

0.0104

[0.0100, 0.0109]

According to Fig. 7.5, the susceptible population is decreasing with respect to time; however, the population of infected people grows exponentially over the first 10–15 days of the outbreak, reaches the maximum size of the active infections in the range between 500–600, and then decays over the time. The outcomes clearly indicate the sensitivity of transmission probability β as there are three distinct curves for its true value, upper and lower boundary of the 95% confidence interval. Fig. 7.4 plot (d) represents the dynamic of the total cases reported in Sri Lanka for the period, that is, the sum of infected and recov-

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FIGURE 7.5 Numerical outcomes of system (7.18)–(7.21), susceptible (a), infected (b), recovered (c), and cumulative cases (d). The values of β are as given in Table 7.1, and the rest of the parameter values are N = 22000000, γ = 1/25, μ = 0.00001, and σ = 1/3 [6,9,12].

ered patients; however, deaths are not included as there are only 11 fatalities over the time horizon considered for this study, which is about 0.56%. Now the system in Eqs. (7.1)–(7.4) is solved to estimate the parameters β, E0 , and I0 matching with reported COVID-19 cases in Sri Lanka in the first 80 days of its outbreak. Numerical schemes presented in [24–26,39] are coupled with the Runge-Kutta method of order four to carry out the simulation of the problem in optimization. The algorithm stops once the termination condition ||J (um+1 ) − J (um )|| < ERR is satisfied. The outcome of this simulation is given in Fig. 7.6. The blue dotted line represents the reported COVID-19 cases, whilst the red thick line stands for the simulated curve. A local minimum is obtained for the optimization problem in (7.23), and the estimated parameter values are β = 0.3582, E0 = 17, and I0 = 0, suggesting that there were around a significant number of exposed cases yet no infections initially. It is clearly observed from Fig. 7.6 that during about first 25 days, the simulated curve and the actual data fit closely; however, there are two major deviations as a result of a sudden rise in COVID-19 cases: firstly, due to the outbreak in Naval Bases (after 45 days), and secondly, due to imported infected cases of migrant Sri Lankan workers repatriated from the Middle East countries (after 75 days).

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FIGURE 7.6 Numerical outcomes of system (7.1)–(7.4) matching with reported COVID-19 cases during the first 80 days of the outbreak. The parameter values are N = 22000000, γ = 1/25, μ = 0.00001, and σ = 1/3 [6,9,12].

7.3 Mathematical models with heterogeneity of cases In this section, a more extended version of the SEIR (Susceptible-Exposed-InfectedRecovered) compartmental model structure is developed to study this dynamic further [9,12–14]. In Sri Lanka, the health authorities treated all the symptomatic COVID-19 cases in government hospitals rather than advising them to be self-isolated. However, recent international travelers and close contacts of the identified COVID-19 patients are isolated in government managed quarantine centers in different parts of the island [6]. If patients from those groups are identified, then they are immediately taken to the hospitals and treated. It is also found that a reasonable number of individuals tested positive while they were asymptomatic [4,6]. Based on this policy structure in Sri Lankan context, seven population compartments are considered for the model: susceptible (S), exposed (E), infected asymptomatic (IA ), infected with mild symptoms (IM ), isolated in designated hospitals (IH ), patients with critical conditions treated in intensive care units (IC ), and patients who were clinically determined as recovered (R) [9]. Following the compartmental transition schematic diagram illustrated in Fig. 7.7, the seven-dimensional differential system describing the COVID-19 transmission is given by dS = −(β1 E + β2 IA + β3 IM )S − qS dt dE = k + (β1 E + β2 IA + β3 IM )S − σ E dt

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dIA dt dIM dt dIH dt dIC dt dR dt

= φσ E − δ1 IA − γ1 IA = (1 − φ)σ E − δ2 IM

(7.32)

= δ1 IA + δ2 IM − ηIH − γ2 IH = ηIH − γ3 IC − μIC = γ 1 I A + γ2 I H + γ3 I C ,

where β1 , β2 , and β3 represent the transmission rates from the exposed, infected and asymptomatic, and infected and symptomatic, respectively, while q is the rate of isolation of the susceptible individuals due to lockdown, k is the rate of imported exposed cases, σ is the rate at which the exposed cases become infected, φ is the percentage of exposed individuals who become asymptomatic, δ1 is the rate at which the asymptomatic cases are tested and hospitalized, δ2 is the rate at which the symptomatic cases are tested and admitted to hospitals, η is the rate of patients whose condition becomes severe and who require intensive care treatments, γ1 is the recovery rate of asymptomatic cases who are not in hospitals, γ2 is the recovery rate of mild symptomatic cases who are in general wards in hospitals, γ3 is the recovery rate of critically sick patients, and μ is the death rate of the disease. The initial conditions for model (7.32) are as follows: S(0) = S 0 , E(0) = E 0 , IA (0) = IA0 , 0 , I (0) = I 0 , I (0) = I 0 , and R(0) = R 0 . IM (0) = IM H H C C We let the set of solutions to the system of nonlinear differential equations in (7.32), denoted by , be as follows:  = {(S, E, IA , IM , IH , IC , R) ∈ R7+ : S + E + IA + IM + IH + IC + R ≤ 1, S, E, IA , IM , IH , IC , R ≥ 0}.

FIGURE 7.7 Schematic diagram of COVID-19 transmission.

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7.3.1 Analysis of the model 7.3.1.1 Basic reproduction number Using assumptions A(1)–A(5) enables us to partition the matrix Df (x0 ). This is given by the following lemma. Lemma 7.2. If x0 is a DFE of system (7.10) and fi (x) satisfies A(1)–A(5), then the derivatives DF(x0 ) and DV(x0 ) are partitioned as  DF(x0 ) =

F 0

0 0



 , DV(x0 ) =

0 J4

V J3

 ∂F

 ,

  ∂V  i (x0 ) and V = (x0 ) with ∂xj ∂xj 1 ≤ i, j ≤ m. Further, F is nonnegative, V is a nonsingular M-matrix, and all eigenvalues of J4 have a positive real part.

where F and V are the m × m matrices defined by F =

i

∂Fi (x0 ) = 0 if either i > m or j > m. Similarly, ∂xi ∂Vi A(2) and A(4) give that if x ∈ X0 then Vi (x) = 0 for i ≤ m. This provides (x0 ) = 0 for i ≤ m ∂xi and j > m. This shows the stated partition and zero blocks. The nonnegativity of F follows from A(1) and A(4). Let ej be the Euclidean basis vectors. That is, ej is the j th column of the n × n identity matrix. Then, for i = 1, . . . , m,

Proof. Let x0 ∈ X0 be a DFE. By A(3) and A(4),

 ∂V  i

∂xi

 V (x + he ) − V (x )  i 0 j i 0 . h h→0+

(x0 ) = lim

To show that V is a nonsingular M-matrix, note that if x0 is a DFE, then using A(2) and A(4), Vi (x0 ) = 0 for i = 1, . . . , m, and if i = j , then the j th component of x0 + hej = 0 and ∂Vi Vi (x0 + hej ) ≤ 0 by A(1) and A(2). Therefore, ≤ 0 for i ≤ m and j = i and V has the Z ∂xj sign pattern [15]. Furthermore, by A(5), all eigenvalues of V have positive real parts. These two conditions provide that V is a nonsingular M-matrix [15]. Finally, A(5) also implies that the eigenvalues of J4 have a positive real part. This completes the proof. Now we aim to compute the basic reproduction number for system (7.32). The method of next generation matrix is used to derive R0 . For this purpose, we now define the new vector of only infected variables X = (E, IA , IM ) containing the classes that are responsible for transmitting the virus in the population. It is assumed that the classes of IH and IC are fully isolated, and it is unlikely that the virus is transmitted to the society anymore. Hence

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we establish the following system of differential equations [15–17]: dE = k + (β1 E + β2 IA + β3 IM )S − σ E dt dIA = φσ E − δ1 IA − γ1 IA dt dIM = (1 − φ)σ E − δ2 IM . dt

(7.33)

To apply the next generation matrix method, the necessary matrices F and V are obtained as follows [15,16]: ⎛

β1 S 0 F =⎝ 0 0

β2 S 0 0 0

⎞ β2 S 0 0 ⎠ 0

(7.34)

and ⎛

0 (δ1 + γ1 ) 0

σ V =⎝ −φσ −(1 − φ)σ

⎞ 0 0 ⎠. δ2

(7.35)

Now, the next generation matrix system is ⎛

S 0 φβ2 S 0 β1 S 0 β3 (φ − 1) − + ⎜ σ δ2 δ1 + γ 1 F V −1 = ⎜ ⎝ 0 0

S 0 β2 δ1 + γ 1 0 0

⎞ S 0 β3 δ2 ⎟ ⎟. 0 ⎠ 0

(7.36)

So, the basic reproduction number is the spectral radius ρ of the matrix F V −1 . Thus we obtain R0 = S 0

β

1

σ

+

(1 − φ)β3 φβ2  . + δ2 δ1 + γ 1

(7.37)

The expression for R0 reveals very useful information about the dynamic of COVID-19 transmission such that the expected number of secondary infection is the addition of infections due to the exposed, asymptomatic, symptomatic cases, respectively [30,31]. As φ goes to 1, the secondary infections are not produced by the cases with mild symptoms as they have been tested and isolated early. Mathematically, it can be very easily shown that lim R0 = S 0

φ→1

β

1

σ

+

β2  . δ1 + γ 1

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7.3.1.2 Stability analysis of the disease-free equilibrium Let us first obtain matrix M such that ⎛

S 0 β1 − σ M =F −V =⎝ −φσ −(1 − φ)σ

S 0 β2 (δ1 + γ1 ) 0

⎞ S 0 β3 0 ⎠. δ2

(7.38)

Now define s(M) = max{Re(α) : α is an eigenvalue of M}. Note that s(M) is a simple eigenvalue of M with a positive eigenvector. In relation to R0 we can establish the following equivalences: R0 > 1 if and only if s(M) > 0 and R0 < 1 if and only if s(M) < 0. Let us now define the set of solution to system (7.12) by 1 = {(E, IA , IM ) ∈ R3+ : E + IA + IM ≤ 1, E, IA , IM ≥ 0}. Theorem 7.3. If R0 < 1, then the DFE, E0 is locally asymptotically stable on 1 . Proof. To prove this, we need to apply assumptions A(1)–A(5) and A(1)–A(4) are easily verified. For A(5), we need to show that the matrix   M 0 J E0 = −J3 J4 has negative real parts with J3 = −F , ⎛ −σ J4 = ⎝ φσ (1 − φ)σ

0 −(δ1 + γ1 ) 0

⎞ 0 0 ⎠. −δ2

We then compute the eigenvalues of J4 , which yields s(J4 ) = max{−δ2 , −σ, −(δ1 + γ1 )} < 0. Thus, if R0 < 1, then the DFE E0 is locally asymptotically stable.

7.3.2 Introducing optimal control measures It is very clear that the only possible strategy to combat the novel corona virus is to control its spread over the population as per the current development. Controlling can be achieved by reducing the transmission rates [18]. In our model, in system (7.32), the spread of the virus is mainly due to three population compartments: exposed, infected asymptomatic, and infected with mild symptoms, and none of the three groups is isolated until the individuals are being clinically tested. The asymptomatic cases have been a very serious concern for the public health system across the globe including Sri Lanka. It has been estimated that around 20% of the cases may be asymptomatic, hence they are undetected, yet with the potential of spreading the virus over the population. In this section, we introduce control measures to system (7.32). The model is modified addressing the dynamic of transmission and necessary mathematical derivations, and analysis will be carried out.

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7.3.2.1 Mathematical model with control parameters In the model with control, we introduce the combined factor (1 − u1 ) to reduce the transmission rates β1 , β2 , and β3 respectively from the exposed, infected asymptomatic, and infected with mild symptoms population classes. Thus this u1 measures the effort of personal protection such as wearing face marks, personal hygiene practices, social distancing methods, etc. The control variable u2 measures the rate of identifying asymptomatic cases through contact tracing, testing, and isolating them to treat in designated hospitals. The control variable u3 measures the rate of tracing, testing, and isolating patients with mild symptoms. In this model, we assume that u2 IA and u3 IM are removed from IA and IM compartments, and they are added to the compartment IH . In addition, the critically sick patients who are currently in IH compartment will be transferred to the class of patients in intensive care units at a rate of η. It is further assumed that the asymptomatic cases who are undetected could recover themselves at a rate of γ1 , the patients who are in general wards with mild symptoms recover at a rate of γ2 , and the patients in ICUs recover at a rate of γ3 , and all are added to the recovery compartment. The modified version of system (7.32) can now be established as in system (7.39): dS dt dE dt dIA dt dIM dt dIH dt dIC dt dR dt

= −(1 − u1 )(β1 E + β2 IA + β3 IM )S − qS = k + (β1 E + β2 IA + β3 IM )S − σ E = φσ E − u2 IA − γ1 IA = (1 − φ)σ E − u3 IM

(7.39)

= u2 IA + u3 IM − ηIH − γ2 IH = ηIH − γ3 IC − μIC = γ 1 I A + γ2 I H + γ3 I C .

7.3.2.2 Mathematical analysis of the model It is clear that we have introduced three time-invariant control variables u(t) = (u1 , u2 , u3 ) ∈ U into system (7.32), and these variables are associated with the population compartments S, E, IA , IM , and IH . Further, the control variables are bounded and measurable such that U = {(u1 , u2 , u3 )|uk is Lebesgue measurable on[0, 1], 0 ≤ uk (t) ≤ 1, t ∈ [0, T ], k = 1, 2, 3}.

(7.40)

The objective functional for the control problem in (7.39) is now defined as [10,11]

J (u1 , u2 , u3 ) = 0

T

3   1 Ck u2k dt A1 E + A2 IA + A3 IM + 2 k=1

(7.41)

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subject to (7.39). It is aimed to minimize the cost functional in (7.41), which consists of populations exposed (E), asymptomatic infected (IA ), and mildly infected (IM ) as well as the socioeconomic cost related to wearing masks, sanitizing methods, the cost of social distancing measures, etc. given by C1 u21 , the public health cost on contact tracing, testing, and isolation of asymptomatic cases given by C2 u22 , and the same cost that is for cases with mild symptoms represented by C3 u23 . The constants A1 , A2 , A3 , C1 , C2 , and C3 are the weights and balancing parameters, and they measure the associated relative cost of the interventions over the interval [0, T ]. We find the optimal control measures u∗ = (u∗1 , u∗2 , u∗3 ) such that J (u∗1 , u∗2 , u∗3 ) = min J (u1 , u2 , u3 ).

(7.42)

U

Now, we derive necessary conditions to find the solution for the optimal control problem using Pontryagin’s maximum principle [16,18,20,21]. To show the existence of the control problem, we rewrite system (7.39) in the following form [16,19]: dX = BX + F (X ), where

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ X =⎜ ⎜ ⎜ ⎜ ⎝ ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ B =⎜ ⎜ ⎜ ⎜ ⎝

and

−q 0 0 0 0 0 0

0 σ 0 0 0 0 0

0 0 −(u2 + γ1 ) 0 0 0 0 ⎛

⎜ ⎜ ⎜ ⎜ ⎜ F (X ) = ⎜ ⎜ ⎜ ⎜ ⎝

S(t) E(t) IA (t) IM (t) IH (t) IC (t) R(t)

0 0 0 −u3 0 0 0

(7.43)

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

0 0 0 0 −(η + γ2 ) 0 0

0 0 0 0 0 −(μ + γ3 ) 0

−(1 − u1 )(β1 E + β2 IA + β3 IM )S k + (β1 E + β2 IA + β3 IM )S φσ E (1 − φ)σ E u2 IA + u3 IM ηIH γ1 I A + γ2 I H + γ3 I C

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

0 0 0 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟, ⎟ ⎟ ⎟ ⎠

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and dX is the derivative of X with respect to time. To show the uniform Lipschitz continuity, we let G(X ) = BX + F (X ).

(7.44)

The function F (X ) in Eq. (7.44) satisfies |F (X1 ) − F (X2 )| ≤ Z1 |S1 − S2 | + Z2 |E1 − E2 | + Z3 |IA1 − IA2 | + Z4 |IM1 − IM2 | + Z5 |IH 1 − IH 2 | + Z6 |IC1 − IC2 | + Z7 |R1 − R2 |. Now, choose Z > 0 such that Z = max(Z1 , Z2 , Z3 , Z4 , Z5 , Z6 , Z7 ). Thus we have |F (X1 ) − F (X2 )| ≤ Z(|S1 − S2 | + |E1 − E2 | + |IA1 − IA2 | + |IM1 − IM2 | + |IH 1 − IH 2 | + |IC1 − IC2 | + |R1 − R2 |). Further, we have |G(X1 ) − G(X2 )| ≤ Z|X1 − X2 | with Z = Z1 + Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + K < ∞. Therefore the function G(X ) is uniformly Lipschitz continuous. Hence we can state that the solution of the control system in (7.39) exists. Theorem 7.4. Given the objective functional J (u1 , u2 , u3 ) according to (7.41), where the control set U given by (7.40) is measurable subject to (7.39) with the initial condition for the problem at t = 0, then there exists an optimal control u∗ = (u∗1 , u∗2 , u∗3 ) such that J (u∗1 , u∗2 , u∗3 ) = min{J (u1 , u2 , u3 ), (u1 , u2 , u3 ) ∈ U}. Proof. It is noted that the state variables and the control variables in problem (7.39) are nonempty, and the set U containing the control variables is closed and convex. The righthand side of system (7.39) is continuous, bounded above and can be written as a linear function of u with time-invariant coefficients depending on the state. There exist constants l1 , l2 > 0 and m > 1 such that the intergrand L(y, u, t) of the objective functional J is convex and satisfies L(y, u, t) ≥ l1 (|u1 |2 + |u2 |2 + |u3 |2 )m/2 − l2 . The state variables and the set of control U are clearly bounded and nonempty. The solutions are bounded and convex. Thus the system is bilinear in control variables as the solutions are bounded. Now, the following is verified so that 1 A1 E + A2 IA + A3 IM + (C1 u21 + C2 u22 + C3 u23 ) ≥ l1 (|u1 |2 + |u2 |2 + |u3 |2 )m/2 − l2 , 2 where A1 , A2 , A3 , C1 , C2 , C3 , l1 , l2 > 0 and m > 1 [22,23]. Now we discuss the method of obtaining the solution to problem (7.39). For this, it is necessary to define the Lagrangian and Hamiltonian for the optimal control problem (7.39). Thus the Lagrangian L is stated as 1 L(E, IA , IM , u1 , u2 , u3 ) = A1 E + A2 IA + A3 IM + (C1 u21 + C2 u22 + C3 u23 ), 2

(7.45)

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and for the Hamiltonian H , we let X = (S, E, IA , IM , IH , IC , R), U = (u1 , u2 , u3 ) and λ = (λ1 , λ2 , λ3 , λ4 , λ5 , λ6 , λ7 ), and we write H (X, U, λ) = L(E, IA , IM , u1 , u2 , u3 ) + λ1 (−(1 − u1 )(β1 E + β2 IA + β3 IM )S − qS) + λ2 (k + (β1 E + β2 IA + β3 IM )S − σ E) + λ3 (φσ E − u2 IA − γ1 IA ) + λ4 ((1 − φ)σ E − u3 IM )

(7.46)

+ λ5 (u2 IA + u3 IM − ηIH − γ2 IH ) + λ6 (ηIH − γ3 IC − μIC ) + λ7 (γ1 IA + γ2 IH + γ3 IC ), where λj , j ∈ {1, 2, 3, 4, 5, 6, 7} are the adjoint variables. The next derivation is the necessary conditions for the Hamiltonian H given in (7.26). Theorem 7.5. Given an optimal control u∗ = (u∗1 , u∗2 , u∗3 ) and a solution X ∗ = (S ∗ , E ∗ , IA∗ , Im∗ , IH∗ , IC∗ , R ∗ ) with respect to system (7.39), there exist adjoint variables λj , j ∈ {1, 2, 3, 4, 5, 6, 7} satisfying dλ1 dt dλ2 dt dλ3 dt dλ4 dt dλ5 dt dλ6 dt dλ7 dt

= (u1 − 1)(β1 E + β2 IA + β3 IM )(λ2 − λ1 ) + qλ1 = −A1 + Sβ1 (u1 − 1)(λ2 − λ1 ) + σ (λ2 − λ3 φ + λ4 (φ − 1)) = −A2 + u2 (λ3 − λ5 ) + (u1 − 1)Sβ2 (λ2 − λ1 ) + γ1 (λ3 − λ7 ) = −A3 + Sβ3 (u1 − 1)(λ2 − λ1 ) + u3 (λ4 − λ5 )

(7.47)

= γ2 (λ5 − λ7 ) + η(λ5 − λ6 ) = γ3 (λ6 − λ7 ) + μλ6 =0

with the transversality conditions λj (tf ) = 0, j ∈ {1, 2, 3, 4, 5, 6, 7}.

(7.48)

In addition, the optimal control functions u∗1 , u∗2 , u∗3 are given by   S ∗ (β E ∗ + β I ∗ + β I ∗ )(λ − λ )  1 2 A 3 M 2 1 u∗1 = min 1, max 0, C1   I ∗ (λ − λ )  3 3 u∗2 = min 1, max 0, A C2

(7.49)

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Proof. The control system (7.39) is obtained by taking the derivative dX ∂H (t, u∗1 , u∗2 , u∗3 , λ1 , λ2 , λ3 , λ4 , λ5 , λ6 , λ7 ) = , dt ∂λ and the adjoint system (7.47) is obtained taking dλ −∂H (t, u∗1 , u∗2 , u∗3 , λ1 , λ2 , λ3 , λ4 , λ5 , λ6 , λ7 ) , = dt ∂X ∗ and the optimal control measures can be derived using 0=

∂H (t, u∗1 , u∗2 , u∗3 , λ1 , λ2 , λ3 , λ4 , λ5 , λ6 , λ7 ) . ∂U

7.3.3 Numerical results In this section, we obtain the numerical solutions for the problem without control (7.32) and for the control problem (7.39). The Runge-Kutta algorithm of order four is implemented in MATLAB to solve the problem without control, and the numerical schemes presented in [24–26] are coupled with the Runge-Kutta method of order four to carry out the simulation for the problem with control.

7.3.3.1 Algorithm for the optimal control problem STEP 0: Guess an initial estimation to control parameters u and tf . STEP 1: Use the initial conditions S(0), E(0), IA (0), IM (0), IH (0), IC (0), and R(0) and the stocked values by u and tf . ∗ , I ∗ , I ∗ , and R ∗ which iterate forward in the Find the optimal states S ∗ , E ∗ , IA∗ , IM H C control problem (7.39)–(7.49). STEP 2: Use the stocked values by u and the transversality conditions λj (tf ) for j = 1, 2, 3, 4, 5, 6, 7 while searching the constant λ7 (tf ) using the scant-method. Find the adjoint variables λj (tf ) for j = 1, 2, 3, 4, 5, 6, 7, which iterate backward in the control problem (7.39)–(7.49). STEP 3: Update the control utilizing new state variables S, E, IA , IM , IH , IC , R, and λj (tf ) for j = 1, 2, 3, 4, 5, 6, 7 in the characterization of optimal u∗ given in (7.49). STEP 4: Test the convergence. If the values of the sought variables in this iteration and the final iteration are sufficiently small, check out the recent values as solutions. If the values are not small, go back to STEP 1 [27–29].

7.3.4 Simulation of the COVID 19 dynamic system without control Fig. 7.8 shows the simulation results of the problem without control measures given in (7.32). It has been found recently that there are a significant number of asymptomatic cases

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within the populations who are also carriers of the virus. In the public health perspectives, it is critical to clinically identify these cases through aggressive testing and isolate them if they are found to be positive for the virus. The outcome of this task depends on how many cases are asymptomatic as a proportion. Therefore we aim to assess the sensitivity of this proportion at the parameter level. Thus we let φ to be varying and consider the vector of values φ = (0.1, 0.25, 0.35, 0.4, 0.45, 0.5) for this simulation. The rest of the parameters are β1 = 0.5, β2 = 0.6, β3 = 0.45, γ1 = 0.5, γ2 = 0.2, γ3 = 0.05, δ1 = 0.15, δ2 = 0.25, η = 0.005, μ = 0.04, σ = 1/5, φ = 0.25, k = 0.00405, and q = 0.0004. The initial conditions for the dimensionless form of the problem are S(0) = 0.85, E(0) = 0, IA (0) = 0, IM (0) = 0, IH (0) = 0, IC (0) = 0, and R(0) = 0 [9,12]. No control measures u1 , u2 , and u3 are inactive in this case.

FIGURE 7.8 Simulated solution curves for the exposed (a), asymptomatic (b), symptomatic with mild (c), isolated in hospitals (d), treated in ICUs (e), and recovered (f) as given in (7.32) with varying parameter φ = (0.1, 0.25, 0.35, 0.4, 0.45, 0.5).

It is very clearly seen from Fig. 7.8 that, as φ increases, the number of asymptomatic cases also increases, and this critical early diagnostic strategy has helped to reduce the number of hospitalizations (IH ) and that of severely sick patients (IC ). The solution trajectories of the exposed E population onto the asymptomatic IA , symptomatic with mild IM , isolated in hospitals IH , and critically sick IC are presented respectively in Fig. 7.9(a)–(d).

7.3.5 Simulation of the optimal control problem In this section, we evaluate the efficacy of our three control measures: personal protection and social distancing, diagnostic and isolation of asymptomatic cases, and diagnostic

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FIGURE 7.9 Solution trajectories (E, IA ), (E, IM ), (E, IH ), and (E, IC ) for problem (7.32) with fixed parameter φ = 0.25.

and isolation of mild symptomatic cases (that is, u1 , u2 , and u3 are all nonzero). First, we simulate the problem in (7.39) considering nonoptimal control measures. We consider three combinations (u1 = 0.75, u2 = 0.5, u3 = 0.5), (u1 = 0.5, u2 = 0.3, u3 = 0.3), and (u1 = 0.25, u2 = 0.2, u3 = 0.2). The simulated results are given in Fig. 7.10. According to Fig. 7.10, it is clearly seen that when the control measures are increased, the curves flatten and the peak occurs with a delay so that the public health system and hospitals can be prepared to handle the outbreak. The cost functional given in (7.41) is used to compute the associate cost for the government if nonoptimal control measures are introduced. The cost incurred if u1 = 0.75, u2 = 0.5, u3 = 0.5 is 4.9214 × 106 , if u1 = 0.5, u2 = 0.3, u3 = 0.3 is 4.0192 × 106 , and if u1 = 0.25, u2 = 0.2, u3 = 0.2 is 3.6519 × 106 . The main goal of the optimal control problem presented in (7.39)–(7.49) is to minimize the number of exposed (E), asymptomatic infected cases (IA ), and mild symptomatic infected cases (IM ). In the public health point of view, it is aimed to reduce the number of patients who are in the community and able to transmit the virus and to isolate them in designated hospitals. The simulation of the optimal control problem (7.39)–(7.49) is performed over three scenarios based on the relative importance of the three control measures. The parameters are β1 = 0.5, β2 = 0.6, β3 = 0.45, γ1 = 0.5, γ2 = 0.2, γ3 = 0.05, η = 0.005,

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FIGURE 7.10 Simulated solution curves for the exposed (a), asymptomatic (b), symptomatic with mild (c), isolated in hospitals (d), treated in ICUs (e), and recovered (f) as given in problem (7.39) considering combinations of nonoptimal control measures (u1 = 0.75, u2 = 0.5, u3 = 0.5), (u1 = 0.5, u2 = 0.3, u3 = 0.3), and (u1 = 0.25, u2 = 0.2, u3 = 0.2).

μ = 0.04, σ = 1/5, φ = 0.25, k = 0.00405, and q = 0.0004. The initial conditions for the problem are S(0) = 0.85, E(0) = 0, IA (0) = 0, IM (0) = 0, IH (0) = 0, IC (0) = 0, and R(0) = 0.

7.3.5.1 Scenario 1 We assume that the social distancing and personal protection measures are highly important, while the costs on the diagnostic and isolation of the two types of cases are equal. The simulated outcomes for each population E, IA , IM , IH , IC , and R are presented in Fig. 7.11, while the time-invariant functions u1 (t), u2 (t), and u3 (t) are illustrated in Fig. 7.12. It is seen from Fig. 7.11 that the control interventions are effective since the number of cases for each E, IA , and IM population has reduced compared to that for the problem without control in (7.32). Further, it is seen that the peak of each curve has been reduced and it is delayed. Thus the optimal control measures have helped to flatten the curve. The control functions in Fig. 7.12 suggest that tracing, testing, and isolation of both asymptomatic and symptomatic infections are required for the entire period of time considered for the simulation.

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FIGURE 7.11 Simulated solution curves for the exposed (a), asymptomatic (b), symptomatic with mild (c), isolated in hospitals (d), treated in ICUs (e), and recovered (f) for the optimal control problem given in (7.39)–(7.49) with A1 = 50, A2 = 75, A3 = 60, C1 = 8, C2 = C3 = 2. It is assumed that the relative cost for social distancing and personal protection is high.

7.3.5.2 Scenario 2 We assume that tracing, testing, and isolating asymptomatic cases are more critical. The simulated outcomes for each population E, IA , IM , IH , IC , and R are presented in Fig. 7.13, while the time invariant functions u1 (t), u2 (t), and u3 (t) are illustrated in Fig. 7.14. It is also seen from Fig. 7.11 that the control interventions are effective since the number of cases for each of E, IM , IH , IC populations has reduced compared to that for the problem without control. All three control interventions are needed in their full capacity during the initial stage of the outbreak, according to Fig. 7.14.

7.3.5.3 Scenario 3 We assume that social distancing with personal protection and tracing, testing, and isolating mild asymptomatic cases are equally critical. The simulated outcomes for each population E, IA , IM , IH , IC , and R are presented in Fig. 7.15, while the time invariant functions u1 (t), u2 (t), and u3 (t) are illustrated in Fig. 7.16. According to Fig. 7.15, if the health system focuses equally on social distancing and personal protection and tracing of asymptomatic cases, then the peak of the exposed, asymptomatic, symptomatic, hospitalized, and ICU treated cases can be minimized; on

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FIGURE 7.12 Optimal control profiles u1 (t), u2 (t), and u3 (t) with A1 = 50, A2 = 75, A3 = 60, C1 = 8, C2 = C3 = 2.

FIGURE 7.13 Simulated solution curves for the exposed (a), asymptomatic (b), symptomatic with mild (c), isolated in hospitals (d), treated in ICUs (e), and recovered (f) for the optimal control problem given in (7.39)–(7.49) with A1 = 50, A2 = 75, A3 = 60, C1 = 5, C2 = 8, and C3 = 2. It is assumed that the relative cost for tracing and testing asymptomatic cases is high.

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FIGURE 7.14 Optimal control profiles u1 (t), u2 (t), and u3 (t) with A1 = 50, A2 = 75, A3 = 60, C1 = 5, C2 = 8, C3 = 2.

FIGURE 7.15 Simulated solution curves for the exposed (a), asymptomatic (b), symptomatic with mild (c), isolated in hospitals (d), treated in ICUs (e), and recovered (f) for the optimal control problem given in (7.39)–(7.49) with A1 = 50, A2 = 75, A3 = 60, C1 = 9, C2 = 9, and C3 = 3. It is assumed that the relative cost for tracing and testing symptomatic cases is high while less importance is given for social distancing and personal protection.

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FIGURE 7.16 Optimal control profiles u1 (t), u2 (t), and u3 (t) with A1 = 50, A2 = 75, A3 = 60, C1 = 9, C2 = 9, C3 = 3.

the other hand, each peak can be delayed. Therefore it can be stated that this control strategy is successful as the government needs to encourage more social distancing and personal protection practices together with effective tracing, testing, and isolation strategy for the patients who do not show symptoms. The algorithm for the optimal control problem was iterated 100 times until the optimal solutions were found. The cost functional given in (7.41) was evaluated in each iteration, and the behavior of execution is given in Fig. 7.17. The convergence of cost to its optimal value of 9.035 × 105 units for Scenario 1 can be clearly seen, while it is 11.88 × 105 units for Scenario 2; however, for Scenario 3, the cost is as small as 1.95 × 103 .

7.4 Mathematical model with imported cases This model is an extension to the model discussed in Section 7.3 where heterogeneity of the cases due to comorbidity is considered. Thus it was assumed that clinically tested positive patients for COVID-19 are not homogeneous, but their age, gender, and history of chronic diseases are considered as those may worsen severity and lead to ICU treatment [32]. Again, a group of patients Ic is added to the model representing the severe cases of COVID-19. These severe patients are absorbed by the available ICU beds until they reach capacity. Similar to the model in Section 7.3, the recovery can happen in two ways. First, the patients in Im class may show mild symptoms of COVID-19, and eventually all of them recover fully. Second, the condition may become severe based on the patients’ age, gender, lifestyle (smoking or alcohol addicted), and presence of chronic diseases. However,

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FIGURE 7.17 The behavior of cost functional given in (7.41) with respect to iterations.

they recover due to the ICU treatment they receive, while a small proportion die. While all other parameters are the same, new parameters are introduced to the model: δ is the rate at which a patient’s level becomes critical. This should be in a measurable functional form with respect to several demographic variables. Further, γ1 is the recovery rate of the patients who show mild symptoms, while γ2 is the recovery rate of the severely ill patients who are treated in ICUs, where γ1 > γ2 . In this model, the parameter u ∈ (0, 1) is the control parameter, and it depends on aggregated public health interventions and social distancing control variables. It is further assumed that the patients in ICU are isolated from the public and they are not responsible for the community transmission [12,32]. In the context of Sri Lanka, the island got the virus mainly from the people who arrived from overseas. Thus we introduce a new compartment A representing the class of people who arrived to the island from overseas. Further, we let λ be the rate of new arrivals through the airport, k1 be the net rate of returned individuals who are cleared as negative to the virus and become again susceptible, and k2 be the net rate of returned individuals who are tested and clinically identified as infected. After March 17, 2020, the government introduced the mandatory quarantine requirement for all the travelers returning from overseas immediately after their arrival to the country. Thus we assume that their ability to transmit the virus to the community is negligible. The dynamic can now be represented according to the following schematics diagram. (See Fig. 7.18.) The dynamic of this transmission can now be represented by the following differential system: dS = −(1 − u)βSIm + k1 A dt

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FIGURE 7.18 Schematic diagram of the model with imported cases.

dA dt dE dt dIm dt dIc dt dR dt

= λ − k1 A − k2 A = (1 − u)βSIm − σ E

(7.50)

= σ E + k2 A − γ1 Im − δIm = δIm − γ2 Ic − μIc = γ 1 I m + γ2 I c

7.4.1 Sensitivity of the control measures (NPIs) The aim of this model is to assess the efficacy of NPIs generally considered as combinations of control measures; however, the timing of control measures is not included in the simulation. We vary the control parameter that addresses the combined impact of government social distancing control measures [14]. The outcome of this simulation is given in Fig. 7.19. It can be seen that as the efficacy of the control measure increases, the epidemic curve flattens, and also it is possible to delay the peak of the outbreak so that the national health system is able to treat patients without getting overwhelmed [32]. Fig. 7.20 shows how it is possible to reduce the maximum of the curve representing the patients who show mild symptoms and the maximum of the curve representing critically ill patients for the given period of time. The figure clearly shows that the maximum number of mild disease and critical disease cases could have declined when the NPIs were increased up to 80%. Fig. 7.21 shows how useful these NPIs were in averting mild infections and critically ill cases compared with no such NPIs in place. If the combined measures had been implemented up to 80%, then there would have been 97% mild and critical cases prevented. On the other hand, if those measures were effective only at 20%, then only 15% and 17% mild and critical cases could have been averted, respectively. Fig. 7.22 demonstrates the impact of changes in NPIs from one level to another in the reducing basis. The highest impact can be seen when the NPIs were decreased from 60% to 50% and again from 50% to 40%. In each case, the number of mild and critical cases has been more than double.

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FIGURE 7.19 The simulation of model (7.50) considering the varying level of the control parameter u. The rest of the parameter values used for the simulation are β = 0.7, γ1 = 0.24, γ2 = 0.05, μ = 0.02, σ = 1/4, δ = 0.025/3, λ = 0.000205, k1 = 0.6, and k2 = 0.4 [32].

FIGURE 7.20 The change in the peak of mild cases and critical cases with respect to the combined control parameter u.

7.4.2 Sensitivity of the control with overseas exposed cases The government of Sri Lanka decided to minimize the rate of imported exposed cases to the island five days after the first local COVID-19 patient was identified though a policy decision to reduce international arrivals. However, the authorities were flexible, and Sri Lankan citizens were permitted to enter the country from selected countries for about

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FIGURE 7.21 Percentage of mild and critical COVID-19 cases averted compared with no NPIs.

FIGURE 7.22 Impact of reducing NPIs between different levels on mild and critical cases.

three more days. Thereafter, the decision was made to shut down the international airport for all arrivals to the country. The impact of this decision was also analyzed using the model through parameter λ. We defined this parameter applicable during the very early stage of the outbreak as a step function in time such that ⎧ ⎪ ⎪ ⎨0.041505, if t ≤ 5; λ(t) = 0.006105, if 5 ≤ t < 8; ⎪ ⎪ ⎩0.000003, if t ≥ 8c,

(7.51)

where t is the number of days after the first Sri Lankan case of COVID-19 was identified by the health authorities in the country [6]. The outcome of this simulation is given in Fig. 7.23. Fig. 7.24 shows how the peaks of the infected curve can be reduced with respect

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to the decision made to stop international arrivals of those who can be exposed to the virus.

FIGURE 7.23 The simulation of model (7.50) considering the varying level of the control parameter u. The rest of the parameter values used for the simulation are β = 0.7, γ1 = 0.24, γ2 = 0.05, μ = 0.02, σ = 1/4, δ = 0.025/3, λ = 0.000205, k1 = 0.6, and k2 = 0.4 [1,5,12,32].

7.4.3 Sensitivity of the timing of implementing combined control measures It is critical to introduce social distancing control measures timely to maximize the health benefits of such measures [5]. Most of the European Union (EU) countries found to have misjudged the scheduling of these measures; eventually, they ended up stressing their health system due to a large influx of critically ill cases that required intensive care treatments. In this simulation, we varied the time in days from the date when the first local case was identified until the government decided to impose drastic social distancing control measures (80% efficacy). It is assumed that until this date, the authorities were very flexible, and they are at a mild level of restrictions (20% efficacy) [1,5]. Fig. 7.25 shows how the dynamic has changed with respect to this time parameter denoted by (T r). Fig. 7.26 demonstrates how vital that policy decision was in the point of view of the country’s healthcare system. According to Fig. 7.25, it is very clear that this threshold value in time is critical to minimize the burden to this highly contagious COVID-19. It suggests that if the government had introduced 80% social distancing control measures within five days of the first local case was identified, we may have prevented the steeper growth of cases and reduced the size of the peak significantly. The model also demonstrates that if we wait nearly for 30

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FIGURE 7.24 The change in the peak of mild cases and critical cases with respect to NPIs governed by parameter u and the decision made to stop overseas exposed cases informed through λ.

FIGURE 7.25 The simulation of model (7.50) considering the varying threshold time to impose strong social distancing control measures.

days to impose control measures, then we are likely to experience a large peak of cases. Fig. 7.26 illustrates the relationship between time to introduce measures and the peak of cases showing how rapidly it grows with each delay. Fig. 7.27 shows the percentage of mild and critical cases averted for different delays it took the authorities to implement the con-

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trol restriction compared with the case that they were only imposed after waiting for 30 days.

FIGURE 7.26 The change in the peak of mild cases and critical cases with respect to the threshold time to impose strong social distancing control measures.

FIGURE 7.27 Percentage of mild and critical cases averted compared with if restrictions were delayed for further 30 days from now.

7.5 Conclusion The wide spread transmission of COVID-19 is the major public health, social, and global economic concern at the moment. The entire world is transforming into new norms to live with this novel virus. At the time of writing, several vaccine candidates have been identi-

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fied and clinically tested as successful in fighting against the virus; however, widespread vaccination has not been started due to lack of production, cost of storage and distributions, and various other policy-related issues. In addition, scientists have not been yet completely aware of the entire efficacy of these vaccines and how likely the population can be protected against re-infections [41]. As long as neither herd immunity through infections nor the vaccine-based immunization for the whole world population is not expected to be achieved in the foreseeable future, the entire world has to wait a long time to become free from COVID-19. This means that people may have to learn how to live with frequent outbreaks in the future. Due to these facts, it is very important to continue with social distancing and public health interventions so that the disease transmission can be reduced. Thus various health policies introduced by the governments could be seen for a few more years. In addition, mathematical models may be extensively used, and they would be supportive to the policy makers to evaluate, modify, and maintain their public health decisions. This chapter focused on a retrospective analysis of the Sri Lankan response to the COVID-19 outbreak in its initial phrase using three types of mathematical models that are in the SEIR (Susceptible-Exposed-Infected-Recovered) structure. However, they were extended adjusting them to the epidemiological context and the country’s profile. The first model was calibrated to infection data in Sri Lanka during the first 82 days of the epidemic using optimization techniques aiming to estimate the average overall transmission rate at the beginning and to find an approximation to the initial size of the epidemic. The results show that the optimal average transmission rate, the initial amount of the exposed cases, and the amount of the infected cases are estimated respectively as β = 0.3582, E0 = 17, and I0 = 0. The second model was a theoretical optimal control problem considering the symptomatic and asymptomatic nature of the disease transmission while addressing the cascade of disease progression; that is, mild cases, hospitalized cases, and critical cases. Three types of control parameters were introduced to the model; (1) personal protection and social distancing, (2) diagnostic and isolation of asymptomatic cases, and (3) diagnostic and isolation of mild symptomatic cases. The aim was to find optimal time-varying allocation of resources of disease control over these three categories of control measures. It should be noted that the control variables have not been clearly defined in their functional form with respect to multiple epidemiological, health, and other economic factors at this stage. This was mainly due to the limitation of data availability; thus the implementation was hypothetical in numerous aspects. The third model considered for the study accounted for the importation of cases from overseas at the very early stage of the outbreak in Sri Lanka, and the model also addressed the homogeneity of the population through demographic variability. Critical parameters, such as control measures, bundled as a collection of NPIs, the timing of the introduction of NPIs, and the importation of cases, were considered mainly for the sensitivity analysis. Projecting the dynamic and investigating the disease progression trend were important aspects included in the Sri Lankan COVID-19 response from the beginning due to foreseen

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challenges that the country’s health system would have experienced. This study revealed that increasing NPIs as an early response to combat the virus could have been beneficial because the size of the mild and critical case peaks was declining in return. Compared with a baseline of no NPIs, an early 80% level of NPIs in the society could have averted 97% of mild and critical cases each, whereas 20% of combined control measures would have only averted 15% and 17% of mild and critical cases, respectively. The impact of changing the level of gross interventions was higher when they range between 60% to 40%. The public health recommendation by the global health scientists was not only limited on the packaging the interventions but as to impose them at optimal time points during the initial phase. It was found that the more introducing measures was delayed, the more health disadvantage through rising of mild and critical cases was observed. In Sri Lanka, the model suggested that immediate measures that were put in place could have prevented 48% of mild infections and 51% of critical infections compared with the case that they were implemented with a 30-day waiting time. A delay up to 25 days could have dropped respective averted benefits to 20% and 21%, implying a large health cost as a consequence of failure to act on time. This study focused on mathematical modeling applications to best understand the dynamic of immature COVID-19 epidemic in Sri Lanka during it first 82 days mainly. Thus the limitations due to lack of knowledge about the disease and information on parameters, health implications as well as providing policy lead interpretations of model-based outcomes have been obvious. However, the models developed can still be adopted to a range of COVID-19 epidemiological contexts emerging from time to time. For example, determining the optimal vaccine roll out strategy, implications on complexities such as waning immunity, evidences on possible reinfections, evaluation of trade-off between vaccineinduced immunity and management of NPIs, and combating the outbreaks due to novel COVID-19 strains could be potential future applications using these mathematical models. It is well known that the pandemic has damaged most of the economies across the globe, we as well as the world have accepted that foreseeable future will be different to pre-pandemic era [42]. Therefore balancing the economy with the public health well-being would be challenging. Future research developments in policy decision-making would then be interesting and worth if the economic, financial, management, and educational models were to couple with mathematical models in epidemiology in this class.

References [1] P. Boldog, T. Tekeli, Z. Vizi, Risk assessment of novel coronavirus Covid-19 outbreaks outside China, Journal of Clinical Medicine, MDPI 9 (2020) 571, https://doi.org/10.3390/jcm9020571. [2] F. Zhou, T. Yu, R. Du, F. Fan, Clinical course and risk factors for mortality of adult inpatients with Covid-19 in Wuhan, China: a retrospective cohort study, The Lancet 2020 (2020) 1054–1062, https:// doi.org/10.1016/S0140-6736(20)30566-3. [3] https://www.worldometers.info/coronavirus/. [4] WHO, DRAFT landscape of COVID-19 candidate vaccines – 20 April 2020, https://www.who.int/ blueprint/priority-diseases/key-action/novel-coronavirus-landscape-ncov.pdf. [5] J. Hellewell, S. Abbott, N. Bosse, Feasibility of controlling Covid-19 outbreaks by isolation of cases and contacts, Lancet Global Health 2020 (2020), https://doi.org/10.1016/S2214-109X(20)30074-7.

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[28] I. Abouelkheir, F.E. Kihal, M. Rachik, Time needed to control an epidemic with restricted resources in SIR model with short-term controlled population: a fixed point method for a free isoperimetric optimal control problem, Mathematical and Computational Applications, MDPI 23 (2018) 64, https:// doi.org/10.3390/mca23040064. [29] O. Onyejekwe, Z. Kebede, Epidemiological modeling of measles infection with optimal control of vaccination and supportive treatment, Applied and Computational Mathematics 4 (4) (2015) 264–274. [30] P.V. Driessche, Reproduction numbers of infectious disease models, Advanced Research Evolving Science 2 (2017) 288–303. [31] P.V. Driessche, J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences 180 (2002) 29–48. [32] N.M. Ferguson, D. Laydon, Impact of non-pharmaceutical interventions (NPIs) to reduce Covid-19 mortality and healthcare demand, Imperial College Covid-19 Response Team, 2020. [33] Q. Lin, S. Zhao, A conceptual model for the coronavirus disease 2019 (Covid-19) outbreak in Wuhan, China with individual reaction and governmental action, International Journal of Infectious Diseases 93 (2020) 211–216. [34] K. Prem, Y. Liu, The Effect of Control Strategies to Reduce Social Mixing on Outcomes of the COVID-19 Epidemic in Wuhan, China: a Modelling Study, vol. 20, Elsevier Ltd, 2020. [35] P. Boldog, T. Tekeli, Risk Assessment of Novel Coronavirus COVID-19 Outbreaks Outside China, Journal of Clinical Medicine 9 (2020). [36] C. Kirkeby, T. Halasa, Methods for estimating disease transmission rates: evaluating the precision of Poisson regression and two novel methods, Scientific Reports 7 (2017) 1–11. [37] S. Sanche, Y.T. Lin, High contagiousness and rapid spread of severe acute respiratory syndrome coronavirus 2, Emerging Infectious Diseases 26 (7) (2020) 1470–1477. [38] B. Ridenhour, J.M. Kowalik, Unraveling R0: considerations for public health applications, American Journal of Public Health 104 (2014) 32–41. [39] M. Raissi, N. Ramezani, On parameter estimation approaches for predicting disease transmission through optimization, deep learning and statistical inference methods, Letters in Biomathematics (2019) 1–26. [40] A.A. Ding, H. Wu, Estimation of ordinary differential equation parameters using constrained local polynomial regression, HHS Public Access 24 (2014) 1613–1631. [41] C. Rio, P. Malan, COVID-19 in 2021—Continuing Uncertainty, American Medical Association, 2021. [42] A.W. Bartik, M. Bertrand, The impact of Covid-19 on small business outcomes and expectations, Proceedings of the National Academy of Sciences of the United States of America 117 (30) (2020) 17656–17666.

8 Global stability of a diffusive HTLV-I infection model with mitosis and CTL immune response Noura H. AlShamrania and Ahmed M. Elaiwb,c a Department of Mathematics, Faculty of Science, University of Jeddah, Jeddah, Saudi Arabia b Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah, Saudi Arabia c Department of

Mathematics, Faculty of Science, Al-Azhar University, Assiut Branch, Assiut, Egypt

List of abbreviations HIV HTLV-I HAM/TSP ATL DNA RNA CTL PDEs

Human immunodeficiency virus Human T-lymphotropic virus type I HTLV-associated myelopathy/tropical spastic paraparesis Adult T-cell leukemia/lymphoma Deoxyribonucleic acid Ribonucleic acid Cytotoxic T lymphocyte Partial differential equations

8.1 Introduction Human T-lymphotropic virus type I (HTLV-I) is a provirus that targets uninfected CD4+ T cells. HTLV-I can spread to uninfected CD4+ T cells from cell-to-cell through the virological synapse [1]. HTLV-I is a single-stranded ribonucleic acid (RNA) retrovirus that reverse transcribes its RNA genome into a proviral deoxyribonucleic acid (DNA) copy, which in turn reaches the host chromatin and integrates into the DNA of the host genome, at which point the virus is referred to as a provirus. Later, a cell infected with this virus enters a latent period, and it is not capable to produce DNA and infects uninfected cells. Although latent HTLV-infected cells can survive for a long time, they may be suddenly activated by antigen and become able to infect uninfected cells. During the primary infection stage of HTLV-I, the proviral load can reach a high level, approximately 30%–50% [2]. HTLV-I infects the human body and can lead to two diseases, HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP) and adult T-cell leukemia/lymphoma (ATL). Only a small percentage of infected individuals develop the disease, 2%–5% percent of HTLV-I carriers develop symptoms of ATL, and another 0.25%–3% develop Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00013-2 Copyright © 2023 Elsevier Inc. All rights reserved.

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HAM/TSP [3]. Still, there is no cure or licensed vaccine or therapeutics or effective treatment available for HTLV-I-associated pathologies-infection is life-long [4], [5]. The discovery of the first human retrovirus HTLV-I dates back to 1980, and after three years the human immunodeficiency virus (HIV) was determined [6]. HTLV-I is a global epidemic that infects about 10–25 million persons [7]. The infection is endemic in the Caribbean, southern Japan, the Middle East, South America, parts of Africa, Melanesia, and Papua New Guinea [8]. HTLV-I can be transmitted from an infected individual to an uninfected one through sexual, parenteral transmission (blood transfusions, organ transplantation, and via infected sharp objects), and from mother to child (breast-feeding and during delivery).

8.1.1 Mathematical models Mathematical models of HTLV-I dynamics are helpful tools to understand the pathogenesis of the HTLV-I infection within the host and to recognize the risk factors of the progression of HAM/TSP and ATL.

8.1.1.1 Model with latent HTLV-infected cells Stilianakis and Seydel [9] formulated an HTLV-I model to describe the interaction of uninfected CD4+ T cells, latent HTLV-infected cells, Tax-expressing HTLV-infected cells, and leukemia cells (ATL cells) as follows: ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

dS(t) dt = ρ − αS(t) − ηS(t)Y (t), dE(t) dt = ηS(t)Y (t) − (ψ + ω) E(t), dY (t) dt = ψE(t) − (ϑ + δ)Y  (t),  dZ(t) Z(t) dt = ϑY (t) + Z(t) 1 − Zmax − θ Z(t),

(8.1)

where S(t), E(t), Y (t), and Z(t) are the concentrations of uninfected CD4+ T cells, latent HTLV-infected cells, Tax-expressing HTLV-infected cells, and ATL cells at time t, respectively. The transmission of HTLV-I to uninfected CD4+ T cells can be only from cell-to-cell, that is, the HTLV-I virions can only survive inside the host CD4+ T cells and cannot be detectable in the plasma. Uninfected CD4+ T cells are created at rate ρ and die at rate αS. The rate at which new infection appears by cell-to-cell contact between Tax-expressing HTLV-infected cells and uninfected CD4+ T cells is assumed to be ηSY . The natural death rate of latent HTLV-infected cells, Tax-expressing HTLV-infected cells, and ATL cells is represented by ωE, δY , and θ Z, respectively. The term ψE accounts for the rate of latent HTLV-infected cells become Tax-expressing HTLV-infected cells. ϑY is the transmission rate at which  Tax-expressing HTLV-infected cells convert to ATL cells. The logistic term Z Z 1 − Zmax denotes the proliferation rate of ATL cells, where Zmax is the maximal concentration that ATL cells can grow. The parameter is the maximum proliferation rate constant of ATL cells. Many researchers have been concerned to study mathematical modeling and analysis of HTLV-I infection in several works. In 2002 Wang et al. [10] carried out a complete

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mathematical analysis for the global dynamics of model (8.1). A mathematical model that takes into account both infectious and mitotic routes of viral transmission as well as the role of HTLV-specific cytotoxic T lymphocytes (CTLs) immune response was constructed by Gómez-Acevedo and Li in 2005 [11]. Li and Lim extended the model given in [11] by incorporating the role of the viral protein Tax in the persistence of HTLV-I infection [12]. The model given in [12] was shown to be a cooperative system if a sharp condition was satisfied. This means that the global dynamics of the model was studied under the validity of a sharp condition. If this condition was not satisfied, the authors left it as an open problem. After that, Vargas-De-León [13] provided a complete global analysis of the dynamics of the HTLV-I infection given in [12], which completely solved the open problem left before. Song and Li [14] developed model (8.1) by considering a saturation response of the infection rate and presented a complete mathematical analysis of the global dynamics of the model. The force of infection was assumed to be a function in a general form by Cai et al. in 2011 [15].

8.1.1.2 Model with delay It has been reported in [2] and [16] that the biological rationale for introducing a delay at the stage when the cell mutates from uninfected to latent lies in the fact that there is a waiting period between the time when the virus contacts the cell and the time when the viral RNA is incorporated into the DNA of the host genome. Since the proviral load varies from patient to patient and since there is a direct correlation between the development of disease and the proviral load, it would be logical to consider that during the delayed time some mechanism, by chance, intervenes to prevent or enhance the speed with which the cell will become latent infected. Based on this fact, Katri and Ruan [16] incorporated a discrete-time delay in an HTLV-I dynamics model (8.1) to describe the time between the emission of contagious particles by Tax-expressing HTLV-infected cells and the infection of uninfected CD4+ T cells as follows: ⎧ dS(t) ⎪ dt = ρ − αS(t) − ηS(t)Y (t), ⎪ ⎪ ⎪ ⎪ ⎨ dE(t) ˆ (t − θ ) Y (t − θ ) − (ψ + ω) E(t), dt = ηS (8.2) dY (t) ⎪ ⎪ dt = ψE(t) − (ϑ + δ)Y (t), ⎪   ⎪ ⎪ ⎩ dZ(t) = ϑY (t) + Z(t) 1 − Z(t) − Z(t), dt Zmax where θ is a positive constant representing the length of the delay and ηˆ > 0.

8.1.1.3 Model with mitosis As mentioned before, HTLV-I has two modes of transmission, the first is the horizontal transmission via direct cell-to-cell contact [1], and the second is the vertical transmission through mitotic division of Tax-expressing HTLV-infected cells [4]. Tax-expressing HTLVinfected cells proliferate faster than uninfected CD4+ T cells and latent HTLV-infected cells. This leads to an increase in the proviral load. Therefore vertical mitotic transmission plays

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an important role in the persistence of HTLV-I infection [4]. Li and Lim [12] formulated an HTLV-I dynamics model that takes into account both horizontal and vertical routes of transmission as follows: ⎧ dS(t) = ρ − αS(t) − ηS(t)Y (t), ⎪ ⎪ ⎨ dt   dE(t) ∗ Y (t) 1 − S(t)+E(t) − (ψ + ω) E(t), (8.3) = ϕηS(t)Y (t) + κr dt K ⎪ ⎪ ⎩ dY (t) ∗ dt = ψE(t) − δ Y (t). The fraction ϕ ∈ (0, 1) is the probability that new HTLV infections via horizontal transmission could enter a latent period. The other route of transmission for HTLV is vertical caused by selective expansion of Tax-expressing CD4+ T cells that are driven into proliferation by  S+E ∗ + HTLV Tax gene at rate r Y 1 − K , where K is the CD4 T cells carrying capacity. The  S+E ∗ term κr Y 1 − K accounts for HTLV-infected cells being latent and therefore escaping from the immune system, where κ ∈ (0, 1). Asquith and Bangham [17] reported that even in the presence of rapid selective mitotic division target cell populations were less than the total CD4+ T cells carrying capacity, i.e., S + E < K. Therefore Lim and Maini [7] replaced  S+E ∗ the logistic term r Y 1 − K by an exponential growth term r ∗ Y .

8.1.1.4 Model with CTL immune response It is well known that CTLs are considered a significant component of the human immune response against viral infections [20]. CTLs inhibit viral replication and kill the cells that are infected by viruses. It has been reported in [4], [17–19] that CTLs play an effective role in controlling the HTLV-I infection. CTLs can recognize and kill Tax-expressing HTLV-infected cells; moreover, they can reduce the proviral load. The within-host HTLV-I dynamics model with CTL-mediated immune response is given by [20] ⎧ dS(t) = ρ − αS(t) − ηS(t)Y (t), ⎪ ⎪ ⎨ dt dY (t) Y (8.4) dt = ϕηS(t)Y (t) − δY (t) − μC (t)Y (t), ⎪ ⎪ Y ⎩ dC (t) Y Y = σ C (t)Y (t) − πC (t), dt where C Y (t) is the concentration of HTLV-specific CTLs at time t. The newly HTLV-infected cells experience irreparable destruction by the strong immune response. Consequently, a small fraction ϕ ∈ (0, 1) of the newly infected CD4+ T cells survive after the immune attack and become Tax-expressing, while the remaining part 1 − ϕ dies off. The term μC Y Y is the killing rate of Tax-expressing HTLV-infected cells due to their specific immunity. The proliferation rate for HTLV-specific CTLs is given by σ C Y Y . The natural death rate of HTLVspecific CTLs is represented by πC Y . In the literature, several mathematical models have been proposed to describe the dynamics of HTLV-I under the effect of CTL-mediated immune response (see, e.g., [8], [21–31]). Gomez-Acevedo and colleagues [23] investigated an HTLV-I infection mathematical model by incorporating the response of the HTLV-I specific CTLs. They performed a rigorous mathematical analysis including stability analysis for viral infection and CTL response. They also derived the threshold quantities 0 and 1 for

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the HTLV-I infection and the CTL response, respectively. Lang and Li [24] studied a mathematical model with a more general form for the CTL response and investigated that their model undergoes complex behavior by introducing a sigmoidal response function. Their model exhibits stable and unstable periodic behaviors due to the occurrence of Hopf bifurcation. Li and Shu [25] studied a delayed model for HTLV-I infection, and they performed a Hopf bifurcation analysis, where they also derived threshold quantities 0 and 1 . In another study, Li and Shu [26] presented a model to describe the transmission dynamics of HTLV-I infection with delayed CTL response, and they performed a global stability analysis by formulating a proper Lyapunov function. They also derived threshold quantities 0 and 1 for HTLV-I infection and the CTL response, respectively. In 2020 Wang et al. [28] studied the global dynamics of an HTLV-I infection model with intracellular distributed time delay, nonlinear incidence rate of infection, nonlinear lytic and nonlytic CTL immune responses, and immune impairment. In 2013 Muroya et al. [29] applied Lyapunov functional techniques to a delayed HTLV-I infection model with a nonlinear incidence rate of the form ηSG(Y ) and an HTLV-specific CTL immune response. Nakata et al. [30] performed the global stability analysis for an HTLV-I infection model with intracellular latent delay and CTL response. It is well known that the CTL immune response could be broadly classified into lytic and nonlytic components. Wang and his colleagues [31] developed a model of HTLV-I infection with two time delays: an intracellular delay and a CTL immune response delay.

8.1.1.5 Model with mitosis and CTL immune response To include both the mitotic division of Tax-expressing HTLV-infected cells and the CTL immune response in the HTLV-I dynamics, Lim and Maini [7] proposed the following model: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

dS(t) dt = ρ − αS(t) − ηS(t)Y (t), dE(t) ∗ dt = ηS(t)Y (t) + κr Y (t) − (ψ + ω) E(t), dY (t) ∗ Y dt = ψE(t) − δ Y (t) − μC (t)Y (t), dC Y (t) = σ Y (t) − πC Y (t). dt

(8.5)

Here the proliferation rate of CTLs is represented by linear function σ Y . The vertical transmission of HTLV-I is caused by selective expansion of Tax-expressing CD4+ T cells that are driven into proliferation by the HTLV Tax gene at rate r ∗ Y . The authors performed local and global stability analysis of model (8.5) by constructing suitable Lyapunov functions. Li ηSY and Zhou [32] developed (8.5) by replacing the bilinear incidence ηSY by the form 1+ S. Li and Ma [33] generalized model (8.5) by including a delay CTL immune response and a general incidence rate in the form F (S, Y )Y . In 2021 Khajanchi et al. [34] extended model (8.5) by assuming that the proliferation of HTLV-specific CTLs occurs at a bilinear rate in the form σ2 C Y Y . The authors performed the global stability for chronic infection steady state by using the Lyapunov method and geometric approach. The local and global stability conditions have been derived by using the threshold parameters 0 and 1 .

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8.1.1.6 Model with diffusion All of the above-mentioned HTLV dynamics models did not include the diffusion of the cells. Wang and Ma [35] introduced a diffusive HTLV-I infection model with mitotic division of Tax-expressing infected cells and CTL immune response as follows: ⎧ ∂S(x,t) = dS S(x, t) + ρ − αS(x, t) − ηS(x, t)Y (x, t), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎨ ∂E(x,t) = dE E(x, t) + ϕηS(x, t)Y (x, t) + r ∗ Y (x, t) − (ψ + ω) E(x, t), ∂t (8.6) ∂Y (x,t) ⎪ = dY Y (x, t) + ψE(x, t) − δ ∗ Y (x, t) − μC Y (x, t)Y (x, t), ⎪ ∂t ⎪ ⎪ ⎪ ⎩ ∂C Y (x,t) = dC Y C Y (x, t) + σ Y (x, t) − πC Y (x, t), ∂t ∂2 is the Laplacian operator, while dU is ∂x 2 the diffusion coefficient corresponding to component U of the model. In model (8.6), the proliferation rate of CTLs is given by a linear function σ Y . The aim of this chapter is to formulate and analyze a diffusive HTLV-I infection model with bilinear proliferation rate of CTLs. We show that the model is well posed by establishing that the solutions of the model are nonnegative and bounded. We derive two threshold parameters 0 and 1 , which govern the existence and stability of the three steady states of the model. Global stability of all steady states is proven by formulating Lyapunov functions and utilizing the Lyapunov-LaSalle asymptotic stability theorem. We perform some numerical simulations to illustrate the theoretical results.

where x = (x1 , x2 , ..., xm ) is the position and  =

8.2 Model formulation We consider the following factors: (F1) Bilinear proliferation rate of CTLs in the form σ C Y Y ; (F2) HTLV-I can be transmitted via two routes: (i) horizontal transmission by direct cellto-cell touch via virological synapse and (ii) vertical transmission by mitotic division of Tax-expressing HTLV-infected cells; (F3) Spatial diffusion for all compartments. Taking into account factors (F1)–(F3), we propose the following partial differential equations (PDEs) model: ⎧ ∂S(x,t) = dS S(x, t) + ρ − αS(x, t) − ηS(x, t)Y (x, t), ⎪ ⎪ ⎪ ∂t ⎪ ⎪ ∂E(x,t) ⎨ = dE E(x, t) + ϕηS(x, t)Y (x, t) + κr ∗ Y (x, t) − (ψ + ω) E(x, t), ∂t (8.7) ∂Y (x,t) ⎪ = dY Y (x, t) + ψE(x, t) + (1 − κ) r ∗ Y (x, t) − δ ∗ Y (x, t) − μC Y (x, t)Y (x, t), ⎪ ∂t ⎪ ⎪ ⎪ ⎩ ∂C Y (x,t) = dC Y C Y (x, t) + σ C Y (x, t)Y (x, t) − πC Y (x, t), ∂t where x ∈ , t > 0. The term (1 − κ) r ∗ Y refers to the cells that stay in the Tax-expressing HTLV-infected cells compartment, while κr ∗ Y accounts for the HTLV infected cells be-

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ing latent and therefore escaping from the immune system [32]. The proliferation rate for HTLV-specific CTLs is represented by σ C Y Y . All remaining parameters have the same biological meaning as explained in the previous section. The spatial domain  ⊂ Rm , m ≥ 1 is connected and bounded with a smooth boundary ∂. The initial conditions are given by S(x, 0) = G1 (x),

E(x, 0) = G2 (x),

Y (x, 0) = G3 (x),

C Y (x, 0) = G4 (x),

¯ x ∈ ,

(8.8)

where Gi (x), i = 1, ..., 4, are nonnegative and continuous functions. In addition, we consider the following homogeneous Neumann boundary conditions: ∂S ∂E ∂Y ∂C Y = = = = 0, ∂ V ∂ V ∂ V ∂ V

t > 0,

x ∈ ∂,

(8.9)

∂ is the outward normal derivative on the boundary ∂. The Neumann bound∂ V ary conditions can represent a natural dispersal barrier and indicate that viruses and cells cannot cross the isolated boundary [36]. In [7] it is assumed that r ∗ < υ ∗ = min {α, ω, δ ∗ }, which corresponds to experimental evidence indicating that the proliferation rate of HTLV-infected cells is generally lower than the rate of removal due to natural death. Since r ∗ < δ ∗ and 0 < κ < 1, then (1 − κ) r ∗ < δ ∗ and where

δ ∗ − (1 − κ) r ∗ > 0. Let δ = δ ∗ − (1 − κ) r ∗ and r = κr ∗ . Then system (8.7) will take the following form of PDEs: ⎧ ∂S(x,t) = dS S(x, t) + ρ − αS(x, t) − ηS(x, t)Y (x, t), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ∂E(x,t) ⎨ = dE E(x, t) + ϕηS(x, t)Y (x, t) + rY (x, t) − (ψ + ω) E(x, t), ∂t (8.10) ∂Y (x,t) ⎪ = dY Y (x, t) + ψE(x, t) − δY (x, t) − μC Y (x, t)Y (x, t), ⎪ ∂t ⎪ ⎪ ⎪ ⎩ ∂C Y (x,t) = dC Y C Y (x, t) + σ C Y (x, t)Y (x, t) − πC Y (x, t). ∂t We mention that when the proliferation rate of CTLs is given by linear function σ Y and κ = 1, model (8.10) leads to model (8.6). Moreover, when the immune response is linear and there is no diffusion (i.e., dS = dE = dY = dC Y = 0), model (8.10) tends to the model presented in [7].

8.3 Well-posedness of solutions For mathematical convenience, we assume that uninfected CD4+ T cells, latent HTLVinfected cells, Tax-expressing HTLV-infected cells, and HTLV-specific CTLs have the same ˜ diffusion, i.e., dS = dE = dY = dC Y = d. Lemma 8.1. Under the above assumption, model (8.10) has a unique, nonnegative, and bounded solution defined on ¯ × [0, +∞) for any given initial data satisfying (8.8).

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 ¯ R4 be the set of all bounded and uniformly continuous functions Proof. Let X = BU C ,  ¯ R4+ ⊂ X. Then the positive cone X+ induces a partial from ¯ to R4 , and let X+ = BU C , order on X. Suppose that the norm is defined by θ X = sup |θ (x)|, where |·| is the Euclidean x∈¯

norm on R4 . This implies that the space (X, · X ) is a Banach lattice [37], [38]. For any initial data G = (G1 , G2 , G3 , G4 )T ∈ X+ , we define H = (H1 , H2 , H3 , H4 )T : X+ → X by H1 (G)(x) = ρ − αG1 (x) − ηG1 (x)G3 (x), H2 (G)(x) = ϕηG1 (x)G3 (x) + rG3 (x) − (ψ + ω) G2 (x), H3 (G)(x) = ψG2 (x) − δG3 (x) − μG4 (x)G3 (x), H4 (G)(x) = σ G4 (x)G3 (x) − πG4 (x). It is easy to note that H is locally Lipschitz on X+ . We can rewrite system (8.10) with initial conditions (8.8) and boundary conditions (8.9) as the following abstract functional differential equation:

¯ dU ¯ ¯ dt = U + H (U ), t > 0, ¯ U(0) = G ∈ X+ , where U¯ = (S, E, Y, C Y )T and U¯ = (dS S, dE E, dY Y, dC Y C Y )T . It is possible to show that 1 lim dist (G(0) + hH (G), X+ ) = 0, ∀ G ∈ X+ . + h→0 h It follows from [37–39] that, for any G ∈ X+ , system (8.10) with (8.8)–(8.9) has a unique nonnegative mild solution (S (x, t) , E (x, t) , Y (x, t) , C Y (x, t)) defined on ¯ × [0, Tm ), where [0, Tm ) is the maximal existence time interval on which the solution exists. In addition, this solution also is a classical solution for the given problem. To prove the boundedness of all state variables, we define  (x, t) = S (x, t) +

1 μ Y C (x, t) . [E (x, t) + Y (x, t)] + ϕ ϕσ

˜ then using system (8.10) we obtain Since dS = dE = dY = dC Y = d, ∂(x, t) ω μπ Y (δ − r) ˜ − d(x, t) = ρ − αS(x, t) − E(x, t) − Y (x, t) − C (x, t). ∂t ϕ ϕ ϕσ We have δ − r = δ ∗ − r ∗ > 0. Hence ∂(x, t) ˜ ω μπ Y (δ ∗ − r ∗ ) − d(x, t) = ρ − αS(x, t) − E(x, t) − Y (x, t) − C (x, t) ∂t ϕ ϕ ϕσ  μ Y 1 C (x, t) = ρ − φ(x, t), ≤ ρ − φ S(x, t) + {E(x, t) + Y (x, t)} + ϕ ϕσ

Chapter 8 • Global stability of a diffusive HTLV-I infection model

239

where φ = min{α, ω, δ ∗ − r ∗ , π}. Thus (x, t) satisfies the following system: ⎧ ∂(x,t) ˜ ⎪ − d(x, t) ≤ ρ − φ(x, t), ⎪ ∂t ⎪ ⎨ ∂ = 0, ⎪ ⎪ ∂ V ⎪ ⎩ (x, 0) = G (x) + 1 [G (x) + G (x)] + μ G (x) ≥ 0. 1 2 3 ϕ ϕσ 4

(t) be a solution to the following ordinary differential equation system: Let  ⎧

(t), ⎨ d dt(t) = ρ − φ 

(0) = max(x, 0). ⎩  x∈¯

 ρ

This gives that  (t) ≤ max φ , max(x, 0) . According to the comparison principle [40], we 

x∈¯

(t). Then we get have (x, t) ≤ 

 ρ , max(x, 0) , (x, t) ≤ max φ x∈¯ 

which implies that S (x, t), E (x, t), Y (x, t), and C Y (x, t) are bounded on ¯ × [0, Tm ). We deduce from the standard theory for semilinear parabolic systems that Tm = +∞ [41]. This  shows that solution S (x, t) , E (x, t) , Y (x, t) , C Y (x, t) is defined for all x ∈ , t > 0 and also is unique and nonnegative.

8.4 Steady state analysis In the following lemma, we derive two threshold parameters which guarantee the existence of the steady states of the model. Lemma 8.2. There exist two nonnegative threshold parameters 0 and 1 satisfying the condition 1 ≤ 0 such that system (8.10) has three steady states as follows: (i) If 0 ≤ 1, then the system has a unique steady state Ð0 ; (ii) If 1 ≤ 1 < 0 , then the system has only two steady states Ð0 and Ð1 ; (iii) If 1 > 1, then the system has three steady states Ð0 , Ð1 , and Ð2 . Proof. Let (S, E, Y, C Y ) be any steady state of system (8.10) satisfying the following equations: 0 = ρ − αS − ηSY,

(8.11)

0 = ϕηSY + rY − (ψ + ω) E,

(8.12)

0 = ψE − δY − μC Y,

(8.13)

0 = (σ Y − π) C .

(8.14)

Y

Y

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The straightforward calculation finds that system (8.10) admits three steady states. (i) Infection-free steady state Ð0 = (S0 , 0, 0, 0), where S0 = ρ/α. This case describes the situation of healthy state where the HTLV is absent. (ii) Persistent infection steady state with ineffective HTLV-specific CTL immune response Ð1 = (S1 , E1 , Y1 , 0), where S1 =

S0 αδ α , E1 = (0 − 1) , Y1 = (0 − 1) , 0 ηψ η

and 0 is the basic infection reproduction number for system (8.10), and it is defined as 0 =

ϕηψS0 . (δ − r) ψ + δω

The parameter 0 decides whether or not a persistent infection can be established in the absence of CTL immune response. At the steady state Ð1 , the HTLV infection persists with inactive CTL immune response. (iii) Persistent infection steady state with effective HTLV-specific CTL immune response Ð2 = (S2 , E2 , Y2 , C2Y ), where σρ π [r (πη + ασ ) + ηρϕσ ] , , E2 = πη + ασ σ (ψ + ω) (πη + ασ ) π (δ − r)ψ + δω (1 − 1), Y2 = , C2Y = σ μ(ψ + ω) S2 =

and 1 is the HTLV-specific CTL immunity reproduction number, and it is stated as 1 =

ψσρϕη . [(δ − r)ψ + δω] (πη + ασ )

The parameter 1 determines whether or not the HTLV-specific CTL immune response can be activated. Moreover, it is easy to see that 1 =

0 πη . 1 + ασ

Therefore 1 ≤ 0 , and this completes the proof.

8.5 Global stability analysis In this section, we prove the global asymptotic stability of all steady states by constructing a Lyapunov functional following the method presented in [42–44]. We will use the arithmetic-geometric mean inequality   n n  1 n χi ≥  χi , χi ≥ 0, i = 1, 2, ..., n, n i=1

i=1

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241

which yields SY Ej EYj Sj + + ≥ 3, j = 1, 2. S Sj Yj E Ej Y Let a function j (S, E, Y, C Y ) be defined as  ˆ j (t) = j (x, t) dx, 

(8.15)

j = 0, 1, 2.



Let ϒj be the largest invariant subset of

 ˆj d  ϒj = (S, E, Y, C Y ) : = 0 , j = 0, 1, 2. dt We define a function (υ) = υ − 1 − ln υ. From the divergence theorem and Neumann boundary conditions (8.9), we obtain    0= ∇U · V dx = div(∇U) dx = U dx, ∂        U U 2 1 1  dx ∇U · V dx = div( ∇U) dx = − 0= U U U2 ∂ U   for U = S, E, Y, C Y . Thus we obtain  U dx = 0, 





U dx = U

 

U 2 dx U2

for

U = S, E, Y, C Y .

(8.16)

For convenience, we drop the input notation, i.e., (S, E, Y, C Y ) = (S(x, t), E(x, t), Y (x, t), C Y (x, t)). Theorem 8.1. If 0 ≤ 1, then Ð0 is globally asymptotically stable. Proof. Define 0 (x, t) as 

S 0 (x, t) = S0  S0

 +

1 ψ +ω μ (ψ + ω) Y E+ Y+ C . ϕ ϕψ ϕψσ

ˆ 0 (S0 , 0, 0, 0) = 0. We calculate ˆ 0 (S, E, Y, C Y ) > 0 for all S, E, Y, C Y > 0, and  Clearly,  along the solutions of model (8.10) as follows:   S0 1 ∂0 = 1− [dS S + ρ − αS − ηSY ] + [dE E + ϕηSY + rY − (ψ + ω) E] ∂t S ϕ

∂0 ∂t

242

Advances in Epidemiological Modeling and Control of Viruses   μ (ψ + ω)  ψ +ω  dY Y + ψE − δY − μC Y Y + dC Y C Y + σ C Y Y − πC Y ϕψ ϕψσ   δ (ψ + ω) μπ (ψ + ω) Y S0 r Y− C = 1− (ρ − αS) + ηS0 Y + Y − S ϕ ϕψ ϕψσ   S0 dE dY (ψ + ω) μdC Y (ψ + ω) + dS 1 − S + E + Y + C Y . S ϕ ϕψ ϕψσ

+

Using S0 = ρ/α, we obtain ∂0 (S − S0 )2 (δ − r) ψ + δω μπ (ψ + ω) Y = −α + C (0 − 1) Y − ∂t S ϕψ ϕψσ   S0 dE dY (ψ + ω) μdC Y (ψ + ω) S + E + Y + C Y . + dS 1 − S ϕ ϕψ ϕψσ Consequently, we calculate

ˆ0 d dt

as follows:

  ˆ0 d (S − S0 )2 [(δ − r) ψ + δω] (0 − 1) = −α dx + Y dx dt S ϕψ       S0 μπ (ψ + ω) S dx 1− − C Y dx + dS ϕψσ S      dE dY (ψ + ω) μdC Y (ψ + ω) + E + Y dx + C Y dx. ϕ  ϕψ ϕψσ  

(8.17)

Using equality (8.16), Eq. (8.17) is reduced to the following form:   ˆ0 d (S − S0 )2 [(δ − r) ψ + δω] (0 − 1) = −α dx + Y dx dt S ϕψ     S 2 μπ (ψ + ω) Y − C dx − dS S0 dx. ϕψσ S2   ˆ

ˆ

0 0 Therefore ddt ≤ 0 for all S, Y, C Y > 0 and ddt = 0 when S = S0 and Y = C Y = 0. The solu

tions of system (8.10) converge to ϒ0 [45]. The set ϒ0 includes elements with S = S0 and Y = C Y = 0, and then ∂Y ∂t = Y = 0. The third equation of system (8.10) implies

0=

∂Y = ψE. ∂t

This yields E = 0. Hence ϒ0 = {Ð0 }, and by applying the Lyapunov-LaSalle asymptotic stability theorem, we get that Ð0 is globally asymptotically stable [46–48]. Theorem 8.2. If 1 ≤ 1 < 0 , then Ð1 is globally asymptotically stable. Proof. Let 1 (x, t) be defined as       1 ψ +ω μ (ψ + ω) Y S E Y + E1  + + Y1  C . 1 (x, t) = S1  S1 ϕ E1 ϕψ Y1 ϕψσ

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We calculate

∂1 ∂t

243

as follows:

    S1 E1 ∂1 1 = 1− 1− [dS S + ρ − αS − ηSY ] + [dE E + ϕηSY + rY − (ψ + ω) E] ∂t S ϕ E     μ (ψ + ω)  ψ +ω Y1  + 1− dY Y + ψE − δY − μC Y Y + dC Y C Y + σ C Y Y − πC Y ϕψ Y ϕψσ   E1 r E1 ψ + ω r δ (ψ + ω) S1 − Y + E1 − Y = 1− (ρ − αS) + ηS1 Y + Y − ηSY S ϕ E ϕ E ϕ ϕψ   S1 ψ + ω Y1 δ (ψ + ω) μ (ψ + ω) Y μπ (ψ + ω) Y E + Y1 + C Y1 − C + dS 1 − S − ϕ Y ϕψ ϕψ ϕψσ S     dE E1 dY (ψ + ω) Y1 μdC Y (ψ + ω) + 1− E + 1− Y + C Y . ϕ E ϕψ Y ϕψσ Using the steady state conditions for Ð1 ρ = αS1 + ηS1 Y1 , ηS1 Y1 + we obtain

r ψ +ω δ (ψ + ω) Y1 = E1 = Y1 , ϕ ϕ ϕψ

    S1 S1 r Y E1 SY E1 ∂1 = 1− − ηS1 Y1 − Y1 (αS1 − αS) + ηS1 Y1 1 − ∂t S S S1 Y1 E ϕ Y1 E r EY1 EY1 r r − Y1 + ηS1 Y1 + Y1 + ηS1 Y1 + Y1 − ηS1 Y1 ϕ E1 Y ϕ E1 Y ϕ     S1 dE E1 μ (ψ + ω) Y μπ (ψ + ω) Y C Y1 − C + dS 1 − S + 1− E + ϕψ ϕψσ S ϕ E   Y1 μdC Y (ψ + ω) dY (ψ + ω) 1− Y + C Y + ϕψ Y ϕψσ   S1 SY E1 EY1 (S − S1 )2 + ηS1 Y1 3 − − − = −α S S S1 Y1 E E1 Y   EY1 r π Y Y E1 μ (ψ + ω)  − Y1 − C + Y1 2 − + ϕ Y1 E E1 Y ϕψ σ       S1 dE E1 dY (ψ + ω) Y1 S + 1− E + 1− Y + dS 1 − S ϕ E ϕψ Y μdC Y (ψ + ω) + C Y ϕψσ   S1 r (Y E1 − EY1 )2 SY E1 EY1 (S − S1 )2 − + ηS1 Y1 3 − − − = −α S ϕ EE1 Y S S1 Y1 E E1 Y     S dE E1 μ (ψ + ω) (ασ + ηπ) 1 S + 1− E + (1 − 1) C Y + dS 1 − ϕψησ S ϕ E   dY (ψ + ω) Y1 μdC Y (ψ + ω) + 1− Y + C Y . (8.18) ϕψ Y ϕψσ

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ˆ 1 (t) along the positive solutions of (8.10), and we Therefore we take the time derivative of  use equality (8.16) to find 

 r (S − S1 )2 (Y E1 − EY1 )2 dx − dx S ϕ  EE1 Y      S1 SY E1 EY1 μ (ψ + ω) (ασ + ηπ) (1 − 1) 3− − − dx + + ηS1 Y1 C Y dx S S1 Y1 E E1 Y ϕψησ      S 2 E 2 Y 2 dE E 1 dY Y1 (ψ + ω) − dS S 1 dx − dx − dx. ϕ ϕψ S2 E2 Y2   

ˆ1 d = −α dt

Thus, if 1 ≤ 1, then using inequality (8.15) we get ˆ1 d dt

ˆ1 d dt

≤ 0 for all S, E, Y, C Y > 0. Moreover,

= 0 when S = S1 , E = E1 , Y = Y1 , and C Y = 0. The solutions of system (8.10) converge

to ϒ1 . Thus ϒ1 = {Ð1 }, and by applying the Lyapunov-LaSalle asymptotic stability theorem, we get that Ð1 is globally asymptotically stable. Theorem 8.3. If 1 > 1, then Ð2 is globally asymptotically stable. Proof. Define 2 (x, t) as 

S 2 (x, t) = S2  S2 Calculate

∂2 ∂t



      CY 1 ψ +ω μ (ψ + ω) Y E Y + E2  + + . Y2  C2  ϕ E2 ϕψ Y2 ϕψσ C2Y

as follows:   S2 ∂2 = 1− [dS S + ρ − αS − ηSY ] ∂t S   1 E2 + 1− [dE E + ϕηSY + rY − (ψ + ω) E] ϕ E    ψ +ω Y2  + 1− dY Y + ψE − δY − μC Y Y ϕψ Y    C2Y  μ (ψ + ω) dC Y C Y + σ C Y Y − πC Y + 1− Y ϕψσ C   E2 r E2 S2 r − Y = 1− (ρ − αS) + ηS2 Y + Y − ηSY S ϕ E ϕ E ψ + ω Y2 δ (ψ + ω) δ (ψ + ω) ψ +ω E2 − Y− E + Y2 + ϕ ϕψ ϕ Y ϕψ μ (ψ + ω) Y μπ (ψ + ω) Y μ (ψ + ω) Y C Y2 − C − C2 Y + ϕψ ϕψσ ϕψ

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    μπ (ψ + ω) Y S2 dE E2 C2 + d S 1 − S + 1− E ϕψσ S ϕ E     C2Y Y2 μdC Y (ψ + ω) dY (ψ + ω) 1− Y + 1 − Y C Y . + ϕψ Y ϕψσ C

+

Using the steady state conditions for Ð2 π , σ r ψ +ω δ (ψ + ω) μ (ψ + ω) Y E2 = Y2 + C2 Y2 , ηS2 Y2 + Y2 = ϕ ϕ ϕψ ϕψ ρ = αS2 + ηS2 Y2 ,

Y2 =

we obtain

    S2 r Y E2 S2 SY E2 ∂2 = 1− − ηS2 Y2 − Y2 (αS2 − αS) + ηS2 Y2 1 − ∂t S S S2 Y2 E ϕ Y2 E r EY2 EY2 r r − Y2 + ηS2 Y2 + Y2 + ηS2 Y2 + Y2 − ηS2 Y2 ϕ E2 Y ϕ E2 Y ϕ       S2 dE E2 dY (ψ + ω) Y2 S + 1− E + 1− Y + dS 1 − S ϕ E ϕψ Y   CY μdC Y (ψ + ω) + 1 − 2Y C Y ϕψσ C   2 S2 SY E2 EY2 (S − S2 ) + ηS2 Y2 3 − − − = −α S S S2 Y2 E E2 Y       Y E2 S2 EY2 dE E2 r − + dS 1 − S + 1− E + Y2 2 − ϕ Y2 E E2 Y S ϕ E     CY Y2 μdC Y (ψ + ω) dY (ψ + ω) 1− Y + + 1 − 2Y C Y ϕψ Y ϕψσ C   r (Y E2 − EY2 )2 SY E2 EY2 S2 (S − S2 )2 − + ηS2 Y2 3 − − − = −α S ϕ EE2 Y S S2 Y2 E E2 Y       S2 dE E2 dY (ψ + ω) Y2 S + 1− E + 1− Y + dS 1 − S ϕ E ϕψ Y   CY μdC Y (ψ + ω) + 1 − 2Y C Y . ϕψσ C

ˆ 2 (t) along the positive solutions of (8.10) and using equalTaking the time derivative of  ity (8.16), we obtain   ˆ2 r d (S − S2 )2 (Y E2 − EY2 )2 = −α dx − dx dt S ϕ  EE2 Y     S2 SY E2 EY2 3− − − dx + ηS2 Y2 S S2 Y2 E E2 Y 

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 S 2 E 2 dE E 2 dx − dx 2 ϕ S E2     μdC Y C2Y (ψ + ω) Y 2 C Y 2 dY Y2 (ψ + ω) − dx −  2 dx. ϕψ ϕψσ Y2   CY

− dS S2

ˆ2 d Y dt ≤ 0 for all S, E, Y, C > 0. More ˆ 2 over, ddt = 0 when S = S2 , E = E2 , Y = Y2 , and C Y = C2Y . Hence ϒ2 = {Ð2 }. Applying the Lyapunov-LaSalle asymptotic stability theorem, we get Ð2 is globally asymptotically sta-

Hence, if 1 > 1, then using inequality (8.15) we get

ble.

8.6 Numerical simulations In this section, we illustrate the results of Theorems 8.1–8.3 by performing numerical simulations. To solve system (8.7) numerically, we choose the spatial domain as  = [0, 2] with a step size 0.02. The step size for time is given by 0.1. Further, we choose the following initial conditions for system (8.7):     S(x, 0) = 500 1 + 0.4 cos2 (πx) , E(x, 0) = 30 1 + 0.5 cos2 (πx) ,     Y (x, 0) = 0.4 1 + 0.5 cos2 (πx) , C Y (x, 0) = 0.2 1 + 0.5 cos2 (πx) , x ∈ [0, 2]. (8.19) In addition, we consider the homogeneous Neumann boundary conditions: ∂S ∂E ∂Y ∂C Y = = = = 0, ∂ V ∂ V ∂ V ∂ V

t > 0,

x = 0, 2.

(8.20)

Furthermore, we fix the values of some parameters (see Table 8.1), and others will be varied. We select different values of η and σ under the above initial and boundary conditions, which leads to the following scenarios: Table 8.1 The data of model (8.7). Parameter

Value

Parameter

ρ

10

μ

Value 0.2

α

0.01

σ

Varied

η

Varied

ω

0.01

ϕ

0.2

ψ

0.003

κ

0.9

dS

0.1

r∗

0.008

0.01

δ∗

dE

0.2

dY

0.2

π

0.1

dC Y

0.2

Scenario 1 (Stability of Ð0 ): η = 0.003 and σ = 0.05. For this set of parameters, we have 0 = 0.70 < 1. Fig. 8.1 displays that the trajectories of system (8.7) reach the steady state

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FIGURE 8.1 The behavior of solution trajectories of system (8.7) when 0 ≤ 1.

Ð0 = (1000, 0, 0, 0). This shows that Ð0 is globally asymptotically stable according to Theorem 8.1. In this situation there is no infection. Scenario 2 (Stability of Ð1 ): η = 0.007 and σ = 0.05. Then we calculate 1 = 0.68 < 1 < 1.63 = 0 . Hence the numerical results illustrate that Ð1 = (611.43, 60.28, 0.91, 0) exists. Fig. 8.2 demonstrates that the trajectories of system (8.7) tend to Ð1 . Thus the numerical results are consistent with Theorem 8.2. This situation leads to a persistent HTLV infection but with ineffective CTL immune response. Scenario 3 (Stability of Ð2 ): η = 0.007 and σ = 0.3. Then we calculate 0 = 1.63 > 1 and 1 = 1.32 > 1. According to these data, the steady state Ð2 = (810.81, 29.29, 0.33, 0.32) exists. In Fig. 8.3 we show that the trajectories of system (8.7) tend to Ð2 and then it is globally asymptotically stable, which agrees with Theorem 8.3. Hence a persistent HTLV infection with effective HTLV-specific CTL immune response is attained.

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FIGURE 8.2 The behavior of solution trajectories of system (8.7) when 1 ≤ 1 < 0 .

8.7 Conclusion and discussion In this work, we have developed and analyzed an HTLV-I infection model which describes the interactions between CD4+ T cells, latent HTLV-infected cells, Tax-expressing HTLVinfected cells, and HTLV-specific CTLs. The model incorporates spatial diffusion for all types of cells. HTLV-I can be transmitted via two routes, (i) horizontal transmission via direct cell-to-cell touch via virological synapse and (ii) vertical transmission by mitotic division of Tax-expressing HTLV-infected cells. The well-posedness of solutions, including the existence of global solutions and the boundedness, was justified. We derived the basic infection reproduction number 0 and the HTLV-specific CTL immunity reproduction number 1 , which determine the existence and stability of the three steady states of the model. We constructed suitable Lyapunov functions and utilized the Lyapunov-LaSalle asymptotic stability theorem to establish the global asymptotic stability of all steady states. We proved that when 0 ≤ 1, the infection-free steady state is globally asymptotically sta-

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FIGURE 8.3 The behavior of solution trajectories of system (8.7) when 1 > 1.

ble, which biologically means that the HTLV-I is cleared up. When 1 ≤ 1 < 0 , the persistent infection steady state with ineffective HTLV-specific CTL immune response is globally asymptotically stable. This situation leads to a chronic HTLV-I infection, but with the ineffective CTL immune response. When 1 > 1, the persistent infection steady state with effective HTLV-specific CTL immune response is globally asymptotically stable. This case leads to a chronic HTLV-I infection but with an effective CTL immune response. We conducted numerical simulations to support and clarify our theoretical results. Model (8.10) assumed that once the uninfected CD4+ T cells are contacted by Taxexpressing HTLV-infected cells, they become latently infected instantaneously. However, such a process needs time. The effect of intracellular time delay on the dynamics of HTLV infection has significant importance. Moreover, immune response delay plays an important role in HTLV-I dynamics. Delayed HTLV-I infection models have been formulated and analyzed in many articles (see, e.g., [26], [28], [31], and [33]). Therefore model (8.10) can be extended and developed by taking into account the following assumptions:

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(A1) The Tax-expressing HTLV-infected cell contacts an uninfected CD4+ T cell at time t − τ1 , the cell becomes HTLV-infected at time t; (A2) The latent HTLV-infected cell takes τ2 time units to transmit to Tax-expressing HTLVinfected cells; (A3) Activation of immune response is not instantaneous, but mediated by some time lag τ3 . Model (8.10) under assumptions (A1)–(A3) becomes: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

∂S(x,t) = dS S(x, t) + ρ − αS(x, t) − ηS(x, t)Y (x, t), ∂t ∂E(x,t) = dE E(x, t) + ϕηS(x, t − τ1 )Y (x, t − τ1 ) + κr ∗ Y (x, t) − (ψ ∂t

+ ω) E(x, t),

∂Y (x,t) = dY Y (x, t) + ψE(x, t − τ2 ) + (1 − κ) r ∗ Y (x, t) − δ ∗ Y (x, t) − μC Y (x, t)Y (x, t), ∂t ∂C Y (x,t) = dC Y C Y (x, t) + σ C Y (x, t − τ3 )Y (x, t − τ3 ) − πC Y (x, t). ∂t

Our model can be useful to study the dynamics of different viruses such as coronavirus [49], [50] and Ebola virus [51].

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[37] Z. Xu, Y. Xu, Stability of a CD4+ T cell viral infection model with diffusion, International Journal of Biomathematics 11 (5) (2018) 1850071. [38] Y. Zhang, Z. Xu, Dynamics of a diffusive HBV model with delayed Beddington-DeAngelis response, Nonlinear Analysis: Real World Applications 15 (2014) 118–139. [39] H.L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, 1995. [40] M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations, Prentice Hall, Englewood Cliffs, 1967. [41] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer-Verlag, New York, 1993. [42] A.M. Elaiw, A.D. AlAgha, Global dynamics of reaction-diffusion oncolytic M1 virotherapy with immune response, Applied Mathematics and Computation 367 (2020) 124758. [43] K. Hattaf, Spatiotemporal dynamics of a generalized viral infection model with distributed delays and CTL immune response, Computation 7 (2) (2019), 21. [44] A.M. Elaiw, A.D. AlAgha, Analysis of a delayed and diffusive oncolytic M1 virotherapy model with immune response, Nonlinear Analysis: Real World Applications 55 (2020) 103116. [45] J.K. Hale, S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. [46] E.A. Barbashin, Introduction to the Theory of Stability, Wolters-Noordhoff, Groningen, 1970. [47] J.P. LaSalle, The Stability of Dynamical Systems, SIAM, Philadelphia, 1976. [48] A.M. Lyapunov, The General Problem of the Stability of Motion, Taylor & Francis, Ltd., London, 1992. [49] A.M. Elaiw, A.D. AlAgha, Global dynamics of Sars-CoV-2/cancer model with immune responses, Applied Mathematics and Computation 408 (2021) 126364. [50] A. Al Agha, S. Alshehaiween, A. Elaiw, M. Alshaikh, A global analysis of delayed Sars-CoV-2/cancer model with immune response, Mathematics 9 (11) (2021) 1283. [51] Z. El Rhoubari, K. Hattaf, N. Yousfi, A class of Ebola virus disease models with post-death transmission and environmental contamination, in: K. Hattaf, H. Dutta (Eds.), Mathematical Modelling and Analysis of Infectious Diseases, Studies in Systems, Decision and Control, vol. 302, Springer, Cham, 2020, pp. 295–321.

9 Mathematical tools and their applications in dengue epidemic data analytics Wickramaarachchillage Pieris Tharindu Mihiruwan Wickramaarachchia , Kaluhath Karunathilaka Withanage Hasitha Erandib , and Shyam Sanjeewa Nishantha Pererab a Department of Mathematics, The Open University of Sri Lanka, Nawala, Nugegoda, Sri Lanka b Research and Development Center for Mathematical Modeling, Department of Mathematics, University of Colombo, Colombo, Sri Lanka

9.1 Introduction Communicable diseases caused by various pathogens are emerging. They are very common and frequent in the modern world, leaving millions of people vulnerable. The transmission of these diseases among the world population has increased significantly in the modern era due to urbanization, increased air travel, and climate change [1]. The consequences of these epidemic diseases range from minor to serious. Some of the diseases are life-threatening, while other diseases may create a variety of social and economic problems for communities, and people’s well-being is challenged. These infectious diseases should be managed very carefully, minimizing the burden for the public health of communities and also causing minimal damage to societal lifestyles and economies of countries. This may be done through early diagnostics of cases, efficient patient management such as isolation, quarantine, and providing treatments and immunizations. In the case of vector-borne diseases such as dengue or malaria, it is very critical to destroy the mosquitoes through continuous social awareness and environmental management. Effective disease management could be achieved through a better data collection mechanism and data analytic tools. Epidemiological time series are generally noisy, and sometimes they cannot be perfectly reliable for multiple reasons, such as the timing of case reporting, the accuracy of case diagnostics, and their definitions, and due to human error. Therefore classical statistical methods, such as statistical correlations, may not be suitable to analyze these data since pattern recognition and identification of relationships with respect to external drivers of disease transmission should be done before the epidemic is predicted statistically. For instance, dengue transmission is extremely sensitive to climate Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00014-4 Copyright © 2023 Elsevier Inc. All rights reserved.

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variations. Thus it is very important to identify the time-dependent relationships between climate factors and disease cases. Such analyses may reveal both time-invariant correlations and lead time differences from one variable to another. The revealed time series spectral properties may be vital for mathematical and statistical models at their parameter level. Fluctuations in epidemiological time series data usually consist of multiple periodic components whose cycles and trends need to be delineated and adjusted. Appropriate identification of the relationship between disease outbreaks and associated risks at certain time scales has been a major methodological issue in infectious disease research. The Fourier transform is one of the mathematical tools for analyzing the patterns of nonstationary data [2]. Other than the Fourier transform, a number of spectral analysis methods, including autoregressive, moving average, and autoregressive moving average methods, have recently been developed to estimate the spectral properties of data series [3,4]. Since real time climate and dengue data show complex nonlinear dynamics with strong seasonality, multiyear oscillations, and nonstationarity, conventional statistical methods may be inadequate in extracting the hidden information. Considering the time shift invariant and the capability of capturing the nonrepetitive waveform [5,6], we consider the classical Fourier transform method as a tool to identify the periodic patterns of dengue-related data [7]. At first, most of the concepts in Fourier analysis were introduced to heat flow [2]. Then, it was used as a tool to solve practical problems in the area of frequency analysis [8], fast convolution [9], image processing [10], and signal processing [11]. Furthermore, the Fourier transform was used as an important tool in financial economics as well [12,13]. Moreover, the Fourier transform was used as a tool to identify perceived differences in sound between violins and drums, sonic booms, and the mixing of colors [2]. Later, a number of researchers investigated the possibility of combining the Fourier transform with other convolution and assimilation operations, such as spectral-pooling and the ensemble Kalman filter, to improve their efficiency [14,15]. A few studies also used the Fourier transform as a tool to identify the periodic pattern of epidemiological data [16–20]. However, it should be noted that the Fourier transform comes with its own limitations. One of the major disadvantages has been its inability to identify spectral properties of a noisy time series data. That is, although Fourier analysis has potential to quantify constant periodic components in a time-series, it is not able to characterize signals whose frequency content varies with time. Further, the Fourier transform does not create a representation of the signal in both the time and frequency domains, thereby not allowing efficient access of localized information about the signal [1,21]. Wavelet analysis resolves this problem of nonstationarity in time series data through performing a local time-scale decomposition of the signal. This helps us to track how the different scales related to periodic components of the signal vary with respect to time [22]. Cross wavelets and wavelet phase difference analysis can be used to investigate the dependencies between two signals. The wavelet theory has a number of applications in a range of areas in real world problems such as climate, epidemiology, biosciences, finance, and

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physics. Josué M. et al. [26] applied wavelet analysis to capture the most fundamental information contained in multivariate climate time series, and they found that the theory is suitable to investigate correlation among climate time series in a multivariate context. Kim C. Raath et al. [27] studied the relationship among water and energy prices using wavelet theories, and they found that water prices, at certain time horizons, lead energy prices during specific localized economic events. Their research also helped investors to make financial decisions by correctly identifying the dynamic interactions of water and energy commodities. The wavelet theory’s powerful ability to identify the hidden relationships in time series has encouraged researchers in epidemiology and infectious diseases modeling to apply this type of analysis to support decision-making, model parameter estimation, and determination of correct timing of control intervention. A study that was done using wavelets to analyze dengue incidents in China suggests that periods of elevated temperatures can lead the occurrence of dengue outbreaks across Chaozhou area during 2014–17, and the risk of transmission was identified to be the highest when the temperature is between 25°C and 28°C [28]. The multifactor, complex, and uncertain transmission nature of dengue disease has resulted in that classical statistical models alone do not provide optimal forecasting capability. Perhaps the information revealed from these models is not sufficient to establish time-dependent relationships among critical variables. Widely applied Fourier techniques are useful to capture the seasonal relationships; however, this method does not provide information about powerful spectral properties among noisy time series that are very common in disease transmission problems, which resulted in wavelet theories being demanded to fill these gaps. The aim of this chapter is to demonstrate the potential of the Fourier transform and wavelet analysis tools in identifying unseen relationships between dengue transmission dynamic and its external drivers, such as climate variation and the mobility of the population. The ability to apply the techniques in a local context is emphasized through using these methods to analyze dengue transmission patterns in three countries: Sri Lanka, Indonesia, and Thailand. In this chapter, the first section describes the theoretical background of the Fourier transformation, highlighting its own advantages over statistical methods in time series data analysis in the epidemiological context. Then, the results are presented and discussed based on the case studies done applying Fourier transform techniques to analyze dengue cases and climate variation in three geographic regions: Colombo (Sri Lanka), Jakarta (Indonesia), and Bangkok (Thailand). Next, the benefits of wavelet theory over the Fourier transform are discussed with illustrative case studies and their findings from Colombo, Sri Lanka in relation to dengue outbreaks, climate fluctuations, and human mobility while discussing wavelet’s ability to reveal critical spectral information that is highly useful as an input to accurate decision-making processes.

9.2 Fourier transformation Baron Jean Baptiste Joseph Fourier (1768–1830) introduced a mechanism to decompose a function in space or time into sinusoidal components with different frequencies, ampli-

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tudes, and phases. The mechanism was named Fourier transform, and it represents the amplitude and phase of each sinusoid corresponding to a specific frequency [31]. The Fourier transform of a function of time is itself a complex-valued function of frequency where magnitude represents the amount of that frequency present in the original function and the phase offset of the basic sinusoid in that frequency. That is, the Fourier transform establishes a simultaneous dual view of the phenomena between time domain and frequency domain representations. The process of the Fourier transform can be illustrated by the black box concept as follows. (See Fig. 9.1.)

FIGURE 9.1 Process of the Fourier transform.

If the original function is continuous, then the Fourier transform decomposes the original function into sinusoidal functions at all frequencies, which are combined by means of an operation called the Fourier integral. There are several different variations of the Fourier transform equation. Considering a continuous function f , one form of the Fourier transform of f can be defined as follows:  ∞ f (t) exp (−i2πωt)dt, (9.1) F (ω) = −∞

where F is a function of real variable ω and |F (ω)| is called the amplitude spectrum of f . Here i is the imaginary number and i 2 = −1. Applying the same transform to F (ω) gives  ∞ f (s) = F (ω) exp (−i2πωt)dt. (9.2) −∞

If f (t) is an even function of t, then f (s) = f (t), and if f (t) is an odd function of t, then f (s) = f (−t). If f (t) is neither even nor odd, f (t) often can be divided into even or odd components. Notice that most of the dengue-related data have been recorded in the form of time series, which is a sequence of observations recorded at a succession of time intervals. To discover the underlying periodic patterns of the data series, we need to extract spectral properties. For that purpose, we consider the discrete Fourier transform, which is suitable to analyze the data series recorded in the discrete time domain.

9.2.1 Discrete time Fourier transform The discrete Fourier transform (DFT) is a formula for evaluating the N Fourier coefficients from a sequence of N numbers [31]. The DFT for a finite time series {xn } is obtained by

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decomposing a sequence of values into components of different frequencies [2], and the DFT for the time series {xn } is defined as D(k) =

N −1  n=0

  2π exp −i kn xn , N

k = {0, . . . , N − 1},

(9.3)

where N is the number of data points in time series {xn }. For the purpose of comparison, the relative Fourier transform can be computed by dividing all the elements of the spectrum by the highest Fourier amplitude in the spectrum. The algorithm of the efficient computation of DFT is called fast Fourier transform (FFT).

9.2.2 Fast Fourier transform The publication of the FFT algorithm for calculating DFT was proposed by J. W. Cooley and J. W. Tukey in 1965 [32], and it was a turning point in digital signal processing and in certain areas of numerical analysis. To have an idea of the computational cost of the straightforward method of DFT, let us consider a time series with N number of data points. Assuming that N is a multiple of 2, the summation over N in Eq. (9.3) can be decomposed into two separate sums, one over the even-numbered values of N and the other one over the odd-numbered values of N as follows: N 2 −1

D(k) =

 n=0

N     2 −1  2π 2π exp −i k(2n) x(2n) + exp −i k(2n + 1) x(2n+1) . N N

(9.4)

n=0

It can be noticed that the first sum is the DFT of the sequence {x(2n) } of length N/2, and the second sum is the sequence {x(2n+1) } of length N/2. Define these sequences as {yn } = {x(2n) } and {zn } = {x(2n+1) }.

(9.5)

If N is a power of two, then the algorithm can be repeated, giving N log2 N operations. Hence the FFT methods require only of the order of N log2 N complex multiplications, and the computational cost is reduced by the order of log2 N/N factor. Therefore this algorithm reduces the computational complexity from N 2 to N log N . Perhaps more than any other factor, it was the invention of FFT by Cooley and Tukey in the 1960s that has made possible many of the applications of digital signal processing. Now let us illustrate the idea of FFT using an example function. Consider a sine wave creating five waves in 1 second with a frequency of 100. Then the function for the sine wave x at time t/100 is given by   x = sin 5(2π)t . (9.6) Fig. 9.2 illustrates the sine wave and the Fourier spectrum. From the Fourier spectrum, we can identify information about the frequencies of the waves.

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FIGURE 9.2 Sine spectrum and fast Fourier transform of sine wave.

9.2.3 Dengue epidemic data analysis Several previous works used FFT to extract the spectral properties of dengue and rainfall data [16,17,19]. Moreover, often works used different versions of FFT to forecast or confirm the periodicity of climate data [33–35]. To demonstrate the method’s usability for dengue and climate data analysis, we use a simple version of FFT. Since the Fourier decomposition determines all the spectral components embedded in the data series, it can be used to quantify the time-frequency content of dengue cases and related data.

9.2.4 Study areas For our illustration, we used dengue cases and climate data from Colombo, Jakarta, and Bangkok. Colombo is the financial center and the most densely populated city in Sri Lanka. The first dengue outbreak in Sri Lanka occurred between 1965 and 1966. Several dengue outbreaks have occurred since then. The worst outbreak year was 2017, with 186,101 cases reported, and Colombo reported 25% of the country’s annual dengue cases. The city is influenced by two monsoon seasons: the Northeast monsoon season from December to January and the Southwest monsoon season from May to September. Furthermore, the temperature in urban Colombo ranges from 22◦ C to 33◦ C. Jakarta is a global city in the global economic network and the world’s second most populous urban area. As a result, Jakarta has a direct impact on global socioeconomic affairs and can easily spread the disease to other cities linked to the global economic network. Dengue was first reported in 1968, and since then the most severe outbreaks have been reported, with the disease now reaching endemic status in Jakarta. Jakarta has a tropical

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climate with temperatures ranging from 23◦ C to 33◦ C and an average annual rainfall of 2000 mm, with the highest annual rain falling between December and February. Bangkok is the major international transportation hub and one of the world’s most popular tourist destinations. As a result, during outbreak periods, it eventually becomes a source of disease transmission. The first dengue case in Thailand was discovered in 1949, and the first outbreak occurred in 1958; since then Bangkok has reported dengue outbreaks almost every 1–2 years. Bangkok has a tropical savanna climate with temperatures ranging from 22◦ C to 35.4◦ C. Bangkok is influenced by two monsoon seasons: the Southwest monsoon season, which lasts from May to July, and the Northeast monsoon season, which lasts from October to February.

9.2.5 Numerical results and discussion We considered dengue cases, rainfall, minimum temperature, and maximum temperature data in Colombo, Sri Lanka from 2008 to 2015, weekly reported dengue cases and rainfall data from 2008 to 2015 in Jakarta, Indonesia, and monthly dengue cases and rainfall data in Bangkok, Thailand from 2008 to 2015. All the analysis was implemented using MATLAB . First, we have examined the raw data. Fig. 9.3 shows dengue cases in Colombo, Jakarta, and Thailand. From Fig. 9.3 it can be observed that dengue cases in Colombo have two peaks each year: one peak at the beginning of the year and one in the middle of the year. Moreover, it can be noticed that dengue cases in Jakarta and Bangkok have one peak each

FIGURE 9.3 Dengue cases in (a) Colombo, (b) Jakarta, and (c) Bangkok from 2008 to 2015.

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year. In Jakarta the occurring time location of the highest peak varies from year to year for the considered time period. For example, the highest peak in 2008 was in the middle of the year, while the highest peak in 2010 was at the beginning of the year. Then we illustrate weekly average rainfall data in Colombo and Jakarta, and monthly average rainfall data in Bangkok (Fig. 9.4) to get an idea of the rainfall data in each city. Rainfall data in Colombo and Bangkok in Figs. 9.4(a) and 9.4(c) show that both cities have two rainfall seasons each year. Moreover, from Fig. 9.4(b) it can be observed that rainfall in Jakarta has one peak each year.

FIGURE 9.4 Weekly average rainfall data in (a) Colombo and (b) Jakarta from 2008 to 2015, (c) monthly rainfall data in Bangkok from 2008 to 2015.

To examine the periodic pattern of each time series, we plotted the Fourier spectrum of the reported dengue cases and rainfall data from January, 2008 to December, 2015, and some observations were of particular interest. According to Fig. 9.5, dengue cases in urban Colombo exhibit a 26-week periodic pattern. In other words, it illustrates that dengue cases increase every six and a half months. In addition, a Fourier amplitude related to a two-and-a-half-year periodic pattern can be observed with 0.7 relative amplitude. That is related to the dengue outbreak cycle of 2–3 years [36]. Then we evaluated the relative Fourier spectrum for rainfall data for the same time period, which is depicted in Fig. 9.6. From Fig. 9.6 it can be noticed that rainfall data from urban Colombo shows a 26-week periodic pattern. According to the study [37], Colombo is influenced by two monsoon seasons, and there are two friendly climate seasons for mosquitoes per year. Thus the 26-week period of rainfall underscores its correlation with dengue cases.

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FIGURE 9.5 Relative Fourier spectrum of weekly reported dengue cases in Colombo.

FIGURE 9.6 Relative Fourier spectrum of weekly average rainfall data in Colombo.

Similarly, we examined the periodic pattern of maximum temperature and minimum temperature from 2008 to 2015 (see Fig. 9.7 for raw data), which is illustrated in Figs. 9.8 and 9.9. According to Figs. 9.8 and 9.9, it can be observed that weekly average temperature exhibits a 52-week periodic pattern. In other words, the results indicate that the maximum temperature and the minimum temperature in urban Colombo have an annual pattern.

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FIGURE 9.7 (a) Weekly average maximum temperature, (b) Weekly average minimum temperature in Colombo from 2008 to 2015.

FIGURE 9.8 Relative Fourier spectrum value of maximum temperature in CMC area.

In addition, a Fourier amplitude related to 26-week with 0.5 relative amplitude can be observed in Fig. 9.8. Since seasonal fluctuation of temperature depends on rainfall [38], the rainfall pattern may have contributed to the maximum temperature pattern.

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FIGURE 9.9 Relative Fourier spectrum value of minimum temperature in CMC area.

FIGURE 9.10 Relative Fourier spectrum of weekly reported dengue cases in Jakarta.

Now, consider dengue cases and rainfall data in Jakarta. We computed the relative Fourier spectrum for dengue cases and rainfall data for the same time period. According to Figs. 9.10 and 9.11, dengue cases and rainfall from Jakarta illustrate a 52week periodic pattern. Further, from Fig. 9.10 it can be observed that there is a Fourier amplitude related to a two-and-a-half-year periodic pattern. In other words, it shows that in Jakarta dengue outbreaks have a two-and-a-half year cycle. In practice, it is natural to

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FIGURE 9.11 Relative Fourier spectrum of weekly average rainfall data in Jakarta.

expect a similar pattern for dengue as rainfall. However, for the Jakarta data, it can be observed that, though there is a Fourier amplitude related to the 26-week periodic pattern in rainfall with 0.5 relative amplitude, dengue cases do not show a significant Fourier amplitude related to the 26-week periodic pattern. Now, let us examine the periodic patterns of dengue and rainfall data in Bangkok. From Fig. 9.12 it can be observed that the highest Fourier amplitude was related to a 32-month period and the second highest amplitude was related to a 12-month period. In other words, the dengue outbreak in Bangkok has a 3-year periodic pattern, and the reported dengue cases have an annual pattern. From Fig. 9.13 it can be observed that the highest Fourier amplitude of rainfall data in Bangkok is related to the 12-month periodic cycle. Though Bangkok is influenced by two monsoon seasons [39], the Fourier amplitude related to the 6-month period has less than 0.5 relative amplitude. It illustrates that Bangkok has an annual rainfall pattern. Notice that the highest Fourier amplitude of dengue cases in each city emphasizes that dengue incidence has a correlation with rainfall. Therefore it is useful to analyze the crosscorrelation between rainfall and dengue cases to calculate the time delay between rainfall and dengue. To calculate the time delay, the Pearson correlation formula [40] has been used. The Pearson correlation formula for two time series X and Y can be represented by ρ(X, Y ) =

cov(X, Y ) . σ (X)σ (Y )

(9.7)

Here cov(X, Y ) denotes the covariance of the variables X and Y . The standard deviation of X and Y is denoted by σ (X) and σ (Y ). The correlation measure between dengue cases and rainfall was plotted with time lags from 0 to 20 weeks. Here we assumed that dengue

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FIGURE 9.12 Relative Fourier spectrum of monthly reported dengue cases in Bangkok.

FIGURE 9.13 Relative Fourier spectrum of monthly reported rainfall data in Bangkok.

cases are reported within or soon after the rainy season. Fig. 9.14 represents the correlation measures between rainfall data and dengue cases for each city. From Fig. 9.14 it can be observed that the highest correlation for Colombo occurs with a 10-week delay and the highest correlation for Jakarta occurs with an 11-week delay. For Bangkok, the highest correlation occurs with a 2-month delay. In other words, the corre-

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FIGURE 9.14 Cross-correlation between reported dengue cases and rainfall in (a) Colombo, (b) Jakarta, and (c) Bangkok.

lation between rainfall and dengue illustrates that rainfall has a delayed effect of approximately two months on dengue cases. Although the classical FFT of a function tells us precisely the periodicity and the size or the amplitude of the component of frequency, it does not precisely tell us the temporal information. For example, we observed that the dengue cases in Colombo have a 26-week periodicity. However, the spectrum does not give any information about when that period starts exactly. Moreover, the classical FFT formula cannot be used to analyze the crosscorrelation between two time series. To overcome these problems, we consider wavelet analysis.

9.3 Wavelet analysis Wavelet has been a powerful and useful tool in mathematics applied in many scientific domains such as engineering, epidemiology, ecology, finance, etc. This theory has been very popular in the field of time-frequency analysis and its applications. The theory of wavelets is very rich in mathematical content, which encourages researchers to explore it for various applications [23]. The Fourier transformation has been a very powerful tool mathematically, and it is still used widely in many scientific applications. However, the Fourier transform fˆ(ω) = (Ff )(ω) does not reveal much useful information in most of the applications. First, it takes infinite amount of time to extract spectral information fˆ(ω) using both past and future information of the signal. Further, the Fourier transform does not have a flexible timefrequency window that automatically narrows at high center-frequency and widens at low

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center-frequency. In contrast, wavelet analysis allows using long time windows when more precise low frequency information is required and shorter when high frequency information is needed [24]. In summary, the drawbacks of Fourier analysis can be stated as follows [41]: 1. The Fourier transform can be computed for only one frequency at a time; 2. The exact representations cannot be computed in real time; 3. The Fourier transform provides information only in the frequency domain but does not provide any information in the time domain. Since Fourier analysis fails to provide both time and frequency localization simultaneously, many researchers choose short-time Fourier transform (STFT) to solve their problems. STFT is defined by the integral transform  ∞ STw f (t, ω) = f (τ )w(τ − t)exp(−iωτ )dτ, (9.8) −∞

where the function w(t) is called the window function and the over line indicates the complex conjugation [27,44]. This STFT is sometimes called the windowed Fourier transform (WFT). STFT transforms a univariate function into a bivariate function [24]. It gives both time and frequency information of the function f (t) on the time-frequency plane. The complex spectrum STw f (t, ω) provides approximated spectral properties of the function around the time localization t. The mathematical background and the development of wavelet theory to overcome FFT using rich mathematical concepts can be widely found in many real world applications in the literature (for example, [23–25,27,41,44]) where wavelet transformation has been used. Now, let L2 (0, 2π) denote the set of all measurable functions f defined on (0, 2π) with  2π |f (x)|2 dx < ∞. 0

It can be verified that L2 (0, 2π) is a vector space and any function f ∈ L2 (0, 2π) has a Fourier series representation given by f (x) =

∞ 

cn exp(inx),

(9.9)

n=−∞

√ where i = −1, and the constants cn denote the Fourier coefficients of the function f and are defined as  2π 1 cn = f (x)exp(−inx)dx. (9.10) 2π 0 One important feature in (9.9) is that wn (x) =exp(inx), n = ..., −1, 0, 1, ...,

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is an orthonormal basis of L2 (0, 2π). The other feature of (9.9) is that the orthonormal basis {wn } is generated by dilation of a single function w(x) =exp(ix). This means wn (x) = w(nx) ∀n ∈ Z and is called the integral dilation. The basis function w(x) = exp(ix) = cos(x) + i sin(x) is a sinusoidal wave. For any n ∈ Z with large absolute value, the wave wn (x) = w(nx) has high frequency, and for n with small absolute value, the wave has low frequency. The orthonormal property of {wn }, the Fourier series representation in (9.9), satisfies the Parseval identity given by [42,44] 1 2π



2π

∞ 

|f (x)|2 dx =

0

|cn |2 .

(9.11)

n=−∞

It should be clear that the two function spaces L2 (0, 2π) and L2 (R) are different to each other. In L2 (0, 2π), a single function w(x) = exp(ix) generates the entire space. Similarly, we take a single function ψ to generate all of L2 (R). In the case where the wavelet ψ has a very fast decay, we shift ψ along R. We do this by defining ψ as ψ(x − k), k ∈ Z. It should be noted that the waves with different frequencies must be considered for the sinusoidal situation. Hence we define the small waves as   ψ 2j x − k ,

j, k ∈ Z,

which is obtained from a single wavelet function ψ(x) with a binary dilation of 2j and a dyadic translation of k/2j . Let the functions f, g ∈ L2 (R). The inner product is defined as  ∞ f, g = f (x)g(x)dx, −∞

and the norm is defined as f 2 = f, f 1/2 . The result above gives that, for any j, k ∈ Z,   f 2j x − k 2 =



∞ −∞

  | f 2j x − k |2 dx

1/2 = 2−j/2 f 2 .

Now, if a function ψ ∈ L2 (R) has a unit length, then all the functions ψj,k , defined by   ψj,k = 2j/2 ψ 2j x − k ,

j, k ∈ Z,

have a unit length given as ψj,k 2 = ψ 2 = 1,

j, k ∈ Z.

(9.12)

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The Kronecker symbol [41] is defined on Z × Z, and it is given by  1, for j = k; δj,k = 0, for j = k.

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(9.13)

A function ψ ∈ L2 (R) is said to be an orthogonal wavelet (or o.n. wavelet) if the family {ψj,k }, as defined in (9.12), is an orthonormal basis of L2 (R); that is, ψj,k , ψl,m  =δj,l .δk,m ,

j, k, l, m ∈ Z,

and any f ∈ L2 (R) can be written as ∞ 

f (x) =

(9.14)

cj,k ψj,k (x),

j,k=−∞

and the series converges in L2 (R). The series representation of the function f in (9.14) is called a wavelet series. The wavelet coefficients cj,k , analogous to the notion of Fourier coefficients in (9.10), are given by cj,k =f, ψj,k .

(9.15)

Now, we define the integral transform Wψ on L2 (R) as  x −b dx, f (x)ψ a −∞



−1/2

(Wψ f )(b, a) =|a|



∞

f ∈ L2 (R),

(9.16)

then the wavelet coefficients in (9.15) can be written as [42,44]   k 1 cj,k =(Wψ f ) j , j . 2 2

9.3.1 Wavelet transform Wavelets are finite energy functions which are capable of representing time-frequency localization of a transient signal with only a small finite number of coefficients. If ψ ∈ L2 (R) satisfies the admissibility condition given by  Cψ =

∞

−∞

2 ˆ |ψ(ω)| dω |ω|

< ∞,

(9.17)

then ψ is called a basic wavelet and, with respect to any basic wavelet ψ, the integral/continuous wavelet transform (IWT/CWT) of function f ∈ L2 (R) is defined as [23,24,27,44] 1

(Wψ f )(ba) =|a|− 2





∞

−∞

f (t)ψ

 t −b dt, a

(9.18)

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where a, b ∈ R with a = 0. Wavelet

∞implies a small wave, thus the area under the graph of the wavelet ψ(t) is zero. That is, −∞ ψ(t)dt = 0. By setting   1 t −b , (9.19) ψba (t) =|a|− 2 ψ a the IWT defined in (9.18) can be represented as (Wψ f )(ba) =f, ψba .

(9.20)

We recall the STFT defined in (9.8), which provides STw f (t, ω) =f (t), w(t − b)exp(iωt).

(9.21)

The only difference between IWT and STFT, which can be seen from (9.20) and (9.21), is the kernels. We can interchange IWT and STFT if we switch w(t − b)exp(iωt) ↔ ψba (t). Any original signal f ∈ L2 (R) can be uniquely recovered by the inverse transform defined as  ∞ ∞  da 1 f (t) = (9.22) (Wψ f )(ba) ψba 2 db, Cψ −∞ −∞ a where ψba is as in (9.19).

9.3.2 Basic wavelet functions 9.3.2.1 Haar wavelet This is one of the simplest basic wavelet functions. We let the Haar scaling function φ(t), which is a box function, be specified as [42]  1, if 0 ≤ t ≤ 1; φ(t) = 0, otherwise. The Haar wavelet function ψ(t) is defined as ψ(t) = φ(2t) − φ(2t − 1), and we obtain ⎧ ⎪ if 0 < t ≤ 1/2; ⎪ ⎨1, ψ(t) = −1, if 1/2 < t ≤ 1; ⎪ ⎪ ⎩0, otherwise. The family

  ψm,n (t) =2−m/2 ψ 2−m t − n ,

m, n ∈ Z,

produces an orthonormal basis for L2 (R). It should be noted that the Haar wavelet function is not continuous. The Haar wavelet is illustrated in Fig. 9.15.

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FIGURE 9.15 The Haar wavelet function.

FIGURE 9.16 The scaling function φ is given in the left-hand side and the Meyer wavelet function ψ is given in the right-hand side.

9.3.2.2 Meyer wavelet The Meyer wavelet function and the scaling function both are defined in the frequency ˆ domain [43]. The Fourier transform φ(ω) of the scaling function φ(t) is defined as [42,43] ⎧ −1/2 , ⎪ ⎪ ⎨(2π) 3 ˆ φ(ω) = (2π)−1/2 cos( π2 v( 2π |ω| − 1)), ⎪ ⎪ ⎩0,

if |ω| ≤ if

2π 3

2π 3 ;

≤ |ω| ≤

4π 3 ;

otherwise,

where v is a function such that v(α) = α 4 (35 − 84α + 70α 2 − 20α 3 ) and ˆ Fourier transform ψ(ω) of the wavelet function ψ(t) is defined as ⎧ −1/2 exp(iω/2) sin( π v( 3 |ω| − 1)), ⎪ ⎪ ⎨(2π) 2 2π 3 ˆ ψ(ω) = (2π)−1/2 exp(iω/2) cos( π2 v( 4π |ω| − 1)), ⎪ ⎪ ⎩0,

if if

2π 3 4π 3

≤ |ω| ≤ ≤ |ω| ≤

α ∈ [0, 1]. The

4π 3 ; 8π 3 ;

otherwise.

It should be noted that ψˆ is compactly supported (finite and nonzero duration), and it has at least k derivatives (k is arbitrary and it may be ∞). Thus the inverse Fourier transform can be used to obtain the wavelet function ψ. An illustrative form of the Meyer wavelet function is given in Fig. 9.16.

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9.3.2.3 Morlet wavelet The Morlet wavelet is defined as   ψ(t) =Kexp(−i2ω0 t).exp −t 2 /2 , where ω0 represents the central angular frequency of the wavelet. In order for ψ to have unit energy, the normalization constant K [1,27,30] is selected such that K = π −1/4 . The relationship between the frequencies f and wavelet scales can be derived as 4πa 1  = . f ω0 + 2 + ω02 It is obviously seen that if ω0 ≈ 2π then f ≈ 1/a. An example of the Morlet wavelet is given in Fig. 9.17.

FIGURE 9.17 The Morlet wavelet function.

9.3.3 Wavelet power spectrum The wavelet transform can be considered as a generalization of the classical Fourier transform so that the spectral properties of any time series x(t), t ∈ R can be visualized. The wavelet power spectrum of the wavelet transform (Wψ x)(ba) of the time series x(t) with the basic wavelet ψ is defined as (Sψ x)(ba) = (Wψ x)(ba) 2 .

(9.23)

The global wavelet power spectrum (S ψ x)(a), which is comparable with the Fourier spectrum of a signal, can be defined as the averaged energy of all wavelet coefficients of the same scale a and given by (S ψ x)(a) =

σx2 T

 0

T

(Wψ x)(ba) 2 db,

(9.24)

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where σx is the standard deviation of the time series x(t) and T is the duration of the time series [27,42,44]. Averaging the scale components gives the mean variance of each time location defined as σ 2 π 1/4 b1/2 (S ψ x)(b) = x Cψ



∞

a 1/2 (Wψ x)(ba) 2 da,

(9.25)

0

where Cψ as in (9.17).

9.3.4 Wavelet coherency and phase difference It is useful to quantify the statistical relationship between two signals if we have two nonstationary time series. The wavelet coherence function measures the correlation between two time series x(t) and y(t) [1]. The wavelet cross spectrum of the two time series x(t) and y(t) can be defined as (Wψ xy)(ba) =(Wψ x)(ba)(Wψ y)(ba)∗ .

(9.26)

Here ‘∗’ denotes the complex conjugate. The cross spectrum normalized by the spectrum of each signal gives the wavelet coherence, which is defined as [27,44] (Rψ xy)(ba) =

(Wψ xy)(ba) . (Wψ xx)(ba) 1/2 (Wψ yy)(ba) 1/2

(9.27)

The notation ‘ stands for the smoothing operator in both time and scale parameters [1]. The wavelet coherency is similar to simple statistical correlation but 0 ≤ (Rψ xy)(ba) ≤ 1. This measure equals to 1 and implies a perfect relationship between the two signals in both time and scale; however, this goes to 0 if the two time series are independent. As with the complex wavelets (for example, Morlet wavelet), the local phase ζ is proportional to the ratio between the imaginary part (I) and the real part (R) of the wavelet transform defined as [1,44] (ζψ x)(ba) = tan−1

I[(Wψ x)(ba)] . R[(Wψ x)(ba)]

(9.28)

The phase difference of two time series x(t) and y(t) is useful in identifying antiphase relationships, and it is defined as (ζψ xy)(ba) = tan−1

I[(Wψ xy)(ba)] . R[(Wψ xy)(ba)]

(9.29)

Figs. 9.18 and 9.19 show the wavelet power spectra of a sine wave. The Morlet wavelet is used as the basic wavelet function. Figure (9.18) corresponds to a sine wave with a single frequency, and Figure (9.19) corresponds to a sine wave with multiple frequencies.

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FIGURE 9.18 The wavelet power spectrum of a sine wave with a single frequency.

FIGURE 9.19 The wavelet power spectrum of a sine wave with multiple frequencies.

9.3.5 Statistical significance Similarly, in time series methods, we need to assess the statistical significance of the patterns identified by the wavelet approach. The bootstrapping methods, simply resampling procedures, are employed to evaluate the statistical significance. The observed time series data, which share some properties with the original series, are used to construct the time series defined under the null hypothesis. A procedure based on a resampling of the observed data with a Markov process scheme that preserves only the short temporal correlations is used. We focus on testing whether the wavelet power spectra or the wavelet coherence observed at a particular position on the time-scale plane have not resulted from

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a random process with the same Markov transitions (time order) as the original time series. The procedure of computing the bootstrapped series is discussed in [1,21]. The significant regions are covered in a black line inside the power spectrum (for example, see Fig. 9.18).

9.3.6 Cone of influence Wavelet transformations are used for the time series that are short and noisy. The values of the wavelet transform are generally corrupted as the wavelet approaches the edges of the noisy time series, producing a boundary effect. This affected area increases as the size of the scale parameter a increases. This region is said to be the cone of influence, and it is generally represented as a cone in the wavelet power spectra (for example, see Fig. 9.18).

9.4 Wavelet analysis in epidemiology and dengue Epidemiological time series are often generated by complex systems of which information we know is limited. Globalization, unsystematic urbanization, and climate change have forced disease spreads to be more complex. Predictable behavior in such systems, such as trends and periodicity, is therefore of great interest for the researchers. Most traditional mathematical methods that examine periodicity in the frequency domain, such as Fourier analysis as discussed previously in this chapter, have implicitly assumed that the underlying processes are stationary in time [45]. However, epidemiological time series are extremely nonstationary. Thus classical time series models are not capable of handling these noisy epidemiological data accurately, or sometimes they fail to provide useful information such as periodicity and spectral properties. Wavelet analysis is widely used in epidemiology and environmental sciences due to its specific nature of identifying spectral properties of nonstationary time series. Bernard and Mario [1] analyzed weekly measles cases in the city of York in the United Kingdom (UK) for the prevaccination era, 1944–66. Before 1960, a two-year mode dominated, but after 1960 the wavelet power spectrum has shown more aperiodic dynamics. An association between cholera dynamics and ENSO (El Nino Southern oscillation) in Ghana has been found using cross wavelet and wavelet coherence analysis [1]. Using wavelet analysis, it is suggested that the cholera cases in Ghana are weakly coherent with ENSO except for 1980–85 in the 4-year periodic band. Yan and Gary [46] used wavelet analysis together with genetic algorithms to find the seasonality effect of climate to dengue outbreaks in Singapore. Bernard and Mario [44] analyzed monthly data for Thailand from 1983 to 1997 using wavelet approaches, and they have noted a strong association between monthly dengue cases in Thailand and the dynamics of El Nino for the 2–3 year periodic mode. Anna M. Stewart-Ibarra et al. in [29] applied wavelet techniques to analyze dengue cases in Ecuador with respect to climate periodicity and social-ecological risk factors in 2010. They highlighted the importance of geospatial information in dengue surveillance and the potential to develop a climate-driven spatio-temporal prediction model supporting their results through the wavelet analysis.

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9.4.1 Analysis of dengue incidents in urban Colombo The weekly dengue cases were obtained from Colombo Municipal Council (CMC) and the Ministry of Health, Sri Lanka from 2006 to 2017. We used wavelet analysis to identify any patters of dengue cases in Colombo city considered as one geographic area.

9.4.2 Effect of climate Climate variations play an important role in occurring dengue outbreaks in Colombo. After the rainy season, a dengue outbreak generally follows in Colombo. Therefore it is important to identify the patterns of dengue outbreaks and the relationship between climate variation and dengue cases in Colombo to set up early warning systems [25]. We obtained daily climate data in Colombo for various parameters such as temperature, rainfall, pressure, humidity, and wind speed from the Department of Meteorology, Sri Lanka. These daily climate data were then transformed into weekly data by taking the average. Then, we used cross wavelet analysis to identify the relationships between dengue cases and climate variables such as temperature and rainfall in Colombo. Generally, other climate factors such as humidity and pressure vary similar to temperature in Colombo, so they were not considered in this study.

9.4.3 Analysis of dengue incidents in urban Colombo The weekly dengue cases were analyzed from 2006 to 2017 using wavelet transformation approach discussed above. Fig. 9.20 shows the power spectrum of dengue cases time series in Colombo city. According to this figure, dengue cases show some seasonal oscillations, and these fluctuations are in 16–32 weekly band period, and this region is shown in orange (light gray in print version) and red (gray in print version) color. Climate and dengue cases

FIGURE 9.20 The wavelet power spectrum of dengue cases in Colombo Municipal Council from 2006 to 2017.

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FIGURE 9.21 Top: Cross wavelet power spectrum of weekly dengue cases versus average maximum temperature. Bottom: Cross wavelet power spectrum of weekly dengue cases versus average rainfall in Colombo from 2006 to 2017.

time series data were analyzed using wavelet theory to identify any available phase relationships. The cross wavelet power spectra obtained from a MATLAB program are given in Fig. 9.21. The cross wavelet power spectrum that was obtained for maximum temperature versus dengue cases shows an antiphase relationship (arrows pointing left). However, Fig. 9.21 suggests this relationship is not significant at 5% level. In contrast, there exists a significant antiphase relationship between dengue cases and rainfall with an ∼ 16 period band. This suggests a possible lead time of rainfall for an outbreak of dengue in Colombo during the period, and such information related to the time delay would be critical in parameter estimation in mathematical models.

9.4.4 Effect of human mobility: a case study Human mobility can also have a direct influence on the dengue transmission since the larger the human movements and their gathering, the larger the contact rates. Data on population mobility were limited, and they were only obtained during the year 2011 by perusing the newspapers and internet. Information on school holidays, major public and cultural events in the city, international exhibitions, large scale international sport tournaments, the New Year festival, the Vesak (Buddhist cultural) festival, and school celebrations was considered because during these events the contact rate of people (i.e., aggregation of people) tends to be higher than average, while school vacations were considered as the period of time where the contact rate of people is relatively lower. A weekly mobility index MI (t) in (9.30) was constructed for the year by assigning weights for each of the above events held in the city of Colombo, assuming the arbitrary power of the contact rate of humans. We included a noise component to the proposed index to incorporate the uncer-

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FIGURE 9.22 The Colombo district map (source: http://www.colombo.dist.gov.lk/).

tainty. A higher value for the index means a higher mobility, and a lower value for the index means a lower mobility MI (t) =max

 m 

wi ni (t) + ζ, 0 ,

wi , ni (t) ∈ R, ∀i ∈ {1, 2, ..., m}

and ξ ∼ N (0, 1).

i=1

(9.30) Further, wi denotes the weight assigned to each social event i, ni (t) denotes the number of events in each category of event i, and m is the total number of categories of events. We used this mobility index to identify any patterns of weekly dengue cases in Colombo with respect to human mobility for the year 2011 using cross wavelet analysis. Furthermore, the city of Colombo is surrounded with four major suburban regions, namely, Dehiwala, Kolonnawa, Kotte, and Nugegoda (Fig. 9.22). A large number of people travel from these suburban areas to Colombo and come back daily. This human movement may be responsible for spreading the dengue virus to these surrounding areas supposing that Colombo city is the potential source for dengue. We used cross wavelet analysis to find any relationships between the variation of human mobility index constructed for Colombo city and the spread of dengue cases with in the city itself. The mobility and the dengue cases show a statistically significant relationship during the middle of the year 2011 (Fig. 9.23). Fig. 9.24 represents the cross wavelet power spectra of weekly mobility index in Colombo versus weekly dengue cases in Dehiwala, Kolonnawa, Kotte, and Nugegoda divisions, respectively. All four divisions show significant regions in each power spectrum against the red (gray in print version) noise at 5% level during the weeks 15–25 (around the middle of the year). In Kolonnawa, Kotte, and Nugegoda, it can be seen an in-phase relationship with the arrows pointing right most of the weeks in the above period. In Kotte

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FIGURE 9.23 The cross wavelet power spectrum of weekly dengue cases versus mobility index variation in Colombo during the year 2011.

FIGURE 9.24 The cross wavelet power spectra of the weekly mobility index in Colombo and weekly dengue cases in suburban bordering areas (top left: Dehiwala, top right: Kotte, bottom left: Kolonnawa, bottom right: Nugegoda) of Colombo city limits during the year 2011.

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it can be seen a reasonably high power in the 2–3 weeks band in the period from weeks 20–25.

9.5 Conclusion Data generated through epidemiological processes are understood to be very noisy and complex. This may be due to multiple reasons such as the uncertainty and the multifactor nature of disease transmission, different disease diagnostic methods used, the method of case classification, and the level of accuracy of data collection. Therefore these data should be analyzed using the techniques that are able to extract hidden information regarding the disease dynamics and time-invariant interdependencies among numerous exogenous drivers of the underlying dynamics. Information pulled out may be used to develop a wide range of statistical and mathematical models in forecasting, models in intervention assessment, decision-making and policy evaluations, and budgeting. This chapter discussed some nontraditional but powerful and effective tools in mathematics that can be used to critically analyze dengue epidemic time series data. We have mainly focused on two techniques: fast Fourier transformation (FFT) and wavelet analysis. In the first phase of this chapter we introduced FFT as a mathematical tool to identify the periodic behavior of dengue and related climate data critically discussing its advantages over statistical methods in time series analysis. We assessed the applicability of this technique using a case study that investigates the influence of climate variation to dengue transmission dynamic in Colombo, Jakarta, and Bangkok. Then, we examined the delay effect of climate drivers on dengue using the Pearson correlation coefficient. Our results demonstrated that the periodic pattern of dengue cases followed the periodic pattern of rainfall data with an approximate delay of two months. Wavelet theory was then introduced with sufficient details on its mathematical development, explaining its clear signal processing benefits over FFT. We applied wavelet transformation to investigate the spectral properties of dengue spread in Colombo during the time period from 2006 to 2017. The multivariate patterns of dengue disease transmission in Colombo were then analyzed using the cross wavelet transformation considering climate variables (temperature and rainfall) and time-dependent human mobility index constructed. A likely existence of a lead time from weekly averaged rainfall to dengue spread was identified having an ∼ 16 period band. The impact of varying mobility in Colombo city to the dynamic of dengue epidemic profile in four suburban adjacent areas (Dehiwala, Kolonnawa, Kotte, and Nugegoda) was also investigated. The statistical tests each for wavelet and cross wavelet transformations were at 5% level of significance. All four divisions considered for the study showed significant regions in each power spectrum against the red (gray in print version) noise during the weeks 15–25 (around the middle of the year of investigation). It is worth noting that both FFT and wavelet tools are not capable of forecasting any patterns or relationships among variables to the future. The projections would be achieved through well-established statistical or mathematical models. The FFT and wavelet tech-

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niques can be useful in detecting patterns of noisy dengue time series data in relation to multiple external drivers of the epidemic. Such a finding may be considered as an input to accurately estimate the parameters and to correctly establish the association of the variables in governed mathematical equations that are ultimately used for simulations and projections.

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[20] T. Zhang, M. Yang, X. Xiao, Z. Feng, C. Li, Z. Zhou, Q. Ren, X. Li, Spectral analysis based on fast Fourier transformation (FFT) of surveillance data: the case of scarlet fever in China, Epidemiology and Infection 142 (3) (2014) 520–529. [21] B. Cazelles, M. Chavez, D. Berteaux, F. Ménard, J.O. Vik, S. Jenouvrier, N.C. Stenseth, Wavelet analysis of ecological time series, Oecologia 156 (2) (2008) 287–304. [22] S.A.A. Karim, M.H. Kamarudin, M.K. Hasan, Wavelet transform and fast Fourier transform for signal compression: a comparative study, in: 2011 International Conference on Electronic Devices, Systems and Applications (ICEDSA), IEEE, 2011. [23] S. Jemai, M. Ellouze, Variability of precipitation in arid climates using the wavelet approach: case study of watershed of gabes in South-East Tunisia, Atmosphere (2017). [24] D. Komorowski, S. Pietraszek, The use of continuous wavelet transform based on the Fast Fourier transform in the analysis of multi-channel electrogastrography recordings, Journal of Medical Systems (2016). [25] W.P.T.M. Wickramaarachchi, S.S.N. Perera, Modelling and analysis of dengue disease transmission in urban Colombo: a wavelets and cross wavelets approach, Journal of the National Science Foundation of Sri Lanka (2015). [26] J.M. Polanco-Martínez, J. Fernández-Macho, Dynamic wavelet correlation analysis for multivariate climate time series, Nature Scientific Reports (2020). [27] K.C. Raath, K.B. Ensor, Time-varying wavelet-based applications for evaluating the water-energy nexus, Frontiers in Energy Research (2020). [28] Q. Zhang, Y. Chen, Epidemiology of dengue and the effect of seasonal climate variation on its dynamics: a spatio-temporal descriptive analysis in the Chao-Shan area on China’s southeastern coast, BMJ Open 2 (2019). [29] A.M. Stewart-Ibarra, Á.G. Muñoz, Spatiotemporal clustering, climate periodicity, and socialecological risk factors for dengue during an outbreak in Machala, Ecuador, in 2010, in: BMC Infectious Diseases, 2014. [30] P.S. Addison, Introduction to redundancy rules: the continuous wavelet transform comes of age, Philosophical Transactions of the Royal Society (2018). [31] M.T. Heideman, D.H. Johnson, C.S. Burrus, Gauss and the history of the fast Fourier transform, Archive for History of Exact Sciences 34 (3) (1985) 265–277. [32] J.W. Cooley, J.W. Tukey, An algorithm for the machine calculation of complex Fourier series, Mathematics of Computation 19 (90) (1965) 297–301. [33] R.I.N. Juárez, W.T. Liu, FFT analysis on NDVI annual cycle and climatic regionality in Northeast Brazil, International Journal of Climatology: A Journal of the Royal Meteorological Society 21 (14) (2001) 1803–1820. [34] A. Kalauzi, M. Cukic, H. Millán, S. Bonafoni, R. Biondi, Comparison of fractal dimension oscillations and trends of rainfall data from Pastaza Province, Ecuador and Veneto, Italy, Atmospheric Research 93 (4) (2009) 673–679. [35] A. Zakaria, The generation of synthetic sequences of monthly cumulative rainfalls using FFT and least squares method, in: Seminar Hasil Penelitian & Pengabdian Kepada Masyarakat, September 2008, University Lampung, 2008. [36] P.D.N.N. Sirisena, F. Noordeen, Evolution of dengue in Sri Lanka–changes in the virus, vector, and climate, International Journal of Infectious Diseases 19 (2014) 6–12. [37] H.K.W.I. Jayawardene, D.U.J. Sonnadara, D.R. Jayewardene, Trends of rainfall in Sri Lanka over the last century, Sri Lanka Journal of Physics 6 (2005). [38] W.A.J.M. De Costa, Climate change in Sri Lanka: myth or reality? Evidence from long-term meteorological data, Journal of the National Science Foundation of Sri Lanka 36 (2008). [39] N. Thanvisitthpon, S. Shrestha, I. Pal, Urban flooding and climate change: a case study of Bangkok, Thailand, Environment and Urbanization ASIA 9 (1) (2018) 86–100. [40] T.R. Derrick, J.M. Thomas, Time series analysis: the cross-correlation function, in: Innovative Analyses of Human Movement, 2004, pp. 189–205. [41] C.K. Chul, Wavelet Analysis and Its Applications, Department of Mathematics, Texas A and M University College, Station, Texas, USA, 1995. [42] D.T. Lee, A. Yamamoto, Wavelet analysis: theory and applications, Hewlett-Packard Journal 45 (1994) 44–52.

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10 Covid-19 pandemic model: a graph theoretical perspective Fadekemi Janet Osayea and Alex Alochukwub a Alabama State University, Montgomery, AL, United States b University of Johannesburg, Johannesburg,

South Africa

10.1 Introduction A graph, often called a network, is a structure that consists of entities (usually called vertices or nodes) and edges representing interactions or connections between the entities. Graph theory, a branch of mathematics, is the study of graphs which are mathematical structures for modeling pairwise relations between objects. It plays a role in modeling a variety of disciplines, ranging from 5G and routing networks in telecommunications, to protein-protein interaction networks, and disease modeling in biology. We may consider a COVID-19 network with nodes representing people and edges formed by those within two meters apart. Such a contact network is an indicator of how contagions like COVID19 spread. Other examples can be seen in arrays of interlinked processors in computers, transportation networks such as the train stations (the stations being the entities, the links being the rails), the world wide web (the web pages being the entities, the links being the clickable hyper links on the web pages), or in social networks, with people in a particular group being the entities, and friendships between them being the links. Generally, almost all academic disciplines can be modeled as graphs for prediction purposes especially in cases where there are connections or links between entities (or individuals). To have a clear understanding of an outbreak using graphs, one needs to study the underlying network to see how connected the individual networks are. A network is said to be connected if any individual (or node) can be reached from any other individual by following the network links; epidemiologically, this is equivalent to an infection being able to reach the entire population from any individual. Research in graph theory has provided a vast knowledge of quantitative tools and approaches for describing different real life networks, many of them having epidemiological applications. In recent times, researchers and epidemiologists have focused their attention to the use of networks due to their ease and critical role for modeling new infectious diseases such as HIV [12], SARS virus, Ebola and Asian bird flu, which are diseases that have become threats to human lives [10]. COVID-19, or coronavirus disease 2019, is also one of such deadly diseases. It is a respiratory disease caused by a severe acute syndrome Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00015-6 Copyright © 2023 Elsevier Inc. All rights reserved.

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coronavirus (SARS-CoV-2), which has not been previously observed within the human population [8] before it started in 2019 in Wuhan, China. The coronavirus disease spreads through droplets when an infected person sneezes or coughs, which can then be transmitted from the infected person to an uninfected person through close contacts. According to the World Health Organization, there are presently over 76 million confirmed cases and 1.7 million deaths in 222 countries, areas, or territories [23]. Methods used for the prevention and reduction in the spread of COVID-19 include regular washing of hands, use of hand sanitizers, social distancing, movement restrictions, and in some cases complete lockdown. To combat the spread of the disease, some pharmaceutical companies such as Pfizer/BioNTech and Moderna have produced vaccines that are now authorized for emergency use in the USA by the Food and Drug Administration [5]. At the time of preparing this publication, these vaccines are limited in supplies and not everyone will be able to get vaccinated as soon as possible. One may want to know the policies in place to prevent further spread pending sufficient vaccines. Thus there is a need for further research on efficient mathematical models in understanding the spread of the disease. Hence we present a discrete time mathematical model for COVID-19 from a graph theoretical perspective and give a sufficient condition that guarantees the possible end or continuation of a COVID-19 pandemic at any given time t. This chapter is organized as follows. In Section 10.2, we define some of the most important graph theory terminology and notation that will be used throughout this chapter, as well as some epidemiological terminology in relation to network theory. In Section 10.3, we present a review of some epidemic and network models on COVID-19. We also review previous results for modeling infectious diseases from a graph theoretical perspective, which motivated our model approach. Section 10.4 describes in detail our graph theoretical discrete time model for the spread of COVID-19 and establishes some rules that will be useful in the proofs of our main results. In Section 10.5, we present some of our results based on the proposed model. In the concluding Section 10.6, we discuss the significance of the results obtained in Section 10.5 and recommend other solution approaches.

10.2 Preliminaries 10.2.1 Graph theory terminology A graph G = (V , E) consists of a nonempty set of elements V referred to as vertices and a set (possibly empty) E of 2-element subsets of V called edges. We often write V for V (G) and E for E(G). Hence, for each 2-element e = {u, v} ∈ E, we say that vertices u and v are adjacent or e is incident with u and v if both are joined by the edge e. Herein, we consider only finite graphs. The order of a graph G is the number of vertices in G denoted by n or |V (G)|, while the size of G is the cardinality of the edge set E denoted by m or |E(G)|. A vertex coloring of a graph G is an assignment of colors to the vertices of G, one color to each vertex. Mathematically, a vertex coloring is a map c : V → S such that c(u) = c(v)

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whenever u and v are adjacent. For the purpose of this chapter, there is a possibility for any two adjacent vertices to have the same color. The degree of a vertex v of G denoted by degG (v) (or deg(v)) is the number of edges incident with v. The minimum and maximum degree of a graph G denoted as δ = δ(G) and  = (G), respectively, is the minimum and maximum of the degrees of vertices in G. A vertex of degree one is called an end vertex or a leaf. A graph is said to be k-regular if the degree of every vertex in G is k. A walk W in a graph G is an alternating sequence of vertices and edges W : v0 , e1 , v1 , e2 , v2 , . . . , vk−1 , ek , vk or simply W : v0 , v1 , . . . , vk such that for each i ∈ 0, 1, · · · , k, ei = vi vi+1 . W is said to be a path if all vi s are distinct. The path P : v0 , v1 , . . . , vk is often referred to as a (v0 , vk )-path since it begins at vertex v0 and ends at vk . The distance dG (u, v) between two vertices u, v of a graph G is the length of a shortest (u, v)-path in G. For a subset S ⊆ V and a vertex v of G, the distance d(v, S) between v and S is the minimum value of d(x, i), i ∈ S. The eccentricity ex(v) of v is the largest distance between v and any other vertex in G. A graph is connected if there is a path between every pair of vertices in G. A connected graph is a network. For the purpose of this research, we consider only simple, connected, and undirected graphs. The neighborhood NG (v) of a vertex v ∈ V is the set of all vertices adjacent to v in G, while the closed neighborhood N [v] is the union of {v} and its neighborhood. Hence |NG (v)| = degG (v) and |NG [v]| = |NG (v) ∪ {v}| = degG (v) + 1. A graph of order n is said to be a complete graph if all the vertices are pairwise adjacent. A complete graph of order n is denoted by Kn . A tree is a connected graph without cycles. A spanning tree is a spanning subgraph of G, which is a tree. A k-regular tree is a tree for which all internal vertices have a degree k and the leaves have degree 1. A graph is bipartite if V (G) can be partitioned into two nonempty subsets V1 and V2 such that every edge of G joins a vertex of V1 to a vertex of V2 . If each vertex of V1 is joined to every vertex of V2 , then G is called a complete bipartite graph and is denoted as Kn,m , where n = |V1 | and m = |V2 |, or vice versa. The star graph Sn−1,1 of order n, sometimes simply known as an n-star, is a tree on n vertices and m edges with one vertex having degree n − 1 and the other n − 1 vertices having degree 1. For basic definition and further reading in graph theory, we refer the reader to [4] and [3] for definitions related to distances in graphs.

10.2.2 Epidemiological terminology Most of the mathematical models for modeling infectious disease are based on the compartmentalization of the population into small groups according to individual’s disease status. The compartmental model was first discovered by Lowell Reed and Wade Hampton Frost of John Hopkins University in the 1920s [11], a work that remained unpublished. It is a model used in analyzing the extent to which a disease will spread through a population. The Susceptible-Infectious-Recovered (SI R) model and the Susceptible-InfectiousSusceptible(SI S) models are the two most common compartmental models, where S, I , and R refer to the number of susceptible, infected, and recovered individuals, respectively, from a population of size N .

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While an outbreak is a causally connected cluster of cases that, by chance or because the transmission probability is low, dies out before spreading to the population at large, an epidemic, on the other hand, is such that the infection escapes the initial group of cases into the community at large and results in population-wide incidence of the disease. An epidemic that spreads across several countries is a pandemic. The basic reproduction number R0 of an infection is the expected number of secondary cases that is generated as a result of just one case in the population, assuming that every other individual is susceptible to the infection. Mathematically, R0 = βτ = cˆ · t · τ , where β is the infection producing contacts per unit time, τ is the mean infectious period, t is the transmissibility (i.e., probability of infection between a susceptible and an infected person), cˆ is the average rate of contact between the infected and a susceptible. R0 < 1 implies that the outbreak will die out at some point, and R0 > 1 means the outbreak will increase with time. R0 = 1 is a critical value that determines if an outbreak will occur or not in a compartmental model. For a detailed systematic review of epidemiological and network theory, see the paper by Keeling and Eames in [9].

10.3 A survey of mathematical models on diseases Many factors are considered when deciding the most efficient mathematical models to be used for modeling a particular disease. Aside from compartmental models, statistical/machine learning models have also been used due to their predictive abilities. While the compartmental models assume equal likelihood of interaction between individuals, that is, every individual has equal likelihood of transmitting the disease, the statistical models usually do not depict the entire story of phenomenon since the data might not be homogeneous. A more recently developed contact network model, invented by Grassberger in 1986 but later improved by M.E.J. Newman in the early 2000s [13], uses graph theory approach and helps address some of the concerns from previous models. In this section we present some previous results on these models.

10.3.1 Epidemic/pandemic models on Covid-19 Recent mathematical models on COVID-19 have included the use of differential equations ([14], [24], [20], [25]) and statistical or machine learning methods ([6],[7], [21]). A compartmental model for COVID-19 epidemic was presented by Nyabadza et al. in [14] by fitting a Susceptible-Exposed-Infected-Recovered-(SEI R) model on available data (the cumulative number of infected cases). The model which considered the effects of social distancing on the transmission dynamics of COVID-19 used South Africa as a case study. The model described in (10.1) assumes a constant immigration rate of  in which a proportion p of this immigration rate is assumed to be exposed, or otherwise susceptible

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with recruitment rate (1 − p). dS dt dE dt dI dt dR dt

= Srec − λS = Erec + λS − κE

(10.1)

= κE − αI = αI

subject to S(0) = S0 > 0, E(0) = E0 ≥ 0, I (0) = I0 ≥ 0, R(0) = R0 ≥ 0, where κ is the progression rate of individuals being exposed to infected, α is the recovery rate, and  (1 − p), t0 ≤ t < tlock Srec = (10.2) 0, t ≥ tlock ,  Erec =  λ=

p, t0 ≤ t < tlock 0,

βI N (t) , βρI N (t) ,

t ≥ tlock , t0 ≤ t ≤ tlock t ≥ tlock

.

(10.3)

(10.4)

Note that t0 represents the initial time when infection was identified, tlock is the time when lockdown occurs, β is the rate of infection, and ρ is the rate of incorporating social distance in the model. This model shows that a relaxation of social distancing can significantly increase the number of confirmed cases during lockdown. Serhani and Labbardi [20] proposed a similar model to that of Nyabadza et al. [14], but a modified compartmental SI R model with the assumption that infected individuals were all originally asymptomatic. By using Morocco as their case study, the modified model, described in Fig. 10.1, is then described by a dynamic system enclosing a closed population with Susceptible-Infected-Asymptomatic-Quarantined-Recovered-Dead (SI AQRD). Here β is the transmission or contact rate with the other parameters defined as described in Fig. 10.1. The authors established a relationship between the basic reproduction number R0 and the containment control coefficient c0 and showed that a high c0 is crucial in controlling the spread of the disease. Zeb et al. [24] and Ahmed et al. [1], however, modeled COVID-19 outbreak using numerical approach by incorporating isolation class. While Ahmed et al. [1] used both fractional ordinary differential equations (FODEs) and ordinary differential equations to approximate the basic reproduction number R0 with Nigeria as a case study, Zeb et al. [24]

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FIGURE 10.1 Diagram of the transition between compartments (Sehani and Labbardi). [20].

used nonstandard finite difference (NSFD) scheme and Runge-Kutta fourth order method for the proposed model. Both studies presented different graphical results, which shows agreement between numerical and theoretical results, useful in minimizing the spread of infection. Various statistical models have also been used to model COVID-19. For example, Olivia et al. [17] developed a simple Markov chain model for the COVID-19 epidemic as shown in Fig. 10.2. The diagram illustrates how everyone in the population can transition between different states starting from being healthy to becoming infected, and possibly being contagious before becoming symptomatic, which can lead to being hospitalized and then death or recovery. The authors assumed that for j ∈ {1, · · · , 7}, the transitional probabilities pj follow from an Erlang distribution denoted ξ which can be scaled when multiple outcomes from a state are possible. They further expressed the basic reproduction number R0 in terms of the transitional probability p0 , where

R0 = p0

∞ ∞ ∞    (1 − p2 − p5 )i + p0 p2 (1 − p2 − p5 )i (1 − p3 − p6 )i . i=0

i=0

j =0

In order to ensure that their proposed model mimics some of the clinical observations, the following were reported. R0 = 2.75, p1 ∼ ξ(10, 4), p2 ∼ (1 − ω5 )ξ(3, 1.5), p3 ∼ (1 − ω6 )ξ(14, 2), p4 ∼ (1 − ω7 )ξ(4, 1), p5 ∼ ω5 ξ(5, 0.5), p6 ∼ ω6 ξ(10, 1), p7 ∼ ω7 ξ(6, 1), with ω5 = 0.85, ω6 = 0.9, ω7 = 0.95. To predict recovered and deceased cases, Theerthagiri et al. [21] use different machine learning classification algorithms for predicting the possibility of occurrence of COVID19 based on their characteristics. The dataset includes two inputs (X) as age and gender, while the output is whether the patient recovered or died. Prediction results from logistic regression (LR), K-nearest neighbor (KN N), decision tree (DS), support vector machines (SV M), and multilayer perceptron (MLP ) were compared. However, the KN N algorithm shows 80.4% prediction accuracy and a lowest error rate of 0.19. For other statistical or machine learning results on COVID 19, see [7], [18], and [26].

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FIGURE 10.2 Markov chain’s model describing transition between states after an individual is infected, Olivia et al. [17].

10.3.2 Network models Research on network models and the application of graph theory to epidemic or pandemic models have been a subject of study in recent years in order to address some of the concerns with compartmental and statistical models [22]. Bhapkar et. al. [2] provided a graph theoretical model for the COVID-19 pandemic by describing virus graph of different types and studying the exponential growth of infected people over some period of days for different countries. L. Ancel Meyers and her colleagues in [10] compared the predicted R0 from compartmental models on severe acute respiratory syndrome (SARS) to that obtained using contact networks. In their model, each person represents a vertex, while contact between two people represents an edge. Using degree distribution Pdeg (k) of the network defined by the fraction of vertices (or nodes) in the graph with degree k, they showed that for SARS, the network transmissibility t is connected to the basic reproduction number by  k 2   −1 , R0 = t k  2 where k and k are the mean and mean square degrees of the network, respectively. To obtain R0 , they used parameters of the Poisson network given by Pk = (e−μ )

μk k!

with mean contact number μ = 19.6 and power law network given by  0 for k = 0, Pk = k Ck −ψ e κ for k > 0,

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with parameters κ = 94.2 and ψ = 2, where Pk is the probability that a randomly selected person in the community network has degree k. Their model indicated that for a single value of R0 , any two outbreaks may have completely different epidemiological outcomes even under the same settings. A very common way of describing connectivity in networks is the use of adjacency matrix denoted as A. Supposed the vertices (individuals) of a network represented with a graph G are labeled as v1 , v2 , . . . , vN , then the adjacency matrix of G is an N × N matrix whose (i, j )th entry is 1 if there is an edge joining vertices vi and vj , and 0 otherwise. This implies that infection can pass from person vi to person vj whenever Aij = 1. Several useful network properties can be obtained from the adjacency matrix of a network [9]. For example, the average number of contacts per person is n¯ =

1  1 Aij = trace(A2 ). N N ij

The powers of A are also used to calculate a measure of network clustering  defined by =

trace(A3 ) ||A2 || − trace(A2 )

.

Epidemiologically, a connected graph implies an infection can be transmitted to any individual in the network by following the network paths (links), which mathematically means ∞ m m m=1 A (or equivalently limm∞ A ) has no zero terms. Using this same notion of graph connectivity, Seibold and Callender [19] used an agentbased modeling platform to compare predictions from mathematical epidemiology to results obtained from simulations of disease transmission on a regular tree graph. A tree graph is a rooted-tree centered on a root vertex, and it contains no closed loops (or cycles). The regular tree graph is a tree graph with the same number of branches connected to every non-end vertex. Mainly, properties such as network diameter (largest eccentricity of G) and the density D of a network, defined as D=

2|E| , |V |(|V | − 1)

were used in studying the duration on an outbreak by Seibold and Callender [19]. Specifically, they proved that the number of vertices N is a function of the tree height λ and the degree of the root vertex d. Theorem 10.1 ([19]). Let N be the number of vertices in a regular tree graph. Then N=

λ 

di.

i=0

A more theoretical application of graph theory to modeling diseases was first used by Mukwembi [12] for studying the effects of the rate of replacement of dead cells by either

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healthy cells or by infected cells on HIV infection dynamics. Here a vertex set of G is the set of sites of the lymph nodes, while an edge indicates that the corresponding sites in the lymph node are neighbors. The transition rule in [12] is such that a cell is either healthy, infected, or dead, and a healthy cell becomes infected if it is adjacent to at least one infected cell. Moreover, he proved that for n = |V (G)|, if there exists a vertex whose number of infected neighbor is n − 1, then it will take a linear time t move for every cell to become healthy. That is, if R ∗ is the number of infected neighbor of a vertex, S is the set of vertices of V that were infected at initial configuration t = 0, s = |S|, r(t) is the number of vertices that were infected at each discrete time t, and ki = |{x ∈ V |d(x, S) = i}| with i = 0, 1, 2, . . . , e, then the following theorem follows: Theorem 10.2 ([12]). Let the number of vertices in G be n and S = V . If R ∗ = n − 1, then all vertices can become healthy in a linear time moves of order 3(n−s+δ) δ+1 . Moreover, r(t) =

kt 0

if t < e, otherwise,

unless s = 1, and S contains a vertex of degree R ∗ , in which case r(0) = 1, r(1) = n − 1, r(2) = 1 and r(t) = 0 for all t ≥ 3, where e is the maximum value of d(x, S), x ∈ V . A similar graph theoretical approach was used in [15] and [16]. Nyabadza, Mukwembi, and Rodrigues in [15] used a graph theoretical technique to categorize contacts and links that can possibly lead to a tuberculosis (TB) transmission in any specific community. Particularly, an individual at any given time t in a community containing a total of N people can be in any of the following categories: S for susceptible, L for latently infected (exposed to TB but not infectious), or I for infectious. In this case, each individual represents a vertex, and a contact between two people corresponds to an edge. A threshold value α ∗ = 1 − n1 was proposed for reasonable infective R0 (i.e., infected individuals who are deliberate about not spreading the disease), while a proportion 1 − α ∗ are considered to be unreasonable infective I0 − R0 , where I0 is the set of infected individuals at the initial configuration t = 0. Theorem 10.3 ([15]). Let the number of vertices in G be n and e = maxx∈V d(x, I0 ). 1. If α ∗ ≥ 1 − n1 , then there exists a set V0 ⊂ V with |V0 | ≥ e such that every vertex in V0 is susceptible for all t. 2. If α ∗ < 1− n1 , then there is an integer m and graphs Gm,n of order n such that for all time t ≥ t ∗ (m, n) and for all v ∈ Gm,n , every vertex in Gm,n is either latently infected or infectious. The model suggests that the proportion of the reasonable infective in a population that can help prevent an epidemic is a function of α ∗ , where the epidemic dies out if α ∗ ≥ 1 − n1 but grows if otherwise. To verify Theorem (10.3), numerical simulations were carried out to generate a local network of interactions among the L and I classes using a graph adjacency

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matrix Gij given by  Gij = and

 Ij =

1 if and only if i, j are connected, 0 otherwise

1 if and only if individual j is infectious, 0 otherwise,

so that the rate at which a susceptible individual i, κi , becomes infectious is the product of the rate of transmission t and the number of infectious contacts: κi = t



Gij Ij .

j

By using South Africa as a case study, Nyabadza, Mukwembi, and Rodrigues in [16] examined the spread of drug use wherein individuals in the population are classified into two categories: users in treatment and drug users not in treatment. They described a drug use epidemic by a finite number of states and transition rules governing the dynamics in each discrete time step using a graph theoretical perspective. A neighborhood prevalence of each individual in the population was considered. The neighborhood in this sense for an individual vi refers to the proportion of vi ’s drug user contacts who are not in treatment amongst all of vi ’s contacts. The authors further introduced three different parameters α ∗ , β ∗ , and γ ∗ , which are dependent on the neighborhood prevalence, and examined how slight changes in these parameters will affect the system dynamics. A critical term R, simn−2 ilar to a reproduction number, was defined to be (n−1)α ∗ . It was shown that if R < 1, then the following theorem is valid. Theorem 10.4 ([16]). Let G = (V , E) be a graph with n vertices. If R < 1, then there exists a set F ⊂ V such that ct (F ) = green, where ct (F ) = green implies that F is the set of susceptible individuals. Moreover, |F | = e(I0 ), where e and I0 are as defined in Theorem 10.3. In other words, R < 1 guarantees the existence of a drug-free subpopulation. However, if R > 1, then the following theorem holds. Theorem 10.5 ([16]). Let G = (V , E) be a connected graph with n vertices. Assume that α ∗ ≤ 1 1 1 ∗ ∗ ∗ n−1 , β ≤ n−1 , and γ ≤ n−1 . Then, for all t ≥ t , there exists D ⊆ V such that all vertices in D are drug users who are either under treatment or not under treatment, unless G is a path. This shows that, irrespective of the values of α ∗ , β ∗ , and γ ∗ , R > 1 guarantees a drug epidemic within the subpopulation. To verify the graph theoretical results, the transitional rules for the spread of drug use in each discrete time step from one state to another were described, and the simulations presented validate the theoretical results.

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10.4 SEIRD model on Covid-19 In this work we use a graph theoretical approach to investigate the spread of COVID-19 by describing extremal cases with algebraic formulas in terms of the number of infected or exposed persons of given order and at any discrete time. We also find a sufficient condition that guarantees the possible end or continuation of a COVID-19 pandemic at a given time. We consider a discrete time SEI RD model (see Fig. 10.3) which considers the effect of social distancing on a finite population of n individuals. The population at any given time is partitioned into the following classes: • • • • •

Susceptible, S(t): Individuals with no coronavirus in their system but are at risk of being infected when in close contact with an infected person. Exposed, E(t): Individuals who have come in contact with an infected person and may be asymptomatic. Infected, I (t): Individuals with either mild or clinical symptoms who can transmit the virus to the healthy individuals resulting in successful infection. Recovered, R(t): Individuals who can recover either naturally due to robust immune responses or due to available supportive clinical treatment. Dead, D(t): Individuals who die as a result of the coronavirus infection.

Thus, at any given time, an individual in the population under consideration is in one of the following states: susceptible S(t), exposed E(t), infected I (t), or recovered R(t). We further assume that the population is constant, that is, the ratio of the growth rate to removal rate is one. Let |S(t)| = st , |E(t)| = et , |I (t)| = it , |R(t)| = rt , and |D(t)| = dt . Then n = st + et + it + rt .

FIGURE 10.3 Transition models of susceptible (blue) S to exposed (orange) E, from exposed to infected (red) I , infected to recovered (green) R or death (black) D.

We model the COVID-19 pandemic by a graph G as follows. The vertex set V of G is the set of individuals, while the edge set is the set of close contact or any social interactions between any two individuals in the population. By a contact, we mean two individuals are within a physical distance of at most 6 feet (or 2 meters). To model the effect of social distancing on the population, we arbitrarily color the state of an individual at any given time. (See Fig. 10.3.)

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For simplicity purpose, we arbitrarily color the vertices of G with blue if the corresponding individual is susceptible, orange if the corresponding individual is exposed, red if the corresponding individual is infected, green if recovered, and black if dead. We will denote the color of v at time t by ct (v), and nt (v, i) denotes the number of neighbors of v with color i at time t. For a subset A ⊆ V , we will denote the set {ct (v) : v ∈ A} by ct (A). The following definition and assumption will be useful when defining the transition rules. p

Definition 10.1. Let v be a vertex of G. Then the neighborhood prevalence Nt (v) of v at time t is defined as nt (v, orange) + nt (v, red) p . Nt (v) = degG (v) p

Thus, for an individual v, Nt (v) is simply the proportion of v’s contacts who are either exposed or infected with the virus. It is observed that COVID-19 as a respiratory infection can either stay or be cleared within a few days. Thus, for the purpose of our model, we may assume the day as the unit of time. The COVID-19 spreading process initially begins with a susceptible population, a small amount of infected individuals, and individuals who are exposed to the infection as a result of contact or social interaction with infected individuals. Hence we start our modeling by initializing t = 0 and then randomly assign color red to a vertex and color blue to the remaining vertices (see Figs. 10.4 and 10.5). We assume that the new cases of the virus were as a result of contacts between susceptible individuals with a proportion of infected (symptomatic) or exposed (asymptomatic) individuals. An exposed individual progresses to either becoming infected after testing positive or showing symptoms; or remain exposed if tested negative during an incubation period (or simply quarantine), typically between 2 to 14 days at a constant infectious rate β. An exposed individual who neither shows any symptoms or test negative is assumed to be susceptible. Now, based on the current state of each individual in the population, the next state of vertices in each discrete time step is determined using the following transitional rules: T1: A susceptible individual becomes exposed if at least one of its neighbors is colored red or orange, and its neighborhood prevalence is at least the exposure rate α, otherwise the individual remains susceptible. Precisely, if ct (v) = blue, then  p orange if Nt (v) ≥ α, ct+1 (v) = p blue if 0 ≤ Nt (v) < α. After the incubation period, an exposed individual either becomes infected if positive or susceptible if negative with infection rate zero,  red if Pt+p (v) = β, 0 < β ≤ 1, ct+p+1 (v) = blue if Pt+p (v) = β = 0,

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where Pt+p (v) = β is the probability of an individual getting infected and (2 ≤ p ≤ 14) is assumed to be the incubation period. T3: An infectious individual can either recover after a period of time or die. If we assume that the expected recovery rate and death rate conditional on being infected are φ and ψ, respectively, then the transition from the infectious state to recovery or death state is defined as follows: if ct (v) = red, then  green if 0 < φ ≤ 1 , ct+1 (v) = black if φ = 0. The recovery rate φ lies between 0 ≤ φ ≤ 1 such that φ = 0 implies no recovery (death ψ) and φ = 1 implies complete recovery. T4: An individual who recovers as a result of robust immune responses or due to clinical supportive treatment is assumed to either remain recovered or susceptible after a period of time with possibility of reinfection. On the other hand, an individual who dies is assumed dead in subsequent discrete time steps. Precisely, a death state is an absorbing state. Hence, if ct (v) = green, then  green if Pt (v) = φ < 1, ct+1 (v) = blue if Pt (v) = φ = 1. Furthermore, if ct (v) = black, then

ct+1 (v) = black

if Pt (v) = ψ = 1 or φ = 0.

If the transition window between recovery and susceptible states is excluded from the model, then recovery state is also an absorbing state. The above rules are designed in such a way that no random vertices and edges are added to the population. For our finite population on n individuals, we give examples using graph theory with Fig. 10.4 and Fig. 10.5, how parameters α, β, and φ affect the dynamics of the spread. Example 1: b

b r

g

b

b

b

b b

b

b

b

t =0

t =1

p

t =2

FIGURE 10.4 Pandemic G1 = S3,1 ; Nt (v) = 13 , α = 12 ; β = 16 ; φ = 1; r = red, b = blue g = green.

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At t = 0 with φ = 1, the infected vertex colored red recovers and changes to green by T 3. p The center vertex adjacent to the red vertex remains blue since Nt (v) = 13 < α = 12 by T 1. At t = 1, the vertex colored green becomes blue by T 4 at t = 2 and the remaining vertices remains blue by T 1. Example 2: b r

b

g

o

g

o

t =1

g

r g

r

b

b t =0

o

b

g

t =2

g

g g

r t =3

t =4

p

FIGURE 10.5 Pandemic G2 = S3,1 ; Nt (v) = 13 , α = 13 ; β = 12 ; φ = 14 ; r = red, b = blue, g = green, o = orange.

At t = 0 with φ = 14 , the infected vertex colored red recovers and changes to green by T 3. The center vertex adjacent to the red vertex at t = 0 changes to orange at t = 1 by T 1 since p Nt (v) = α = 13 , while the remaining vertices remain blue by T 1. Subsequent discrete time steps t follow the same procedure depending on the values of α, β, and φ.

10.5 Some results n−2 Here we denote the term 2α(n−1) by R and prove that it is critical in determining the dynamics of the COVID-19 spread. We state that this critical term bears a resemblance to the reproduction number defined in epidemic models whose dynamics are driven by ordinary differential equations [14]. We assume that the graph representing a pandemic is connected since the virus is spread across the population through close contacts and interactions. This assumption is reasonable since in any subpopulation it is natural to expect that not all individuals are in contact with initially infected or exposed individuals. p

Proposition 10.1. Let n ≥ 2 be the order of G. If there exists a vertex v ∈ V such that Nt (v) ≥ 1 n−2 deg(v) , then R = 2α(n−1) is a critical term of G. p

1 Proof. If there exists a vertex v in G with Nt (v) ≥ deg(v) , then v is adjacent to at least a vertex, say u, where ct (u)= red or orange. Since the degree of any vertex in G is 1 ≤ deg(v) ≤ n − 1 for all v ∈ G, it follows that p

Nt (v) ≥

1 1 ≥ , deg(v) n − 1

1 where n−1 is the proportion of infected/exposed among the infectious class at t = 0. Hence 1 . the proportion of noninfectious or nonexposed individuals at t = 0 is 1 − n−1

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Now, by rule T 1, the exposure rate α is a threshold for determining if a vertex colored blue changes color to orange or red. Since an individual who eventually becomes infected must transition two steps: blue to orange and orange to red, it follows that the infectious and exposed are equally likely to infect a susceptible individual. Thus 1 1 n−2 α = (1 − ) =⇒ = 1. 2 n−1 2α(n − 1) n−2 Let R = 2α(n−1) , then R = 1, R < 1, and R > 1 are conditions for determining if a pandemic dies out or grows.

We now examine the implications of R < 1 and R > 1.

10.5.1 Effects of R < 1 on a network To prove the subsequent theorem, we make use of the following notations: Let I0 be a subset of V consisting of vertices of G colored red or orange at the initial stage t = 0. Let ex = e(I0 ) be the maximum value of d(x, I0 ), x ∈ V . For each integer i = 0, 1, 2, . . . , e, let Ni denote the set Ni = {x ∈ V : d(x, I0 ) = i} so that for t > e, Nt = ∅. For each i = 0, 1, 2, . . . , e − 1, let Fi ⊆ Ni be the set of all vertices with at least one neighbor in Ni+1 with the assumption that e > 1. We show that if R < 1, then there is a subpopulation that has a reasonably large number of infection-free individuals. Theorem 10.6. Let G = (V , E) be a graph with vertices n ≥ 2. If R < 1, then there exists a set V0 ⊂ V with |V0 | ≥ ex such that ct (V0 ) = blue for all t.   1 . Let V0 : F1 ∪ (∪ex Proof. If R < 1, then α > 12 1 − n−1 i=2 Ni ). By the definition of Ni , it is easy to see that ∪ex i=0 Ni = V (G) such that for i = j , Ni ∩ Nj = ∅. Using this definition and the fact that ex > 1, F1 = ∅; hence V0 = ∅. Thus, since |Ni | ≥ 1 for all i = 0, 1, . . . , ex, we have |V0 | = |F1 | +

ex 

|Ni | ≥ ex.

i=2

This proves the first part of the theorem. We now show that if v ∈ V0 , then ct (v) = blue for all t,

(10.5)

which follows from the first part of the theorem. To prove (10.5), it suffices to show that no vertex v ∈ V0 is such that ct (v) ∈ {orange, red, green} for some time t. Suppose to the contrary that v is a vertex in V0 such that ct (v) ∈ {orange, red, green}. Let v be chosen so that t is the smallest nonnegative integer with this property. Then, for all j < t − 1, cj (u) = blue for all u ∈ V0 .

(10.6)

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Since v ∈ V0 by (10.6), ct−2 (v) = blue. By applying T 1 and T 2, we have ct (v) = {orange, blue, red} but not green. Assume that ct (v) = red. Since ct−2 (v) = blue, together with an application of T 1 and T 2 implies that Npt−2 (v) ≥ α and β ≥ 12 , respectively. It follows that Npt−2 =

nt−2 (v, red) + nt−2 (v, orange) nt−2 (v, red) + nt−2 (v, orange) + nt−2 (v, green) + nt−2 (v, blue)

≥α 1  1 . > 1− 2 n−1 Denote nt−2 (v, red), nt−2 (v, blue), nt−2 (v, green), and nt−2 (v, orange) by r, s, p, q, respectively. Then we obtain 1 1  r +q > 1− , r +s +p+q 2 n−1 implying that n(s + p) < n(r + q) + 2(s + p). This, in conjunction with r + s + p + q ≤ n − 1, yields s + p < n1 − 2 < 1. Since s and p are whole numbers, s = 0 and p = 0, i.e., at time t − 2 all the neighbors of v are either red or orange. It follows that v has a neighbor, say v0 , in V0 . Hence ct−2 (v) = red, orange; a contradiction to (10.6). This completes the proof of the theorem.

10.5.2 Effects of R > 1 on a network We have seen that for any network the condition R < 1 guarantees the existence of a pandemic-free subpopulation. In this subsection we will need the following family of graphs first constructed in [15]. Definition 10.2. Let n, m be integers satisfying n = mm−1−1 for some integer k. Let Wi , i = {1, m, m2 , m3 , . . . , md } be sets with |Wi | = i. We define a graph Gn,m as the graph with vertex  set Wi in which for j = 0, 1, . . . , k − 1, each vertex in Wmj is joined to m unique vertices in Wmj +1 (see Fig. 10.6). k+1

Theorem 10.7. Let the number of vertices in G be n ≥ 2. For fixed β and φ, if R > 1, then there exists an integer m and graphs Gn,m of order n such that for all t ≥ ln(n(m+1)+1) − 1 and ln m for all v in Gn,m , we have ct (v) = orange, red, or green. Proof. Assume that R > 1. Then α
1 guarantees a pandemic irrespective of the values of β and φ. This abstract model, as is the case with several mathematical models, has its own limitations. The noninclusion of some epidemiological parameters (such as birth rate, death rate, duration of infection, etc.) presents some shortcomings and is currently the object of study in the extension of this work. We note that a numerical simulation that is in agreement with the theoretical result obtained in this book chapter in comparison with other compartmental models on various countries is also currently a subject of study for a new

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research article. An interesting open problem is the case when R = 1 since it requires some complicated analysis based on the network structure.

References [1] I. Ahmed, G.U. Modu, A. Yusuf, P. Kuman, I. Yusuf, A mathematical model of coronavirus disease containing asymptomatic and symptomatic cases, Results Phys. (2021) 103776. [2] H.R. Bhapkar, P.N. Mahalle, P.S. Dhotre, Virus graph and Covid-19 pandemic: a graph theory approach, in: A.E. Hassanien, N. Dey, S. Elghamrawy (Eds.), Big Data Analytics and Artificial Intelligence Against COVID-19: Innovation Vision and Approach. Studies in Big Data, vol. 78, Springer, Cham, 2020. [3] F. Buckley, F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, California, 1990. [4] G. Chartrand, L. Lesniak, P. Zhang, Graphs and Digraphs, 7th edition, CRC Press, Taylor & Francis Group, Chapman & Hall, Book, USA, 2016. [5] Food and Drug Administration on Coronavirus, available at https://www.fda.gov/emergencypreparedness-and-response/coronavirus-disease-2019-covid-19/covid-19-vaccines. (Accessed 23 December 2020). [6] F.T. Fernandes, T.A. de Oliveira, C.E. Teixeira, et al., A multipurpose machine learning approach to predict Covid-19 negative prognosis in São Paulo, Braz. Sci. Rep. 11 (2021) 3343, https://doi.org/10. 1038/s41598-021-82885-y. [7] A. Hamed, A. Sobhy, H. Nassah, Accurate classification of Covid-19 based on incomplete heterogenous data using a KNN variant algorithm, Res. Square (2020). [8] D. Isaacs, D. Flowers, J.R. Clarks, H.B. Valman, M.R. MacNaughton, Epidemiology of coronavirus respiratory infections, Arch. Dis. Child. 58 (7) (1983) 500–503. [9] M.J. Keeling, K.T.D. Eames, Networks and epidemic models, J. R. Soc. Interface 2 (2005) 295–307. [10] L. Ancel Meyers, B. Pourbohloul, M.E.J. Newman, D.M. Skowronski, R. Brunham, Network theory and Sars: predicting outbreak diversity, J. Theor. Biol. 232 (2005) 71–81. [11] L. Ancel Meyers, Contact network epidemiology: bond percolation applied to infectious disease prediction and control, Bull. Am. Math. Soc. 44 (1) (2007) 63–86. [12] S. Mukwembi, A note on the effects of replenishment of depleted cells on HIV infectious dynamics: a graph theoretical approach, Physica A 387 (2008) 1200–1204. [13] M.E.J. Newman, The spread of epidemic disease on networks, Phys. Rev. E 66 (2002) 016128. [14] F. Nyabadza, F. Chirove, W. Chukwu, M.V. Visaya, Modelling the potential impact of social distancing on the Covid-19 epidemic in South Africa, Comput. Math. Methods Med. (2020) 5379278, https:// doi.org/10.1155/2020/5379278. [15] F. Nyabadza, S. Mukwembi, B.G. Rodrigues, A tuberculosis model: the case of reasonable and unreasonable infective, Physica A 388 (2009) 1995–2000. [16] F. Nyabadza, S. Mukwembi, B.G. Rodrigues, A graph theoretical perspective of a drug abuse epidemic model, Physica A 390 (2011) 1723–1732. [17] O.B. Cano, S.C. Morales, C. Bendtsen, Covid-19 modeling: the effects of social distancing, medRxiv, 2020. [18] G. Pinter, I. Felde, A. Mosavi, P. Ghamisi, R. Gloaguen, Covid-19 pandemic prediction for Hungary; a hybrid machine learning approach, Mathematics 8 (6) (2020) 890, https://doi.org/10.3390/ math8060890. [19] C. Seibold, H.L. Callender, Modeling epidemics on a regular tree graph, Lett. Biomath. 3 (1) (2016) 59–74, https://doi.org/10.1080/23737867.2016.1185979. [20] M. Serhani, H. Labbardi, Mathematical modeling of Covid-19 spreading with asymptomatic infected and interacting peoples, J. Appl. Math. Comput. (2020). [21] P. Theerthagiri, I.J. Jacob, A.U. Ruby, Y. Vamsidhar, Prediction of Covid-19 possibilities using KNN classification algorithm, research square, Preprint, available at https://doi.org/10.21203/rs.3.rs-70985/ v2. [22] M. Watson, Modelling the Spread of Diseases Through a Population, Honors Thesis, University of Texas at Austin, Texas, 2009.

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[23] Health World, Organization on coronavirus, available at https://www.who.int/emergencies/ diseases/novel-coronavirus-2019. (Accessed 23 December 2020). [24] A. Zeb, E. Alzahrani, V.S. Erturk, G. Zaman, Mathematical model for coronavirus disease 2019 (COVID 19) containing isolation class, BioMed Res. Int. 2020 (2020) 3452402, https://doi.org/10.1155/2020/ 3452402. [25] Z. Zhang, A. Zeb, O.F. Egbelowo, V.S. Erturk, Dynamics of a fractional order mathematical model for Covid-19 epidemic, Adv. Differ. Equ. 2020 (1) (2020) 1–16. [26] Y. Zoabi, S. Deri-Rozov, N. Shomron, Machine learning-based prediction of Covid-19 diagnosis based on symptoms, npj Digit. Med. 4 (2021) 3, https://doi.org/10.1038/s41746-020-00372-6.

11 Towards nonmanifest chaos and order in biological structures by means of the multifractal paradigm Maricel Agopa and Alina Gavrilu¸tb a Department of Physics, Gheorghe Asachi Technical University from Ia¸si, Ia¸si, Romania b Faculty of

Mathematics, Alexandru Ioan Cuza University from Ia¸si, Ia¸si, Romania

11.1 Introduction Complex systems are large interdisciplinary research topics that have been studied by means of a combination of basic theory derived especially from physics and computer simulations. Such kinds of systems are composed of many interacting entities that are called “agents” (structural units). Examples of complex systems can be found in human societies, the brain, the internet, ecosystems, biological evolution, stock markets, economies, and many others (Mitchell [1], Badii [2], Mandelbrot [3]). On the same topic, probably one of the most intriguing complex systems in nature is the DNA, which creates cells by means of a simple but very elegant language. It is responsible for the remarkable way in which individual cells organize into complex systems such as organs, and these organs form even more complex systems such as the organism (Winfree [4]). The way in which such a system manifests cannot be predicted only by the behavior of individual elements or by adding their behavior; rather, it is determined by the manner in which the elements relate to influence the global behavior. Among the most significant properties of complex systems are emergence, self-organization, adaptability, etc. (Mitchell [1], Badii [2], Mandelbrot [3]). Usually models used to describe complex system dynamics are based on the uncertain hypothesis that the variables describing it are differentiable (Luis [5], Deville and Gatski [6], Nottale [7]). The success of these models must be understood gradually on domains in which differentiability is still valid. However, the differential procedures are not suitable when describing processes related to complex system dynamics, which imply nonlinearity and chaos (it is reminded that this is the de facto case) (Nottale [7], Merche¸s and Agop [8], Agop and P˘aun [9]). Since the nondifferentiability appears as a universal property of complex systems, it is necessary to construct a nondifferentiable physic. In such a conjecture, by considering that the complexity of interaction processes is replaced by nondifferentiability, it is no longer Advances in Epidemiological Modeling and Control of Viruses. https://doi.org/10.1016/B978-0-32-399557-3.00016-8 Copyright © 2023 Elsevier Inc. All rights reserved.

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necessary to use the entire classical “arsenal” of quantities from the standard physics (differentiable physics). Therefore, to describe complex system dynamics by remaining faithful to the differentiable mathematical procedures, it is necessary to employ a multifractal paradigm, which explicitly introduces scale resolutions, both in the expression of variables and in the fundamental equations which govern complex system dynamics. This means that instead of “working” with a single variable described by a strict nondifferentiable function, it is possible to “work” only with approximations of this mathematical function obtained by averaging them on different scale resolutions. As a consequence, any variable purposed to describe complex system dynamics will perform as the limit of a family of mathematical functions, this being nondifferentiable for null scale resolutions and differentiable otherwise (Nottale [7], Merche¸s and Agop [8], Agop and P˘aun [9]). In the present chapter, considering the multifractal paradigm as being functional (in the form of multifractal theory of motion), a nondifferentiable model describing the complex system dynamics in the form of biological structures is proposed.

11.2 Mathematical model 11.2.1 Short reminder on the multifractal theory of motion The fundamental hypothesis postulates that the dynamics of structural units of any biological structure are described by continuous but nondifferentiable curves (multifractal curves). Indeed, such an assumption is sustained by the following example related to the collision processes in a complex fluid: between two successive collisions, the trajectory of the complex fluid structural unit is a straight line that becomes nondifferentiable at the impact point. Considering that all the collision impact points form an uncountable set of points, it results that the trajectories of the complex fluid structural units become continuous and nondifferentiable curves, i.e., fractal curves. Obviously, the reality is much more complicated, taking into account both the diversity of the particles that compose a complex fluid and the various interactions between them in the form of double/triple collisions, etc. Then the complex fluid becomes multifractal. Extrapolating the previous reasoning for any biological structure, it results that it can be assimilated to a multifractal. In such a context, the dynamics of the biological structure’s structural units become operational in the multifractal paradigm through the multifractal theory of motion. The fundamental hypothesis of the multifractal theory of motion is that the dynamics of the biological structure are described through multifractal curves. This leads to the following consequences (Nottale [7], Merche¸s and Agop [8], Agop and P˘aun [9]): (I) Any multifractal curve is explicitly scale δt dependent. More precisely, its length tends to infinity when δt tends to zero (Lebesgue theorem, Merche¸s and Agop [8], Agop and P˘aun [9]). Moreover, the space becomes a multifractal in the Mandelbrot sense;

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(II) The dynamics of the biological structure are related to the behavior of a set of functions during the zoom operation of δt. Then δt ≡ dt through the functionality of the substitution principle; (III) The dynamics of the biological structure’s structural units are described through multifractal variables. Then two derivatives of the variable field Q (t, dt) can be defined as follows: dQ+ Q (t, t + t) − Q(t, t) = limt→0 , dt t dQ− Q (t, t) − Q (t − t, t) = limt→0 . dt t

(11.1)

The sign “+” corresponds to the forward processes, while the sign “−” corresponds to the backward ones; (IV) The differential of the spatial coordinate field has the form d± X i (t, dt) = d± x i (t) + d± ξ (t, dt) .

(11.2)

The differentiable part d± x i (t) is scale resolution independent, while the nondifferentiable part d± ξ (t, dt) is scale resolution dependent; (V) The nondifferentiable part of the spatial coordinate field satisfies the nondifferentiable equation d± ξ

i

  2 i f (α) −1 , (t, dt) = λ± (dt)

(11.3)

where λi± are constant coefficients associated with the differentiable-nondifferentiable transition, f (α) is the singularity spectrum of order α of fractal dimension, and α is the singularity index. There are many modes, and thus a varied selection of definitions of fractal dimensions: more precisely, the fractal dimension in the sense of Kolmogorov, the fractal dimension in the sense of Hausdorf-Besikovici, etc. (Merche¸s and Agop [8], Agop and P˘aun [9], Cristescu [10], Jackson [11]). Selecting one of these definitions and operating in the biological structure dynamics, the value of the fractal dimension must be constant and arbitrary for the entirety of the dynamic analysis. For example, it is regularly found that DF < 2 for correlative processes, DF > 2 for noncorrelative processes, etc. In such a conjecture, through (11.3) it is possible to identify not only the “areas” of the biological structure dynamics that are characterized by a certain fractal dimension, but also the number of “areas” whose fractal dimensions are situated in an interval of values. More than that, through the singularity spectrum f (α) it is possible to identify classes of universality in the biological structure dynamics laws, even when regular or strange attractors have different aspects (Nottale [7], Merche¸s and Agop [8], Agop and P˘aun [9], Mazilu, Agop, and Merche¸s [12]);

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(VI) The differential time reflection invariance of any variable is recovered by means of the operator:     1 d+ + d− i d+ − d− dˆ = − . (11.4) dt 2 dt 2 dt This is a natural result of Cresson’s theorem (Adda and Cresson [13]). Applying the operator (11.4) to X i , the complex velocity field yields ˆ i dX Vˆ i = = VDi − VFi dt

(11.5)

with VDi =

1 d+ X i + d− X i 1 d+ X i − d− X i , VFi = , i = 1, 2, 3. 2 dt 2 dt

(11.6)

The real part of VDi is scale resolution independent, while the imaginary one VFi is scale resolution dependent; (VII) Since multifractalization implies stochasticization (Nottale [7], Merche¸s and Agop [8], Agop and P˘aun [9], Mazilu, Agop, and Merche¸s [12]), the whole statistic “arsenal” in the form of averages, variances, covariances, etc., becomes operational. Thus, for the average of d± X i , let the following functionality be chosen:   d± X i ≡ d± x i (11.7) with

  d± ξ i = 0.

(11.8)

The previous relation (11.8) implies that the average of the nondifferential part of the spatial coordinate field is null. (VIII) The complex fluid dynamics can be described through the scale covariant derivative given by operator (11.9) and (11.10): 

dˆ 1 = ∂t + Vˆ i ∂i + (dt) dt 4

2 f (α)

 −1

D lk ∂l ∂k ,

(11.9)

where lk

lk

D lk = d lk − id , d lk = λl+ λk+ − λl− λk− , d = λl+ λk+ + λl− λk− .

(11.10)

For Markov-type stochastic processes, λi+ λl+ = λi− λl− = 2λδ ie ,

(11.11)

f (α) ≡ DF ,

(11.12)

and for

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where λ is a specific coefficient associated with the fractal-nonfractal transition, the scale covariant derivation becomes 

dˆ = ∂t + Vˆ l ∂l − iλ(dt) dt

2 DF



−1

∂l ∂ l .

(11.13)

In the particular case of motions on Peano-type curves, which implies DF = 2, the scale covariant derivative (11.13) takes the standard form from the scale relativity theory dˆ (11.14) = ∂t + Vˆ l ∂l − iD∂l ∂ l , dt where λ ≡ D is the diffusion coefficient associated with the fractal-nonfractal transition. Therefore this model generalizes all the results of Nottale’s theory (i.e., scale relativity theory) (Nottale [7]). Now, accepting the functionality of the scale covariance principle, i.e., applying the operator (11.9) to the complex velocity fields (11.5), in the absence of any external constraint, the motion equation (i.e., the geodesic equation on a multifractal space) takes the following form: 

dˆ Vˆ i 1 = ∂t Vˆ i + Vˆ l ∂l Vˆ i + (dt) dt 4

2 f (α)



−1

D lk ∂l ∂k Vˆ i = 0.

(11.15)

This means that the multifractal acceleration ∂t Vˆ i , the multifractal convection Vˆ l ∂l Vˆ i , and the multifractal dissipation D lk ∂l ∂k Vˆ i make their balance in any point of the multifractal curve. Particularly, for (11.11) and (11.12), the motion Eq. (11.15) becomes   2 dˆ Vˆ i i l ˆi DF −1 ˆ ˆ ∂l ∂ l Vˆ i = 0. (11.16) = ∂t V + V ∂l V − iλ(dt) dt

11.2.2 Stationary nonlinear behaviors through Schrödinger-type “regimes” as “synchronization modes” For irrotational motions of the biological structure’s structural units, the complex velocity (11.5) takes the form 

Vˆ i = −2iλ(dt)

2 f (α)

 −1 i

∂ lnΨ ,

(11.17)

where lnΨ is the scalar potential of velocity fields and Ψ is the multifractal state function. Then, substituting (11.17) in (11.16) and using the method from (Agop and Merche¸s [14], Agop and P˘aun [9]), the motion equations become  2

λ (dt)

4 f (α)

 −2 l



∂ ∂l + iλ(dt)

2 f (α)



−1

∂t = 0,

(11.18)

i.e., the geodesic equation in the form of Schrödinger multifractal type (i.e., at various scale resolutions).

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In the one-dimensional stationary case, (11.18) takes the form d 2 + k02 = 0 dx 2

(11.19)

with E

k02 = 2m0



λ2 (dt)

4 f (α)

 . −2

(11.20)

In (11.20) E is the multifractal energy of the biological structure’s structural unit and m0 is the rest mass of the biological structure’s structural unit. The solution of (11.19) can be written in the form (x) = hei(k0 x+θ) + he−i(k0 x+θ) ,

(11.21)

where h is the complex amplitude, h is the complex conjugate of h, and θ is a phase. Thus h, h, and θ label each structural unit from an eventual biological structure that has, as a “fundamental property”, the same k0 . Eq. (11.19) has a “hidden” symmetry by means of a homographic group of multifractal type. Indeed, the ratio ε of two independent linear solutions of (11.19) is a solution of Schwartz’s differential equation of multifractal type (for the classical case, see Mazilu and Agop [15]): d {ε, x} = dx ε˙ =

    ε¨ 1 ε¨ 2 − = 2k02 ε˙ 2 ε˙

dε d 2ε , ε¨ = 2 . dx dx

(11.22) (11.23)

The left part of (11.22) is invariant with respect to the homographic transformations of multifractal type ε ↔ ε =

aε + b cε + d

(11.24)

with a, b, c, and d real parameters (of multifractal type). Relation (11.24) corresponding to all possible values of these parameters defines the group SL (2R) of multifractal type (Mazilu, Agop, and Merche¸s [12], Mazilu and Agop [15]). Thus, all the biological structure’s structural units having the same k0 are in biunivocal correspondence with the transformations of the group SL (2R) of multifractal type. This allows the construction of a “personal” parameter of multifractal type ε for each biological structure’s structural unit separately. Indeed, as a “guide”, a general form of the solution of (11.22) is chosen, which is written as ε  = l + mtan (k0 x + θ ).

(11.25)

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Thus, through l, m, and θ , it is possible to characterize any structural unit of the biological structure. In such a conjecture, identifying the phase from (11.25) with the one from (11.21), the “personal” parameter of multifractal type becomes ε =

h + hε , h = l + im, h = l − im, ε ≡ e2i(k0 x+θ) . 1+h

(11.26)

The fact that (11.25) is also a solution of (11.22) implies, by making (11.24) explicit, the group of SL (2R) multifractal type (Mazilu and Agop [15], Barbilian [16,17]) h =

ah + b ah + b ch + d , k = , h= k. ch + d ch + d ch + d

(11.27)

Therefore the group (11.27) works as “synchronization modes” among the various structural units of any biological structure, process in which the amplitudes and phases of each of them obviously participate, in the sense that they are also connected. More precisely, through the group (11.27), the phase of k is only moved with a quantity depending on the amplitude of the biological structure’s structural units at the transition among various structural units of any biological structure. But not only that, the amplitude of the structural unit of any biological structure is also affected from a homographic perspective. The usual “synchronization” manifested through the delay of amplitudes and phases of the structural units of any biological structure must represent here only a totally particular case. The structure of group (11.27) is typical of SL (2R) one, which is taken in the standard form [A1 , A2 ] = A1 , [A2 , A3 ] = A3 , [A3 , A1 ] = −2A2 ,

(11.28)

where Ak , k = 1, 2, 3, are the infinitesimal generators of the group. Because the group is simple transitive, these generators can be easily found as the components of the Cartan coframe of multifractal type (for the classical case, see Cartan [18]) from the relation (see Mazilu and Agop [15], Agop and Merche¸s [14]) 

 ∂  ∂f ∂ + dx k = ω1 h2 ∂h + h¯ 2 ∂∂h¯ + h − h¯ k ∂k d (f ) = ∂x k



 (11.29) ∂ ∂ + h¯ ∂∂h¯ + ω3 ∂h + ∂∂h¯ (f ) , 2ω2 h ∂h where ωk are the components of the Cartan coframe of multifractal type to be found from the system

 2 dh = ω1 h2 + 2ω2 h + ω3 , dh = ω1 h + 2ω2 h + ω3 , dk = ω1 k h − h .

(11.30)

Thus both the infinitesimal generators and the coframe of multifractal types are immediately obtained by identifying the right-hand side of (11.29) with the standard dot product of SL (2R) algebra of multifractal type ω1 A3 + ω3 A1 − 2ω2 A2 ,

(11.31)

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so that A1 =

 ∂ ∂ ∂ ∂ ∂ ∂ ∂ + + h¯ , A3 = h2 + h¯ 2 , A2 = h + h − h¯ k ∂h ∂ h¯ ∂h ∂h ∂k ∂ h¯ ∂ h¯

(11.32)

and dk dh − dh h + h dk hdh − hdh hhdk , ω3 = − +

ω1 =

 , 2ω2 =  . h−h h−h k h−h h−h k h−h k

(11.33)

In real terms from (11.26), these last equations can be written as A1 =

∂ ∂ ∂ ∂ ∂ ∂ , A2 = l + m , A3 = l 2 − m2 + 2lm + 2m ∂l ∂l ∂m ∂l ∂m ∂θ

(11.34)

l mdl − ldm dθ dm l 2 + m2 , ω2 = − dθ, ω2 = dθ + . (11.35) 2m m m 2m m It should be mentioned that, in Barbilian [16,17], Mazilu and Agop [15], it does not work with the previous differential forms, but with the absolute invariant differentials   dk dh + dh dh kdh 1 2 ω =

, ω3 = (11.36) −  , ω = −i k h−h k h−h h−h ω1 =

or, in real terms, exhibiting a three-dimensional Lorentz structure of this space

1 = ω1 = dθ +

dm dm dl dl dl , 2 = cos θ + sin θ , 3 = − sin θ + cos θ . m m m m m

(11.37)

The advantage of this representation is that it makes obvious the connection with the Poincaré representation of the Lobachevsky plane. Indeed, the metric here is ds 2 2 2 = ω − 4ω1 ω2 = g



dk dh + dh − k h−h

2

dhdh + 4

2 h−h

(11.38)

or in real terms   2 2 2 ds 2 dl 2 dl 2 + dm2 1 2 3 − =− + + = − dθ + + , g m m2

(11.39)

where g is a constant. These metrics reduce to those of Poincaré in case when ω2 = 0 or 1 = 0, which defines the variable θ as the “angle of parallelism” (in Levi-Civita sense) of the hyperbolic plane of multifractal type (the connection of multifractal type). In fact, recalling that in modern terms dl m represents the connection form of the hyperbolic plane of multifractal type, relations (11.37) then represent a general Bäcklung transformation of multifractal type in that plane.

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11.2.3 Nonstationary nonlinear behaviors through Schrödinger-type “regimes” as “synchronization modes” Now, new data regarding the behavior of biological system dynamics, complementary to the class of solutions associated with Eq. (11.18) (generated through initial and boundary conditions), can also be given on the base of transformation groups (which leave invariant Eq. (11.18)) (Mazilu, Agop, and Merche¸s [12]). These transformation groups constitute, in the most general case of the one-dimensional drug release dynamics, a realization of the Lie group SL (2R) through the action (Mazilu, Agop, and Merche¸s [12]) t →

αt + β X , X → , γt +δ γt +δ

(11.40)

where α, β, γ , and δ are real elements. Let us consider that, in accordance with general mathematical procedures from (Mazilu, Agop, and Merche¸s [12]), the biological system dynamics may be generally described with the help of a 2 × 2 matrix with real elements. In a biological system, it is obvious that the problem revolves around a family of such matrices, each of them describing the dynamics of a biological system entity (structural unit). The interactions between the biological system entities can then be expressed through relations between the representative matrices. These relations must contain certain parameters that characterize the structure of the biological system, adequate to the description of the biological structure dynamics. Therefore the matrix which generates nonharmonic curve (Mazilu, Agop, and Merche¸s [12]) is a 2 × 2 matrix with real elements, written in the form   = α β . M (11.41) γ δ The elements of this matrix contain, in an unspecified form, both the physical parameters of the complex biological and the possible initial conditions of the biological system dynamics. More precisely, the elements of matrix (11.4) depend on the scale resolution in the sense of the multifractal theory of motion (the motion curves are continuous and nondifferentiable, i.e., multifractal curves) (Merche¸s and Agop [8], Agop and P˘aun [9]). In such a conjecture, the results to be obtained will also be in a sense of the previously-mentioned theory. A set of such matrices, with variable elements, may be admitted as relevant for the biological system dynamics; for example, by means of a fundamental spinor set, given by 2 X 2 matrices which describe the release dynamics. This description is analogous to the spinor description of space-time (Penrose [19]). In such a situation, any 2×2 matrix of form (11.4) can be written as a linear combination with real coefficients, which implies two special matrices, namely, the unity matrix Uˆ and a null-trace matrix I(from involution), meaning ˆ M=λ Uˆ +μIˆ.

(11.42)

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The involution Iˆ has some important properties such as its squared form is a multiple of ˆ Uˆ and the fixed points of its homographic action are the ones of matrix M. In (11.42) there exists the liberty to choose a parameterization in which the squared form of Uˆ can be the unity matrix, up to a sign. In this case the elements of Iˆ may be expressed with the help of only two parameters that represent the asymptotic directions of ˆ If the asymptotic directions are complex, being of the form u ± iv, the represenmatrix M. tation of the matrix Iˆ through asymptotic directions is of a spherical type. Then, satisfying the above-mentioned properties a) and b) implies, for the matrix Iˆ, the form   1 −u −u2 − v 2 Iˆ = , Iˆ = −Uˆ . (11.43) 1 u v Such a representation of the biological system dynamics has an important advantage. When analyzing the physics of the problem, the model allows an explicit differential description of the biological system dynamics through matrix geometry, identic to the metric geometry of space at a certain moment, the hyperbolic geometry of second type (in a Barbilian sense (Barbilian [16,17])). The representation of biological system dynamics through 2 × 2 matrices leads to a natural matrix of the matrices’ space, for example the Killing-Cartan metric of SL(2R)-type algebra of these matrices (Mazilu and Agop [15]). The basic covectors of such geometry are, in the general case of matrix (11.41), given by the external differential forms ω1 =

αdβ−βdα , 

−γ dα ω2 = αdγ  , ω3 = 2  = αγ − β .

βdγ −γ dβ 

(11.44)

In the parameterization given through (11.42) and (11.43), (11.44) becomes 1 du dv ω1 = d + sin2  2 − sin  cos  2 v v v u udu+vdv vdu − udv ω2 = 2 d + 2 sin2  + 2 sin  cos  2 v v v2 2 + v2 2 − v 2 )du + 2uvdv u (u d + sin2  ω3 = v v2 2uvdu − (u2 − v 2 )dv + sin  cos  , v2

(11.45)

where μ tan  = . λ Related to these covectors, the metric is given by the squared form  ds 2 =ω1 ω3 −

ω2 2

2 =dΦ 2 −sin2 

du2 +dv 2 . v2

(11.46)

(11.47)

As such, for as long as the biological system is represented defined by the core property that physics admits as being essential (which is the biological system dynamic), its description

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mode is a metric geometry. In this case, the metric is given through (11.47), where  is an arbitrary “phase”, and u and v are “coordinates” obtained from the (local) dynamic of the biological system in the way previously described. Such a metric approach for the biological dynamics can be certainly delegated to harmonic maps, from the biological system to the space. As soon as the mapping mode of the biological system on the space available at its disposal is solved, the quantities , u, and v (and the elements of the matrix family which represent the biological system) are obtained. In principle, a “position” function will be sufficient to correctly define a specific quantity of the biological system. The difficulty of representing the biological system in this form is overcome though the 

harmonic map X = (X1 , X2 , X3 ) → ξ = ξ 1 , ξ 2 , ξ 3 , which can provide a set of quantities as functions of spatial coordinates. Let the functional corresponding to the harmonic mapping principle be considered (for details, see (Misner et al. [20], Xi [21])):   1 ∂ξ μ ∂ξ ν J= gμν (ξ ) , (11.48) d 3 X |h|hil (X) 2 ∂X i ∂X l where h is the space metric and g is the associated metric of the biological system. Canceling the first degree variation of this functional, in relation to the spatial coordinates, gives the sought harmonic map. Taking into account the fact that the space is Euclidean and using (11.46) for the metric tensor associated with the polymer-drug complex system, for the integrand of (11.48), the expression will be     sin  2  ∂ξ μ ∂ξ ν il l l l |h|h (X) ∂ g ∂ − u∂ u + ∂ ν∂ ν , ≡∂ (ξ ) μν l l l ∂X i ∂X l v

(11.49)

where the usual notation ∂l denotes the gradient. The Euler equations corresponding to functional (11.49) are as follows: l

l

l v∂ v ∂l ∂ l +sin cos  ∂l u∂ u+∂ =0 2 v l l l v∂ v = 0. ∂i ∂vl2u = 0, ∂i ∂vl2v + ∂l u∂ u+∂ v2

(11.50)

The last two equalities of (11.50) represent a harmonic map from the Euclidean space to the hyperbolic plane (Lobacevsky plane) in the Beltrami-Poincaré representation. As a consequence of these equalities, there will be   ∂l u∂ l u+∂l v∂ l v ∂i = 0. (11.51) v2 This means that the scalar quantity under the gradient ∂i is constant in space, but at the same time positive, being a sum of two squares of a real quantity. Let this quantity be denoted as m2 , where m is real. The first equality in (11.50) becomes the three-dimensional sine-Gordon-type equation (in a multifractal sense) ∂l ∂ l +m2 sin  cos  = 0.

(11.52)

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Its extension on a space-time manifold (in a multifractal sense) through a generalized mapping principle of (11.48) type implies the functionality of a four-dimensional sineGordon-type equation (in a multifractal sense) ∇+m2 sin  cos  = 0,

(11.53)

where ∇ is the d’Alembert-type operator (in a multifractal sense). For the standard sineGordon equation, see (Jackson [11]). A solution of Eq. (11.50) can be given relatively simply, if it is assumed that the quantities depend on the “localization” on the space-time manifold by means of the linear form (the ansatz in a multifractal sense, see in the usual case (Skyrme [22]) ξ =a1 X1 +a2 X2 +a3 X3 +a4 X4 =V0 t,

(11.54)

where X1 , X2 , X3 are the spatial coordinates and X4 is the time. As such, (11.53) becomes d 2  m2 + sin  cos  = 0, dξ 2 a 2

(11.55)

a 2 =a12 +a 22 +a 23 +a 24 .

(11.56)

where

d dξ ,

after which it can be integrated and leads to the Further on, (11.55) is multiplied by expression   d 2 m2 2 + 2 sin  =b2 , dξ a where b is an integration constant which is assumed to be real. As such, ξ is an elliptic integral of first kind (see (Anghelu¸ta˘ [23])  arcsin Z dc 

b (ξ −ξ0 ) = ± (11.57) 

 0 1−z2 1−s 2 z2 of modulus m2 , 0≤s≤1, (11.58) b2 a 2 where z = sin , ξ0 is an integration constant assumed to be real, while  becomes the sn Jacobi elliptic function of the same modulus (11.58) s2=

=sn [b (ξ −ξ0 ) ;s] .

(11.59)

The elliptic function (11.59) degenerates both in the limit s → 0, situation in which it implies the periodic mode (in a multifractal sense) →sin [b (ξ −ξ0 ) ; s → 0],

(11.60)

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as well as in the limit s → 1, situation in which it implies the kink mode (in a multifractal sense)  → tanh [b (ξ − ξ0 ) ; s → 1] .

(11.61)

In such a context, admitting that  is a measure of the nonlinear behavior of biological system dynamics, though (11.59), it results that the process dynamics in any biological system is dictated by synchronization modes (i.e., those described through the Jacobi elliptical function). The complexity of dynamic processes can be expressed through nonlinear behaviors of the biological system, the explicit presentation of these nonlinearities being dictated through the modulus s of elliptical functions sn. Therefore the linear behaviors of the dynamic process (induced by weak-type interactions between the composing components of the biological system) can be expressed through periodic-type release modes (11.60), while the nonlinear behaviors induced by strong interactions between the components of the biological system can be expressed through kink-type release modes (11.61). Without giving up of the ansatz in a multifractal sense, the sine-Gordon-type equation (in a multifractal sense), both through separation and intervention of some normalized coordinates (in the sense η = a1 X1 , τ = a4 X4 ), all of these based on specific operational procedures (Mazilu, Agop, and Merche¸s [12]), also implies the following special solutions (for details in the usual case, see (Jackson [11]): (I) soliton (+) and antisoliton (−) modes (in a multifractal sense):    τ − αη , 0 ≤ α < 1; 1 (η, τ, α) = 4arctan exp ± √ 1 − α2 (II) breather modes-localized oscillating modes (in a multifractal sense):   √   sin 1−qτ q 2 (η, τ, q) = 4arctan , 0 < q 0. dt This together with S(0) = S0 ≥ shows that S(t) ≥ 0, t ∈ [0, T ). Note that (A1 ) implies that co[h(I )]={0} and h(I ) is continuous at 0. By the continuity of ϕ at I = 0, there exists a positive constant δ such that when |I | < δ, ϕ(I ) is continuous, and the differential inclusion in (12.3) becomes the following differential equation with continuous right-hand side: dI = λSI − (d + γ + )I − ϕ(I )I = I [λS − (d + γ − ) − ϕ(I )]. dt

(12.5)

Now, if I0 = 0, it follows from (12.5) that I (t) = 0 for all t ∈ [0, T ). If I0 > 0, we claim that I (t) > 0 for all t ∈ [0, T ). Otherwise, let t1 = inf {t : I (t) = 0}. Then t1 > 0 and I (t1 ) = 0. It follows from the continuity of I (t) on [0, T ) that there exists a positive constant θ such that t1 − θ and 0 < I (t) < δ for t ∈ [t1 − θ, t1 ). Then integrating (12.5) from t1 − θ > 0 to t1 leads to 0 = I (t1 ) = I (t1 − θ )e

 t1

t1 −θ [λS(ξ )−(d+γ −ε)−φ(ξ )]dξ

> 0,

which is a contradiction. Therefore I (t) > 0 for all t ∈ [0, T ). The next result addresses the global existence and boundedness of solutions to model (12.2). Proposition 12.2. Suppose that (A1 ) is satisfied. Then, for any S0 ≥ 0 and I0 ≥ 0, there is at least one solution (S(t), I (t)) to model (12.2) satisfying S(0) = S0 and I (0) = I0 . Furthermore, any such solution is bounded and exists for all t ∈ [0, +∞). Proof. Note that the map (S, I ) → (A − dS − βSI, βSI − (d + γ + )I − co[h(I )]) is an upper semicontinuous set-valued map with nonempty compact convex values. By the existence theorem of solution of differential inclusion ([17], p. 77, Thm. 1), there exists a solution (S(t), I (t)) of model (12.2) on [0, t0 ) for some t0 > 0 satisfying the initial condition S(0) = S0 , I (0) = I0 . By Proposition 12.1, we know that S(0) ≥ 0 and I (0) ≥ 0 for t ∈ [0, t0 ). From (12.3) we have d(S + I ) ∈ A − d(S + I ) − (γ + )I − co[h(I )]. dt Choose any v ∈ co[h(I )]; when S + I >

A d , we have that A − d(S

+ I ) − (γ + )I − v < 0.

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 Therefore 0 ≤ S + I ≤ max Ad , S0 + I0 , that is, the solution (S(t), I (t)) is bounded on [0, t0 ). Using the boundedness and by virtue of the continuation theorem ([17], p. 78, Thm. 2), we conclude that the solution (S(t), I (t)) indeed exists on the time interval [0, +∞) and is bounded. This proof is completed.

12.1.3 Equilibria and their stability By an equilibrium of model (12.2), we mean a constant solution of model (12.2), (S(t), I (t)) = (S ∗ , I ∗ ), t ∈ [0, +∞). Clearly, (S ∗ , I ∗ ) is an equilibrium of model (12.2) if and only if  0 = A − dS ∗ − λS ∗ I ∗ , (12.6) 0 ∈ λS ∗ I ∗ − (d + γ + )I ∗ − co[h(I ∗ )]. Thus, if (S ∗ , I ∗ ) is an equilibrium of model (12.2), then there exists a constant ξ ∗ ∈ co[h(I ∗ )] such that  A − dS ∗ − λS ∗ I ∗ = 0, (12.7) λS ∗ I ∗ − (d + γ + )I ∗ − ξ ∗ = 0. Obviously, such a constant ξ ∗ is unique, being given by ξ ∗ = S ∗ I ∗ − (d + γ + )I ∗ ∈ co[h(I ∗ )]. Suppose that (A1 ) holds. In order to obtain the equilibria of model (12.2), we need to solve the following inclusion:  0 = A − dS − λSI, (12.8) 0 ∈ λSI − (d + γ + )I − co[ϕ(I )]I. Obviously, the disease-free equilibrium E0 = (A/d, 0) always exists. An endemic equilibrium satisfies  0 = A − dS − λSI, (12.9) 0 ∈ λS − (d + γ + ) − co[ϕ(I )]. Solving the first equation of (12.9) for S in terms of I gives S = A/(d + λI ). Substituting this into the second equation (inclusion), we have Aλ − (d + γ + ) ∈ co[ϕ(I )] = [ϕ(I − ), ϕ(I + )]. d + λI

(12.10)

Denote g(I ) =

Aλ − (d + γ + ), d + λI

and let R0 =

Aλ , d(d + γ +  + ϕ(0))

(12.11)

which is the basic reproductive number of model (12.2). We will see in the sequel that the existence of an endemic equilibrium is closely related to the size of R0 .

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Lemma 12.1. If R0 > 1, then inclusion (12.10) has a unique positive solution I˜ satisfying Aλ − d(d + γ + ) I˜ ≤ . λ(d + γ + ) Proof. We first show the existence of a positive solution I˜ of inclusion (12.10). Note that R0 > 1 implies that g(0) > ϕ(0) ≥ 0. Also note that g(I ) is decreasing in I and ϕ(I ) is nondecreasing in I . Moreover, g(I ) ≤ 0 when I ≥ [Aλ − d(d + γ + )]/[λ(d + γ + )]. Thus the set  I : g(I ) ≥ ϕ(I + ), I > 0 is bounded. Let  I˜ = sup I : g(I ) ≥ ϕ(I + ), I > 0 . Then it is obvious that g(I˜) ≥ ϕ(I˜− ) and 0 < I˜ ≤ [Aλ − d(d + γ + )]/[λ(d + γ + )]. We claim that g(I˜) ∈ [ϕ(I˜− ), ϕ(I˜+ )]. Otherwise, g(I˜) ≥ ϕ(I˜+ ) = limI →I˜+ ϕ(I ). By (A1 ), there exists δ > 0 such that g(I˜ + δ) > ϕ(I˜ + δ) = ϕ((I˜ + δ)+ ). This contradicts the definition of I˜. Therefore g(I˜) ∈ [ϕ(I˜− ), ϕ(I˜+ )], that is, I˜ is a positive solution of inclusion (12.10). We next show that I˜ is the unique positive solution of (12.10). Set I1∗ = I˜ and assume that I2∗ = I1∗ is another positive solution of (12.10). Then there exist η1∗ ∈ co[ϕ(I1∗ )] and η2∗ ∈ co[ϕ(I2∗ )] such that  Aλ = (d + λI1∗ )(η1∗ + d + γ + ), (12.12) Aλ = (d + λI2∗ )(η2∗ + d + γ + ). From the monotonicity of ϕ (see (A1 )) it follows that H=

η1∗ − η2∗ ≥ 0. I1∗ − I2∗

Subtraction of the two equations in (12.12) results in 0 =d(η1∗ − η2∗ ) + λ(d + γ + )(I1∗ − I2∗ ) + λ(I1∗ η1∗ − I2∗ η2∗ ) =d(η1∗ − η2∗ ) + λ(d + γ + )(I1∗ − I2∗ ) + λ(I1∗ η1∗ − I1∗ η2∗ + I1∗ η2∗ − I2∗ η2∗ ) =[(d + λI1∗ )H + λ(d + γ +  + η2∗ )](I1∗ − I2∗ ). This further leads to (d + λI1∗ )H + λ(d + γ +  + η2∗ ) = 0. On the other hand, I1∗ > 0 and η2∗ ≥ 0 imply (d + λI1∗ )H + λ(d + γ +  + η2∗ ) > 0, which is a contradiction. Therefore (12.10) has the unique positive solution I˜, and the proof the lemma is completed. A direct consequence of Lemma 12.1 is the following uniqueness theorem for an endemic equilibrium. Theorem 12.1. Suppose that (A1 ) holds. If R0 > 1, then model (12.2) has a unique endemic equilibrium E ∗ = (S ∗ , I ∗ ) with I ∗ = I˜ being the unique positive solution of inclusion (12.10) as is shown in Lemma 12.1 and S ∗ = A/(d + λI ∗ ).

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By (A1 ) and Remark 12.1, we can analyze the stability of the model at the disease-free equilibrium E0 by investigating the eigenvalues of the Jacobian matrix of (12.2) at E0 . This matrix is calculated as

 −d −λ Ad 0 J (P ) = . 0 λ Ad − (d + γ + ) − φ(0) It is clear that the stability of E0 is fully determined by the sign of the term λ Ad − (d + γ + ) − ϕ(0): E0 is asymptotically stable if λ Ad − (d + γ + ) − ϕ(0) < 0; it is unstable if λ Ad − (d + γ + ) − ϕ(0) > 0. The above stability criteria can be stated in terms of R0 . Theorem 12.2. Assume that (A1 ) holds. Then the disease-free equilibrium E0 is asymptotically stable if R0 < 1, and it becomes unstable when R0 > 1. Now we turn to the stability of the unique endemic equilibrium E ∗ = (S ∗ , I ∗ ) which exists if R0 > 1. We can show that R0 > 1 is actually also a necessary condition for the existence of the endemic equilibrium E ∗ = (S ∗ , I ∗ ). Indeed, assuming that E ∗ exists, it follows from (12.9) that λA = (η∗ + d + γ + )(d + λI ∗ ), where η∗ ∈ co[ϕ(I ∗ )]. Thus, by the monotonicity of ϕ, we have d (R0 − 1), λ which implies that R0 > 1. Therefore R0 > 1 is a sufficient and necessary condition for the existence of the unique endemic equilibrium E ∗ = (S ∗ , I ∗ ). Assume that R0 > 1 and that ϕ is differentiable at I ∗ . Then the Jacobian matrix of (12.2) at the endemic equilibrium E ∗ can be calculated as 

−λS ∗ −d − λI ∗ . J∗ = λI ∗ −ϕ (I ∗ )I ∗ 0 < I∗ ≤

Note that tr(J ∗ ) = −d − λI ∗ − ϕ (I ∗ )I ∗ < 0, det (J ∗ ) = (d + λI ∗ )ϕ (I ∗ )I ∗ + λ2 S ∗ I ∗ > 0. Based on the above, we have the following theorem. Theorem 12.3. Suppose that (A1 ) holds. If R0 > 1 and ϕ is differentiable at I ∗ , then the endemic equilibrium E ∗ is asymptotically stable. The above stability results are local. Moreover, for E ∗ it is assumed that ϕ is differentiable at I ∗ . In the sequel, we show that E0 is indeed globally asymptotically stable when R0 ≤ 1; and E ∗ is globally asymptotically stable when R0 > 1, regardless of whether or not ϕ is differentiable at I ∗ . To this end, we need to apply the Lyapunov theory for discontinuous systems (see, e.g., [7,8]). We begin by introducing a LaSalle-type invariance principle. Consider a system described by the following differential inclusion: x(t) ˙ ∈ F (x),

(12.13)

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where F is an upper semicontinuous set-valued map from R n to R n with compact and convex values. We also assume 0 ∈ F (0), that is, 0 is an equilibrium of (12.13). A Lyapunov function for (12.13) is a smooth function V : R n → R satisfying the following conditions: (L1) Positive definiteness: V (x) > 0 for all x = 0; in addition, V (0) = 0. (L2) Properness: the sublevel set {x ∈ R n : V (x) ≤ a} is bounded for every a ≥ 0; (L3) Strong infinitesimal decrease: max < ∇V (x), v >≤ 0, ∀x = 0.

v∈F (x)

A set W is said to be weakly invariant for (12.13) if for any x0 ∈ W there is at least one solution x(t) satisfying x(0) = x0 such that x(t) ∈ W for all t at which x(t) exists. Let V be a Lyapunov function for (12.13). For any l > 0, by (L1) and the continuity of V , the level set {x ∈ R n : V (x) ≤ l} contains a neighborhood of 0. Denote by Vl the largest connected component of the level set {x ∈ R n : V (x) ≤ l} that contains 0. The following LaSalle-type invariance principle is from Theorem 3 in [8]. Lemma 12.2. Assume that V : R n → R is a Lyapunov function for (12.13), and let  ZV = x ∈ R n : ∃v ∈ F (x), < ∇V (x), v >= 0 . Denote by M the largest weakly invariant subset of ZV Ll . Let x0 ∈ Ll and x(t) be any solution with x(0) = x0 . Then dist (x(t), M) → 0 as t → +∞. In particular, if M = 0 and l = +∞, then 0 is globally asymptotically stable for (12.13). Now, we are in the position to state and prove the following global stability result for the disease-free equilibrium. Theorem 12.4. Suppose that assumption (A1 ) is satisfied. If R0 ≤ 1, then the disease-free equilibrium E0 is globally asymptotically stable. Proof. In order to apply Lemma 12.2, we shift the disease-free equilibrium E0 to the origin by letting x = S − Ad . Then (12.3) is transformed to the following form:  dx λA dt = −dx − λxI − d I, (12.14) dI λA dt ∈ λxI + [ d − (d + γ + )]I − co[ϕ(I )]I. Let x2 A + I, 2 d which is obviously a smooth function. It is easy to verify that (L1) and (L2) are satisfied for V1 . Denote the right-hand side of (12.14) by G(x, I ), that is, 

I −dx − λxI − λA d . G(x, I ) = λxI + [ λA d − (d + γ + )]I − co[ϕ(I )]I V1 (x, I ) =

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By (A1 ), it is easy to see that map G is an upper semicontinuous set-valued map with nonempty compact convex values. For any v = (v1 , v2 ) ∈ G(x, I ), there exists a corresponding function η(t) ∈ co[ϕ(I )] such that

v=

−dx − λxI − λA d I λA λxI + [ d − (d + γ + )]I − η(t)I

 .

From this, we can calculate ∇V1 (x, I ) · v as follows:

 A I −dx − λxI − λA d ∇V1 (x, I ) · v = (x, ) λxI + [ λA d d − (d + γ + )]I − η(t)I λA A ]I. = −dx 2 − λx 2 I − [d + γ +  + η(t) − d d λA When R0 ≤ 1, by the monotonicity of ϕ, we have d +γ + +η(t)− λA d ≥ d +γ + +ϕ(0)− d ≥ 0, and hence ∇V1 (x, I ) · v ≤ 0. This verifies (L3), and hence V1 is a Lyapunov function for (12.14). Furthermore, when R0 < 1,

 ZV1 = (x, I ) ∈ R 2 : ∃v ∈ G(x, I ), < ∇V (x, I ), v >= 0 = {(0, 0)} .

 When R0 = 1, ZV1 = {(0, 0)} {(0, I ) : η(t) = ϕ(0), I = 0}. Note that if x ≡ 0, it follows from the first equation of (12.14) that I = 0. Therefore, for any l > 0, the largest weakly invariant subset of ZV1 Ll is the singleton M = {(0, 0)}. By Lemma 12.2, (0, 0) is globally asymptotically stable for (12.14) if R0 ≤ 1; that is, E0 is globally asymptotically stable for (12.2) if R0 ≤ 1, and the proof is completed. The following theorem deals with the global asymptotic stability of the endemic equilibrium E ∗ when R0 > 1. Theorem 12.5. Suppose that (A1 ) is satisfied. If R0 > 1, then model (12.2) has a unique endemic equilibrium E ∗ = (S ∗ , I ∗ ), which is globally asymptotically stable. Proof. The existence and uniqueness of E ∗ have been confirmed in Theorem 12.1. We now prove the global asymptotic stability of E ∗ . Let x = S − S ∗ and y = I − I ∗ . Then (12.3) is transformed to  dx ∗ ∗ dt = −dx − λx(I + y) − (d + γ +  + η )y, (12.15) dy ∗ ∗ ∗ ∗ dt ∈ λx(I + y) + (η − co[ϕ(I + y)])(I + y), where η∗ = λA/(d + λI ∗ ) − (d + γ + ) ∈ co[ϕ(I ∗ )]. Consider the function V2 (x, y) =

x 2 d + γ +  + η∗ I∗ + y + (y − I ∗ ln ). 2 λ I∗

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This is a smooth function with respect to (x, y). It is easy to verify that (L1) and (L2) are satisfied. Denote 

−dx − λx(I ∗ + y) − (d + λ +  + η∗ )y H (x, y) = λx(I ∗ + y) + (η∗ − co[ϕ(I ∗ + y)])(I ∗ + y) It is easy to see that the map H (x, y) is an upper semicontinuous set-valued map with nonempty compact convex values. For any v = (v1 , v2 ) ∈ H (x, y), there exists a corresponding function η(t) ∈ co[ϕ(I ∗ + y)] such that

 −dx − λx(I ∗ + y) − (d + γ +  + η∗ )y v= . λx(I ∗ + y) + (η∗ − η(t))(I ∗ + y) The gradient of V2 (x, y) is given by ∇V2 (x, y) = (x,

d + γ +  + η∗ y ). λ I∗ + y

Thus

 d + γ +  + η∗ y −dx − λx(I ∗ + y) − (d + γ +  + η∗ )y ) ∇V2 (x, y) · v = (x, λx(I ∗ + y) + (η∗ − η(t))(I ∗ + y) λ I∗ + y d + γ +  + η∗ = −dx 2 − λx 2 (I ∗ + y) − (η(t) − η∗ )y. λ The monotonicity of ϕ implies that (η(t) − η∗ )y ≥ 0, and hence dx 2 + λx 2 (I ∗ + y) +

d + γ +  + η∗ (η(t) − η∗ )y ≥ 0. λ

That is, ∇V2 (x, I ) · v ≤ 0 verifying (L3). Thus V2 is a Lyapunov function for (12.15).  Letting ∇V2 (x, I ) · v = 0, we can obtain ZV2 = {(0, 0)} {(0, y) : η(t) = η∗ , y = 0}. If x ≡ 0, then by the first equation of (12.15) we obtain y = 0. Therefore, for any l > 0, the largest weakly invariant subset of ZV 2 Ll for (12.15) is M = {(0, 0)}. By Lemma 12.2, (0, 0) is globally asymptotically stable for (12.15); that is, E ∗ is globally asymptotically stable for (12.2). The proof is completed.

12.1.4 Global convergence in finite time One important feature of discontinuous ODE systems that a smooth ODE system cannot have is that convergence to equilibrium in finite time is possible under some conditions. This topic has been particularly explored recently by Forti et al. [18] for discontinuous neural network models. Motivated by this, in the sequel, we investigate the possibility of convergence to equilibrium in finite time for our model (12.2). To this end, we need to apply nonsmooth Lyapunov functions as was done in Forti et al.[18], which requires a generalization of the notion of gradient.

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A function f : R n → R is said to be regular at x if the following hold: (i) it is locally Lipschitz near x; (ii) for all direction v ∈ R n , there exists the usual one-sided directional derivative f (x, v) = lim

ρ→0+

f (x + ρv) − f (x) , ρ

and f (x, v) = f o (x, v), where f o (x, v) =

lim y→x

ρ→0+

f (y + ρv) − f (y) , ρ

is the generalized directional derivative of f at x in the direction of v. A function f is said to be regular in R n if it is regular at every x ∈ R n . Let f : R n → R be locally Lipschitz in R n . Then, by Rademacher theorem, f is differentiable at almost all (a.a.) x ∈ R n in the sense of Lebesgue measure (see, e.g., [14,18]). For such a function, the Clarke generalized gradient (see [14]), denoted by ∂f , is defined by 

 ∂f (x) = co

lim ∇f (xi ) : xi → x, xi = f ,

i→+∞

where f is the set of measure zero in which the gradient of f is not differentiable. Lemma 12.3 (Chain rule [13,18]). If f (x) : R n → R is regular and x(t) : [0, +∞) → R n is absolutely continuous on any compact subinterval of [0, +∞), then x(t) and f (x(t)) : [0, +∞) → R are differentiable for a.a. t ∈ [0, +∞) and d f (x(t)) =< γ (t), x(t) ˙ >, ∀γ (t) ∈ ∂f (x(t)). dt We start with the endemic equilibrium E ∗ = (S ∗ , I ∗ ). If ϕ(I ) is continuous at I ∗ , then the full model system (12.2) is continuous at the neighborhood of E ∗ , and hence it is unlikely for the system to allow convergence to E ∗ in a finite time. This observation motivates the following assumption: (A2 ) Assume that R0 > 1 and that ϕ(I ) has a jump discontinuity at I ∗ where I ∗ is the unique positive solution of (12.10). Moreover, η∗ = λS ∗ −(d +γ +) ∈ (ϕ(I ∗− ), ϕ(I ∗+ )), where S ∗ = A/(d + λI ∗ ). If ϕ does not have a jump discontinuity at I ∗ , then the solution can only reach equilibrium in infinite time, because once it gets sufficiently close to I ∗ , the discontinuities of ϕ will never again be encountered and so the subsequent evolution is effectively governed by a classical system.  Under (A2), δ:=min ϕ(I ∗+ ) − η∗ , η∗ − ϕ(I ∗− ) > 0, and we have the following theorem confirming global convergence to E ∗ in a finite time.

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Theorem 12.6. Suppose that (A1 ) and (A2 ) are satisfied. Then every solution of model (12.2) with the initial condition S(0) = S0 ≥ 0 and I (0) = I0 ≥ 0 converges to E ∗ in a finite time. More precisely, (S(t), I )(t) = (S ∗ , I ∗ ) for all t ≥ t∗ =

λ2 B(S0 , I0 ) , dδ 2

(12.16)

where B(S0 , I0 ) =

(S0 − S ∗ )2 d + γ +  + η∗ I0 + (I0 − I ∗ − I ∗ ln ∗ ) + 2d/λ2 2 λ I x(t) = S(t) − S ∗



I0 −I ∗ 0

ϕ(ρ + I ∗ ) − η∗ dρ. ρ + I∗

y(t) = I (t) − I ∗ .

and Then by (12.15) we know that there exists a Proof. Let measurable function η(t) ∈ co[ϕ(I ∗ + y(t))] such that  dx ∗ ∗ dt = −dx − λx(I + y) − (d + λ +  + η )y, (12.17) dy ∗ ∗ ∗ dt = λx(I + y) + (η − η(t))(I + y). Construct the following Lyapunov function: V3 (x, y) =

x 2 d + γ +  + η∗ I∗ + y + (y − I ∗ ln )+α 2 λ I∗

 0

y

ϕ(I ∗ + ρ) − η∗ dρ, I∗ + ρ

(12.18)

where α is a positive constant to be specified later. It can be easily verified that V3 (x, y) is a regular function in (x, y). Moreover, V3 (x, y) > 0 for (x, y) = 0, V3 (0, 0) = 0, and V3 (x, y) → +∞ as x → +∞ or y → +∞. Note that   co[ϕ(I ∗ + y)] − η∗ d + λ +  + η∗ y + α . ∂V3 (x, y) = x, λ I∗ + y I∗ + y By Lemma 12.3, we know that for a.a. t ≥ 0, dV3 (x(t), y(t)) =< ξ(t), (x(t), ˙ y(t)) ˙ >, ∀ξ(t) ∈ ∂V3 (x(t), y(t)). dt In particular, for   d + λ +  + η∗ y co[ϕ(I ∗ + y)] − η∗ ξ(t) = x(t), + α ∈ ∂V3 (x(t), y(t)), λ I∗ + y I∗ + y we obtain dV3 (x(t), y(t)) 4dα − λ2 α 2 ≤− (η(t) − η∗ )2 . dt 4d Choose α > 0 sufficiently small such that 4d −λ2 α > 0. It follows from (A2 ) that [η(t)−η∗ ]2 ≥ δ 2 if (x(t), y(t)) = (0, 0). Therefore, for almost all t ∈ {t : (x(t), y(t)) = (0, 0)}, dV3 (x(t), y(t)) 4dα − λ2 α 2 ≤ −δ 2 . dt 4d

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Integrating both sides of the above inequality from 0 to t, we obtain 0 ≤ V3 (x(t), y(t)) ≤ V3 (x(0), y(0)) − δ 2

4dα − λ2 α 2 t. 4d

This implies that V3 (x(t), y(t)) reaches 0 at t = t ∗ , where t∗ =

4dV3 (S0 , I0 ) , δ 2 (4dα − λ2 α 2 )

and retains 0 after t ∗ as well (see [18] for a detailed argument of this type). Thus we have proved that (x(t), I (t)) = 0 for t ≥ t ∗ , or equivalently, (S(t), I (t)) = (S ∗ , I ( ∗)) for t ≥ t ∗ . Note that the above argument is valid for all α ∈ (0, 4d/λ2 ). Choosing α = 2d/λ2 at which the term 4dλ − λ2 α 2 attains its maximum value 4d 2 /λ2 , one obtains the value t∗ =

λ2 V3 (S0 , I (0)) λB(S0 , I0 ) = . dδ 2 dδ 2

The proof is completed. Notice that if ϕ(I ∗+ ) → ϕ(I ∗− ), then t ∗ → ∞, because δ = 0 for continuous ϕ, and so we must have δ → 0 as the discontinuous jump is closed. A more meaningful and desirable situation is the global convergence to the disease-free equilibrium E0 in a finite time. Since (A1 ) assumes the continuity of the treatment function h(I ) at I = 0, finite time convergence to E0 is impossible under (A1 ). Thus discontinuity is required for h(I ) at I = 0, as is stated in the following assumption: (A3 ) h : R+ → R+ is nondecreasing and has at most a finite number of jump discontinuities in every compact interval. Moreover, h(0) = 0 and h(I ) is discontinuous at I = 0. A typical treatment function satisfying (A3 ) is the following:  0, I = 0; h(I ) = γ , I > 0. This corresponds to an immediate response to the occurrence of a disease with the constant γ being an effort strength. Under (A3 ), by (12.7), we know that (A/d, 0) is the disease-free equilibrium of model (12.2). Let x = S − A/d. Then (12.3) is transformed to  dx λA dt = −dx − λxI − d I, (12.19) dy λA dt ∈ λxI + [ d − (d + γ + )]I − co[h(I )]. From (12.4) there exists a measurable function η ∈ co[h(I )] corresponding to (x(t), I (t)) such that  dx λA dt = −dx − λxI − d I, (12.20) dI λA f or a.a. t ∈ [0, +∞). dt = λxI + [ d − (d + γ + )]I − η(t)

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Theorem 12.7. Suppose that (A3 ) holds. If d + γ +  − λA/d ≥ 0, then every solution of model (12.2) with initial condition S(0) = S0 ≥ 0 and I (0) = I0 ≥ 0 converges to E0 in a finite time, i.e., the disease dies out in a finite time. More precisely, (S(t), I (t)) = (A/d, 0) for t ≥ t∗ =

dQ(S0 , I (0)) , Ah(0+ )

(12.21)

where Q(S0 , I0 ) =

(S0 − Ad )2 A + I0 . 2 d

Proof. Let V1 (x, I ) be the same Lyapunov function as in the proof of Theorem 12.4. Then evaluating V1 (x(t), I (t)) along system (12.19) gives λA A dV1 (x(t), I (t)) A ≤ −dx 2 − λx 2 I − (d + γ +  − )I − η(t). dt d d d By (A3 ) we know that η(t) ≥ h(0+ ). Note that d + γ +  −

λA d

≥ 0. Then we have

dV1 (x(t), I (t)) −Ah(0+ ) ≤ . dt d Integrating both sides of the above inequality from 0 to t, we obtain 0 ≤ V1 (x(t), I (t)) ≤ V1 (x(0), I (0)) −

Ah(0+ ) Ah(0+ ) t = Q(S0 , I0 ) − t. d d

This implies that V1 (x(t), I (t)) = 0 for t ≥ t ∗ , which means (x(t), I (t)) = (0, 0), and hence (S(t), I (t)) = (A/d, 0) for t ≥ t ∗ . The theorem is proved. Remark 12.3. Under (A3 ), the basic reproductive number of model (12.2) is given by R0 =

λA , d(d + γ + )

thus condition d + γ +  − λA/d ≥ 0 is equivalent to R0 ≤ 1.

12.1.5 Conclusion and discussion We have revisited the SIR model with treatment considered by [21]. But unlike in [21] where treatment function is assumed to be continuous, our main concern here is the impact of the adoption of a discontinuous treatment function. Our results on the disease-free equilibrium E0 show that when the basic reproductive number R0 of the model is less than one, as is expected, the disease-free equilibrium is globally asymptotically stable. Note that under assumption (A1 ), R0 depends on ϕ(0) by (12.11), which is a decreasing function of ϕ(0) (the initial treatment rate). Thus a larger

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initial treatment rate will help eliminate the disease, and formula (12.11) determines how large R0 should be. Under (A1 ) and when R0 > 1, the disease-free equilibrium becomes unstable. However, the existence, uniqueness, and global asymptotic stability of an endemic equilibrium all follow, regardless of whether I ∗ is a continuous or discontinuous point of ϕ(I ). What we believe is most interesting and most novel in this chapter are the results on the convergence to an equilibrium in a finite time. This is impossible if a smooth treatment function is adopted. Therefore it presents a true advantage of discontinuous treatments. We are also able to establish an estimation of the precise time (finite) it takes for a solution to settle to the equilibrium. This is particulary important and useful for designing treatment strategies aiming to eliminate the disease in a finite time. From expressions (12.16) and (12.21), one may easily see how the model parameters as well as the initial values and the initial treatment strength will affect the (finite) time it takes to eradicate the disease. Taking (12.21) as an example, we find that t ∗ is increasing in the magnitude of the initial infectious population I0 and decreasing in the initial treatment strength h(0+ ), which is all reasonable and natural. Since most, if not all, existing works on disease models with treatment assume continuous treatment functions, our results here on the finite time convergence suggest that it should be worthwhile to reconsider those models by incorporating discontinuous treatment functions. This section focuses on Guo’s work in reference [24].

12.2 Global stability of an SIS epidemic model with discontinuous treatment strategy 12.2.1 Introduction AIDS (acquired immune deficiency syndrome) is an infectious disease caused by HIV (human immunodeficiency virus). HIV damages the human immune system by infecting and killing helper T cells, which leads to the loss of humoral and cellular immune functions and makes the body open to opportunistic infections. HAART (highly active antiretroviral therapy) is composed of RTI (reverse transcriptase inhibitors) and PI (protease inhibitors). RTI can prevent the RNA information of HIV from retrogressing into DNA information. PI can inhibit the protease hydrolysis of virus and prevent the infected CD4+ T cells from producing infectious virus. Therefore HAART can inhibit HIV infection. In recent years some experts and scholars have constructed mathematical models of HIV, immune system, and drug treatment and carried out theoretical and numerical analysis on the models, which has made a great contribution to the prevention and control of HIV infection. In 2003 Culshwa [25] established a dynamic model of intercellular HIV infectious with time delay and discussed the mechanism of intercellular HIV infection and the stability of the model. In 2004 Lou [26] studied the pathogenesis of AIDS, the interaction between HIV and the immune system, and established the ODE model, DDE model, and ODE optimal control model with drug treatment. In 2008 Zhou [27] established a differential equation model of HIV infected CD4+ T cells with cure rate, discussed the stable

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conditions of the model, and verified the theoretical results by numerical simulation. After that, Wang et al. [28] established a viral dynamics model with immune response and antiretroviral therapy and discussed the equilibrium point and stability of the model. In 2007 Darlai et al. [29] established an HIV time delay model with logistic growth and antiretroviral therapy, discussed the Andronov Hopf branch of the model, and conducted sensitivity analysis on the parameters in the model, indicating that increasing antiretroviral therapy and virus clearance rate can reduce the infection rate or regeneration rate of the virus. With the continuous deterioration of the AIDS epidemic, people are more aware of the importance of treatment, and the intensity of treatment will increase significantly, forming a significant leap. Therefore it is of great significance to consider the infectious disease model with discontinuous treatment strategy [8,13,14,30,31]. For example, Zhang et al. [32] studied the dynamic behavior of an SEIR infectious disease model with discontinuous treatment. On the basis of article [27], a kind of HIV infection model with discontinuous treatment measures is established. The positive definiteness, boundedness, and stability of the equilibrium point of the discontinuous model are discussed. The numerical simulation of the model and the comparison of the treatment effect under different baselines are carried out.

12.2.2 Model and preliminaries In article [27], T is the healthy CD4+ T cell, I is the infected CD4+ T cell, V is the virus particle, A is the rate of producing healthy CD4+ T cell by precursor cell, d is the death rate of healthy CD4+ T cell, a is the intrinsic growth rate of healthy CD4+ T cell, Tmax is the capacity of CD4+ T cell, β is the infection rate of healthy CD4+ T cell, δ is the death rate of infected CD4+ T cell, ρ is the cure rate of infected CD4+ T cell, q is the average rate of virus production by infected cells, and c is the virus clearance rate. The following HIV infection model is established: ⎧ dT T ⎪ ⎪ ⎨ dt = A − dT + aT (1 − Tmax ) − βT V + ρI, dI (12.22) dt = βT V − δI − ρI, ⎪ ⎪ ⎩ dV = qI − cV dt

due to the presence of 98 percent of HIV CD4+ T cells in the host brain and lymph nodes. In brain and lymph node tissues, HIV is more serious in cell-cell infection than in cell-free virus infection [26]. In this chapter, according to the references [24,33], we ignore the infection of virus particles, but consider that the continuous treatment and discontinuous treatment exist at the same time in the practical sense, so we add the continuous treatment factors and then establish a kind of HIV infection between cells with discontinuous treatment measures:  dS dt = A − dS − βI S + γ I + h(I ), (12.23) dI dt = βI S − (d + γ )I − h(I ),

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where A is the growth rate of populations, d is the natural mortality rate, β is the contact rate, γ is the rate of naturally leaving the infected stage, respectively. The function h(I ) = ψ(I )I denotes the treatment rate. ψ(I ) satisfies the following assumptions. Obviously, the treatment rate should be nondecreasing as the number of infectious individuals is increasing. The following assumption will be needed throughout the paper. (A1 ) ψ : [0, ∞) → [0, ∞) is nondecreasing and has at most a finite number of jump discontinuities in every compact interval. There is no less of generality in assuming ψ is continuous at I = 0, otherwise we define ψ(0) to be ψ(0+ ). Here ψ(0+ ) denotes the right limit of ψ(I ) as I →0+ . By the definition of solutions for differential equations with discontinuous right-hand sides in [3–14,17,24,30,32,33], we call (S(t), I (t)) a solution with initial condition (S(0), I (0)) = (S0 , I0 ), S0 , I0 ≥ 0,

(12.24)

of model (12.23) on [0, T ), 0 < T ≤ ∞, if it is absolutely continuous on any compact subinterval of [0, T ), and almost everywhere on [0, T ) (abbreviated to a.e. on (0, T )) satisfies the following differential inclusion: 

dS dt dI dt

∈ A − dS − βI S + γ I + co[h(I )], ∈ βI S − (d + γ )I − co[h(I )],

(12.25)

where co[h(I )]=[h(I − 0), h(I + 0)]. Here h(I − 0), h(I + 0) denote the left limit and the right limit of the function h(I ) at I , respectively. From (A1 ) it is clear that the set map (S, I ) → (A − dS − βI S + γ I + co[h(I )], βI S − (d + γ )I − co[h(I )]) is an upper semicontinuous set-valued map [7] with nonempty compact convex values. By the measurable selection theorem ([30], p. 17,Theorem 2.2.5), if (S(t), I (t)) is a solution of model (12.25) on [0, T ), then there is a measurable function m(t) ∈ co[h(I )] such that 

dS dt dI dt

= A − dS − βI S + γ I + m(t), = βI S − (d + γ )I − m(t),

(12.26)

where m(t) is uniquely determined by (S(t), I (t)) up to a set of measure zero in [0, T ); and m(t) is continuous for all t ∈ [0, T ) if and only if (S(t), I (t)) is continuously differentiable for all t ∈ [0, T ).

12.2.3 Positivity and boundedness In this section, we prove that the solutions exist on [0, +∞) and are nonnegative. The main result is as follows.

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Theorem 12.8. Suppose that assumption (A1 ) holds, and let (S(t), I (t)) be the solution with initial condition (12.25) of model (12.23) on [0, T ). Then (S(t), I (t)) is nonnegative and bounded on [0, T ). Proof. By the definition of a solution of (12.23) in the sense of Filippov, (S(t), I (t)) must be a solution to differential inclusion (12.25). From the first equation of (12.23) we have dS |S=0 = A + γ I + h(I ) > 0. dt This together with S(0) = S0 ≥ 0 shows that S(t) ≥ 0, t ∈ [0, T ). Note that (A1 ) implies co[h(I )] and h(I ) is continuous at I = 0. Combining the continuity of ψ at I = 0, it may be concluded that there exists a positive constant δ such that ψ(I ) is continuous as |I | < δ. For this reason, the differential inclusion (12.25) becomes the following system of differential equation as |I | < δ, ψ(I ) is continuous and the differential inclusion in (12.25) becomes the following differential equation with continuous righthand side: dI = (βS − d − γ − ψ(I ))dt. I Hence  t I = I (0) exp( (βS − d − γ − ψ(I ))dt) ≥ 0. 0

Therefore I (t) ≥ 0 for all t ∈ [0, T ). From the first equation in (12.23) for all t ∈ [0, T ), because the sum of the cure rate and the natural recovery rate of infected cells is less than or equal to the infection rate, therefore h(I ) + γ I ≤ βI S, we have dS ≤ A − dS. dt Let A − dS = 0. It follows clearly that S0 =

A d . Hence

lim sup S(t) ≤ S0 .

t→∞

Let F = S + I , we have dS = A − dS − dI = A − dF. dt Then

 t  t  t F = exp( −ddt)[ A exp( ddt)dt + F (0)]. 0

0

0

(12.27)

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Hence F is bound on [0, T ). By (12.27) it is easy to know that I (t) has an upper bound M, we concluded that the solution (S(t), I (t)) exists on [0, +∞). This shows that the model can be studied in the feasible region

  = (S, I ) ∈ R2+ : S ≤ S0 , I ≤ M . It can be verified that  is positively invariant with respect to (12.23). For this reason, we have 0 ≤ S ≤ S0 , 0 ≤ I ≤ M, that is, (S(t), I (t)) is bounded and nonnegative. This completes the proof.

12.2.4 Stability of equilibria In this section, we show the stability of the equilibria for model (12.23). We first discuss the existence of the equilibria as follows. An equilibrium of model (12.23) is, by definition, a constant solution of (12.23), (S(t), I (t)) = (S ∗ , I ∗ ), where P ∗ = (S ∗ , I ∗ ) satisfies the following system:  0 ∈ A − dS ∗ − βI ∗ S ∗ + γ I ∗ + co[h(I ∗ )], 0 ∈ βI ∗ S ∗ − (d + γ )I ∗ − co[h(I ∗ )].

(12.28)

Since h(0) = 0, it follows that there always exists a disease-free equilibrium P 0 = ( Ad , 0). Next, we will consider the existence of an endemic equilibrium. By the second equation of (12.28), we conclude that S∗ =

A − I ∗. d

(12.29)

Substituting (12.28) into the second inclusion of (12.28) yields βA − (d + γ ) − βI ∗ ∈ co[ψ(I ∗ )] = [ψ(I ∗− ), ψ(I ∗+ )]. d

(12.30)

Write g(I ) =

βA − (d + γ ) − βI, d

and let R0 =

βA . d(d + γ + ψ(0))

We next will determine the existence of an endemic equilibrium. Theorem 12.9. Suppose that assumption (A1 ) holds. If R0 > 1, then there exists a unique positive endemic equilibrium Isatisfying Aβ − d(d + γ ) . I≤ βd

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Proof. We first show the existence of a positive solution I of inclusion (12.30). Note that g(0) > ϕ(0) ≥ 0. Also note that g(I ) is decreasing in I and ϕ(I ) is nondecreasing in I .  Moreover, g(I ) ≤ 0 when I ≥ (Aβ − d(d + γ ))/βd. Thus the set I : g(I ) ≥ ψ(I + ), I > 0 is bounded. Let  I= sup I : g(I ) > ψ(I + ), I > 0 . − ) and 0 < I ≤ Aβ − d(d + γ )/βd. Then it is obvious that g(I) ≥ ϕ(I − We claim that g(I) ∈ [ϕ(I ), ϕ(I + )]. Otherwise, g(I) > ϕ(I + ) = lim ϕ(I ). + I →I

By (A1 ), there exists δ > 0 such that g(I+ δ) > ϕ(I+ δ) = ϕ((I+ δ)+ ). This contradicts − ), ϕ(I + )], that is, I  is a positive solution of incluthe definition of I. Therefore g(I) ∈ [ϕ(I sion (12.30). We next show that I is the unique solution of (12.25). Set I1∗ = I and assume that I2∗ = I1∗ is another positive solution of (12.25). Then there exist η1∗ ∈ co[ϕ(I1∗ )] and η2∗ ∈ co[ϕ(I2∗ )] such that  βA ∗ ∗ d − (d + γ ) − βI1 = η1 , (12.31) βA ∗ ∗ d − (d + γ ) − βI2 = η2 . From the monotonicity of ϕ, it follows that H=

η1∗ − η2∗ ≥ 0. I1∗ − I2∗

Subtraction of the two equations in (12.31) results in η1∗ − η2∗ = −β < 0, I1∗ − I2∗ which is a contraction. Therefore (12.30) has the unique positive solution I, and the proof of the theorem is completed. A direct consequence of (12.28) is the following uniqueness theorem for endemic equilibrium. Theorem 12.10. Assume that (A1 ) holds. The disease-free equilibrium P 0 is locally asymptotically stable if R0 < 1 and is unstable if R0 > 1. Proof. We analyze the stability of the disease-free equilibrium by investigating the eigenvalues of the Jacobian matrix of model (12.23) at P 0 . The matrix is

J (P ) = 0

−d 0

−β Ad + γ + ψ(0) β Ad − γ − ψ(0)

 .

(12.32)

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Matrix (12.32) has two roots λ1 , λ2 , where λ1 = − d < 0, A λ2 =β − d − γ − ψ(0) d =(d + γ + ψ(0))(R0 − 1).

(12.33)

From (12.33) it is easily seen that the disease-free equilibrium is locally asymptotically stable when R0 < 1. We have shown that there exists a positive endemic equilibrium in Theorem 12.11. Here we establish its local stability. Theorem 12.11. Suppose that assumption (A1 ) holds. If R0 > 1, then the endemic equilibrium P ∗ of system (12.25) is locally asymptotically stable. Proof. The Jacobian matrix of (12.25) at the endemic equilibrium P ∗ = (S ∗ , I ∗ ) is ∗



J (P ) =

−βI ∗ − d βI ∗

−βS ∗ + γ + ψ (I ∗ )I ∗ + ψ(I ∗ ) βS ∗ − γ − d − ψ (I ∗ )I ∗ − ψ(I ∗ )

 .

(12.34)

The characteristic equation of J (P ∗ ) is λ2 + C1 λ + C2 = 0, where C1 = βI ∗ + d − βS ∗ + d + γ + ψ (I ∗ )I ∗ + ψ(I ∗ ) = βI ∗ + d − (βS ∗ − d − γ − ψ(I ∗ )) + ψ (I ∗ )I ∗ = βI ∗ + d + ψ (I ∗ )I ∗ ≥ 0. C2 = (βI ∗ + d)(−βS ∗ + d + γ + ψ (I ∗ )I ∗ + ψ(I ∗ )) + βI ∗ (βS ∗ − γ − ψ (I ∗ )I ∗ − ψ(I ∗ )) = βI ∗ d − dβS ∗ + d(d + γ + ψ (I ∗ )I ∗ + ψ(I ∗ )) = βI ∗ d − d(βS ∗ − d − γ − ψ(I ∗ )) + dψ (I ∗ )I ∗ = βI ∗ d + dψ (I ∗ )I ∗ ≥ 0. Hence all of the Routh-Hurwitz criteria are satisfied. Thus it follows that the endemic equilibrium P ∗ of (12.23) is always locally asymptotically stable. The proof is completed. Theorem 12.12. Suppose that assumption (A1 ) holds. If R0 < equilibrium P 0 of (12.23) is globally asymptotically stable.

d d+γ +ψ(0) , then the disease-free

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Proof. To prove the global asymptotic stability of P 0 , we use the method of Lyapunov function. Define S (12.35) L = d(S − S0 − S0 ln ) + dI. S0 Because L ≥ 0, L = 0 if and only if (S, I ) = (S0 , 0), that is, A = dS0 , then dL dS S0 dS dI =d( − )+d dt dt S dt dt S − S0 (A − βSI − dS + γ I + co[h(I ∗ )]) + d(βSI − dI − γ I − co[h(I )]) =d S d d (S − S0 )2 − dβ(S − S0 )I + γ I (S − S0 ) + (S − S0 )co[h(I )] =−d S S S + dβSI − d 2 I − γ I d − dco[h(I )] (S − S0 )2 S − S0 S − S0 + dβS0 I − d 2 I + γ I d( − 1) + dco[ψ(I )]I ( − 1) S S S S − S0 S − S0 (S − S0 )2 + dI (βS0 − d) + γ I d( − 1) + dco[ψ(I )]I ( − 1). =−d S S S

=−d

When βS0 − d = (d + γ + ψ(0))R0 − d < 0, we know dL ≤ 0. dt By LaSalle’s invariance principle [31], P0 is globally asymptotically stable in , completing the proof. The following theorem states the global stability of the endemic equilibrium P ∗ . Theorem 12.13. Suppose that assumption (A1 ) holds. If R0 > 1, then the endemic equilibrium P ∗ of (12.23) is globally asymptotically stable. Proof. To study the global stability of the endemic equilibrium, we make use of a Lyapunov function W of the form S (12.36) W = S − S0 − S ln ∗ . S Then dW dS S ∗ dS = − dt dt S dt S − S∗ (A − βSI − dI + γ I + ψ(I )I ) = S ∗ S −S (12.37) = (βS ∗ I + dS ∗ − γ I − ψ(I )I − dS + γ I + ψ(I )I ) S S − S∗ 2 ) (βI + d) = −( S ≤ 0.

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Therefore lim W = 0.

t→∞

Hence lim S(t) = S ∗ .

(12.38)

y = I − I ∗.

(12.39)

t→∞

Let

Then dy dI dI ∗ = − dt dt dt = βSI − dI − γ I − ψ(I )I − βS ∗ I ∗ + dI ∗ + γ I ∗ + ψ(I ∗ )I ∗ = β(SI − S ∗ I ∗ ) − d(I − I ∗ ) − γ (I − I ∗ ) − ψ(I )I + ψ(I ∗ )I ∗ ≤ β(MS0 − S ∗ I ∗ ) − (d + γ )y + ψ(I ∗ )I ∗ = ε − (d + γ )y,

(12.40)

where ε = β(MS0 − S ∗ I ∗ ) + ψ(I ∗ )I ∗ . Hence

 t  t  t y ≤ exp( −(d + γ )dt)[ ε(exp( (d + γ )dt)dt + y(0))]. 0

0

(12.41)

0

dy dI dI ∗ = − dt dt dt = β(SI − S ∗ I ∗ ) − (d + γ )(I − I ∗ ) − ψ(I )I + ψ(I ∗ )I ∗ = βSI − βS ∗ I ∗ + (d + γ )I ∗ + ψ(I ∗ )I ∗ − (d + γ )I ∗ − (d + γ )y − ψ(I )I = βSI − (d + γ )I ∗ − (d + γ )y − ψ(I )I

(12.42)

≥ −[(d + γ )I ∗ + ψ(M)M] − (d + γ )y = −μ − (d + γ )y, where μ = (d + γ )I ∗ + ψ(M)M. Hence

 t  t  t y ≥ exp( −(d + γ )dt)[ μ(exp( (d + γ )dt)dt + y(0))]. 0

0

(12.43)

0

By (12.39), (12.40), we know y = 0 when t → ∞, therefore lim I = I ∗ .

t→∞

(12.44)

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dy dW dy It follows from (12.36) to (12.42) that dW dt ≤ 0, dt ≤ 0. Thus dt , dt is negative definite with ∗ ∗ respect to P . The global stability of P follows from the classical stability theorem of Lyapunov.

12.2.5 Simulation We have considered an SIS epidemic model that incorporates the discontinuous treatment strategies. Unlike in previous SIS epidemic models, we are interested in finding the impact of the adoption of a discontinuous treatment function. The basic reproductive number R0 is derived some reasonable assumptions on the discontinuous treatment function. It is a sharp threshold parameter which completely determines the global dynamics of model (12.23) and whether the disease goes to extinction d , the disease-free equilibrium is globally stable so that the disor not. When R0 < d+γ +ψ(0) ease always dies out, and when R0 > 1, the endemic equilibrium emerges as the positive equilibrium and it is globally stable. For making the utility of the concepts more visibly apparent, we present numerical simulations of solutions of model (12.23). We give a treatment function satisfying (A1 ) as follows:  c1 I, I ≤ I0 ; h(I ) = c2 I, I > I0 , where 0 ≤ c1 ≤ c2 . It can be used to describe the following case: when the infective individuals attain some threshold I0 , we need to strengthen the treatment rate. Let A = 100, d = 0.02, β = 0.00002, γ = 0.02, c1 = 0, c2 = 0.2, I0 = 280, one could easily see that R0 = 2.5 > 1 by using R0 = d(d+γβA +ψ(0)) . We obtain Fig. 12.1. Fig. 12.1 shows that the infective will go to a differential level. In addition, the basic reproductive number R0 is independent of c2 , but the different values of c2 can affect the stability level of the infective. That is to say, larger values of c2 can lead to a lower stability level of the infective. It shows that strengthening the treatment rate after the number of infective individuals has increased to some high level is also a beneficial disease control. Let A = 15, d = 0.02, β = 0.00002, γ = 0.02, c1 = 0, c2 = 0.2, I0 = 280, one easily sees that d = 0.5 by using R0 = d(d+γβA R0 = 0.375 < d+γ +ψ(0) +ψ(0)) . We obtain Fig. 12.2.

12.2.6 Conclusion We have considered an SIS epidemic model with discontinuous treatment strategies. Under some reasonable assumptions on the discontinuous treatment function, we confirmed the well-posedness of the model, described the structure of possible equilibria, and established the stability/instability of the equilibria. Most interestingly, we found that in the viewpoint of eliminating the disease from the host population, discontinuous treatment strategies would be superior to continuous ones. The methods we have used to obtain the mathematical results are the generalized Lyapunov theory for discontinuous differential equations and some results on nonsmooth analysis.

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FIGURE 12.1 Endemic equilibrium.

FIGURE 12.2 Disease-free equilibrium.

12.3 Global stability of an SEIR epidemic model with infectious force in latent period and infected period under discontinuous treatment strategy 12.3.1 Introduction Infectious diseases have always been a great enemy to human health. The epidemics of infectious diseases have brought great disasters to human survival and national livelihood. Hence how to prevent or effectively control the transmission of infectious diseases is an important problem to be considered and solved. Methods of mathematical modeling and

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differential equations play an important role in the study of disease transmission, prevention, and control. The deterministic model for communicable disease, termed an SIR model, was initially proposed by Kermack and McKendrick. From then on, a lot of theoretical and empirical studies have shown that the establishment of mathematical models described by differential equations is a very effective method to study the dynamics of infectious diseases. Disease control researchers applied many epidemic models to study the mechanisms of disease transmission (see [32,34–37]). In [37] Hattaf and Yousfi applied optimal control to establish exogenous reinfection of ordinary differential equations in a tuberculosis disease model. There are many methods for control of infectious diseases, such as treatment, isolation, immunity, and so on. Treatment plays a very important role in controlling the spread of infectious diseases. In recent years, some mathematical models incorporating treatment have been studied by many researchers (see [9,15,18,19,21,23,24,38–41]). In [38] Wang and Ruan proposed an epidemic model to simulate the limited resources for the treatment of patients, which can occur because patients have to be hospitalized but there are limited beds in hospitals, or there is not enough medicine for treatments. In [39] Li and his collaborators constructed an SIR epidemic model with nonlinear incidence and treatment. The results show that a backward bifurcation occurs if the capacity is small. Recently, Guo and others considered an SIR epidemic model with discontinuous treatment strategies. The results show that discontinuous treatment strategies would be superior to continuous ones [18,24,33,42]. On the other hand, there are a lot of infectious diseases which have latent periods. That is, a susceptible individual first goes through a latent period after infection before becoming infectious. When we use mathematical models to analyze the above-mentioned diseases, omitting the latent period will lead to some inaccurate results on their transmission law. Under this circumstance, the epidemic models with latent periods play a very important role in epidemiology. Some models with latent periods can be of SEI, SEIS, SEIR, or SEIRS type, respectively [43,44]. In [44] the authors studied the global dynamics of an SEIR model with this saturating infection rate. The latent period is the period between the invasion of pathogenic microorganisms and the earliest appearance of clinical symptoms. Pathogenic microorganisms do not cause disease as soon as they invade the human body. They need a certain amount and virulence to make the human body sick. Many diseases are not only infectious in the infected period, the latent period is also infectious, but the infectivity is weaker than the onset. In other words, once a susceptible person is infected with the virus, they become contagious before they become ill (that is, during the latent period), and when they become ill, they remain contagious (see [45]). In recent years, many researchers have studied the models of infectious diseases with latent period, and more attention has been paid to the infectivity of infected people. In fact, many diseases are infectious during the latent period, but less infectious than during the infected period (see [42]). Since the emergence of the first pneumonia cases in Wuhan, China in December, 2019, the novel coronavirus (2019-nCov) infection has been quickly spreading out to other provinces and neighboring countries. This novel coronavirus has made it clear that it can also transmit the virus to oth-

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ers during the latent period (see [46,47]). In 2020, Wang and his collaborators considered a novel coronavirus latent person infection, and believed that COVID-19 infected people in the incubation period may expel the virus outside the body and become the source of infection (see[47]). In [24,33] Guo and Zhang studied discontinuous treatment strategies for infectious diseases, the resulting model is a discontinuous system. Some nonsmooth analysis techniques [7,8,17,29–31] are used for this system. This chapter is based on references [24,33,42], we study the global stability of an SEIR model with infectious force in the latent period and the infected period under discontinuous treatment strategy. This section is organized as follows. In the next section, we construct the model and introduce the main assumptions for a discontinuous treatment function. In Section 12.3.3, positivity and boundedness of the solution in the sense of Filippov for the model are clearly discussed. We obtain the existence of a possible equilibrium, the basic reproductive number, and the stability of equilibrium in Section 12.3.4. In Section 12.3.5, we discuss that the solutions converge to the disease-free equilibrium in finite time. Finally, we make simulations to conform to our analytical results.

12.3.2 Model and preliminaries We consider a model with state variables S, E, I , and R, which represent the number of susceptible, exposed, infected, and removed individuals, respectively. The infected individuals with infectious force in the latent period and the infected period will be given a discontinuous treatment. Consider the following SEIR model with discontinuous treatment: ⎧ dS ⎪ = A − dS − ρS − β1 ES − β2 I S, ⎪ ⎪ dt ⎪ ⎨ dE = β ES + β I S − dE − ωE, 1 2 dt (12.45) ⎪ dI = ωE − (d + α + δ)I − h(I ), ⎪ dt ⎪ ⎪ ⎩ dR dt = δI + h(I ) + ρS − dR, where A is the growth rate of population, d is the natural mortality rate, α is the death rate induced by disease, the bilinear incidence β1 ES is found in the latent population, and the infectivity is relatively weak, the bilinear incidence of β2 I S is found in the infected population, β1 , β2 is the infection rate, ρ is effective vaccination rate and has permanent immunity after inoculation, ω and δ are the rates of naturally leaving the latent stage and the infected stage, respectively. The function h(I ) = φ(I )I denotes the treatment rate. φ(I ) satisfies the following assumptions. Obviously, the treatment rate should be nondecreasing as the number of infectious individuals is increasing. In reality, the treatment strategy usually is not smooth and even not continuous due to limited resources, usually there are some restrictions on treatment strengths. The following assumption will be needed throughout the chapter. (A1 ) φ : [0, ∞) → [0, ∞) is nondecreasing and has at most a finite number of jump discontinuities in every compact interval. There is no loss of generality in assuming φ is

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continuous at I = 0; otherwise we define φ(0) to be φ(0+ ). Here φ(0+ ) denotes the right limit of φ(I ) as I →0+ . Since the variable R does not appear in the first three equations of model (12.45), we only need to study the first three equations of model (12.45), thereby lowering the dimension of the system to be studied, i.e., ⎧ dS ⎪ ⎪ ⎨ dt = A − dS − ρS − β1 ES − β2 I S, dE (12.46) dt = β1 ES + β2 I S − dE − ωE, ⎪ ⎪ ⎩ dI = ωE − (d + α + δ)I − h(I ). dt

By the definition of solutions for differential equations with discontinuous right-hand sides in [17,31], we call (S(t), E(t), I (t)) a solution with initial condition (S(0), E(0), I (0)) = (S0 , E0 , I0 ), S0 , E0 , I0 ≥ 0

(12.47)

of model (12.46) on [0, T ), 0 < T ≤ ∞, if it is absolutely continuous on any compact subinterval of [0, T ) and almost everywhere on [0, T ) (abbreviated to a.e. on [0, T )) satisfies the following differential inclusion: ⎧ dS ⎪ ⎪ ⎨ dt = A − dS − ρS − β1 ES − β2 I S, dE (12.48) dt = β1 ES + β2 I S − dE − ωE, ⎪ ⎪ ⎩ dI ∈ ωE − (d + α + δ)I − co[h(I )], dt

where co[h(I )]=[h(I − 0), h(I + 0)]. Here h(I − 0) and h(I + 0) denote the left limit and the right limit of the function h(I ) at I , respectively. From (A1 ) it is clear that the set map (S, E, I ) →(A − β1 SE − β2 SI − (d + ρ)S, β1 SE + β2 SI − (d + ω)E, ωE − (d + α + δ)I − co[h(I )]) is an upper semicontinuous set-valued map with nonempty compact convex values. By the measurable selection theorem ([31], p.17, Theorem 2.2.5), if (S(t), E(t), I (t)) is a solution of model (12.46) on [0, T ), then there is a measurable function m(t) ∈ co[h(I )] such that ⎧ dS ⎪ ⎪ ⎨ dt = A − dS − ρS − β1 ES − β2 I S, dE (12.49) a.e. on [0, T ) dt = β1 ES + β2 I S − dE − ωE, ⎪ ⎪ dI ⎩ = ωE − (d + α + δ)I − m(t). dt

12.3.3 Positivity and boundedness In this section, we prove that the solutions exist on [0, +∞) and are nonnegative. The main result is as follows.

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Theorem 12.14. Suppose that assumption (A1 ) holds, and let (S(t), E(t), I (t)) be the solution with initial condition (12.47) of model (12.46) on [0, T ). Then (S(t), E(t), I (t)) is nonnegative and bounded on [0, +∞). Proof. By the definition of a solution of (12.46) in the sense of Filippov, (S(t), E(t), I (t)) must be a solution to differential inclusion (12.48). From the first equation of (12.48) we have  t  τ S(t) =[S0 + A exp( β1 E(s) + β2 I (s) + (d + ρ)ds)dτ ] 0 0 (12.50)  t β1 E(s) + β2 I (s) + (d + ρ)ds) > 0 × exp(− 0

for all t ∈ (0, T ). According to (A1 ), we have co[h(I )]={0} and h(I ) is continuous at I = 0. Combining the continuity of φ at I = 0, it may be concluded that there exists a positive constant δ such that φ(I ) is continuous as |I | < δ. For this reason, the differential inclusion (12.48) becomes the following system of differential equations as |I | < δ:  dE dt = β1 ES + β2 I S − dE − ωE, (12.51) dI dt = ωE − (d + α + δ + φ(I ))I. We divide this into four cases to discuss the positivity of the solutions for (12.48). (i) E0 = I0 = 0. From (12.51) we see that E(t) = I (t) = 0 for all t ∈ [0, T ). (ii) E0 > 0, I0 = 0. By the continuity of E(t) at t = 0 and dI dt |t=0 = ωE0 > 0, we conclude E(t) > 0 and I (t) > 0 for all t ∈ [0, T ). If it is not true, then we can set t1 = inf {t : E(t) = 0 or I (t) = 0} ∈ (0, T ).

(12.52)

If E(t1 ) = 0, then from dE dt ≥ −(d + ω)E for 0 ≤ t ≤ t1 we deduce that E(t1 ) ≥ E0 exp(−(d + ω)t1 ) > 0. This is a contradiction. If I (t1 ) = 0, then there is θ such that t1 − θ > 0 and 0 < I (t) < δ on [t1 − θ, t1 ). Therefore the second equation of (12.51) implies dI ≥ −(d + α + δ + φ(I ))I. dt This gives

 I (t1 ) ≥ I (t1 − θ ) exp(−

t1

t1 −θ

(d + α + δ + φ(I (ξ )))dξ ) > 0.

This is also a contradiction. Hence E(t) and I (t) are positive for all t ∈ (0, T ). The same conclusion can be drawn for the following cases:

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(iii) E0 = 0, I0 > 0. (iv) E0 > 0, I0 > 0. Our next goal is to prove the boundedness of the solutions of model (12.48). Under the basis of the above proof, adding the three equations of (12.48) yields d(S + E + I ) ∈ A − d(S + E + I ) − (α + δ)I − ρS − co[h(I )]. dt Fix any v ∈ co[h(I )]. When S + E + I >

(12.53)

A d , we must have

A − d(S + E + I ) − (α + δ)I − ρS − v < 0.

(12.54)

 For this reason, we have 0 ≤ S + E + I ≤ max Ad , S0 + E0 + I0 , that is, (S(t), E(t), I (t)) is bounded on [0, T ). By the boundedness and the continuation theorem, we conclude that the solution (S(t), E(t), I (t)) exists on [0, +∞), namely, the solution is global existence. This completes the proof.

Remark 12.4. For any solution of (12.48) with nonnegative initial value (12.47), we have the following detailed statements (i), (ii), and (iii). (i) The solution (S(t), E(t), I (t)) of (12.48) exists on [0, +∞) and S(t) > 0 (t > 0), E(t) ≥ 0 (t > 0), I (t) ≥ 0 (t > 0). (ii) If E(0) = 0 and I (0) = 0, then the solution (S(t), E(t), I (t)) of (12.48) exists on [0, +∞), S(t) > 0 (t > 0), E(t) ≡ 0 (t ≥ 0), I (t) ≡ 0 (t ≥ 0). (iii) If one of E(0) and I (0) is greater than zero, then the solution (S(t), E(t), I (t)) of (12.48) exists on [0, +∞) and S(t) > 0 (t > 0), E(t) > 0 (t > 0), I (t) > 0 (t > 0).

12.3.4 Stability of equilibrium In this section, we show the stability of the equilibrium for model (12.46). We first discuss the existence of the equilibrium as follows. The equilibrium of model (12.46) is, by definition, a constant solution of (12.46), (S(t), E(t), I (t)) = (S ∗ , E ∗ , I ∗ ), where P ∗ = (S ∗ , E ∗ , I ∗ ) satisfies the following system: ⎧ ∗ ∗ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎨0 = A − dS − ρS − β1 E S − β2 I S , 0 = β1 E ∗ S ∗ + β2 I ∗ S ∗ − dE ∗ − ωE ∗ , ⎪ ⎪ ⎩0 ∈ ωE ∗ − (d + α + δ)I ∗ − co[h(I ∗ )].

(12.55)

A Since h(0) = 0, it follows that there always exists a disease-free equilibrium P 0 = ( d+ρ , 0, 0). Next, we will consider the existence of an endemic equilibrium. By the three equations of

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(12.55) and h(I ) = φ(I )I , we conclude that (d + ω)(d + α + δ + φ(I ∗ )) , β2 ω + β1 (d + α + δ + φ(I ∗ )) A(β2 ω + β1 (d + α + δ + φ(I ∗ )) − (d + ρ)(d + α + δ + φ(I ∗ )))(d + ω) E∗ = , (d + ω)(β2 ω + β1 (d + α + δ + φ(I ∗ ))) ωA(β2 ω + β1 (d + α + δ + φ(I ∗ )) − ω(d + ρ)(d + α + δ + φ(I ∗ )))(d + ω) I∗ = , (d + α + δ + φ(I ∗ ))(d + ω)(β2 ω + β1 (d + α + δ + φ(I ∗ )))

S∗ =

(12.56)

and let R0 =

β1 β2 ω A ( + ). d + ρ d + ω (d + ω)(d + α + δ + φ(0))

(12.57)

We next claim that R0 is the basic reproductive number for model (12.46) which will determine the existence of an endemic equilibrium. Theorem 12.15. Assume that (A1 ) holds. The disease-free equilibrium P 0 is locally asymptotically stable if R0 < 1 and is unstable if R0 > 1. Proof. We analyze the stability of the disease-free equilibrium by investigating the eigenvalues of the Jacobian matrix of model (12.46) at P 0 . The matrix is ⎞ ⎛ β1 A β2 A − d+ρ −(d + ρ) − d+ρ ⎟ ⎜ β1 A β2 A (12.58) J (P 0 ) = ⎝ 0 ⎠. d+ρ − (d + ω) d+ρ 0 ω −(d + α + δ + φ(0)) The characteristic equation of J (P 0 ) has three roots λ1 = −(d + ρ) < 0, λ2 , and λ3 , where λ2 + λ3 = − (d + α + δ + φ(0) + d + ω −

β1 A ) d +ρ

d + α + δ + φ(0) + 1 − R0 + d +ω β2 Aω ](d + ω). (d + ω)(d + ρ)(d + α + δ + φ(0)) β2 ωA β1 A λ2 λ3 = (d + ω − )(d + α + δ + φ(0)) − d +ρ d +ρ = (d + ω)(d + α + δ + φ(0))(1 − R0 ). =−[

(12.59)

From (12.59) it is easily seen that both the real parts of λ2 and of λ3 are negative when R0 < 1. When R0 > 1, one of λ2 and λ3 is a number with a positive real part. Thus the diseasefree equilibrium is locally asymptotically stable if R0 < 1 and unstable if R0 > 1. Thus these results establish the theorem. Theorem 12.16. Suppose that assumption (A1 ) holds. If R0 > 1, the endemic equilibrium P ∗ of system (12.46) is locally asymptotically stable.

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Proof. The Jacobian matrix of (12.46) at the endemic equilibrium P ∗ = (S ∗ , E ∗ , I ∗ ) is ⎛

−(d + ρ + β1 E ∗ + β2 I ∗ ) ∗ J (P ) = ⎝ β1 E ∗ + β2 I ∗ 0

⎞ −β1 S ∗ −β2 S ∗ β1 S ∗ − (d + ω) β2 S ∗ ⎠ . ω −

(12.60)

Here − stands for −(d + α + δ) − φ (I ∗ )I ∗ − φ(I ∗ ), and replacing β1 S ∗ − (d + ω) by − β2 ωS ∗ d+α+δ+φ(I ∗ ) = − gives ⎛

−(d + ρ + β1 E ∗ + β2 I ∗ ) ∗ J (P ) = ⎝ β1 E ∗ + β2 I ∗ 0

−β1 S ∗ − ω

⎞ −β2 S ∗ β2 S ∗ ⎠ . −

(12.61)

The characteristic equation of J (P ∗ ) is λ3 + a1 λ2 + a2 λ + a3 = 0, where a1 =d + ρ + β1 E ∗ + β2 I ∗ +  + , a2 =(d + ρ + β1 E ∗ + β2 I ∗ )( + ) + β1 S ∗ (β1 E ∗ + β2 I ∗ ) + φ (I ∗ )I ∗ , a3 =ωβ2 S ∗ (β1 E ∗ + β2 I ∗ ) + φ (I ∗ )I ∗ (d + ρ + β1 E ∗ + β2 I ∗ ) + β1 S ∗ (β1 E ∗ + β2 I ∗ ). Since φ is nondecreasing, φ (I ∗ ) ≥ 0. This implies a1 > 0, a2 > 0, a3 > 0. Then a1 a2 − a3 =(d + ρ + β1 E ∗ + β2 I ∗ + )(d + ρ + β1 E ∗ + β2 I ∗ )( + ) + (d + ρ + β1 E ∗ + β2 I ∗ + )β1 S ∗ (β1 E ∗ + β2 I ∗ ) + 2 (d + ρ + β1 E ∗ + β2 I ∗ ) + φ (I ∗ )I ∗  + (d + α + δ + φ(I ∗ ))(d + ρ) + φ (I ∗ )I ∗ (d + ρ + β1 E ∗ + β2 I ∗ ) + φ (I ∗ )I ∗ 2 > 0. Hence all of the Routh-Hurwitz criteria are satisfied. Thus it follows that the endemic equilibrium P ∗ of (12.46), which exists if R0 > 1, is always locally asymptotically stable. The proof is completed. We next prove global stability of the disease-free equilibrium and endemic equilibrium. We need to use the LaSalle-type invariance principle for the differential inclusion (Theorem 3 in [8]) to prove their global stability. A Let x = S − d+ρ . We obtain the following system analogous to (12.48): ⎧ A A ⎪ ⎪ dx dt = −(d + ρ)x − β1 xE − β2 xI − β1 d+ρ E − β2 d+ρ I, ⎨ A A dE dt = β1 xE + β2 xI + β1 d+ρ E + β2 d+ρ I − (ω + d)E, ⎪ ⎪ ⎩ dI ∈ ωE − (d + α + δ)I − co[φ(I )]I. dt

(12.62)

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Set V1 (x, E, I ) = and

A (d + ω)A x2 + E+ I 2 d +ρ (d + ρ)ω

(12.63)

⎛

⎞ A A E − β2 d+ρ I −(d + ρ)x − β1 xE − β2 xI − β1 d+ρ ⎜ ⎟ A A G(x, E, I ) = ⎝ β1 xE + β2 xI + β1 d+ρ E + β2 d+ρ I − (ω + d)E ⎠ . ωE − (d + α + δ)I − co[φ(I )]I

For any v = (v1 , v2 , v3 )T ∈ G(x, E, I ), there exists η(t) ∈ co[φ(I )] such that ⎛

⎞ A A E − β2 d+ρ I −(d + ρ)x − β1 xE − β2 xI − β1 d+ρ ⎜ ⎟ A A v = ⎝ β1 xE + β2 xI + β1 d+ρ E + β2 d+ρ I − (ω + d)E ⎠ . ωE − (d + α + δ)I − η(t)I

(12.64)

Hence ⎛

⎞ A A −(d + ρ)x − β1 xE − β2 xI − β1 d+ρ E − β2 d+ρ I A(d + ω) ⎜ A ⎟ A A , ) ⎝ β1 xE + β2 xI + β1 d+ρ ∇V1 (x, E, I ) · v = (x, E + β2 d+ρ I − (ω + d)E ⎠ d + ρ (d + ρ)ω ωE − (d + α + δ)I − η(t)I

=−

(d + α + δ + η(t))A(d + ω) A β1 β2 ω (1 − ( + ))I ω(d + ρ) d + ρ d + ω (d + ω)(d + α + δ + η(t))

(12.65)

− (d + ρ)x 2 − β1 x 2 E − β2 x 2 I. When R0 ≤ 1, the nondecreasing of φ implies A β1 β2 ω (d + α + δ + η(t))A(d + ω) (1 − ( + ))I ω(d + ρ) d + ρ d + ω (d + ω)(d + α + δ + η(t)) (d + α + δ + φ(0))A(d + ω) (1 − R0 )I ≤ 0. ≤− ω(d + ρ)

−

It shows that V1 is a Lyapunov function of (12.62). Furthermore, when R0 < 1, we have

 ZV1  (x, E, I ) ∈ 3 : ∇V1 (x, E, I ) · v = 0, v ∈ G(x, E, I ) = {(0, E, 0) : E ≥ 0} . When I = 0 and x = 0, we have l > 0, we set

dE dt

=

β1 A d+ρ E − (d

+ ω)E, which implies lim E(t) = 0. For any

 V1l  (x, E, I ) ∈ 3 : V1 (x, E, I ) ≤ l .

t→∞

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Hence, the largest weakly invariant subset of ZV1 When R0 = 1, we have ZV1 = {(0, E, 0) : E ≥ 0}





V1l is the singleton M={(0, 0, 0)}.

{(0, E, I ) : η(t) = φ(0), I > 0} .

From the first equation of (12.62) and x = 0, it may be concluded that I = 0. Therefore we see that the largest weakly invariant subset of ZV1 V1l is also the singleton M={(0, 0, 0)}. By the LaSalle-type invariance principle, the equilibrium (0, 0, 0) of (12.62) is globally asymptotically stable as R0 ≤ 1. Summarizing the above analysis, we obtain the following theorem. Theorem 12.17. Suppose that assumption (A1 ) holds. If R0 ≤ 1, then the disease-free equilibrium P 0 of (12.48) is globally asymptotically stable. The following theorem states the global stability of the endemic equilibrium P ∗ . Theorem 12.18. Suppose that assumption (A1 ) holds. If R0 > 1, then the endemic equilibrium P ∗ of (12.48) is globally asymptotically stable. Proof. Let S E ) + (E − E ∗ − E ∗ ln ∗ ) S∗ E d +ω I (I − I ∗ − I ∗ ln ∗ ). + ω I

V2 (S, E, I ) =(S − S ∗ − S ∗ ln

(12.66)

Write η∗ = and

1 (ωE ∗ − (d + α + δ)I ∗ ) ∈ co[φ(I ∗ )] I∗

⎞ A − β1 SE − β2 SI − (d + ρ)S ⎠. H (S, E, I ) = ⎝ β1 SE + β2 SI − (d + ω)E ωE − (d + α + δ)I − co[φ(I )]I ⎛

(12.67)

For any v = (v1 , v2 , v3 )T ∈ H (S, E, I ), there exists η(t) ∈ co[φ(I )] such that ⎞ A − β1 SE − β2 SI − (d + ρ)S v = ⎝ β1 SE + β2 SI − (d + ω)E ⎠ . ωE − (d + α + δ)I − η(t)I ⎛

The gradient of V2 is given by ∇V2 (S, E, I ) = (1 −

E∗ d + ω I∗ S∗ ,1 − , (1 − )). S E ω I

(12.68)

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Hence

⎞ ⎛ A − β1 SE − β2 SI − (d + ρ)S S∗ E∗ d + ω I∗ ⎝ ∇V2 (S, E, I ) · v = (1 − , 1 − , (1 − )) β1 SE + β2 SI − (d + ω)E ⎠ S E ω I ωE − (d + α + δ)I − η(t)I

S∗ + β1 S ∗ E + β2 S ∗ I + (d + ρ)S ∗ − β1 SE ∗ − S I I∗ d + ω (d + α + δ + η(t))(I − I ∗ ) β2 SE ∗ + (d + ω)E ∗ − (d + ω)E − E I ω d +ω ∗ S S∗ (12.69) (η − η)(I − I ∗ ) ≤ (d + ρ)S ∗ (2 − ∗ − ) + S S ω S ∗ EI ∗ SE ∗ I − + β1 S ∗ E ∗ (3 − − ∗ ∗) ∗ S IE S EI ∗ ∗ ∗I S SE EI + β2 S ∗ I ∗ (3 − − ). − S I E ∗ S ∗ EI ∗ The monotonicity of φ implies (η∗ − η)(I − I ∗ ) ≤ 0. Thus ∇V2 (S, E, I ) · v ≤ 0. This shows that V2 is a Lyapunov function of (12.48). Define

 ZV2  (S, E, I ) ∈ 3 : ∃v ∈ H (S, E, I ), ∇V2 (S, E, I ) · v = 0     E I = (S ∗ , E ∗ , I ∗ ) (S ∗ , E, I ) : ∗ = ∗ , η∗ = η(t) . E I =A − (d + ρ)S − A

If S = S ∗ , then the first equation of (12.48) implies I = I ∗ . This gives E = E ∗ . Consequently, for any l > 0, the largest weakly invariant subset of ZV2 V2l of (12.48) is the singleton (S ∗ , E ∗ , I ∗ ). Here

 V2l = (S − S ∗ , E − E ∗ , I − I ∗ ) ∈ 3 : V2 (S, E, I ) ≤ l . Therefore P ∗ is globally asymptotically stable if R0 > 1. This completes the proof. Remark 12.5. From Theorems 12.15–12.18, we can claim that the basic reproductive number R0 is a sharp threshold value and that the global dynamical behaviors of system (12.48) and the outcome of the disease are completely determined. In other words, when R0 ≤ 1, the disease-free equilibrium P 0 is globally stable so that the disease goes to extinction, while if R0 > 1, the endemic equilibrium P ∗ is globally stable so that the disease remains endemic.

12.3.5 Global convergence in finite time For a system of ordinary differential equations (ODE) with continuous right-hand side, any solution cannot attain an equilibrium in a finite time. But an ODE system with discontinuous right-hand side can have possibility that a solution converges to an equilibrium in a finite time. Next, we will discuss that the solutions converge to the disease-free equilibrium.

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Hence we must assume that the treatment function h(I ) in model (12.45) is discontinuous at I = 0. Here we give the following assumption. (A2 ) h(I ) : [0, ∞) → [0, ∞) is nondecreasing and has at most a finite number of jump discontinuities in every compact interval. Furthermore, h(0) = 0 and h(I ) is discontinuous at I = 0. A Under assumption (A2 ), it is easily seen that ( d+ρ , 0, 0) is the disease-free equilibrium

of (12.48). Let x = S −

A d+ρ . Then (12.48) changes into

⎧ dx A A ⎪ ⎪ ⎨ dt = −(d + ρ)x − β1 xE − β2 xI − β1 d+ρ E − β2 d+ρ I, A A dE dt = β1 xE + β2 xI + β1 d+ρ E + β2 d+ρ I − (ω + d)E, ⎪ ⎪ ⎩ dI ∈ ωE − (d + α + δ)I − co[h(I )].

(12.70)

dt

From (12.49) there exists a measurable function η(t) ∈ co[h(I )] such that ⎧ A A ⎪ ⎪ dx dt = −(d + ρ)x − β1 xE − β2 xI − β1 d+ρ E − β2 d+ρ I, ⎨

dE A A dt = β1 xE + β2 xI + β1 d+ρ E + β2 d+ρ I − (ω + d)E, f or a.e. t ∈ [0, ∞). ⎪ ⎪ ⎩ dI = ωE − (d + α + δ)I − η(t). dt

(12.71)

Theorem 12.19. Suppose that (A2 ) holds. If β2 Aω β1 A + ≤ 1, (d + ρ)(d + ω) (d + ρ)(d + ω)(d + α + δ) then every solution of model (12.46) with the initial condition S(0) = S0 ≥ 0 and I (0) = I0 ≥ 0 converges to P 0 in a finite time, i.e., the disease dies out in a finite time. More precisely, A (S(t), E(t), I (t)) = ( d+ρ , 0, 0) for t ≥ t∗ = where Q(S0 , E0 , I0 ) =

A 2 (S0 − d+ρ ) 2

+

(d + ρ)2 ωQ(S0 , E0 , I0 ) , A((d + ρ)(d + ω) − β1 A)h(0+ )

A d+ρ E0

+

(d+ω)A (d+ρ)ω I0 .

Proof. Let V1 (x, E, I ) be the same Lyapunov function as (12.63). Its derivative along the solutions of (12.71) is dV1 (d + ω)A(d + α + δ) = − (d + ρ)x 2 − β1 x 2 E − β2 x 2 I − dt (d + ρ)ω β2 Aω β1 A − ]I × [1 − (d + ρ)(d + ω) (d + ρ)(d + ω)(d + α + δ) A((d + ω)(d + ρ) − β1 A) − η(t). (d + ρ)2 ω

Chapter 12 • Global stability of epidemic models

By (A2 ) we know that η(t) ≥ h(0+ ). When

β1 A (d+ρ)(d+ω)

+

β2 Aω (d+ρ)(d+ω)(d+α+δ)

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≤ 1, we have

dV1 A((d + ρ)(d + ω) − β1 A) h(0+ ). ≤− dt (d + ρ)2 ω

(12.72)

Integrating the differential inequality (12.72) from 0 to t gives 0 ≤ V1 (x, E, I ) ≤ V1 (x(0), E(0), I (0)) −

A((d + ρ)(d + ω) − β1 A) h(0+ )t. (d + ρ)2 ω

(12.73)

Hence, when t > t ∗ , V1 (x, E, I ) = 0, this implies (x, E, I ) = (0, 0, 0), where t∗ =

(d + ρ)2 ωQ(S0 , E0 , I0 ) . A((d + ω)(d + ρ) − β1 A)h(0+ )

Q(S0 , E0 , I0 ) =

(S0 −

A 2 d+ρ )

2

+

(d + ω)A A E0 + I0 . d +ρ (d + ρ)ω

A Consequently, every solution of (12.48) (S(t), E(t), I (t)) = ( d+ρ , 0, 0) for t ≥ t ∗ . ∗ ∗ From the form of t we find that t is increasing in the initial exposed and infective individuals but decreasing in the initial treatment rate h(0+ ). If we take more effective control measures for infectious diseases at the initial time of the disease spread, then the diseases β1 A β2 Aω go to extinction more quickly. The number (d+ρ)(d+ω) + (d+ρ)(d+ω)(d+α+δ) is just the basic reproductive number of the SEIR model without treatment. The above analysis shows that the disease can go to extinction in a finite time under a discontinuous treatment strategy.

12.3.6 Simulations We have considered an SEIR epidemic model that incorporates the discontinuous treatment strategies. Unlike in previous SEIR epidemic models, we are interested in finding the impact of the adoption of a discontinuous treatment function. The basic reproductive number R0 is derived under some reasonable assumptions on the discontinuous treatment function. It is a sharp threshold parameter which completely determines the global dynamics of model (12.48) and whether the disease goes to extinction or not. When R0 ≤ 1, the disease-free equilibrium is globally stable so that the disease always dies out, and when R0 > 1, the disease-free equilibrium is unstable, while the endemic equilibrium emerges as the positive equilibrium and it is globally stable. For making the utility of the concepts more visibly apparent, we present numerical simulations of solutions of model (12.45). We give a treatment function satisfying (A1 ) as follows:  c1 I, I ≤ I0 ; h(I ) = c2 I, I > I0 ,

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FIGURE 12.3 Endemic equilibrium.

FIGURE 12.4 Disease-free equilibrium.

where 0 ≤ c1 ≤ c2 . It can be used to describe the following case: when the infective individuals attain some threshold I0 , we need to strengthen the treatment rate. Let A = 240, d = 0.02, α = 0.01, β1 = 0.001, β2 = 0.002, ρ = 0.1, ω = 0.5, δ = 0.5, c1 = 0, c2 = 0.2, I0 = 280, one could easily see that R0 = 11.1031 > 1 by using (12.57). Figs. 12.3 and 12.5 show that the infective will go to an endemic level. In addition, the basic reproductive number R0 is independent of c2 , but the different values of c2 can affect the stability level of the infective. That is to say, larger values of c2 can lead to a lower stability level of the infective. It shows that the strengthening of the treatment rate after the number of infective individuals has increased to some high level is also a beneficial disease control. Let A = 240, d = 0.02, α = 0.01, β1 = 0.00005, β2 = 0.0001, ρ = 0.1, ω = 0.5, δ = 0.5, c1 = 0, c2 = 0.2,

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FIGURE 12.5 Curves of the endemic equilibrium point with coefficient c2 .

FIGURE 12.6 Curves of the disease-free equilibrium point with coefficient c2 .

I0 = 280, one easily sees that R0 = 0.5552 < 1 by using (12.57). We obtain Figs. 12.4 and 12.6. This shows that the disease goes to extinction. This numerical verification supports Theorem 12.17. In addition, we find that different values of c2 can affect the peak values of the infection. Figs. 12.5 and 12.6 reflect that larger values of c2 can reduce the peak values of the infection. Therefore we can also prevent the spread of disease by increasing the treatment rate after the number infective individuals reaches some high level. From the numerical simulations, strengthening the treatment rate after the infective individu-

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als reach some level is also effective for disease control, even though we do not take any treatment measures at the initial time of the disease outbreak.

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Index A Amplitude spectrum, 256 Analysis bifurcation, 61 dengue epidemic data, 258 efficacy, 170, 183 mathematical, 208 model, 65, 196, 205 of dengue incidents, 276 stability, 33, 44, 170, 207, 234 global, 70, 76, 81, 240 local, 70, 75 steady state, 239 wavelet, 266, 275 Antivirals, 56 Antivirus treatment, 58 Artificial immunity, 102, 104, 107, 109, 112, 122, 124 period, 97, 103, 127, 135, 141 Artificial intelligence, 319 Asymptomatic, 203, 207, 213 infected, 203, 207, 208 ATL cells, 232 B B-cells, 56 Bacterial infections, 29 Basic reproduction number, 31, 34, 47, 51, 58, 59, 65, 68, 72, 89, 91, 145, 148, 151, 205 Basic wavelet, 269 functions, 270 Bifurcation, 24, 61, 152 analysis, 61 Hopf, 147–149, 152, 154, 155, 161, 164, 165 parameter, 61 Biological dynamics, 315, 317 structure, 306, 307, 311

dynamics, 307, 313, 319 Box function, 270 Branches of solutions, 24 C CD4+ T cells, 56 CD8+ T cells, 56 Cells ATL, 232 CD4+ T, 56 CD8+ T, 56 dynamics, 65 infected, 56–58, 63, 64, 293, 338, 340 Chain-binomial models, 97 Chaos, 318 Chronic diseases, 194, 219 hepatitis, 55 infection, 56, 235 Classical solution, 4 Colombo Municipal Council (CMC), 276 Communicable diseases, 191, 253, 348 Compartmental models, 95, 96, 192, 287, 288, 291, 301 epidemic, 96 Complete bipartite graph, 287 Cone of influence, 275 Confidence interval, 126, 127, 137 Contact rate, 289 Continuous time Markov chain (CTMC), 96 Continuous treatment, 324, 338 Control measures, 207, 214, 221 combined, 224 efficacy, 191 optimal, 170, 178–180, 187, 194, 199, 207 parameters, 208 reinfection, 177, 186

365

366

Index

Coronavirus disease 2019 (COVID-19), 145, 285, 286 Correlated vaccination, 116, 124 COVID-19, 191, 296 dynamic system, 212 epidemic models, 288, 290 network, 285 pandemic models, 288, 295 SEIRD model, 295 CTLs, 56, 233–235 immune response, 234–236, 240, 247, 249 delay, 235 CTMC epidemic models, 96 Cytotoxic T-lymphocytes (CTLs), 56, 233 D Decomposition joint state, 102 one-state-at-a-time, 101 Dengue, 258, 275 cases, 260, 263, 265 disease, 255 epidemic data analysis, 258 incidents, 255, 276 outbreaks, 255, 258–260, 264, 275, 276 transmission, 253, 255, 277, 280 virus, 278 Deoxyribonucleic acid (DNA), 231 Diffusion coefficient, 236 model, 236 Discontinuous treatment, 323, 338, 349 function, 324, 336, 337, 346, 349, 359 strategy, 324, 337, 338, 346–349, 359 Discrete Fourier transform (DFT), 256 Discrete time intervals, 99 Discrete time Markov chain (DTMC), 96 Disease chronic, 194, 219 communicable, 191, 253, 348 dengue, 255 dynamics, 95, 98, 109, 110, 169, 323 in population, 106 eradication, 141

infectious, 30, 97, 191, 253, 285–287, 323, 337, 338, 347, 348 liver, 55 management, 253 models, 61, 337 outbreak, 47, 104, 105, 107, 108, 123, 141, 254 progression, 194, 227 respiratory, 285 rotavirus, 169, 185 spread, 135, 275, 359 SVEIRS, 104 transmission, 29, 34, 98, 170, 192, 227, 253, 255, 259, 280, 292, 323, 348 Disease-free equilibrium, 33, 68, 70, 72, 76, 89, 207 Dissipativity, 65 Distributed time delay, 147, 149, 165 Drug epidemic, 294 DTMC epidemic models, 96, 97 SVEIRS, 98 Dynamics biological, 315, 317 structure, 307, 313, 319 cell, 65 disease, 95, 98, 109, 110, 169, 323 SVEIRS, 104, 110 epidemic, 96 HCV, 64 local, 46, 62 population, 2 virus, 57 E Edges, 286 Effect of climate, 276 of human mobility, 277 psychological, 29, 30, 32, 146, 148, 149, 151, 165 Efficacy, 169, 177, 213, 221 analysis, 170, 183 index, 183 of control measures, 191 of treatment, 58 strategy, 183

Index

End vertex, 287 Endemic equilibrium, 31, 50, 69, 75, 81, 173, 175 existence, 69 point, 76, 174 state, 42–45, 184 Epidemic COVID-19, 288, 290 drug, 294 dynamics, 96 modeling, 171 models, 96, 97, 141, 146, 147, 298, 348 CTMC, 96 on COVID-19, 288 SEIR, 347 SIR, 97, 145, 147, 149, 152, 165 SIS, 337 stochastic, 96 SVEIRS, 98, 99, 103, 111, 115, 116, 119, 122, 135, 141 parameters, 34 problem, 32 Epidemiology, 275 Equilibria, 65, 327 disease-free, 33, 68, 70, 72, 76, 89, 207 endemic, 31, 50, 69, 75, 81, 173, 175 existence, 65, 170, 172 level, 53 stability, 174, 324, 341, 352 unstable, 62 Escaping infection, 123, 125, 135 Existence condition, 178 equilibria, 65, 170, 172 endemic, 69 optimal control, 178 theorem, 326 Exogenous reinfection, 348 Exponential stability result, 2 F Fast Fourier transform (FFT), 257, 280 Finite delay times, 101, 102 Fourier integral, 256 Fourier transformation, 255

367

Fractional ordinary differential equations (FODE), 289 Free virus particles, 57, 58 Function basic wavelet, 270 box, 270 Haar scaling, 270 Lyapunov, 60, 77, 78, 235, 236, 248 switching, 30 treatment, 30, 31 discontinuous, 324, 336, 337, 346, 349, 359 unimodal, 2 window, 267 G Global convergence in finite time, 332, 357 to the homogeneous solution, 3 Global stability, 57, 59, 70, 76, 81, 235, 236, 337, 347 analysis, 240 Globally stable, 346, 357, 359 Graph, 285 complete bipartite, 287 k-regular, 287 H Haar scaling function, 270 Haar wavelet, 270 Harmonic mapping, 315 HCV dynamics, 64 HCV infection, 55, 64 Hepatitis, 29, 55, 56, 91 chronic, 55 virus, 55 Hepatitis B virus (HBV), 29, 51, 57 Hepatitis C virus (HCV), 55, 57 Heterogeneity of cases, 203 Highly active antiretroviral therapy (HAART), 337 HIV infection, 337, 338 dynamics, 293 Hopf bifurcation, 147–149, 152, 154, 155, 161, 164, 165 HTLV infection, 234, 240

368

Index

HTLV-I, 231–233, 235 infection, 232–236 HTLV-I-associated myelopathy/tropical spastic paraparesis (HAM/TSP), 231 Human immune response, 234 Human immunodeficiency virus (HIV), 57, 232, 337 Human population, 98, 99, 103, 286 Human T-lymphotropic virus type I (HTLV-I), 231 Hyperbolic equilibrium point, 59 I Immune response, 1, 64, 92, 234, 237, 250 CTL, 233–236, 240, 247, 249 delay, 235, 249 human, 234 Imported cases, 219 Incidence rate nonlinear, 146 nonmonotonic, 30 Incubation period, 98, 101, 103, 107, 117, 122, 126, 131, 134, 141, 148, 166, 296, 297 rotavirus disease, 169 Infected asymptomatic, 203, 207, 208 cells, 56–58, 63, 64, 293, 338, 340 hepatocytes, 58, 63–65, 89, 91 individuals, 31, 53, 72, 104, 134, 170, 231, 289, 293, 296, 349 people, 117, 137, 139–141, 185, 201, 348 period, 347, 349 person, 98, 104, 117, 126, 286, 288, 295 population, 31, 170, 171, 176, 177, 183, 184, 186, 187, 194, 199, 349 Infection, 122 bacterial, 29 chronic, 56, 235 coefficient, 324 escaping, 123, 125, 135 force, 30, 31, 50, 170 nonmonotonic, 31 HCV, 55, 64 HIV, 337, 338 intercellular, 337

HTLV, 234, 240 HTLV-I, 232–236 liver, 29 process, 169 rate, 91, 116, 119, 124, 133–136, 140, 141, 169, 170, 233, 296, 338, 340, 349 constant, 59 respiratory, 296 rotavirus, 169, 171, 177, 187 secondary, 196, 206 viral, 58, 59, 66, 91, 234 virus, 58 Infectious agent, 98, 103, 116 class, 98, 124, 298 contacts, 294 diseases, 30, 97, 191, 253, 285–287, 323, 337, 338, 347, 348 epidemic outbreaks, 95 mathematical model, 70 SEIR, 338 force, 349 latent period, 347 individuals, 101, 106, 116, 120, 122–126, 135, 297, 339, 349 people, 101, 116, 117, 120, 137 period, 101, 122, 132, 135, 141, 288 person, 116, 126 population, 126, 337 state, 97, 98, 105–108, 125, 297 virus, 337 Infectiousness, 98, 103, 104 Initial parameters, 198 Integral dilation, 268 Fourier, 256 Intercellular HIV infection, 337 Intracellular delay, 235 J Jacobian matrix, 59, 71, 72, 75, 79, 81, 329, 342, 343, 353, 354 Joint state decomposition, 102 K k-regular graph, 287

Index

L Latent period, 231, 234, 348, 349 Leaf, 287 Li and Muldowney’s geometric approach, 62 Li-Muldowney global stability criterion, 59, 89 Linearization, 59 Liver diseases, 55 Liver infection, 29 Local dynamics, 46, 62 Local stability, 59, 70, 343 analysis, 70, 75 Lyapunov function, 60, 77, 78, 235, 236, 248 M Markov chain, 98, 110, 114, 115, 118, 119 SVEIRS, 112 Maximum map, 21, 22 Media awareness, 170, 171 Meyer wavelet, 271 Mild solution, 4 Mild symptoms, 203, 206–209, 219–221 Model analysis, 65, 196, 205 chain-binomial, 97 compartmental, 95, 96, 192, 287, 288, 291, 301 construction, 63 disease, 61, 337 epidemic, 96, 97, 141, 146, 147, 298, 348 DTMC, 96, 97 SIR, 97, 145, 147, 149, 152, 165 stochastic, 96 SVEIRS, 99, 115, 119, 122, 141 formulation, 31, 236 SVEIRS, 98, 99, 116, 119, 124, 129, 135 prototype, 122, 136 with CTL immune response, 234 with delay, 233 with diffusion, 236 with latent HTLV-infected cells, 232 with mitosis, 233 with mitosis and CTL immune response, 235 Modeling, 1, 169–171, 228, 232, 285–288, 292 Morlet wavelet, 272

369

Multifractal energy, 310 paradigm, 306 theory of motion, 306 type, 310–312, 319 N n-star, 287 Natural death rate, 58, 171, 232, 234, 324 Network, 285 COVID-19, 285 Nodes, 285 Nonlinear incidence rate, 146 Nonmonotonic incidence rate, 30 Nonmonotonic infection force, 31 NPIs, 192, 194, 221, 227, 228 Numerical results, 200 Numerical simulation, 3, 18, 21, 22, 25, 31, 57, 89, 164, 183, 236, 246, 249, 346, 359, 361 O One-state-at-a-time decomposition, 101 Optimal control, 170, 178–180, 187, 194, 199, 207 existence, 178 input, 170 problem, 176, 178, 209, 212–214, 219 unconstrained, 181 variables, 178 Optimization, 199 Ordinary differential equations (ODE), 357 Outbreak, 30, 98, 103, 169, 191, 192, 201, 202, 214, 216, 221, 223, 228, 285, 288, 292 disease, 47, 104, 105, 107, 108, 123, 141, 254 period, 98 Overseas exposed cases, 222 P Pandemic, 288, 298 COVID-19, 295 models on COVID-19, 288 Parameter bifurcation, 61 control, 208 initial, 198

370

Index

Partial differential equations (PDE), 236 Period artificial immunity, 97, 103, 127, 135, 141 infected, 347, 349 infectious, 101, 122, 132, 135, 141, 288 latent, 231, 234, 348, 349 outbreak, 98 Periodic solution, 152 Periodicity, 258, 266 Persistent infection, 240 HTLV, 247 steady state, 240, 249 Phase difference, 273 Pontryagin’s maximum principle (PMP), 180 Population dynamics, 2 growth, 2 human, 98, 99, 103, 286 infected, 31, 170, 171, 176, 177, 183, 184, 186, 187, 194, 199, 349 infectious, 126, 337 level, 323 susceptible, 105, 112, 125, 130, 135, 171, 176, 201, 296 SVEIRS, 98, 99, 101, 105–107, 109, 110, 122, 125 vaccinated, 105, 107, 171, 177, 185–187 Problem epidemic, 32 optimal control, 176, 178, 209, 212–214, 219 Proportionality constant, 31 Protease inhibitors (PI), 337 Prototype SVEIRS model, 122, 136 Provirus, 231 Psychological effect, 29, 30, 32, 146, 148, 149, 151, 165 R Rainfall data, 258–260, 263–265, 280 Reaction-diffusion, 1–3, 25 Recovery rate, 31, 32, 171, 204, 220, 289, 297 Recurrent epidemic waves, 145, 146, 148, 165, 166 Reinfection, 228, 297 control, 177, 186

exogenous, 348 process, 177 Respiratory disease, 285 Respiratory infection, 296 Retrovirus, 231 Reverse transcriptase inhibitors (RTI), 337 Ribonucleic acid (RNA), 231, 233 Rota Teq (RV5), 169 Rotarix (RV1), 169 Rotavirus, 169 disease, 169, 185 incubation period, 169 infection, 169, 171, 177, 187 S Saturated type incidence, 30 Schrödinger multifractal type, 309 Schrödinger-type regimes, 309, 313 Secondary infections, 196, 206 SEIR epidemic model, 347 SEIR infectious disease, 338 SEIRD model on COVID-19, 295 Sensitivity, 98, 122, 124, 129, 135, 148, 166, 194, 201, 213, 221, 222, 224, 338 Severe acute respiratory syndrome (SARS), 291 Severe acute syndrome coronavirus (SARS-CoV-2), 286 Simulation, 49, 123 numerical, 3, 18, 21, 22, 25, 31, 57, 89, 164, 183, 236, 246, 249, 346, 359, 361 SIR epidemic model, 97, 145, 147, 149, 152, 165 SIS epidemic model, 337 Social distancing, 192, 213, 215, 216, 219, 227, 286, 288, 289, 295 measures, 191, 209 control, 224 Solution classical, 4 mild, 4 periodic, 152 Spatiotemporal pattern formation, 20 Spectral properties, 254, 258, 272 Spectrum amplitude, 256 wavelet power, 272

Index

Stability, 327 analysis, 33, 44, 170, 207, 234 criteria, 329 equilibria, 174, 324, 341, 352 global, 57, 59, 70, 76, 81, 235, 236, 337, 347 local, 59, 70, 343 Statistical significance, 274 Steady state analysis, 239 persistent infection, 240, 249 Stochastic epidemic models, 96 Subpopulations, 294, 299, 301, 323, 324 Susceptible class, 105, 124 individual contacts, 116, 120 individual meets, 120 people, 122, 129, 130, 132, 133, 135, 137, 138, 140, 141 person, 113, 116–120, 127, 136, 137, 194, 348 population, 105, 107, 112, 125, 130, 135, 171, 176, 201, 296 human, 186, 187 state, 98, 100, 105, 107–109, 112, 113, 115, 124, 126, 128, 297 subpopulations, 324 Susceptible-Exposed-Infected-Recovered (SEIR), 192, 194, 203, 227 Susceptible-Infected-Recovered (SIR), 95, 191, 287 Susceptible-Infected-Susceptible (SIS), 287 Susceptible-Vaccinated-Infected (SVI), 170 SVEIRS disease, 104 dynamics, 104, 110 epidemic, 97, 103 epidemic models, 98, 99, 103, 111, 115, 116, 119, 122, 135, 141 DTMC, 98 Markov chain, 112 model, 98, 99, 116, 119, 124, 129, 135 prototype, 122, 136 population, 98, 99, 101, 105–107, 109, 110, 122, 125 stochastic process, 110 SVIR epidemic model, 96

371

Switching function, 30 Synchronization modes, 309, 311, 313, 318–320 T T-cells, 56 Transmission, 289 dengue, 253, 255, 277, 280 disease, 29, 34, 98, 170, 192, 227, 253, 255, 259, 280, 292, 323, 348 tuberculosis, 293 Treatment antivirus, 58 continuous, 324, 338 discontinuous, 323, 338, 349 function, 30, 31 Tuberculosis (TB) transmission, 293 Typhoid conjugate vaccine (TCV), 95, 104 Typhoid fever, 95, 97, 98, 103, 104, 116, 119, 141 epidemic, 103 U Unconstrained optimal control, 181 Uncorrelated vaccination, 136 Uniform persistence, 78, 80, 88 Unimodal function, 2 Uninfected hepatocytes, 58, 64, 65, 89 person, 286 Universal Immunization Programme (UIP), 187 Unstable equilibrium, 62 V Vaccinated population, 105, 107, 171, 177, 185–187 Vaccination, 122, 141, 169 age, 96 correlated, 116, 124 individual, 171 rate, 31, 98, 125, 130–132, 135, 140, 141 state, 105 strategies, 96 uncorrelated, 136 Vertices, 285

372

Index

end, 287 Viral infections, 58, 59, 66, 91, 234 Virus clearance rate, 338 dengue, 278 dynamics, 57 hepatitis, 55 human immunodeficiency, 57, 232, 337 infection, 58 infectious, 337 natural death rate, 58 particles, 57, 338 population dynamics, 1 production, 92, 338 rate, 91 replication, 64 Virus-immune system interaction, 57, 63

W Wavelet analysis, 266, 275 basic, 269 coherency, 273 Haar, 270 Meyer, 271 Morlet, 272 power spectrum, 272 series, 269 theory, 254, 255, 280 transform, 269 Well-posedness of solutions, 237 Widespread, 169 Window function, 267 Windowed Fourier transform (WFT), 267 World Health Organization (WHO), 95, 169, 186