Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics: Selected works presented at the BIOMAT Consortium Lectures, Morocco 2018 [1st ed. 2019] 978-3-030-23432-4, 978-3-030-23433-1

Table of contents :
Front Matter ....Pages i-xi
Mathematical Modeling of Thrombin Generation and Wave Propagation: From Simple to Complex Models and Backwards (Alexey Tokarev, Nicolas Ratto, Vitaly Volpert)....Pages 1-22
Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-Containing Capsids and Cure Rate (Adil Meskaf, Karam Allali)....Pages 23-33
Dynamics of a Generalized Model for Ebola Virus Disease (Zineb El Rhoubari, Hajar Besbassi, Khalid Hattaf, Noura Yousfi)....Pages 35-46
Bifurcations in a Mathematical Model for Study of the Human Population and Natural Resource Exploitation (I. M. Cholo Camargo, G. Olivar Tost, I. Dikariev)....Pages 47-58
Mathematical Analysis of the Dynamics of HIV Infection with CTL Immune Response and Cure Rate (Sanaa Harroudi, Karam Allali)....Pages 59-70
Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell Population in the Light of Suns and Stars (Y. Elalaoui, L. Alaoui)....Pages 71-85
The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps (Driss Kiouach, Yassine Sabbar)....Pages 87-105
Global Dynamics of a Generalized Chikungunya Virus (Hajar Besbassi, Zineb El Rhoubari, Khalid Hattaf, Noura Yousfi)....Pages 107-117
Differential Game Model for Sustainability Multi-Fishery (Nadia Raissi, Chata Sanogo, Mustapha Serhani)....Pages 119-137
Towards a Thermostatistics of the Evolution of Protein Domains Through the Formation of Families and Clans (Rubem P. Mondaini, Simão C. de Albuquerque Neto)....Pages 139-152
Analysis of Tumor/Effector Cell Dynamics and Decision Support in Therapy (S. Sabir, N. Raissi)....Pages 153-164
Optimal Control of an HIV Infection Model with Logistic Growth, CTL Immune Response and Infected Cells in Eclipse Phase (Jaouad Danane, Karam Allali)....Pages 165-176
Khinchin–Shannon Generalized Inequalities for “Non-additive” Entropy Measures (Rubem P. Mondaini, Simão C. de Albuquerque Neto)....Pages 177-190
Application of the Markov Chains in the Prediction of the Mortality Rates in the Generalized Stochastic Milevsky–Promislov Model (Piotr S̀liwka)....Pages 191-208
Modelling the Role of Vector Transmission of Aphid Bacterial Endosymbionts and the Protection Against Parasitoid Wasps (Sharon Zytynska, Ezio Venturino)....Pages 209-230
The Incorporation of Fractal Kinetics in the PK Modeling of Chemotherapeutic Drugs with Nonlinear Concentration-Time Profiles (Tahmina Akhter, Sivabal Sivaloganathan)....Pages 231-254
Mathematical Modeling of Inflammatory Processes (O. Kafi, A. Sequeira)....Pages 255-269
Modeling the Memory and Adaptive Immunity in Viral Infection (Adnane Boukhouima, Khalid Hattaf, Noura Yousfi)....Pages 271-297
Optimal Temporary Vaccination Strategies for Epidemic Outbreaks (K. Muqbel, A. Dénes, G. Röst)....Pages 299-307
On the Reproduction Number of Epidemics with Sub-exponential Growth (D. Champredon, Seyed M. Moghadas)....Pages 309-320
Numerical Simulations for Cardiac Electrophysiology Problems (Alexey Y. Chernyshenko, A. A. Danilov, Y. V. Vassilevski)....Pages 321-329
Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory Optimization Hypothesis (Abdoulaye Mendy, Mountaga Lam, Jean Jules Tewa)....Pages 331-358
A Fractional Order Model for HBV Infection with Capsids and Cure Rate (Moussa Bachraou, Khalid Hattaf, Noura Yousfi)....Pages 359-371
Modeling Anaerobic Digestion Using Stochastic Approaches (Oussama Hadj Abdelkader, A. Hadj Abdelkader)....Pages 373-396
Alzheimer Disease: Convergence Result from a Discrete Model Towards a Continuous One (M. Caléro, I. S. Ciuperca, L. Pujo-Menjouet, L. M. Tine)....Pages 397-431
Back Matter ....Pages 433-437

Citation preview

Rubem P. Mondaini Editor

Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics Selected works presented at the BIOMAT Consortium Lectures, Morocco 2018

Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics

Rubem P. Mondaini Editor

Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics Selected works presented at the BIOMAT Consortium Lectures, Morocco 2018

123

Editor Rubem P. Mondaini President, BIOMAT Consortium – International Institute for Interdisciplinary Sciences Rio de Janeiro, Brazil Federal University of Rio de Janeiro Rio de Janeiro, Brazil

ISBN 978-3-030-23432-4 ISBN 978-3-030-23433-1 (eBook) https://doi.org/10.1007/978-3-030-23433-1 Mathematics Subject Classification (2010): 92B05, 97M10, 97M60, 49J20, 93A30 © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The present book is a collection of papers which have been accepted for publication after a peer-review evaluation by the Editorial Board of the BIOMAT Consortium (http://www.biomat.org) and international referees ad hoc. These papers have been presented on technical sessions of the BIOMAT 2018 International Symposium, the 18th symposium of the BIOMAT series which was held at the Faculty of Sciences and Technology, University Hassan II, Mohammedia, Morocco, from 29 October to 2 November 2018. On behalf of the BIOMAT Consortium, we thank the members of the BIOMAT 2018 Local Organizing Committee, Karam Allali (chair), Khalid Hattaf, Ahmed Taik, Noura Yousfi, Noureddine Moussaid, Saida Amine, Mustapha Kabil, Chakib Abchir, and Driss Karim, for their professional expertise at following the guidelines and fine tradition of the BIOMAT Consortium and preserving the excellency of the BIOMAT Symposium series on this first BIOMAT Conference in Africa. We are so much indebted to all these colleagues as well as to research collaborators and Ph.D. students of the faculty for their invaluable help since the opening session on Monday morning to the closing session on Friday evening. The financial support in terms of lunches, coffee breaks for all the participants, and accommodation for the invited keynote speakers have been provided by the Faculty of Sciences and Technology. The BIOMAT Consortium has succeeded once more in its fundamental mission of enhancing the interdisciplinary scientific activities of mathematical and biological sciences on developing countries with the organization of the BIOMAT 2018 International Symposium. The authors of papers from Western and Eastern Europe, Africa, and South America had the usual opportunity of exchanging scientific feedback of their research fields with their colleagues from Morocco and other 16 countries: Algeria, Brazil, Cameroon, Canada, Chad, Colombia, France, Germany, Hungary, Italy, Mali, Nigeria, Poland, Portugal, Russia, and Senegal. The editor of the book and president of the BIOMAT Consortium is very glad for the continuous collaboration and critical support of his wife, Carmem Lucia, on the editorial work, from the reception of submitted papers for the peer-review procedure of BIOMAT Consortium Editorial Board to the ultimate publication of the

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scientific programme. He also thanks his former research student Dr. Simão C. de Albuquerque Neto from Federal University of Rio de Janeiro for his computational skills and enlightening discussions during the preparation of our presentations to the BIOMAT 2018. Mohammedia, Morocco November 2018

Rubem P. Mondaini

Editorial Board of the BIOMAT Consortium

Rubem Mondaini (Chair) Adelia Sequeira Alain Goriely Alan Perelson Alexander Grosberg Alexei Finkelstein Ana Georgina Flesia Anna Tramontano Avner Friedman Carlos Condat Charles Pearce Denise Kirschner David Landau De Witt Sumners Ding-Zhu Du Dorothy Wallace Eduardo Massad Eytan Domany Ezio Venturino Fernando Cordova-Lepe Fernando R. Momo Fred Brauer Frederick Cummings Gergely Röst Guy Perriére Gustavo Sibona Helen Byrne Jacek Miekisz Jack Tuszynski Jaime Mena-Lorca Jane Heffernan

Federal University of Rio de Janeiro, Brazil Instituto Superior Técnico, Lisbon, Portugal University of Arizona, USA Los Alamos National Laboratory, New Mexico, USA New York University, USA Institute of Protein Research, Russia Universidad Nacional de Cordoba, Argentina University of Rome, La Sapienza, Italy Ohio State University, USA Universidad Nacional de Cordoba, Argentina University of Adelaide, Australia University of Michigan, USA University of Georgia, USA Florida State University, USA University of Texas, Dallas, USA Dartmouth College, USA Faculty of Medicine, University of S. Paulo, Brazil Weizmann Institute of Science, Israel University of Torino, Italy Catholic University del Maule, Chile Universidad Nacional de Gen. Sarmiento, Argentina University of British Columbia, Vancouver, Canada University of California, Riverside, USA University of Szeged, Hungary Université Claude Bernard, Lyon, France Universidad Nacional de Cordoba, Argentina University of Nottingham, UK University of Warsaw, Poland University of Alberta, Canada Pontifical Catholic University of Valparaíso, Chile York University, Canada vii

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Jean Marc Victor Jerzy Tiuryn John Harte John Jungck José Fontanari Karam Allali Kazeem Okosun Kristin Swanson Kerson Huang Lisa Sattenspiel Louis Gross Ludek Berec Michael Meyer-Hermann Nicholas Britton Panos Pardalos Peter Stadler Pedro Gajardo Philip Maini Pierre Baldi Rafael Barrio Ramit Mehr Raymond Mejía Rebecca Tyson Reidun Twarock Richard Kerner Ryszard Rudnicki Robijn Bruinsma Rui Dilão Sandip Banerjee Seyed Moghadas Siv Sivaloganathan Somdatta Sinha Suzanne Lenhart Vitaly Volpert William Taylor Yuri Vassilevski Zhijun Wu

Editorial Board of the BIOMAT Consortium

Université Pierre et Marie Curie, Paris, France University of Warsaw, Poland University of California, Berkeley, USA University of Delaware, Delaware, USA University of Sao ˜ Paulo, Brazil University Hassan II, Mohammedia, Morocco Vaal University of Technology, South Africa University of Washington, USA Massachusetts Institute of Technology, MIT, USA University of Missouri-Columbia, USA University of Tennessee, USA Biology Centre, ASCR, Czech Republic Frankfurt Inst. for Adv. Studies, Germany University of Bath, UK University of Florida, Gainesville, USA University of Leipzig, Germany Federico Santa Maria Technical University, Valparaíso, Chile University of Oxford, UK University of California, Irvine, USA Universidad Autonoma de Mexico, Mexico Bar-Ilan University, Ramat-Gan, Israel National Institutes of Health, USA University of British Columbia, Okanagan, Canada University of York, UK Université Pierre et Marie Curie, Paris, France Polish Academy of Sciences, Warsaw, Poland University of California, Los Angeles, USA Instituto Superior Técnico, Lisbon, Portugal Indian Institute of Technology Roorkee, India York University, Canada Centre for Mathematical Medicine, Fields Institute, Canada Indian Institute of Science Education and Research, India University of Tennessee, USA Université de Lyon 1, France National Institute for Medical Research, UK Institute of Numerical Mathematics, RAS, Russia Iowa State University, USA

Contents

Mathematical Modeling of Thrombin Generation and Wave Propagation: From Simple to Complex Models and Backwards . . . . . . . . . . . Alexey Tokarev, Nicolas Ratto, and Vitaly Volpert

1

Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-Containing Capsids and Cure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Adil Meskaf and Karam Allali

23

Dynamics of a Generalized Model for Ebola Virus Disease . . . . . . . . . . . . . . . . . Zineb El Rhoubari, Hajar Besbassi, Khalid Hattaf, and Noura Yousfi

35

Bifurcations in a Mathematical Model for Study of the Human Population and Natural Resource Exploitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I. M. Cholo Camargo, G. Olivar Tost and I. Dikariev

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Mathematical Analysis of the Dynamics of HIV Infection with CTL Immune Response and Cure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sanaa Harroudi and Karam Allali

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Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell Population in the Light of Suns and Stars . . . . . . . . . . . . . . . Y. Elalaoui and L. Alaoui

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The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Driss Kiouach and Yassine Sabbar

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Global Dynamics of a Generalized Chikungunya Virus . . . . . . . . . . . . . . . . . . . . . 107 Hajar Besbassi, Zineb El Rhoubari, Khalid Hattaf, and Noura Yousfi Differential Game Model for Sustainability Multi-Fishery. . . . . . . . . . . . . . . . . . 119 Nadia Raissi, Chata Sanogo, and Mustapha Serhani

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Contents

Towards a Thermostatistics of the Evolution of Protein Domains Through the Formation of Families and Clans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 Rubem P. Mondaini and Simão C. de Albuquerque Neto Analysis of Tumor/Effector Cell Dynamics and Decision Support in Therapy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 S. Sabir and N. Raissi Optimal Control of an HIV Infection Model with Logistic Growth, CTL Immune Response and Infected Cells in Eclipse Phase . . . . . . . . . . . . . . . 165 Jaouad Danane and Karam Allali Khinchin–Shannon Generalized Inequalities for “Non-additive” Entropy Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 Rubem P. Mondaini and Simão C. de Albuquerque Neto Application of the Markov Chains in the Prediction of the Mortality Rates in the Generalized Stochastic Milevsky–Promislov Model . . . . . . . . . . . 191 ` Piotr Sliwka Modelling the Role of Vector Transmission of Aphid Bacterial Endosymbionts and the Protection Against Parasitoid Wasps . . . . . . . . . . . . . . 209 Sharon Zytynska and Ezio Venturino The Incorporation of Fractal Kinetics in the PK Modeling of Chemotherapeutic Drugs with Nonlinear Concentration-Time Profiles . . . 231 Tahmina Akhter and Sivabal Sivaloganathan Mathematical Modeling of Inflammatory Processes . . . . . . . . . . . . . . . . . . . . . . . . . 255 O. Kafi and A. Sequeira Modeling the Memory and Adaptive Immunity in Viral Infection . . . . . . . . . 271 Adnane Boukhouima, Khalid Hattaf, and Noura Yousfi Optimal Temporary Vaccination Strategies for Epidemic Outbreaks . . . . . 299 K. Muqbel, A. Dénes, and G. Röst On the Reproduction Number of Epidemics with Sub-exponential Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 D. Champredon and Seyed M. Moghadas Numerical Simulations for Cardiac Electrophysiology Problems . . . . . . . . . . 321 Alexey Y. Chernyshenko, A. A. Danilov, and Y. V. Vassilevski Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory Optimization Hypothesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331 Abdoulaye Mendy, Mountaga Lam, and Jean Jules Tewa A Fractional Order Model for HBV Infection with Capsids and Cure Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359 Moussa Bachraou, Khalid Hattaf, and Noura Yousfi

Contents

xi

Modeling Anaerobic Digestion Using Stochastic Approaches . . . . . . . . . . . . . . . 373 Oussama Hadj Abdelkader and A. Hadj Abdelkader Alzheimer Disease: Convergence Result from a Discrete Model Towards a Continuous One . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 M. Caléro, I. S. Ciuperca, L. Pujo-Menjouet, and L. M. Tine Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433

Mathematical Modeling of Thrombin Generation and Wave Propagation: From Simple to Complex Models and Backwards Alexey Tokarev, Nicolas Ratto, and Vitaly Volpert

1 Introduction Blood coagulation is one of the most important defense systems of the organism. This system controls the formation of a polymeric fibrin clot at the sites of vessel wall damage, thus protecting us from blood loss. From the medical point of view, the disruptions of coagulation system can lead to many dangerous states—from bleedings (insufficient hemostatic reaction) to thromboses (redundant hemostatic reaction). Thromboses and bleedings are the leading cause of death in many diseases and conditions such as atherosclerosis, myocardial infarction, stroke, sepsis, cancer, snakebites, frostbites, burns, surgery, as well as hemophilias and other inherited and acquired diseases. That dictates the need in understanding of how this system can be diagnosed and influenced in medical care.

The author “A. Tokarev” was working in the institute “Dmitry Rogachev” at the time of presentation of this work in a session of the BIOMAT 2018 in Morocco. A. Tokarev Peoples’ Friendship University of Russia (RUDN University), Moscow, Russian Federation Dmitry Rogachev National Research Center of Pediatric Hematology, Oncology and Immunology, Moscow, Russian Federation N. Ratto · V. Volpert () Peoples’ Friendship University of Russia (RUDN University), Moscow, Russian Federation Institute Camille Jordan, UMR 5208 CNRS, University Lyon 1, Villeurbanne, France INRIA, Universit de Lyon, Universit Lyon 1, Institute Camille Jordan, Villeurbanne, France e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_1

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Extensive biochemical, biophysical, and biomedical research is undertaken in this direction during last decades. Any progress on this way needs in-depth knowledge of biochemistry of coagulation and neighbor systems, which are really extremely complex and still have reactions to be discovered and mechanisms to be investigated. Functioning of coagulation on the interface of flowing blood and vessel wall (or inside the growing platelet thrombus) needs application of modern and developing new biophysical approaches. And necessity to understand and consider the patient specificity of coagulation, and final task to push all results into medical practice make all this work biomedical. Thrombin generation is the central process of coagulation as thrombin is a central enzyme of this system (see the Sect. 2 below). Thus, the terms “thrombin generation,” “coagulation,” and “clotting” are often used as synonyms. However, one of the most informative tests of coagulation system in which thrombin concentration is monitored in time in homogeneous conditions is also called thrombin generation [1, 2]. Another approach is to monitor thrombin and fibrin formation in an essentially non-homogeneous, spatially distributed, non-stirred conditions, referred as trombodynamics [3–5]. Finally, a number of experimental assays exists to study coagulation (by monitoring the fibrin or/and thrombin formation) in blood or plasma flow conditions [6, 7]. Needless to say that developing and using of these and all other experimental methods of studying coagulation demand corresponding mathematical models. Mathematical modeling is a well-known and indispensable tool to understand complex biophysical and biological phenomena. Models of various detalization and even completely new computational approaches were developed in this area. Advances in modeling the coagulation alone have been recently reviewed in [8–13], and complex multiscale models of thrombosis which include integrated models of coagulation, platelet aggregation, and blood flow have been reviewed in [14–17]. The main task of the present survey is to show how coagulation models having different background, complexity, and properties can be linked together into one general picture. In most cases, the reaction-diffusion-convection equations are used to describe time- and space-varying concentrations of coagulation substances—zymogens, activated factors, inhibitors, and their complexes: ∂c + (v.∇)c = R(c, k) − ∇(−D∇c) ∂t

(1)

with corresponding initial and boundary conditions. Here, vector notation is used for concentrations, c; rates of chemical reactions, R(c, k); kinetic constants, k; diffusion coefficients, D; and fluid flow velocity field, v; the latter should be specified analytically or be the solution of fluid-phase (for example, Navier–Stokes) equations. In non-stirred conditions v = 0, and Eq. (1) turns into the reactiondiffusion problem. In the fully stirred (i.e., homogeneous) conditions, the diffusion terms vanish, too, so the system (Eq. (1)) turns into ODE system. The level of model detalization determines the dimension and components of the vector c (number

Mathematical Modeling of Thrombin Generation and Wave Propagation: From. . .

3

and list of substances) and the actual form of kinetic functions Ri . The latter are based either on the knowledge/assumptions of detailed mechanisms of biochemical reactions and measured/estimated value of kinetic parameters (in detailed models and in reduced models obtained from detailed ones), on the general reaction scheme and some fitted values of kinetic parameters (in simulation models), or may even be in some unusual, hypothetical form if some system property is proposed and being tested (in phenomenological models).

2 Biochemical Structure of the Coagulation System Coagulation system (see Fig. 1) is a cascade of proteolytic enzymatic reactions with each level consisting of two processes: zymogen (coagulation factor, F) activation to the active enzyme (activated coagulation factor, Fa) followed by its rapid irreversible trapping by inhibitors always circulating in blood. On each level, the short-living coagulation factor catalyzes the reaction of activation on the next cascade level. The final product of coagulation cascade is fibrin (Fn) which rapidly polymerizes into

Fig. 1 General scheme of the coagulation cascade [18]

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3-dimensional mesh (gel); Fn mesh in vivo slows blood flow and glues aggregating platelets and other blood cells into the clot. There are two pathways of coagulation activation [19, 20]. Activation by the extrinsic pathway begins when blood comes in contact with tissue factor (TF): this transmembrane protein is expressed by the majority of cells except those normally being in contact with blood. TF binds with FVIIa, which circulates in tiny amounts (1% of total FVII), making FVIIa able to cleave FIX and FX to their activated forms. Intrinsic activation pathway [21] is initiated by the contact of blood with any “foreign” surface. Upon adsorption on this surface, FXII becomes activated due to conformational changes and then stimulates its own formation both autocatalytically and by activating prekallikrein to kallikrein: kallikrein activates its cofactor high molecular weight kininogen and FXII. Generated FXIIa activates FXI, FXIa activates FIX, and FIXa activates FX. Both pathways unite at the activation of FX to FXa. FXa cleaves prothrombin (FII) to thrombin (FIIa)—the central coagulation enzyme. In addition to cleavage of fibrinogen (Fg) to fibrin (Fn), thrombin controls at least three positive feedback loops activating FV, FVIII, and FXI which are located above in the cascade. Two of these loops lead to the activation of cofactors FVa and FVIIIa which bind with FXa and FIXa, respectively, forming prothrombinase and intrinsic tenase complexes having activities 104 –105 times larger than free enzymes have. Therefore, upon the initial activation by any pathway, local thrombin concentration increases in a dramatically non-linear manner leading to full Fg conversion to Fn. Thrombin also controls negative feedback loop of protein C (pC) activation. This reaction needs thrombin reversible binding to thrombomodulin (Tm) expressed by intact endothelium cells. Formed pCa is an inhibitor of FVa (and, possibly, of FVIIIa, which is unstable by itself). And yet another negative feedback of coagulation is extrinsic tenase (TF-VIIa) complex inhibition by tissue factor pathway inhibitor (TFPI) which depends on the TFPI-Xa binding into the final TFPI-Xa-VIIa-TF complex [22].

3 Phenomenological Modeling of Blood Coagulation: Understanding the General Principles 3.1 Cascade Backbone with Inhibition Control on Every Step: Triggering General agreement on Roman nomenclature of blood coagulation factors [23] and understanding the architecture of coagulation system as a cascade of proenzymeenzyme transformations [24] opened the era of mathematical modeling of blood coagulation. First considerations [24, 25] attempted to understand coagulation as a biochemical amplifier, as it was noted the increase of quantity of enzymes produced on each consequent stage of coagulation cascade. Thus, Levine [25] obtained the

Mathematical Modeling of Thrombin Generation and Wave Propagation: From. . .

S1

E1

E2 E3

S2

E4

S3

E2

S1

5

I2 E3

S2 S3

etc.

I3 E4

I4

etc.

Fig. 2 Schemes of the ‘open’ and ‘damped’ cascades which explain burst-like and trigger-like system response to any activation [26]

analytical solution for an amplifier gain after a limited stimulus time as a function of parameters (kinetic constants and zymogens’ initial concentrations) and pointed out to its sensitivity to perturbations, i.e. to the significant overall effect which can be caused by even small but simultaneous shifts in the rate constants and initial conditions. Hemker [26] revised this concept and argued that the cascade should be viewed as a trigger from zero (no clotting) to upper (full clotting) state. In particular, in the model of an “open” (without inhibition, see Fig. 2) n-stages enzyme cascade the solution for the nth stage had the form Pn (t) ∼ t n /n! (actual formula depends on particular biochemical mechanisms, etc.). This means that increasing n makes P (t) dependence more steep, with more evident lag-time, after which the burst of product formation happens. In the model of a “damped” cascade (all intermediate enzymes are rapidly inhibited, see Fig. 2) this burst is limited, and the equation describing the transition rate and upper level of Pn was also derived [25, 26]. Moro and Martorana [27] extended this approach by including “negative feedbacks” of thrombin inhibition due to thrombin adsorption on generated fibrin strands and by products of fibrin degradation. They obtained the explicit solution for thrombin, fibrin, and products of fibrin degradation, and concluded that the latter cannot influence the stability of the initial state.

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3.2 Positive Feedback Loops: Thresholds Linearity of the above-mentioned models suggests they have single—clotted— steady state, and initial (non-clotted) state is unstable. In fact, blood is liquid, which means that non-clotted state is also stable, and thus coagulation is a threshold process. This issue was addressed by Semenov and Khanin who considered two nonlinear qualitative models of coagulation cascade capable of explaining its threshold properties [28, 29]. The first model [29] was based on the scheme of extrinsic pathway of coagulation and included one positive feedback loop of FV activation by thrombin (see Fig. 3). This scheme resulted in non-linearity in the equation for thrombin: ⎧ d[VIIa] ⎪ = αK1 − H1 [VIIa], ⎪ dt ⎪ ⎪ ⎪ ⎨ d[Xa] = K2 [VIIa] − H2 [Xa], dt (2) d[Va] ⎪ = K3 [IIa] − H3 [Va], ⎪ dt ⎪ ⎪ ⎪ ⎩ d[IIa] = K [Xa] [Va] − H [IIa]. 4 4 dt Ka +[Va]

Extrinsic stimulation

1

VII

VIIs

2

X

Xs Vs

4

II

IIs

5

I

Is

V

3

Fig. 3 Scheme having threshold properties due to accounting of the positive feedback loops [29]

Mathematical Modeling of Thrombin Generation and Wave Propagation: From. . .

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Here, α is a stimulation intensity (triggering signal) assumed to be constant as only initial time was considered. This system was reduced using the Tikhonov’s theorem, yielding two equations for concentration of FVa and FIIa:  dX dτ dY dτ

= Y − X,

(3)

X = α 1+X − bY.

H4 [IIa] Here, X = [Va] Ka ; Y = K3 H3 Ka ; τ = H3 t; b = H3 . Phase-plane analysis of this system shows that, depending on α, it can have one or two steady states in the (H1 H2 H3 H4 ) area of positive concentrations (X, Y ≥ 0). At α < αthres = Ka (K , the 1 K2 K3 K4 ) only steady state is zero, and it is a stable node; after any disturbance, the system returns to this state: blood remains liquid. At α > αthres , there are two steady states, unstable zero (saddle) and stable upper (node), and any disturbance leads to the transition of the system from zero to the upper state: blood clots. Extending their analysis, Semenov and Khanin considered the scheme of intrinsic coagulation pathway including two positive feedback loops—FVIII and FV activation by thrombin [28]:

⎧ dx1 ⎪ dt = k1 α − K1 x1 , ⎪ ⎪ ⎪ dx ⎪ 2 ⎪ ⎪ dt = k2 x1 − K2 x2 , ⎪ ⎪ ⎪ 3 ⎪ ⎪ dx = k3 x2 − K3 x3 , ⎪ ⎨ dt dx4 dt = k4 x7 − K4 x4 , ⎪ ⎪ dx ⎪ ⎪ 5 = k5 x3 x4 − K5 x5 , ⎪ dt Ka +x4 ⎪ ⎪ ⎪ dx6 ⎪ ⎪ = k x 6 7 − K6 x6 , ⎪ dt ⎪ ⎪ ⎩ dx7 x6 dt = k7 x5 K"a +x6 − K7 x7 ,

XIIa, XIa IXa VIIIa

(4)

Xa Va IIa

Here, xi is the concentration of an active factor formed on the ith cascade stage (specified opposite to the equation), and again α is a stimulation intensity. Using the Tikhonov’s theorem, this system was reduced to two equations for concentration of FXa and FIIa:  y dx dτ = 1+y − x, (5) dy y dτ = ax b+y − cy, x5 K1 K2 K3 K5 x7 K4 α k1 k2 k3 k4 k5 k7 α k1 k2 k3 k5 ; y = Ka k4 ; τ = K5 t; a = Ka K1 K2 K3 K4 Ks2 ; b = 7 c= K K5 . Again, phase portrait of the system depends on α.    Ka K6 k4 1/2 2 K3 K4 K5 Ka K7 1 + , only one—zero— < αthres = K1kK  k7 K k6 K4 1 k2 k3 k4 k5

where x = Ka K6 k4 Ka k6 K4 ;

At α

a

stable point exists. At α > αthres , there are three steady states: two stable nodes separated by the saddle and its separatrix. In order for the system to move to an

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upper stable steady state, the initial conditions must be in the region of attraction of this state, which is separated from the region of attraction of the zero steady state by a separatrix. As the activating intensity increases, the saddle approaches zero steady state, and the conditions for the coagulation activation become alleviated. In reality, tiny amounts of all active factors always circulate in blood [30–34]— coagulation “idles,” so that slow steady conversion of fibrinogen into fibrin occurs. Therefore, the bottom stable stationary point at which the unactivated system is located should be slightly shifted from the zero position. This can be achieved by accounting for the reactions of activation of prothrombin and factor X by the free factors Xa and IXa, respectively [35]. The position of this point on the phase plane is determined by the level of the activation signal. Upon sufficiently large stimulation intensity, due to the instantaneous displacement of the isoclines, the system located in the bottom stable steady state turns into the region of attraction of the upper steady state, and blood coagulates. Jesty and Beltrami [36, 37] studied the threshold properties of four modeling enzymatic systems progressively approaching the structure of the coagulation cascade, from 1-enzyme autocatalytic formation to 4-enzyme system with three feedback loops (representing FV, VIII, and XI activation by thrombin). First, they obtained analytical conditions for steady states lose their stability, particularly depending on the feedback loops activities. Then, they extended their analysis using direct numerical simulations. Their systems have many common properties with the actual coagulation system: the threshold behavior, the dependence of the lagperiod duration on the concentration of zymogens, the weak dependence of the system behavior on the activator level (if the latter exceeds the threshold); longrange feedback loop (reaction of FXI activation by thrombin) slightly reduces the activation threshold and slightly enhances the response. In line with this approach, later they supposed that major role of inhibitors of coagulation (native and clinically used) is to control the stability of the bottom “idling” steady state of coagulation system rather than to gradually reduce its response to activation [32].

3.3 Long-Range Feedback in Spatial Case: Blood as an Active Media Understanding the threshold and bistable properties of coagulation system and considering fibrin clot formation as a principally spatial task led Ataullakhanov and Guria to the hypothesis that coagulation should be some new example of autowave systems: thrombin should be able to autocatalytically sustain the spreading of its own formation in space, i.e. in blood plasma [38]. Biochemical background of this hypothesis were (raw at that time) data on the existence of the reaction of FXI activation by thrombin—long-range feedback, looping the coagulation cascade from the very bottom to the very top (see Fig. 1). In the same article, Ataullakhanov and Guria hypothesized that spreading of thrombin autowave is limited by the

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second autowave, which is the wave of protein C activation (see Fig. 1); the latter hypothesis wasn’t supported in future studies. Nevertheless, the first hypothesis [38] and corresponding phenomenological 2-autowave mathematical model [39]: ⎧ ⎪ ⎨ ∂u ∂t = ⎪ ⎩

∂v ∂t

K1 u2 u+K2

− K3 uv + Du,   2  − K8 uv + Dv, 1 + Kv7 = K5 u 1 − Kv6

(6)

started extensive studies of spatial aspects of blood coagulation. Besides influence on the experimental studies, they stimulated the appearance of a long list of mathematical models of various kind and complexity: detailed and reduced, homogeneous and spatial, which we review below in the Sects. 4 and 5 together with the insights obtained with their help.

4 Detailed Modeling of Blood Coagulation: Solving the Biomedical Questions 4.1 Simulating the Thrombin Generation: Many-Parameter Fitting Is Useless and Is Not Really Necessary Extensive search for an adequate coagulation test that can reflect thrombotic/hemorrhagic shifts of the coagulation system has led prof. Hemker’s group to the development of thrombin generation test (TGT), in which the non-linear, explosive kinetics of thrombin formation in plasma followed by thrombin decline due to depletion of prothrombin and inhibition of all coagulation factors—thrombin generation (TG) curve—is detected [1, 2]. This research posed a question of how TG can be simulated mathematically. Such “simulation” model was proposed in 1991 in the same group [40]. Broadly speaking, this was the first quantitative blood coagulation model, as it was based on real (in general) reaction scheme, used experimentally measured kinetic constants (Michaelis–Menten mechanism for all enzymatic reactions was supposed), and was verified by comparison with experimental kinetic curves for three basic coagulation factors. However, model trigger was FXa time-decaying influx and experimental trigger was TF; kinetic constants used were measured in purified system and experiments were done in plasma. That is why this model was indeed simulation rather than detailed. Model study showed that coagulation system has a “threshold” for total triggering FXa concentration, i.e. the prothrombinase activity is central for coagulation to be complete, and the effects of other reactions on this “threshold” were studied. Later analysis resulted in the conclusion that uncertainty in real reaction constants and reaction mechanisms (and also the possibility of shifting to diffusional control in the result of fibrin jellification) make modeling of “simple” TG curve with “complex” mathematical model impractical or even non-scientific [41]. This results from the

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A. Tokarev et al. Factor V 3. Km, kcat

Factor X 1. Km, kcat Factor Vlla (trigger)

Factor Xa + Factor Va

Kd Prothrombinase

2. Km, kcat

Prothrombin

4. Km, kcat Thrombin + Anthithrombin 5. kdec

TAT

Fig. 4 ‘Wagenvoord model’—a minimal, ‘mock’ model which parameters can be fitted to describe any thrombin generation curve [41]

fact that any experimental TG curve is 4-parametric (may be fitted by 4-parametric function, or the sum of two such functions in PRP case), while even the simplest possible coagulation model is based on six reactions and have 14 free parameters, so multiple sets of parameters exist that result in perfect agreement with experiment, even in the case of simulating the influence of substance which model actually lacks. This simple 6-reaction model [41] is sometimes called the Wagenvoord model (Fig. 4). The authors underline that Wagenvoord model is a “mock-model,” i.e. it is improbable simple and shouldn’t ever be used as a realistic predictor of thrombin generation [42]. They stress that for such complex system as coagulation even successful simulation of experimental data does not validate the underlying model assumptions, and propose the strategy of real successful use of simulation techniques [11, 41]. To study the threshold and stability properties of coagulation intrinsic pathway, prof. Ataullakhanov’s group in 1995 built similar simulation mathematical model [43, 44]. They suggested constant [FXIa] as a trigger to mimic the intrinsic pathway and S-shaped [Ca2+]-dependence of tenase and prothrombinase complexes formation rates to describe the threshold-like coagulation response to [Ca2+] [45]; they varied some unknown constants to fit the experimental AMC kinetics with the model (AMC is a fluorogenic substance produced by thrombin from the specially added synthetic substrate), and ended with two sets of best-fit model parameters and corresponded threshold [FXIa] values. Later this model underwent at least two variants of reduction considered below in Sect. 5.1. To study the processes of coagulation in standard laboratory clotting tests— prothrombin time, PT, and activated partial thromboplastin time, APPT, Khanin

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and coworkers constructed qualitative models describing coagulation triggering by extrinsic and intrinsic pathways, respectively, in these tests [46, 47]. Models’ outputs were PT and APPT, and no kinetic parameter fitting was made. The effect of coagulation factors’ deficiencies on clotting times was studied; low sensitivity to moderate deficiencies was demonstrated, and the deficiencies which can be sensed by these tests were determined. All these examples show that even non-fitted models as well as models having several sets of chosen parameters can be helpful in studying robust systems.

4.2 Homogeneous Detailed Model of Reconstituted Coagulation Can be Modified and Used by Others Instead of the native plasma, prof. Mann’s group in 1994 studied the kinetics of coagulation in a reconstituted system prepared by controlled mixing of purified coagulation factors and phospholipid vesicles [48]; to describe these experiments, they constructed the mathematical model by explicitly accounting for all involved enzyme complexes (instead of proposing the Michaelis–Menten kinetics), and obtained good agreement with experimental kinetics [49]. They concluded that such model “assembly” from equations quantitatively describing each separate coagulation reaction gives a new useful research tool. Soon after that, this model was modified in DuPont Merck group to evaluate the effects of exogenous inhibitors of thrombin and other coagulation proteases for the antithrombotic therapy [50]. Later, Hockin and Mann included other sets of reactions into their model: TFPI pathway, ATIII action, VII/VIIa competition for TF, feedback FVII activation by FIXa, FXa and thrombin, etc., although not FXI activation [51]. The resulting model seems to be the most popular model of coagulation to date. This model was used to estimate the effect of “idling” coagulation enzymes in circulating blood in the TFindependent coagulation [34], to study the independence of advanced (following the lag-period) stages of TF-triggered coagulation on the TF activity [52], to study the impact of uncertainty of, i.e., model sensitivity to, the values of rate constants of coagulation reactions [53], to study the consequences of variations of coagulation zymogens’ concentrations in normal and pathological ranges [54], to differentiate the coagulation phenotypes in familial protein C deficiency [55], to investigate the very complex pathway of FVa inhibition by pCa [56], etc. Also, Mitrophanov’s group in the US Army Medical Research Command modified this model to estimate the effects of hypothermia [57] and diluted plasma supplementation with promising concentrates of coagulation factors [58] on thrombin generation.

4.3 Spatially Distributed Models: Blood Is an Active Media The first detailed model aimed to investigate the spatio-temporal dynamics of blood coagulation was proposed in prof. Ataullakhanov’s group in 1996 [59, 60]. The

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intrinsic pathway triggered by constant FXIa influx (from one of the boundaries) was considered. Similar to [43], the S-shaped [Ca2+]-dependence of tenase and prothrombinase complexes formation rate was used, and some unknown model constants were fitted. The model had three general properties: (1) solution in the form of a running thrombin peak, (2) the threshold response to FXIa influx, and (3) the ability of the long-range feedback loop of FXI activation by thrombin to sustain the autowave spatial propagation of thrombin peak. Later, this model was used in two different ways. First, to provide some biochemical ground to the previously proposed two-autowave phenomenological model [39] (see the Sect. 3.3 above), it was supplemented with hypothetical reaction block of thrombin switching from procoagulant into anticoagulant state, and severely reduced to three equations forming another 2-autowave model (considered in the Sect. 5.2 below). Second, it was supplemented with a more accurate description of the intrinsic pathway and with the extrinsic tenase influx to simulate the extrinsic pathway, and resulting model was intensively studied in both homogeneous and spatially distributed conditions [61]. This line of research was continued by Panteleev, who constructed the new spatio-temporal model of TF-triggered coagulation [62]. Instead of gathering a huge amount of reactions at once, this model was constructed in several sequential steps. First, some blocks were simulated separately (TFPI/extrinsic tenase; intrinsic tenase). Second, they were combined together, and other reaction routes were added one-by-one with simulations of the corresponding experiments. Model assumptions included: prothrombinase reactions proceed in analogy to intrinsic tenase ones; lipid-dependent reactions—by analogy to platelet-dependent ones; platelet activation by thrombin happen in one single step. This model complemented the experimental data of Ovanesov who graduated from the same laboratory and studied spatial dynamics of coagulation in details. Together, model and experiments showed that (1) spatial propagation of coagulation is regulated by the intrinsic tenase and, for example, hemophilias are diseases of impaired clot growth (with normal clot growth initiation by TF); (2) clot propagation is regulated by protein C pathway, and Tm action can terminate the clot growth. Panteleev’s model (modified accordingly to experimental conditions) was used in studies of other important questions of spatio-temporal dynamics of blood coagulation: regulation of coagulation initiation and propagation by the activity of FIX (in hemophilia B and its treatment) [18]; regulation of thrombin autowave propagation and termination (in the presence of added phospholipids) by the feedback loops of FXI, FVIII, and pC activation by thrombin [3]; regulation of coagulation sensitivity to shear flow by the feedback loop of TF-VII activation by FXa [63]; regulation of coagulation sensitivity to surface TF distribution pattern by the feedback loops of TF-VII activation by FXa and FV activation by thrombin [64]; regulation of coagulation threshold by fibrin polymerization and coagulation triggering by FV activation by thrombin [65]. The latter study concluded that coagulation network has several distinct tasks/properties: threshold (ability to not clot at low activation), triggering (narrow activation interval of fibrin gel formation), control by blood flow (switching-off above some shear rate), spatial propagation

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(ability to form a 3-dimensional clot of a nonzero size), and localization (prevention against clotting the entire blood)—and each of them is fulfilled/controlled by the corresponding sub-network; however, there exists partial overlapping of these subnetworks. Recently this model was supplemented by sub-network of fibrinolysis reactions to study spatial dynamics of clot lysis [4]. It is unclear why despite repeated citations, Panteleev’s model has not been used by any other research group.

4.4 Clotting in Flow: The Main Actors Are Platelets Completely different approach to study the hemostasis was used in prof. Fogelson’s group: from the beginning they aimed to study blood coagulation system combined with platelet activation and flow. They started with the platelet activation/aggregation model in flow [66] and extension of Jesty and Beltramy’s model by introducing membrane binding sites supporting enzymatic reactions [67]. In the latter case, they showed that variations in surface-binding site densities can serve as a “switch,” drastically altering the responsiveness of the system. Their model [68] accounted for influence on coagulation of both platelets (by introducing densities of platelet binding sites for coagulation factors) and flow (by introducing kf low · (cout − c) terms into ODEs, where cout is plasma concentration of a given protein or platelet), as well as simple submodel of platelet activation. The main conclusions of this work were: (1) existence of a threshold-like dependence of thrombin response to the surface densities of TF and binding sites for factors and (2) platelet adhesion to subendothelium can block TF-VIIa complex activity, and it can impact the reduction of thrombin production in hemophilias. Later they combined platelet and plasma coagulation submodels into the “complete” model of mixed platelet-fibrin thrombus formation in flow [69, 70] and performed its intensive sensitivity analysis [71]. Their model was used, for example, in Neeves’s lab to estimate the thrombin generation kinetics and relative impact of intrinsic/extrinsic tenases on FXa generation inside the growing thrombus as a function of [FVIII] [72], and in Mitrophanov’s group to investigate the effects of coagulation factors supplementation on thrombus resistance in flow [73]. A number of other groups have proposed their own detailed mathematical models of coagulation. For example, Jordan and Chaikof [74] studied the synergistic effect of multiple TF sites on coagulation activation in flow.

5 Reduction: Back to Low-Dimensional Models Complex models are hard to understand and study. Thus, reduction—lowering model dimension and simplification of the right-hand side (RHS) of equations—is a well-known way to treat complex models. Reduced models should be distinguished from the phenomenological ones as the latter are written to express some relation or

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even hypothesis and may not have a known mechanistic (biochemical) background in the “hypothetical” part. In contrast, reduced model should inherit the mechanistic (biochemical) background from the parent full model. A number of reduced models of blood coagulation were proposed to study qualitatively various aspects of regulation of this system. The most intriguing questions are mechanisms of spatial spreading of coagulation and its termination. As we will see below, these studies made impact not only to biophysics of coagulation, but also to the theory of active media.

5.1 Homogeneous Reduced Models: Easy Transition to the Spatial Case At least two attempts to reduce the dimension of the first detailed model of coagulation [43, 44] were proposed. The reductions were based on the separation of variables of different time scales and applying the Tikhonov’s theorem to the subsystem of fast variables. The fastest variables were the concentrations of complexes of intrinsic tenase and prothrombinase: the rate constants of the reactions of their formation and dissociation are two orders of magnitude higher than other rate constants of the system. Pokhilko considered thrombin concentration as a fast variable, too [44]. Introducing the new variable m = Z + y8 , where Z is the prothrombinase concentration and y8 is the concentration of free FVIIIa, suggesting the closeness of the kinetic constants of activation and inhibition/decay for FVa and VIIIa, and neglecting the inhibition of FVa by pCa, she obtained the following system of equations for FIXa, FXa, and FVIIa pool: ⎧ dx ⎪ = k9 C − h9 x, ⎪ ⎨ dt dy xz dt = k10 x + p1 .f (Ca ). 1+f (Ca )x − h10 y, ⎪ ⎪ ⎩ dz dt = y2 − h8 z.

(7)

Here, x = [IXa], y = [Xa], z = m/k8, y2 = p2 .y + p3 .f (Ca ). 1+fyz (Ca )y is the quasistationary concentration of thrombin, C is the constant concentration of FXIa [Ca ]a (which is an activator), f (Ca ) = b+[C a is the dependence of the assembly rate a] of the intrinsic tenase and prothrombinase complexes on the [Ca 2+ ], ki , hi are kinetic constants, pi are combinations of kinetic constants, a and b are parameters. This model describes well the experimental homogeneous kinetics of coagulation activated by the intrinsic pathway and the threshold on the concentration of calcium ions. Right away, this model was transferred to the spatially distributed conditions. Pokhilko [44] showed that taking into account the activation of FXI by thrombin does not affect the kinetics of the process at rate constant values ranged from 0 to 10−4 min−1 (the exact value of this constant was unknown, but literature data

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showed it was below 10−4 min−1 ). However, in a spatially distributed system, existence of this reaction should lead to the ability of coagulation to spread from the activating surface into the plasma in a self-sustaining way. Thus, the system (7) was supplemented by the kinetic term for FXI activation by thrombin and by diffusion terms of all three substances, and solved numerically; FXIa influx from the left wall was considered as a trigger. The threshold behavior of the system for FXIa influx and the FXIa activation constant by thrombin was obtained. Above these thresholds, thrombin peak velocity rapidly reached the steady value. These results were later reproduced in all qualitative models of Ataullakhanov’s lab (see the Sect. 4.3 above). Molchanova also performed the extensive study of time-scales separation in the model [43, 44] and reduced it (using Tikhonov’s theorem) to three variables corresponding to thrombin, pCa, and FXa concentration [75]. She found at least four types of dynamics regimes in the reduced system: bistable (nonzero bottom state), monostable, self-oscillating (limit cycle), and coexistence of the stable state with the limit cycle, and suggested their 3-dimensional model to be quasi-2-dimensional, i.e. that it exists a limiting surface containing all the steady states and the limit cycles. Panteleev also performed reduction of his quantitative model of coagulation in the limits of initial reaction time and low TF density in the full-stirred conditions [65]. His reduction was based on identification and removing of nonessential components followed by time-scale separation analysis, and resulted in 3-equation system for TF pool, thrombin, and FVa concentrations, respectively: ⎧ dx 3p ⎪ ⎪ ⎨ dt = −b2 x3p , ⎪ ⎪ ⎩

dx2 dt dx5 dt

= b3 x3p (a3 + a4 x5 ) − a5 x2 ,

(8)

= a6 x2 ,

which had the analytical solution. Phase-plane and time-dependent analysis of this system showed the necessity of FV feedback loop for the explosive fibrin formation.

5.2 Reduced Spatial Models: Variety of Autowave Regimes Taken together, discovery of variety of dynamic regimes in homogeneous model [75] and ability of coagulation to propagate in space in an autowave manner [44, 59, 60] raises the question: are there any other dynamic regimes possible for spatially distributed coagulation? The previously proposed two-autowave phenomenological model [39] (see the Sect. 3.3 above) illustrated the regime of coagulation propagation followed by termination. To provide some biochemical ground to that phenomenological model, and especially to the suggested inhibitor autocatalysis (responsible for the termination mode), Zarnitsina et al. [76] supplemented the detailed model [59, 60] with the hypothetical reaction block of thrombin switching from the procoagulant (T1) into the anticoagulant (T2) state (namely thrombin activates pC with cleaving the peptide P, which catalyzes the conversion T1 →

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T2 but not T2 → T1, and T2 activates pC more rapidly than T1), and then severely reduced the resulting model. The reduction was based on the suggestions: rapid equilibrium T1 ⇐⇒ T2, rapid inhibition and quasistationarity of P, quasistationarity of FIXa, Xa, Va, and VIIIa, and some simplification of the RHS of equations. The resulting model consisted in three equations for dimensionless concentrations of thrombin, pCa and FXIa: ⎧ ⎪ ⎨ ⎪ ⎩

du dt dv dt dw dt

2u = K1 uw(1 − u) 1+K 1+K3 v − u + Du,

= K5 u2 − K6 v + Dv,

(9)

= u − K4 w + Dw,

with all diffusion coefficients assumed to be equal. Extensive studies of model (9) followed. In the homogeneous case, this system has 1 to 3 positive steady states and various dynamic regimes (including chaotic oscillations), which however were not studied in details. In the spatial case, this model has solutions in the form of a running excitation pulse, and several regimes of this pulse were found: disappearing at some distance from the activator, stationary propagation with constant or oscillating amplitude, trigger wave, and stopping with transformation into the stationary stand-alone peak [76]. Lobanova conducted comprehensive studies of this model and found regimes of trigger waves with oscillating rear part and running replicating (dividing) trigger waves [77], multihumped pulses [78], and second regime of stand-alone peak formation [79].

5.3 Reduced Models in Flow: Scenarios of Coagulation Termination Ermakova et al. studied model (9) in the conditions of a steady parabolic flow in a two-dimensional channel, taking into account the wall-immobilized thrombomodulin which is a cofactor of protein C activation by thrombin. They found two possible scenarios of termination of the spatial propagation of coagulation: in the narrow channel, it can be achieved by the thrombomodulin activity, and in a wider channel it can be achieved by the flow [80].

6 Conclusions and Future Directions Four main types of mathematical models have been used to study blood coagulation both in homogeneous and spatially distributed conditions: phenomenological, simulating, detailed (quantitative), and reduced. Each of these types is extremely powerful for its own tasks and conditions, but limited for other tasks and conditions: (1) Phenomenological models are used to express mathematically and visualize some idea/hypothesis and show its principal possibility by itself or in some

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context. However, due to lack of mechanistic background, these models cannot be used to prove the hypothesis (only a combination of quantitative model with experimental data is able to confirm hypothesis). On the contrary, a hypothesis can be refused, depending on the specific problem and task, using any model— phenomenological, simulating, quantitative, or reduced. (2) Simulation models are needed just after the phenomenological ones—when all main processes in the system are known (for example, at least the skeleton of the biochemical reaction network), but without precise information about all parameter values (for example, kinetic constants, concentrations, etc.). These models are used to investigate rough model properties at a semi-quantitative level: existence of thresholds, possible solution types and their stability, etc. But inaccuracies of these models should be always kept in mind and underlined in the conclusions. (3) Detailed, quantitative models are ideal in the case all mechanisms are understood and their parameters can be determined. Along with obvious advantages, the desire for quantitative modeling entails a number of problems: (a) for a number of reasons (see the Sects. 4.1, 4.2), detailed knowledge of all mechanisms and parameters of the biological process of interest isn’t always possible; understanding of this uncertainty often requires serious immersion into the literature; therefore, some parts of the quantitative models remain simulating or even phenomenological, calling remarks of the previous paragraphs (2) and (1). (b) The natural patient-specificity precludes one-to-one comparison between experimental data and even the most detailed mathematical model, as well as immediate usage of this comparison in diagnostics. Special approaches to overcome this difficulty are urgently necessary to be created. For example, each mathematical model should be equipped with a “shell”— a module which extracts meaningful characteristics of the system (for example, response time, amplitude, thresholds, velocity of spatial propagation, etc.), analyzes the sensitivity of these characteristics to all model input parameters, and performs other tasks specific to the given model and system. (c) Despite the fact that increased computer performance practically removed previous restrictions on the number of model variables and parameters, a new difficulty has arisen: difficulty of sharing and interchange of complex models between research groups. This problem may be partially solved using special model exchange formats like SBML [81]. (4) Reduced model can be created if a detailed model has been built, and some parts of this detailed model turned out to be redundant on certain time/spatial scales: • there are slow variables that can be put to constants; • there are fast variables, and kinetics of their relaxation to the quasistationarity is not of interest;

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• there exist whole model blocks that do not participate in the processes of interest or have little effect on the results. In these cases, creation of the reduced model is extremely useful for: (a) qualitative analysis of the system behavior; (b) analysis of the role of certain system parts in its functioning (by comparing reduced and detailed models); (c) embedding into the model of a higher scale instead of the full model (for adjustment, or for simplicity). Regarding the point (4a), we must note that the set of regimes of a reduced model solution can be much richer than of the parental quantitative model. Therefore, this reduced model (and methods of its research) can appear to be interesting even separated from the parent system—for example, for neighboring (or far) research area. However, caution should be taken in transferring new regimes found in the reduced model to the real system. The only known successful example of this kind for coagulation is the prediction of its ability for spatial propagation in a selfsustained manner due to the activation of FXIa by thrombin and for control of this process by the pC/Tm subsystem. Apparently, this modeling pipeline is common to mathematical biology and biophysics. This review shows implementation of this pipeline in the fundamental studies of structure and regulation mechanisms, as well as in the development of experimental methods of diagnostics and correction of the blood coagulation system. Acknowledgements This work was partially supported by the “RUDN University Program 5100” to A.T. and V.V. and by the Dynasty Foundation Fellowship to A.T.

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Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-Containing Capsids and Cure Rate Adil Meskaf and Karam Allali

1 Introduction Hepatitis B virus (HBV) can engender liver disease, leading to the infection of the healthy hepatocyte cells [1]. Contracting this life-threatening pathology stems from body fluid contact. In the last couple of decades, a number of mathematical models describing HBV dynamics were elaborated [2–4]. Other HBV infection models, including HBV DNA-containing capsids effects, have been advanced and studied in recent works [5–7]. It is known that HBV DNA-containing capsids are released from the infected cells under the mature virions form after being enveloped by cellular membrane lipids and viral envelope proteins [8, 9]. More recently, research studies have been conducted on the optimal control of HBV viral infection through combining both pegylated interferon (PEG IFN) and lamivudine (LMV) therapies [10]. This present work aims at determining an optimal control of HBV infection through administering both PEG IFN and LMV drugs. For this purpose, we will consider the following nonlinear differential equations:

A. Meskaf () Department of SEG, Polydisciplinary Faculty, University Chouaib Doukkali, El Jadida, Morocco K. Allali Department of Mathematics, Faculty of Sciences and Technologies, University Hassan II, Mohammedia, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_2

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⎧ dH ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dI ⎪ ⎪ ⎨ dt dD ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dV ⎪ ⎩ dt

= s − μH (t) − (1 − u1 )kH (t)V (t) + rI (t), = ke−λτ (1 − u1 )H (t − τ )V (t − τ ) − (δ + r)I (t), (1.1) = (1 − u2 )aI (t) − βD(t) − δD(t), = βD(t) − uV (t),

Susceptible host cells (H ) are produced at a rate s, die at a rate μ, and become infected by virus at a rate k. Infected cells (I ) die at a rate δ. In addition, through therapy, a part of infected cells may also revert to the uninfected state at a rate r [11, 12]. The constant λ is assumed to be the death rate of infected but not yet virus-producing cells. The intracellular delay, τ , represents the time needed for infected cells to produce virions after viral entry. The term e−λτ is the probability of surviving from time t − τ to time t. The intracellular HBV DNA-containing capsids (D) are produced at a rate a, they are transmitted to blood at a rate β and die at a rate δ. The virions (V ) grow in blood at a rate β, decay at a rate u. Finally, u1 and u2 denote the efficiency of PEG IFN and LMV drugs, respectively. It is noteworthy to mention the role of PEG IFN drug is to block the new infections of the healthy hepatocytes in the liver, while the prime function of the second drug (LMV) is to inhibit viral production. In order to study the system of delayed differential equation (1.1), the initial functions need to be first stated and the functional framework needs to be specified. First, let X = C([−τ, 0]; R4 ) be the Banach space of continuous mapping from [−τ, 0] to R4 equipped with the sup-norm ϕ = sup ϕ(t). −τ ≤t≤0

Assume that the initial functions verify (H (θ ), I (θ ), D(θ ), V (θ )) ∈ X).

(1.2)

As usual, for biological reasons, these initial functions H (θ ), I (θ ), D(θ ), and V (θ ) have to be non-negative: H (θ ) ≥ 0, I (θ ) ≥ 0, D(θ ) ≥ 0, V (θ ) ≥ 0, for θ ∈ [−τ, 0].

(1.3)

For any initial conditions (H (θ ), I (θ ), D(θ ), V (θ ) satisfying (1.2) and (1.3), the system (1.1) has a unique solution, moreover, this solution is non-negative and bounded for all t ≥ 0. The paper is organized as follows. The next section is devoted to the optimization analysis of the viral infection model. In Sect. 3, we construct an appropriate numerical algorithm and give some numerical simulations. The last section sums up our research work.

Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-. . .

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2 The Optimal Control Analysis 2.1 The Optimization Problem In order to state the optimization problem, we first consider that u1 and u2 vary with time. The problem (1.1) becomes ⎧ dH ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dI ⎪ ⎪ ⎨ dt dD ⎪ ⎪ ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dV dt

= s − μH (t) − (1 − u1 (t))kH (t)V (t) + rI (t), = ke−λτ (1 − u1 (t))H (t − τ )V (t − τ ) − (δ + r)I (t), (2.1) = (1 − u2 (t))aI (t) − βD(t) − δD(t), = βD(t) − uV (t),

For this problem, we have the following result Theorem 2.1 For any initial conditions (H (θ ), I (θ ), D(θ ), V (θ )) satisfying (1.2) and (1.3), the system (2.1) has a unique solution; in addition, this solution is nonnegative and bounded for all t ≥ 0. The optimization problem that we consider is to maximize the following objective functional

J (u1 , u2 ) =

tf



H (t) −

0

A

1 2 u1 (t) +

2

A2 2  u (t) dt, 2 2

(2.2)

where tf is the period of treatment and the positive constants A1 and A2 are based on the benefits and costs of the treatment u1 , u2 , respectively. The two control functions, u1 (t) and u2 (t), are assumed to be bounded and Lebesgue integrable. Our target is to maximize the objective functional defined in Eq. (2.2) by increasing the number of the uninfected cells, decreasing the viral load, and minimizing the cost of treatment. In other words, we are seeking optimal control pair (u∗1 , u∗2 ) such that J (u∗1 , u∗2 ) = max{J (u1 , u2 ) : (u1 , u2 ) ∈ U },

(2.3)

where U is the control set defined by U = {(u1 (t), u2 (t)) : ui (t) measurable, 0 ≤ ui (t) ≤ 1, t ∈ [0, tf ], i = 1, 2}. (2.4)

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2.2 Existence of an Optimal Control Pair The existence of the optimal control pair can be directly obtained using the results in Refs. [13, 14]. More precisely, we have the following theorem Theorem 2.2 There exists an optimal control pair (u∗1 , u∗2 ) ∈ U such that   J u∗1 , u∗2 = max(u1 ,u2 )∈U J (u1 , u2 ).

(2.5)

Proof To use the existence result in Ref. [13], we must check the following properties: (P1 ) The set of controls and corresponding state variables is nonempty. (P2 ) The control U set is convex and closed. (P3 ) The right-hand side of the state system is bounded by a linear function in the state and control variables. (P4 ) The integrand of the objective functional is concave on U. (P5 ) There exist constants c1 , c2 > 0, and α > 1 such that the integrand L(H, u1 , u2 ) of the objective functional satisfies  α2 L(H, u1 , u2 ) ≤ c2 − c1 | u1 |2 + | u2 |2 .

(2.6)

Where 

A1 2 A2 2 u (t) + u (t) . L(H, u1 , u2 ) = H (t) − 2 1 2 2

(2.7)

In order to verify these conditions, we use a result by Lukes [14] to give the existence of solutions of system (1.1), which gives condition (P1 ). The control set is convex and closed by definition, which gives condition (P2 ). Since our state system is bilinear in u1 , u2 , the right- hand side of system (1.1) satisfies condition (P3 ), using the boundedness of the solutions. Note that we have the Hessian matrix of L in (u1 , u2 ) is  H ess(u1 , u2 ) =

−A1 0 0 −A2

,

then, det (H ess(u1 , u2 )) = A1 A2 ≥ 0,

∀(u1 , u2 ) ∈ U,

(2.8)

Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-. . .

27

So, that the integrand of our objective functional is concave. Also, we have the last needed condition  (2.9) L(H, u1 , u2 ) ≤ c2 − c1 | u1 |2 + | u2 |2 ,   where c2 depends on the upper bound on H and c1 = min A21 , A22 > 0. We conclude that there exists an optimal control pair (u∗1 , u∗2 ) ∈ U such that   J u∗1 , u∗2 = max(u1 ,u2 )∈U J (u1 , u2 ).

 

2.3 Optimality System Pontryagin’s minimum principle given in Ref. [15] provides necessary conditions for an optimal control problem. This principle converts (1.1), (2.2), and (2.3) into a problem of maximizing an Hamiltonian, T , pointwisely with respect to u1 and u2 :  A1 2 A2 2 u1 + u2 − H + λi fi 2 2 5

T (t, H, I, D, V , Hτ , vτ , u1 , u2 , λ) =

i=0

with ⎧ ⎪ ⎪ f1 ⎨ f2 ⎪ f ⎪ ⎩ 3 f4

= s − μH (t) − (1 − u1 )kH (t)V (t) + rI (t), = ke−λτ (1 − u1 )H (t − τ )V (t − τ ) − (δ + r)I (t), = (1 − u2 )aI (t) − βD(t) − δD(t), = βD(t) − uV (t),

(2.10)

By applying Pontryagin’s minimum principle with delay in state [15], we obtain the following theorem Theorem 2.3 For any optimal controls u∗1 , u∗2 , and solutions H ∗ , I ∗ , D ∗ , and V ∗ of the corresponding state system (1.1), there exist adjoint variables, λ1 , λ2 , λ3 , and λ4 , satisfying the equations ⎧     λ1 (t) = 1 + λ1 (t) μ + 1 − u∗1 (t) kV ∗ (t) ⎪ ⎪      ⎪ ⎪ ∗ t + τ − 1 ke−λτ V ∗ (t), ⎪ t + τ u (t)λ +χ [0,t −τ ] 2 ⎪ f 1 ⎪    ⎨ λ2 (t) = λ2 (t)(δ + r) − λ3 (t) 1 − u∗2 (t) a − λ1 r,   ⎪ λ3 (t) = λ3 (t) δ + β − βλ4 (t) ⎪   ⎪ ⎪ ⎪ λ4 (t) = λ1 (t) k(1 − u∗1 (t))H ∗ (t) + λ4 (t)u − λ4 (t) ⎪ ⎪   ⎩ +χ[0,tf −τ ] (t)λ2 (t + τ ) ke−λτ (u∗1 (t + τ ) − 1)H ∗ (t)

(2.11)

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with the transversality conditions λi (tf ) = 0, i = 1, . . . , 5.

(2.12)

Moreover, the optimal control is given by   

k −λτ ∗ ∗ ∗ ∗ λ2 (t)e Vτ Hτ − λ1 (t)V (t)H (t) =min 1, max 0, A1   1 λ3 (t)aI ∗ (t) . u∗2 =min 1, max 0, A2 u∗1

(2.13)

Proof The adjoint equations and transversality conditions can be obtained by using Pontryagin’s minimum principle with delay in state [15], such that ⎧ ∂T ∂T ⎪ λ1 (t) = − ∂H (t) − χ[0,tf −τ ] (t) ∂H (t + τ ), ⎪ τ ⎪ ⎨  ∂T λ2 (t) = − ∂I (t), ∂T ⎪ (t), λ3 (t) = − ∂D ⎪ ⎪ ⎩ λ (t) = − ∂T (t) − χ ∂T [0,tf −τ ] (t) ∂Vτ (t + τ ), 4 ∂V

λ1 (tf ) = 0, λ2 (tf ) = 0, λ3 (tf ) = 0, λ4 (tf ) = 0,

(2.14)

The optimal control u∗1 and u∗2 can be solved from the optimality conditions, ∂T (t) = 0, ∂u1

∂T (t) = 0. ∂u2

∂T (t) =A1 u1 (t) + kλ1 (t)v(t)H (t) − kλ2 (t)Vτ Hτ e−λτ = 0, ∂u1 ∂T (t) =A2 u2 (t) − aλ3 (t)I (t) = 0. ∂u2

(2.15)

(2.16)

By the boundedness of the two controls in U , it is easy to obtain u∗1 and u∗2 in the form of (2.13), respectively. If we substitute u∗1 and u∗2 in the systems (1.1) and (1.1), we obtain the following optimality system: dH ∗ dt dI ∗ dt dD ∗ dt dV ∗ dt

= s − dH ∗ (t) − k(1 − u∗1 (t))V ∗ (t)H ∗ (t) + rI ∗ (t), = ke−λτ (1 − u∗1 (t))V ∗ (t − τ )H ∗ (t − τ ) − (δ + r)I ∗ (t), = (1 − u∗2 (t))aI ∗ (t) − δD ∗ (t) − βD ∗ (t) = βD ∗ (t) − uV ∗ (t),

Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-. . .

then, ⎧     λ1 (t) = 1 + λ1 (t) μ + 1 − u∗1 (t) kV ∗ (t) ⎪ ⎪      ⎪ ⎪ ⎪ +χ[0,tf −τ ] (t)λ2 t + τ u∗1 t + τ − 1 ke−λτ V ∗ (t), ⎪ ⎪   ⎨  λ2 (t) = λ2 (t)(δ + r) − λ3 (t) 1 − u∗2 (t) a − λ1 (t)r,   ⎪ λ3 (t) = λ3 (t) δ + β − βλ4 (t) ⎪  ⎪ ⎪ λ (t) = λ (t) k(1 − u∗ (t))H ∗ (t) + λ (t)u − λ (t) ⎪ 1 4 4 ⎪ 1 ⎪   ⎩ 4 +χ[0,tf −τ ] (t)λ2 (t + τ ) ke−λτ (u∗1 (t + τ ) − 1)H ∗ (t) ,   

k −λτ ∗ ∗ ∗ ∗ λ2 (t)e Vτ Hτ − λ1 (t)V (t)H (t) =min 1, max 0, A1   1 ∗ ∗ λ3 (t)aI (t) . u2 =min 1, max 0, A2

29

(2.17)

u∗1

λi (tf ) = 0, i = 1, . . . , 5.

(2.18)

(2.19)  

3 Numerical Simulations In order to perform the numerical simulations, the optimization system will be solved numerically using a discretized scheme based on forward and backward finite difference approximation method [16–18]. Indeed, we will have the following numerical algorithm: The parameters of our numerical simulations are taken from Refs. [19, 20]; i.e. s = 2.6 × 107 , k = 1.67 × 10−12 , μ = 0.01, δ = 0.053, a = 150, β = 0.87, u = 3.8, τ = 5, λ = 1.1 × 10−2 , A1 = 50,000, and A2 = 5000. The role of these two last positive parameters A1 and A2 is to balance the terms size in the equations. The new parameter of our problem, the cure rate will be r = 0.01 [11, 12]. Figure 1 shows the uninfected cells during the first weeks of observation. It can be seen, that after the treatments (with control), the uninfected cells population grows significantly comparing with the curve representing the no-treatment case. From Fig. 2, we clearly observe the role of an optimal control in reducing the infected cells. We remark by making a zoom on the first 100 days of observation (see right-hand side of the figure) that the number of infected cells at time t = 100 (days) is 40 in the case with control and 3100 without control which means that administrating the good therapy can help the patient by a significant reduce of the infected cells number. Figure 3 shows that after introducing good therapy, the number of capsids load declines towards zero. However, the number of capsids remains at a very high level in the case without any control strategy.

30

A. Meskaf and K. Allali

Algorithm 1 The Numerical Algorithm Step 1: for i = −m, . . . , 0, do: Hi = H0 , Ii = I0 , Di = D0 , Vi = V0 , ui1 = 0, ui2 = 0. end for for i = n, . . . , n + m, do: λi1 = 0, λi2 = 0, λi3 = 0, λi4 = 0, λi5 = 0. end for Step 2: for i = 0, . . . , n − 1, do: Hi+1 = Hi + h[s − μHi − k(1 − ui1 )Vi Hi + rIi ], Ii+1 = Ii + h[ke−λτ (1 − ui1 )Vi−m Hi−m − (δ + r)Ii ], Di+1 = Di + h[(1 − ui2 )aIi − δDi − βDi ], Vi+1 = Vi + h[βDi − uVi ], λn−i−1 = λ1n−i − h[1 + λ1n−i (μ + (1 − ui1 )kVi+1 )] 1 +χ[0,tf −τ ] (tn−i )λn−i+m (ui+m − 1)ke−λτ Vi+1 ], 2 1 n−i−1 n−i n−i n−i = λ2 − h[λ2 (δ + r) − λ3 (1 − ui2 )a − λ1n−i r], λ2 n−i−1 λ3 = λ3n−i − h[λ3n−i (δ + β) − λ4n−i β], n−i−1 λ4 = λ4n−i − h[λ1n−i (1 − ui1 )kHi+1 + λ4n−i u +χ[0,tf −τ ] (tn−i )λn−i+m (ui+m − 1)ke−δτ Hi+1 ], 2 1 i+1 n−i−1 −λτ e Vi−m+1 Hi−m+1 − λn−i−1 Vi+1 Hi+1 ) R1 = (k/A1 )(λ2 1 aI , R2i+1 = (1/A2 )λn−i−1 i+1 3 = min(1, max(R1i+1 , 0)), ui+1 1 = min(1, max(R2i+1 , 0)), ui+1 2 end for Step 3: for i = 1, . . . , n, write H ∗ (ti ) = Hi , I ∗ (ti ) = Ii , D ∗ (ti ) = Di , V ∗ (ti ) = Vi , u∗1 (ti ) = ui1 , u∗2 (ti ) = ui2 . end for

3

x 109

Uninfected Cells

2.5

Without Control With Control

2 1.5 1 0.5 0

0

100

200

300

400

500

Time Fig. 1 The uninfected cells as function of time

600

700

800

900

Optimal Control of a Delayed Hepatitis B Viral Infection Model with DNA-. . . 6 x 10

8

Without Control With Control

10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0

Without Control With Control

Infected Cells

Infected Cells

5 4 3 2 1 0

31

0

100

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400

500

600

700

800

900

0

10

20

30

40

50

60

70

80

90

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Time

Fig. 2 The infected cells (left) as function of time and a zoomed region (right) 9 x 10

10

6

2.5 x 10 Without Control With Control

8

Without Control With Control

2

6

Capsids

Capsids

7 5 4 3 2

1.5 1 0.5

1 0 0

100

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500

600

700

800

0

900

0

10

20

30

40

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80

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Time

Time

Fig. 3 The capsids (left) as function of time and a zoomed region (right) 2.5 x 10

10

5

3.5 x 10 Without Control With Control

2

Without Control With Control

3

1.5

Virus

Virus

2.5

1

2 1.5 1

0.5

0.5

0

0 0

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900

0

10

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50

60

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90

100

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Fig. 4 The virions (left) as function of time and a zoomed region (right)

The role of therapy control is also observed in Fig. 4. It was shown that with control, the number of HBV virions decreases significantly after the first weeks of therapy, while without control it stays equal to a high level. This indicates the impact of the administrated therapy in controlling viral replication. Finally, Fig. 5 represents the optimal controls u1 and u2 in blocking new infection and inhibiting viral production. The two curves present the drug administration schedule during the period of treatment. Both controls start from zero and become nearly constant. This means that the two treatments should be administrated with a constant manner and without any interruption.

32

A. Meskaf and K. Allali 0.7

0.03

0.6

0.025

0.5

Control u 2

Control u1

0.035

0.02 0.015 0.01 0.005

0.4 0.3 0.2 0.1

0

0 0

100

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300

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500

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600

700

800

900

0

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500

600

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900

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Fig. 5 The optimal control u1 (left) and the optimal control u2 (right) versus time

4 Conclusion In this paper, we have studied an HBV infection model with intracellular HBV DNA-containing capsids. The considered model includes four differential equations describing the interaction between the uninfected cells, infected cells, capsids, and HBV virus. Both the treatments, the intracellular delay and the cure rate of infected cells, are incorporated into the model. We have proved the existence and uniqueness of the optimal controls using Pontryagin’s maximum principle. The problem was solved numerically using backward and forward finite-difference scheme. It was shown that, with the two optimal treatments, the number of the healthy hepatocytes increases remarkably, whereas the number of the infected hepatocytes decreases significantly. In addition, it was also observed that, with the control strategy, the viral load decreases considerably compared with the model without control, which can improve the patient’s life quality.

References 1. S.M. Ciupe, R.M. Ribeiro, P.W. Nelson, A.S. Perelson, Modeling the mechanisms of acute hepatitis B virus infection. J. Theor. Biol. 247(1), 23–35 (2007) 2. L. Min, Y. Su, Y. Kuang, Mathematical analysis of a basic virus infection model with application to HBV infection. Rocky Mt. J. Math. 38, 1573–1585 (2008) 3. M.A. Nowak, S. Bonhoeffer, A.M. Hill, R. Boehme, H.C. Thomas, H. McDade, Viral dynamics in hepatitis B virus infection. Proc. Natl. Acad. Sci. 93(9), 4398–4402 (1996) 4. K. Wang, A. Fan, A. Torres, Global properties of an improved hepatitis B virus model. Nonlinear Anal. Real World Appl. 11(4), 3131–3138 (2010) 5. K. Manna, S.P. Chakrabarty, Chronic hepatitis B infection and HBV DNA-containing capsids: modeling and analysis’. Commun. Nonlinear Sci. Numer. Simul. 22, 383–395 (2015) 6. K. Manna, S.P. Chakrabarty, Global stability and a non-standard finite difference scheme for a diffusion driven HBV model with capsids’. J. Differ. Equ. Appl. 21, 918–933 (2015) 7. K. Manna, S.P. Chakrabarty, Global stability of one and two discrete delay models for chronic hepatitis B infection with HBV DNA-containing capsids’. Comput. Appl. Math. 36, 525–536 (2017) 8. V. Bruss, Envelopment of the hepatitis B virus nucleocapsid. Virus Res. 106, 199–209 (2004)

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9. D. Ganem, A.M. Prince, Hepatitis B virus infection: natural history and clinical consequences. N. Engl. J. Med. 350, 1118–1129 (2004) 10. J. Danane, A. Meskaf, K. Allali, Optimal control of a delayed hepatitis B viral infection model with HBV DNA-containing capsids and CTL immune response. Optimal Control Appl. Methods 39(3), 1262-1272 (2018) 11. K. Hattaf, N. Yousfi, Dynamics of HIV infection model with therapy and cure rate. Int. J. Tomogr. Stat. 16(11), 74-80 (2011) 12. X. Zhou, X. Song, X. Shi, A differential equation model of HIV infection of CD4+ T-cells with cure rate. J. Math. Anal. Appl. 342,(2), 1342–1355 (2008) 13. W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Springer, New York, 1975) 14. D.L. Lukes Differential Equations: Classical to Controlled. Mathematics in Science and Engineering (Academic Press, New York, 1982), p. 162 15. L. Göllmann, D. Kern, H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints. Optimal Control Appl. Methods 30, 341– 365 (2009) 16. K. Hattaf, N. Yousfi, Optimal control of a delayed HIV infection model with immune response using an efficient numerical method. ISRN Biomath. 2012 (2012). https://doi.org/10.5402/ 2012/215124 17. H. Laarabi, A. Abta, K. Hattaf, Optimal control of a delayed SIRS epidemic model with vaccination and treatment. Acta Biotheor. 63(2), 87–97 (2015) 18. L. Chen, K. Hattaf, J. Sun, Optimal control of a delayed SLBS computer virus model. Phys. A 427, 244–250 (2015) 19. K. Manna, Global properties of a HBV infection model with HBV DNA-containing capsids and CTL immune response. Int. J. Appl. Comput. Math. (2016). https://doi.org/10.1007/s40819016-0205-4 20. A. Meskaf, K. Allali, Y. Tabit. Optimal control of a delayed hepatitis B viral infection model with cytotoxic T-lymphocyte and antibody responses. Int. J. Dyn. Control (2016). https://doi. org/10.1007/s40435-016-0231-4

Dynamics of a Generalized Model for Ebola Virus Disease Zineb El Rhoubari, Hajar Besbassi, Khalid Hattaf, and Noura Yousfi

1 Introduction Ebola virus disease (EVD) is caused by virus Ebola which was identified in 1976 in the Democratic Republic of Congo near a river called Ebola. There are five known species of Ebola: Zaire, Bundibugyo, Sudan, Reston, and Taï Forest. The first three, Bundibugyo ebolavirus, Zaire ebolavirus, and Sudan ebolavirus have been associated with large outbreaks in Africa. The virus causing the 2014–2016 West African outbreak belongs to the Zaire ebolavirus species [1]. The EVD dominated international news in 2014, and the World Health Organization (WHO) reported more than 28,000 cases worldwide and over 11,000 deaths [2]. Many mathematical models have been proposed and developed in order to understand the dynamics of EVD. Most of these models suppose that Ebola can be transmitted only from an infected individual who is still alive [3–10]. However, based on statistics in Ref. [11], Weitz and Dushoff [12] estimated that 10–30% of cases are caused by post-death transmission. In addition, the African practices such as washing of deceased individuals during burial ceremonies can favorise the transmission of Ebola from dead to the living during funerals. For these reasons, we propose a new generalized model of EVD given by the following nonlinear system

Z. El Rhoubari () · H. Besbassi · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Science Ben M’sik, Hassan II University, Sidi Othman, Casablanca, Morocco K. Hattaf Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Science Ben M’sik, Hassan II University, Sidi Othman, Casablanca, Morocco Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Derb Ghalef, Casablanca, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_3

35

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⎧ dS ⎪ dt = A − μS − f (S, I )I − g(S, D)D, ⎪ ⎪ ⎪ ⎨ dI = f (S, I )I + g(S, D)D − (μ + d + r)I, dt dR ⎪ ⎪ dt = rI − μR, ⎪ ⎪ ⎩ dD dt = (μ + d)I − bD,

(1)

where S(t), I (t), R(t), and D(t) are the numbers of susceptible, infectious, recovered, and deceased individuals at time t, respectively. The parameter A is the recruitment rate, and μ is the natural death rate of the population. The susceptible individuals can acquire infection after effective contact with infectious individuals at rate f (S, I )I or with deceased human individual at rate g(S, D)D. Hence, the term f (S, I )I +g(S, D)D denotes the total infection rate of susceptible individuals. Moreover, infectious individuals experience death due to the EVD at rate d and they are recovered at rate r. On the other hand, deceased human individuals can be buried directly during funerals at rate b. Then b1 represents the caring duration of deceased human individuals. The schematic representation of model (1) is illustrated in Fig. 1. Since the third equation of system (1) is decoupled from the other equations, it suffices to study the following system ⎧ dS ⎪ = A − μS − f (S, I )I − g(S, D)D, ⎪ ⎨ dt dI dt = f (S, I )I + g(S, D)D − (μ + d + r)I, ⎪ ⎪ ⎩ dD = (μ + d)I − bD.

(2)

dt

As in Ref. [13], we assume that the general incidence f and g are continuously differentiable in the interior of R2+ and satisfy the following properties:

Fig. 1 Schematic representation of model (1)

Dynamics of a Generalized Model for Ebola Virus Disease

37

∂f ∂f (S, I ) > 0, (S, I ) ≤ 0 for all I, S ≥ 0. ∂S ∂I ∂g ∂g (S, D) > 0, (S, D) ≤ 0 for all S, D ≥ 0. (H2 ) g(0, D) = 0, ∂S ∂D The rest of the paper is organized as follows. In the next section, we prove the well-posedness of our model. Also, we give the basic reproduction number and discuss the existence of equilibria. The local and global stability of equilibria are investigated in Sects. 3 and 4. We support our study by numerical simulations in Sect. 5. Finally, some conclusions are shown in Sect. 6.

(H1 ) f (0, I ) = 0,

2 Preliminary Results In this section, we prove that the model presented by system (2) is well-posed. After, we discuss the existence of equilibria. For biological reasons, we assume that the initial conditions of system (2) satisfy: S(0) = S0  0, I (0) = I0  0 and D(0) = D0  0. First, we have the following result. Theorem 2.1 The first quadrant R3+ is positively invariant with respect system (2). Furthermore, all solutions of (2) are uniformly bounded in the compact subset  = (S, I, D) ∈

R3+

 A A(μ + d) : S+I ≤ , D ≤ . μ bμ

Proof Clearly, R3+ is positively invariant with respect system (2). It remains to show that all solutions of (2) are   uniformly bounded. Let S(t), I (t), D(t) be any solution with non-negative initial conditions (S0 , I0 , D0 ). Summing up the first two equations of system (2), we have d (S + I ) = A − μS − (μ + d + r)I ≤ A − μ(S + I ). dt Then   A lim sup S(t) + I (t) ≤ . μ t→∞ Similarly, from the third equation of (2), we obtain lim sup D(t) ≤ t→∞

A(μ + d) . bμ

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Thus, all solutions of (2) are uniformly bounded in the region . This completes the proof.  Now, we calculate the basic reproduction number and discuss the existence of equilibria.  Obviously, system (2) has always one disease-free equilibrium A Ef μ , 0, 0 . Then we define the basic reproduction number of (2) as follows: R0 =

bf

A



μ,0

+ (μ + d)g

A



u,0

(μ + d + r)b 

(3)

,



f A μ ,0 μ+d+r



(μ+d)g A μ ,0 (μ+d+r)b



which can be rewritten as R01 +R02 , where R01 = and R02 = . The difference between R01 and R02 is that the first corresponds to Ebola infection by infected individuals and the second to Ebola infection by contact with infected corpses. The other equilibrium of (2) satisfies the following equations A − μS − f (S, I )I − g(S, D)D = 0,

(4)

f (S, I )I + g(S, D)D − (μ + d + r)I = 0,

(5)

(μ + d)I − bD = 0.

(6)

By (4)–(6), we have   A − μS (μ + d)(A − μS) bf S, + (μ + d)g S, = b(μ + d + r). μ+d +r b(μ + d + r)

(7)

A A . So, there is no equilibrium when S > . μ μ   by Hence, we define a function ψ on the interval 0, A μ Since I =

A−μS μ+d+r

≥ 0, we have S ≤

  A − μS (μ + d)(A − μS) ψ(S) = bf S, + (μ + d)g S, − b(μ + d + r). μ+d +r b(μ + d + r) We have ψ(0) = −b(μ + d + r) < 0, ψ ψ  (S) = b



∂f ∂f μ − ∂S μ + d + r ∂I

 A = b(μ + d + r)(R0 − 1) and μ



 + (μ + d)

∂g μ(μ + d) ∂g − ∂S b(μ + d + r) ∂D

> 0.

Thus,for R0 > 1, there exists a unique endemic equilibrium E ∗ (S ∗ , I ∗ , D ∗ ) with A , I ∗ > 0 and D ∗ > 0. S ∗ ∈ 0, μ

Dynamics of a Generalized Model for Ebola Virus Disease

39

The above discussions can be summarized in the following theorem. Theorem 2.2 Let R0 defined by Eq. (3) (i) If R0 ≤ 1, then system (2) has a unique disease-free equilibrium of the form Ef (S0 , 0, 0), where S0 = A μ. (ii) If R0 > 1, the disease-free equilibrium is still present and system (2)  has a unique endemic equilibrium of the form E ∗ (S ∗ , I ∗ , D ∗ ) with S ∗ ∈ 0, A μ , ∗ ∗ I > 0 and D > 0.

3 Local Stability In this section, we evaluate the local stability of both equilibria Ef and E ∗ . Theorem 3.3 (i) If R0 < 1, then the disease-free equilibrium Ef is locally asymptotically stable. (ii) If R0 > 1, then Ef becomes unstable and the infection equilibrium E ∗ is locally asymptotically stable. Proof The characteristic equation at any equilibrium is given by    −I ∂f −D ∂g −μ −λ  ∂f ∂g −f (S, I ) − I −D −g(S, D)   ∂S ∂S ∂I ∂D   ∂f ∂g ∂f ∂g  I ∂S + D ∂S I ∂I +f (S, I )−(r +μ+d) − λ D ∂D +g(S, D)  = 0     0 μ+d −b − λ (8) Evaluating (8) at Ef , we have    A 2 ,0 + b(μ + d + r)(1 − R0 ) = 0. (μ + λ) λ + λ μ + d + r + b − f μ (9) Clearly, Eq. (9) has three roots that are: λ1 = −μ,  √ f A , 0 − (r + μ + d + b) − δ μ , λ2 = 2  √ f A μ , 0 − (r + μ + d + b) + δ λ3 = . 2

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  2 where δ = r + μ + d + b − f A + 4b(r + μ + d)(R0 − 1). Obviously, λ1 μ,0 and λ2 are negative. However, λ3 is negative if R0 < 1 and positive if R0 > 1. Evaluating (8) at E ∗ , we get λ3 + a1 λ2 + a2 λ + a3 = 0,

(10)

where ∂f ∗ ∂f ∗ ∂g ∗ D∗ I − I + D + g(S ∗ , D ∗ ) ∗ , a1 = μ + b + ∂S ∂I ∂S I  ∗  ∂f D ∂g ∗ ∗ ∗ ∗ ∗ ∗ a2 = f (S , I ) + g(S , D ) ∗ I + D I ∂S ∂S  ∗ ∂f ∗ ∗ ∗ D +μ b − I + g(S , D ) ∗ ∂I I  ∂f ∗ ∂f ∗ ∂g ∗ ∂g (D ∗ )2 , I − I + D − +b ∂S ∂I ∂S ∂D I ∗     ∂g ∗ ∂f ∂f ∗ ∂g ∗ a3 = b μ + d + r I + D − bμ I ∗ − μ μ + d D . ∂S ∂S ∂I ∂D By (H1 ) and (H2 ), we deduce that a1 , a2 , and a3 are positive. Further, we have   D∗ a1 a2 − a3 = (a1 − μ − b)a2 + μ + b f (S ∗ , I ∗ ) + g(S ∗ , D ∗ ) ∗ I   ∗ ∂f ∗ ∂f ∗ ∂g ∗ 2 ∗ ∗ D × I + D +μ b− I + g(S , D ) ∗ ∂S ∂S ∂I I  ∗ 2 ∂f ∗ ∂g ∗ ∂f ∗ ∂g (D ) I + D − I − +b2 ∂S ∂S ∂I ∂D I ∗  ∗ ∂f ∗ ∂g ∗ ∂f ∗ ∗ ∗ D > 0. +bμ b + g(S , D ) ∗ + I + D − I I ∂S ∂S ∂I It follows from the Routh–Hurwitz theorem [14] that all roots of (10) have negative real parts. Therefore, the endemic equilibrium E ∗ is locally asymptotically stable if R0 > 1. 

4 Global Stability In this section, we focus on the global stability of equilibria.

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41

Theorem 4.1 The disease-free equilibrium Ef is globally asymptotically stable if R0  1. Proof Consider the following Lyapunov functional  V (t) = bI + g

A , 0 D. μ

The time derivative of V along the positive solution of (2) is given by   A V˙ (t)|(2) = bf (S, I )−b(r +μ+d)+(μ+d)g ,0 I μ   A , 0 D. +b g(S, D)−g μ A , which implies that all omega limit points satisfy μ t→∞ A S(t) ≤ A μ . Hence, it is sufficient to consider solutions for which S(t) ≤ μ . From the expression (3) of R0 and (H1 )–(H2 ), we get We have lim sup S(t) ≤

   A A V˙ (t)|(2) ≤ bf , 0 − b(r + μ + d) + (μ + d)g ,0 I μ μ   ≤ (μ + d + r) R0 − 1 I. Consequently, V˙ (t)|(2) ≤ 0 for R0 ≤ 1. Furthermore, it is not hard to check that the largest compact invariant set in {(S, I, D)| V˙ (t) = 0} is the singleton {Ef }. By LaSalle invariance principle [15], we deduce that Ef is globally asymptotically stable if R0 ≤ 1.  Next, we study the global stability of the endemic equilibrium E ∗ by assuming that R0 > 1 and the functions f and g satisfy, for all S, I, D > 0, the following hypothesis 

 f (S, I ) f (S, I ∗ ) I 1− − ∗ ≤ 0, f (S, I ∗ ) f (S, I ) I   f (S ∗ , I ∗ )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ ) D 1− − ∗ ≤ 0. f (S, I ∗ )g(S ∗ , D ∗ ) f (S ∗ , I ∗ )g(S, D) D

(H3 )

Theorem 4.2 Assume that (H3 ) holds. If R0 > 1, then the endemic equilibrium E ∗ is globally asymptotically stable. Proof Define a Lyapunov functional as follows: W (t) = S(t)−S ∗ −

S(t) S∗

  g(S ∗ , D ∗ ) ∗ f (S ∗ , I ∗ ) I (t) D(t) ∗ dX+I D + . H H f (X, I ∗ ) I∗ b D∗

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where H (X) = X − 1 − ln X, X > 0. Obviously, H : (0, +∞) → [0, +∞) attains its strict global minimum at X = 1 and H (1) = 0. Then H (X) ≥ 0 and the functional W is non-negative. In order to simplify the presentation, we will use the following notations: X = X(t) for any X ∈ {S, I, D}. The time derivative of W along the positive solution of system (2) is given by    I ∗ dI g(S ∗ , D ∗ ) D ∗ dD f (S ∗ , I ∗ ) dS + 1 − + . 1 − W˙ (t)|(2) = 1 − f (S, I ∗ ) dt I dt b D dt Using A = μS ∗ + f (S ∗ , I ∗ )I ∗ + g(S ∗ , D ∗ )D ∗ = μS ∗ + (μ + d + r)I ∗ and (μ + d)I ∗ = bD ∗ , we obtain   S f (S ∗ , I ∗ ) ∗ ˙ 1− W (t)|(2) = μS 1 − ∗ S f (S, I ∗ )  f (S ∗ , I ∗ ) f (S, I )I I ∗ ∗ ∗ + +f (S , I )I 2 − − ∗ f (S, I ∗ ) f (S, I ∗ )I ∗ I  f (S ∗ , I ∗ ) f (S ∗ , I ∗ )g(S, D)D D +g(S ∗ , I ∗ )D ∗ 3 − + − ∗ ∗ f (S, I ) f (S, I ∗ )g(S ∗ , D ∗ )D ∗ D D∗I g(S, D)DI ∗ − + . g(S ∗ , D ∗ )D ∗ I DI ∗ Thus,   S f (S ∗ , I ∗ ) ∗ ˙ 1− W (t)|(2) = μS 1 − ∗ S f (S, I ∗ )  I f (S, I )I f (S, I ∗ ) + +f (S ∗ , I ∗ )I ∗ − 1 − ∗ + I f (S, I ) f (S, I ∗ )I ∗ +g(S ∗ , D ∗ )D ∗  f (S ∗ , I ∗ )g(S, D)D D f (S, I ∗ )g(S ∗ , D ∗ ) + × −1− ∗ + D f (S ∗ , I ∗ )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ )D ∗    

f (S ∗ , I ∗ ) f (S, I ) f (S, I ∗ ) ∗ ∗ ∗ −f (S , I )I H +H +H f (S, I ∗ ) f (S ∗ , I ∗ ) f (S, I )     ∗ ∗ ∗ f (S , I ) g(S, D)DI ∗ D I ∗ ∗ ∗ +H +H −g(S , D )D H f (S, I ∗ ) DI ∗ g(S ∗ , D ∗ )D ∗ I

 f (S, I ∗ )g(S ∗ , D ∗ ) . +H f (S ∗ , I ∗ )g(S, D)

Dynamics of a Generalized Model for Ebola Virus Disease

43

Since the function f (S, I ) is strictly monotonically increasing with respect to S, we have   f (S ∗ , I ∗ ) S 1− ≤ 0. 1− ∗ S f (S, I ∗ ) From (H3 ), we obtain −1 −

  I f (S, I ∗ ) f (S, I )I I f (S, I ∗ ) f (S, I ) ≤0 + − + = 1 − I∗ f (S, I ) f (S, I ∗ )I ∗ f (S, I ∗ ) f (S, I ) I∗

and D f (S ∗ , I ∗ )g(S, D)D f (S, I ∗ )g(S ∗ , D ∗ ) + + ∗ ∗ ∗ D f (S , I )g(S, D) f (S, I ∗ )g(S ∗ , D ∗ )D ∗   f (S, I ∗ )g(S ∗ , D ∗ ) D f (S ∗ , I ∗ )g(S, D) − ∗ ≤ 0. = 1− f (S, I ∗ )g(S ∗ , D ∗ ) f (S ∗ , I ∗ )g(S, D) D

−1 −

Since H (X) ≥ 0 for X > 0, we have W˙ |(2) ≤ 0 with equality if and only if S = S ∗ , I = I ∗ , and D = D ∗ . It follows from LaSalle invariance principle that E ∗ is globally asymptotically stable. 

5 Numerical Simulations In this section, we carry out some numerical simulations in order to illustrate the β1 S and g(S, D) = theoretical results. For simplicity, we choose f (S, I ) = 1 + α1 I β2 S . Hence, system (2) becomes 1 + α2 D ⎧ β2 SD β1 SI ⎪ − , = A − μS − ⎪ dS ⎪ ⎨ dt 1 + α1 I 1 + α2 D β SI SD β 1 2 dI ⎪ dt = 1 + α I + 1 + α D − (μ + d + r)I, ⎪ ⎪ 1 2 ⎩ dD dt = (μ + d)I − bD,

(11)

where β1 and β2 are the infection rates caused by infectious and deceased human individuals, respectively. The non-negative constants α1 and α2 measure the saturation effect. The other parameters have the same biological meanings as in system on above analysis, system (11) has one disease-free equilibrium  (2). Based A , 0, 0 , and a unique endemic equilibrium E ∗ (S ∗ , I ∗ , D ∗ ) when R0 = Ef μ

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(μ + d)Aβ2 Aβ1 + > 1. On the other hand, it is not hard to show μ(μ + d + r) bμ(μ + d + r) that the hypotheses (H1 )–(H3 ) are satisfied. From Theorems 3.3, 4.1, and 4.2, we deduce the following result. Corollary 5.1 (i) If R0 ≤ 1, then the disease-free equilibrium Ef of system (11) is globally asymptotically stable. (ii) If R0 > 1, then the disease-free equilibrium Ef becomes unstable and the endemic equilibrium E ∗ of (11) is globally asymptotically stable. Now, we check the above main results with numerical simulations. First, we simulate system (11) with the following parameter values: A = 50, μ = 0.5, β1 = 0.005, β2 = 0.001, α1 = 0.01, α2 = 0.01, d = 0.05, r = 0.06, and b = 0.8. By simple computation, we get R0 = 0.9324. Then system (11) has one disease-free equilibrium Ef (100, 0, 0). It follows from Corollary 5.1(i) that Ef is globally asymptotically stable and the solution of (11) converges to Ef (see Fig. 2). In this case, the disease dies out. Next, we choose β2 = 0.02 and do not change the other parameter values. We have R0 = 3.0738 > 1. Hence, system (11) has a unique endemic equilibrium E ∗ (43.9694, 45.9240, 31.5879). From (ii) of Corollary 5.1, E ∗ is globally asymptotically stable, which means that the Ebola virus persists in the population and the disease becomes endemic. Figure 3 illustrates this result.

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6 Conclusions In this work, we have proposed a new mathematical model for Ebola virus disease with two general incidence functions that cover various types of incidence rate existing in the literature. Under some assumptions on these general incidence functions, we have proved that the global stability of the proposed model is completely determined by the basic reproduction number R0 . More precisely, the disease-free equilibrium is globally asymptotically stable if R0 ≤ 1, which means the extinction of the disease. However, when R0 > 1, the disease-free equilibrium becomes unstable and the model has an endemic equilibrium which is globally asymptotically stable. In this case, the disease persists in the population. Since experimental treatments for Ebola are under development, it would be interesting to add to our present model a vaccination parameter. We leave this for future work.

References 1. WHO, Ebola virus disease (2018), http://www.who.int/en/news-room/fact-sheets/detail/ebolavirus-disease 2. WHO, Ebola situation report (2015), http://apps.who.int/iris/bitstream/handle/10665/190067/ ebolasitrep_21Oct2015_eng.pdf?sequence=1 3. X.-S. Wang, L. Zhong, Ebola outbreak in West Africa: real-time estimation and multiple-wave prediction. Math. Biosci. Eng. 12(5), 055–1063 (2015) 4. A. Rachah, D.F.M. Torres, Mathematical modelling, simulation, and optimal control of the 2014 Ebola outbreak in West Africa. Discret. Dyn. Nat. Soc. 2015, Article ID 842792 (2015)

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5. A. Rachah, D.F.M. Torres, Dynamics and optimal control of Ebola transmission. Math. Comput. Sci. 10(3), 331–342 (2016) 6. A. Rachah, D.F.M. Torres, Predicting and controlling the Ebola infection. Math. Methods Appl. Sci. 40(17), 6155–6164 (2017) 7. C.L. Althaus, Estimating the reproduction number of Ebola (EBOV) during outbreak in West Africa. PLOS Curr. 6 (2014), https://doi.org/10.1371/currents.outbreaks. 91afb5e0f279e7f29e7056095255b288 8. G. Chowell, N.W. Hengartner, C. Castillo-Chavez, P.W. Fenimore, J.M. Hyman, The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda. J. Theor. Biol. 229(1), 119–126 (2004) 9. E.V. Grigorieva, N. Khailov, Optimal intervention strategies for a SEIR control model of Ebola epidemics. Mathematics 3, 961–983 (2015) 10. I. Area, H. Batarfi, J. Losada, J.J. Nieto, W. Shammakh, A. Torres, On a fractional order Ebola epidemic model. Adv. Differ. Equ. 2015, 278 (2015) 11. WHO Ebola Response Team, Ebola virus disease in West Africa–the first 9 months of the epidemic and forward projections. N. Engl. J. Med. 371(16), 1481–1495 (2014) 12. J.S. Weitz, J. Dushoff, Modeling post-death transmission of Ebola: challenges for inference and opportunities for control. Sci. Rep. 5, 8751 (2015) 13. 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, 1–9 (2013) 14. I.S. Gradshteyn, I.M. Ryzhik, Routh-Hurwitz theorem, in Tables of Integrals, Series, and Products (Academic Press, San Diego, 2000) 15. J.P. LaSalle, The stability of dynamical systems, in Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, 1976)

Bifurcations in a Mathematical Model for Study of the Human Population and Natural Resource Exploitation I. M. Cholo Camargo, G. Olivar Tost, and I. Dikariev

1 Introduction Water, energy, and food are vital resources for human well-being, poverty reduction, and sustainable development [1, 2]. Zhang, in Ref. [3], performs a bibliographical review of the water-energy-food nexus, and proposes that these resources are closely related, given that water is used for power generation, via turbine movement, as well as fossil fuel extraction. Water is additionally used in food production in fishing, forestry, and agricultural activities. Power is necessary for water extraction, procurement, and treatment, and generates energy in the form of biofuels, such as sugar cane and corn. This relationship demands that, if action is taken for one of these, the remaining elements will be affected [2, 4, 5]. As the global population grows, so too do water, energy, and food consumption [6], and resource exploitation in general. Humans burn fossil fuels such as coal, petroleum, and natural gas, in order to generate energy. However, as these burn, carbon dioxide is released. In large quantities, carbon dioxide contributes to global warming, which, in turn causes climate change [7]. Climate change entails high temperatures, which lead to water evaporation, more powerful hurricanes, forest fires, and changes in precipitation. These produce rising in sea levels and lake and river flooding, which cause fresh water to combine with salt water [8, 9]. Other

I. M. Cholo Camargo () · G. Olivar Tost Department of Mathematics and Statistics, Universidad Nacional de Colombia - Manizales, Campus La Nubia, Manizales, Colombia Arizona State University, Tempe, AZ, USA e-mail: [email protected]; [email protected] I. Dikariev Brandenburg University of Technology, Cottbus, Germany © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_4

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human actions, including logging and burning forests in order to plant oil palms, favor CO2 , as they destroy the trees which absorb it. The same effect is seen on marine ecosystem destruction. These actions are performed unchecked, and without regard for the environmental effects generated thereby [10]. Humans both generate the greenhouse effect and eliminate the resources which control it [11]. La Guajira is one of the departments, which, according to the Instituto de Hidrología, Meteorología y Estudios Ambientales, presented an index of greater than 0.6 in 2014. This is quite a high dryness index, which is linked to high water scarcity, and as such, food scarcity. Consequently, the department’s economic and social development is limited [12]. The Ranchería River is the main source of water for the above-mentioned department, and provides the majority of municipalities with water for mining, fishing, livestock farming, and agriculture. The same body of water receives all of the waste produced by these human and economic activities [13–15]. Mining causes water contamination from the waste that it produces, and this contamination intensifies water scarcity [1, 16]. These economic activities may negatively affect the environment and humans generally, but are vital to the economy. Mining contributes 56% of the GDP, and coal extraction, among other mining activities, represent 93% of the GDP. Their MPI, however, is higher than the national MPI. Mathematical models have been proposed to study the interaction between natural resources and the human population. Volterra and Lotka separately formulated similar interaction models, which are now known as the predator-prey model [17, 18]. Brander employs a model similar to that of Volterra–Lotka, and uses humans as predators and resources as prey [19], in order to show the excessive exploitation of natural resources and their negative consequences on the human population. Yongzhen and their collaborators use this model to study fish [20], a resource for which Quin II has performed a bibliographical review of fishing models [21], and have additionally presented bioeconomical models in both free and reserved areas [22, 23]. D’Alessandro proposes a production-consumption model, in which the relationship between the human population and proposed system dynamics of the forest, climate change, and technology are analyzed [24]. Misra proposes that resources decrease when population and industrialization increase, but that they are not eliminated when proper technology use is employed together with economic efforts [25]. It is also revealed that, if the population’s demand for resources is controlled with economic incentives, the density of forest resources balances out [26]. Misra and Kusum further reinforce this idea, proposing a model for the conservation of forest resources, considering the technological and economic incentive areas [27]. Klausemir, in 1999, modeled the interaction between water and vegetation in semi-arid zones, and concluded that, with this model, vegetation patterns may be predicted [28]. Their model has been studied, and various modifications have been made thereto [29]. In this study, a mathematical model which studies the interaction between the human population and the resources they exploit, including the forest, fish, and water, is proposed. Numerical simulations are performed using the MATCONT

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software and theoretical parameter values, with which Hopf-type bifurcations which indicate the conditions under which humans may survive with said resources, such that they also prevail, are indicated. The present document is organized as follows: Sect. 2 presents the base and extended models, the latter of which will be utilized here. Section 3 details certain numerical simulations, and Sect. 4 contains the conclusions.

2 Mathematical Model 2.1 Base Model D’Alessandro considers a small population which exploits two resources: the land and forest, from which corn and wood are obtained. Where • • • • • • •

L is the number of people in the community. S is the number of trees in the forest. β is the proportion of people who work in the forest. 1 − β is the proportion of people who work in agriculture. ρ is the rate of natural forest growth. K1 is the maximum capacity. K2 is the minimum capacity. Falling below this value causes the forest to vanish entirely (Allee effect).

Considering a logistical model for forest growth with the Allee effect, the following is obtained:   S S ˙ S = ρS 1 − −1 K1 K2 Wood exploitation requires a certain number of people βL with technological factor α in order to measure the velocity at which trees are cut. αβLS Therefore, considering natural growth and forest exploitation, the tree variation is obtained.   S S ˙ − 1 − αβLS (1) S = ρS 1 − K1 K2 In accordance with D’Alessandro, population variation depends on total obtained production P = PT + PS and the consumption of the population σ L˙ = γ (P − σ L)

(2)

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where 0 < γ < 1 is a calorie factor, PT is land production, and PS is forest production PT = λ(1 − β)δ Lδ

PS = αβLS

where 0 < δ < 1 affects land production and λ is land fertility. By replacing total production P in Eq. (2), and adding conversion factor φ, the following is obtained:   L˙ = γ λ(1 − β)δ Lδ + φαβLS − σ L As such, the system below results: ⎧     ⎨S˙ = ρ S − 1 1 − S − αβL S K2 K1 ⎩L˙ = γ λ (1 − β)δ Lδ−1 + φαβS − σ  L

2.2 Extended Model Using the above ideas, the extended model is proposed. A population dedicated to land and forest exploitation, which also engages in mining and fishing, and consumes water, power, and food, is considered. It is assumed that the fish population has a growth rate of ρ2 , and that its growth rate is logistical. ⎧ ⎪ L˙ ⎪ ⎨ S˙ ⎪ ⎪ ⎩F˙

  = γ φ1α1 β1δ Lδ +  φ2 α2 β 2 LS + φ3 α3 β3 LF + φ4 α4 β4 L − σ L = ρ1 S kS2 − 1 1 − kS1 − α2 β2 LS  = ρ2 F 1 − kF3 − α3 β3 LF

(3)

Additionally, it is considered that agriculture is among the activities with the greatest water consumption [8] and that its production depends on this resource. Using the analytical solution of the logistical equation, δ(W ) =

δ0 δm (δm − δ0 )e−δw W + δ0

(4)

Analogically, this is performed with forest growth factor ρ1 , as more intensive growth causes greater water consumption [30]. ρ1 (W ) =

ρ0 ρm (ρm − ρ0 )e−ρw W + ρ0

(5)

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where • ρ0 , ρm are initial and maximum forest growth rates, respectively. • δ0 , δm are initial and maximum land production factors, respectively. Water may be extracted from natural resources such as rivers, dams, or aquifers, and the availability of said resource depends on precipitation and rain water on the surface. Precipitation and rain water depend on water evaporation and soil infiltration capacity [31]. Water growth is given by precipitation μ, it evaporates at a constant rate , and is absorbed by the soil. Said water is used for mining activities and human consumption. In accordance with Klausmeier’s framework, the below equation is obtained: W˙ = μ − W − γ1 α1 β1δ Lδ − γ4 α4 β4 L

(6)

Finally, the following system is achieved: ⎧ ⎪ ⎪L˙ ⎪ ⎪ ⎪ ⎨S˙ ⎪ F˙ ⎪ ⎪ ⎪ ⎪ ⎩˙ W

  = γ φ1α1 β1δ Lδ + φ α β LS + φ α β LF + φ α β L − σ L 2 2 2 3 3 3 4 4 4  = ρ1 S kS2 − 1 1 − kS1 − α2 β2 LS  = ρ2 F 1 − kF3 − α3 β3 LF

(7)

= μ − W − γ1 α1 β1δ Lδ − γ4 α4 β4 L L > 0,

S > 0,

F > 0,

W > 0.

3 Numerical Simulations Numerical simulations were performed using the MATCONT software, into which certain initial conditions and parameter values were entered. Certain working values were extracted from D’Alessandro [24] and Kar [23]: φ1 = 1, φ2,3,4 = 3, α1 = 20, α2 = 0.0001, α3 = 0.0003, α4 = 0.0001, β1 = 0.3, β2 = 0.3, β3 = 0.2, β4 = 0.2, σ = 1.8, k1 = 12000, k2 = 700, k3 = 200000, ρ2 = 0.08, γ = 0.1,  = 0.7, μ = 4848, γ1,4 = 1, ρ0 = 0.0000043, ρm = 0.025 = ρw , δ0 = 0.00012 = δw , δm = 0.7 with initial conditions (L0 , S0 , F0 , W0 ) = (200, 12000, 5000, 10000) Figure 1 shows the behavior of state variables with the parameter values defined from the beginning. This behavior indicates that, initially, the fish population grows

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more rapidly than the human population, while water and forest resources diminish. Water is the first to reach equilibrium. Figure 2 shows that, with a higher land production factor, human populations increase to a certain point, as they exploit resources. When the human population begins to decrease, the forest begins to recuperate, but the fish population disappears. However, with a lower fertility index, fish numbers are recuperated, see Fig. 3. Observation of Fig. 4 reveals that when the tree growth factor is very small, they disappear rapidly. The human population and fish do not disappear, but do take time to stabilize.

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Certain parameters are important because they affect the dynamics of the system surrounding the internal equilibrium point. E ∗ = (1302.5, 10685, 4620, 36915.5) For this reason, bifurcation diagrams were created for said parameters. First is the α2 technology parameter, for which points that characterize system dynamics, a Fold-type bifurcation point which indicates the value at which equilibrium points appear and disappear, and two Hopf-type bifurcation points, in which periodic oscillations are created, are found, Fig. 5. One of the Hopf points presents

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Fig. 5 Human population diagram, with respect to the α2 parameter

when α2 = 0.00024, and is unstable given that l1 = 3.227510e−05. L∗ = 1317.5, S ∗ = 6321.5, F ∗ = 2366.1, W ∗ = 6912.6 the other Hopf point α2 = 0.000119 and is stable, given that l1 = −2.226773e−05. L∗ = 1275.9, S ∗ = 2273.9, F ∗ = 8608.7, W ∗ = 6912.6 The Fold point appears when α2 = 0.00024 with normal coefficient a = 1.338144e−06 and L∗ = 1317, S ∗ = 6252.9, F ∗ = 2447.5, W ∗ = 6912.6 Likewise, these results are obtained when the proportion of people dedicated to forest exploitation parameter is varied, such that the Hopf bifurcation point is obtained when β2 = 0.72. This bifurcation is stable, given that l1 = −4.094039e−06, and the other point is when β2 = 0.35, which is unstable, l1 = 1.120292e−05, see Fig. 6. The technology applied to fishing may be an important factor in the search for population stability, as indicated by the Hopf points encountered on variance of the α3 parameter, and may be visualized in Fig. 7. These occur when α3 = 0.000115 with l1 = −3.674869e−06 L∗ = 3151.1, S ∗ = 5951.5, F ∗ = 18,224.8, W ∗ = 6912.5 and when α3 = 0.000196 with l1 = 2.058293e−05, the following is obtained: L∗ = 1903.5, S ∗ = 2781.4, F ∗ = 13,114.7, W ∗ = 6912.6

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It must be mentioned that, in addition to generating a Fold-type bifurcation when α = 0.000115 with normal coefficient, a = 1.088520e−0 in L∗ = 3165.6, S ∗ = 6235.7, F ∗ = 17,905.9, W ∗ = 6912.5 These same points are obtained β3 = 0.0769 and β3 = 0.13 with l1 = 3.635798e−05 and l1 = −5.700484e−06, respectively, for the Hopf-type bifurcations and the saddle-node case, which occurs when β3 = 0.0766 with a = −1.409051e−05.

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4 Conclusions A model was proposed, with the goal of finding strategies for an equilibrium between the human population and forest, fish, and water resources. By way of numerical simulations, it was found that chemical products, including fertilizers, may negatively affect the aquatic ecosystem, which coincides with that proposed in Ref. [32]. If forest exploitation is high and the growth rate is very low, the number of trees decreases. If the established threshold is exceeded, the forest simply disappears. Thus, not only is the proportion of individuals who work in an activity important for objective achievement, so too is adequate technology implementation in the activity. For example: in fishing, technological improvements may represent a threat, as more and more fish will be extracted, to the point where there are so few that they cannot reproduce at the same rate at which they are captured [33, 34]. Similar threats exist in the forest, however, as indicated by Misra, appropriate technological administration in forest exploitation may help to conserve this resource. Acknowledgements Ingrid M. Cholo acknowledges Colciencias for its partial support, under Grant 645-2014:Convocatoria Nacional Jóvenes Investigadores e Innovadores 2014 and Grant Convocatoria Doctorados Nacionales No. 727 de 2015 Colciencias. Gerard Olivar acknowledges the Universidad Nacional de Colombia for its partial support under Grant Modelamiento avanzado de mercados de energia electrica para toma de decisiones de inversion y establecimiento de politicas, DIMA project number 35467 and Colciencias under Grant Modelado y Simulacion del Metabolismo Urbano de Bogota D.C. contract 022-2017. He also acknowledges CEMarin for its support in this research.

References 1. FAO, The Water-Energy-Food Nexus: A New Approach in Support of Food Security and Sustainable Agriculture (Food and Agriculture Organization of the United Nations, Roma, 2014), http://www.fao.org/3/a-bl496e.pdf 2. D.L. Keairns, M. El-Halwagi, K.M. Ng, Editorial overview: process systems engineering: a chemical engineering perspective of the food energy water nexus. Curr. Opin. Chem. Eng. 18, iii–iv (2017) 3. C. Zhang, X. Chen, Y. Li, W. Ding, G. Fu, Water-energy-food nexus: concepts, questions and methodologies. J. Clean. Prod. 195, 625–639 (2018) 4. D. Garcia, F. You, Systems engineering opportunities for agricultural and organic waste management in the food-water-energy nexus. Curr. Opin. Chem. Eng. 18, 23–31 (2017) 5. L.G. Carmona, K. Whiting, A. Carrasco, The water footprint of heavy oil extraction in Colombia: a case study. Water 9, 340 (2017) 6. D.J. Garcia, F. You, The water-energy-food nexus and process systems engineering: a new focus. Comput. Chem. Eng. 91, 49–67 (2016) 7. IPCC, Technical summary, in Climate Change 2007: The Physical Science Basis, ed. by S. Solomon, D. Qin, M. Manning, Z. Chen, M. Marquis, K.B. Averyt, M. Tignor, H.L. Miller. Contribution of Working Group I to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change (Cambridge University Press, Cambridge, 2007)

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8. N. Mancosu, R. Snyder, G. Kyriakakis, D. Spano, Water scarcity and future challenges for food production. Water 7, 975–992 (2015) 9. W.J.M. Martens, R. Slooff, E.K. Jackson, Climate change, human health, and sustainable development. WHO Bull. OMS 75, 583–588 (1997) 10. INVEMAR, Informe del estado de los ambientes y recursos marinos y costeros en Colombia: Año 2011. Serie de Publicaciones Periódicas No. 8. Santa Marta (2012), p. 203, http://www. invemar.org.co/redcostera1/invemar/docs/IER_2011.pdf 11. PNUD, Objetivos de Desarrollo Sostenible, Colombia: Herramientas de aproximación al contexto local (2016), http://www.co.undp.org/content/dam/colombia/docs/ODM/undp-coODSColombiaVSWS-2016.pdf 12. J. Bonet-morón, L.W. Hahn-de-castro, La mortalidad y desnutrición infantil en La Guajira. Banco de la República 52(255), 1–49 (2017), http://www.banrep.gov.co/docum/Lectura_ finanzas/pdf/dtser_255.pdf 13. C. Doria Argumedo, L.J. Vivas Aguas, Fuentes terrestres de contaminacin en la zona costera de La Guajira, Colombia. Rev. Investig. Agrar. Ambient. 7, 123–138 (2016) 14. A. Londoño, Seminario Taller Buenas Prácticas Agrícolas. Unión Temporal Universidad Tecnológica de Pereira Comité de Cafeteros de Risaralda (Convenio -Ministerio de Comercio Industria y turismo - Fomipyme y Gobernacin de Risaralda, Risaralda, 2006) 15. J. Romero, Calidad del Agua. Escuela Colombiana de Ingeniería. ISBN: 958-8060-53-2. Bogotá (2005) 16. FAO, WWC. Towards a Water and Food Secure Future: Critical Perspectives for PolicyMakers (Natural Resources and Environment Department, 2015), http://www.fao.org/3/ai4560e.pdf 17. V. Volterra, Variazioni e fluttuazioni del numero d’individui in specie animali conviventi. Mem. R. Accad. Naz. dei Lincei 5, 113 (1926) 18. A.J. Lotka, Elements of Mathematical Biology (Dover, New York, 1956) 19. J.A. Brander, M.S. Taylor, The simple economics of Easter Island: a ricardo-malthus model of renewable resource use. Am. Econ. Rev. 88, 119–138 (1998) 20. P. Yongzhen, L. Yunfei, Y. Rong, A prey-predator model with harvesting for fishery resource with reserve area. Appl. Math. Model. 37, 3048–3062 (2013) 21. T.J. Quinn II, Rumminations on the development and future of population dynamics models in fisheries. Nat. Resour. Model. 16, 341–392 (2003) 22. H. Matsuda, T.K. Kar, A bioeconomic model of a single species fishery with a marine reserve. J. Environ. Manag. 86, 171–180 (2008) 23. T.K. Kar, K. Chakraborty, Economic perspective of marine reserves in fisheries: a bioeconomic model. Math. Biosci. 240, 212–222 (2012) 24. S. D’Alessandro, Non-linear dynamics of population and natural resources: the emergence of different patterns of development. Ecol. Econ. 62, 473–481 (2007) 25. J.B. Shukla, K. Lata, A.K. Misra, Modeling the depletion of a renewable resource by population and industrialization: effect of technology on its conservation. Nat. Resour. Model. 24, 242– 267 (2011) 26. A.K. Misra, K. Lata, J.B. Shukla, Effects of population and population pressure on forest resources and their conservation: a modeling study. Environ. Dev. Sustain. 16, 361–374 (2014) 27. A.K. Misra, K. Lata, A mathematical model to achieve sustainable forest management. Int. J. Model. Simul. Sci. Comput. 6, 1550040 (2015) 28. C.A. Klausmeier, Regular and irregular patterns in semiarid vegetation. Science 284, 1826– 1828 (1999) 29. J.A. Sherrat, An analysis of vegetation stripe formation in semi-arid landscapes. J. Math. Biol. 51, 183 (2005) 30. Organizacin de las Naciones Unidas para la Alimentacin y la Agricultura. Los bosques y el agua, http:www.fao.org/sustainable-forest-management/toolbox/modules/forestand-waterbasic-knowledgees

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31. R. González-Bravo, M. Sauceda-Valenzuela, J. Mahlknecht, E. Rubio-Castro, J.M. PonceOrtega. Optimization of water grid at macroscopic level analyzing water-energy-food nexus. ACS Sustain. Chem. Eng. 6(9), 12140–12152 (2018) 32. J.A. Blanco, Bosques, suelo y agua: explorando sus interacciones. Rev. Econ. 26(2), 1–9 (2017) 33. Universitat Autónoma de Barcelona, Las mejoras sin restricciones en tecnología pesquera amenazan el futuro del pescado salvaje (2017), https://www.uab.cat/web/sala-de-prensa/ detalle-noticia/las-mejoras-sin-restricciones-en-tecnologia-pesquera-amenazan-el-futurodel-pescado-y-el-marisco-salvaje-1345667994339.html?noticiaid=1345722216945 34. E.D. Galbraith, D.A. Carozza, D. Bianchi. A coupled human-earth model perspective on longterm trends in the global marine fishery. Nat. Commun. 8, 14884 (2017)

Mathematical Analysis of the Dynamics of HIV Infection with CTL Immune Response and Cure Rate Sanaa Harroudi and Karam Allali

1 Introduction Human Immunodeficiency Virus (HIV) is a worldwide public health problem and more than 35 million people have died of this disease [1]. In the last decades, many mathematical models have been developed in order to better describe and understand the dynamics of the HIV disease [2–5]. Mathematical models describing the interaction between the HIV viruses, CD4+ T cells, and the cytotoxic T lymphocytes (CTL) taking into account antiretroviral treatment regimes have been carried out in Ref. [6]. For the importance of the optimization techniques and optimal control in the study of HIV, we refer the reader to Refs. [7, 8] and references therein. In this paper, we extend the investigation of Ref. [7] by introducing a cure rate of the infected cells to the susceptible host cells before the viral genome is integrated into the genome of the lymphocyte [9, 10]. The dynamics of HIV infection with CTL responses and cure rate that we consider is given by the following nonlinear system of differential equations:

S. Harroudi () · K. Allali Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies, University Hassan II of Casablanca, Mohammedia, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_5

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⎧ dx(t) ⎪ ⎪ = λ − dx(t) − βx(t)v(t) + ρy(t), ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dy(t) ⎪ ⎪ = βx(t)v(t) − (a + ρ)y(t) − py(t)z(t), ⎨ dt ⎪ dv(t) ⎪ ⎪ = aNy(t) − μv(t), ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ dz(t) ⎪ ⎩ = cx(t)y(t)z(t) − hz(t), dt

(1)

the initial conditions x(0) = x0 , y(0) = y0 , v(0) = v0 and z(0) = z0 are given. In this model x(t), y(t), v(t), and z(t) denote the concentrations of uninfected cells, infected cells, HIV virus, and CTL cells, respectively. The healthy CD4+ T cells (x) grow at a rate λ, die at a rate d, and become infected by the virus at a rate βxv. Infected cells (y) die at a rate a, cure at a rate ρ, and killed by the CTLs response at a rate p. Free virus (v) is produced by the infected cells at a rate aN and die at a rate μ, where N is the number of free virus produced by each actively infected cell during its life time. Finally, CTLs (z) expand in response to viral antigen derived from infected cells at a rate c and decay in the absence of antigenic stimulation at a rate h. This paper is organized as follows. First, in next section, we will give a result concerning the well-posedness of solutions. In Sect. 3, we perform a mathematical analysis of the model. Section 4 deals with the numerical simulations. The last section concludes the work.

2 Well-Posedness of Solutions For the problems dealing with cell population evolution, the cell densities should remain nonnegative and bounded. In this section, we will establish the positivity and boundedness of solutions of the model (1). First of all, for biological reasons, the parameters x0 , y0 , v0 , and z0 must be larger than or equal to 0. Hence, we have the following result: Prop 2.1 The solutions of the problem (1) exist. Moreover, they are bounded and nonnegative for all t > 0. Proof The proof is given in Appendix 1.

3 Analysis of the Model In this section, we show that there exists a disease-free equilibrium point and two infection equilibrium points, we study the stability of these equilibrium points.

Mathematical Analysis of the Dynamics of HIV Infection with CTL Immune. . .

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First, the Jacobian matrix of the system (1) is given by ⎛

⎞ −d − βv ρ −βx 0 ⎜ βv −(a + ρ) − pz βx −py ⎟ ⎟. J =⎜ ⎝ ⎠ 0 aN −μ 0 cyz cxz 0 cxy − h

(2)

A straightforward calculation gives the following expression for the basic reproductive number in the model (1): R0 =

aNβλ . dμ(a + ρ)

3.1 Steady States System (1) has an infection-free equilibrium which represent the disease-free equilibrium corresponding to the maximal level of healthy CD4+ T cells.  Ef =

λ , 0, 0, 0 . d

In addition to the disease-free equilibrium, system (1) has three endemic equilibria. The first of them is E1 = (x1 , y1 , v1 , 0), where x1 =

λ , dR0

y1 =

λ(R0 − 1) , aR0

v1 =

λN(R0 − 1) , μR0

The second endemic steady state is E2 = (x2 , y2 , v2 , z2 ), where x2 = y2 = v2 =

2μρh

,

−λμc + aNβh +

(λμc − aNβh)2 + 4μ2 ρcdh

−λμc + aNβh +

(λμc − aNβh)2 + 4μ2 ρcdh , 2cμρ

aN(−λμc + aNβh +

(λμc − aNβh)2 + 4μ2 ρcdh , 2cμ2 ρ

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z2 =

2aNβhρ − (a + ρ)(−λμc + aNβh + p(−λμc + aNβh +

(λμc − aNβh)2 + 4μ2 ρcdh

(λμc − aNβh)2 + 4μ2 ρcdh)

.

3.2 The Stability Analysis Here, we will analyze locally asymptotical stability of the three equilibrium points.

3.2.1

Stability of the Disease-Free Equilibrium

We begin by studying the stability of the disease-free equilibrium Ef . The following result holds. Prop 3.1 (1) The disease-free equilibrium, Ef , is locally asymptotically stable for R0 < 1. (2) The disease-free equilibrium, Ef , is unstable for R0 > 1. Proof The proof is given in Appendix 2.

3.2.2

Stability of the Endemic Equilibria

We start by studying the local stability of the infected-equilibrium E1 . Prop 3.2 (1) If R0 < 1, then the point E1 does not exist and E1 = Ef when R0 = 1. (2) If R0 > 1 and R1 ≤ 1, then E1 is locally asymptotically stable. (3) If R1 > 1, then E1 is unstable. Here R1 =

aNβλμc(a + ρ) . a 3 N 2 β 2 h + cdμ2 (a + ρ)2

Proof The proof is given in Appendix 2. For the second endemic-equilibrium point E2 , we have the following result Prop 3.3 (1) If R2 < 1, then the point E2 does not exist. (2) If R2 > 1, then E2 is locally asymptotically stable. Here R2 =

2aNβhρ (a + ρ)(−λμc + aNβh (λμc − aNβh)2 + 4μ2 ρcdh)

Proof The proof is given in Appendix 2.

.  

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4 Numerical Simulations For our numerical simulations, we have used the Euler finite-difference scheme method in order to discretize the four equations. The parameter values or ranges used are presented in Table 1. Figure 1 shows the evolution of the infection during the first 500 days of observation. We can see that the curves converge towards the disease-free steady state Ef = (10, 0, 0, 0). In this case the basic reproduction number is less than unity (R0 = 5.952 × 10−1 < 1) which supports our theoretical result about the stability of Ef . Figure 2, shows the behavior of the infection during the first 500 days of observation. All the curves converge towards the endemic equilibrium E1 = (8.266, 5.777 × 10−1 , 86.66, 0) and R0 = 1.209 > 1 and R1 = 4.389 × 10−1 < 1. We can see that the virus persists in the absence of the cellular immunity. Figure 3, shows the evolution of the infection during the first 500 days of observation. We can observe that the curves converge to the endemic equilibrium E2 = (11.53, 9.102, 1.365×103 , 1.225×102 ) and R2 = 1.395 > 1. Finally, Fig. 4, shows the importance of adding the cure rate into our suggested viral dynamics model for the same parameters as in Fig. 2. Indeed, by increasing the value of the cure rate parameter, the basic reproduction number decreases and becomes less than unity, which means that the disease becomes not endemic.

Table 1 Parameters, their symbols and meaning, and default values used in HIV literature Parameters λ d ρ β

p

Meaning Source rate of CD4+ T cells Decay rate of healthy cells Cure rate of infected cells Rate at which CD4+ T cells become infected Death rate of infected CD4+ T cells, not by CTL Clearance rate of virus Number of virions produced by infected CD4+ T cells Clearance rate of infection

c h

Activation rate of CTL cells Death rate of CTL cells

a μ N

Value 1–10 cells µl−1 days 0.007–0.1 days−1 0.01 days−1 0.00025–0.5 µl virion−1 days−1 0.2–0.3 days−1

References [11] [11] [9] [11]

2.06–3.81 days−1 6.25–23599.9 virion−1

[12] [5, 13]

1 × 10−4 –4.048 × 10−4 ml virion days−1 0.0051–3.912 days−1 0.004–8.087 days−1

[5, 14]

[11]

[5] [5]

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Uninfected cells

10

Infected cells

9 8 7 6 5

0

100

200

300

400

0.5

0 0

500

100

200

300

400

500

400

500

Time (days)

Time (days) 2 1.5

CTLs

Virus

100

50

1 0.5

0

0

100

200

300

400

0

500

0

100

Time (days)

200

300

Time (days)

Fig. 1 Behavior of the infection during the first days for λ = 1; d = 0.1; β = 0.00025; p = 0.001; h = 0.2; a = 0.2; c = 0.03; N = 750; μ = 3; and ρ = 0.01 which correspond to the stability of the disease-free equilibrium Ef

1

Infected cells

Uninfected cells

9 8 7 6 5

0

100

200

300

400

0.8 0.6 0.4

500

0

100

Time (days)

200

300

400

500

400

500

Time (days) 2 1.5

CTLs

Virus

100

50

1 0.5

0

0

100

200

300

Time (days)

400

500

0

0

100

200

300

Time (days)

Fig. 2 Behavior of the infection during the first days for λ = 1; d = 0.1; β = 0.00025; p = 0.001; h = 1.2; a = 0.2; c = 0.03; N = 1500; μ = 3; and ρ = 0.01 which correspond to the stability of the endemic-equilibrium E1

Mathematical Analysis of the Dynamics of HIV Infection with CTL Immune. . . 20

Infected cells

Uninfected cells

25 20 15 10 5

0

100

200

300

400

15 10 5 0

500

0

100

200

300

400

500

400

500

Time (days)

3000

150

2000

100

CTLs

Virus

Time (days)

1000 0

65

50

0

100

200

300

400

0

500

Time (days)

0

100

200

300

Time (days)

Fig. 3 Behavior of the infection during the first days for λ = 5; d = 0.1; β = 0.00025; p = 0.001; h = 3.15; a = 0.3; c = 0.03; N = 1500; μ = 3; and ρ = 0.01 which correspond to the stability of the endemic equilibrium E2 1.3 1.2 1.1 1 0.9 0.8 0.7 0

0.02 0.04 0.06 0.08

0.1

0.12 0.14 0.16 0.18

0.2

Fig. 4 The reproduction number R0 (solid line) compared to the constant value one (dashed line) for λ = 1; d = 0.1; β = 0.00025; p = 0.001; h = 1.2; a = 0.2; c = 0.03; N = 1500; and μ = 3

5 Conclusion In this work we gave a mathematical model that describes HIV infection with cure rate. The considered model includes four differential equations describing the interaction between the uninfected cells, infected cells, free HIV viruses, and CTL immune response cells. First, the well-posedness of solutions result is proved. Then, We have studied the local stability of the disease-free equilibrium and the endemic

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equilibria. Finally, numerical simulations are performed in order to strengthen our theoretical findings and to show the effectiveness of the cure rate in reducing the severity of the disease.

Appendix 1: Well-Posedness of Solutions First, we will show that the nonnegative orthant R4+ = {(x, y, v, z) ∈ R4 : x ≥ 0, y ≥ 0, v ≥ 0 and z ≥ 0} is positively invariant. Indeed, for (x(t), y(t), v(t), z(t)) ∈ R4+ we have: x˙ |x=0 = λ ≥ 0, y˙ |y=0 = βxv ≥ 0, v˙ |v=0 = aNy ≥ 0, and z˙ |z=0 = 0 ≥ 0. Therefore, all solutions initiating in R4+ are positive. Next, we will prove that these solutions remain bounded. Remark that by adding the two first equations in (1), we have x˙1 = λ − dx − ay − pyz, thus x1 (t) ≤ x1 (0)e−δt +

λ (1 − e−δt ), δ

where x1 (t) = x(t) + y(t) and δ = min(d; a), since 0 ≤ e−δt ≤ 1 and 1 − e−δt ≤ 1. λ We deduce that x1 (t) ≤ x1 (0) + . Therefore x and y are bounded. δ From the equation v˙ = aNy − μv, we have v(t) ≤ v(0)e−μt + aN

t

y(ξ )e(ξ −t)μ dξ

0

then, v(t) ≤ v(0) +

aN y ∞ (1 − e−μt ). μ

aN y ∞ . Therefore v is bounded. μ Now, we prove the boundedness of z(t). From the fourth equation of (1), we have

Since 1 − e−μt ≤ 1, we have v(t) ≤ v(0) +

z˙ (t) + hz(t) = cx(t)y(t)z(t). Moreover, from the second equation of (1), it follows that z˙ (t) + hz(t) =

c x(t) (βx(t)v(t) − (a + ρ)y(t) − y(t)) ˙ . p

Thus, by integrating over time, we have z(t) = z(0)e−ht +

0

t

c x(s) (βx(s)v(s) − (a + ρ)y(s) − y(s)) ˙ eh(s−t) ds. p

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67

From the boundedness of x, y, and v, and by using integration by parts, it follows the boundedness of z(t).

Appendix 2: Analysis of the Model For the local stability result (3.1), the proof is stated as follows. At the disease-free equilibrium, Ef , the Jacobian matrix is given as follows: ⎞ βλ − 0 ⎟ ⎜ d ⎟ ⎜ ⎜ 0 −(a + ρ) βλ 0⎟ =⎜ ⎟. d ⎟ ⎜ ⎝ 0 aN −μ 0⎠ 0 0 0 −h ⎛

JEf

−d ρ

The characteristic polynomial of the Jacobian matrix (4) at Ef is given by ζ 2 + a1 ζ + a2 = 0,

(3)

where a1 = a + ρ + μ, a2 = (a + ρ)μ(1 − R0 ). The two roots of (3) are: ζ1 = ζ2 =

−(μ + a + ρ) −

(μ + a + ρ)2 − 4μ(a + ρ) (1 − R0 ) , 2

−(μ + a + ρ) +

(μ + a + ρ)2 − 4μ(a + ρ) (1 − R0 ) . 2

It is clear that ξ1 is negative. Moreover, ξ2 is negative when R0 < 1, which means that Ef is locally asymptotically stable. The local stability proof of the endemic point E1 (3.2) is given as follows. It is clear that when R0 < 1, the point E1 does not exist and when R0 = 1, this point is equal to the free equilibrium (E1 = Ef ). Assume now that R0 > 1, the Jacobian matrix at E1 is ⎞ ρ −βx1 0 −d − βv1 ⎜ βv1 −(a + ρ) βx1 −py1 ⎟ ⎟. J1 = ⎜ ⎠ ⎝ 0 aN −μ 0 0 0 0 cx1 y1 − h ⎛

(4)

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The characteristic equation associated with (4) is given by (cx1 y1 − h − ζ )(ζ 3 + b1 ζ 2 + b2 ζ + b3 ) = 0, where b1 = a + ρ + d + μ + βv1 , b2 = μd + (d + μ)(a + ρ) + (a + μ)βv1 − aNβx1 , b3 = dμ(a + ρ) + μβv1 − aNdβx1 . Simple calculation of the first eigenvalue leads to cx1 y1 − h =

hD z (R1 − 1) R1

with Dz =

λμc(a + ρ) . a 2 Nβh

The sign of this eigenvalue is negative if R1 < 1. Moreover, from the Routh– Hurwitz theorem [15], the other eigenvalues of the above matrix have negative real parts. Consequently, E1 is locally asymptotically stable when R0 > 1 and R1 ≤ 1. Finally, for the local stability of the equilibrium point E2 (3.3), we have the following result. The endemic equilibrium point E2 can be rewritten as follows:  E2 =

aNA (a + ρ)(R2 − 1) 2μρh A , , , A 2cμρ 2cρμ2 p

(5)

with A = −λμc + aNβh +

!

(λμc − aNβh)2 + 4μ2 ρcdh.

From (5), it is clear that the endemic-point E2 does not exist if R2 < 1. ⎞ −d − βv2 ρ −βx2 0 ⎜ βv2 −(a + ρ) − pz2 βx2 −py2 ⎟ ⎟. J2 = ⎜ ⎠ ⎝ 0 aN −μ 0 cx2 z2 0 cx2 y2 − h cy2 z2 ⎛

(6)

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The characteristic equation associated to (6) is given by ξ 4 + c1 ξ 3 + c2 ξ 2 + c3 ξ + c3 = 0, where c1 = d + μ + a + ρ + βv2 + pz2 , c2 = d(μ + a + ρ) + μ(a + ρ) + (a + μ)βv2 + (d + μ)pz2 + βv2 pz2 − aNβx2 − phz2 , c3 = dμ(a + ρ) + βaμv2 + d(μ + h)pz2 + (μ + h)βv2 pz2 + cpρy22 z2 − daNβx2 , c4 = dhμpz2 + hμβv2 pz2 + cρμpz2 y22 − aNβhpz2 y2 .

From the Routh–Hurwitz theorem applied to the fourth order polynomial, the eigenvalues of the Jacobian matrix (6) have negative real parts since we have c1 c2 > c3 and c1 c2 c3 > c32 + c12 c4 . Consequently, we obtain the asymptotic local stability of the endemic point E2 .

References 1. World Health Organization HIV/AIDS Key facts, http://www.who.int/mediacentre/factsheets/ fs360/en/index.html. Accessed July 2017 2. A.S. Perelson, P.W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo. SIAM Rev. 41(1), 3–44 (1999) 3. M. Nowak, R. May, Mathematical biology of HIV infection: antigenic variation and diversity threshold. Math. Biosci. 106(1), 1–21 (1991) 4. D. Kirschner, Using mathematics to understand HIV immune dynamics, Notices Amer. AMS Notices 43(2), 191–202 (1996) 5. M.S. Ciupe, B.L. Bivort, D.M. Bortz, P.W. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models. Math. Biosci. 200(1), 1–27 (2006) 6. K. Allali, Y. Tabit, S. Harroudi, On HIV model with adaptive immune response, two saturated rates and therapy. Math. Model. Nat. Phenom. 12(5), 1–14 (2017) 7. K. Allali, S. Harroudi, D.F.M. Torres, Analysis and optimal control of an intracellular delayed HIV model with CTL immune response. Math. Comput. Sci. 12(2), 111–127 (2018) 8. M. Ciupe, B. Bivort, D. Bortz, P. Nelson, Estimating kinetic parameters from HIV primary infection data through the eyes of three different mathematical models. Math. Biosci. 200(1), 1–27 (2006) 9. X. Zhou, X. Song, X. Shi, A differential equation model of HIV infection of CD4+ T cells with cure rate. J. Math. Anal. Appl. 342(2), 1342–1355 (2008) 10. X. Liu, H. Wang, W. Ma, Global stability of an HIV pathogenesis model with cure rate. Nonlinear Anal Real World Appl. 12(6), 2947–2961 (2011) 11. R. Culshaw, S. Ruan, R. Spiteri, Optimal HIV treatment by maximising immune response. J. Math. Biol. 48(5), 545–562 (2004)

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12. A.S. Perelson, A.U. Neumann, M. Markowitz, J.M. Leonard, D.D Ho, HIV-1 dynamics in vivo: virion clearance rate, infected cell life-span, and viral generation time. Science. 271(5255), 1582–1586 (1996) 13. Y. Wang, Y. Zhou, F. Brauer, J.M. Heffernan, Viral dynamics model with CTL immune response incorporating antiretroviral therapy. J. Math. Biol. 67(4), 901–934 (2013) 14. K.A. Pawelek, S. Liu, F. Pahlevani, L. Rong, A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data. Math. Biosci. 235(1), 98–109 (2012) 15. I.S. Gradshteyn, I.M. Ryzhik, Routh Hurwitz Theorem, in Tables of Integrals, Series, and Products (Academic Press, San Diego, 2000)

Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell Population in the Light of Suns and Stars Y. Elalaoui and L. Alaoui

1 Introduction The cell is the most important basic unit in all living organisms. It is recognized that a cell transits through the cell cycle corresponding to phases G1-S-G2-M (first gap, synthesis phase, second gap, mitosis, respectively). In order to maintain a living organism, the cell possesses different abilities such as the fact to retain, recover, and translate the genetic information. Among these processes: the cell replication which occurs at phase M, where at each division the hereditary information is passed on from a mother cell to its daughter by a copy of its genome. It is known that the duplication occurs by the use of one DNA strand which is antiparallel as a template for the formation of a complementary strand. From the biochemical characteristics of the DNA polymerases, it is known that a labile primer is required to initiate unidirectional synthesis such that each duplex would be shortened on the 5’ end of the daughter DNA strand and it should proceed smoothly to the end of its template. However the 3’ end of the parental strand is left incompletely copied which is converted to a double stranded deletion in the subsequent generation. Thus, the inability of DNA polymerase to fully replicate the template is termed the end-replication problem. The division is not identical where a gradual loss of chromosomal ends known as telomeres is accumulated at each replication. This loss of DNA leads to cell cycle exit [25, 26]. Indeed cellular aging and the replicative senescence are linked together [22] where the loss of telomere in primary human cell would accumulate to reach Y. Elalaoui () Faculty of Sciences, Department of Mathematics, Mohammed V University, Agdal, Rabat, Morocco L. Alaoui International University of Rabat, Sala Al-Jadida, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_6

71

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the Hayflick limit and contributes to the programmed death of the organism which demonstrates that such cells have a finite lifetime. These facts impose a hierarchy based on how differentiated cells are. In order to understand this phenomenon, multiple models have been deeply studied, such as in Refs. [10, 18, 19, 23, 24, 27, 28]. The Olovnikov assumption [25, 26] plays an important role in some models: the authors in Ref. [8] explain rigorously the cell aging process where they demonstrate the polynomial growing of cells in time such that the cells with least telomeres are the cells which grow fastly. In Ref. [9] a linear age structured model was investigated where the overlapping of generations is allowed, they gave result about the polynomial growth of the telomere with the stabilization of the highest class which is due to that in each division a daughter is in the same telomere state as the mother cell and one daughter cell is in lower telomeric state. Further, several extensions of the model have been investigated such as the work by Dyson et al. by imposing a logistic loss term for every class which knows a polynomial growing and resulting in extinction. The model proposed here tries to describe the interactions between the several classes of cells based on how many mutations they accumulated and how long their telomeres are, only by tracking their lineages. This paper is composed as follows. In Sect. 2, we introduce some results on the machinery of suns and stars. In Sect. 3, we reformulate the model as an abstract integral equation via some transformation. Finally, in Sect. 4 we give the results about the asynchronous exponential growth property associated to the model.

2 Preliminaries A family of bounded linear operators (T (t))t≥0 on some Banach space X such as the following holds: (i) T (0) = IX , where I is the identity operator. (ii) T (t + s) = T (t)T (s), for t, s ≥ 0 (the semigroup property). is called a semigroup on X. In addition to (i) and (ii), if lim T (t)x − x X = 0,

t→0+

for all x ∈ X

we say that (T (t))t≥0 is a strongly continuous semigroup on X. It is known that the semigroup is connected to the abstract Cauchy problem [21] of the form d (T (t)x) = A(T (t)x) dt

Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell. . .

73

where the operator A is called the infinitesimal generator of the semigroup (T (t))t≥0 and it is given by Ax = lim

t→0+

T (t)x − x t

(1)

for x ∈ D(A) := {x ∈ X, the limit in (1) exists in X}. It is known that A is a closed densely defined operator [21]. In the following, we are considering the Banach space of the type X = L1 ([−r, 0], F ) where 0 < r ∈ R is finite and F is some Banach space, endowed with the norm

||f ||1 :=

0 −r

||f (s)||F ds

for f ∈ X

Definition 2.1 The strongly continuous semigroup (T (t))t≥0 on X which is defined by the (left) translation T (t)f (s) := f (t + s)

if t + s < 0

is called a semigroup of translation. Let the adjoint semigroup (T ∗ (t))t≥0 on the dual space X∗ such that T ∗ (t) := (T (t))∗ and (T ∗ (t))t≥0 satisfies the condition T ∗ (0) = I and the semigroup property. Since (T (t))t≥0 is a strongly continuous semigroup on some no reflexive space X it is not necessary that the semigroup T ∗ is strongly continuous. Indeed it is continuous with respect to the weak*-topology [29]. The adjoint operator A∗ is the infinitesimal generator associated with the semigroup (T ∗ (t))t≥0 which is weakly*-densely defined in X∗ such that the norm closure X = D(A∗ ) is the maximal subspace on which (T ∗ (t))t≥0 is strongly continuous. It is given by X := {x ∗ ∈ X∗ : lim ||T ∗ (t)x ∗ − x ∗ ||X∗ = 0} t→0+

By taking the restriction to the invariant subspace X , the semigroup T  := T ∗ |X is a strongly continuous semigroup. With this latter, we associate its generator A defined by D(A ) = {x  ∈ D(A∗ ) : A∗ x  ∈ X } A x  = A∗ x 

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We restart this procedure of construction by duality, so from T  we obtain T ∗ on X∗ associated to the generator A∗ . Analogously, we repeat the construction to obtain T  , X , and A such that the adjoint operator (T ∗ (t))t≥0 is strongly continuous on X = D(A∗ ). Therefore, we can use the pairing with X and X to define an embedding j : X → X∗ such that < j x, x  > =< x  , x >,

x ∈ X, x  ∈ X

Obviously, j (X) ⊂ X . In the case where j (X) = X , we say that X is reflexive with respect to the strongly continuous semigroup (T (t))t≥0 . On the level of the generator, it is well known that if we perturb A0 of a some strongly continuous semigroup (T0 (t))t≥0 on some -reflexive space X with some bounded linear operator  from X into X∗ . Then, we can construct some perturbed semigroup (T (t))t≥0 on X solution of the variation constant formula [11]:

t

T (t)x = T0 (t)x + 0

T0∗ (t − s)T (s)xds

(2)

The following result say more about the integral in Eq. (2) which is considered as a weak* integral "t Proposition 2.2 ([11]) Let F : t → 0 T ∗ (t − s)f (s)ds be the weak* integral where f : R+ → X∗ . Then the following holds (a) If f is weakly* continuous, then F is weakly* continuous with F (t) ∈ X∗ . (b) If f is norm continuous, then F is norm continuous where F is an X -valued function. Since we investigate A∗ , A , and A∗ we can find a characterization of the generator A of perturbed semigroup (T (t))t≥0 (see Ref. [16]). Indeed we have the following important result on the characterization of the generators Proposition 2.3 ([13]) ∗ = A∗ +  (1) D(A∗ ) = D(A∗ 0 ) and A 0 ∗ (2) A is the part of A in X, such that

D(A) = {x ∈ X : x ∈ D(A∗ ), A∗ x ∈ X} Ax = A∗ x + x

Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell. . .

75

3 The Model The subject of our investigation is the following model ∂ ∂ uj,i (a, t) + uj,i (a, t) = −(μj,i (a) + βj,i (a))uj,i (a, t) ∂t ∂a

A n   uj,i (0, t) = 2 pj,k,i βk,i (a)uk,i (a, t)da 0

k=j



A

+ qj,k,i−1

βk,i−1 (a)uk,i−1 (a, t)da 0

j = 1, 2, . . . , n, i = 0, 1, . . . , m − 1

(3)

which describes the interaction between proliferate cells of densities uj,i (a, t) of age a at time t. These cells experience several states of differentiation through tracking their telomeric state where the index j corresponds to the number of telomeres of a cell. Furthermore, these cells acquired a multiple levels of mutations where the index i is the number of mutations a cell has accumulated. The cells divide and die respectively with rates βj,i (a), μj,i (a) (the index i, j correspond to the telomeric class j and mutation class i). The transitions from the mother cells in two distinct classes of telomeric state j = 1, 2, . . . , n and mutation state i = 0, . . . , m − 1 to their daughters obeys the two rules established by the probabilities pj,k,i (which corresponds to the probability that a cell in the telomeric state k and mutation state i produce by division a cell in the telomeric state j and mutation state i) and qj,k,i−1 (which corresponds to the probability that a cell in the telomeric state k and mutation state i − 1 produce by acquiring mutation a cell in the telomeric state j and mutation state i). These rules are interpreted by the following

(Hp,q )

⎧ 1 ⎪ ⎪ pj,j,i = ∀1 ≤ j ≤ n, 0 ≤ i ≤ m − 1 ⎪ ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎪ p = 0 for j > k, ∀2 ≤ j ≤ n, 0 ≤ i ≤ m − 1 ⎪ ⎨ j,k,i qj,k,i = 0 for j > k, ∀2 ≤ j ≤ n, 0 ≤ i ≤ m − 1 ⎪ ⎪ ⎪ ⎪ n n ⎪ ⎪  1 ⎪ ⎪ ⎪ pj,k,i + qj,k,i = ∀1 ≤ j ≤ n, 0 ≤ i ≤ m − 1 ⎪ ⎩ 2 k=j

k=j

We consider the state space X = L1 (; RN ) where  = [0, a] ˜ endowed with norm ||f|| =

N 

a˜

|fk (s)|ds,

f = [f0 , f1 , . . . , fN ]tr ∈ X

k=0 0

with 0 < a˜ < +∞ as the maximal age at which division occurs.

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We consider the transformation defined by (χf )(s) := Z(s)f (s),

s ∈ (0, a) ˜

(4)

which is the fundamental matrix of the differential problem d Z(a) = −M(a)Z(a) da Z(0) = I Since det Z(a) = 0 the matrix Z −1 exists and it is the diagonal matrix given by  Z −1 (a) = Diag Zi−1 (a)

(5)

0≤i≤m−1

with ⎛ Zi−1 (a) := ⎝z¯ j,k,i :=

⎧ " ⎨ e− 0a˜ Diag (μj,i (τ )+βj,i (τ )

if k = j

⎩0

if k = j

⎞ ⎠

(6)

In the following, we will use the method developed in Refs. [11–14] to prove the existence and uniqueness of solution associated to the model (3). In fact, the suns and stars machinery was proved to be equally significant as renewal equation in Refs. [15, 17] and [1–7, 20]. We can rewrite the model (3) in a more compact and general form. Let’s consider the transformation of the column vector of densities as follows: u(a, t) := (Ui (a, t))0≤i≤m−1 where Ui is the column vector given by Ui (a, t) = (uj,i (a, t))1≤j ≤n . Then we obtain the following system of partial differentials equation ∂ ∂ u(t, a) + u(t, a) = −M(a)u(t, a) ∂t ∂a

a˜ u(t, 0) = B(a)u(t, a)da

(7)

0

The diagonal matrix M is of order N × N and is given by M(a) Diag(Mi )0≤i≤m−1 where Mi is given as follows: Mi (a) := Diag(μj,i (a) + βj,i (a))1≤j ≤n

:=

Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell. . .

77

The matrix B(a) is as follows ⎛

P0 ⎜ Q1 ⎜ ⎜ B(a) = ⎜ 0 ⎜ . ⎝ .. 0

0 P1 Q2 .. .

0 0 P2 .. .

··· ··· ··· .. .

⎞

0 0 0 .. .

⎟ ⎟ ⎟ ⎟ ⎟ ⎠

0 · · · Qm−1 Pm−1

where the 0s are blocks with entries equal to zero, and the matrix Pi which describes the dynamic of the cellular proliferation within the ith mutation, is given by ⎛

β1,i (a) 2β2,i (a)p1,2,i · · · ⎜ .. ⎜ 0 . β2,i (a) ⎜ ⎜ .. . . . . Pi (a) = ⎜ . . . ⎜ ⎜ . . .. ⎝ .. 0 ··· ···

· · · 2βn,i (a)p1,n,i .. . .. .. . . .. . 2β (a)p n,i

0

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n−1,n,i

βn,i (a)

However, the matrix which describes the mutation dynamic between the (i − 1)th class and the ith class is given by ⎛

2β1,i−1 (a)q1,1,i−1 2β2,i−1 (a)q1,2,i−1 · · · ⎜ . ⎜ 0 2β2,i−1 (a)q2,2,i−1 . . ⎜ ⎜ .. .. .. Qi (a) = ⎜ . . . ⎜ ⎜ .. . .. ⎝ . 0

···

· · · 2βn,i−1 (a)q1,n,i−1 .. . .. .. . . .. . 2β (a)q

··· 0

n,i−1

⎞

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ n−1,n,i−1

2βn,i−1 (a)qn,n,i−1

In order to study our model, we need some biological significant assumptions on the parameters β and μ: ⎧ i. βj,i ∈ L1 (0, a) ˜ ∩ L∞ (0, a), ˜ there exists some constant β¯ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ such that 0 < βj,i (a) ≤ β¯ for 1 ≤ j ≤ n, 0 ≤ i ≤ m − 1 ⎪ ⎪ ⎨ (Hβ,μ ) a.e in (0, a), ˜ ⎪ ⎪ ⎪ ⎪ ii. μ ∈ L1 (0, a) ⎪ ˜ there exists some constant μ¯ such that j,i ⎪ ⎪ ⎪ ⎩ 0 ≤ μj,i (a) ≤ μ¯ for 1 ≤ j ≤ n, 0 ≤ i ≤ m − 1 a.e in (0, a), ˜ We set the transformation v(t, a) := χ (u(t, .))(a) = Z(a)u(t, a),

for a ∈ (0, a) ˜

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then the density v(t, .) satisfies the following ∂ ∂ v(t, a) + v(t, a) = 0 ∂t ∂a

A v(t, 0) = k(a)v(t, a)da := φ(v(t, .))

(8)

0

v(0, a) = v0 (a) := Z(a)u0 (a) where the kernel matrix k given by k(a) = B(a)Z −1 (a) = (kr,s )0≤r,s≤m−1 with the matrix Z −1 is given by (5) and (6). More precisely, ⎛

W0 ⎜ Y1 ⎜ ⎜ k(a) = ⎜ 0 ⎜ . ⎝ ..

0 W1 Y2 .. .

0 0 W2 .. .

··· ··· ··· .. .

0 0 0 .. .

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(9)

0 · · · 0 Ym−1 Wm−1 with  Wi =

βk,i z¯ k,k,i

if k = j

2pj,k,i βk,i z¯ k,k,i

if k > j

2qk,k,i βk,i z¯ k,k,i

if k = j

2qj,k,i βk,i z¯ k,k,i

if k > j

(10)

and  Yi =

(11)

It is known if φ = 0 in (8) the solution is given by the semigroup of translation {T (t)}t≥0 :  (T (t)f )(a) =

f (a − t)

if a − t > 0

0

if a − t ≤ 0

(12)

for t ≥ 0 and a ∈ . Since the chosen state space is X, the generator A of T is given by D(A) = {f ∈ X : f is absolutely continuous, f (0) = 0} Af = −f 

Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell. . .

79

The space L1 ([0, a], ˜ Rn ) can be embedded into the space NBV([0, a), ˜ Rn ) via the integration j : L1 ([0, a], ˜ Rn ) → NBV(([0, a), ˜ Rn )

a (jf )(a) := f (s)ds, a ∈ [0, a) ˜ 0

By integration of Eq. (8) from a to 0, one obtains d v(t) = A∗ j v(t) + φ(v(t))H dt

(13)

v(0) = f where the operator A∗ 0 is defined as follows D(A∗ ) = {ϕ ∈ X∗ , ϕ(a) =

a

ψ(s)ds ∀a ∈ [0, a] ˜ and ψ ∈ X∗ }

0 ∗

A

ϕ = −ϕ



and H (a) = {e1 , e2 , . . . , eN } is the standard basis of RN such that  Hi (a) =

ei

for a ∈ (0, a) ˜

0

for a = 0

The problem (13) yields by integrating the abstract equation v(t) = T (t)f + j −1



t

T ∗ (t − s)φ(v(s))H ds

(14)

0

where T ∗ is again translated to the right with extension by zero. Theorem 3.1 Assume that (Hp,q ) and (Hβ,μ ) are satisfied. Then for all f ∈ X the (14) has a unique solution v(t) on [0, T ), for some T > 0. Proof Since assumption (Hβ,μ ) hold, we obtain that φ ∈ L(X; RN ). Then, by easy contractions mapping arguments the existence and uniqueness of solution v(., f ) is obtained such that the formula Tφ (t)f = v(t, f ),

t ≥ 0, f ∈ X

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Y. Elalaoui and L. Alaoui

defines a strongly continuous semigroup on X such that the unbounded operator Aφ from X into X is as follows: tr  d d Aφ f := − f1,0 , · · · , − fn,m−1 da da tr  where f = f1,0 , . . . , fn,m−1 , tr denotes the transpose of the vector, and the domain of the differential operator Aφ is defined by D(Aφ ) = {f ∈ X : fj,i is absolutely continuous on , f(0) = φ(f)}  

is its infinitesimal generator.

4 Analysis of the Model The purpose of this section is to demonstrate that the solution exhibits the asynchronous exponential growth property. So we need to prove that the solution satisfy several spectral properties. First, we give a result about the generational compactness which consists of an important tool in determining the asymptotic behavior of the perturbed semigroup {Tφ (t)}t≥0 . Recall that given a strongly continuous semigroup {S(t)}t≥0 on some Banach space E, we say that {S(t)}t≥0 is eventually compact if there exists some t0 ≥ 0 such that the operator S(t) is compact on E for all t ≥ t0 . If we define v0 (t, f ) = T (t)f we can rewrite (14) as the following v = v0 + Fv where (Fv)(t) := j −1 we obtain

"t 0

T (t − s)φ(v(s))H ds. So by successive approximations

v = v0 +

∞ 

F n v0

k=1

This first lemma gives a compactness result on the first cell’s generation. Lemma 4.1 For t > 0 the mapping f → j −1

0

from X into RN is compact.

t

T (t − s)φ(T (s)f )H ds

Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell. . .

"0

Proof one has Fv0 =

−a˜

81

k(a)f (t − a)da where k is given by (9). Then

||Fv0 (x) − Fv0 (y)||Rn ≤ Mx,y ||f ||X which tends to 0 as |x − y| → 0. Therefore, by Arzela Ascoli argument we obtain the result.   Using the same argument as the result above, we can find a compactness criterion for the nth cell’s generation Lemma 4.2 For t > 0 and k ∈ N. The mapping f → vk (t, f ) from X into X is compact. Proposition 4.3 The strongly continuous semigroup (Tφ (t))t≥0 is eventually compact. Proof Since for t > 2a, ˜ v0 = 0 then Tφ is equal to the finite sum of compact operators which gives the result.   Now, we set the operator π(λ)f := j −1 R(λ, A∗ )φHf . Lemma 4.4 ([17]) Let λ ∈ C and ϕ ∈ X∗ . Then

 j −1 R(λ, A∗ ϕ (t) = 0

0

t ∈ [−θ, 0]

eλ(t−s) ϕ(ds),

t

Therefore, by the Lemma 4.4 we have that λ ∈ σ (Aφ ) if and only if

a˜

1 = π(λ)f =

e−λa k(a)f (a)da

0

since R(λ, Aφ ) = R(λ, A) + j −1 R(λ, A∗ )φH R(λ, Aφ ) In the following, we need two ingredients: the growth bound which is defined by ω(T (t)) = ω(A) := inf{w : ||T (t)|| ≤ Mewt , M ≥ 0} and the spectral bound as  s(A) =

sup{eλ : λ ∈ σ (A)}

if σ (A) = ∅

−∞

if σ (A) = ∅

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The following proposition gives a characterization of the generator of the semigroup solution of (14) Proposition 4.5 The spectrum of Aφ satisfies σ (Aφ ) = σp (Aφ ) = {λ ∈ C, 1 ∈ σ (π(λ))}  

A " a − 0 λ+μj,i (τ )+βj,i (τ )dτ e βj,i (a)da = 1 = λ ∈ C, 0

The spectral bound of Aφ satisfies s(Aφ ) = ω(Aφ ), is a simple pole and a dominant eigenvalue of Aφ , and is the unique real solution λ0 of the equation r(φ¯ λ ) = 1, i.e., of the equation

A

e−

"a 0

λ+μj,i (τ )+βj,i (τ )dτ

βj,i (a)da = 1

(15)

0

Proof Since the eventual compactness holds σ (Aφ ) = σp (Aφ ) = {λ ∈ C, 1 ∈ σ (π(λ))}. By easy computation of det(φ¯ λ − ζ IN ) the eigenvalues of r(π(λ)) are of the form  A "  a wλ := e− 0 λ+μj,i (τ )+βj,i (τ )]dτ βj,i (a)da 1≤j ≤n

0

which proves the first assertion. Since π is strictly decreasing on R and limλ→−∞ r(π(λ)) = +∞ and limλ→+∞ r(π(λ)) = 0, then r(π(λ)) = 1 admits a unique real root λ˜ = s(Aφ ) and by the eventual compactness of the semigroup s(Aφ ) = ω(Aφ ) gives us a pole and dominant eigenvalue of Aφ . Therefore, there exists a unique real solution λ˜ which satisfies r(π(lambda)) = 1 and there exists δ > 0 such that other complex solution λ of the equation r(π(λ)) = 1 satisfy |eλ − eλ0 | ≤ δ.   Proposition 4.6 The operator π(λ) is irreducible. ∗ ) ∈ RN such Proof Let X := (x1 , x2 , . . . , xN ) ∈ RN and X∗ := (x1∗ , x2∗ , . . . , xN that

< π(λ)X, X∗ >=

A 0

e−λa

n m−1   i=0

2βk,i z¯ k,k,i xk

 n

pj,k,i xj∗

j =1

k=1

+

n  j =1

qj,k,i xj∗+n



da

Since z¯ k,k,i > 0, β > 0 and pj,k,i > 0 for k > j we obtain that π(λ) is irreducible.  

Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell. . .

83

Theorem 4.7 (Tφ (t))t≥0 has the AEG property. More precisely, we have e−λ0 t Tφ (t) − P ≤ Me−δt , for some constants δ > 0, M ≥ 1 and for all t ≥ 0, and P is given by Pf = C(f )(eλ0 · ⊗ Xλ0 ),

(16)

where C(f ) =

Xλ∗0 , φ(θ →

"θ 0

e−λ0 (θ−s) f (s)ds)

Xλ∗0 , φ(θ → θ e−λ0 θ ⊗ Xλ0 )

,

f ∈ E,

(17)

where Xλ0 and Xλ∗0 are two positive eigenvectors respectively of φ¯ λ0 and φ¯ λ∗0 associated with the eigenvalue r(φ¯ λ0 ) = 1. Proof It is a direct consequence of Propositions 4.3, 4.5, 4.6 and Remarks [7, Remarks 2.5, 2.6]. The following result gives the equivalence of the solution semigroup of (3) and the translation semigroup (Tφ (t))t≥0 where G(t)f = χ −1 Tφ (t)χf, for all f ∈ E Theorem 4.8 The semigroup (G(t))t≥0 has the AEG property. More precisely there exists a projection P˜ of rank one and there exists δ > 0 such that e−λ0 t G(t) − P˜ ≤ Me−δt ,

t ≥0

and P˜ is such that P˜ f := χ −1 P (χf ), i.e., P˜ f = C(χf )(eλ0 · Z −1 (·) ⊗ Xλ0 ),

f ∈E

where χ , P , and C are respectively given by (4), (16) and (17), and Xλ0 is as given in Theorem 4.7.

References 1. L. Alaoui, Population dynamics and translation semigroups. Dissertation, University of Tübingen, 1995 2. L. Alaoui, Generators of translation semigroups and asymptotic behavior of the Sharpe-Lotka model. Diff. Int. Equ. 9, 343–362 (1996) 3. L. Alaoui, A cell cycle model and translation semigroups. Semigroup Forum 54(1), 135–153 (1997)

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Qualitative Analysis of a PDE Model of Telomere Loss in a Proliferating Cell. . .

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The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps Driss Kiouach and Yassine Sabbar

1 Introduction Hethcote et al. [1] introduced the following epidemic model: ⎧. ⎪ S = A − μS − βSI, ⎪ ⎪ ⎪ ⎨I. = βSI − (μ + γ + δ + r )I, 2 . ⎪ Q = δI − (μ + k + r3 )Q, ⎪ ⎪ ⎪ ⎩. R = γ I + kQ − μR,

(1)

where S(t), I (t), Q(t), and R(t) denote the number of susceptible, infective but not quarantined, quarantined infective, and removed individuals at time t, respectively. The parameter A is the recruitment rate of susceptible corresponding to births and immigration. μ is the natural death rate. δ is the rate for individuals leaving the infective compartment I to the quarantine compartment Q. r2 and r3 represent the disease-related death rate in classes I and Q, respectively. γ and k denote the recovery rate from group I and Q to R, respectively. β denotes the average number of adequate contacts. The parameters involved in system (1) are all positive constants. Those authors presented the basic reproduction number of system (1) βA as follows: R0 = μ(μ+δ+γ +r2 ) . If R0 ≤ 1, the deterministic SIQR model (1) has only the disease-free equilibrium P 0 (S0 , 0, 0, 0) = ( A μ , 0, 0, 0) which is globally asymptotically stable in the invariant set:

D. Kiouach · Y. Sabbar () MSTI Team, High School of Technology, Ibn Zohr University, Agadir, Morocco e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_7

87

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 = (S, I, Q, R) ∈

R+ 4

 A . :S+I +Q+R ≤ μ

If R0 > 1, P 0 becomes unstable and there exists a globally asymptotically stable endemic equilibrium: ∗

∗

∗

∗

∗

P (S , I , Q , R ) =



δI ∗ A μ(R0 − 1) , , , μR0 β k + μ + r3 

R0 − 1 kδ γ+ . k + μ + r3 β

As we know, the incidence rate of a disease is the number of new cases per unit time and it plays an important role in the mathematical epidemiology. Zhien et al. [2] analyzed SIQR model with three types of incidence: simple mass action f (S, I ) = βSI , standard incidence f (S, I ) = βSI /N, and quarantine adjusted incidence f (S, I ) = βSI /(N − Q). Also, in many previous epidemic models, the bilinear incidence rate is frequently used (see Refs. [3–5]). Due to homogeneously mixed contact between individuals, we have used the mass action rate βSI as an incidence rate of our epidemic model. In reality, the spread of a disease is characterized by randomness due to the unpredictable character of the human contacts. Also, environmental fluctuations have important effects on the growth and propagation of an epidemic disease. Consequently, there are many authors who studied stochastic epidemic models. They introduced random effects into models by different techniques (see Refs. [6, 7]). Both from biological and mathematical perspectives, our main concern is to study the effect of a randomly fluctuating environment on the disease outbreak by analyzing the dynamical behavior of our stochastic model. To include stochastically effects in the model (1), we use the approach mentioned by Ref. [8]. We assume that the random perturbations are directly proportional to S(t), I (t), Q(t), and R(t). Then, we obtain the following SIQR epidemic model with perturbation stochastic: ⎧ ⎪ ⎪ ⎪dS(t) = [A − μS(t) − βS(t)I (t)]dt + σ1 S(t)dB1 (t), ⎪ ⎨dI (t) = [βS(t)I (t) − (μ + γ + δ + r )I (t)]dt + σ I (t)dB (t), 2 2 2 ⎪dQ(t) = [δI (t) − (μ + k + r3 )Q(t)]dt + σ3 Q(t)dB3 (t), ⎪ ⎪ ⎪ ⎩ dR(t) = [γ I (t) + kQ(t) − μR(t)]dt + σ4 R(t)dB4 (t),

(2)

where Bi (t) (i = 1, 2, 3, 4) are independent Brownian motions defined on a complete probability space (, F, P) with a filtration {Ft }t≥0 satisfying the usual conditions, and σi > 0 (i = 1, 2, 3, 4) are their intensities. For the epidemic model (2), we have determined the value of the stochastic basic reproduction number as follows:

The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps

R0s =

89

 βA σ 2  1 − 2 . μ + γ + δ + r2 μ 2

The dynamic of model (2) is established by the stochastic threshold value R0s . That is, if R0s ≤ 1, the disease dies out almost surely; and if R0s > 1, the disease persists in the mean with probability one. The random perturbations can suppress the spread of disease. That is to say, the stochastic threshold in white noise environment is less than that without white noise (R0s < R0 ). This phenomenon is largely mentioned in the previous works. For example, Ji et al. [9] investigated the threshold of SIR epidemic models with stochastic perturbation. Liu et al. [10] established interesting results on the threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching. Zhao and Jiang [11] investigated the threshold of a stochastic SIRS epidemic model with saturated incidence and then considered the threshold of a stochastic SIS epidemic model with vaccination [12, 13]. Furthermore, they studied the threshold of a stochastic SIRS epidemic model with varying population size [14]. Zhang et al. [15] presented the threshold of a stochastic SIQS epidemic model with standard bilinear incidence, which determines the extinction and persistence of the disease. When the noise is small, they obtained a threshold of the stochastic model which determines the extinction and persistence of the disease. Besides, they proved that large noise will suppress the epidemic from prevailing. However, the effects due to unexpected environmental shocks (tsunami, floods, earthquakes, hurricanes, whirlwinds, etc.) have been neglected. The stochastic system (2) has not taken into consideration the impact of these phenomena on the disease outbreak. The proposed solution to this issue is introducing a jump process into the underlying population dynamics. Many works have introduced Lévy jumps process into their models. For example, Bao et al. [16] produced pioneering works on this approach. They first studied a stochastic Lotka–Volterra population systems with Lévy jumps. From then on, many interesting studies on the epidemic models with Lévy jumps have been reported (see Refs. [17, 18]). Motivated by the abovementioned works, in this contribution, we consider the following SIQR epidemic model with both white noise and Lévy jumps perturbations: ⎧ ⎪ dS(t) = [A − μS(t) − βS(t)I (t)]dt + σ1 S(t)dB1 (t) ⎪ ⎪ ⎪ " ⎪ ⎪ # ⎪ + Z η1 (u)S(t − )N(dt, du), ⎪ ⎪ ⎪ ⎪ ⎪ dI (t) = [βS(t)I (t) − (μ + γ + δ + r2 )I (t)]dt + σ2 I (t)dB2 (t) ⎪ ⎪ " ⎪ ⎨ # + η2 (u)I (t − )N(dt, du), Z

⎪ dQ(t) = [δI (t) − (μ + k + r3 )Q(t)]dt + σ3 Q(t)dB3 (t) ⎪ ⎪ ⎪ " ⎪ ⎪ # du), + Z η3 (u)Q(t − )N(dt, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ dR(t) = [γ I (t) + kQ(t) − μR(t)]dt + σ4 R(t)dB4 (t) ⎪ ⎪ " ⎪ ⎩ # du), + Z η4 (u)R(t − )N(dt,

(3)

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where S(t − ), I (t − ), Q(t − ), and R(t − ) are the left limits of S(t), I (t), Q(t), # and and R(t), respectively. N is a Poisson counting measure with compensator N characteristic measure ν on a measurable subset Z of (0, ∞) satisfying ν(Z) < ∞. # We assume that ν is a Lévy measure such that N(dt, du) = N (dt, du) − ν(du)dt. Bi (t) (i = 1, 2, 3, 4) are independent of N . The function η : Z ×  → R is bounded, B(Z) × Ft -measurable and is continuous with respect to ν. The aim of this paper is to investigate the asymptotic properties of the new improved stochastic epidemic model (3) and establish the threshold which determines the extinction and the persistence in the mean of the disease. This threshold value coincides with the deterministic threshold in absence of white noise and Lévy jumps. Mathematical tools and required assumptions are given in the next section. Well-posedness of a stochastic model (3) is given in Sect. 3. The sufficient condition for the extinction of the disease is given in Sect. 4. The condition for the persistence of the disease is obtained in Sect. 5. Numerical simulations are presented in Sect. 6 in order to illustrate and confirm the theoretical results. The paper ends with a brief discussion.

2 Preliminaries and Assumptions Throughout this paper, we let (, F, P) be a complete probability space with a filtration {Ft }t≥0 satisfying these conditions: right continuous and F0 contains all P-null sets. Now, we consider the d-dimensional stochastic differential equation of Itô type dx(t) = f (x(t), t)dt + g(x(t), t)dB(t)

for all t ≥ t0 ,

(4)

with initial value x(0) = x0 ∈ Rd . B(t) denotes n-dimensional standard Brownian motion defined on the complete probability space (, F, {Ft }t≥0 , P). C 2,1 (Rd × [t0 , ∞]; R+ ) is the family of all nonnegative functions V (x, t) defined on Rd × [t0 , ∞] such that they are continuously twice differentiable in x and once in t. The differential operator L of Eq. (4) is defined by [19] L=

d d   ∂2 ∂ ∂ 1  T + g (x, t)g(x, t) fi (x, t) + . ij ∂xi ∂xj ∂t ∂xi 2 i=1

i,j =1

If L acts on a function V ∈ C 2,1 (Rd × [t0 , ∞]; R+ ), then   1 LV (x, t) = Vt (x, t) + Vx (x, t)f (x, t) + trace g T (x, t)Vxx (x, t)g(x, t) . 2

The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps

91

By Itô’s formula, if x(t) ∈ Rd , we have dV (x(t), t) = LV (x(t), t)dt + Vx (x(t), t)g(x(t), t)dB(t). Suppose x(t) ∈ R, we now present the 1-dimensional Itô formula for the following Itô-Lévy process:

# η(t, u)N(dt, du).

dx(t) = f (ω, t)dt + g(ω, t)dB(t) + Z

Let V ∈ C 2,1 (R2 × [t0 , ∞]) and define Y (t) = V (x(t), t)). Then dY (t) =

  ∂V ∂V (x(t), t)dt + (x(t), t) f (ω, t)dt + g(ω, t)dB(t) ∂t ∂x 1 ∂ 2V + g 2 (ω, t) 2 (x(t), t)dt 2 ∂ x

 V (x(t − ) + η(t, u), t) − V (x(t − ), t) + Z

 ∂ (x(t − ), t)η(t, u) ν(du)dt ∂x

$ % #(dt, du). + V (x(t − ) + η(t, u), t) − V (x(t − ), t) N −

Z

For the purpose of well analyzing our model (3), it necessary that we make the following assumptions: (A1 ) For the jump diffusion coefficient, we assume that for a given ζ > 0, there exists a constant Lζ > 0 such that

|Fi (x, u) − Fi (x, u)|2 ν(du) < Lζ |x − y|2 ,

∀ |x| ∨ |y| ≤ ζ,

Z

where F1 (x, u) = xη1 (u), F2 (x, u) = xη2 (u), F3 (x, u) = xη3 (u) and F4 (x, u) = xη4 (u). (A2 ) For all u ∈ Z, we assume that 1 + ηi (u) > 0, i = 1, 2, 3, 4 and

  ηi (u) − ln(1 + ηi (u)) ν(du) < ∞. Z

Biologically, if ηi (u) > 0 the Lévy jumps increase the quantity of the host population. Otherwise, if −1 < ηi (u) < 0, the number of individuals is minimized gradually.

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(A3 ) We suppose there exists a constant κ > 0, such that



2 ln(1 + ηi (u)) ν(du) < κ.

Z

Biologically, it is shown that the intensity of Lévy jumps cannot exceed the environmental carrying capacity. For proving the next lemmas, we add the following supplementary assumption: (A4 ) There exists  > 1 such that 1  σ 2 − λ > 0, μ − ( − 1)# 2 2 where # σ 2 = σ12 ∨ σ22 ∨ σ32 ∨ σ42 , and λ=

 (1 + η1 (u) ∨ η2 (u) ∨ η3 (u) ∨ η4 (u))p − 1(u) Z

 − (η1 (u) ∧ η2 (u) ∧ η3 ∧ η4 (u)) ν(du). Biologically, the assumptions (A1 )–(A4 ) mean that the intensities of Lévy jumps are not very large. Now, we list some proven results used in our analysis [20]. Lemma 2.1 Let M = {Mt }t≥0 be a real-valued continuous local martingale vanishing at t = 0. Then lim M, Mt = ∞ a.s.

t→∞

"⇒ lim

Mt

t→∞ M, Mt

= 0 a.s.

and also lim sup t→∞

M, Mt < ∞ a.s. t

"⇒ lim

t→∞

Mt = 0 a.s. t

where M, Mt denotes the quadratic variation of M. Lemma 2.2 Let M = {Mt }t≥0 be a local martingale vanishing at time 0. Define

t

ρM (t) := 0

dMs , (1 + s)2

t ≥ 0,

where Mt = M, Mt is Meyers angle bracket process. Then lim

t→∞

Mt = 0 a.s. t

provided that

lim ρM (t) < ∞ a.s.

t→∞

The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps

93

Under the assumptions (A1 )–(A4 ), we have the following lemmas: Lemma 2.3 Let (S(t), I (t), Q(t), R(t)) be the solution of model (3) with any initial value (S(0), I (0), Q(0), R(0)) ∈ R4+ . Then lim

t→∞

S(t) + I (t) + Q(t) + R(t) =0 t

a.s.

Moreover, S(t) = 0, t→∞ t lim

I (t) = 0, t→∞ t lim

Q(t) R(t) = 0 and lim =0 t→∞ t t→∞ t lim

a.s.

Lemma 2.4 Let (S(t), I (t), Q(t), R(t)) be the solution of model (3) with any initial value (S(0), I (0), Q(0), R(0)) ∈ R4+ . Then "t "

"t "

# η1 (u)S(s − )N(ds, du) lim =0 t→∞ t "t " # η3 (u)Q(s − )N(ds, du) =0 lim 0 Z t→∞ t 0

#(ds, du) η2 (u)I (s − )N =0 t→∞ t "t " #(ds, du) η4 (u)R(s − )N lim 0 Z =0 t→∞ t

Z

lim

0

Z

a.s. Moreover, "t lim

t→∞

lim

t→∞

0

"t 0

S(s)dB1 (s) =0 t Q(s)dB3 (s) =0 t

"t lim

t→∞

lim

t→∞

0

"t 0

I (s)dB2 (s) =0 t R(s)dB4 (s) =0 t

a.s.

3 Well-Posedness of Stochastic Model To analyze the asymptotic properties of a population system, the first concern is whether the solution of the system is unique, positive, and global. In order to guarantee a stochastic differential equation (SDE) has a unique global positive solution, the coefficients of the equation should verify the linear growth condition and the local Lipschitz condition. However, the coefficients of (3) do not satisfy the linear growth condition, so the solution may explode at a finite time. In this section, under assumptions (A1 )–(A4 ), we shall verify that the unique solution of system (3) is global and positive.

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Theorem 3.1 For any initial value (S(0), I (0), Q(0), R(0)) ∈ R4+ , there exists a unique positive solution (S(t), I (t), Q(t), R(t)) of system (3) on t ≥ 0, and the solution will remain in R4+ with probability one. That is to say, the solution (S(t), I (t), Q(t), R(t)) ∈ R4+ for all t ≥ 0 almost surely. Proof By the assumption (A1 ), the coefficients of system (3) are locally Lipschitz continuous. Then for any initial value (S(0), I (0), Q(0), R(0)) ∈ R4+ there is a unique local solution (S(t), I (t), Q(t), R(t)) on [0, τe ), where τe is the explosion time. To show that the solution is global, we only need to prove that τe = ∞ a.s. Let 0 > 0 such that S(0), I (0), R(0) > 0 . For each integer  ≤ 0 , we define the following stopping times: τ = inf {t ∈ [0, τe ) : S(t) ≤  or I (t) ≤  or Q(t) ≤  or R(t) ≤ } , τ = lim τ = inf {t ∈ [0, τe ) : S(t) ≤ 0 or I (t) ≤ 0 or or Q(t) ≤ 0 R(t) ≤ 0} . →0

Define a C 2 -function V : R4+ → R+ by 

S V = S − α − α ln α

+ (I − 1 − ln I ) + (Q − 1 − ln Q) + (R − 1 − ln R),

where α > 0 is a positive constant to be determined later. Obviously, this function is nonnegative which can be seen from x − 1 − ln x > 0 for x > 0. Using Itô’s formula, we compute that   α 1 dV = LV dt + 1 − σ1 SdB1 (t) + 1 − σ2 I dB2 (t) S I   1 1 σ3 QdB3 (t) + 1 − σ4 RdB4 (t) + 1− Q R

# η1 (u)S(t − ) − α ln(1 + η1 (u))N(dt, du) + Z

+

#(dt, du) η2 (u)I (t − ) − ln(1 + η2 (u))N

Z

+

# η3 (u)Q(t − ) − ln(1 + η3 (u))N(dt, du)

Z

+

Z

#(dt, du), η4 (u)R(t − ) − ln(1 + η4 (u))N

The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps

95

where, αA + αβI + αμ − βS S δI γI kQ + (μ + γ + δ + r2 ) − + (μ + k + r3 ) − − +μ Q R R

LV = A − μS − (μ + r2 )I + (μ + r3 )Q − μR −

σ2 σ2 σ12 + 2 + 3 2 2 2

σ42 + αη1 (u) − α ln(1 + η1 (u))ν(du) + 2 Z

η2 (u) − ln(1 + η2 (u))ν(du) + +

Z



η3 (u) − ln(1 + η3 (u))ν(du) +

+

η4 (u) − ln(1 + η4 (u))ν(du)

Z

Z

≤ A − (μ + r2 )I + αβI + αμ + 3μ + γ + δ + r2 + k + r3 σ2 σ2 σ12 + 2 + 3 2 2 2

2 σ αη1 (u) − α ln(1 + η1 (u))ν(du) + 4 + 2 Z

η2 (u) − ln(1 + η2 (u))ν(du) +

+

Z



η3 (u) − ln(1 + η3 (u))ν(du) +

+

η4 (u) − ln(1 + η4 (u))ν(du).

Z

Z

Given the fact that x − ln(1 + x) ≥ 0 for all x ≥ 0 and the hypothesis (A2 ), we define

η2 (u) − ln(1 + η2 (u))ν(du) M1 = + αη1 (u) − α ln(1 + η1 (u))ν(du) +

Z

η3(u) − ln(1 + η3 (u))ν(du) +

+ Z

To simplify, we choose α =

Z

η4 (u) − ln(1 + η4 (u))ν(du). Z

μ+r2 β .

One can see that

LV ≤A + +αμ + 3μ + γ + δ + r2 + k + r3 + ≡ M2 .

σ2 σ2 σ2 σ12 + 2 + 3 + 4 + M1 2 2 2 2

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Therefore, one can obtain that EV (S(τ ∧T ), I (τ ∧T ), Q(τ ∧T ), R(τ ∧T )) ≤ V (S(0), I (0), Q(0), R(0)) + M2 t. For τ , there is some component of S(τ ), I (τ ), Q(τ ) and R(τ ) equal to . Thus, V (S(τ ), I (τ ), Q(τ ), R(τ )) ≥  − 1 − ln . Hence, V (S(0), I (0), Q(0), R(0)) + M2 T ≥ E(1{τ ≤T } V (S(τ , ω), I (τ , ω), Q(τ , ω), R(τ , ω))) ≥ ( − 1 − ln )P(τ ≤ t). Extending  to 0, we obtain for all t > 0, P(τ ≤ t) = 0. Hence P(τ = ∞) = 1.   Thus, τ = τe = ∞ a.s. which completes the proof of the theorem.

4 The Disease Extinction In this section, our main concern is to determine the conditions for the spread and persistence of an infectious disease. Given that the value of the deterministic threshold R0s characterizes the dynamical behaviors of system (2) and guarantees persistence or extinction of the disease. Similarly, we define the threshold of our stochastic SIQR epidemic model (3) as follows: R0J =



βA σ22 1 − − [η2 (u) − ln(1 + η2 (u))]ν(du) . μ + γ + δ + r2 μ 2 Z

To avoid any ambiguity, we define x(t) =

" 1 t t 0

x(s)ds.

Theorem 4.1 Let assumptions (A1 )–(A4 ) hold and let (S(t), I (t), Q(t), R(t)) be the solution of system (3) with any initial value (S(0), I (0), Q(0), R(0)) ∈ R4+ . If R0J < 1, then lim sup t→∞

 ln I (t) ≤ (μ + γ + δ + r2 ) R0J − 1 < 0 a.s. t

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Furthermore, we have that lim S(t) =

t→∞

A a.s. μ

lim Q(t) = 0 a.s.

t→∞

lim R(t) = 0 a.s.

t→∞

That is to say, the disease dies out exponentially with probability 1. Proof Integrating from 0 to t on both sides of the first and second equations of system (3), gives S(t) − S(0) σ1 = A − μS(t) − βS(t)I (t) + S(s)dB1 (s) t t

1 t # + η1 (u)S(s − )N(ds, du), t 0 Z I (t) − I (0) σ2 = βS(t)I (t) − (μ + δ + γ + r2 )I (t) + I (s)dB2 (s) t t

t 1 # + η2 (u)I (s − )N(ds, du). t 0 Z Noticeably, we get S(t) =

A μ + δ + γ + r2 − I (t) + ψ1 (t), μ μ

(5)

where  1 S(t) − S(0) I (t) − I (0) σ1 σ2 + − S(s)dB1 (s) − I (s)dB2 (s) ψ1 (t) = − μ t t t t 

t

t 1 1 − # − # η1 (u)S(s )N(ds, du) − η2 (u)I (s )N (ds, du) . − t 0 Z t 0 Z By Lemmas 2.3 and 2.4, we have lim ψ1 (t) = 0

t→∞

a.s.

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Making use of Itô’s formula to (3), one can conclude that 

1 d ln I (t) = βS(t)−(μ + δ + γ + r2 )− σ22 − η2 (u)−ln(1 + η2 (u))ν(du) dt 2 Z

# + σ2 dB2 (t) + ln(1 + η2 (u))N(dt, du). Z

Integrating from 0 to t on both sides, we get

ln I (t) 1 2 = βS(t) − (μ + δ + γ + r2 ) − σ2 − η2 (u) − ln(1 + η2 (u))ν(du) t 2 Z

σ2 B2 (t) 1 t ln I (0) # + + . ln(1 + η2 (u))N(dt, du) + t t 0 Z t Substituting the result (5) into the previous equation, implies that  ln I (t) A μ + δ + γ + r2 1 =β − I (t) + ψ1 (t) − (μ + δ + γ + r2 ) − σ22 t μ μ 2

ln I (0) σ2 B2 (t) − + η2 (u) − ln(1 + η2 (u))ν(du) + t t Z

t 1 # ln(1 + η2 (u))N(ds, du) + t 0 Z ≤

βA 1 ln I (0) σ2 B2 (t) βψ1 (t) − (μ + δ + γ + r2 ) − σ22 + + μ 2 t t

t 1 #(ds, du). η2 (u) − ln(1 + η2 (u))ν(du) + ln(1 + η2 (u))N − t 0 Z Z

Let k1 (t) = have

"t " 0

Z

# ln(1 + η2 (u))N(ds, du). According to the hypothesis (H3 ), we

k1 , k1 (t) = t

[ln(1 + η2 (u))]2 ν(du) < κt. Z

By Lemma 2.4 and the large number theorem for martingales, we get lim

t→∞

k1 (t) =0 t

and

lim

t→∞

B2 (t) =0 t

a.s.

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Consequently, if the condition R0J < 1 holds, we have lim sup t→∞

 ln I (t) ≤ (μ + γ + δ + r2 ) R0J − 1 < 0 a.s. t

Therefore, lim I (t) = 0 a.s.

(6)

t→∞

By (5) and (6), one can see that lim supS(t) = t→∞

A a.s. μ

Next, we shall show that lim Q(t) = 0 and lim R(t) = 0 a.s. t→∞ t→∞ From system (3), we have

Q(t) − Q(0) σ3 t = δI (t) − (μ + k + r3 )Q + Q(s)dB3 (s) t t 0

1 t #(ds, du), η3 (u)Q(s − )N + t 0 Z and R(t) − R(0) σ4 = γ I (t) + kQ(t) − μR(t) + t t

t 1 # η4 (u)R(s − )N(ds, du). + t 0 Z

t

R(s)dB4 (s) 0

Making use of Lemmas 2.1–2.4 and (6), we get the desired result.

 

5 The Disease Persistence In this section, we shall establish the condition for persistence of the disease. Firstly, we shall present the definition of persistence in the mean as follows: Definition 5.1 The disease in epidemic model (2) is said to be persistent in the mean if lim infI (t) > 0 a.s. t→∞

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Theorem 5.2 Let assumptions (A1 )–(A4 ) hold and let (S(t), I (t), Q(t), R(t)) be the solution of system (3) with any initial value (S(0), I (0), Q(0), R(0)) ∈ R4+ . If R0J > 1, then lim I (t) = I# > 0 a.s.

t→∞

S > 0 a.s. lim S(t) = #

t→∞

# > 0 a.s. lim Q(t) = Q

t→∞

# > 0 a.s. lim R(t) = R

t→∞

where  μ J R0 − 1 , I# = β

  (μ + δ + γ + r2 ) R0J − 1 A # , S= − μ β  J  δμ #= Q R −1 , β(μ + k + r3 ) 0   # = (γ + kδ)μ R J − 1 . R 0 β(μ + k + r3 ) That is to say, the disease will prevail if R0J > 1. Proof According to the Itô’s formula, one can see that  ln I (t) A μ + δ + γ + r2 =β − I (t) + ψ1 (t) − (μ + δ + γ + r2 ) t μ μ

σ2 B2 (t) 1 2 ln I (0) − η2 (u) − ln(1 + η2 (u))ν(du) + − σ2 + 2 t t Z

t 1 # + ln(1 + η2 (u))N(ds, du) t 0 Z  μ + δ + γ + r2 I (t) + βψ1 (t) = (μ + γ + δ + r2 ) R0J − 1 − β μ

ln I (0) σ2 B2 (t) 1 t #(ds, du). + + + ln(1 + η2 (u))N t t t 0 Z Using Lemmas 2.2–2.4, we get lim I (t) =

t→∞

 μ J R0 − 1 > 0 a.s. β

The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps

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From (5), one can conclude that lim S(t) =

t→∞

A μ + δ + γ + r2 − lim I (t). t→∞ μ μ

Consequently,   A (μ + δ + γ + r2 ) R0J − 1 lim S(t) = − > 0 a.s. t→∞ μ β On the other hand, we have established that

Q(t) − Q(0) σ3 t = δI (t) − (μ + k + r3 )Q + Q(s)dB3 (s) t t 0

1 t #(ds, du), η3 (u)Q(s − )N + t 0 Z and R(t) − R(0) σ4 = γ I (t) + kQ(t) − μR(t) + t t

t 1 # η4 (u)R(s − )N(ds, du). + t 0 Z

t

R(s)dB4 (s) 0

Thus, lim Q(t) =

 J  δμ R0 − 1 > 0 a.s. β(μ + k + r3 )

lim R(t) =

 (γ + kδ)μ  J R0 − 1 > 0 a.s. β(μ + k + r3 )

t→∞

and

t→∞

This completes the proof.

 

6 Numerical Simulations In this section, we shall use Euler numerical approximation to illustrate the rigor of our analytical results. The two examples given below concern the results obtained in Theorems 4.1 and 5.2.

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Example 6.1 Choose A = 4, β = 0.0176, μ = 0.2, γ = 0.02, δ = 0.01, k = 0.01, r2 = 0.1, r3 = 0.2, σ1 = 0.11, σ2 = 0.2, σ3 = 0.12, σ4 = 0.18, η1 = 0.02, η2 = 0.08, η3 = 0.01, η4 = 0.02, Z = (0, ∞), and ν(Z) = 1. Then, βA = 1.0697 > 1, μ(μ + γ + δ + r2 ) 

βA σ22 1 R0s = − = 1.0091 > 1, μ + γ + δ + r2 μ 2 

βA σ22 1 − − [η2 (u) − ln(1 + η2 (u))]ν(du) R0J = μ + γ + δ + r2 μ 2 Z R0 =

= 0.9699 < 1. The computer simulations shown in Fig. 1 (left) support the result of Theorem 4.1. That is to say, the disease in system (3) dies out exponentially with probability one. Although the disease in systems (1) and (3) is persistent. If we decrease β to 0.16, we get βA = 0.9697 < 1, μ(μ + γ + δ + r2 ) 

βA σ22 1 R0s = − = 0.9091 < 1, μ + γ + δ + r2 μ 2 

βA σ22 1 − − [η2 (u) − ln(1 + η2 (u))]ν(du) R0J = μ + γ + δ + r2 μ 2 Z R0 =

= 0.8999 < 1.

3 3

Deterministic Without jumps With jumps

2.5

2

1.5

I(t)

I(t)

2

1.5

1

1

0.5

0.5

0

Deterministic Without jumps With jumps

2.5

0

20

40

60

80 100 120 140 160 180 200 time

0

0

50

100

150

200 time

250

300

350

400

Fig. 1 Computer simulation of the paths I (t) for the SIQR epidemic model (3) (with jump), the trajectory of I (t) for the system (2) (without jump), and the solution I (t) of the corresponding deterministic system (1)

The Threshold of a Stochastic SIQR Epidemic Model with Lévy Jumps

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By the Theorem 4.1, the disease will tend to zero exponentially with probability one (see Fig. 1(right)). Example 6.2 Let β = 0.021 and other parameters be the same as the previous example. Then βA = 1.2121 > 1, μ(μ + γ + δ + r2 ) 

βA σ22 1 s − = 1.1970 > 1, R0 = μ + γ + δ + r2 μ 2 

βA σ22 1 J − − [η2 (u) − ln(1 + η2 (u))]ν(du) R0 = μ + γ + δ + r2 μ 2 Z R0 =

= 1.1968 > 1. We can conclude, by Theorem 5.2, the solution (S(t), I (t), Q(t), R(t)) of (3) obeys that lim I (t) = 2.548 > 0 a.s.

t→∞

lim S(t) = 18 > 0 a.s.

t→∞

lim Q(t) = 0.127 > 0 a.s.

t→∞

lim R(t) = 0.387 > 0 a.s.

t→∞

This means the disease persists almost surely. The computer simulations shown in Figs. 2 and 3 support these results clearly.

40

10 Deterministic Without jumps With jumps

35

Deterministic Without jumps With jumps

9 8

30 7 6 I(t)

S(t)

25 20

5 4

15

3 10 2 5 0

1 0

50

100

150

200 time

250

300

350

400

0

0

50

100

150

200 time

250

300

Fig. 2 Computer simulation of the paths (S(t), I (t)) for the SIQR epidemic model (3)

350

400

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2.5

2 Deterministic Without jumps With jumps

2

Deterministic Without jumps With jumps

1.8 1.6 1.4

1.5 R(t)

Q(t)

1.2

1

1 0.8 0.6

0.5

0.4 0.2

0

0

50

100

150

200 time

250

300

350

400

0

0

50

100

150

200 time

250

300

350

400

Fig. 3 Computer simulation of the paths (Q(t), R(t)) for the SIQR epidemic model (3)

7 Conclusions This work is concerned with the persistence and extinction of a stochastic SIQR epidemic model with Lévy jumps. To begin with, we proved the global existence and uniqueness of the positive solution to the system (3) with any positive initial value. Then we established the criteria for persistence and extinction of the disease. Furthermore, we showed that a Lévy-jumps noise (e.g., tsunami, volcanoes, avian influenza, hurricanes, earthquakes, toxic pollutants, etc.) in reality could be in order the epidemic to be suppressed so that it dies out. Results showed that noises have important effects on the persistence and extinction of the disease. Some interesting topics deserve further investigation. We may analyze more realistic but complex models, such as considering the effects of regime switching with Lévy jumps on the stochastic system (3). We leave these works as our future contribution. Authors Contributions The authors declare that the study was conducted in collaboration with the same responsibility. All authors read and approved the final manuscript.

References 1. H. Hethcote, M. Zhien, L. Shengbing, Effects of quarantine in six endemic models for infectious diseases. Math. Biosci. 180, 141–160 (2002) 2. M. Zhien, Y. Zhou, J. Wu, Modeling and Dynamics of Infectious Diseases. (Higher Education Press, Beijing, 2009) 3. D. Kiouach, Y. Sabbar, Stability and threshold of a stochastic sirs epidemic model with vertical transmission and transfer from infectious to susceptible individuals. Discrete Dynam. Nat. Soc. 2018, 7570296 (2018) 4. J. Li, Z. Ma, Qualitative analysis of sis epidemic model with vaccination and varying total population size. Math. Comput. Model. 35, 1235–1243 (2002)

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5. J. Li, Z. Ma, Stability analysis for sis epidemic models with vaccination and constant population size. Discrete. Contin. Dyn. Syst. Ser. 4, 635–642 (2004) 6. L. Allen, An introduction to stochastic epidemic models. Math. Epidem. 144, 81–130 (2008) 7. A. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, A stochastic differential equation sis epidemic model. SIAM J. Appl. Math. 71, 876–902 (2011) 8. K. Bao, Q. Zhang, Stationary distribution and extinction of a stochastic sirs epidemic model with information intervention. Adv. Diff. Equ. 352, 1–19 (2017) 9. C. Ji, D. Jiang, Threshold behaviour of a stochastic sir model. Appl. Math. Model. 38, 5067– 5079 (2014) 10. Q. Liu, D. Jiang, N. Shi, Threshold behavior in a stochastic SIQR epidemic model with standard incidence and regime switching. App. Math. Comput. 316, 310–325 (2018) 11. Y. Zhao, D. Jiang, The threshold of a stochastic sirs epidemic model with saturated incidence. Appl. Math. Lett. 34, 90–93 (2014) 12. Y. Zhao, D. Jiang, The threshold of a stochastic sis epidemic model with vaccination. Appl. Math. Comput. 243, 718–727 (2014) 13. Y. Zhao, D. Jiang, Dynamics of stochastically perturbed sis epidemic model with vaccination. Abstr. Appl. Anal. 243, 439–517 (2013) 14. Y. Zhao, D. Jiang, X. Mao, The threshold of a stochastic sirs epidemic model in a population with varying size. Discrete Contin. Dyn. Syst. Ser. B 20(2), 1289–1307 (2015) 15. X. Zhang, H. Huo, H. Xiang, Q. Xiang, Q. Shi, D. Li, The threshold of a stochastic siqs epidemic model. Phys. A 482, 362–374 (2017) 16. J. Bao, X. Mao, G. Yin, C. Yuan, Competitive lotka–volterra population dynamics with jumps. Nonlinear Anal. 74, 6601–6616 (2011) 17. Q. Liu, D. Jiang, T. Hayat, B. Ahmed, Analysis of a delayed vaccinated sir epidemic model with temporary immunity and levy jump. Nonlinear Anal. Hybrid Syst. 27, 29–43 (2018). 18. Q. Liu, D. Jiang, T. Hayat, B. Ahmed, Dynamics of a stochastic delayed sir epidemic model with vaccination and double diseases driven by levy jumps. Phys. A 27, 29–43 (2018). 19. X. Mao, Stochastic Differential Equations and Applications (Chichester, Horwoodl, 1997) 20. Y. Zhou, W. Zhang, Threshold of a stochastic sir epidemic model with levy jumps. Phys. A 446, 204–216 (2016)

Global Dynamics of a Generalized Chikungunya Virus Hajar Besbassi, Zineb El Rhoubari, Khalid Hattaf, and Noura Yousfi

1 Introduction Chikungunya virus (CHIKV) is an alphavirus and is transmitted from human to human by the bites of infected female mosquitoes. Most commonly, the mosquitoes involved are Aedes aegypti and Aedes albopictus, two species which can also transmit other mosquito-borne viruses, including dengue. The most common symptoms of CHIKV infection are fever and joint pain. Other symptoms include muscle pain, headache, nausea, fatigue, and rash [1]. CHIKV was first recognized as a human pathogen during the 1950s in Africa, and since then, cases have been identified in many countries in Africa and Asia [2]. On 9 December 2013, the Pan American Health Organization (PAHO) has issued an alert about the transmission of CHIKV in the Americas [3]. Since then, the transmission of CHIKV was confirmed in 44 countries and territories in the region, with more than 2 million reported cases and 403 deaths. Nowadays, many researchers investigate the dynamical behavior of CHIKV infection by constructing mathematical models. Most of these models describe the transmission dynamics of CHIKV in human and mosquito populations [4–9]. However, the mathematical modeling of the dynamics of the CHIKV within host is very few. Therefore, Wang and Liu formulated and analyzed a model of population

H. Besbassi () · Z. El Rhoubari · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Science Ben M’sik, Hassan II University, Casablanca, Morocco K. Hattaf Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Science Ben M’sik, Hassan II University, Casablanca, Morocco Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_8

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growth of CHIKV within human body [10]. In 2018, Elaiw et al. [11] improved the model of Wang and Liu [10] by proposing a class of CHIKV infection models. Above within-host CHIKV dynamics models are all based on the assumption that the cell infection is caused only by contact with free virus. In reality, the virus can spread by two fundamental modes, one by virus-to-cell infection through the extracellular space and the other by cell-to-cell transfer involving direct cell-to-cell contact [12–14]. For these reasons, we propose the following model: ⎧ T˙ ⎪ ⎪ ⎨˙ I ⎪ V˙ ⎪ ⎩ B˙

= λ − dT − f (T , I, V )V − g(T , I )I, = f (T , I, V )V + g(T , I )I − aI, = kI − μV − qBV , = η + cBV − hB,

(1)

where T (t), I (t), V (t), and B(t) denote the concentrations of uninfected cells, infected cells, CHIKV particles, and B cells at time t, respectively. The parameter λ is the recruitment rate of uninfected cells and k is the production rate of free CHIKV particles by infected cells. The CHIKV particles are attacked by the B cells at rate qBV . The B cells are created at rate η and proliferated at rate cBV . The parameters d, a, μ, and h are, respectively, the death rates of uninfected cells, infected cells, free CHIKV particles, and B cells. In addition, healthy cells become infected either by free virus at rate f (T , I, V )V or by direct contact with an infected cell at rate g(T , I )I . Hence, the term f (T , I, V )V +g(T , I )I represents the total infection rate of uninfected cells. As in Refs. [15, 16], the incidence functions f (T , I, V ) and g(T , I ) for both modes are continuously differentiable and satisfy the following hypotheses:  ∂g (H0 ) g(0, I ) = 0, for all I ≥ 0; (T , I ) ≥ 0 or g(T , I ) is a strictly monotone ∂T  ∂g (T , I ) ≤ 0, for all increasing function with respect to T when f ≡ 0 and ∂I T ≥ 0 and I ≥ 0. (H1 ) f (0, I, V ) = 0, for all I ≥ 0 and V ≥ 0.  (H2 ) f (T , I, V ) is a strictly monotone increasing function with respect to T or ∂f (T , I, V ) ≥ 0 when g(T , I ) is a strictly monotone increasing function with ∂T  respect to T , for any fixed I ≥ 0 and V ≥ 0. (H3 ) f (T , I, V ) is a monotone decreasing function with respect to I and V . The organization of this paper is as follows: In Sect. 2, we present some basic results about the properties of solutions and the existence of equilibria. The stability analysis of the model is investigated in Sect. 3. An application of our main results is presented in Sect. 4. Some conclusions are given in the last section.

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2 Basic Results In this section, we first show the positivity and boundedness of solutions for system (1). Theorem 2.1 All solutions of system (1) starting from positive initial value (T0 ,I0 ,V0 ,B0 ) remain bounded and positive for all t > 0. Proof Since T˙ |T =0 = λ > 0, I˙|I =0 = 0 ≥ 0, ˙ B=0 = η > 0, V˙ |V =0 = kI ≥ 0 for all I ≥ 0, B|

(2)

we deduce that R4+ is positively invariant with respect (1). It remains to prove that all solutions of system (1) are bounded. Let N(t) = T (t) + I (t) +

aq a V (t) + B(t). 2k 2kc

Then a aq ˙ N˙ (t) = T˙ (t) + I˙(t) + V˙ (t) + B(t) 2k 2kc aμ aqη aqh a V (t) + − B(t) = λ − dT (t) − I (t) − 2 2k 2kc 2kc aqη ≤ λ+ − δN (t), 2kc a where δ = min{ , d, μ, h}. Hence, 2 lim sup N(t) ≤ t→∞

aqη λ + , δ 2kcδ

which implies that all solutions of system (1) are bounded. This completes the proof.  Now, we discuss the existence of equilibria. By a simple computation, system (1) η λ has always one infection-free equilibrium of the form Ef ( , 0, 0, ). Hence, we d h define the basic reproduction number of our model as follows: R0 =

kf

λ

  + μ + q hη g( dλ , 0)   . a μ + q hη 

d , 0, 0

(3)

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To find the other equilibria of (1), we solve the following system: λ − dT − f (T , I, V )V − g(T , I )I = 0,

(4)

f (T , I, V )V + g(T , I )I − aI = 0,

(5)

kI − μV − qBV = 0,

(6)

η + cBV − hB = 0.

(7)

η μ(h − cV ) + qη ,I = V = ϕ1 (V ), h − cV k(h − cV )   λ − aϕ1 (V ) = ϕ2 (V ), and k(h − cV )f ϕ2 (V ), ϕ1 (V ), V +[μ(h − cV ) + T = d η qη]g(ϕ2 (V ), ϕ1 (V )) = a[μ(h − cV ) +qη)]. Since B = ≥ 0, we have h − cV h h V < . Hence, there is no biological equilibrium when V ≥ . So, we consider c c the function ψ defined on [0, hc ) by From (4) to (7), we get B =

    ψ(V ) = k(h − cV )f ϕ2 (V ), ϕ1 (V ), V + [μ(h − cV ) + qη] g ϕ2 (V ), ϕ1 (V ) −a [μ(h − cV ) + qη] . We have ϕ2 (0) =

λ d

> 0,

lim

V →( hc )−

a ϕ2 (V ) = −∞, and ϕ2 (V ) = − ϕ1 (V ) < 0 with d

μ(h − cV )2 + qηh > 0. Then the equation ϕ2 (V ) = 0 admits a unique k(h − cV )2 # ∈ (0, h ). Thus, B # = η > 0 and ψ(V #) = −a[μ(h − cV #) + qη] < 0. solution V # c h−cV #) Since ψ(0) = a(μh + qη)(R0 − 1) > 0, we deduce that there exists a V ∗ ∈ (0, V such that ψ(V ∗ ) = 0. From (6) and (7), we obtain ϕ1 (V ) =

B∗ =

η μ + qB ∗ ∗ ∗ V > 0. > 0, I = h − cV ∗ k

Substituting V = V ∗ and I = I ∗ in (4) and define a function ϕ3 as ϕ3 (T ) = λ − dT − f (T , I ∗ , V ∗ )V ∗ − g(T , I ∗ )I ∗ . Since ϕ3 (0) = λ > 0, ϕ3 ( dλ ) = −f ( dλ , I ∗ , V ∗ )V ∗ − g( dλ , I ∗ )I ∗ < 0, and ϕ3 is a strictly decreasing function of T , then there exists a unique T ∗ ∈ (0, dλ ) such that ϕ3 (T ∗ ) = 0. Therefore, model (1) has a unique chronic infection equilibrium E ∗ (T ∗ , I ∗ , V ∗ , B ∗ ) when R0 > 1. The previous discussions can be summarized in the following result.

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Theorem 2.2 (i) When R0 ≤ 1, model (1) has one infection-free equilibrium Ef ( dλ , 0, 0, hη ). (ii) When R0 > 1, model (1) has a unique chronic infection equilibrium E ∗ (T ∗ , I ∗ , V ∗ , B ∗ ) with T ∗ ∈ (0, dλ ), I ∗ > 0, V ∗ > 0, and B ∗ > 0.

3 Stability Analysis of Equilibria In this section, we focus on the stability of both equilibria of system (1). Note that the characteristic equation of system (1) is given by     ∂g ∂f ∂g ∂f ∂f  −d −v  −I −ξ −V −I −g(T , I ) −V −f (T , I, V ) 0   ∂T ∂T ∂I ∂I ∂V     ∂f ∂g ∂f ∂g ∂f   V +I V +I +g(T , I )−a −ξ V +f (T , I, V ) 0   = 0. ∂T ∂T ∂I ∂I ∂V     0 k −μ−qB −ξ −qV        0 0 cB cV −h−ξ 

(8) Firstly, we have the following result. Theorem 3.1 The free-infection equilibrium Ef is locally asymptotically stable if R0 < 1 and becomes unstable if R0 > 1. Proof Evaluated (8) at Ef , we get 

   λ η η 2 (1 − R0 ) = 0. ,0 ξ +a μ + q (ξ + d)(ξ + h) ξ + μ + a + q − g h d h Hence, the roots of this equation are ξ1 = −d, ξ2 = −h,   √  − a + μ + q hη − g dλ , 0 −  , ξ3 = 2   √  − a + μ + q hη − g dλ , 0 +  ξ4 = , 2 where  = (μ + a + q hη − g( dλ , 0))2 − 4a(μ + q hη ))(1 − R0 ). Clearly, ξ1 , ξ2 , and ξ3 are negative. However, ξ4 is negative if R0 < 1 and it is positive if R0 > 1. Therefore, Ef is locally asymptotically stable if R0 < 1 and unstable if R0 > 1. 

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The following theorem characterizes the global stability of the free-infection equilibrium Ef when R0 ≤ 1. Theorem 3.2 The free-infection equilibrium Ef is globally asymptotically stable when R0 ≤ 1. Proof Construct a Lyapunov functional as follows: L(t) = I (t) +

f ( dλ , 0, 0) V (t). μ + q hη

Calculating the time derivative of L along the solutions of (1), we get   λ dL μ + qB f = f (T , I, V ) − , 0, 0 V dt d μ + q hη ' &     kf dλ , 0, 0 + μ + q hη g(T , I ) −1 I +a a(μ + q hη )   λ ≤ f (T , 0, 0) − f , 0, 0 V + a(R0 − 1)I d ≤ a(R0 − 1)I. it is not hard to show that the largest Since R0 ≤ 1, we have dL dt (t) ≤ 0. Also, % $ = 0 is the singleton {Ef }. By the LaSalle’s invariant set in (T , I, V , B)| dL dt invariance principle [17], Ef is globally asymptotically stable for R0 ≤ 1.  Now, we focus on the global stability of the chronic infection equilibrium E ∗ by assuming that R0 > 1 and the functions f and g satisfy, for all T , I, V > 0, the following hypothesis:   f (T , I, V ) f (T , I ∗ , V ∗ ) V 1− ≤ 0, − f (T , I ∗ , V ∗ ) f (T , I, V ) V∗   f (T ∗ , I ∗ , V ∗ )g(T , I ) f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) I 1− − ∗ ≤ 0. f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) f (T ∗ , I ∗ , V ∗ )g(T , I ) I

(H4 )

Theorem 3.3 Assume that (H4 ) holds. If R0 > 1, then the chronic infection equilibrium E ∗ is globally asymptotically stable. Proof Consider the following Lyapunov functional:  f (T ∗ , I ∗ , V ∗ ) I (t) ∗ dX + I  ∗ ∗ I∗ T ∗ f (X, I , V )   qf (T ∗ , I ∗ , V ∗ )V ∗ ∗ f (T ∗ , I ∗ , V ∗ )V ∗ ∗ V (t) B(t) + , + V  B  kI ∗ V∗ ckI ∗ B∗

W (t) = T (t) − T ∗ −

T

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where (x) = x − 1 − ln x, x > 0. So, the time derivative of W along the positive solutions of (1) satisfies   f (T ∗ , I ∗ , V ∗ ) λ − dT − f (T , I, V )V − g(T , I )I W˙ (t)|(1) = 1 − f (T , I ∗ , V ∗ )   I∗ f (T , I, V )V + g(T , I )I − aI + 1− I   ∗ V∗ f (T , I ∗ , V ∗ )V ∗ 1− kI − μV − qBV + kI ∗ V   B∗ qf (T ∗ , I ∗ , V ∗ )V ∗ 1− η + cBV − hB . + ckI ∗ B Using λ = dT ∗ + f (T ∗ , I ∗ , V ∗ )V ∗ + g(T ∗ , I ∗ )I ∗ , kI ∗ = μV ∗ + qB ∗ V ∗ , and η = hB ∗ − cB ∗ V ∗ , we get   T f (T ∗ , I ∗ , V ∗ ) ∗ ˙ 1− W (t)|(1) = dT 1 − ∗ T f (T , I ∗ , V ∗ )  f (T ∗ , I ∗ , V ∗ ) f (T , I, V )V +f (T ∗ , I ∗ , V ∗ )V ∗ 3 − + f (T , I ∗ , V ∗ ) f (T , I ∗ , V ∗ )V ∗ V f (T , I, V )V I ∗ IV ∗ − − − f (T ∗ , I ∗ , V ∗ )V ∗ I V∗ I ∗V  f (T ∗ , I ∗ , V ∗ ) f (T ∗ , I ∗ , V ∗ )g(T , I )I + +g(T ∗ , I ∗ )I ∗ 2 − f (T , I ∗ , V ∗ ) f (T , I ∗ , V ∗ )g(T ∗ , I ∗ )I ∗ 2  qηf (T ∗ , I ∗ , V ∗ )V ∗ I g(T , I ) ∗ − B − B − ∗− . I g(T ∗ , I ∗ ) ckI ∗ B ∗ B Then

  T f (T ∗ , I ∗ , V ∗ ) ∗ ˙ 1− W (t)|(1) = dT 1 − ∗ T f (T , I ∗ , V ∗ )  V f (T , I, V )V f (T , I ∗ , V ∗ ) + +f (T ∗ , I ∗ , V ∗ )V ∗ − 1 − ∗ + V f (T , I ∗ , V ∗ )V ∗ f (T , I, V )  ∗ ∗ ∗ ∗ ∗ f (T , I, V )V I ∗ f (T , I , V ) f (T , I , V ) − − +f (T ∗ , I ∗ , V ∗ )V ∗ 4 − f (T , I ∗ , V ∗ ) f (T , I, V ) f (T ∗ , I ∗ , V ∗ )V ∗ I  ∗ ∗ ∗ ∗ f (T , I , V )g(T , I )I I IV + g(T ∗ , I ∗ )I ∗ − 1 − ∗ − − ∗ I V I f (T , I ∗ , V ∗ )g(T ∗ , I ∗ )I ∗  f (T ∗ , I ∗ , V ∗ ) f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) ∗ ∗ ∗ + g(T 3− + , I )I ∗ ∗ ∗ f (T , I , V )g(T , I ) f (T , I ∗ , V ∗ )  2 f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) g(T , I ) qηf (T ∗ , I ∗ , V ∗ )V ∗ ∗ − − − B − B . f (T ∗ , I ∗ , V ∗ )g(T , I ) g(T ∗ , I ∗ ) ckI ∗ B ∗ B

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Hence,   T f (T ∗ , I ∗ , V ∗ ) + f (T ∗ , I ∗ , V ∗ ) 1− W˙ (t)|(1) = dT ∗ 1 − ∗ T f (T , I ∗ , V ∗ )  f (T , I, V )V f (T , I ∗ , V ∗ ) V + ×V∗ − 1 − ∗ + + g(T ∗ , I ∗ ) V f (T , I ∗ , V ∗ )V ∗ f (T , I, V )  I f (T ∗ , I ∗ , V ∗ )g(T , I )I f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) + × I∗ − 1 − ∗ − I f (T , I ∗ , V ∗ )g(T ∗ , I ∗ )I ∗ f (T ∗ , I ∗ , V ∗ )g(T , I )    f (T , I ∗ , V ∗ ) f (T ∗ , I ∗ , V ∗ ) +  − f (T ∗ , I ∗ , V ∗ )V ∗  f (T , I ∗ , V ∗ ) f (T , I, V ) 

   ∗ ∗ ∗ ∗ ∗ f (T , I, V )V I IV ∗ , I ∗ )I ∗  f (T , I , V ) + − g(T + ∗ I V f (T ∗ , I ∗ , V ∗ )V ∗ I f (T , I ∗ , V ∗ ) 

 g(T , I ) f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) + + f (T ∗ , I ∗ , V ∗ )g(T , I ) g(T ∗ , I ∗ )  2 qηf (T ∗ , I ∗ , V ∗ )V ∗ ∗ . B −B − ckI ∗ B ∗ B

By (H2 ), we deduce that   T f (T ∗ , I ∗ , V ∗ ) 1− ∗ 1− ≤ 0. T f (T , I ∗ , V ∗ ) By (H4 ), we obtain  f (T , I, V ) f (T , I, V )V f (T , I ∗ , V ∗ ) V = 1 − + + V∗ f (T , I ∗ , V ∗ )V ∗ f (T , I, V ) f (T , I ∗ , V ∗ )  V f (T , I ∗ , V ∗ ) − ∗ ≤0 × f (T , I, V ) V

−1−

and I f (T ∗ , I ∗ , V ∗ )g(T , I )I f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) − + I∗ f (T , I ∗ , V ∗ )g(T ∗ , I ∗ )I ∗ f (T ∗ , I ∗ , V ∗ )g(T , I )   f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) I f (T ∗ , I ∗ , V ∗ )g(T , I ) − ∗ ≤ 0. = 1− f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) f (T ∗ , I ∗ , V ∗ )g(T , I ) I −1 −

Since (x) ≥ 0, we have W˙ (t)|(1) ≤ 0 with equality if and only if T = T ∗ , I = I ∗ , V = V ∗ , and B = B ∗ . From LaSalle’s invariance principle, we conclude that the chronic infection equilibrium E ∗ is globally asymptotically stable when R0 > 1. 

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4 Application The aim of this section is to apply our main results to the following model: ⎧ ⎪ T˙ ⎪ ⎪ ⎪ ⎪ ⎨ I˙ ⎪ ⎪ ⎪ ⎪ V˙ ⎪ ⎩ B˙

β1 T V β2 T I − , 1 + α1 V 1 + α2 I β2 T I β1 T V + − aI, = 1 + α1 V 1 + α2 I = kI − μV − qBV , = η + cBV − hB,

= λ − dT −

which is a special case of system (1) by letting f (T , I, V ) = β2 T 1+α2 I ,

(9)

β1 T 1+α1 V

and g(T , I ) =

where α1 and α2 are non-negative constants that measure the saturation effect, β1 is the virus-to-cell infection rate, and β2 is the cell-to-cell transmission rate. Obviously, the assumptions (H0 )–(H3 ) hold, and we have   f (T , I ∗ , V ∗ ) f (T , I, V ) −α1 (V − V ∗ )2 V 1− = − ≤0 f (T , I ∗ , V ∗ ) f (T , I, V ) V∗ V ∗ (1 + α1 V )(1 + α1 V ∗ ) and   f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) I f (T ∗ , I ∗ , V ∗ )g(T , I ) − ∗ 1− f (T , I ∗ , V ∗ )g(T ∗ , I ∗ ) f (T ∗ , I ∗ , V ∗ )g(T , I ) I =

−α2 (I − I ∗ )2 ≤ 0. I ∗ (1 + α2 I )(1 + α2 I ∗ )

Therefore, the assumption (H4 ) is satisfied. From Theorems 3.1, 3.2, and 3.3, we have the following result. Corollary 4.1 (i) If R0 ≤ 1, then the infection-free equilibrium Ef of system (9) is globally asymptotically stable. (ii) If R0 > 1, then the infection-free equilibrium Ef becomes unstable and the chronic infection equilibrium E ∗ of system (9) is globally asymptotically stable.

5 Conclusions In this work, we have proposed and qualitatively analyzed a generalized within-host CHIKV infection model with humoral immunity and two modes of transmission. The incidence rate for both modes of transmission is modeled by two general nonlinear functions which cover many special cases existing in the literature. We first proved the existence, positivity, and boundedness of solutions which ensures

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that the proposed model is well-posed. Under some hypotheses on the general incidence functions, the global dynamics of the model are completely determined by the basic reproduction number R0 . More precisely, we have proved that the infection-free equilibrium is globally asymptotically stable if R0 ≤ 1, which leads to the eradication of virus in the host. When R0 > 1, then the infection-free equilibrium becomes unstable and a chronic infection equilibrium exists and is globally asymptotically stable, which means that the virus persists in the host. Furthermore, the models and results presented in Refs. [10, 11] are extended and improved. According to the above main results, we deduce a strategy to control the CHIKV infection. This strategy is based on the reduction of the value of R0 to make it less than or equal to one. Therefore, the results obtained from this work can be useful to determine an effective treatment that reduced the value of R0 to a threshold of less than or equal to one in order to eradicate the virus.

References 1. WHO, Chikungunya: fact sheet (2017). http://www.who.int/mediacentre/factsheets/fs327/en/ 2. M.C. Robinson, An epidemic of virus disease in Southern Province, Tanganyika territory, in 1952–53. Trans. R. Soc. Trop. Med. Hyg. 49(1), 28–32 (1955) 3. PAHO, Epidemiological alert, Chikungunya Fever (2013). http://www.paho.org/hq/index.php? option=com_docman&task=doc_view&Itemid=270&gid=23806&lang=en 4. X. Liu, P. Stechlinski, Application of control strategies to a seasonal model of chikungunya disease. Appl. Math. Model. 39(12), 3194–3220 (2015) 5. C.A. Manore, K.S. Hickmann, S. Xu, H.J. Wearing, J.M. Hyman, Comparing dengue and chikungunya emergence and endemic transmission in A. aegypti and A. albopictus. J. Theor. Biol. 356, 174–191 (2014) 6. F.B. Agusto, S. Easley, K. Freeman, M. Thomas, Mathematical model of three age-structured transmission dynamics of chikungunya virus. Comput. Math. Methods Med. 2016, 4320514 (2016) 7. N. Baez-Hernandez, M. Casas-Martinez, R. Danis-Lozano, J.X. Velasco-Hernandez, A mathematical model for Dengue and Chikungunya in Mexico, bioRxiv (2017). https://doi.org/10. 1101/122556 8. S.M. Sabiu, N. Hussaini, Modelling the dynamics of Chikungunya in human and mosquito populations. Bayero J. Pure Appl. Sci. 10(1), 634–641 (2017) 9. Y. Dumont, J.M. Tchuenche, Mathematical studies on the sterile insect technique for the chikungunya disease and Aedes albopictus. J. Math. Biol. 65(5), 809–854 (2012) 10. Y. Wang, X. Liu, Stability and Hopf bifurcation of a within-host chikungunya virus infection model with two delays. Math. Comput. Simul. 138, 31–48 (2017) 11. A.M. Elaiw, T.O. Alade, S.M. Alsulami, Analysis of within-host CHIKV dynamics models with general incidence rate. Int. J. Biomath. (2018). https://doi.org/10.1142/ S1793524518500626 12. W. Mothes, N.M. Sherer, J. Jin, P. Zhong, Virus cell-to-cell transmission. J. Virol. 84, 8360– 8368 (2010) 13. P. Zhong, L.M. Agosto, J.B. Munro, W. Mothes, Cell-to-cell transmission of viruses. Curr. Opin. Virol. 3, 44–50 (2013) 14. Q. Sattentau, Avoiding the void: cell-to-cell spread of human viruses. Nat. Rev. Microbiol. 6, 815–826 (2008)

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15. K. Hattaf, N. Yousfi, A generalized virus dynamics model with cell-to-cell transmission and cure rate. Adv. Differ. Equ. 2016, 174 (2016) 16. K. Hattaf, N. Yousfi, Qualitative analysis of a generalized virus dynamics model with both modes of transmission and distributed delays. Int. J. Differ. Equ. 2018, 9818372 (2018) 17. J.P. LaSalle, The Stability of Dynamical Systems. Regional Conference Series in Applied Mathematics (SIAM, Philadelphia, 1976)

Differential Game Model for Sustainability Multi-Fishery Nadia Raissi, Chata Sanogo, and Mustapha Serhani

1 Introduction Starting almost 50 years ago, C.W. Clark has had a fundamental enduring impact on fisheries economics, he articulated what was previously a vague concept of optimal fisheries management in a precise mathematical model rooted in modern capital theory. In 1975, Clark [8] built what was called the fundamental fishery model, a classical optimal control problem, whose analysis leads to a bang–bang control solution and was useful to preserve North Pacific whales. A recent literature follows this work and constitutes what was known as Clark school. This occurred thanks to the development of dynamic optimization during the 1980s and early 1990s. In 2017 [10], Clark and Munro underline the objectives of almost all fishery management project which are a stocks conservation as well as necessity of the “pur et dur” rebuilding program for overexploited capture fishery resources. The authors recommended the application of game theory, even in an Exclusive Economic Zone (E.E.Z.) management. Differential games theory was first introduced by Isaacs [15] in 1965, it represents a generalization of optimal control to the cases where there is more than one controller, see Refs. [5, 6, 16, 22]. This fact was behind the growth request of

N. Raissi () Mohammed V University in Rabat, Faculty of Sciences, Rabat, Morocco e-mail: [email protected] C. Sanogo Université des Sciences des Techniques et des Technologies de Bamako, Faculté des Sciences et Techniques, Bamako, Mali M. Serhani () Department of Economy, TSI Team, FSJES, University Moulay Ismail, Toulal, Meknes, Morocco e-mail: [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_9

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this theory to tackle the problems of fishery management, since the hypothesis of a single decision-maker is no more available. In the literature, there are several works on the integration of differential games theory to model the fishery resource management problems, see, for instance, Refs. [2, 4, 8, 9, 13]. Clark [8] considers a problem of exploitation with two fleets. The author establishes a guesswork for the determination of optimal effort in the competitive case. The strategy is piecewise built, according to the biomass position with respect to the bionomic equilibria. He shows in the competitive case that the least efficient fleet is forced to leave the competition for the benefit of the most efficient one. Raissi [18] considers a regulator aiming to maintain both fleets as long as possible in the fishing. The analysis of the model via two-level optimization gives the same conclusion as in Clark [8]. Doyen and Pereau [14] consider a model combining the viability approach and differential games. The analysis shows, on the one hand, that when the agents can form coalitions, the stability of big coalition is possible for high quantities of the stock. On the other hand, for low levels, the most effective fleet obliges the others to leave the fishery. Also the main result of Mullon and Mullon [17] is that the least financially constrained player can readily exclude another from the competition and they suggested to support local financial systems in order to reach coexistence. In the present study, we consider a resource exploited by two fleets. Each of them tries to maximize the income generated by its exploitation. However, the income doesn’t depend only on fleet exploitation strategy but also on that adopted by the competitor. This problem is formulated as a differential game between two players on infinite horizon. The Pontryagin maximum principle is used to identify a competitive Nash equilibrium insuring, under some reasonable assumptions, the sustainable cohabitation for both fleets. Then, we draft an optimal strategy built through this Nash equilibrium. To prove the optimality of this strategy, the inductive method based on the verification functions and Hamilton–Jacobi theory is invoked. The crucial role played by the value function as being the unique solution of the HJB equation (under constrained initial condition) permits us to prove the truth of our conjecture. The conclusion highlighted in this study may be useful for the decision-makers worried about the durability of both fleet activity. This paper spreads out on three sections. The mathematical model is formulated in the next section. The third section is devoted to analysis of the model and to the determination of Nash equilibrium. In the last section, Hamilton–Jacobi equation is invoked to prove the optimality of the candidate identified by means of the maximum principle.

2 Mathematical Model We consider two fleets in competition, fishing the same resource. The mathematical model is formulated as a differential game with infinite horizon. Each player aims

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to maximize his benefit while taking account of the strategy adopted by the other player. The system obeys the following dynamic: x(t) ˙ = F (x(t)) − q2 E2 (t)x(t) − q1 E1 (t)x(t), x(0) = x0 ; 0 ≤ Ei (t) ≤ Eimax ∀t ≥ 0, i = 1, 2,

(1)

where x(t) stands for the biomass level of the stock at time t, F (.) is the intrinsic growth function of the resource, qi is the catchability rate of the fleet i, Ei is the fishery effort for fleet i, Eimax is the maximum fishery effort for fleet i. The aim of the player i (i = 1, 2) is to maximize his income generated by the exploitation of the resource . Ji =

+∞

e−δi t (pi qi x(t) − ci )Ei (t)dt, i = 1, 2.

(2)

0

For the fleet i, δi > 0 denotes the discount rate, pi is the resource unit price, and ci is the unit cost of the fishery effort. We suppose that the players don’t cooperate. A pair of admissible control (E1∗ , E2∗ ) is said to be a Nash equilibrium for the non-cooperative differential game (1)–(2), if the following property holds (see, for instance, Refs. [2, 7, 15, 16]): 

J1 (E1∗ , E2∗ ) ≥ J1 (E1 , E2∗ ); for all admissible E1 , J2 (E1∗ , E2∗ ) ≥ J2 (E1∗ , E2 ); for all admissible E2 .

(3)

In other words, Ei∗ (.) is an optimal control of the following problem Pi , associated with the player i; i, j = 1, 2, j = i,

+∞

max Ei

e−δi t (pi qi x(t) − ci )Ei (t)dt,

0

with x(t) ˙ = F (x(t)) − qi Ei (t)x(t) − qj Ej∗ (t)x(t), 0 ≤ Ei (t) ≤ Eimax , x(0) = x0 .

(4)

3 Model Analysis 3.1 Nash Equilibrium In order to build an optimal strategy, we identify bionomic equilibrium for each fleet, as used in several works dealing with fishery problems as in Refs. [8, 14, 18]. We look for a common equilibrium to both players that guaranties their sustainability. To do this, we apply an adequate version of Pontryagin maximum principle at Nash equilibrium (see Refs. [8, 9, 11]).

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Define the current value of pseudo-Hamiltonian, H˜ i , for the player i; i = 1, 2   H˜ i (x, Ei , Ej∗ , μi ) = (pi − μi )qi x − ci Ei + μi F (x) − qj Ej∗ x .

(5)

If (E1∗ , E2∗ ) is a Nash equilibrium and x ∗ is the associated state, there exists for all t ≥ 0, a costate μi , Lagrange multipliers mi ; ni ≥ 0 such that ∂ H˜ i = −μ˙ i + δi μi , ∂x ∂ H˜ i = ni − mi , ni (Eimax − Ei ) = 0; and mi Ei = 0 i = 1, 2. ∂Ei

(6) (7)

 ci be the switch function associated with problem Pi , Let σi := (pi − μi ) − qi x it follows that  ∗ Ei = Eimax if σi > 0, (8) Ei∗ = 0 if σi < 0. If σi ≡ 0 on an interval Ii , we obtain a singular control. In this case, we have μi = pi −

ci ∀ t ∈ Ii , i = 1, 2. qi x

(9)

Differentiating (9) with respect to t and taking account of Eq. (6) leads after some basic algebraic calculus to   ci ci ci ∗  ∗ (F (x) − qj Ej ) = δi pi − . (F (x) − qj Ej x) + pi − qi x qi x qi x 2

(10)

This last equation is equivalent to d dx

  ci ci pi − (F (x) − qj Ej∗ x) = δi pi − , qi x qi x

(11)

Set for each i, j = 1, 2 with j = i d ψi (x, Ej ) := dx



ci pi − qi x



(F (x) − qj Ej∗ x)



 ci . − δi pi − qi x

(12)

From this analysis, we could identify a unique equilibrium candidate which will be helpful to determine Nash equilibrium.

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 x . Under the following hypotheTheorem 3.1 Assume that F (x) := rx 1 − K sis: • (H0 ) pi qi − ci ≥ 0 for each i = 1, 2 • (H1 ) r ≤ qi Eimax for i = 1, 2 • (H2 ) K is large, (K $ 1), then there exists a unique (x ∗ , E1∗ , E2∗ ) ∈]0, K[×[0, E1max ] × [0, E2max ] solution of the following system: 

F (x ∗ ) = (q1 E1∗ + q2 E2∗ )x ∗ ψi (x ∗ , Ej∗ ) = 0 for each i, j = 1, 2 with j = i.

(13)

Proof Assume that both equations ψi (x, Ej∗ ) = 0 for each i, j = 1, 2 with j = i have x as solution. Then Ej∗ =

pi qi x − ci ln(x) F (x) − δi , i, j = 1, 2; i = j. qj x qj (pi qi x − ci )

(14)

On the other hand, at the equilibrium, Eq. (4) states that   F (x ∗ ) = q1 E1∗ x ∗ + q2 E2∗ x ∗ .

(15)

Substituting, E1∗ and E2∗ by its expressions (14), yields to F (x ∗ ) − δ2

p2 q2 x ∗ − c2 ln(x ∗ ) ∗ p1 q1 x ∗ − c1 ln(x ∗ ) ∗ x − δ x = 0. 1 p2 q2 x ∗ − c2 p1 q1 x ∗ − c1

(16)

Thus the existence of solution of this last equation in ]0,K[ establishes the existence statement in the theorem. Let us prove this existence in the case of the logistic growth. Let f be the continuous function defined on ]0,K] by  x (p2 q2 x − c2 )(p1 q1 x − c1 ) − δ2 (p2 q2 x − c2 ln(x)) f (x) = r 1 − K × (p1 q1 x − c1 ) − δ1 (p1 q1 x − c1 ln(x))(p2 q2 x − c2 ). Then we have lim f (x) = +∞

x→0+

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and thanks to hypothesis (H0 ) and (H2 ) f (K) < 0. By a direct application of intermediate value theorem, we proof the existence of at least one root x ∗ ∈ ]0, K[ of f , and thus, x ∗ is a solution of (16). Let us now turn to the unicity of such equilibrium point. From a straight calculus we show that for each E there exists a unique x solution of ψi (x, E) = 0; in the case of the logistic, each function ψi is decreasing with respect to x on ]0,K], indeed r d ci ψi (x, E) = − δi − p i < 0 2 dx K qi x therefore and change its sign on this interval. The theorem will be established as soon as we proof the admissibility of the fishing efforts (Ei∗ ). From equilibrium equation (13) and hypothesis (H1 ), we have qi Ei∗

 x∗ =r 1− − qj Ej∗ ≤ r ≤ qi Eimax for each i = 1, 2. K

This occurs with the positivity of each Ei∗ .

 

Remark 3.1 The fishing effort at the Nash equilibrium, for each player, is equal to his optimal effort when he exploits singly the resource, less a term which is inversely proportional at his catchability and depends on the second player parameters. This fact highlights the interaction between the fleets in competition. With this Nash equilibrium candidate, it remains to give a way to reach it starting for a giving initial stocks level, and to proof its optimality, this is the goal of the next section.

4 Optimal Strategy In this section we invoke Hamilton–Jacobi–Bellman (HJB) equation to prove the ˆ built using Nash equilibrium identified in the optimality of a candidate (x, ˆ E), previous section.

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4.1 Hamilton–Jacobi–Bellman Equation Consider (x, τ ) ∈ [0, K] × R+ , we perturb the problem Pi with respect to initial condition (τ, x). Hence we construct the following value function Vi for i = 1, 2:

Vi (x, τ ) = max

+∞

e−δi t (pi qi x(t) − ci )Ei (t)dt.

(17)

τ

Our study is based on the two following lemmas. Lemma 4.1 For i = 1, 2, if Vi is C 1 , then it fulfills the HJB equation   ∂Vi (x, τ ) ∂Vi (x, τ ) + max x(τ ˙ ) + li (τ, x, Ei ) = 0, ∀ (τ, x) ∈ [0, +∞[×[0, K]. Ei ∂t ∂x (18)

where li (t, x, Ei ) = e−δi t (pi qi x − ci )Ei . Lemma 4.2 If there exists ϕ : [0, +∞[×[0, K] −→ R of class C 1 fulfilling HJB equation (18), then Ei at which the maximum in HJB is achieved is an optimal control for the problem (4). The proofs are simple adaptation of those given in Refs. [11, 12, 19] and are based on the Bellman principle. We know that in several cases the value function Vi is the unique solution of the HJB equation with an appropriate boundary conditions (see Refs. [3, 12, 19–21]). In this study, taking into account the particularity of the model, in which the explicit time dependence is modeled by the term e−δt , the above HJB equation can be rewritten in a version. Lemma 4.3 Set V i (x) = eδi τ Vi (x, τ ), so V i satisfy the autonomous HJB equation x

x

op

δi V i (x) − V i (x)F (x) − (pi qi x − ci )Ei + V i (x)(qi Ei x + qj Ej x) ≥ 0, ∀x, ∀Ei , (19) op

with equality for all path (x op , Ei ) which is optimal for the player i. x op Note that Ej is an optimal control of the player j and V i (x) stands for the derivative of V i (x) with respect to x. x The functions V i , V i depend only on the variable x. (x,τ ) Proof Firstly, we have ∂Vi∂t = −δi e−δi τ V i (x) and Hence by multiplying Eq. (18) by eδi τ we obtain x

∂Vi (x,τ ) ∂x

˙ ) + li1 (τ, x, Ei )} = 0, −δi V i (x) + max{V i (x)x(τ Ei

x

= e−δi τ V i (x).

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where li1 := eδi τ li . It follows that x

˙ ) + li1 (t, x, Ei ) ≤ 0, ∀Ei , ∀x, − δi V i (x) + V i (x)x(τ

(20)

op

and along the optimal path Ei

x

op

−δi V i (x) + V i (x)x˙ op (τ ) + li1 (τ, x, Ei ) = 0, ∀x. The substitution, in (20), of li1 and x(τ ˙ ) by its values, leads to autonomous version of the HJB equation. x

δi V i (x) − V i (x)F (x) − (pi qi x − ci )Ei x

(21)

op

+ V i (x)(qi Ei x + qj Ej x) ≥ 0, ∀x, Ei , op

with equality along the optimal path (x op , Ei ).

 

4.2 Optimal Strategy 4.2.1

Conjecture

The purpose now is to make a conjecture concerning an optimal strategy based on the Nash equilibrium. We use the inductive method developed by Clarke [11] based on the verification theory to confirm this conjecture. The idea is to build a verification (test) function inspired from the value function and then verify whether this test function is a solution of HJB equation (21). If it is the case, then the conjecture is correct according to Lemmas 4.2 and 4.1; otherwise, an adjustment of the conjecture for the case where it fails to be optimal is required. As conjecture, we propose the following feedback control ⎧ max max ⎨ (E1 , E2 ) if x > x ∗ , Eˆ = (Eˆ 1 , Eˆ 2 ) = E ∗ = (E1∗ , E2∗ ) if x = x ∗ , ⎩ (0, 0) if x < x ∗ .

(22)

ˆ Let xˆ be the associated trajectory to E. As signaled in the previous section, we drop the bionomic equilibria strategy, to adopt the idea of single Nash equilibrium x ∗ , in order to keep both fleets in the competition. To achieve this aim, we will assume that the natural growth of the resource is a logistic and the parameters of the model fulfill hypothesis (H0 ) and (H1 ): The presumed optimal strategy is described as follows: if x > x ∗ the biomass is higher than x ∗ , then the better choice for the control is to take Ei = Eimax for

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i = 1, 2. Hence, according to hypothesis (H1 ), the biomass x(t) decreases until achieving (x = x ∗ ). At this level, to stay thereafter at the equilibrium, we choose Ei = Ei∗ (x ∗ ) for i = 1, 2. Both fleets exploit the resource simultaneously, each of them takes account on the presence of the other player as carrying out by the expressions of the fishing efforts Ei∗ (x ∗ ). Now, if x < x ∗ , the biomass level is lower than x ∗ , the better choice for the control is to take Ei = 0 for i = 1, 2, permitting hence to biomass x(t) to growth at x = x ∗ .

4.2.2

Optimality Proof

In the first step, we build a verification function. As discussed above, we invoke the value function V¯i and the conjecture. The chosen verification function is equal to V¯i along the path (Eˆ i , x) ˆ because we hopefully obtain the same optimality result in Lemma 4.1 for this verification function. Definition 4.1 Let ϕi be a verification function defined as

+∞

ϕi (x) =

ˆ e−δi t (pi qi x(t) ˆ − ci )E(t)dt.

0

We obtain the following characterization of ϕi Lemma 4.4 

C0 (τ ∗ (x)) if x ≤ x ∗ , C0 (τ  (x)) + C1 (x, τ  (x)) if x > x ∗ ,

ϕi (x) =

(23)

where functions C0 , C1 and implicit functions τ  , τ ∗ of x are given by e−δi τ (pi qi x ∗ − ci )Ei∗ , ∀ τ ≥ 0, δi

C0 (τ ) =

τ ∗ (x) is given implicitly by the Cauchy problem ⎧ x˙ = F (x(t)), ∀ t ≥ 0, ⎪ ⎪ ⎪ ⎪ ⎪ x(0) = x, ⎪ ⎨ x(τ ∗ ) = x ∗ ⎪ and ⎪ ⎪ ⎪ ⎪ −1 ⎪ ⎩ τ ∗ (x) = , F (x)

C1 (x, τ ) = 0

τ

e−δi t (pi qi x(t) ˆ − ci )Eimax dt, ∀ τ ≥ 0,

(24)

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and where τ  (x) is an implicit solution of the following Cauchy problem: ⎧ x˙ = F (x(t)) − qj Ejmax x(t) − qi Eimax x(t), ∀ t ≥ 0, ⎪ ⎪ ⎨ x(0) = x, −1 ⎪ ⎪ x(τ  ) = x ∗ , and (τ  ) (x) = . ⎩ max F (x) − qi Ei x − qj Ejmax x

(25)

Proof We know that

+∞

ϕi (x) =

ˆ e−δi t (pi qi x(t) ˆ − ci )E(t)dt.

0

• For x ≤ x ∗ the calculus of ϕi (x) gives

ϕi (x) =

τ ∗ (x)

e

−δi t

(pi qi x(t) ˆ − ci )0dt +

0

=

e−δi τ δi

∗ (x)

+∞ τ ∗ (x)

e−δi t (pi qi x ∗ − ci )Ei∗ dt,

(pi qi x ∗ − ci )Ei∗ ,

= C0 (τ ∗ (x)).

(26) (27)

• For x > x ∗ we have

ϕi (x) = 0

+

τ  (x)

e−δi t (pi qi x(t) ˆ − ci )Eimax dt

+∞ τ  (x)

e−δi t (pi qi x ∗ − ci )Ei∗ dt,

= C0 (τ  (x)) + C1 (x, τ  (x)). Which achieves the proof of the lemma.

(28)  

The next step consists to prove that the verification function given by Lemma 4.4 fulfills the HJB equation. Remark firstly that ϕi is continuously differentiable. The implicit functions theorem assure that τ ∗ (x) and τ  (x) are twice continuously differentiable; hence, according to the fact that C0 is of class C ∞ and C1 is of class C 1 with respect to (x, τ ), since x(t, x0 ) is C 2 with respect to the initial condition x0 (dependence of solutions with respect to initial condition [1]). We conclude that ϕi ∈ C 1 . It remains to prove that ϕi satisfy the HJB equation. Theorem 4.1 The verification function ϕi given by Eq. (23) fulfills the HJB inequality (19) with equality along the path (Eˆ i , x) ˆ given by the conjecture, for i = 1, 2.

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Proof To prove that ϕi fulfills HJB inequality (19), is equivalent to the proof of the following inequality: δi ϕi (x) − ϕix (x)(F (x) − qi Ei x − qj Eˆ j x) − (pi qi x − ci )Ei ≥ 0 ∀ x, ∀ Ei , (29) with the equality along Eˆ i , i.e., δi ϕi (x) − ϕix (x)(F (x) − qi Eˆ i x − qj Eˆ j x) − (pi qi x − ci )Eˆ i = 0,

(30)

where Eˆ i and Eˆ j are given by the conjecture. Reformulate the inequality (29) under the new form δi ϕi (x) − ϕix (x)F (x) − ((pi − ϕix (x))qi x − ci )Ei + qj ϕix (x)Eˆ j x ≥ 0, ∀x, Ei . (31) • For x > x ∗ , we proceed by steps: Step 1. ϕix (x) > 0: We have ϕi (x) = C0 (τ  (x)) + C1 (x, τ  (x)). Hence ϕix (x) = C0 (τ  (x))(τ  (x)) + ≥ C0 (τ  (x))(τ  (x)) +

∂C1 (x, τ  (x)) ∂C1 (x, τ  (x))  + (τ (x)) , ∂x ∂t ∂C1 (x, τ  (x))  (τ (x)) , ∂t

∂C1 (x, τ  (x)) ≥ 0, according to fact that x → τ  (x) is nondecreasing and by ∂x the way x → C1 (x, τ  (x)) is nondecreasing. Hence, since

e−δi τ (x) (pi qi x ∗ − ci )(Eimax − Ei∗ ) . −F (x) + (qi Eimax + qj Ejmax )x 

ϕix (x) ≥

(32)

which proves that ϕix (x) > 0,

(33)

according to assumptions H2 − H3 . Step 2. Prove the HJB equality along the candidate path (30): We know that

+∞

+∞ d −δi t ϕi (x) = (e e−δi t (pi qi x(t) ˆ − ci )Eˆ i (t)dt = − ϕi (x(t)))dt. ˆ dt 0 0 (34)

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Equation (34) can be rewritten as

+∞

e−δi t (pi qi x(t) ˆ − ci )Eˆ i (t)dt =

0

0

+∞

˙ˆ e−δi t (δi ϕi (x(t)) ˆ − ϕix (x(t)) ˆ x(t)dt, (35)

which can be rewritten as

+∞ ˙ˆ − (pi qi x(t) e−δi t (δi ϕi (x(t)) ˆ − ϕix (x(t)) ˆ x(t) ˆ − ci )Eˆ i (t))dt = 0.

(36)

0

The integrand is sign unchanged. Indeed, ϕi (x(t)) ˆ is nonnegative, ϕix (x(t)) ˆ is ∗ ˙ nonnegative according to inequality (33), x(t) ˆ is nonpositive since x > x , finally (pi qi x(t) ˆ − ci ) is nonnegative. Hence, we conclude that ˙ˆ − (pi qi x(t) ˆ − ϕix (x(t)) ˆ x(t) ˆ − ci )Eˆ i (t) = 0, ∀ t ≥ 0. δi ϕi (x(t))

(37)

˙ˆ by its value and taking t = 0, we obtain By substituting x(t) δi ϕi (x) − ϕix (x)(F (x) − qi Eˆ i x − qj Eˆ j x) − (pi qi x − ci )Eˆ i = 0, ∀ t ≥ 0. (38) which proves the HJB equality along the candidate path, as required in (30). Step 3. Proof of the HJB inequality (29): Our hope is to evaluate the expression ϕi (x) − ϕi (x + ) = C0 (τ  (x)) + C1 (x, τ  (x)) − C0 (τ  (x + )) −C1 (x + , τ  (x + )). To do this, calculate the closed line (curve) integral ( I= γ

  ci ci (F (x) − qj Ejmax x)dt − e−δi t pi − dx, e−δi t pi − qi x qi x

(39)

where γ is the closed boundary of D constituted by trajectories x(t)(starting ˆ at x), xˆ (t) (starting at x +), the segment [x, x +], and the segment [τ  (x), τ  (x +)] and pointed in the anticlockwise direction (Fig. 1). Consider k > 0 as an arbitrary constant, γ can be piecewise described as γ = γ1 ∧ γ2 ∧ γ3 ∧ γ4 ,

Differential Game Model for Sustainability Multi-Fishery Fig. 1 The closed path γ and the domain D

x+Δ

131

γ = γ1 Λ γ2 Λ γ3 Λ γ4

γ4

∧

γ3 = x (t ; x+Δ) Δ

x ∧

γ1 = x(t ; x)

D x*

τ#(x)

γ2

τ# (x+Δ)

where γ1 : [ 0 , τ  (x)] −→ R2 α −→ (t (α), x(α)) = (α, x(α)), ˆ γ2 : [ τ  (x) , τ  (x + )] −→ R2 α −→ (t (α), x(α)) = (α, x ∗ ), γ3 : [τ  (x + ) , 2τ  (x + )] −→ R2 α −→ (t (α), x(α)) = (2τ  (x + ) − α, xˆ (2τ  (x + ) − α)), γ4 : [2τ  (x + ) , 2τ  (x + ) + k] −→ R2     α −→ (t (α), x(α)) = 0, − α + x +  + 2 τ (x + ) . k k It’s clear that γ is closed. According to line integral calculus, the integral I of Eq. (39) becomes I = A + B + C + D,

(40)

where

A= 0

τ  (x)

e−δi α

ci ˆ (F (x(α)) ˆ − qj Ejmax x(α)) qi x(α) ˆ

 ci ˙ x(α) ˆ dα. − pi − qi x(α) ˆ

 pi −

(41)

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B=



ci ∗ ∗ ∗ e−δi α pi − (F (x ) − q E x ) dα. j j qi x ∗

τ  (x)

2τ  (x+) 

C=−

e

−δi (2τ  (x+)−α)

τ  (x+)

 pi −

(42)

ci qi xˆ (2τ  (x + ) − α)



(43)

×(F (xˆ (2τ  (x + ) − α)) − qj Ejmax xˆ (2τ  (x + ) − α))

−e−δi (2τ

 D= k

 (x+)−α)

2τ  (x+)+k

 pi −



ci ˙ˆ (2τ  (x + ) − α) dα. x qi xˆ (2τ  (x + ) − α)

& pi −

2τ  (x+)

qi

 k

ci

'

 dα. α + x +  + 2 k τ  (x + )

(44)

Observe that A = C1 (x, τ  (x))

(45)

and  1 ci   (F (x ∗ ) − qj Ejmax x ∗ ), B = − (e−δi τ (x+) − e−δi τ (x) ) pi − δi qi x ∗ = C0 (τ  (x + )) − C0 (τ  (x)).

(46)

To compute C, consider the variable change ξ = 2τ  (x + ) − α, then

C=

τ  (x+) 

e

−δi ξ

 pi −

0

−e

−δi ξ

ci (F (xˆ (ξ )) − qj Ejmax xˆ (ξ )) qi xˆ (ξ )

 pi −



ci ˙xˆ (ξ ) dξ, qi xˆ (ξ )

= −C1 (x + , τ  (x + )). Compute now D, consider the variable change λ=

 (−α + 2τ  (x + )) + x + . k

(47)

(48)

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133

It follows k D=− k 

 ci pi − dλ, qi λ x+

x+ 

= x

x

ci pi − dλ. qi λ

(49)

We conclude according to equalities (45), (46), (48), and (49) that I = C1 (x, τ  (x)) − C1 (x + , τ  (x + ))

x+  ci pi − dλ, −C0 (τ  (x + )) + C0 (τ  (x)) + qi λ x

(50)

which leads to x+ 

I = ϕi (x) − ϕi (x + ) +

ci pi − dλ. qi λ

x

(51)

On the other hand, the line integral I of Eq. (39) can be computed using Green’s formula as a double integral on the domain D bounded by the closed path γ . Indeed   ci ci e−δi t pi − (52) (F (x) − Ejmax qj x)dt − e−δi t pi − dx, qi x qi x γ   

 ci ci −δi t −δi t d max δi e pi − −e pi − (F (x) − Ej qj x) dtdx. = qi x dx qi x D (

I =

Hence

e−δi t ψi (x)dtdx,

I =−

(53)

D

where ψ is given by the hypothesis H1 . But, from the fact that ψi is nonincreasing, we obtain ∀ x > x ∗ , ψi (x) < ψi (x ∗ ) = 0,

(54)

I > 0.

(55)

it follows that

Hence, according to Eq. (51), we get

I = ϕi (x) − ϕi (x + ) +

x+ 

pi −

x

ci dλ > 0. qi λ

(56)

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By differentiating by  and tending it towards 0, we obtain  ci ≥ 0, − ϕix (x) + pi − qi x

(57)

(pi − ϕix (x))qi x − ci ≥ 0.

(58)

so

Recall the HJB inequality (29) which we would prove, according to inequality (58), we have δi ϕi (x) − ϕix (x)F (x) − ((pi − ϕix (x))qi x − ci )Ei + qj ϕix (x)Eˆ j x

(59)

≥ δi ϕi (x) − ϕix (x)F (x) − ((pi − ϕix (x))qi x − ci )Eimax + qj ϕix (x)Eˆ j x = 0, the last vanishing equality leads according to step 2, since for x > x ∗ we have Eˆ i = Eimax . This proves the HJB equality for all Ei in the case where x > x ∗ . • For x < x ∗ : Remark firstly that the HJB equality along the candidate path (30) is fulfilled by the ˙ˆ same reasoning at step 2, in x > x ∗ case, by remarking that x(t) is nonnegative ∗ since x < x . Prove now the HJB inequality. We have according to the conjecture that Eˆ j = 0 and ϕix (x) =

∂C0 (τ ∗ (x)) = −δi ϕi (x)τ ∗ (x), ∂x

but τ ∗ (x) =

−1 , F (x)

hence ϕix (x) =

δi ϕi (x) . F (x)

(60)

Then, δi ϕi (x) − ϕix (x)F (x) = 0; hence, the inequality (31) becomes −((pi − ϕix (x))qi x − ci )Ei ≥ 0 ∀ x, ∀ Ei . It suffices to prove (pi − ϕix (x))qi x(t) − ci < 0 ⇔ ϕix (x) > pi −

ci . qi x

(61)

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By substituting ϕix (x) by its value (60) in (61), the inequality to prove, it becomes,  ci δi ϕi (x) > pi − F (x), qi x

(62)

and by substituting δi ϕi (x) by its value in (26), the inequality becomes e−δi τ

Use now the fact that

e

∗ (x)

 ci F (x). (pi qi x ∗ − ci )Ei∗ > pi − qi x

F (x ∗ ) F (x ∗ ) qj ∗ ≥ Ei∗ = − Ej , the above inequality becomes ∗ qi x qi x ∗ qi

−δi τ ∗ (x)

 F (x ∗ ) ci F (x). (pi qi x − ci ) > pi − qi x ∗ qi x ∗

By remarking that τ ∗ (x ∗ ) = 0 and hence e−δi τ e

−δi τ ∗ (x)

(63)

∗ (x ∗ )

= 1, we can write

  ci ci ∗ −δi τ ∗ (x ∗ ) pi − F (x ) > e pi − F (x). qi x ∗ qi x

(64)

 ci pi − F (x), to prove (64) it suffices to prove that φi (x) qi x is nondecreasing. By the standard calculus, we lead to

Let φi (x) = eδi τ

φi

(x)

=

∗ (x)

∗ eδi τ (x)



d dx

  ci ci pi − (F (x) − qj Ej x) − δi pi − . qi x qi x (65)

Remark that φi (x) = eδi τ

∗ (x)

ψi (x, Ej )

where ψi is given in Theorem 3.1 . To prove that φi is nondecreasing goes back to prove that ψi is nonnegative for x < x∗. According to Theorem 3.1 , ψi (., Ej ) is decreasing and admit x ∗ as a root, hence ψi (x, Ej ) > ψi (x ∗ , Ej ) = 0, ∀ x < x ∗ , consequently ψi (x, Ej ) ≥ 0, ∀x < x ∗ .

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It follows that (64) is fulfilled for x < x ∗ , and hence the Hamilton–Jacobi–Bellman inequality (31) holds in the case where x < x ∗ . The HJB equation is then fulfilled in all cases, leading to the truthfulness of our conjecture.  

5 Conclusion This study intended to answer the question: is it possible that two fleets harvest durably the same resource while maximizing their income every time? An optimal strategy of exploitation around a Nash equilibrium was developed, allowing both fleets to coexist in the exploitation while maximizing their incomes. The Pontryagin maximum principle was used to identify a Nash equilibrium, while the HJB theory and inductive method were used to confirm its optimality. One of the perspectives would be to integrate more decision-makers having ecological objectives as well as economic ones in the model. Another perspective would be to consider a model in which the decision-makers cooperate in a way to reach a common goal.

References 1. V. Arnold, Ordinary Differential Equations (Springer, Berlin, 1992) 2. J.P. Aubin, Mathematical Methods of Game and Economic Theory (Dover Publications Inc/Northwestern University, New York/Amsterdam, 2007) 3. M. Bardi, I.C. Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations (Birkhäuser, Boston, 1997) 4. H. Benchekroun, N.V. Long, Transboundary fishery: a differential game model. J. Econ. 69(274), 207–221(2002) 5. A. Bressan, From optimal control to non-cooperative differential games: a homotopy approach. J. Control Cybern. 38(4A), 1081–1106 (2009) 6. R. Buckdahn, P. Cardaliaguet, M. Quincampoix, Some recent aspects of differential game theory. J. Dyn. Games Appl. 1(1), 74–114 (2011) 7. P. Cardaliaguet, S. Plaskacz, Existence and uniqueness of a Nash equilibrium feedback for a simple non-zero-sum differential game. Int. J. Game Theory 32, 33–71 (2003) 8. C.W. Clark, Mathematical Bioeconomics: The Optimal Management of Renewable Resources (Wiley, New York, 1990) 9. C.W. Clark, Mathematical Bioeconomics: The Mathematics of Conservation, 3rd edn. (Wiley, New Jersey, 2010) 10. C.W. Clark, G.R. Munro, Capital theory and the economics of fisheries: implications for policy. Mar. Resour. Econ. 32(2), 123–142 (2017) 11. F.H. Clarke, Method of Dynamic and Nonsmooth Optimization, CBMS-NSF (SIAM, Philadelphia, 1990) 12. F.H. Clarke, Y.S. Ledyaev, R.J. Stern, P.R. Wolenski, Nonsmooth Analysis and Control Theory. Graduate Texts in Mathematics (Springer, New York, 1998) 13. E.J. Dockner, S. Jorgensen, N.V. Long, G. Sorger, Q. Mary, Differential Games in Economics and Management Science (Cambridge University Press, Cambridge, 2001)

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Towards a Thermostatistics of the Evolution of Protein Domains Through the Formation of Families and Clans Rubem P. Mondaini and Simão C. de Albuquerque Neto

1 Introduction: The Construction of the Sample Space for Statistical Analysis We choose to work with the Protein Family Database (Pfam) [1–3] for characterizing the sample space of the statistical analysis to be undertaken in the present contribution. The evolution of the Pfam database can be seen from Table 1. Let us now proceed heuristically to identify protein domain families from protein domains. We consider the sequences of amino acids below to represent fictitious proteins and the domains depicted on them (Fig. 1). We then proceed to the formation of protein domain families according to the following scheme which satisfies the requirements for alignment of domains (Fig. 2). The longest common number of rows on the example above is m = 5 and the longest common number of columns is n = 4. We then consider at least one array (5 × 4) as a representative of each protein domain family. In general, we construct m (rows) × n (columns) arrays, and we associate at least one array (m × n) as a representative of each protein domain family. In order to obtain these arrays, we have to take into consideration that if a protein domain family has the structure of m rows with nl , l = 1, 2, . . . , m, amino acids on the l-th row, in order to obtain a representative m × n array as a member of the sample space, we have to discard all the rows with nl < m. For the rows with nl > m, then (nl − n) amino acids should be discarded.

R. P. Mondaini () · S. C. de Albuquerque Neto Federal University of Rio de Janeiro, Centre of Technology, COPPE, Rio de Janeiro, RJ, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_10

139

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Table 1 The evolution of Pfam database Pfam database Version Year 18.0 2005 19.0 2005 20.0 2006 21.0 2006 22.0 2007 23.0 2008 24.0 2009 25.0 2011 26.0 2011 27.0 2013 28.0 2015 29.0 2015 30.0 2016 31.0 2017 32.0 2018

No. of families class. into clans 7973 8183 8296 8957 9318 10,340 11,912 12,273 13,672 14,831 16,230 16,295 16,306 16,712 17,929

No. of families 1181 1399 1560 1683 1815 2016 3132 3439 4243 4563 4939 5282 5423 5996 7001

Fig. 1 Protein domains in 04 fictitious proteins

Fig. 2 Protein domain families formed from the protein domains of Fig. 1

No. of clans 172 205 239 262 283 303 423 458 499 515 541 559 595 604 628

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Fig. 3 A generic structure of a protein domain family and the construction of a (m × n) representative array of the sample space

The (m × n) array obtained is a representative of the sample space. A generic structure of the arrays to be considered to represent protein domain families on the sample space is depicted in Fig. 3. Protein domains in red and remaining protein domains in blue should be discarded as explained above. We now consider a “Poissonization” process in which a discrete binomial distribution with discrete time is embedded in a Poisson process of continuous time [4]. From the probability of occurrence of finding Nj (a) a-amino acids (a = A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y, according to the one-letter code) in the j -th column (j = 1, 2, . . . , n) or pj (a) =

 Nj (a) "⇒ pj (a) = 1 , ∀j m a

(1)

we consider the probability of finding a-amino acids in the j -th column on a time t     Nj (a, t)  "⇒ pj Nj (a, t), t = pj Nj (a, t), t = 1 , ∀j m a

(2)

Let σ be the probability per unit time that a transition of one a-amino acid from the j -th column to another column. Then σ t is the probability of this transition to occur on the interval of time t, and (1 − σ t) is the probability that no transition will occur during the same interval of time. We can then write for the probability of finding an a-amino acid at time t + t,       p Nj (a, t + t), t + t = σ t · p Nj (a, t) − 1, t + (1 − σ t)p Nj (a, t), t (3)

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The usual procedure is to start from this master equation to derive the Fokker– Planck equation driving the process of transmutation of amino acids. We address the reader to Ref. [4] and from now on, we omit the dependence on t on all subsequent equations of this work.

2 The Free Energies Associated with a Protein Domain Family The mean energy of the amino acid distribution per column of a protein domain family is then given by: Uj =



pj (a)Ej (a)

(4)

a

where Ej (a) is the mean energy of the a-amino acid in the j -th column. We now suppose that all external constraints are kept fixed and that all changes in the mean energy Uj of the distribution of amino acids on each column of the (m × n) array representing a protein domain family (Eq. (4)) are due only to heat flux [4, 5]. By partial differentiation of Uj w.r.t. T , the absolute temperature, we then get the heat capacity of this distribution or Cj =

∂Uj ∂T

(5)

Since there are no mechanical work to be done on the amino acids, we can write from Eq. (5) and the definition of entropy, dUj = Cj dT = T dSj = T

∂Sj dT ∂T

(6)

From Eqs. (5), (6), we get ∂Sj 1 ∂Uj = ∂T T ∂T

(7)

And we have after integrating this equation: 1 Sj = Uj + T

Uj dT + Kj T2

where Kj is a function of the external constraints alone [6].

(8)

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From the 3rd law of thermodynamics, the entropy should tend to a constant value as T → 0. This value should not depend on the external constraints and we then require that Kj ≡ 0. Equation (8) can be then written as:

Fj ≡ Uj − T Sj = −T

Uj dT T2

(9)

where Fj is the Helmholtz free energy of the j -th column of amino acids’ distribution of the representative (m × n) array associated with a protein domain family. Usually we take μ1 ≡ kB1T , with kB the Boltzmann constant, kB = 1.38 × 10−23 J/◦ K. In the present contribution we shall use units such that kB = 1. We write from Eq. (9): F j ≡ Uj −

1 1 Sj = μ1 μ1

Uj dμ1

(10)

3 The Maximization of Entropy Principle: The Case of Gibbs–Shannon Entropy We take for granted that the reader knows the elementary procedure of looking for extrema of a constrained function. This is applied to the entropy maximization process [5, 7, 8]. We then write the augmented Lagrangian & Lj = Sj + λ1 1 −



'

&

pj (a) + μ1 Uj −

a



' pj (a)Ej (a)

(11)

a

where Sj = −



pj (a) log pj (a)

(12)

a

is the Gibbs–Shannon entropy per column of amino acids in the (m × n) representative arrays, and λ1 , μ1 stand for the Lagrange multipliers. We then require for the first variation, δLj =

∂Lj δpj (a) = 0 ∂pj (a)

(13)

or 

 − log pj (a) − 1 − λ1 − μ1 Ej (a) δpj (a) = 0

(14)

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Since Sj is a convex function, the probability distribution pj (a) in Eq. (14) will correspond to its maximum: pj (a) = e−(1+λ1 ) e−μ1 Ej (a)

(15)

From Eq. (1) we write e(1+λ1 ) ≡ Zj ≡



e−μ1 Ej (b)

(16)

b

where Zj is the partition function of the amino acids’ distribution corresponding to the j -th column of the (m × n) array. From Eqs. (4), (15), (16) we then get for the mean energy, ) Uj =

a

Ej (a) e−μ1 Ej (a) = Zj

)

Ej (a) e−μ1 Ej (a) ∂ =− log Zj ) −μ E (b) 1 j ∂μ e 1 b

a

(17)

We can write for the Helmholtz free energy, Eq. (10), F j ≡ Uj −

1 1 Sj = − log Zj μ1 μ1

(18)

4 Generalized Entropy Measures: (1) Havrda–Charvat Entropy Our proposal in the present section is the rederivation of the fundamental results associated with the construction of a thermostatistics in Sect. 2 and the expression of thermodynamic variables in terms of a partition function. We now take into consideration an old generalization of the Gibbs–Shannon entropies, the Havrda– Charvat set of entropies. We stress the connectedness of the formulae with the calculation of the limits for recovering the results already obtained in Sect. 2. Analogously to Sect. 2, we write the constrained Lagrangian for a generalized one-parameter Havrda–Charvat entropy [9], Hjs . & Ljs = Hjs + λs 1 −



'

&

pj (a) + μs Ujs −



a

' pˆ j (a)Ej (a)

(19)

a

where 1 Hjs = − 1−s

&

 s pj (a) 1− a

and s is a non-dimensional positive definite parameter.

' (20)

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As an entropy function, Hjs should be a concave function and since we have  s−2 ∂ 2 Hjs 0. The symbols pˆ j (a) will stand for s-escort probabilities [8] associated with the usual probabilities of occurrence given by Eq. (1) of Sect. 1, through s  pj (a) pˆ j (a) = )  s pj (b)

(22)

b

The mean energy of a column of amino acids of a (m × n) array is now given by:  Uj s = pˆ j (a)Ej (a) (23) a

We then have for the first variation of the Lagrangian, Eq. (19), δLjs =

∂Ljs δpj (c) = 0 ∂pj (c)

which means that + *   ∂ pˆ j (a)  s−1 s − λs δac − μs δac pj (a) Ej (a) δpj (a) = 0 1−s ∂pj (c) a a

(24)

(25)

where s−1 s−1  s   pj (a) δac s pj (a) s pj (c) ∂ pˆ j (a) = ) s −   s 2 ) ∂pj (c) pj (b) p (b) j b

(26)

b

From Eqs. (25), (26), we then get *

 + s−1  s−1 pj (a) Ej (a) − Ujs s  pj (a) − λs − μs δpj (a) = 0 s )  1−s b pj (b)

(27)

From the arbitrariness of the variation, the square bracket should be also zero. We then have, after multiplying it by pj (a):   s  s s pj (a) Ej (a) − Ujs s  pj (a) − λs pj (a) − μs =0 s )  1−s b pj (b)

(28)

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After summing up in a, we get for the parameter λs , λs =

s (Zjs )1−s 1−s

(29)

where (Zjs )1−s ≡



pj (a)

s

(30)

a

and Zjs is an auxiliary partition function whose implicit form will be derived in the following lines. After substituting Eqs. (29), (30) into Eq. (28), we get   s−1  s−1 Ej (a) − Ujs s pj (a) s  s 1−s pj (a) − − μs =0 (Zjs ) 1−s 1−s (Zjs )1−s

(31)

From the requirement of concavity, s > 0 and Eq. (31) we can choose 1>s>0 &

 ' 1 (1 − s) Ujs − Ej (a) 1−s 1 + μs (Zjs )1−s

1 pj (a) = Zjs

(32)

The escort probabilities (Eq. (22)) are then given by:  pˆ j (a) =

) b



(1−s) Ujs −Ej (a) 1 + μs (Zjs )1−s



s 1−s

  s  1−s (1−s) Ujs −Ej (b) 1 + μs 1−s (Z )

(33)

js

The mean energy of the amino acids belonging to the j -th column (Eq. (23)) is now given by: ) Ujs =

a

  s  1−s (1−s) Ujs −Ej (a) Ej (a) 1 + μs 1−s (Z ) js

  s  1−s ) (1−s) Ujs −Ej (b) 1 + μs 1−s (Z )

(34)

js

b

The auxiliary partition function, Eq. (30), will be given implicitly by: Zjs =

 a

&

 ' s (1 − s) Ujs − Ej (a) 1−s 1 + μs (Zjs )1−s

(35)

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The Gibbs–Shannon results for pj (a) (Eqs. (15), (16)) and Uj (Eq. (17)) of Sect. 2 will be recovered through:

lim pˆ j (a) =

s→1

  s  1−s (1−s) Ujs −Ej (a) lim 1 + μs (Z )1−s js

s→1

  s  1−s ) (1−s) Ujs −Ej (b) 1−s b lim 1 + μs (Z ) s→1

lim e



μs

s Ujs −Ej (a) (Zjs )1−s

s→1

=

)

js

e−μ1 Ej (a)  = ) −μ E (b) = pj (a) 1 j Ujs −Ej (b) be



lim e

b



μs

s

(36)

(Zjs )1−s

s→1

and lim

lim Ujs =

)

s→1 a

  s  1−s (1−s) Ujs −Ej (a) Ej (a) 1 + μs 1−s (Z ) js

  s  1−s ) (1−s) Ujs −Ej (b) lim b 1 + μs 1−s (Z )

s→1

s→1

) =

μs

a

Ej (a) lim e s→1 

) b

=−

js



lim e

μs

s Ujs −Ej (a) (Zjs )1−s

s Ujs −Ej (b) (Zjs )1−s



 ) =

Ej (a)e−μ1 Ej (a) ) −μ E (b) 1 j be

a

s→1

∂ log Zj = Uj ∂μ1

(37)

We should note that the partition function given by Eq. (16) is not obtained by taking the limit of the Zjs function, Eq. (35), since we have according to Eq. (16), lim Zjs = eμ1 Uj Zj

s→1

(38)

However, if we write, analogously to Eq. (18), the corresponding Helmholtz free energy: F j s ≡ Uj s −

1 Hj μs s

(39)

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where Hjs is the Havrda–Charvat entropy given by Eq. (20) or Hjs =

(Zjs )1−s − 1 1−s

(40)

we get lim

lim Fjs =

)

s→1

  s  1−s (1−s) Ujs −Ej (a) E (a) 1 + μ s a j (Z )1−s

s→1

lim

s→1

= Uj − =−



) b

js





(1−s) Ujs −Ej (b) 1 + μs (Zjs )1−s

s 1−s

& − lim

s→1

1 (Zjs )1−s − 1 μs 1−s

'

 1 1 log lim Zjs = Uj − log(eμ1 Uj Zj ) s→1 μ1 μ1

1 log Zj = Fj μ1

(41)

where we have used Eqs. (37), (38). We can then define, for the Helmholtz free energy [8], F js ≡ −

1 (Zj )1−s − 1 μs 1−s

(42)

and Zj is the partition function introduced by Eq. (16) of Sect. 2. The mean energy of the amino acids’ distribution per column of a (m × n) array (Eq. (34)) can be also written as: ∂ Uj s ≡ − ∂μs

&

(Zj )1−s − 1 1−s

' (43)

Fjs and Ujs from Eqs. (42), (43), respectively, will satisfy an analogous equation to Eq. (10) from Sect. 2 or 1 F js = μs

Ujs dμs

(44)

5 Generalized Entropy Measures: (2) Sharma–Mittal Entropy A maximization process which starts from an analogous equation to Eq. (19) and derives an analogous equation to Eq. (28) will be used in this section.

Towards a Thermostatistics of the Evolution of Protein Domains Through the. . .

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The Lagrangian for a generalized two-parameter Sharma–Mittal entropy [10], (SM)jrs is given by       Ljrs = (SM)jrs + λrs 1 − pj (a) + μrs Ujrs − pˆ j (a)Ej (a) a

(45)

a

where (SM)jrs

1 =− 1−r

& 1−



 s pj (a)

1−r ' 1−s

(46)

a

The concavity of this entropy leads us to require s−r  

 s 1−s  s−2 s(s − r) ∂ 2 (SM)jrs pj (a) pj (a) pˆ j (a) − 1 < 0  2 = s (1 − s)2 ∂ pj (a) a (47) and we can choose 1>r≥s>0 The Havrda–Charvat [9] (Hjs ), Renyi [11] (Rjs ), Landsberg–Vedral [12] ((LV )js ) one-parameter entropy measures could be obtained from Eq. (46) by taking convenient limits. We have Hjs = lim (SM)jrs ; Rjs = lim (SM)jrs ; (LV )js = lim (SM)jrs r→s

r→1

r→2−s

(48)

where Hjs is given by Eq. (20) and Rjs , (LV )js , are given, respectively, by: Rjs =

 s 1 log pj (a) 1−s a

Hj (LV )js = )  s s a pj (a) and analogous formulae to those derived on the last section for Hjs could also be derived for Rjs and (LV )js . The Gibbs–Shannon entropy measure is then obtained by taking the limit s → 1 on all entropies above as    pj (a) log pj (a) (49) lim Hjs , Rjs , (LV )js = − s→1

a

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An analogous equation to Eq. (29) could be also derived. We write λrs =

s (Zjs )1−r , 1−s

λrs = λs if r = s

(50)

where Zjs is also given by Eq. (30). Equations analogous to Eq. (32) (probability of occurrence), Eq. (33) (escort probabilities), Eq. (34) (mean energy of the j -th column of amino acids from a protein domain family), and Eq. (35) (implicit formula for partition function) could be easily derived and we leave the derivation to the reader as an exercise. The results are written below: &  ' 1 (1 − s) Ujrs − Ej (a) 1−s 1 pj (a) = (51) 1 + μrs Zjs (Zjs )1−r  pˆ j (a) =

)



(1−s) Ujrs −Ej (a) 1 + μrs (Zjs )1−r



s 1−s

  s  1−s (1−s) Ujrs −Ej (b) 1 + μ rs 1−r b (Z )

(52)

js

)

  s  1−s (1−s) Ujrs −Ej (a) E (a) 1 + μ rs 1−r a j (Z ) js

Ujrs =

  s  1−s ) (1−s) Ujrs −Ej (b) b 1 + μrs (Z )1−r

(53)

js

Zjs =

 a

&

 ' s (1 − s) Ujrs − Ej (a) 1−s 1 + μrs (Zjs )1−r

(54)

where μrs = μs if r = s. The Helmholtz free energy is now given by: 1 (SM)jrs μrs   s  1−s ) (1−s) Ujrs −Ej (a) a Ej (a) 1 + μrs (Z )1−r

Fjrs ≡ Ujrs −

=

js

  s  1−s ) (1−s) Ujrs −Ej (b) b 1 + μrs (Z )1−r js

−

1 (Zjrs )1−r − 1 μrs 1−r

(55)

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We can then recover formulae (1)–(10) taking the limits lim lim on formus→1 r→s

lae (51)–(55). lim lim pˆ j (a) = pj (a) =

s→1 r→s

e−μ1 Ej (a) Zj

lim lim Ujrs = Uj

(57)

s→1 r→s

lim lim Fjrs = −

s→1 r→s

(56)

1 log Zj = Fj μ1

(58)

where Zj is the partition function given by Eq. (16). We also have lim lim Zjrs = eμ1 Uj Zj

(59)

s→1 r→s

in clear agreement with Eq. (38). We can then define according to Eq. (42), the Helmholtz free energy of this case: Fjrs ≡ −

1 (Zj )1−r − 1 μrs 1−r

(60)

and the mean energy is given analogously to Eq. (43) by: Ujrs

∂ ≡− ∂μrs

&

(Zj )1−r − 1 1−r

' (61)

Equations (60), (61) will satisfy an analogous of Eq. (44), Fjrs =

1 μrs

Ujrs dμrs

(62)

6 Concluding Remarks The belief in the introduction of generalized entropy measures to extend the treatment of statistical physics methods to fundamental theories of soft matter should be more than a fancy dream [13]. The aspect of the resulting equations by adopting a development driven by Havrda–Charvat or Sharma–Mittal entropies does not seem to be promising as can be seen from Sects. 4 and 5. However, the discovery of specific ranges of the non-dimensional r, s parameters to consider the usefulness of these generalized entropies could at least be a great contribution on the statistical classification of protein domain families into clans [1–3]. We have been

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working on this statistical approach as well as on the development of a generalized statistical mechanics to understand the transmutation of amino acids in the process of formation of protein domain families and clans. This work is now proceeding and a few recent results will be published elsewhere.

References 1. S. El-Gebali et al., The PFAM protein families databases in 2019. Nucleic Acids Res. 47, D427–D432 (2018) 2. E.L.L. Sonnhammmer, S.R. Eddy, R. Durbin, Pfam: a comprehensive database of protein domain families based on seed alignments. Proteins Struct. Funct. Genet. 28 405–420 (1997) 3. R.D. Finn et al., The Pfam protein families databases: towards a more sustainable future. Nucleic Acids Res. 44, D279–D285 (2016) 4. R.P. Mondaini, S.C. de Albuquerque Neto, Stochastic assessment of protein databases by generalized entropy measures, in Trends in Biomathematics: Modeling, Optimization and Computational Problems (Springer International Publishing, Cham, 2018), pp. 91–105 5. P.T. Landsberg (ed.), Problems in Thermodynamics & Statistical Physics (Pion Limited, London, 1971) 6. J.J. Binney et al., The Theory of Critical Phenomena (Oxford University Press, Oxford, 1998) 7. E.T. Jaynes, Probability Theory - The Logic of Science (Cambridge University Press, Cambridge, 2003) 8. C. Beck, Generalized information and entropy measures in physics. Contemp. Phys. 50(4), 495–510 (2009) 9. J. Havrda, F. Charvat, Quantification method of classification processes. Concept of structural α-entropy. Kybernetika 3(1), 30–35 (1967) 10. B.D. Sharma, D.P. Mittal, New non-additive measures of entropy in discrete probability distributions. J. Math. Sci. 10, 28–40 (1972) 11. A. Renyi, On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability V.1 (University of California Press, Berkeley, 1961), pp. 547–561 12. P.T. Landsberg, V. Vedral, Distributions and channel capacities in generalized statistical mechanics. Phys. Lett. A 224, 326–330 (1997) 13. B.H. Lavenda, A New Perspective on Thermodynamics (Springer Science+Business Media, Berlin, 2010)

Analysis of Tumor/Effector Cell Dynamics and Decision Support in Therapy S. Sabir and N. Raissi

1 Introduction Because there is not one but several cancers and because each patient is unique, different types of treatments can be proposed to treat a patient. They are prescribed alone or in combination. The choice of treatments is adapted according to each situation. The treatment of tumors by immunotherapy involves mobilizing our own defenses against cancer cells. A major turning point has been crossed with the arrival of new molecules acting on the immune system and effective in certain cancers. The use of these new molecules made it possible to observe tumor regression in some patients. Immunotherapy, therefore, has an increasingly important place alongside conventional treatments, surgery, chemotherapy, and radiotherapy. So, it is very important to understand how cancer evolves to maximize the efficiency of treatments. Cancer is a multi-disciplinary subject that concerns all areas including mathematics, that’s why there are several works that led to very important results. Among these studies is that of Demongeot et al. [8] which is a statistical study that allowed to identify the most common cancers in the region of Valparaiso and suggested appropriate prevention policies. Also Clairambault et al. [7] present a model that describes the cellular dynamics within a tumor microsphere under the effect of cytotoxic and cytostatic drugs, in order to analyze the effects of spatial structure and combination therapies on phenotypic heterogeneity.

S. Sabir () · N. Raissi University Mohammed V in Rabat, Faculty of Sciences, Rabat, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_11

153

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In the same view, numerous mathematical models have been developed to study the interaction between tumor cells and effector cells [3]. For instance, Kuznetsov et al. [6] have developed a two-dimensional dynamical model of a cell-mediated response to a growing tumor cell population. However, it is the first model that takes into account tumor infiltration by effector cells as well as the possibility of inactivation of the effector cells. The analysis of this dynamic system is focused on the qualitative behavior of the system using the bifurcation theory. This analysis was taken up by Jaroudi in her master’s thesis [4] by adding in the model constant parameters corresponding to treatments: chemotherapy and immunotherapy. In this work, these parameters are no more constant and become control variables, in order to determine combined therapy that is to identify a pair of control variables leading to an admissible trajectory, which corresponds to maintain the tumor density below a critical threshold. With this aim in mind, we appeal to viability theory. We built viability kernel which is the set of initial states (tumor cells/effector cells) for which we can define a protocol of combined therapies (chemotherapy and immunotherapy) guaranteed the remains in the set of the state constraints. This corresponds to determining the chances of remission of the patient with respect to his health state at the first screening. The material is organized in five sections. In Sect. 2, the dynamic systems on which we have been based are presented to motivate subsequent discussion and the basic concepts of viability theory are extended to it. In Sect. 3, we establish the existence conditions of a viability kernel and we analytically calculate the viability kernel corresponding to the controlled dynamic system in Sect. 4. Section 5 corresponds to the viability kernel approximation that we determine in Sect. 4 using the Kaviar software.

2 Mathematical Model Starting from Kuznetsov’s model [6] that describes the growth dynamics of two cell populations and their interaction: tumor cells (T ) and immune system effector cells (E). In this dynamical system, the natural growth of the tumor cells is logistic function fT and the growth of the effector cells is affine function fE : fE : R + → R + E → s − dE With s the constant flow of effector immune cells and d the mortality rate of the effector cells.

Analysis of Tumor/Effector Cell Dynamics and Decision Support in Therapy

155

Whereas the function fT is defined as follows: fT : R + → R + T → aT (1 − bT ) Where a is the maximal growth rate of the tumor cells population, and b−1 the maximum number of cells due to competition for resources (oxygen, glucose, etc.). This competition between cells is modeled by −nT E and −mT E. The presence of the tumor stimulates the growth of the effector cells, this phenomenon is modeled by following explicit form F (T , E) that depends on T and E, on a maximum growth rate ρ of the effector cells and a half-saturation constant g, to characterize the rate at which cytotoxic effector cells accumulate in the region of tumor cell localization: F (T , E) : R + × R + → R + (T , E) →

ρT E g+T

The dynamic system of Kuznetsov is therefore:  T˙ = a(1 − bT )T − nET E E˙ = s − dE + ρT g+T − mT E

(1)

Then, Jaroudi inserts in this model into her master’s thesis [4], parameters (c, i) corresponding to treatments (chemotherapy/immunotherapy) and obtained the following autonomous dynamical system:  T˙ = a(1 − bT )T − μcT − ET (2) E E˙ = σ − dE + ρT g+T − βT E − hcE + i i describes the positive and direct effect of immunotherapy on effector cells and c the concentration of chemotherapy that eliminates tumor cells with μ rate and effector cells with h rate. The results of analysis of this dynamical system illustrated by numerical simulations allowed to confirm the efficacy of combination therapy compared to single therapy. In the present work, we generalize this model by setting treatments as a variable of time which leads to controlled system where c and i are no more constant and become variable control. So, we put c = c(t) and i = i(t) and replacing them in the previous system we obtain the following controlled system: ⎧ ⎪ ⎨ T˙ = a(1 − bT )T − μc(t)T − ET E E˙ = σ − dE + ρT g+T − βT E − hc(t)E + i(t) ⎪ ⎩ E(0) = E , and T (0) = T 0 0

(3)

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Since chemotherapy and immunotherapy have adverse side effects, the dose of these treatments is very sensitive and should not exceed a maximum threshold that depends on several criteria related to each patient. We therefore admit that the concentration of chemotherapy c and immunotherapy i is limited as follows: 0 ≤ c ≤ cmax and 0 ≤ i ≤ imax The set U constraints for the control is defined as: U = [0, cmax ] × [0, imax ] The survival state of the patient is related to the density of the tumor and the size of the effector cells to fight against this tumor. This is expressed by state constraints: 0 ≤ T ≤ Tmax and Emin ≤ E ≤ Emax So, the set of viability constraints K is defined by: K = [0, Tmax ] × [Emin , Emax ] We have to deal with a controlled system with constraints on the state, so the appropriate conceptual framework for analyzing such a problem is the viability theory. This theory consists in studying the existence of controls generating states verifying viability constraints, these states are called viable states or viable trajectories. The set of all states initial condition from which at least one viable evolution starts is called the viability kernel. We return to our system, in this case the determination of the viability kernel corresponds to the identification of the chances of survival of a patient on treatment according to the state of his cells at the first diagnosis. This is the subject of the next section.

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3 Viability Kernel At the heart of this theory, the concept of viability kernel is fundamental. The model of the study corresponds to the evolution: ⎧  ⎨ X = f (X, u) X(0) = X0 , and u(t) ∈ U ⎩ contraint to X(t) ∈ K, ∀t ≥ 0 For this controlled system, the viability kernel exists if the function f satisfies the following conditions [1]: • f is continuous. • f and U are convex. • f has linear growth. In this case, the viability kernel can be identified as the set of initial conditions X0 for which a viable trajectory exists (X(t) ∈ K, ∀t ≥ 0), where K is the set of constraints. Back to our system and putting the vector X = (T , E), the function f is defined by: f (T , E, c, i)=(a(1−bT )T −μc(t)T −ET , σ −dE+

ρT E −βT E−hc(t)E+i(t)) g+T

In the case of our system (3), it is clear that the multifunction f is continuous because it was built on a summation of continuous functions including the function ρT E g+T which is continuous on K since the denominator g + T never cancels on K. The set U = [0, cmax ] × [0, imax ] is compact then convex. As for convexity of the multifunction f it can be done by different methods, it’s sufficient to proof the Graph of f is convex or just seeing that f is composed of two convex functions. Moreover, f is linearly growing and satisfies the following inequality: if ∃b > 0 such as ∀X ∈ K : sup f (X, c, i) ≤ b( X +1) After having verified the sufficient conditions for the existence of the viability kernel (possibly empty) corresponding to our system, we will prove in the following paragraph the non-emptiness of the viability kernel. To do this we will demonstrate the existence of at least one viable and stable equilibrium point. The equilibrium is said viable if it satisfies the viability constraints. We are interested in the annihilation equilibrium where the tumor cell density is σ +i zero, (0, d+hc ) is viable if: Emin ≤

σ +i ≤ Emax d + hc

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So, we can define the minimum and maximum values imin and imax of immunotherapy as follows: • imin = Emin (d + hcmin ) − σ • imax = Emax (d + hcmax ) − σ From these results, we can define the new set U constraints for the control, for which the annihilation equilibrium is viable: U = [0, cmax ] × [imin , imax ] Now we can say that the viability kernel associated with the controlled system exists and it is non-empty, we define it as follows: V iab(k) = {(T0 , E0 ) ∈ K|∃(c(.), i(.)) such as ∀t ≥ 0, (T (t), E(t)) ∈ K}

4 Analytical Calculation of the Viability Kernel In this part we will analytically calculate the viability kernel for the two-dimensional model describing the competition between tumor cells and effector cells. To do this we propose to calculate the theoretical boundaries of the viability kernel in the case of our model. We define the curve C1 as the set of points (T , E) satisfying: ⎧  ⎪ ⎨ T = −(a(1 − bT )T − μcT − ET ) E E  = −(σ − dE + ρT g+T − βT E − hcE + i) ⎪ ⎩ E (0) = E = E 1 1 min and T1 (0) = T1 where T1 is chosen such that E  (0) = 0. This is an evolution satisfying the inverse system of the studied system (3) with the choice of control (c, i) = (cmax , imax ) and starting from the initial state (T1 , E1 ). So T1 is the solution of the following equations: σ − dEmin +

ρT1 Emin − βT Emin − hcmax Emin + imax = 0 g + T1

(σ − dEmin − hcmax Emin + imax )(g + T1 ) + ρT1 Emin − βT1 Emin (g + T1 ) = 0

We end up with this second degree equation where T1 is the real positive solution: −βEmin T 2 + (σ − dEmin − hcmax Emin + imax − βEmin + ρEmin )T + (σ − dEmin − hcmax Emin + imax )g = 0

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Then, we define C1 by: C1 = {(T , E)|E = f1 (T ), 0 ≤ T ≤ Tmax } That is, C1 is the solution of the differential equation below, the curve of this solution meets the boundaries of K at a point (T1∗ , E1∗ ): f1 (E) =

T with initial condition T1 E

Now we define the second curve C2 as the set of points (T , E) satisfying: ⎧  ⎪ ⎨ T = −(a(1 − bT )T − μcT − ET ) E E  = −(σ − dE + ρT g+T − βT E − hcE + i) ⎪ ⎩ E (0) = E = E etT 2 2 max 2 Where T2 is chosen such that E  (0) = 0 This is an evolution satisfying the inverse system of the studied system with the choice of control (c, i) = (cmin , imin ) and starting from the initial state (T2 , E2 ). We define C2 by: C2 = {(T , E)|E = f2 (T ), 0 ≤ T ≤ Tmax } So C2 is the solution of the differential equation: f2 (E) =

T with initial condition T2 E

So the viability kernel of our dynamical system is defined by: V iab(K) = {(T , E)|E ≤ f2 (T ) if T ≥ T2 } Figure 1 below illustrates viability kernel in the blue color and the curve limit C2 that meets the boundaries of K at points (T2∗ , E2∗ ) and (T2 , E2 ) with particular parameters. However, the solution C1 is not shown in this figure because, for these parameters, C1 does not meet the set K at any point. The curve C2 characterized the viability kernel which corresponds to the states of the cells during the first diagnosis from which we can define a protocol of a combined therapy: chemotherapy and immunotherapy, to guarantee the remains in the set of constraints and evaluate the chances of remission of a patient. However, the definition or the analytical calculation of the viability kernel remains difficult, that’s why specific algorithms have been developed to approach the viability kernel and that’s the purpose of the next section.

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(T2,E2) K C2

Viab(K)

(T∗2,E∗2) Tmin

Emin Tmax

Fig. 1 The curve C1 does not meet the set of constraints at any point, so the limit curve C2 defines the viability kernel corresponding to the studied system in the case where Emin = 0.4, Emax = 100, cmin = 0 and imin = 0.03

5 Approximate Calculation of the Viability Kernel Given the complexity of the analytical proofs for calculating the exact limits of a viability kernel, many algorithms have been developed to give an approximation of the viability kernel. Although these approximation algorithms require memory and storage space and sometimes they give results that can be difficult to handle, their application provides interesting insights into the viability kernel. Among the different algorithms to approach a viability kernel we find: • Patrick Saint-Pierre’s algorithm uses a discretization of the state space on a regular grid. • Kaviar, the algorithm of Guillaume Deffuant et al. uses Vector Support Machines. • Coquelin et al. propose a dynamic programming approach which does not allow a gain in memory space but in computing time. We also have used Kernel Approximation for viability and Resilience or Kaviar [5] which is a free software, programmed in Java, using Support Vector Machines and is widely used to approximate the viability kernels, catch basins and resilience values. To approach the viability kernel, we first need to write the system with all the necessary initial conditions on Java while defining the set of constraints on states and controls. So, for the approximation we use the values shown in Fig. 2 below: Figure 3 illustrates Kaviar interface that offers more options concerning the SVM [2] (support vector machines) configuration—are supervised learning models

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Fig. 2 Source code of the program written in Java

Fig. 3 The graphical interface of Kaviar

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Fig. 4 Approximation of viability kernel with Kaviar

with associated learning algorithms that analyze data used for classification and regression analysis—and the execution of the program. In this version (version 1.0), we can use either the algorithm with a regular grid or the active learning algorithm, we can define heavy controllers for viability kernel approximation or optimal controllers for capture basins and resilience values approximations. At this stage, we execute the program and obtain the following approximation of the viability kernel corresponding to the system (3), the result can be visualized on 2D or 3D. Figure 4 shows Kaviar’s approximation of the viability kernel with blue color. The red points are the points that don’t belong to the viability kernel while the yellow ones belong to it. So we note that the kernel approximation given by the Kaviar software is suitable for the result of the analytical calculation of the viability kernel corresponding to the studied controlled system, which proves the effectiveness of the software for the dynamic systems of two dimensions. This result is clinically interpreted as being all the densities of the tumor and effector cells at the time of screening from which the patient can be brought back to a state of healing. So, using the mathematical model we have studied (3), we can only bring the patient back to a safe state with a high density of effector cells and a low tumor cell density at the first diagnosis.

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6 Conclusion The theory of viability, whose object is the study of dynamic systems controlled subject to constraints, appears quite relevant in the context of applications in the field of biology. Indeed, a biological problem including cancer, can be expressed in the form of a dynamic system that must be controlled by a treatment, and must meet a number of constraints. The essential concept is the viability kernel, which is the set of initial situations from which there is at least one viable evolution. In this paper, we calculated the viability kernel of the controlled dynamic system: ⎧ ⎪ ⎨ T˙ = a(1 − bT )T − μc(t)T − ET E E˙ = σ − dE + ρT g+T − βT E − hc(t)E + i(t) ⎪ ⎩ E(0) = E , and T (0) = T 0 0

(3)

using the limit curves to determine the exact limits of the viability kernel. The first solution denoted as C1 , has a curve which satisfies the inverse system of the dynamic system (3) with the choice of the minimum control, does not meet the set of constraints at any point. However, the curve of the second solution C2 whose control is maximal meets the set of constraints in the point (T2∗ , E2∗ ), hence the definition of the viability kernel corresponding to our dynamic system. This viability kernel allows to determine the initial states from which a combined therapy protocol can be defined to guarantee the remains in the set of the state constraints. This corresponds to providing doses of the treatment: chemotherapy and immunotherapy with which we can determine the chances of remission of the patient in relation to his state of health during the first screening. In the next work, we are interested in finding the combined therapy that minimizes treatment cost while ensuring patient survival. Maximum principle of Pontryagin allows us to identify achievable equilibrium levels according to the initial conditions and model parameters.

References 1. J.-P. Aubin, Viability Theory (Birkhauser Boston, New York, 2009). 1991 Edition 2. G. Deffuant, S. Martin, L. Chapel, Utiliser des “support vector machines” pour apprendre un noyau de viabilité. MajecSTIC 2005: Manifestation des Jeunes Chercheurs francophones dans les domaines des STIC, IRSA - IETR - lSTI, Nov 2005, Rennes, pp. 195–202 3. L.G. de Pillis, W. Gu, K.R. Fister, T. Head, K. Maples, T. Neal, A. Murugan, K. Yoshida, Chemotherapy for tumors: an analysis of the dynamics and a study of quadratic and linear optimal controls. Math. Biosci. 209, 292–315(2006) 4. R. Jaroudi, G. Baravdish, F. Åström, B.T. Johansson, Source Localization of Reaction-Diffusion Models for Brain Tumors (Springer, Cham, 2016), pp. 414–425 5. Kaviar homepage (2012). http://http://www.kaviar.prd.fr/

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6. V.A. Kuznetsov, I.A. Makalkin, M.A. Taylor, A.S. Perelson, Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56, 295–321 (1994) 7. A. Lorz, T. Lorenzi, J. Clairambault, A. Escargueil, B. Perthame, Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull. Math. Biol. 77(1), 1–22 (2015) 8. C. Taramasco, K. Figueroa, J. Demongeot, Estimation of life expectancy of patients diagnosed with the most common cancers in the Valparaiso Region. Ecancermedicalscience 11, 713 (2017)

Optimal Control of an HIV Infection Model with Logistic Growth, CTL Immune Response and Infected Cells in Eclipse Phase Jaouad Danane and Karam Allali

1 Introduction The human immunodeficiency virus (HIV) is a virus that gradually weakens the immune system. It is considered as the main cause for several deadly diseases after the resulting acquired immunodeficiency syndrome (AIDS) is reached. With 36.7 million people living with HIV, 1.8 million people becoming newly infected with HIV and more than 1 million deaths annually, HIV becomes a major global public health issue [1]. In the last decades, many mathematical models describing HIV dynamics were developed [2–11]. The basic viral infection model with CTLs immune response was first studied in Ref. [3]. In 2012 Buonomo et al. proposed a HIV mathematical model in which they decomposed the infected class into two classes that represent the infected cells in latent stage and others in active stage [4]. More recently the model describing HIV viral dynamics with another fifth compartment representing the cytotoxic T-lymphocytes cells (CTLs) immune system and the infected cells in latent stage is formulated and studied in Ref. [10]. The authors study the global stability of the endemic states and illustrate the numerical simulations in order to show the numerical stability for each problem steady state.

J. Danane () · K. Allali Laboratory of Mathematics and Applications, Faculty of Sciences and Technologies, University Hassan II of Casablanca, Mohammedia, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_12

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This paper will be focused on studying an optimal control problem for HIV infection with CTL immune system. For this purpose, we will consider the following nonlinear differential equations: ⎧ T ⎪ ⎪ x˙ = rx(1 − ) − d1 x − k1 (1 − u1 )xv, ⎪ ⎪ ⎪ Tm ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s˙ = k1 (1 − u1 )xv − d2 s − k2 s, y˙ = k2 s − d3 y − pyz, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v˙ = a(1 − u2 )y − d4 v, ⎪ ⎪ ⎪ ⎪ ⎩ z˙ = cyz − bz.

(1.1)

With T = x + y + s. The initial data are x(0) = x0 ≥ 0, y(0) = y0 ≥ 0, s(0) = s0 ≥ 0, v(0) = v0 ≥ 0, z(0) = z0 ≥ 0. In this model, x, s, y, v, and z denote the concentration of uninfected cells, exposed cells, infected cells, free virus, and CTL cells, respectively. Susceptible host cells CD4+ cells grow at a rate r, die at a rate d1 x, and become infected by virus at a rate k1 xv. The exposed cells die at a rate (d2 + k2 )s. The infected cells increases at rate k2 s and decays at rate d3 y, and are killed by the CTL response at a rate pyz. Free virus is produced by infected cells at a rate ay and decays at a rate d4 v. Finally, CTLs expand in response to viral antigen derived from infected cells at a rate cyz and decay in the absence of antigenic stimulation at a rate bz. u1 and u2 measure the efficacy of reverse transcriptase inhibitor and protease inhibitor, respectively. The paper is organized as follows. Section 2 is devoted to the proof of existence, positivity, and boundedness of solutions. The analysis of the model is described in Sect. 3. Then, in Sect. 4, we do an optimization analysis of the viral infection model. Results obtained by numerical simulations are given in Sect. 5 and we conclude in the last section.

2 Positivity and Boundedness First, it is straightforward to prove that the solutions to the system (1.1) are positive for positive initial conditions. We will examine in section, the boundedness of the solutions to the system (1.1) with non-negative initial conditions. For this purpose we introduce the variable X(t) = x(t) + s(t) + y(t) + pc z(t). Using the first three equations of the system (1.1), we have

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 p x+s+y ˙ − d1 x(t) − d2 s(t) − d3 y(t) − bz(t). X = rx 1 − Tm c Since T = x + s + y ≤ Tm and x ≤ Tm then p X˙ ≤ rTm − d1 x(t) − d2 s(t) − d3 y(t) − bz(t) c ≤ rTm − ρX(t). Where ρ = min(d1 , d2 , d3 , b). Then, lim sup X(t) ≤ t→∞

rTm . ρ

This fact, implies that lim sup x(t) ≤

rTm , ρ

lim sup s(t) ≤

rTm , ρ

rTm ρ

and

lim sup z(t) ≤

rTm , ρ

rTm , ρ

z(t) ≤

t→∞

lim sup y(t) ≤ t→∞

t→∞

t→∞

therefore,we have x(t) ≤

rTm , ρ

s(t) ≤

rTm , ρ

y(t) ≤

and

From the fourth equation of the system (1.1) and y(t) ≤ v˙ ≤

arTm (1 − u2 ) − d4 v, ρ

Then, lim sup v(t) ≤ t→∞

arTm (1 − u2 ) , d4 ρ

Thus, v(t) ≤

arTm (1 − u2 ) , d4 ρ

rTm ρ ,

rTm . ρ

we have

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3 The Optimal Control Problem To state the optimization problem, we first consider that u1 and u2 vary with time. The problem (1.1) becomes  ⎧ x+s+y ⎪ ⎪ x˙ = rx 1 − − d1 x − k1 (1 − u1 (t))xv, ⎪ ⎪ ⎪ Tm ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ s˙ = k1 (1 − u1 (t))xv − d2 s − k2 s, (3.1)

y˙ = k2 s − d3 y − pyz, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ v˙ = a(1 − u2 (t))y − d4 v, ⎪ ⎪ ⎪ ⎩ z˙ = cyz − bz.

With u1 represents the efficiency of drug therapy in blocking new infection, so that infection rate in the presence of drug is (1 − u1 ); while u2 stands for the efficiency of drug therapy in inhibiting viral production, such that the virus production rate under therapy is (1 − u2 ). The optimization problem that we consider is to maximize the following objective functional:

J (u1 , u2 ) =

tf



x(t) + z(t) −

0

A

1 2 u1 (t) +

2

A2 2  u (t) dt, 2 2

(3.2)

where tf is the period of treatment and the positive constants A1 and A2 are based on the benefit-cost of the treatment u1 and u2 , respectively. The two control functions, u1 (t) and u2 (t) are assumed to be bounded and Lebesgue integrable. Our target is to maximize the objective functional defined in Eq. (3.2) by increasing the number of the uninfected cells, maximizing the CTLs immune responses, decreasing the viral load, and minimizing the cost of treatment. In other words, we are seeking an optimal control pair (u∗1 , u∗2 ) such that J (u∗1 , u∗2 ) = max{J (u1 , u2 ) : (u1 , u2 ) ∈ U },

(3.3)

where U is the control set defined by U = {(u1 (t), u2 (t)) : ui (t) measurable, 0 ≤ ui (t) ≤ 1, t ∈ [0, tf ], i = 1, 2}. (3.4)

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4 Existence of an Optimal Control Pair The existence of the optimal control pair can be directly obtained using the results in Refs. [12, 13]. More precisely, we have the following theorem: Theorem 4.1 There exists an optimal control pair (u∗1 , u∗2 ) ∈ U such that J (u∗1 , u∗2 ) =

max

J (u1 , u2 ).

(u1 ,u2 )∈U

(4.1)

Proof To use the existence result in Ref. [12], we must check the following properties: (P1 ) The set of controls and corresponding state variables is nonempty. (P2 ) The control U set is convex and closed. (P3 ) The right hand side of the state system is bounded by a linear function in the state and control variables. (P4 ) The integrand of the objective functional is concave on U . (P5 ) There exists constants c1 , c2 > 0, and α > 1 such that the integrand L(H, Z, u1 , u2 ) of the objective functional satisfies α 2

L(x, z, u1 , u2 ) ≤ c2 − c1 (| u1 |2 + | u2 |2 ) .

(4.2)

Where L(x, z, u1 , u2 ) = x(t) + z(t) −

A

1 2 u1 (t) +

2

A2 2  u (t) . 2 2

(4.3)

In order to verify these conditions, we first use a result by Lukes (see Theorem 9.2.1 in Ref. [13]); the boundedness of the system of state equations under the two controls (3.1) ensures the existence of a solution. We can therefore conclude that the set of controls and the corresponding state variables is nonempty, which gives condition (P1 ). The control set is convex and closed by definition, which gives condition (P2 ). Since our state system is bilinear in u1 , u2 , the right hand side of system (3.1) satisfies condition (P3 ), using the boundedness of the solutions. For the condition (P4 ), the Hessian matrix of the integrand L is given by  HL =

−A1 0 0 −A2

,

then its determinant is given as follows det (HL ) = A1 A2 ≥ 0,

∀(u1 , u2 ) ∈ U,

which implies the concavity of the integrand on U .

(4.4)

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Also, we have the last needed condition L(x, z, u1 , u2 ) ≤ c2 − c1 (| u1 |2 + | u2 |2 ),

(4.5)

  where c2 depends on the upper bound on x, z, and c1 = min A21 , A22 > 0. We conclude that there exists an optimal control pair (u∗1 , u∗2 ) ∈ U such that J (u∗1 , u∗2 ) =

max

(u1 ,u2 )∈U

J (u1 , u2 ).  

5 Optimality System Pontryagin’s Minimum Principle given in Ref. [14] provides necessary conditions for an optimal control problem. This principle converts (3.2) and (3.1) into a problem of maximizing a Hamiltonian, H , pointwisely with respect to u1 and u2 :  A1 2 A2 2 u1 + u2 − x − z + λi fi 2 2 5

H (t, x, s, y, v, z, , u1 , u2 , λ) =

(5.1)

i=0

with  ⎧ x(t) + s(t) + y(t) ⎪ ⎪ f − d1 x(t) − k(1 − u1 (t))x(t)v(t), = rx(t) 1 − 1 ⎪ ⎪ Tm ⎪ ⎪ ⎨ f = k(1 − u (t))x(t)v(t) − (d + k )s(t), 2 1 2 2 ⎪ f3 = k2 s(t) − d3 y(t) − py(t)z(t), ⎪ ⎪ ⎪ ⎪ f = (1 − u2 (t))ay(t) − d4 V (t), ⎪ ⎩ 4 f5 = cy(t)z(t) − bz(t). (5.2) By applying Pontryagin’s minimum principle [14], we obtain the following theorem Theorem 5.1 For any optimal controls u∗1 , u∗2 , and solutions x ∗ , v ∗ , y ∗ , v ∗ , and z∗ of the corresponding state system (3.1), there exists adjoint variables, λ1 , λ2 , λ3 , λ4 , and λ5 satisfying the equations ⎧     ∗  ∗ ∗ λ1 (t) = 1 + λ1 (t) r 2x (t)+yTm(t)+s (t) − 1 + d1 + k1 (1 − u∗1 (t))v ∗ ⎪ ⎪ ⎪ ⎪ ⎪ −λ2 (t)k1 (1 − u∗1 (t))v ∗ ⎪ ⎪ ⎪ rx ∗ (t) ⎪ ⎪ ⎨ λ2 (t) = λ1 (t) Tm + λ2 (t)(d2 + k2 ) − λ3 (t)k2 , ∗

⎪ λ3 (t) = λ1 (t) rxTm(t) + λ3 (t)d3 − λ4 (t)a(1 − u∗2 (t)), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ∗ ∗ ∗ ⎪ ⎪ ⎪ λ4 (t) = λ1 (t)k1 (1 − u1 (t))x (t) − λ2 (t)k1 (1 − u1 (t))x (t) + λ4 (t)d4 , ⎩  ∗ ∗ λ5 (t) = 1 + λ2 (t)py (t) + λ5 (t) b − cy (t)

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with the transversality conditions λi (tf ) = 0, i = 1, . . . , 5.

(5.3)

Moreover, the optimal control is given by   

k u∗1 = min 1, max 0, λ2 (t)x ∗ (t)v ∗ (t) − λ1 (t)x ∗ (t)v ∗ (t) A1   1 ∗ ∗ λ3 (t)ay (t) . u2 = min 1, max 0, A2

(5.4)

6 Numerical Simulations In order to solve our optimization system (3.1), we will use a numerical scheme based on forward and backward finite difference approximation. Hence, we will have the following numerical algorithm. The parameters of our numerical simulations are inspired from Refs. [10, 15]; i.e., r = 0.1, Tm = 10000, d1 = 0.0139, k1 = 0.04, d2 = 0.0495, k2 = 1.1, d3 = 0.5776, a = 100, d4 = 0.6, p = 0.0024, c = 0.15, b = 0.5. We chose as in [16] the two last parameters A1 = 5000 and A2 = 5000. The role of the positive constants A1 and A2 is to balance the terms size in the equations. Figure 1 shows that with control the amount of the uninfected cells population is higher than those observed for without control case. 950 With control Without control

Uninfected cells

900

850

800

750

700 0

10

20

30

40

Fig. 1 The uninfected cells as function of time

50 Days

60

70

80

90

100

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Step 1: x(0) = x0 , y(0) = y0 , s(0) = s0 , v(0) = v0 , z(0) = z0 , u1 (0) = 0, u2 (0) = 0, λ1 (n) = 0, λ2 (n) = 0, λ3 (n) = 0, λ4 (n) = 0, λ5 (n) = 0 Step 2: for i = 0, . . . , n − 1, do:

xi + si + yi i xi+1 = xi + h rxi 1 − − d1 xi − k1 (1 − u1 )vi xi , Tm i si+1 = si + h[k1 (1 − u1 )vi xi − k2 si − d2 si ], yi+1 = yi + h[k2 si − d3 yi − pyi zi ], vi+1 = vi + h[(1 − u22 )ayi − d4 vi ], zi+1 = zi + h[cyi zi− bzi ],   2xi+1 +yi+1 +si+1 n−i r − h 1 + λ − 1 + d1 λ1n−i−1 = λn−i 1 1 Tm  i +(1 − ui1 )k1 vi+1 + λn−i 2 (u1 − 1)k1 vi+1 ,   n−i n−i λ2n−i−1 = λn−i , − h λ1 (t) rxTi+1 + λ (d + k ) − λ k 2 2 2 2 2 3 m  rxi+1 n−i−1 n−i n−i λ3 = λ3 − h λ1 (t) Tm + λ3 (d3 + pzi+1 )  i ) − λn−i cz , −λn−i a(1 − u i+1 2 4 5 i λ4n−i−1 = λn−i − h[λn−i 4 1 (1 − u1 )k1 xi+1 n−i i +λn−i 2 (u1 − 1)k1 xi+1 + d4 λ4 ], n−i−1 n−i n−i λ5 = λ5 − h[1 + λ5 (b − cyi+1 ) + λn−i 3 pzi+1 ], i+1 R1 = (k1 /A1 )(λ2n−i−1 vi+1 xi+1 − λ1n−i−1 vi+1 xi+1 ) R2i+1 = (1/A2 )λ4n−i−1 ayi+1 , ui+1 = min(1, max(R1i+1 , 0)), 1 ui+1 = min(1, max(R2i+1 , 0)), 2 end for Step 3: for i = 1, . . . , n, write x ∗ (ti ) = xi , s ∗ (ti ) = si , y ∗ (ti ) = yi , v ∗ (ti ) = vi , z∗ (ti ) = zi , u∗1 (ti ) = ui1 , u∗2 (ti ) = ui2 . end for

The numerical algorithm From Fig. 2, we observe that the latently infected cells under control converges towards 1.23, while without control it converges towards 12.15, which means that administrating the good therapy amounts can help the patient by significant reduction the latently infected cells number. We notice that with control we observe a dumping oscillating regime and a significant reduction of the latently infected cells.

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20 With control Without control

Latently infected cells

18 16 14 12 10 8 6 4 2 0 0

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Days Fig. 2 The latently infected cells as function of time 50 45

Infected cells

40 35

With control Without control

30 25 20 15 10 5 0 0

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50

60

70

80

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Days Fig. 3 The infected cells as function of time

Figure 3 shows that with control the number of infected cells is significantly reduced after the first days of therapy. However, without control this number remains much higher. The role of therapy control is also observed in Fig. 4. We clearly see that the curve representing virus converges towards 198, 35 with control and converges towards 7.94 × 103 without control. This indicates the impact of the administrated therapy in controlling viral replication.

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With control Without control

7000 6000

HIV

5000 4000 3000 2000 1000 0

0

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With control Without control

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2000 1500 1000 500 0

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50 Days

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Fig. 5 The CTL response as function of time

The CTL cells are clearly affected by the control. This is shown in Fig. 5; indeed the curve of CTL cells converges to 625.17 with control, while without any control it converges towards 1376.28 which reveals the importance of adding the CTL component to HIV viral dynamics. The two optimal controls u1 and u2 corresponding to blocking new infections and inhibiting viral production are represented in Fig. 6. The two curves present the drug administration schedule during the period of treatment. When the first immune boosting drug is administered at full scale, the second drug is at its lowest level and vice versa. In this case the new infection is totally blocked.

Optimal Control of an HIV Infection Model with Logistic Growth, CTL. . . 0.02

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0.015 0.01 0.005 0

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Fig. 6 The optimal control u1 (left) and the optimal control u2 (right) versus time

7 Conclusion In this paper, we have studied the model of the human immunodeficiency virus (HIV) dynamics in the presence of the immune response which is represented by the cytotoxic T lymphocytes (CTL) cells. The considered model includes five differential equations describing the interaction between the uninfected cells, infected cells, HIV virus, and CTL immune response. We have proved the existence and uniqueness of the optimal controls using Pontryagin’s maximum principle. The problem was solved numerically using backward and forward finite difference scheme. It was shown that with the two optimal treatments, the number of the healthy CD4+ T cells increases remarkably while the number of infected CD4+ T cells decreases significantly. In addition, it was also observed that, with the control strategy, the viral load decreases considerably compared with the model without control which can improve the patient’s life quality.

References 1. World Health Organization HIV/AIDS Key facts, (November 2017). http://www.who.int/ mediacentre/factsheets/fs360/en/index.html 2. M.A. Nowak, S. Bonhoeffer, G.M. Shaw, R.M. May, Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations. J. Theor. Biol. 184(2), 203–217 (1997) 3. M.A. Nowak, C.R.M. Bangham, Population dynamics of immune responses to persistent viruses. Science 272, 74–79 (1996) 4. B. Buonomo, C. Vargas-De-León, Global stability for an HIV-1 infection model including an eclipse stage of infected cells. J. Math. Anal. Appl. 385(2), 709–720 (2012) 5. H.L. Smith, P. De Leenheer, Virus dynamics: a global analysis. SIAM J. Appl. Math. 63, 1313– 1327 (2003) 6. E.S. Daar, T. Moudgil, R.D. Meyer, D.D. Ho, Transient highlevels of viremia in patients with primary human immunodeficiency virus type 1. New Engl. J. Med. 324, 961–964 (1991) 7. J.O. Kahn, B.D. Walker, Acute human immunodeficiency virus type 1 infection. New Engl. J. Med. 339, 33–39 (1998) 8. G.R. Kaufmann, P. Cunningham, A.D. Kelleher, J. Zauders, A. Carr, J. Vizzard, M. Law, D.A. Cooper, Patterns of viral dynamics during primary human immunodeficiency virus type 1 infection, The Sydney Primary HIV Infection Study Group. J. Infect. Dis. 178, 1812–1815 (1998)

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9. T. Schacker, A. Collier, J. Hughes, T. Shea, L. Corey, Clinical and epidemiologic features of primary HIV infection. Ann. Int. Med. 125, 257–264 (1996) 10. K. Allali, J. Danane, Y. Kuang, Global analysis for an HIV infection model with CTL immune response and infected cells in eclipse phase. Appl. Sci. 7(8), 861 (2017) 11. Q. Sun, L. Min, Y. Kuang, Global stability of infection-free state and endemic infection state of a modified human immunodeficiency virus infection model. IET Syst. Biol. 9, 95–103 (2015) 12. W.H. Fleming, R.W. Rishel, Deterministic and Stochastic Optimal Control (Springer, Berlin, 1975) 13. D.L. Lukes, Differential Equations. Mathematics in Science and Engineering, vol. 162 (Academic Press, London, 1982) 14. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The Mathematical Theory of Optimal Processes (Wiley, New York, 1962) 15. X. Lai, X. Zou, Modeling cell-to-cell spread of HIV-1 with logistic target cell growth. J. Math. Anal. Appl. 426, 563–584 (2015) 16. J. Danane, A. Meskaf, K. Allali, Optimal control of a delayed hepatitis B viral infection model with HBV DNA-containing capsids and CTL immune response. Optim. Control Appl. Methods 39(3), 1262–1272 (2017). https://doi.org/10.1002/oca.2407

Khinchin–Shannon Generalized Inequalities for “Non-additive” Entropy Measures Rubem P. Mondaini and Simão C. de Albuquerque Neto

1 Introduction In this note, we propose to extend to “non-additive” entropy measures the inequalities obtained by Shannon and Khinchin [1] which give useful information about the synergy of the set of statistical samples to be used on statistical analysis of the occurrence of amino acids in protein domain families (PDF). These samples are given by arrays of m rows and n columns, and we can define the simple and joint probabilities of occurrence by: pj (a) = pj k (a, b) =

nj (a) , m

j = 1, 2, . . . , n j = 1, 2, . . . , n − 1 k = 2, 3, . . . , n

nj k (a, b) , m

(1)

(2)

where nj (a) is the number of occurrences of the a-amino acid in the j -th column and nj k (a, b) is the number of occurrences of the pair of (a, b) amino acids in j -th, k-th columns, respectively. With a, b = A, C, D, E, F, G, H, I, K, L, M, N, P, Q, R, S, T, V, W, Y. We then have   nj (a) = m "⇒ pj (a) = 1 ∀j (3) a

a

R. P. Mondaini () · S. C. de Albuquerque Neto Federal University of Rio de Janeiro, Centre of Technology, COPPE, Rio de Janeiro, RJ, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_13

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pj k (a, b) = 1 ∀j, k

(4)

pk (b)pj k (a|b) = pj k (a, b) = pkj (b, a) = pj (a)pkj (b|a)

(5)

a

nj k (a, b) = m "⇒

a

b

b

The Bayes’ law can be then written as:

where pj k (a|b) is the conditional probability of occurrence of the a-amino acid in the j -th column if the b-amino acid already occurs in column k. Analogously for pkj (b|a). We then have 

pj k (a|b) =



a

pkj (b|a) = 1

(6)

b

and from this last equation and Eq. (5), we get 

pj k (a, b) = pk (b) ,



pj k (a, b) = pj (a)

(7)

1/s pˆ j (a) pj (a) = )  1/s pˆ j (b)

(8)

a

b

We also introduce the escort probabilities s pj (a) pˆ j (a) = )  s , pj (b) 

b

b

 s pj k (a, b) pˆ j k (a, b) = ) )  s , pj k (c, d) c



d

 1/s pˆ j k (a, b) pj k (a, b) = ) )  1/s pˆ j k (c, d) c

(9)

d

where s is a non-dimensional real parameter. From Eqs. (3), (4), we also have 

pˆ j (a) = 1 ∀j

(10)

a

 a

pˆ j k (a, b) = 1

∀j, k

(11)

b

2 The Sharma–Mittal Set of Entropy Measures In this section we introduce the set of “non-additive” Entropy Measures which will be used to characterize the properties of generalized inequalities of Khinchin– Shannon and their application in Information Theory.

Khinchin–Shannon Generalized Inequalities for “Non-additive” Entropy Measures

179

The Sharma–Mittal entropy measures for probabilities pj (a) and pj k (a, b) are given, respectively, by: Sjrs

Sj k rs

1 =− 1−r

1 =− 1−r

&

 s 1−r 1−s pj (a) 1−

' (12)

a

& 1−

 a

pj k (a, b)

s 1−r 1−s

' (13)

b

where r, s are non-dimensional real parameters. We also introduce the Conditional Entropy measure, associated with the conditional probability of occurrence pj k (a|b) Sj |krs

1 =− 1−r

& 1−

 a



pˆ k (b) pj k (a|b)

s 1−r 1−s

' (14)

b

This corresponds to a measure of uncertainty on the knowledge of the distribution of amino acids in column j if the distribution of amino acids in column k is known previously. The restriction on the values of these parameters is given by the requirements of concavity  s−r   s 1−s s−2 s(s − r) ∂ 2 Sjrs p p < 0 "⇒ s (a) (a) p ˆ (a) − 1 c˜g, ˜

g(b − μ) ≥ c(r ˜ − n),

c(r − n) ≥ g(b ˜ − μ)

(4)

cg < c˜g, ˜

g(b − μ) ≤ c(r ˜ − n),

c(r − n) ≤ g(b ˜ − μ).

(5)

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For the feasibility of E4 we need bef ≥ ef μ + cm,

ab + ef m ≥ aμ.

(6)

3.3 Stability The stability conditions are found as follows: for the ecosystem collapse E0 are the opposite ones of both (3) namely: b < μ;

r < n.

(7)

For the infected-aphids-only point E1 bef < ef μ + cm,

cr + gμ ˜ < cn + gb, ˜

(8)

the former of which indicating a transcritical bifurcation with E4 , compare with the first stability condition (6), while the second one shows a possible similar bifurcation with E3 , see the last condition in (4). Similarly, for E2 we obtain er < en + gm,

bg + cn ˜ < μg + cr, ˜

(9)

the second one being the opposite of the second condition in (4), for which a transcritical bifurcation exists for these two equilibria. For E3 the stability condition arises from the explicit eigenvalue, eA3 + ef I3 < f,

cg > c˜g, ˜

(10)

as one of the Routh–Hurwitz conditions on the remaining minor of the Jacobian J#3 is always satisfied, namely −tr(J#3 ) = gA3 + cI3 > 0 while for the other one we need det(J#3 ) = (cg − c˜g)A ˜ 3 I3 > 0, giving the second condition (10). The latter however indicates that only the feasibility conditions (4) provide an achievable point, while (5) lead to an unstable equilibrium. For the point E4 we find an explicit eigenvalue giving the only needed stability condition ˜ 4, r < n + (e + β)W4 + gI

(11)

because the Routh–Hurwitz conditions on the remaining minor J#4 are unconditionally satisfied, namely −tr(J#4 ) = aW4 +cI4 > 0 and det(J#4 ) = (ac+e2 f 2 )I4 W4 > 0.

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4 Interpretation Table 1 contains the possible transitions between the ecosystem states, that can be analytically assessed. Note that it is possible to reach the state in which uninfected and infected aphids coexist, in the absence of wasps, equilibrium E3 by a successful invasion of the pristine environment E0 in two possible ways. At first, if the uninfected aphids establish themselves in the ecosystem, at E2 and then also the infected ones follow; alternatively it could be that the invasion is through infected aphids, equilibrium E1 followed by the appearance also of uninfected aphids. In both cases the final outcome is equilibrium E3 , where both kinds of aphids coexist in the absence of wasps. The latter can only appear in the system if the infected aphids are already present, i.e., from equilibrium E1 the system state moves to E4 . Note that one transition is missing here, namely from E2 to E4 , i.e., when the wasps invade the uninfected-aphids-only environment. This is indeed forbidden by our assumption on the bacteria transmission for which the wasps are the carriers that transmit it to uninfected aphids through the contact between their sting and the aphids cuticle, while deposing the egg. The fact that this assumption is verified in the findings shows the consistency of the model that is proposed. From Table 2 instead it is evident that the aphids in some form, whether affected by the bacteria or free from them, will persist in the ecosystem as long as the wasps are absent, because they appear in all the equilibria E1 –E3 . Even in case of the wasps thriving, we can find the infected aphids to survive, at point E4 . Clearly, in case the coexistence equilibrium is achievable, also in such case the aphids cannot be eradicated. The only point at which the aphids disappear is the origin, which entails the disappearance of the wasps as well. This could be a sacrifice to be accepted, if the real goal is the aphids eradication. However, to achieve it, the necessary conditions involve the birth and death rates of the aphids, and are therefore intrinsic to this species. Namely, they are also quite unachievable in practice, as they simply state that the death rates should exceed the birth rates. These are natural conditions for Table 1 Transcritical bifurcations for the “laboratory” ecosystem (1)–(2) Transition

Threshold condition b >1 μ r >1 n bef >1 ef μ + cm

Interpretation

E1 → E3

cr + gμ ˜ >1 cn + gb ˜

Uninfected and infected aphids established

E2 → E3

bg + cn ˜ >1 μg + cr ˜

Uninfected and infected aphids established

E0 → E1 E0 → E2 E1 → E4

Infected aphids established Uninfected aphids established Wasps and infected aphids established

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Table 2 Summary of “laboratory” ecosystem’s (1)–(2) equilibria with at least one vanishing population Equilibria E0 = (0, 0, 0) E1 = (0, 0, I1 )

Feasibility – b≥μ

E2 = (0, A2 , 0)

r≥n

E3 = (0, A3 , I3 )

cg > c˜g, ˜ g(b − μ) ≥ c(r ˜ − n), c(r − n) ≥ g(b ˜ − μ) cg < c˜g, ˜ g(b − μ) ≤ c(r ˜ − n), c(r − n) ≤ g(b ˜ − μ) bef ≥ ef μ + cm, ab + ef m ≥ aμ

E4 = (W4 , 0, I4 )

Stability b < μ, r < n bef < ef μ + cm cr + gμ ˜ < cn + gb ˜ er < en + gm, bg + cμ ˜ < μg + cr ˜ cg > c˜g, ˜ eA3 + ef I3 < f Unstable r < n + (e + β)W4 + gI ˜ 4

the vanishing of a population, so it is kind of a truism: the wasps play absolutely no role in their possible satisfaction. We now address the issue of the infestation control in the case of the coexistence equilibrium. Recalling the discussion in Sect. 6.1, we consider only the situation when the conic section is an ellipse. The actual position of its intersection with the straight line L depends on the slope of the latter, mL = e(fg − g)[e(e ˜ + β) + ag]−1 . In such situation, an empirical way of trying to have it close to the W axis, so that the infected aphids are close to zero, would be to require I2 ≈ 0 and mL < 0, so # ) with I# < I2 . In this way the I that L must intersect the ellipse at a point (I#, W # would come from (17), population would be controlled. The wasps population W −1 # giving W ≈ ma so that for the value of the uninfected aphids we can also use (16) # ≈ 0 as well. Clearly in this discussion we must understand that these providing A considerations hold for parameters that are intrinsic to the ecosystem and that can hardly be influenced by human interventions. Indeed, negative slope for L means fg < g. ˜ In turn for having I2 > 0 this implies er > en + gm, but close to equality if we impose I2 ≈ 0. Nobody can indeed at the moment predict whether these inequalities, as well as δ > 0, are satisfied in practice.

5 Simulations The data gathered in the laboratory are reported in Table 3. To obtain the daily reproduction rate for aphids, we divide the values of the first row in Table 3 by 14 (days), corresponding to 2 weeks. Furthermore, since the modes does not account for immature aphids, we consider the reproduction lifetime by subtracting the values of the second row from those of the third one. These values over the whole lifetime of the aphids, third row, represent the fractions of time in which the two classes of aphids reproduce. Since in the model reproduction

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Table 3 Laboratory data Data Number of offsprings in 14 days Age of first reproduction (days) Longevity (days) Proportion of parasitized aphids by wasps Mass (weight)

Healthy aphid 56.11 8.13 30.34 0.58 2.00

Infected aphid 53.45 8.53 21.21 0.29 1.88

is a continuous process, we must scale the reproduction rates via these fractions, to obtain the correct number of offsprings over the whole lifetime of a parent individual. Thus r=

56.11 30.34 − 8.13 × = 2.9339, 14 30.34

b=

53.45 21.21 − 8.53 × = 2.2824. 14 21.21 (12)

The mortality rates of these two types of aphids are the reciprocals of their respective longevities. Similarly, for the wasps the average lifespan is about 2 weeks, namely n=

1 = 0.0330, 30.34

μ=

1 = 0.0471, 21.21

m=

1 = 0.0714. 14

(13)

Theoretically, in the absence of bacteria, both classes of aphids would be parasitized in the same way, the fraction would be 0.58. But since only a smaller fraction are parasitized among the infected, the reduction should be ascribed to the action of the bacteria. The rate at which new wasps emerge from uninfected aphids is eAW . From a single wasp then we would find eA newborns, meaning that the fraction of wasp eggs that mature and whose larvae kill the uninfected aphids is e, assuming that all these wasp eggs in the parasitized aphids will mature to produce offsprings, and therefore given by e = 0.58. In case of infected aphids the wasps emerging rate is scaled down by a fraction f , namely ef AW . Hence the “killing rate" of the bacteria on the larvae, the fraction f is calculated as the difference between the proportion of uninfected parasitized aphids and the infected parasitized aphids populations, divided by the proportion of uninfected parasitized aphids. Therefore it should be given by f =

0.58 − 0.29 = 0.5, 0.58

e = 0.58.

(14)

Reference values used for the free parameters in the simulations a = 0.5,

β = 0.8,

g = 0.5,

g˜ = 0.5,

c = 0.5,

c˜ = 0.5.

(15)

A sample of the simulation in which we plot the population values at equilibrium as the parameters a and β vary, is shown in Fig. 1. It is seen that in a narrow

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Fig. 1 The populations W , A, and I , shown as functions of the parameters a and β. The remaining parameter values are given in (12), (13), (14), and (15). Initial conditions: W = 0.07, A = 2.80, I = 0.0. Coexistence of uninfected and infected aphids is possible in a narrow range

range, for 0 < β < 0.1 and 0.8 < a < 1 the three populations coexist at a nonvanishing level. Low level of disease transmission coupled with a sizeable value of the intraspecific competition among the wasps help in keeping a proportion of the aphids uninfected. Specifically for β ≈ 0.07 we find I ≈ 2.5 while A ≈ 1, with W ≈ 1.8; these values change a bit with a of course. But they suggest that it is possible to find roughly 70% of resistant aphids and 30% of uninfected, with a sizeable number of wasps as well. Conversely, for β ≈ 0.02 we have I ≈ 1, A ≈ 2 and W ≈ 2.1, which gives the reverse situation, 70% of uninfected aphids and 30% of infected. Figure 2 shows the same pair of parameters in the larger domain 0 < β < 1 and 0 < a < 1. Outside the region considered above, it is clearly seen that the uninfected aphids vanish altogether. In Fig. 3 wasps decrease while infected aphids increase as a grows. These populations seem essentially not to be influenced by the parameter g. Uninfected aphids vanish, note the very small vertical scale in the middle frame. A similar behavior is shown in the c˜ − a parameter space. Similarly as in Fig. 3, in Fig. 4 wasps decrease while infected aphids increase as a grows and uninfected aphids vanish. Figure 5 is a blow up of the region with the peak in uninfected aphids in Fig. 4.

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Fig. 2 The populations W , A, and I , shown as functions of the parameters a and β. The remaining parameter values are given in (12), (13), (14), and (15). Initial conditions: W = 0.07, A = 2.80, I = 0.0. Here it is shown that for other larger values of the disease transmission the uninfected aphids are wiped out. The corner with the peak in the uninfected aphids corresponds to the picture in Fig. 1

In the a − c parameter space, Fig. 6, wasps are shown to be damaged by an increase in either one of the parameters. Uninfected aphids are wiped out, while the infected ones suffer from low values of a, especially for small values of c. Both wasps and infected aphids decrease with increasing c, while β affects these populations and allows recovery of the uninfected aphids only if it is low and coupled with large c, Fig. 7. In the β − c˜ parameter space, both W and I attain a constant value, while the uninfected aphids disappear. The same behavior is observed in the c˜ − g parameter space as well as in the c˜ − g˜ parameter space. Figure 8 is a blow up of the region with the peak in uninfected aphids in Fig. 7. Both wasps and infected aphids decrease with increasing c and are unaffected by changes in g, uninfected aphids are wiped out, Fig. 9. The same behavior occurs in the c − g˜ parameter space.

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Fig. 3 The populations W , A and I , shown as functions of the parameters a and g. The remaining parameter values are given in (12), (13), (14), and (15). Initial conditions: W = 0.07, A = 2.80, I = 0.0. Wasps decrease while infected aphids increase as a grows. These populations seem essentially not to be influenced by the parameter g. Healthy aphids vanish, note the very small vertical scale in the middle frame

6 A Partial Characterization of Coexistence We address now the issue of finding the conditions for which all the populations in the system (1) with the assumptions (2) thrive.

6.1 Feasibility We could try to discuss the coexistence equilibrium in two ways. Only some partial results will be obtained, though. The first approach is to solve the first two equilibrium equations for A and W in - = e(e + β) + ag, this provides terms of I . Letting  1 [(m − ef I∗ )(e + β) + a(r − n − gI ˜ ∗ )],  1 ˜ ∗ ) − g(m − ef I∗ )], W∗ = [e(r − n − gI  A∗ =

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Fig. 4 The populations W , A, and I , shown as functions of the parameters a and g. ˜ The remaining parameter values are given in (12), (13), (14), and (15). The first two populations seem essentially not to be influenced by the parameter g, except for low values of g˜ and large values of a. Initial conditions: W = 0.07, A = 2.80, I = 0.0

with feasibility conditions I∗ ≤

m(e + β) + a(r − n) , a g˜ + ef (e + β)

e(r − n) − gm ≥ e(g˜ − fg)I∗ .

Substituting into the third equilibrium equation we obtain the quadratic but where all the coefficients are of uncertain sign:

)2

k k=0 Hk I∗ ,

1 2 eβ [e f g˜ − e2 f 2 g + cef ˜ (e + β) + a c˜g] ˜ − 2 [a g˜ + ef (e + β)][f g − g] ˜ − c,   1 H1 = b − μ − [ef (e(r − n) − gm) + c(m(e ˜ + β) + a(r − n))]  β ˜ − (e(r − n) − gm)(a g˜ + ef (e + β))], + 2 [(a(r − n) + m(e + β))e(f g − g)  β H0 = 2 [m(e + β) + a(r − n)][e(r − n) − gm].  H2 =

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Fig. 5 The populations W , A, and I , shown as functions of the parameters a and β. The remaining parameter values are given in (12), (13), (14), and (15). Initial conditions: W = 0.07, A = 2.80, I = 0.0

Even imposing H0 > 0, i.e., er > en + gm, would not guarantee a nonnegative root, as in addition H2 < 0 should be required, if one does not want to discuss all the possible situations that can arise. In terms of the original model parameters, even these relatively simple conditions would provide rather complicated and unusable inequalities. The second approach would solve for A the first equilibrium equation, A=

1 [m − ef I + aW ], e

(16)

so that upon substitution into the remaining ones, the second provides a straight line in the I W phase plane, L:

W =

1 [e(r − n) − gm + e(fg − g)I ˜ ] = mL I + qL e(e + β) + ag

which has a zero at I2 =

e(r − n) − gm e(g˜ − f g)

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Fig. 6 The populations W , A, and I , shown as functions of the parameters a and c. The remaining parameter values are given in (12), (13), (14), and (15). Here wasps are damaged by an increase in either one of the parameters. Healthy aphids are wiped out, while the infected ones suffer from low values of a, especially if c is small. Initial conditions: W = 0.07, A = 2.80, I = 0.0

while the third one gives a conic section (I, W ) = 0 with  βa 2 a c˜ + fβ I W + W (I, W ) = (cf ˜ − c)I 2 − ef + e e  mβ mc˜ I+ W. + b−μ− e e Its intercepts with the axes are the origin, (0, W1 ) and (I 1 , 0) with W1 = −

m < 0, a

I1 =

e(b − μ) − cm ˜ ∈ R. e(c − cf ˜ )

The invariant characterizing the nature of (17) is δ = (cf ˜ − c)

 2 1 a c˜ aβ − ef + + fβ ∈ R. e 4 e

(17)

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Fig. 7 The populations W , A, and I , shown as functions of the parameters β and c. The remaining parameter values are given in (12), (13), (14), and (15). Both wasps and infected aphids decrease with increasing c, β affects these populations and allows recovery of the uninfected aphids if it is low and coupled with large c. Initial conditions: W = 0.07, A = 2.80, I = 0.0

 is an ellipse whenever δ > 0. In such case, if I 1 < 0, no feasible coexistence equilibrium can exist. Indeed in this situation no part of the ellipse lies in the first quadrant. Conversely an intersection between  and L is guaranteed if we require 0 < I2 < I 1 . However, recall that in any case the possibility that δ < 0 exists, and in such case to study the intersections of L with the resulting hyperbola is much more complicated and will not be pursued any further.

6.2 Stability In this case, the diagonal entries of the Jacobian simplify as follows: J11 = −aW∗ ,

J22 = −gA∗ ,

J33 = −cI∗ − β

A∗ W∗ . I∗

From this, clearly the Routh–Hurwitz condition on the trace is immediately satisfied, namely −tr(J (E∗ )) > 0. The remaining conditions are however more involved. From the determinant condition, upon simplification, we find:

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Fig. 8 The populations W , A and I , shown as functions of the parameters β and c. The remaining parameter values are given in (12), (13), (14) and (15). Initial conditions: W = 0.07, A = 2.80, I = 0.0

[agβ + +e(e + β)β]

A∗ W∗ + (e + β)efβW∗ I∗

(18)

+[e2 f 2 g + acg + c(e + β)e]I∗ > ef gβA∗ , while the remaining condition becomes: A∗ W∗2 [aefβ + e(e + β)efβ] + efgβA2∗ W∗ + cefβA∗ I∗ W∗ +efβ

2 2 2 A∗ W∗

I∗

(19)

> A∗ W∗2 [ae(e + β) + a 2 g + 2acβ + e2 f 2 β]

+2acgA∗ I∗ W∗ + A2∗ W∗ [ag 2 + eg(e + β) + 2cgβ + efgβ] +I∗ W∗2 [ae2 f 2 + a 2 c] + I∗2 W∗ [ce2 f 2 + ac2 ] + cg 2 A2∗ I∗ + cg 2 A∗ I∗2 +2agβ

A2 W 3 A3 W 2 A2∗ W∗2 A∗ W∗3 A3 W∗ + a2β + g2β ∗ + aβ 2 ∗ 2 ∗ + gβ 2 ∗ 2 ∗ . I∗ I∗ I∗ I∗ I∗

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Fig. 9 The populations W , A and I , shown as functions of the parameters g and c. The remaining parameter values are given in (12), (13), (14) and (15). Both wasps and infected aphids decrease with increasing c and are unaffected by changes in g. Healthy aphids are wiped out. Initial conditions: W = 0.07, A = 2.80, I = 0.0

These appear of rather difficult use in practice, even knowing the values of the equilibrium populations. In any case, the simulations provide some insight in the cases for which coexistence does appear feasible and stable.

6.3 Using Lab Data A∗ =

1 [(0.0714 − 0.29I∗ )(0.58 + β) + a(2.9009 − gI ˜ ∗ )], 

W∗ =

1 [1.6825 − 0.58gI ˜ ∗ − g(0.0714 − 0.29I∗ )], 

- = e(e + β) + ag = 0.3364 + 0.58β + ag. 

I∗ ≤

0.0414 + 0.0714β + 2.9009a , a g˜ + 0.1682 + 0.29β

1.6825 − 0.0714g ≥ 0.58(g˜ − 0.5g)I∗ .

Modelling the Role of Vector Transmission of Aphid Bacterial Endosymbionts. . .

Substituting into the third equilibrium equation we obtain the quadratic but where all the coefficients are of uncertain sign:

227

)2

k k=0 Hk I∗ ,

1 [0.1682g˜ − 0.0841g + (0.1682 + 0.29β)c˜ + a c˜g] ˜  eβ − 2 [a g˜ + (0.1682 + 0.29β)][0.5g − g] ˜ − c,  1 ˜ + 0.0714β) + 2.9009a)] H1 = 2.2353 − [0.4879 − 0.0207g + c(0.0414  β + 2 [(2.9009a + 0.58(0.0414 + 0.0714β))(0.5g − g) ˜  −(1.6825 − gm)(a g˜ + (0.1682 + 0.29β))], H2 =

H0 =

β [(0.0414 + 0.0714β) + 2.9009a][1.6825 − 0.0714g]. 2

Even imposing H0 > 0, i.e. 1.6825 > 0.0714g, would not guarantee a nonnegative root, as in addition H2 < 0 should be required, if one does not want to discuss all the possible situations that can arise. In terms of the original model parameters, even these relatively simple conditions would provide rather complicated and unusable inequalities. The second approach would solve for A the first equilibrium equation, giving A = 1.7241[0.0714 − 0.29I + aW ], so that upon substitution into the remaining ones, the second provides a straight line in the I W phase plane, L:

W =

1 [1.6825−0.0714g+0.58(0.5g−g)I ˜ ] = mL I +qL 0.3364 + 0.58β + ag

which has a zero at I2 = (1.6825 − 0.0714g)[0.58(g˜ − 0.5g)]−1 while the third one gives a conic section (I, W ) = 0 with (I, W ) = (0.5c˜ − c)I 2 − (0.29 + 1.7241a c˜ + 0.5β) I W

(20)

+1.7241βW + (2.2353 − 0.1232c) ˜ I + 0.1232βW. 2

Its intercepts with the axes are the origin, (0, W1 ) and (I 1 , 0) with W1 = −0.0714a −1 < 0, I 1 = (1.2965 − 0.0714c)[0.58(c ˜ − 0.5c)] ˜ −1 ∈ R. The invariant characterizing the nature of (17) is δ = (0.5c˜ − c)1.7241aβ − 0.25 (0.29 + 1.7241a c˜ + 0.5β)2 ∈ R.  is an ellipse whenever δ > 0. In such case, if I 1 < 0, no feasible coexistence equilibrium can exist. Indeed in this situation no part of the ellipse lies in the first quadrant. Conversely an intersection between  and L is guaranteed if we require 0 < I2 < I 1 .

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6.4 The Field Conditions Here, all the equilibria involving at least one vanishing population disappear, as well as the origin, preventing the system to become aphid-free. The only possible equilibrium is coexistence, that retains also its former stability conditions derived for the laboratory model (1) subject to (2), but whose feasibility is more complicated than for the laboratory system.

7 Discussion In this paper we first explored results in closed systems, which are relevant to manipulation experiment set-ups and closed glasshouse crop environments. While symbiont-infected and uninfected aphids can readily coexist in the absence of parasitoid wasps, the coexistence of infected aphids, uninfected aphids, and parasitoid wasps was achievable but only under a narrow range of parameters. In particular, it required a sizable parasitoid wasp population leading to high intraspecific competition among wasps for the aphid resource; infected aphids suffered when the intraspecific competition between wasps for aphid resources was reduced. The rate of horizontal transmission of the symbiont from infected to uninfected aphids, via wasp oviposition, was also required to be low in order to maintain populations of the uninfected aphids. For example, with β = 0.02 it is estimated that coexistence would occur with 30% of aphids infected by the protective symbiont, whereas if β = 0.07 this would increase the proportion of aphids infected by the symbiont to 70%. Further, intraspecific competition among infected aphids could also influence the coexistence of the aphid types and the dependent wasp population. In the field simulation, where migration of all involved is no longer zero, no situation leads to an aphid-free system. The only equilibrium situation is coexistence, confirming the often found polymorphism of infection in field aphid populations across multiple species of aphid and symbiont [16]. While symbiont-mediated protection of aphids might be of great concern in closed greenhouse systems [14], this can be managed by understanding the mechanisms that can drive coexistence of the aphids and wasps. If symbiont-infected aphids are present in the population, larger populations of parasitoid wasps may need to be maintained. However, if the system is not maintained in the narrow range in which coexistence, and control, can occur then there is potential for driving fixation of the symbiont. Here, only the infected aphids can coexist with a smaller wasp population leading to lower overall control of the aphid population. In this case, a different strategy of control, by using a diverse selection of parasitoid wasps or other natural enemies species that differ in their responses to the symbiont present in the population, may be required [8, 14].

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Our results do suggest that there is great potential for symbiont-mediated aphid protection to be managed in open field systems, since only equilibrium through coexistence is possible. However, open systems will never achieve aphid-free status when a single species of parasitoid is the only control agent present. However, if populations are established they may have the potential to maintain aphid populations below the economic injury level [4]; this is the level above which integrated pest management (IPM) strategies will intervene with additional control methods. With current increases in resistance to chemical pesticides [7], it is important to identify alternative strategies to control aphids. Again, the establishment of a diverse complement of natural enemies (i.e., not just this single parasitoid) may further benefit aphid control. However, it remains to be modelled on how predation by generalist predators (e.g., ladybirds) might disrupt the equilibria determined in this paper. One additional complexity that has not been explored here is the role of additional aphid species in the system that share the parasitoid wasp but differ in their symbionts. The host-preference of the wasps, the inherent resistance of aphid species or genotypes to the wasps [2, 6, 15], and the ability to transfer symbionts across aphid species, could all alter the outcome of the interactions studied. Our work demonstrates that we must consider studying experimental systems with differing proportions of symbiont-infected and uninfected aphids in order to fully comprehend the mechanisms underlying aphid-symbiont interactions in the field. Acknowledgements This research has been partially supported by the The European COST Action: FA 1405—Food and Agriculture: Using three-way interactions between plants, microbes, and arthropods to enhance crop protection and production. The research of Ezio Venturino has been partially supported by the project “Metodi numerici nelle scienze applicate” of the Dipartimento di Matematica “Giuseppe Peano.” Sharon Zytynska was supported by funding from the British Ecological Society (SR16/1069). The paper was written while Ezio Venturino was visiting the Fields Institute during the “Thematic Program on Emerging Challenges in Mathematical Biology”; he gratefully acknowledges the Director Prof. I. Hambleton for his support and Prof. M. Lewis and Prof. H. Eberl for the invitation.

References 1. S.R. Bordenstein, K.R. Theis, Host biology in light of the microbiome: ten principles of holobionts and hologenomes. PLoS Biol. 13, 23 (2015) 2. L. Cayetano, L. Rothacher, J.-C. Simon, C. Vorburger, Cheaper is not always worse: strongly protective isolates of a defensive symbiont are less costly to the aphid host. Proc. Biol. Sci. 282, 20142333 (2015) 3. L. Gehrer, C. Vorburger, Parasitoids as vectors of facultative bacterial endosymbionts in aphids. Biol. Lett. 8, 613–615 (2012) 4. M. Kogan, Integrated pest management: historical perspectives and contemporary developments. Annu. Rev. Entomol. 43, 243–270 (1998) 5. M. Kwiatkowski, C. Vorburger, Modeling the ecology of symbiont-mediated protection against parasites. Am. Nat. 179, 595–605 (2012)

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6. M. Leclair, I. Pons, F. Mahéo, S. Morlière, J.-C. Simon, Y. Outreman, Diversity in symbiont consortia in the pea aphid complex is associated with large phenotypic variation in the insect host. Evol. Ecol. 30, 925–941 (2016) 7. G. Malloch, J. Pickup, F. Highet, S. Foster, M. Williamson, B. Fenton, Assessment of the spread of pyrethroid resistant Sitobion avenae in the UK and an update on changes in the population structure of ıMyzus persicae in Scotland, in Proceedings Crop Protection in Northern Britain (2016), pp. 223–228 8. G.J. Messelink, J. Bennison, O. Alomar, B.L. Ingegno, L. Tavella, L. Shipp, E. Palevsky, F.L. Wäckers, Approaches to conserving natural enemy populations in greenhouse crops: current methods and future prospects. BioControl 59, 377–393 (2014) 9. N.A. Moran, P.H. Degnan, S.R. Santos, H.E. Dunbar, H. Ochman, The players in a mutualistic symbiosis: insects, bacteria, viruses, and virulence genes. Proc. Natl. Acad. Sci. U. S. A. 102, 16919–16926 (2005) 10. K.M. Oliver, A.H. Smith, J.A. Russell, Defensive symbiosis in the real world –advancing ecological studies of heritable, protective bacteria in aphids and beyond. Funct. Ecol. 28, 341– 355 (2014) 11. D. Sanders, R. Kehoe, F.J.F. van Veen, A. McLean, H.C.J. Godfray, M. Dicke, R. Gols, E. Frago, Defensive insect symbiont leads to cascading extinctions and community collapse. Ecol. Lett. 19, 789–799 (2016) 12. A.H. Smith, P. Lukasik, M.P. O’Connor, A. Lee, G. Mayo, M.T. Drott, S. Doll, R. Tuttle, R.A. Disciullo, A. Messina, K.M. Oliver, J.A. Russell, Patterns, causes and consequences of defensive microbiome dynamics across multiple scales. Mol. Ecol. 24, 1135–1149 (2015) 13. A. Sugio, G. Dubreuil, D. Giron, J.-C. Simon, Plant-insect interactions under bacterial influence: ecological implications and underlying mechanisms. J. Exp. Bot. 66, 467–478 (2015) 14. C. Vorburger, Symbiont-conferred resistance to parasitoids in aphids – challenges for biological control. Biol. Control (2017). https://doi.org/10.1016/j.biocontrol.2017.1002.1004 15. C. Vorburger, L. Gehrer, P. Rodriguez, A strain of the bacterial symbiont Regiella insecticola protects aphids against parasitoids. Biol. Lett. 6, 109–111 (2010) 16. S.E. Zytynska, W.W. Weisser, The natural occurrence of secondary bacterial symbionts in aphids. Ecol. Entomol. 41, 13–26 (2016)

The Incorporation of Fractal Kinetics in the PK Modeling of Chemotherapeutic Drugs with Nonlinear Concentration-Time Profiles Tahmina Akhter and Sivabal Sivaloganathan

1 Introduction The word pharmacokinetics means the application of kinetics to a pharmakon (a Greek word means drugs and poisons) [1]. Knowledge about the change of one or more variables as a function of time is known as kinetics. The objective of pharmacokinetics is to study the time course of drug and metabolite concentrations in biological fluids, tissues, and excreta, and also of pharmacological response, and to construct suitable models to interpret such data. The data are analyzed using a mathematical representation of a part or the whole of an organism. Broadly speaking then, the purpose of pharmacokinetics is to reduce data to a number of meaningful parameter values, and to use the reduced data to predict either the results of future experiments or the results of a host of studies which would be too costly and timeconsuming to complete. The effects and the duration of action of the drug are also taken into account, using experimental PK data from humans or animals which are typically a discrete time sequence of drug concentrations obtained from a fixed volume. The data obtained from such studies are useful for the design and execution of subsequent clinical trials, also for the important goals of drug development by the pharmaceutical industry. Clinical pharmacokinetics is the application of pharmacokinetic studies

T. Akhter Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada e-mail: [email protected] S. Sivaloganathan () Department of Applied Mathematics, University of Waterloo, Waterloo, ON, Canada Center for Mathematical Medicine, Fields Institute, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_16

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to clinical practice and to the safe and effective therapeutic management of the individual patient. Identifying appropriate drugs and determining the right dosages are two prerequisites for successful treatment of cancer using chemotherapy. Searching for effective new drugs and evaluating their performance on various tumors is the main focus of preclinical and clinical studies [2]. Furthermore, the correct or optimum dose with minimum toxicity is also a crucial requirement for any effective therapy [2]. This in turn requires extensive experimental and empirical studies (both in vitro and in vivo). In this study, we develop a mathematical model to study a well-known chemotherapeutic cancer drug paclitaxel, in order to facilitate the design of dose administration strategies along with combination schedules. Paclitaxel was the first taxane to gain widespread clinical acceptance and is used now as a chemotherapeutic agent to treat a wide spectrum of tumors. Although paclitaxel was initially thought to display linear pharmacokinetics, Sonnichsen et al. [3] highlighted the fact that the pharmacokinetics is nonlinear by showing that a two compartmental model that incorporated saturable transport and saturable elimination best captured pediatric PK data. A three compartmental model that incorporated saturable transport from the central compartment to peripheral compartments, as well as saturable elimination was used by [4] to capture the concentration/time behavior of paclitaxel, in the case of adults. The data from Giani et al. [4] and Sonnichsen et al. [3] clearly demonstrated that drug distribution saturated and it was also clear from the data that saturation of transport processes occurred at lower plasma concentrations than for elimination. It is often the case that several PK models of varying complexity may be able to capture the behavior of any given data set. The choice of an appropriate model is often guided by physiological considerations and the use of Ockham’s Razor (or the law of parsimony). For drugs that are governed by underlying linear processes, compartmental PK models have provided excellent characterizations for a variety of pharmaceuticals. Similarly, the incorporation of saturable elimination (based on the Michaelis–Menten theory of bimolecular chemical interaction between substrates and enzyme) has led to an even better characterization of the PK of these pharmaceuticals. Although saturable binding and transport are based on sound physiological principles, the processes by which paclitaxel is distributed through tissue suggest that saturable binding is the dominant process, not saturable transport. Thus, a possible fruitful future direction of research might be the investigation of PK models that incorporate saturable binding compared with those that incorporate saturable transport (these latter have tended to predominate PK studies of paclitaxel.) From experimental studies, it is well established that the time-concentration profile of paclitaxel shows nonlinear dose dependence [4–6]. Various models have been developed based on linear and nonlinear timeconcentration profile of paclitaxel, using saturable distribution and elimination for two, three, or more compartments [4–9]. For example, Giani et al. [4] developed a multi-compartmental model which includes a metabolite component. Sonnichsen et al. [3] developed a two compartmental model for children with solid tumors. Both models have assumed two saturable processes, one during the distribution phase

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and the other during the elimination phase. These saturable processes have been incorporated in the models by assuming Michaelis–Menten kinetics [5]. The derivation of Michaelis–Menten kinetics is based on traditional mass action kinetics. If the reaction occurs in dimensionally restricted environments (which occurs specially in vivo), then traditional mass action kinetics fail to capture well the underlying mechanism [10] and resulting behavior of drug concentration levels. Thus to better capture the chemical kinetics, the Michaelis–Menten formalism has to be extended to (what is known in the literature) as fractal Michaelis–Menten kinetics [10]. We extend the existing mathematical model in the literature of Kearns et al. [5] by incorporating fractal kinetics to better capture the saturable distribution and elimination processes for the time-concentration profile of the drug paclitaxel. In Sect. 2, we describe the mathematical formulation of the model along with a description of how the numerical simulations are carried out. In Sect. 4, we briefly discuss the three experimental studies from which we have drawn data in order to validate our model. In Sect. 5, we present the choice of parameter values for our study and compare these with those by Kearns et al. [5] in their model. In Sect. 6, we discuss and appraise our findings.

2 Description of the Mathematical and Computational Models Classical compartmental PK models treat the body as a combination of a number of compartments. This is a popular approach to describe biological systems transport of materials through the body. The usual practice is to consider the kinetic rates as constants. However, the measurement of biological systems present unique challenges where drug molecules interact with membrane interfaces, metabolic enzymes and have to navigate through restricted and crowded micro-environments, where the system is clearly unstirred, heterogeneous, and geometrically fractal [11]. Kopelman [12] first pointed out that classical kinetics are unsatisfactory, especially if the microscopic environment of the reactants are spatially constrained. Fractal order of elementary reactions, rate coefficients with temporal memories, self-ordering, and self-non-mixing criteria have a great impact on the heterogeneous reaction kinetics [12]. A number of publications on diffusion in fractal spaces using fractal calculus have appeared in the literature. These studies show that all classical pharmacokinetics modeling are essentially a subset of fractal pharmacokinetics. Fuite et al. [13] discussed the fractal nature of the liver for the drug miberfradil, which is used to reduce ventricular fibrillation [14]. In this study, we focus on the anticancer drug paclitaxel. It is derived from the Pacific or European Yew tree [15]. In 1962, US National Cancer Institute studied the toxic effects of paclitaxel for the first time. Phase I trials were carried out in 1983, subsequently Phase II trials began in 1988 [16]. Paclitaxel blocks the G2/M phase

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of the cell cycle and as a result, such cells can’t go through normal mitosis [16]. Microtubules are responsible for cell shape, motility of cells, intracellular transport, and mitosis, and paclitaxel is now well known as a mitotic inhibitor as it binds to microtubules [7]. It is now accepted as a most effective anticancer drug and has been used for solid tumors such as breast, ovarian, lung, and head and neck. Usually this drug is administered via intravenous infusion at a high dose [5]. It has a long residence time in the body and is capable of staying in cancer cells for over a week [9]. The characterization of drug transport and absorption has attracted much interest over the last several decades, and remains arguably the central focus of much PK research. Drugs interact with target organs by crossing epithelial membranes either by passive diffusion, pinocytosis, or via carrier-mediated transport. Carriermediated transport of drugs is the primary means of delivering drugs to a variety of tissues ranging from organs such as the kidney, gut, and the choroid plexus in the central nervous system. The mathematical modeling and analysis of this process has been based on classical reaction kinetics (with time-independent rate constants). This has, however, been known to be far from satisfactory for media composed of heterogeneous micro-environments, and especially for reactions that are spatially constrained. Fractal reaction kinetics have been proposed as a better approach to modeling the pharmacokinetics of drug transport in the human body. Work of Mandelbrot [17] and West et al. [18] has shown that organs such as the lungs, kidneys, and anatomical structures such as the circulatory system have a fractal geometry. Other time dependent processes (leading to limit cycle oscillations) have been implicated as leading to nonlinear pharmacokinetics. This suggests that the extension of the Michaelis–Menten formalism, incorporating fractal kinetics may be a better approach to study the time course of a drug in the body.

3 Fractal Michaelis–Menten Kinetics In this section, we briefly discuss the extension of classical Michaelis–Menten formalism to fractal Michaelis–Menten kinetics.

3.1 Enzyme Kinetics Biological and biochemical processes are common characteristic features present in all animals and living organisms. There are complex biochemical reactions catalyzed by proteins (known as enzymes) which react with certain compounds (substrates). In 1913, Michaelis and Menten proposed a characterization of one of the most basic of enzymatic reactions, which has been used as a standard formalism,

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1000

[S]

concentration

900

[P]

substrate

800

enzyme

700

complex

600

product

500

[E]

400 300

[ES]

200 100 0

0

5

10

15

20

25

time

Fig. 1 Michaelis–Menten Kinetics: where we have taken reaction rates. k1 = 0.5e − 3, k2 = 0.5 and k−1 = 0.5e − 4 and initial substrate S(0) = 1000, initial enzyme E(0) = 500, initial product P (0) = 0

since then for describing such reactions. This is represented schematically by (also presented in Fig. 1); k1

k2

S + E  (ES) → E + P . . . (∗) k−1

where a substrate S reacts with an enzyme E to produce a complex (ES) which produces a product P and the enzyme E. k1 , k−1 , and k2 are reaction rate constants associated with particular reactions Let [S], [E], [SE], and [P ] denote the concentrations of reactants in the relation (*). Applying the law of mass action results in differential equation for each reactant, can be described as follows: d[S] dt d[E] dt d[ES] dt dP dt

= −k1 [S][E] + k−1 [ES], = −k1 [S][E] + (k−1 + k2 )[ES], = k1 [S][E] − (k−1 + k2 )[ES], = vp = k2 [ES].

(1)

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The mathematical formulation is completed by a set of initial conditions, corresponding to the start of the whole process of conversion of S to P : S(0) = [S]0 , E(0) = [E]0 , ES(0) = 0, P (0) = 0. On adding the second and third DEs, we obtain [dE] d[ES] + = 0, dt dt

(2)

i.e., E(t) + ES(t) = E0 and using this and substituting for [E] in the third DE (for the enzyme substrate complex), we obtain d[ES] = k1 (E0 − [ES])[S] − (k−1 + k2 )[ES]. dt

(3)

Assuming that the initial formation of the complex [ES] is very rapid (after which it is, for all intents and purposes, at equilibrium), we have k1 (E0 − [ES])[S] − (k−1 + k2 )[ES] ≈ 0,

(4)

from which we can evaluate: [ES] =

k1 [S]E0 , k−1 + k2 + k1 [S]

or [ES] =

E0 [S] , kM + [S]

where kM =

k−1 + k2 k1

is the Michaelis–Menten constant. Since the velocity of the reaction is given by: v = k2 [ES], this implies v=

vmax [S] k2 E0 [S] = , kM + [S] kM + [S]

where vmax = k2 E0 is the maximum velocity of the reaction and kM (the Michaelis– Menten constant) gives substrate concentration at 12 vmax .

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3.2 Batch/Transient Case The underlying assumptions of homogeneity and well-stirred environments lead to time-independent constant reaction rates in classical chemical kinetics. However in the real case scenarios, reactions often occur in restricted geometries where the kinetics can be affected by fluctuations in concentration levels [19]. This situation can be captured mathematically by taking the rates time dependent [20]. If we want to consider the simplest applicable bimolecular elementary reaction: A + B → Products.

(5)

By the law of mass action, this has the macroscopic description [12], dC = −kCA CB , dt

(6)

where C(t) is the reactant concentration and k is the “rate constant” which does not depend on time or concentration. Another underlying mechanism influencing this reaction is diffusion, which arises from differences in drug concentration in different parts of the body. In this case, the system can be described by two fundamental time scales: one is the diffusion time, the time particles require before meeting each other to react and the time particles will take to react with each other [21]. The process is known as reaction limited, when the reaction time is larger than the diffusion time and laws of mass actions are used to define the kinetic rates. On the other hand, law of mass action is not applicable when the diffusion time is larger than the reaction time and the process is known as diffusion limited. As a result, reaction rates are not time independent any more [22]. Now to understand this time dependence for a long time, i.e., the asymptotic reaction situation (t → ∞), studies are done to derive a relation between the microscopic diffusion constant D and the macroscopic rate constant k by stochastic approach [12, 23, 24], which is: k∼D

t → ∞.

(7)

At first the reactant molecules A have been considered by Smoluchowski and other researchers as a random walker (drug molecules) and the reactant molecules of B as a sitter (traps), which idea was later expanded by considering both A and B as random walkers along with the “relative diffusion” (DA + DB ) approach, precisely for different molecules. There are two different methods regarding this idea: one is stochastic, which says that the mean square displacement for the homogeneous system is linear in time where D is a proportionality constant [12]. The other method is based on the first passage time (the time which is taken by a state variable to reach at specified target) and on the mean number of distinct sites S(t) visited on

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the fractal at some resolution [12]. Havlin and Ben-Avraham introduced a scaling hypothesis between the quantity S(t) and the spectral dimension ds as [13]: S(t) ∝ t ds /2 ,

(8)

which means that the random walk steps are proportional to the time [13]. For transient reactions, in both homogeneous and heterogeneous media, the time derivative of quantity S(t) is proportional to the macroscopic constant k, i.e., [12]: k(t) ∝

dS(t) ∝ t −(1−ds /2) , dt

(9)

or k ∝ t −h

(10)

where h=1−

ds , 2

0 ≤ h ≤ 1.

(11)

Now we can write the time dependent rate constant k(t) as: k = k  t −h

(12)

where k  is the time-independent constant and within the reaction medium and the spectral dimension of the path of the random walker is represented ds [12]. If ds = 2, then using the above relation the value of h is 0, which means that k = k  and we will be getting time-independent rate constants. In PK, both compartmental and non-compartmental studies include Eq. (12). For example [25], has incorporated Eq. (12) for the homogeneous and heterogeneous model where they have calculated the overall quality of blood flow. Fuite et al. [13] have studied the fractal compartmental model for liver using the relation (12). In their study, they have used the following relation for the rate of elimination via the liver: ν = k  t −h C.

(13)

and reported that h has a significant impact on the shape of the concentration-time profile. One of the several attempts to incorporate Eq. (12) was carried out by Berry [26], using Monte Carlo simulations in low dimensional media to model enzyme kinetics and have used the equation the Michaelis–Menten formalism to obtain the formula ν=

vmax C kM0 t h + C

(14)

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3.3 Steady State/Steady Source As a special case of (5), the reaction of A and A can be written as: A + A = 2A → Products,

(15)

with the reaction rate equation dCA = −kCA2 . dt

(16)

In the steady state case (16) can be substituted for the steady state rate R as follows: R = kCA2

(17)

where k is the time-independent constant [12]. Anacker and Kopelman [27] have shown that (17) should be replaced by R = k0 CAX ,

(18)

where X ≡1+

2 2−h = ds 2+h

(19)

which is the new interpretation of the reaction order X, i.e.,  X=

1+ 1+

2 ds 4 ds

for

A+A

reaction

for

A+B

reaction

Quintela et al. [28] proposed a new approach to Michaelis–Menten kinetics replacing the kinetic rate constants by the effective kinetic rate constants incorporating the observation scaling factor: kieff = Ai [S]1−D ,

(20)

where D is the fractal dimension of the space. Using these in the reaction equations: k1eff

k2 eff

S + E  (ES)eff → E + P ,

(21)

eff k−1

and relationship between the macroscopic constant k eff and the quantity S(t) proposed by Quintela et al. [28]: k eff -

dS(t) . dt

(22)

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One can obtain the following relation: ν=

eff [S]2−D Vmax eff + [S] kM

(23)

,

eff and k eff are defined in terms of k eff , k eff , and k eff . where the new constants Vmax M 1 2 −1 From (23), we can see that if D = 1 then it will be the form of classical Michaelis– Menten kinetics (the fractal dimension D is less than 1 and greater than 0, i.e., 0 ≤ D ≤ 1). The resulting kinetics will be more and more complex as D deviates from unity [28]. Another approach proposed by Savageau [10] using power law and fractal concentration dependent kinetics for a multi-compartment reaction is given by: n n r r  5  5 dCi g h = αik Cj ij k − βik Cj ij k , dt k=1

j =1

k=1

(24)

j =1

where Cj is the concentration of the species j , α and β are the kinetic rate coefficients, and g and h are the orders of kinetic rates related with each reactant [29]. Rate laws are linearized in terms of concentration or reaction affinities using the power law [30]. Although it is possible to model a number of essential properties using these concepts, it is not applicable for other important biochemical effects like saturation [30]. Savageau [10] argues that although the power law formalism may have complex mathematical form, it nevertheless has significant benefits, as it is capable of capturing fractal phenomena mathematically. Marsh et al. [29] state that the equation proposed by Savageau [10] can be derived by summing over several Michaelis–Menten reactions.

3.4 Dose Dependent Fractal Michaelis–Menten Kinetics Kopleman [12] mentions that, for the transient state, reactants follow random distributions and reactants/walkers (drug molecules), gradually lose their efficiency while they are traversing the fractal space (which has dimension ds ). As a result of this, anomalous kinetics is observed in this state [29]. Whereas in the steady state, there is an inflow of the molecules, which can be treated as well stirred in Euclidean geometry [12]. However, in the fractal case, as self-stirring is unlikely, then it can be a result of extensive fluctuations in the regional concentration along with the growing isolation of molecules, which is known as reaction–diffusion phenomena and as a result of these, at the steady state, distribution molecules can be taken as partially ordered with a reduced reaction rate [29]. Marsh et al. [29] proposed an alternative formulation of fractal kinetics which depends on the dose concentration under steady state conditions. If we consider a

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single compartment model in the steady state, this implies that the drug concentration is constant as a function of time. Even though physiologically, there may be considerable variations in drug concentrations from one location to another, it may still be possible to achieve a steady state scenario if the drug concentration is far higher than enzyme concentration levels. Although drug molecules are lost through the elimination, the heterogeneity of the environment can give rise to re-circulation of released drug molecules trapped long time at various locations. Thus modifying the DEs (3) appropriately, we can rewrite it for a heterogeneous environment as follows [29]: d[ES] = k1 (E0 − [ES])[S]X − (k−1 + k2 )[ES] dt dP = k2 [ES] dt

(25)

where X is the fractal reaction order and using this relation in (5), we obtain the following form for the fractal Michaelis–Menten function: v0 =

vmax S0X km + S0X

,

(26)

where vmax is the maximum velocity of the reaction, km is the Michaelis–Menten constant, and S0 is the initial concentration of S. Using this functional formula in a one-compartmental model with an intravenous infusion, Marsh et al. [29] proposed the following model for the drug miberfradil: vmax S0X dS0 i(t) = + , X dt vd km + S0

(27)

where i(t) is the infusion rate which has the unit mass/time and vd is the volume of distribution which has the unit of volume. In terms of the drug concentration C which can be written as: dC vmax C X i(t) = + . dt km + C X vd

(28)

Following Marsh et al. [31] we want to use this dose dependent steady state fractal Michaelis–Menten kinetics to model the infusion of the drug paclitaxel for the two compartmental case. In the following compartmental model, the blood plasma is taken to be the central compartment where the drug enters through infusion. Elimination is assumed to be a saturable process, the central compartment is connected to a second compartment which could be a tumor, an organ, or a mathematical construct encompassing a number of anatomical structures and physiological processes of interaction,

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Fig. 2 Schematic diagram of three compartmental model with both saturable distribution and elimination from the central plasma compartment

with saturable distribution. The third (peripheral) compartment encompasses other regions and structures (not of direct interest), but where nevertheless linear binding (with the drug in the central compartment) may occur. Kearns et al. [5] used a three compartment model with Michaelis–Menten kinetics (with saturable distribution and elimination) to capture the longtail behavior of paclitaxel’s time-concentration profile. In our study, we extend the Michaelis–Menten kinetics to fractal Michaelis– Menten kinetics for both distribution and elimination for two compartments (we do not consider the peripheral compartment) instead of three (as is done by Kearns et al. [5]) and compare the performance of our model with the model presented by Kearns et al. [5]. Figure 2 is a schematic presentation of the three compartmental model with fractal saturable kinetics for paclitaxel. The model is described by the following system of equations: C˙1 = −

p

d C vmax 1 d kM

p + C1

q

+ k21 C2 − k13 C1 + k31 C3 −

C˙2 =

e C vmax 1 e kM

q + C1

+

i(t) , Vd

(29)

p

d C vmax 1

p

d +C kM 1

− k21 C2 ,

C˙3 = k13 C1 − k31 C3 .

(30) (31)

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Here, superscripts d and e denote distribution and elimination, respectively, i(t) is the infusion rate, and Vd is the volume of distribution. In this model, we consider two different fractal powers p and q, one is for the distribution (p) and the other one is for elimination (q), since these two processes occur in two different organs. In the model described by Kearns et al. [5] p = q = 1 (fractal powers are not considered). In contrast, we consider two compartment model (i.e., k13 = k31 = 0) with fractal kinetics. To solve the problem numerically and illustrate the results graphically, we have used Matlab 2017b. Subroutine ode45 has been used to solve the system of equations for both the fractal and nonfractal models by setting initial concentrations zero at time t = 0 in all three compartments. We consider appropriate time points from the relevant data, for both models. To determine the optimal parameter values, we have used the Matlab built-in optimization algorithm, which uses a genetic algorithm (GA). It is an adaptive heuristic random search algorithm, which is based on the evolutionary ideas of natural selection and genetics and is capable of solving both constrained and unconstrained optimization problems (a brief overview of this algorithm is given in Appendix 1). The ideas underlying this optimization algorithm are derived from evolutionary concepts (from Charles Darwin’s principle of “survival of the fittest”). A population of individual solutions is repeatedly modified by the algorithm. From a current population, GA randomly chooses a set of individuals at each time step and uses those individuals as parents to produce the offspring for the next generation and by repeating the procedure successively, the algorithm is able to find the population for the optimal solution. The simulation gives the opportunity to search for the parameters over a larger space and so avoid being trapped at local extrema. From Kearns et al. [5] we know that all these parameter values should be positive and we have used this as a constraint when running simulations, i.e., we have constrained the search domain from 0 to ∞. In order to run the simulations, the genetic algorithm requires an objective function which can be used to optimize the target parameter values. The goal of the optimization is to minimize the distance between the observed concentration values and the predicted concentration values. We have the plasma concentration values at different time points from the data set, these represent the true values. From our model, we can predict concentrations at the same time points and these represent our predicted values. We use the weighted residual sum square (WRSS) as the objective function, which will minimize the differences between the true and predicted values:

n    (ConDatai − ConP redi )2 WRSS =  ConP red 2 i=1

(32)

i

 where ConDatai are the concentration values from the data set and ConP redi are the predicted concentration values using the model and i represents the discrete time points at which the concentration values have been collected in the experiments.

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The Matlab optimization tool box includes a function, Fmincon, which is used to minimize a scalar function of several variables, with linear constrains and specific bounds. Fmincon uses a gradient-based framework with five algorithm options: “interior-point” (default), “trust-region-reflective,” “sqp,” “sqp-legacy,” and “activeset,” to find local optimum values. After obtaining the optimum parameter values from the GA simulation, we have some idea about the values as well as the range of the parameters. We now run the Fmincon function with the default algorithm (interior-point) and set the initial conditions very close to the parameter values (obtained from GA). This gives us greater confidence that our parameter values are indeed the locally optimum values. To compare the goodness of fit of our model with that of Kearns et al. [5] we use the Akaike information criterion (AIC) (a brief discussion of AIC is presented in Appendix 2), which can be expressed as follows: AIC = Nobs ln(WRSS) + 2Npar ,

(33)

where Nobs represents the number of data points, Npar represents the number of parameters of the model. We know from the definition of AIC that the lower the value of the AIC, the better the fit (according to this criterion).

4 Experimental Data Paclitaxel is used to treat a variety of cancers ranging from ovarian, lung, breast, bladder, melanoma, esophageal, prostate, and many other solid tumor cancers. It has also been used for a different type of cancer known as Kaposi’s sarcoma. It is poorly water soluble, but can be dissolved in organic solvents. To administer the drug to patients, the usual formulation of this drug is: 1:1 blend of cremophor EL (polyethoxylated castor oil commonly known as CrEL) and ethanol which is diluted with 520-fold in normal saline or dextrose solution (5%) [32]. The PK nature of CrEL does not depend on dose, but the infusion time has a great impact on the clearance of the drug [33]. In order to handle some of these side effects clinically, pre-medications are frequently administered. Marsh et al. [6] carried out an analysis to demonstrate the power law behavior of paclitaxel. They utilized 41 sets of data from 20 published papers, and digitized the data in order to study the longtime residence (in the body) of paclitaxel. They reported that there was power law behavior in the tail region using timeconcentration profile data for paclitaxel. Following Marsh et al. [6] we digitized data from three different studies and compared the predictions of our model with that of Kearns et al. [5] A brief description of the studies (from which our data was drawn) is given below.

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4.1 Brown et al. Brown et al. [34] designed their Phase I study for 31 patients of melanoma, lung, colon, head and neck, prostate, kidney, and unknown cancers. Mean age of the patients was 58 years. Taxol was administered for 6 h IV infusion and repeated every 21 days. Blood serum samples were collected before infusion at 5, 15, 30, 60, and 90 min, and 2, 3, 4, 6, 8, 12, 24, and 48 h after infusion. Their study focused on the previously reported toxic effects, which depended on the drug vehicle CrEL and duration of infusion time. As such they were trying to observe the situation by infusing at a rate of 43 cc/h, for the first 30 min, and if there were no major side effects by infusing at a higher rate of 180 cc/h for the next 5.5 h. The doses used in their study were 175, 250, and 275 mg/l. Their conclusions were that pre-medication does not necessarily reduce the toxic side effects. For the PK parameters, nonlinear regression analysis (NONLIN) was used with a weighting of 1/y 2 (where y is the true value from the data). From their investigation, a dose of 225 mg/l was suggested for Phase II study. We extracted plasma concentration data from this study at 11, 13, and 14 time points for a dose 175 mg/l, 250 mg/l, and 275 mg/l, respectively.

4.2 Kearns et al. and Giani et al. Giani et al. [4] studied PK characteristics and toxicity of paclitaxel and 6α hydroxylpaclitaxel in humans, using a four compartmental model. This was carried out for 30 patients. Half of these patients had advanced ovarian cancer, and the other half had advanced breast cancer. The median age of the patients was 54 years. The administered dose for ovarian cancer was 135, 175 mg/m2 and for the breast cancer dose was 225 mg/m2 by either 3 or 24 h infusion. Blood plasma was collected at 1, 2, and 3 h before infusion and at 5, 15, 30 min, 1, 2, 3, 6, and 12 h during infusion, and 21 h post infusion. PK software package ADAPT II was used to fit the time-concentration data for the compartmental analysis and regression models were used to evaluate the pharmacodynamic correlation. Study claimed, the nonlinear disposition of the drug in human specially in the short infusion period and a mathematical model can be a powerful tool in predicting paclitaxel disposition, regardless of the dose and schedule. Using the data from Giani et al. [4], Kearns et al. [5] investigated pharmacokinetics and pharmacodynamics behavior of paclitaxel using three compartmental model. They concluded that hypersensitivity occurred due to the mixing vehicle CrEL and suggested pre-medication to alleviate this situation. The correlation between the neurotoxicity and the pharmacokinetics of the paclitaxel was presented through the study and the recommendation was to use 3 h infusion periods, with pre-medications to reduce neurotoxicity.

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We extracted the concentration data at 11, 11, and 12 time points for doses of 135 mg/m2 , 175 mg/m2 , and 225 mg/m2 respectively.

4.3 Zuylen et al. Zuylen et al. [35] studied the effects of CrEL micelles on the disposition of paclitaxel. Seven solid tumor patients were treated for 3 h infusion on a 3week consecutive cycle of administration with doses of 135 mg/m2 , 175 mg/m2 , and 225 mg/m2 . Patients were 18 years or older with 3 months life expectancy without any previous taxane treatment. Blood plasma samples were collected at the following times before and after the infusion: 1, 2, 3, 3.08, 3.25, 3.5, 3.75, 4, 5, 7, 9, 11, 15, and 24 h. The Siphar package (version 4; SIMED, Creteil, France) was used to evaluate the PK parameters, using a non-compartmental model. The study claimed that the encapsulation of CrEL is one of the reasons that paclitaxel shows nonlinear behavior as it is capable of altering blood distribution. Based on this, they proposed a new pharmacokinetic model for whole blood and also for blood plasma analysis. The model was able to transform nonlinear timeconcentration profiles to linear ones by changing the formulation with the CrEL from the whole blood sample, but not from the blood plasma samples. We extracted the concentration data at 11, 11, and 9 time points for doses of 135 mg/m2 , 175 mg/m2 , and 225 mg/m2 , respectively, from the study, which was reported for the blood plasma samples. Whatever the underlying mechanism for the non-linearity of the paclitaxel timeconcentration profiles, our interest is in developing a model capable of capturing this non-linearity as a function of dose and infusion time, whether it is possible to minimize the toxic side effects of paclitaxel (either via varying the CrEL vehicle mixing proportionality or via change in the schedule of infusion time) is beyond the scope of our present study. We hope to demonstrate that the model recapitulates the behavior of the experimental data to within acceptable margins of error, and that the model can be utilized as a tool in further experimental investigations.

5 Parameter Values and Model Simulation In this section, we present simulations of both our fractal model and the Kearns model and compare both with the experimental data. We also compare the parameter values of both models (in tabular form) using a weighted residual sum square (WRSS) statistical measure, as well as the Akaike information criterion (AIC), in order to assess the goodness of fit of both models. For Fig. 3, data is taken from Kearns et al. [5]. Parameter values for the Kearns model are taken from Kearns et al. [5] and the parameter values for our fractal model are calculated using the Matlab genetic algorithm. Table 1 represents the

Concentration (mM)

Concentration (mM)

Concentration (mM)

The Incorporation of Fractal Kinetics in the PK Modeling of Chemotherapeutic. . .

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6 Exp135 mg/m2

4

Keams135 mg/m2

2 0

Fractal135 mg/m2 0

5

10 15 Time (hour)

20

25

10 Exp175 mg/m2 Keams175 mg/m2

5 0

Fractal175 mg/m2 0

5

10 15 Time (hour)

20

25

15 Exp225 mg/m2

10

Keams225 mg/m2

5 0

Fractal225 mg/m2 0

5

10 15 Time (hour)

20

25

Fig. 3 Comparing Kearns et al. three compartmental model with the proposed fractal two compartmental model (data from Kearns et al.)

Table 1 Optimum parameter values reported by Kearns et al. and the optimum values for the fractal model evaluated by a genetic algorithm (data digitized from Kearns et al.)

Parameters K21 (h) Vdmax (µM/h) Vemax (µM/h) KdM (µM) KeM (µM) Vd (L) K13 (h) K31 (h) p q WRSS AIC

Kearns model 0.68 10.20 18.80 0.32 5.50 4 2.20 0.65 1 1 1.50 20.35

Fractal model 1.97 12.32 13.41 1.67 6.39 4.92 0.0 0.0 1.20 1.70 0.83 12.89

Abbreviation: Vdmax , maximum velocity for distribution; KdM , the concentration associated with one half of the maximum velocity (Vdmax ); Vemax , maximum velocity for elimination; KeM , the concentration associated with one half of the maximum velocity (Vemax ); Vd , volume of distribution at steady state

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Concentration (mM)

Concentration (mM) Concentration (mM)

parameter values for both models. For Fig. 4, data is taken from Van Zuylen et al. [35]. There are no reported parameter values for this data set. The Matlab builtin genetic algorithm has been used to evaluate the parameter values for both the models. Table 2 describes the values for both models. As mentioned earlier, we wanted to apply our model for a different infusion time other than 3 h and for that we chose another data set. Brown et al. [34] applied 6- h infusion times for the doses of 175 mg/l, 250 mg/l, and 275 mg/l in their Phase I study. For Fig. 5, data is taken from Brown et al. [34]. Again there are no reported parameter values for this

3

Exp135 mg/m2

2

Keams135 mg/m2

1 0

Fractal135 mg/m2 0

5

10 15 Time (hour)

20

25

6 Exp175 mg/m2

4

Keams175 mg/m2

2 0

Fractal175 mg/m2 0

5

10 15 Time (hour)

20

25

10

Exp225 mg/m2 Keams225 mg/m2

5 0

Fractal225 mg/m2 0

5

10 15 Time (hour)

20

25

Fig. 4 Comparing Kearns et al. three compartmental model with the proposed fractal two compartmental model (data from Zuylen et al.) Table 2 Optimum parameter values evaluated using genetic algorithm (Matlab) for both Kearns model and the proposed fractal model (data digitized from Zuylen et al.)

Parameters K21 (h) Vdmax (µM/h) Vemax (µM/h) KdM (µM) KeM (µM) Vd (L) K13 (h) K31 (h) p q WRSS AIC

Kearns model 0.47 8.30 22.50 6.16 14.59 9.53 1.44 16.63 1 1 1.45 19.41

Fractal model 0.20 9.36 17.33 3.78 11.84 9.48 0.0 0.0 0.90 0.65 0.52 9.10

Concentration (mM) Concentration (mM) Concentration (mM)

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3 Exp175 mg/m2

2

Keams175 mg/m2

1 0

Fractal175 mg/m2 0

5

10

15

Time (hour) 6 Exp250 mg/m2

4

Keams250 mg/m2

2 0

Fractal250 mg/m2 0

2

4

6

8 10 12 Time (hour)

14

16

18

20

10 Exp275 mg/m2 Keams275 mg/m2

5 0

Fractal275 mg/m2 0

5

10

15 20 Time (hour)

25

30

Fig. 5 Comparing Kearns et al. three compartmental model with the proposed fractal two compartmental model for 6-h infusion (data from Brown et al.) Table 3 Optimum parameter values evaluated using genetic algorithm (Matlab) for both Kearns model and the proposed fractal model (data digitized from Brown et al.)

Parameters K21 (h) Vdmax (µM/h) Vemax (µM/h) KdM (µM) KeM (µM) Vd (L) K13 (h) K31 (h) p q WRSS AIC

Kearns model 1.30 14.72 14.47 7.68 7.77 7.16 10.24 13.19 1 1 6.77 36.39

Fractal model 2.72 15.39 3.38 2.23 0.29 8.45 0.0 0.0 2.05 2.79 1.69 20.96

data set and so again the Matlab built-in genetic algorithm was used to evaluate the parameter values for both models. Table 3 provides the values for both the models along with the WRSS and the AIC.

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6 Discussion Figure 3 shows that the fractal model appears to fit the experimental data much better than the Kearns model. We also observe this from Table 1 based on the WRSS and AIC values. The WRSS value of 0.833 for the fractal model is less than the value of 1.5 for the Kearns model, with a corresponding AIC value of 12.8 for the fractal model compared with an AIC value of 20.34 for the Kearns model. From Fig. 4, it is difficult to distinguish which of the models fits the experimental data better; however, an examination of the WRSS and AIC values for this case (see Table 2) gives a WRSS value of 0.51 for the fractal model compared with a value of 1.45 for the Kearns model. Similarly, we obtain an AIC value of 9.18 for the fractal model compared to 19.41 for the Kearns model. Hence, both WRSS and AIC measures confirm that the fractal model gives a better fit to the data than the Kearns model. Note that the parameter values in this case for both the models have been calculated using GA. Figure 5 shows that the fractal model fits the experimental data better than the Kearns model except for the dose 175 mg/m2 , for which both models fail to capture peak plasma concentration, although they are not too different from the experimental data. However, from Table 3, the WRSS value of 1.69 compared to 6.77 and AIC value of 20.69 compared to 36.39 again confirm that the fractal model provided a better fit to the data than the Kearns model.

7 Conclusion We extended an existing model (that used Michaelis–Menten formalism to model the nonlinear behavior of the chemotherapeutic drug paclitaxel) by incorporating fractal Michaelis–Menten kinetics to better capture the effects of two competitive saturable processes. We have compared our model with the existing model on three different data sets from three different studies. Although it might appear that one shortcoming of our fractal model and that of Kearns et al. is that a different set of parameter values have to be used in both models for each of the three data sets, a closer examination reveals why it may be unreasonable to expect that one set of parameter values should be sufficient to predict the three different scenarios. Firstly, the three data sets have been obtained for patients with very different cancers. Secondly, the age groups of the patients vary significantly for the three data sets. Thirdly, the pre-medication used on the patients are very different for each of the three data sets. Taking into consideration all of these different factors, it is clear that the three data sets obtained are for three very different clinical situations and so clearly we should expect that the parameter values calculated for one data set will not necessarily give the best fit for a data set from a different clinical situation. Naturally, given patients of similar age groups, with the same type of cancer (who received a similar therapeutic drugs), one would expect that

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parameter values extracted previously from a different patient cohort would still give an effective predictive model, which could be used to evaluate the impact of different infusion times and different doses on a new patient cohort. However, it is clear that the model needs to be validated on a broader range of data before it can be evaluated in a clinical setting. Despite all these challenges, we believe that wellvalidated mathematical compartmental models of this nature are indispensable tools in preclinical studies and play a crucial role in the eventual development of effective clinical therapies.

Appendix 1: Genetic Algorithm To find an optimal or best solution, optimization techniques have been used in different areas of science and engineering. These techniques are considering the following factors [36]: • An objective function: the function we want to minimize or maximize, for example, we want to minimize the cost and maximize the profit in manufacturing. • A set of unknown variables: the variables by which the objective function can be effected. • A set of constraints: which can be used to include certain conditions while evaluating values of the unknown parameters. So, an optimization technique is a technique which can be used to find the variables that maximize or minimize the objective function while satisfying the constraints. Genetic algorithm (GA) is a heuristic search algorithm to optimize a problem, inspired by Darwin evolution theory, which uses random search process. The basics of GA can be stated as follows: • It starts by generating a random population of n chromosome which can be think of the solution of the problem. • Calculate f(x), the fitness function of x (chromosome) in the population. • Creates new population by repeating the following processes: – Two different parent chromosomes are selected from the population which we can think as Selection by comparing with evolution process (when the fitness is better, then the chance to be selected is bigger). – Perform a Crossover probability. The offspring (children) will be exact same copy of the parents if there is no crossover (but this does not mean that the new generation is same) and the offspring will be made from parts of parents chromosome, then crossover probability performed. – How often the parts of the chromosome mutated is measured by the Mutation probability and offspring can be taken as is after the crossover if there is no mutation. – Place the new offspring in the new population which is known as Accepting the new population.

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• The new generated population is replaced for the next run of the algorithm. • By testing the constraint conditions it will stop and return the best solution of the current population. • And it will go to the step 2 to continue the loop as long as the tolerance is achieved.

Appendix 2: Akaike Information Criterion (AIC) In statistical study, we are engaging ourselves to estimate the effects for a given variable using certain parameters. While doing this, we certainly may include parameters for which we might lose the physical information we are trying to predict via a mathematical model. And we may also over fit the available data. Form the Occam’s razor philosophical principle we know that if we can describe something with a simple model or with also a more complicated model then we should choose the simple one, which indicates to rely on the simpler model than to the complicated model. Now when we are fitting some observation how do we know that we are including less or more parameters? In 1973, Akaike H., a Japanese Statistician came with a theory which will compare with the models and that comparison will give the idea which model we should choose among the models are available to choose. By doing so, it restricts us from under or over fit the model. Since this is a comparison with the available models we have, it will not tell us that this is the best model rather give us the information that this a better one among the models we have. In statistics this means that a model with less parameters is preferable than a model with more parameters. Again a model with too less parameters will be biased and a model with too many parameters will have low precision [37]. The theory is from the information field theory and named as Akaike information criterion or AIC. Good model would be the one which will minimize the loss of information. Akaike in 1973 proposed an information criterion as follows: AIC = −2(log-likely hood) + 2K where K is the number of estimated parameters used in the model. For a given sets of data log-likelihood can be calculated and from there one can tell about the model, is that it is over or under fit or not (smaller value means worse fit) [37]. If the models are based on conventional least squares regression, then the assumption is that the error obeys Gaussian distribution. And we can compute the AIC formula as follows: AIC = Nobs + ln(WRSS) + 2Npar ; where Nobs is the number of observed data point, Npar is the number of model parameters, WRSS is the weighted residual sum squares [38]. WRSS can be calculated from the following relation:

The Incorporation of Fractal Kinetics in the PK Modeling of Chemotherapeutic. . .

n WRSS = #i=1

253

(Ci − Cˆi )2 2 Cˆi

where Cˆi is the predicted value and Ci is the true value. A lower AIC value indicates a better fit. Idea about the weighted factor is, if at high concentration, i.e., at the beginning of the time plasma profile, data are showing more accuracy than the tail, then the data can be weighted with WRSS ∼ 1 [38]. On the other hand, if the tail end of the profile showing more accuracy, then the data might be weighted with 1/Cˆ2 [38]. Following this idea in our model we have used, the weighted factor for WRSS is 1/Cˆi . Acknowledgements A special thanks to Dr Henry Shum (Assistant Professor, Applied Mathematics, University of Waterloo, Ontario) for his valuable time and thoughtful discussion throughout this study.

References 1. J.G. Wagner, History of pharmacokinetics. Pharmacol. Ther. 12, 537–562 (1981) 2. H. Gurney, How to calculate the dose of chemotherapy. Br. J. Cancer 86, 1297–1302 (2002) 3. D.S. Sonnichsen, C.A. Hurwitz, C.B. Pratt, J.J. Shuster, M.V. Relling, Saturable pharmacokinetics and paclitaxel pharmacodynamics in children with solid tumors. J. Clin. Oncol. 12, 532–538 (1994) 4. L. Gianni, C.M. Kearns, A. Giani, G. Capri, L. Vigano, A. Lacatelli, G. Bonadonna, M.J. Egorin, Nonlinear pharmacokinetics and metabolism of paclitaxel and its pharmacokinetic/pharmacodynamic relationships in humans. J. Clin. Oncol. 13, 180–190 (1995) 5. C.M. Kearns, L. Gianni, M.J. Egorin, Paclitaxel pharmacokinetics and pharmacodynamics. Semin. Oncol. 22, 16–23 (1995) 6. R.E. Marsh, J.A. Tuszy´nski, M.B. Sawyer, K.J.E. Vos, Emergence of power laws in the pharmacokinetics of paclitaxel due to competing saturable processes. J. Pharm. Pharmaceut. Sci. 11, 77–96 (2008) 7. R.E. Marsh, J.A. Tuszyski, M.B. Sawyer, K.J.E. Vos, Emergence of power laws in the pharmacokinetics of paclitaxel due to competing saturable processes. J. Pharm. Pharm. Sci. 11, 77–96 (2008) 8. A. Henningsson, M.O. Karlsson, L. Vigano, L. Gianni, J. Verweij, A. Sparreboom. Mechanism based pharmacokinetic model for paclitaxel. J. Clin. Oncol. 19, 4065–4073 (2001) 9. T. Mori, Y. Kinoshita, A. Watanabe, T. Yamaguchi, K. Hosokawa, H. Honjo, Retention of paclitaxel in cancer cells for 1 week in vivo and in vitro. Cancer Chemother. Pharmacol. 58(5), 665–72 (2006) 10. M.A. Savageau, Development of fractal kinetic theory for enzyme-catalysed reactions and implications for the design of biochemical pathways. BioSystems 47, 9–36 (1998) 11. L.M. Pereira, Fractal pharmacokinetics, Comput. Math. Methods Med. 11(2), 161–184 (2010) 12. R. Kopelman, Rate processes on fractals: theory, simulations, and experiments. J. Stat. Phys. 42, 185–200 (1986) 13. J. Fuite, R. Marsh, J. Tuszy´nski, Fractal pharmacokinetics of the drug mibefradil in the liver. Phys. Rev. E 66, 1–11 (2002). Art. Id. 021904 14. A. Skerjanec, S. Tawfik, Y.K. Tam, Mechanisms of nonlinear pharmacokinetics of mibefradil in chronically instrumented dogs. J. Pharmacol. Exp. Ther. 278, 817–825 (1996)

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15. E.K. Rowinsky, M. Wright, B. Monsarrat, G.J. Lesser, R.C. Donehower, Taxol: pharmacology, metabolism and clinical implications. Cancer Surv. 17, 283–304 (1993) 16. Cancer Care Ontario Canada, https://www.cancercare.on.ca 17. B.B. Mandelbrot, The Fractal Geometry of Nature (Henry Holt and Company, New York, 1983) 18. B.J. West, A.L. Goldberger, Physiology in fractal dimensions. Am. Sci. 75(4), 354–365 (1987). http://www.jstor.org/stable/27854715 19. D.T. Gregory, Fractals in Molecular Biophysics (Oxford Universities Press, New York, 1997) 20. K. Kang, S. Redner, Fluctuation effects in Smoluchowski reaction kinetics. Phys. Rev. A 30, 2833 (1984) 21. D. ben-Avraham, M.A. Burschka, C.R. Doering, Statics and dynamics of a diffusion-limited reaction: anomalous kinetics, nonequilibrium self-ordering, and a dynamic transition. J. Stat. Phys. 60(5/6), 695–728 (1990) 22. C.R. Doering, D. ben-Avraham, Interparticle distribution functions and rate equations for diffusion-limited reactions. Phys. Rev. A 38(6), 3035 (1988) 23. M. Smoluchowski, Mathematical theory of the kinetics of the coagulation of colloidal solutions. Z. Phys. Chem. 19, 129–135 (1917) 24. S. Chandrasekhar, Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15 (1) (1943), https://doi.org/10.1103/RevModPhys.15.1 25. P. Macheras, A fractal approach to heterogeneous drug distribution: calcium pharmacokinetics. Pharm. Res. 13, 663–670 (1996) 26. H. Berry, Monte carlo simulations of enzyme reactions in two dimensions: fractal kinetics and spatial segregation. Biophys. J. 83, 1891–1901 (2002) 27. L.W. Anacker, R. Kopelman, Fractal chemical kinetics: simulations and experiments. J. Chem. Phys. 81, 6402–6403 (1984) 28. M.A. Lopez-Quintela, J. Casado, Revision of the methodology in enzyme kinetics: a fractal approach. J. Theor. Biol. 139, 129–139 (1989) 29. R.E. Marsh, J.A. Tuszy´nski, Saturable fractal pharmacokinetics and its applications, in Mathematical Methods and Models in Biomedicine (Springer, New York, 2013), pp. 339–366, https://doi.org/10.1007/978-1-4614-4178-6_12 30. S. Schnell, T.E. Turner, Reaction kinetics in intracellular environments with macromolecular crowding: simulations and rate laws. Prog. Biophys. Mol. Biol. 85, 235–260 (2004) 31. R.E. Marsh, J.A. Tuszy´nski, Fractal Michaelis Menten kinetics under steady state conditions, application to mibefradil. Pharmaceut. Res. 23, 2760–2767 (2006) 32. A.K. Singla, A. Garg, D. Aggarwal, Paclitaxel and its formulations. Int. J. Pharm. 235, 179–192 (2002) 33. H. Gelderblom, J. Verweij, K. Nooter, A. Sparreboom, Cremophor EL: the drawbacks and advantages of vehicle selection for drug formulation. Eur. J. Cancer 37, 1590–1598 (2001) 34. T. Brown, K. Havlin, G. Weiss, J. Cagnola, J. Koeller, J. Kuhn, J. Rizzo, J. Craig, J. Phillips, D.V. Hoff, A phase I trial of taxol given by a 6-hour intravenous infusion. J. Clin. Oncol. 7, 1261–1267 (1991) 35. L. Van Zuylen, M.O. Karlsson, J. Verweij, E. Brouwer, P. de Bruijn, K. Nooter, G. Stoter, A. Sparreboom, Pharmacokinetic modeling of paclitaxel encapsulation in Cremophor EL micelles. Cancer Chemother. Pharmacol. 47, 309–318 (2001) 36. http://www.obitko.com/tutorials/genetic-algorithms/ga-basic-description.php 37. K.P. Burnham, D.R. Anderson, Model Selection and Multimodel Inference: A Practical Information-Theoretic Approach, 2nd edn. (Springer, New York, 2002) 38. K. Yamaoka, T. Nakagawa, T. Uno, Application of Akaike’s information criterion (AIC) in the evaluation of linear pharmacokinetic equations. J. Pharmacokinet. Biopharm. 6 (2) (1978)

Mathematical Modeling of Inflammatory Processes O. Kafi and A. Sequeira

1 Introduction Since the late 1990s, an important research activity started to emerge in the development of mathematical models of the functioning of the human immune system and of numerical tools for its simulation, allowing the organization of biological information, a better understanding of the complex dynamics involved in the immune response, and simulation and prediction of various realistic scenarios [1]. Therefore, not much is known about these phenomena. We are facing an integrated system with a high level of complexity and variability, spanning the organ level down to the molecular level. The chronic inflammatory response localizes at specific sites of the vascular tree. This site preference is thought to be caused by hemodynamic parameters, especially the wall shear stress (WSS). WSS is a frictional force exerted parallel to the vessel wall [2]. It can be sensed by numerous mechanosensors on the luminal surface of endothelial cells promoting the release of pro-inflammatory mediators giving rise to inflammation. This is a significant motivation for mathematical models to be developed at the microvascular level. The underlying mechanisms of FSI in modeling blood, arterial wall, and atheroma plaque interaction are well understood and, in principle, the phenomena can be analyzed with good accuracy. Unfortunately, there is an important proviso, namely that the actual accuracy is strongly dependent on the correctness of the input data defining geometrical and physical properties. In reality, it is rare to be able to prescribe these with high accuracy in a practical application. In fact, it is even rare

O. Kafi () · A. Sequeira Instituto Superior Técnico, Universidade de Lisboa, Department of Mathematics and CEMAT, Lisbon, Portugal e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_17

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to be able to do so in controlled, laboratory experiments. An abundant literature was devoted to mathematical models dealing with FSI based on several methods such as the level set method [3], the immersed boundary method [4, 5], the fictitious domain method [6, 7], the fully Eulerian formulation [8, 9], and the arbitrary Lagrangian Eulerian (ALE) approach [10–12]. For the model presented in this work, simulating the FSI, the ALE method is used. The reader can also refer to our previous analytical studies [13, 14]. We also developed a mathematical model of hydrodynamics of free-flowing leukocytes. The recruitment and subsequent rolling, activation, adhesion, and transmigration of leukocytes are essential stages of an inflammatory response. Understanding this mechanism is of crucial importance in immunology and in the development of anti-inflammatory drugs [15]. Through the numerical simulations, we analyze the hemodynamic effects of the inlet velocity values. We note that the hydrodynamic forces pushing the cell become higher with increasing inlet velocities. The different sections of this work are presented as follows. In Sect. 2, we introduce the equations used to model the blood–plaque and blood–vessel wall interaction, consisting of 3D FSI approach with more realistic pressure input profile at the inlet and a linear absorbing condition (LAC) at the outlets. A simplified coupled deformation-flow model, as well as its numerical approximation, of an individual leukocyte in a microchannel, derived from a suggested physiological model, where the intracellular viscoelastic fluid is governed by the Oldroyd-B equations, is shown in Sect. 3.

2 Mathematical Model of FSI In this section, we consider the blood flow in a real stenosed carotid bifurcation. Blood is modeled as a non-Newtonian fluid with shear-thinning viscosity behavior verifying the Carreau’s law. Then, we use a coupled approach between the fluid equations and the solid model of the arterial wall and the atheromatous plaque.

2.1 Fluid Equations The blood flow is governed by the generalized Newtonian equations for incompressible fluids: ρf

∂u + ρf (u · ∇)u − ∇ · (2μ(s(u))Du) + ∇p = 0 ∂t ∇.u = 0.

(1)

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This evolution problem involves the blood velocity u = (u1 , u2 , u3 ) and the pressure p as unknowns. The dynamic viscosity μ is a function of the strain rate tensor Du = 12 (∇u + ∇ T u), or more precisely is a function of s(u), the second invariant of the strain rate tensor, defined by: (s(u))2 = 2Du : Du = 2



(Du)ij (Du)j i .

i,j

We consider μ defined by the Carreau–Yasuda law, μ(s(u)) = μ∞ + (μ0 − μ∞ )(1 + (λs(u))2 )(n−1)/2 , with the asymptotic viscosities given by μ0 = 0.0456 Pa s, the viscosity at the lowest shear rate, μ∞ = 0.0032 Pa s, viscosity at the highest shear rate, λ = 10.03 s, ρf = 1060 Kg m−3 , the fluid density, and n = 0.344 (corresponding to a shearthinning viscosity fluid). The values of the parameters are taken from [16]. Figure 1 shows a plot of the pressure waveform at the inlet. At the outlets, we consider typical natural boundary conditions where we fix the stress 2μ(s(u))Du · n − pn = −pn. ¯

(2)

Although it is known that this boundary condition can affect the velocity profile, these spurious effects are localized on the outlet surface and, therefore, far from our region of interest. As it was shown in [17], these boundary conditions, when

9000 8500

Pressure (Pa)

8000 7500 7000 6500 6000 5500 5000 0

0.1

Fig. 1 Inlet pressure waveform

0.2

0.3

0.4

0.5 0.6 Time (s)

0.7

0.8

0.9

1

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used for a FSI simulation, fail to represent the wave propagation phenomena that would be obtained if an extended domain is considered downstream the artificial boundary. In fact, after a single pulse overpasses the artificial boundary, condition (2) introduces high amplitude wave reflections upstream that region (see [17]). To minimize such spurious effect, in [17, 18] the authors coupled the 3D model with a 1D model representing the downstream domain. A less expensive method, the socalled LAC was suggested in[17] for non-Newtonian FSI blood flow simulations, and used successfully in [19] for realistic geometries. Both methods were shown to reduce significantly, yet not totally, the spurious oscillations. Here, we adopt the less expensive method, the use of LAC conditions, given by the expression p¯ = ρf β √ 5/4 Q, where Q is the flow rate and A0 the cross-section reference area at rest. 2A0 The coefficient β is √ related to the mechanical properties of the vessel wall through π h0 E the expression β = , E is the Young modulus, h0 is the wall thickness, and 1 − ν2 ν is the Poisson ratio.

2.2 Structure Equations To simplify, the arterial tissue and the fibrous cap are modeled as compliant isotropic materials, ignoring their different composition and fibrous nature. We consider a 3D nonlinear model of hyperelasticity [20] governed by the following equations: • the equation of motion ρs

∂ 2η − ∇ · σ = F, σ = J −1 FSFT , F = I3 + ∇η, J = det(F), ∂t 2

(3)

• the strain–displacement equations 1 ∇η + (∇η)T + (∇η)T ∇η , ε= 2 η being the displacement vector, σ the Cauchy stress tensor, F the deformation gradient tensor, S the second Piola–Kirchhoff stress tensor, and I3 is the identity matrix. The stress–strain relationship is given by S − S0 = C : (ε − ε0 − ε inel ), where we denote by S0 and ε 0 the initial second Piola–Kirchhoff stress and strain tensors, ε inel the inelastic strain tensor, C the fourth-order stiffness tensor, and “:” represents the inner product of two second-order tensors. In the present study, the vessel wall is considered isotropic in such a way that the stiffness tensor has no preferred direction and the corresponding tensor components are computed internally according to:

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 Cij kl = λs g ij g kl + μs g ik g j l + g il g j k , νE E and μs = are the Lamé coefficients, E is (1 − 2ν)(1 + ν) 2(1 + ν) the Young modulus, ν is the Poisson ratio, and g the metric tensor. Here, we consider ν = 0.45, E = 3 · 105 Pa in the upstream and downstream extended sections and E = 6 · 105 Pa in the vessel wall surrounding the plaque [21], since it has been shown that the presence of a space occupying the plaque is commonly associated with the expansion of the arterial wall [22]. For the atheromatous plaque, the second ∂ω , where ω is the elastic strain Piola–Kirchhoff stress tensor S is computed as S = ∂ε energy density. According to the Mooney–Rivlin model for hyperelastic materials, chosen to model the lipid pool and the fibrous cap, the energy function reads where λs =

2     1  ω = C10 I¯1 − 3 + C01 I¯2 − 3 + κ J¯ − 1 , 2

(4)

where J¯ is the ratio between the current and original volumes and I¯1 , I¯2 are the first two strain invariants, modified to be independent of the volume change (I¯1 = I1 /J¯2/3 and I¯2 = I2 /J¯4/3 , where I1 and I2 are the first two strain invariants). The use of these modified invariants and the last term in Eq. (4) allows the description of nearly incompressible materials. The parameters C10 , C01 , and κ are chosen in agreement with experimental measurements, and their values for the fibrous cap and the lipid pool are taken from [23, 24] (Table 1).

2.3 ALE Formulation Since the vessel wall deforms under the action of the fluid, the mathematical formulation should allow points in the vessel structure to move at every time instant, as well as the points inside the fluid domain. Neither the Lagrangian nor the Eulerian formulations allow this behavior without perturbing the artificial upstream and downstream boundaries. Therefore, an ALE formulation is used. We denote by w the domain velocity. If w is zero, the method is simply an Eulerian approach, otherwise, if w is equal to the velocity of the fluid, then the approach is Lagrangian. w can vary arbitrarily and continuously from one value to another in the fluid field.

Table 1 Parameters used for the plaque components Plaque components Fibrous cap Lipid pool

C10 (N m−2 ) 9200 500

C01 (N m−2 ) 0 0

κ (MPa) 3000 200

Density (kg m−3 ) 1000 1000

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Fig. 2 Medical image of a carotid bifurcation with a stenotic region [27]

We rewrite the equations of the fluid, the generalized Navier–Stokes equations, in the ALE formulation as ρf

∂u |X + ρf ((u − w) · ∇)u − ∇x · (2μ(s(u))Du) + ∇x p = 0 ∂t

(5)

where X is the Lagrangian coordinate and x is the Eulerian coordinate. From a mechanical point of view, the coupling between the systems (5) and (4) is realized by imposing at the interface: • the continuity of the velocity over time: u=

∂η ∂t

• and the equilibrium of the stresses −(2μ(s(u))Du − pI3 )n = S(η) · n, where n is the outward unit vector to the physical interface. This condition should be rewritten on the interface at t = 0 to be prescribed on the structure model. Using the Piola transform, we have  −(det∇0 η)(2μ(s(u))Du − pI3 ) ∇0−T η · n0 = P(η) · n0 , where P = P(η) is the first Piola–Kirchhoff tensor, ∇0 indicates the gradient with respect to the Lagrangian coordinates, and n0 the outward unit vector to the interface at t = 0 (see [25])

2.4 Numerical Simulations In [26], the authors performed 3D FSI numerical simulations for blood flow in an idealized atherosclerotic artery. In order to verify a practical applicability of this approach to a large range of realistic cases, the numerical results were obtained in a 3D realistic geometry reconstructed from a medical image (see Fig. 2).

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Fig. 3 3D meshed geometry. The mesh is more refined in the stenosed region

Fig. 4 Total volume displacement at the peak of the pressure: t = 0.105 s (left); streamlines of the velocity field at the peak of the pressure t = 0.105 s (right)

The realistic geometry was obtained in MeshLab in order to repair and smooth the model surface. Then, in order to design the FSI model, an open-source tool from Politecnico di Milano devised in the finite element library LifeV was used to generate the wall from the vessel lumen surface model previously obtained. After wall model surface decimation in MeshLab and file type conversion in SolidWorks 2010, the model was imported into COMSOL Multiphysics for numerical FSI modeling and simulating [28] (Fig. 3). The preliminary computations presented in this paper (see Fig. 4) were performed with the mesh shown in Fig. 3, which corresponds to a total number of degrees of freedom (DOF) of 2,908,848. The resulting coupled system of motion and fluid equations was solved using a finite element space discretization based on P1-P1 stabilized elements for the fluid, and P2 elements for the structure. The time discretization was based on the BDF of order 2, with a maximum time step of 5 × 10−3 s. The nonlinearities were tackled using Newton’s method and each linear iteration solved with the direct PARDISO solver. The simulations were made using Comsol Multiphysics 4.3b [29] R Xeon(R) CPU ES-2620 v2 @ 2.10 GHz processors. in a workstation with 12 Intel.

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3 Mathematical Model of Free-Flowing Leukocytes 3.1 Model Formulation One of the simplest constitutive relations to model complex flows of homogeneous non-Newtonian fluids is the rate-type Oldroyd-B model which captures the viscoelastic behavior of dilute polymeric solutions under general flow conditions, see, for instance, [30, 31]. Considering incompressible viscoelastic flows, the velocity, pressure, and stress must satisfy the conservation of mass and momentum equations, supplemented with a constitutive equation for the extra stress tensor. The unknowns in the Oldroyd-B model are the fluid velocity u, the pressure γ , and the extra stress contribution τ owing to the polymer behavior which satisfy the following governing equations: ρ

∂u + ρ(u · ∇)u − 2μs div (u) + ∇γ − div τ = ρf, ∂t div u = 0,

 τ +λ

(6) (7)



∂τ + (u · ∇)τ − τ (∇u) − (∇u)T τ − 2μp (u) = 0, ∂t

(8)

where ρ is the fluid density, f is an external force term, μs and μp are the solvent and 1 polymeric viscosities, λ the relaxation time, (u) = (∇u + ∇uT ) the strain rate 2 tensor, and τ (∇u) denotes the matrix–matrix product between ∇u and τ . Obviously, when λ = 0, the Oldroyd-B model reduces to the incompressible Navier–Stokes equations. We now consider two-phase flows, where the computational domain  contains two different immiscible incompressible phases, the intracellular liquid composed of a viscoelastic fluid with density ρc and viscosity μc , immersed in the extracellular liquid which is a Newtonian fluid (e.g., plasma) with density ρec and viscosity μec (see Fig. 5). Thus, for each time t ∈ [0, T ],  is a bounded, connected open set of R2 with boundary ∂, such that c (t) ∩ ec (t) = m (t), (t) = c (t) ∪ ec (t) and c (t) ∩ ec (t) = ∅ where the subscripts c, ec, and m denote cell, extracellular fluid, and membrane, respectively. To simplify, we do not consider in this model the rheology of the cell membrane surface. The physiological model that we propose for the deformation and motion of the cell Fig. 5 Scheme of the flow configuration

Extracellular fluid

Ωc

Ωec

Leukocyte Intracellular fluid

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leads to a coupled problem. Knowing u0 , τ0 , a force term f, the constant solvent and polymer viscosities μec > 0, μc > 0 and the constant relaxation time λ > 0 for the intracellular liquid (λ = 0 for the extracellular liquid), find the velocity u : i × (0, T ) → R2 , the pressure γ : i × (0, T ) → R (i = c, ec) and the extra stress τ : c × (0, T ) → R2×2 sym where the subscript sym denotes the symmetry of the extra stress tensor τ given in two dimensions by τ=

 τ11 τ12 , τ12 τ22

such that ⎧  ∂u ⎪ ⎪ ρec + (u · ∇)u − 2μec div (u) + ∇γ = ρec f, in ec × (0, T ), ⎪ ⎪ ⎪  ∂t ⎪ ⎪ ∂u ⎪ ⎪ ⎪ + (u · ∇)u − 2μc div (u) + ∇γ − div τ = ρc f, in c × (0, T ), ρc ⎪ ⎪ ∂t ⎪ ⎨ div u = (9)  0, in ec × (0, T ) and c × (0, T ), ⎪ ∂τ ⎪ T ⎪ ⎪ τ +λ + (u · ∇)τ − τ (∇u) − (∇u) τ − 2μc (u) = 0, in c × (0, T ), ⎪ ⎪ ∂t ⎪ ⎪ ⎪ ⎪ ⎪ u(·, 0) = u0 , in , ⎪ ⎪ ⎩ τ (·, 0) = τ0 , in c , σ · n = h, on in × (0, T ),

(10)

σ · n = 0, on out × (0, T ),

(11)

u = 0, on w × (0, T ),

(12)

[σ ] · n m = −ς κn m , [u] = 0, on m × (0, T ),

(13)

V m = u · n m , on m × (0, T ),

(14)

(τ · n) · n = 0, on m × (0, T ),

(15)

where in and out are the microchannel inlet and outlet boundaries, respectively, h is a given function, σ is the total stress tensor such that σ (γ , u, τ ) = −γ I + 2μec (u) + τ . At the walls (top and bottom of the microchannel), the fluid velocity is zero. The notation [·] is used to denote the jump of a quantity across m [32]. Let x ∈ m and f be a function defined in a neighborhood of m ; we define the jump across m by [f ] (x) = [f ] m (x) := lim (f (x − hn m (x)) − f (x + hn m (x))), h→0

(16)

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where n m (x) denotes the outward unit normal on m at x. The parameter ς in Eq. (13) is the surface tension coefficient, which is a material property of the twophase system. We restrict ourselves to the model where ς is a constant. In (13), κ denotes the mean curvature. The first condition in (13), the stress balance condition, is due to the conservation of momentum in a (small) material volume that intersects the interface. The second one, the continuity condition, is due to the fact that the fluids have different viscosities. The assumption that there is no change of phase leads to the dynamic interface condition (14). Recently, we proposed the mathematical analysis of a simplified time-dependent Oldroyd-B problem, where the convective term was removed and the density was assumed not to be constant [33].

3.2 Model Implementation In the present study, a level set method has been used to track the interface between the two-phase immiscible flows. The computational domain  is a 2D geometry representing one branch of the bifurcation of a post-capillary venule, of height 35 µm and length L = 100 µm, bounded from the bottom and the top by two curved lines modeling the endothelial surface (Fig. 6). The leukocyte initial shape is a circle of radius R = 3.75 µm [34]. The system (9) coupled with the level set function (17)  ∂φ ∇φ + v · ∇φ = γr ∇ · εr ∇φ − φ(1 − φ) , ∂t |∇φ|

(17)

is solved in Comsol Multiphysics [29], where v (m/s) is the velocity transporting the interface, γr (m/s) and εr (m) are reinitialization parameters, such that εr = hc /4, with hc the characteristic mesh size in the flow region passed by the interface and γr is the maximum flow velocity magnitude occurring in the model. The level set method [35] allows to simulate the flow of two immiscible fluids separated by a moving interface using a smooth continuous function φ. The value φ = 0.5 determines the position of the interface, then it changes smoothly from 0 to 1 in the transition layer near the interface. The region where φ < 0.5 is filled with plasma and φ > 0.5 corresponds to the intracellular region. The level set function is also used to determine the density and the dynamic viscosities as follows:

Fig. 6 Scheme of the computational domain. Attractive forces are applied at the points P1 and P2

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ρ = ρec + (ρc − ρec )φ μ = μec + (μc − μec )φ. The following coupled model is solved using the CFD Module of Comsol Multiphysics [29]  ρ

∂u + u · ∇u = −∇σ + Fst , ∂t div u = 0,

where the surface tension force is Fst = ∇ · [ς {I + (−nnT )}δ], I is the identity matrix, n is the unit outward normal at the interface, δ is a Dirac delta function, and σ denotes the total stress tensor such that: σ (p, u, τ ) = −pI + 2μc (u) + τ, n =

∇φ and δ = 6 |∇φ| |φ(1 + φ)| . |∇φ|

The fluid enters the microchannel by the left-hand side with normal inflow velocity U0 . The outlet is on the right-hand side with null pressure. At the top and the bottom curved boundaries, the flow satisfies no-slip boundary conditions. At the interface between the cell and the plasma, the normal component of the boundary stress contribution from the polymer is equal to zero, (τ · n) · n = 0. For the values of parameters, see Table 2.

3.3 Numerical Results A number of simulations for the resulting coupled system were performed using a finite element method solved in Comsol Multiphysics 4.3b [29], for an individual leukocyte and in the presence of another adherent cell. The computations and results were obtained for the mesh shown in Fig. 7, which corresponds to a total number of degrees of freedom (DOF) of 93,892. Additional simulations for the case of two cells were studied with a total number of 104,231 DOF. A force of attraction was applied in two points shown in Fig. 6. From these simulations, we have observed that Table 2 Parameters used for the numerical simulations provided by the Institute of Molecular Medicine (IMM–FML) in Lisbon, Portugal

Parameters ρec ρc μec μc σst

Value 1000 kg m−3 1063 kg m−3 0.001 Pa s 0.05 Pa s 20 µN/m

Description Plasma density Leukocyte density Plasma viscosity Leukocyte viscosity Surface tension

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Fig. 7 The numerical meshes used for the one cell and two cells simulations

the fluid 2 modeling the cell travels down through the fluid 1 modeling the plasma adheres to the curved plate representing the endothelial surface and starts rolling. Figure 8 shows the cell motion as well as its shape and the resulting disturbed flow, represented by the arrows of the velocity vectors at different time instants. The velocity plays an important role in the motion, deformation, and attraction of the cell to the endothelial wall. In fact, the results show that an increase in the inlet velocity accelerates the cell motion along the endothelial wall, and causes the bonds to break faster after reaching the regions of the attraction forces. We suggest that the hydrodynamic forces pushing the cell get higher as the inlet velocity is increasing. The influence of another cell in the motion, deformation, and attraction of a single leukocyte was also studied. We tested the case where the second leukocyte was already adhering to the endothelial wall. In Fig. 9, we have observed a reciprocal influence of one cell into the other. Whereas the free cell is crawling towards the endothelial surface, the shape of the adherent cell is deforming. On the other hand, the presence of the adherent cell slightly slows the motion at the beginning (t = 0.4 s) compared to the case of the individual leukocyte, then it has practically the same velocity. However, we note that the presence of the adherent cell may help in capturing the free leukocyte. In fact, in Fig. 9 the density of the recirculations, due to the perturbed flow, pushing the cell along the endothelial wall is less important in the case of the presence of the adherent cell, which could be explained by the promotion of the slow rolling that is needed and important to later promote the firm contact with the endothelial cells. In other words, the presence of an adherent cell can favor the slow rolling and consequent adhesion of another leukocyte.

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Fig. 8 Snapshots showing the volume fraction of fluid 2 (cell) and the arrow velocity field at t = 0.4 s, 0.8 s, 2.1 s, and 3.7 s with U0 = 12 µm/s

Fig. 9 The volume fraction of fluid 2 (left: for one cell, right: for two cells) and the streamlines at t = 0.8 s with U0 = 12 µm/s

4 Conclusion The immune system is very complex with various levels of interactions. It can give rise to complex coupled systems of PDEs (or ODEs) that exhibit highly nonlinear behavior, and can often be quite sensitive to small perturbations, hence the importance of using simplified models to capture the dynamics of the immune response

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when they fit well the available experimental data. Simplified mathematical models describing the FSI of blood with the arterial wall and atheroma plaque in a real carotid, on one side, and the coupled deformation-flow of one and two monocytes in a microchannel, on the other side, were developed and analyzed in this work and results have been discussed. Acknowledgements This work was partially supported by the Portuguese FCT—Fundação para a Ciência e a Tecnologia, through the project UID/Multi/04621/2013 of the CEMAT—Center for Computational and Stochastic Mathematics, IST, University of Lisbon.

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16. A.M. Gambaruto, J. Janela, A. Moura, A. Sequeira, Sensitivity of hemodynamics in a patient specific cerebral aneurysm to vascular geometry and blood rheology. Math. Biosci. Eng. 8, 409–423 (2011) 17. J. Janela, A. Moura, A. Sequeira, Absorbing boundary conditions for a 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. Int. J. Eng. Sci. 48, 1332–1349 (2010) 18. L. Formaggia, A. Moura, F. Nobile, On the stability of the coupling of 3D and 1D fluidstructure interaction models for blood flow simulations. ESAIM-Math. Model. Num. 41, 743–769 (2007) 19. S. Ramalho, A. Moura, A.M. Gambaruto, A. Sequeira, Sensitivity to out flow boundary conditions and level of geometry description for a cerebral aneurysm. Int. J. Numer. Method. Biomed. Eng. 28, 697–713 (2012) 20. P.G. Ciarlet, Mathematical Elasticity Three Dimensional Elasticity, vol. 1 (Elsevier, Amsterdam, 2004) 21. S. Le Floc’h, J. Ohayon, P. Tracqui, G. Finet, A.M. Gharib, R.L. Maurice, G. Cloutier, R.I. Pettigrew, Vulnerable atherosclerotic plaque elasticity reconstruction based on a segmentationdriven optimization procedure using strain measurements: theoretical framework. IEEE Trans. Med. Imaging 28, 1126–1137 (2009) 22. S. Glagov, E. Weisenberg, C.K. Zarins, R. Stankunavicius, G.J. Kolettis, Compensatory enlargement of human atherosclerotic coronary arteries. N. Engl. J. Med. 316, 1371–1375 (1987) 23. Z.Y. Li, S. Howarth, R.A. Trivedi, J.M. U-King-Im, M.J. Graves, A. Brown, L. Wang, J.H. Gillard, Stress analysis of carotid plaque rupture based on in vivo high resolution MRI. J. Biomech. 39, 2611–2622 (2006) 24. D. Tang, C. Yang, J. Zheng, P.K. Woodard, G.A. Sicard, J.E. Saffitz, C. Yuan, 3D MRI-based multicomponent FSI models for atherosclerotic plaques. Ann. Biomed. Eng. 32, 947–960 (2004) 25. J. Janela, A. Moura, A. Sequeira, A 3D non-Newtonian fluid-structure interaction model for blood flow in arteries. J. Comput. Appl. Math. 234, 2783–2791 (2010) 26. O. Kafi, N. El Khatib, J. Tiago, A. Sequeira, Numerical simulations of a 3D fluid-structure interaction model for blood flow in an athersclerotic artery. Math. Biosci. Eng. 14, 179–193 (2017) 27. N. El Khatib, Modélisation mathématique de l’athérosclérose, Ph.D thesis, Université Claude Bernard–Lyon 1 (2009). 28. D. Oliveira, Numerical Simulation of Dilatation Patterns of the Ascending Aorta in Aortopathies, MSc thesis, Instituto Superior Técnico, Lisbon (2016) 29. COMSOL Multiphysics, User’s Guide 4.3b, Licence 17073661 (2012) 30. M. Anand, J. Kwack, A. Masud, A new generalized Oldroyd-B model for blood flow in complex geometries. Int. J. Eng. Sci. 72, 78–88 (2013) 31. K.E. Jensen, P. Szabo, F. Okkels, Implementation of the Log-conformation Formulation for Two-dimensional Viscoelastic Flow (2016). Preprint. arXiv:1508.01041v2 32. S. Gross, A. Reusken, Numerical methods for two-phase incompressible flows. Springer Ser. Comput. Math. 40(1) (2011) 33. S. Boujena, O. Kafi, A. Sequeira, Mathematical study of a single leukocyte in microchannel flow. Math. Model. Nat. Phenom. 13 (2018) 34. A.S. Silva-Herdade, G. Andolina, C. Faggio, Â. Calado, C. Saldanha, Erythrocyte deformability – a partner of the inflammatory response. Microvasc. Res. 107, 34–38 (2016) 35. S. Osher, J.A. Sethian, Fronts propagating with curvature-dependent speed: algorithm based on Hamilton-Jacobi formations. J. Comput. Phys. 210, 225–246 (2012)

Modeling the Memory and Adaptive Immunity in Viral Infection Adnane Boukhouima, Khalid Hattaf, and Noura Yousfi

1 Introduction The adaptive immunity is a complex system that involves memory and plays a crucial role to defend our body against viral pathogens [1]. During viral infections, the adaptive immunity is generated by two specific immunities. The first is the humoral immunity based on antibodies which are produced by B lymphocytes and their function is to eradicate the viruses. The second is the cellular immunity involving cytotoxic T lymphocytes (CTL) that are programmed to recognize and kill the infected cells. To better understand the impact of adaptive immunity in viral infections such as human immunodeficiency virus (HIV), hepatitis B virus (HBV), and hepatitis C virus (HCV), several mathematical models have been formulated and studied [2–4]. However, the authors in these studies have not taken into account the latent state of the infected cells that is the period between the infection and the production of virions. Therefore, Elaiw [5] included the latently infected cells into a virus dynamics model with Beddington–DeAngelis functional response but he only considered the humoral response. In [6], the authors incorporated the cure of the infected cells in eclipse stage (latent state) and they replaced the Beddington– DeAngelis functional response by a specific functional response to study the global

A. Boukhouima () · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, Casablanca, Morocco K. Hattaf Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Sciences Ben M’sik, Hassan II University, Casablanca, Morocco Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_18

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stability of an HIV infection model with CTL immune response. All these models cited previously have been restricted to ordinary differential equations (ODEs) in which the memory effect is ignored. Fractional derivative is a generalization of integer derivative to an arbitrary order and it has been extensively applied to model different real phenomena in mechanics, image processing, viscoelasticity, bioengineering, finance, psychology, and control theory [7–13]. In biology, it has been deduced that the membranes of cells of biological organism have a fractional order electrical conductance and they are classified into groups of noninteger models [14]. Fractional differential equations (FDEs) are related to model systems with memory. Precisely, the fractional derivative at the current state t = t1 of system requires all the previous history, in other words, all the states from the initial time t = t0 to t = t1 . Furthermore, another feature of FDEs is that their stability region is larger than ODEs. Recently, many works have been done to study some models in virology by using FDEs. Pinto and Carvalho [15] developed a latency fractional order model for HIV dynamics and they observed that the variation of order in the fractional derivative induces a change in the patient’s epidemic status. In [16], Rihan et al. analyzed a fractional model for HCV dynamics in the presence of interferon-α (IFN) treatment. According to the numerical simulations and the real data, they also confirmed that the FDEs are better descriptors of HCV systems than ODEs. In the study of Arafa et al. [17], the authors compared between the results of the fractional order model, the results of the integer model, and the measured real data obtained from 10 patients during primary HIV infection and they proved that the results of the fractional order model give predictions to the plasma virus load of the patients better than those of the integer model. From the above biological and mathematical considerations, we propose a fractional order model to describe the effect of adaptive immunity in viral infection that is given by ⎧ ⎪ D α x(t) ⎪ ⎪ ⎪ ⎪ ⎪ D α l(t) ⎪ ⎪ ⎪ ⎨D α y(t) ⎪D α v(t) ⎪ ⎪ ⎪ ⎪ ⎪ D α w(t) ⎪ ⎪ ⎪ ⎩ α D z(t)

= λ − dx − f (x, v)v + ρl, = f (x, v)v − (m + ρ + γ )l, = γ l − ay − pyz, = ky − μv − qvw,

(1)

= gvw − hw, = cyz − bz,

where D α denotes the fractional derivative in the sense of Caputo of order α, with 0 < α ≤ 1, defined for an arbitrary function ϕ by 1 D ϕ(t) = (1 − α)

α

0

t



ϕ (u) du. (t − u)α

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The variables x(t), l(t), y(t), v(t), w(t), and z(t) represent the concentrations of susceptible host cells, latently infected cells (infected cells which are not yet able to produce virions), productive infected cells, free virus particles, antibodies, and CTL cells at time t, respectively. Susceptible host cells are assumed to be produced at a constant rate λ, die at the rate dx, and become infected by virus at the rate f (x, v)v. Latently infected cells die at the rate ml, are converted to productive infected cells at a rate γ l, and return to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus at the rate ρl. Productive infected cells die at the rate ay and are killed by CTL cells at the rate pyz. Free virus particles are produced from productive infected cells at the rate ky, cleared at the rate μv and are neutralized by antibodies at the rate qvw. Antibodies are activated against virus at the rate gvw and die at the rate hw. CTL cells develop in response to the productive infected cells at the rate cyz and die at the rate bz. It is important to note when α = 1, system (1) becomes a model with ordinary derivative. In addition, we adopt the Caputo’s fractional derivative for the reason that the initial conditions depend only on integer order derivative. On the other hand, the Hattaf–Yousfi functional response used in (1) has the form βx f (x, v) = , where α0 , α1 , α2 , α3 ≥ 0 are the saturation α0 + α1 x + α2 v + α3 xv factors measuring the psychological or inhibitory effect. This functional response generalizes many types used by several authors. For example, when α0 = 1, we get the specific functional response used in [6]. If α0 = 1 and α3 = α1 α2 , we obtain the Beddington–DeAngelis functional response used in [5]. If α0 = 1 and α1 = α2 = α3 = 0, we get the bilinear incidence rate used in [2]. When α0 = 1, α1 = α3 = 0, or α2 = α3 = 0, we obtain the saturated incidence rate. The rest of this paper is organized as follows: In the next section, we give some basic results concerning the existence, positivity, and boundedness of solutions. The existence of equilibria and their global stability are investigated in Sects. 3 and 4. Numerical simulations of the theoretical results are presented in Sect. 5. This paper ends with a conclusion in Sect. 6.

2 Basic Properties System (1) describes the evolution of cells, then we must prove that the cell-numbers are non-negative and bounded for all t ≥ 0. For biological reasons, we assume that the initial conditions of (1) satisfy x(0) ≥ 0, l(0) ≥ 0, y(0) ≥ 0, v(0) ≥ 0, w(0) ≥ 0, z(0) ≥ 0.

(2)

Theorem 2.1 For any initial conditions satisfying (2), system (1) has a unique solution on [0, +∞). Moreover, this solution remains non-negative and bounded for all t ≥ 0.

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Proof First, system (1) can be written as follows: D α X(t) = F (X), where

⎞ ⎛ ⎞ λ − dx − f (x, v)v + ρl x(t) ⎜f (x, v)v − (m + ρ + γ )l ⎟ ⎜ l(t) ⎟ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ γ l − ay − pyz ⎟ ⎜ ⎜ y(t) ⎟ X(t) = ⎜ ⎟. ⎟ and F (X) = ⎜ ⎟ ⎜ ⎜ v(t) ⎟ ky − μv − qvw ⎟ ⎜ ⎟ ⎜ ⎠ ⎝ ⎝w(t)⎠ gvw − hw cyz − bz z(t) ⎛

Let ⎛ ⎛ ⎞ ⎛ ⎞ −d ρ 0 0 0 0 λ 0000 0 ⎜ 0 −(m + ρ + γ ) 0 0 0 0 ⎟ ⎜0⎟ ⎜0 0 0 0 0 ⎜ ⎜ ⎟ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ γ −a 0 0 0 ⎟ ⎜ 0 ⎜0⎟ ⎜0 0 0 0 0 ζ = ⎜ ⎟,A = ⎜ ⎟,C = ⎜ ⎜ 0 ⎜0⎟ ⎜0 0 0 0 −q 0 k −μ 0 0 ⎟ ⎜ ⎜ ⎟ ⎜ ⎟ ⎝ 0 ⎝0⎠ ⎝0 0 0 0 g 0 0 0 −h 0 ⎠ 0 0000 0 0 0 0 0 0 −b ⎞ ⎛ 00 0 000 ⎜0 0 0 0 0 0⎟ ⎟ ⎜ ⎟ ⎜ ⎜0 0 −p 0 0 0⎟ and D = ⎜ ⎟. ⎜0 0 0 0 0 0⎟ ⎟ ⎜ ⎝0 0 0 0 0 0⎠ 00 c 000 We discuss four cases: • If α0 = 0, F (X) can be formulated as follows: F (X) = ζ + AX +

α0 v(B0 + C)X + zDX, α0 + α1 x + α2 v + α3 xv

where ⎛ β ⎜− α0 ⎜ β ⎜ ⎜ ⎜ α0 B0 = ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0 0

⎞ 0 0 0 0 0⎟ ⎟ ⎟ 0 0 0 0 0⎟ ⎟ ⎟ 0 0 0 0 0⎟ . ⎟ 0 0 0 0 0⎟ ⎟ 0 0 0 0 0⎠ 00000

⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 0⎠ 0

Modeling the Memory and Adaptive Immunity in Viral Infection

Hence, F (X) ≤ ζ + ( A + v B0 + C + z D ) X . • If α1 = 0, we can write F (X) in the form F (X) = ζ + AX + where

α1 x B1 X + vCX + zDX, α0 + α1 x + α2 v + α3 xv ⎛ ⎜0 ⎜ ⎜ ⎜0 ⎜ B1 = ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0 0

β α1 β α1 0 0 0 0

00− 00 0 0 0 0

0 0 0 0

⎞ 0 0⎟ ⎟ ⎟ 0 0⎟ ⎟ ⎟ 0 0⎟ . ⎟ 0 0⎟ ⎟ 0 0⎠ 00

Then F (X) ≤ ζ + ( A + B1 + v C + z D ) X . • If α2 = 0, we have F (X) = ζ + AX +

α2 v B2 X + vCX + zDX, α0 + α1 x + α2 v + α3 xv

where ⎛ β ⎜− α2 ⎜ β ⎜ ⎜ ⎜ α2 B2 = ⎜ ⎜ 0 ⎜ ⎜ 0 ⎜ ⎝ 0 0

0000 0000 00 00 00 00

0 0 0 0

0 0 0 0

⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ . ⎟ 0⎟ ⎟ 0⎠ 0

Consequently, F (X) ≤ ζ + ( A + B2 + v C + z D ) X . • If α3 = 0, we obtain F (X) = ζ + AX +

α3 xv B3 X + vCX + zDX, α0 + α1 x + α2 v + α3 xv

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where ⎛ ⎞ β − ⎜ α3 ⎟ ⎜ β ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ α3 ⎟ ⎜ B3 = ⎜ 0 ⎟ ⎟. ⎜ ⎟ ⎜ 0 ⎟ ⎜ ⎟ ⎝ 0 ⎠ 0 Then F (X) ≤ ζ + ( A + B3 + v C + z D ) X . Thus, the conditions of Lemma 2.4 in [18] are satisfied. Then system (1) has a unique solution on [0, +∞). Next, we show that this solution is non-negative. From (1), we have D α x(t)|x=0 = λ + ρl ≥ 0, D α l(t)|l=0 = f (x, v)v ≥ 0, D α y(t)|y=0 = γ l ≥ 0, D α v(t)|v=0 = ky ≥ 0, D α w(t)|w=0 = 0 ≥ 0, D α z(t)|z=0 = 0 ≥ 0. As in [18, Theorem 2.7], we deduce that the solution of (1) is non-negative. Finally, we prove that the solution is bounded. Defining the function T (t) as T (t) = x(t) + l(t) + y(t) +

aq p a v(t) + w(t) + z(t). 2k 2kg c

So, we have D α T (t) = D α x(t) + D α l(t) + D α y(t) + = λ − dx(t) − ml(t) − ≤ λ − δT (t),

a α aq α p D v(t) + D w(t) + D α z(t) 2k 2kg c

a aμ aqh pb y(t) − v(t) − w(t) − z(t) 2 2k 2kg c

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where δ = min{d, m, a2 , μ, h, b}. Therefore, T (t) ≤ T (0)Eα (−δt α ) +

λ [1 − Eα (−δt α )]. δ

Since 0 ≤ Eα (−δt α ) ≤ 1, we get T (t) ≤ T (0) +

λ . δ 

This completes the proof.

3 Existence of Equilibria It is easy to see that system (1) always has an infection-free equilibrium λ E0 ( , 0, 0, 0, 0, 0). Therefore, the basic reproduction number of our system (1) d is given by R0 =

kβλγ . aμ(m + ρ + γ )(dα0 + λα1 )

Biologically, this number represents the average number of secondary infections produced by one productive infected cell during the period of infection when all cells are uninfected. To investigate the existence of other equilibria, we solve the following system: λ − dx − f (x, v)v + ρl = 0,

(3)

f (x, v)v − (m + ρ + γ )l = 0,

(4)

γ l − ay − pyz = 0,

(5)

ky − μv − qvw = 0,

(6)

gvw − hw = 0,

(7)

cyz − bz = 0.

(8)

From Eqs. (7)–(8), we get w = 0 or v =

h b and z = 0 or y = . g c

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If w = 0 and z = 0. By Eqs. (3)–(6), we have l = v=

kγ (λ − dx) , and aμ(m + γ )

γ (λ − dx) λ − dx ,y = , m+γ a(m + γ )

 aμ(m + ρ + γ ) kγ (λ − dx) . = f x, aμ(m + γ ) kγ

(9)

λ λ . Hence, there is no equilibrium when x > . d d 

λ We define the function h1 on 0, by d

Since l ≥ 0, we have x ≤

h1 (x) = f

 aμ(m + ρ + γ ) kγ (λ − dx) . − x, aμ(m + γ ) kγ

∂f kγ d aμ(m + ρ + γ ) ∂f  We have h1 (0) = − < 0, h1 (x) = − > 0, and kγ ∂x aμ(m + γ ) ∂v  λ aμ(m + ρ + γ ) h1 (R0 − 1). = d kγ Hence  if R0 > 1, there exists an equilibrium E1 (x1 , l1 , y1 , v1 , 0, 0) satisfying λ λ − dx1 γ (λ − dx1 ) kγ (λ − dx1 ) , l1 = x1 ∈ 0, , y1 = , v1 = . d m+γ a(m + γ ) aμ(m + γ ) λ − dx h , If w = 0 and z = 0, we have v = and by Eqs. (3)–(5), we obtain l = g m+γ γ (λ − dx) kγ g(λ − dx) μ y= ,w= − , and a(m + γ ) aqh(m + γ ) q f

 g(m + ρ + γ ) h (λ − dx). = x, g h(m + γ )

Since l ≥ 0, y ≥ 0, and w ≥ 0, we have x ≤

(10)

λ ahμ(m + γ ) − . Hence, there is no d dkgγ

λ ahμ(m + γ ) − . d dkgγ

λ ahμ(m + γ ) We define the function h2 on 0, − by d dkgγ

equilibrium if x >

 h2 (x) = f

h x, g

−

g(m + ρ + γ ) (λ − dx). h(m + γ )

gλ(m + ρ + γ ) ∂f gd(m + ρ + γ )  We have h2 (0) = − < 0, h2 (x) = + > 0, and h(m + γ ) ∂x h(m + γ )   λ ahμ(m + γ ) λ ahμ(m + γ ) h2 − − = h1 . d dkgγ d dkgγ

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Let us introduce the reproduction number for humoral immunity by R1w =

gv1 , h

which represents the average number of the activated antibodies by virus during the period of infection in the absence of the cellular immunity. λ ahμ(m + γ ) and Then, if R1w < 1, we have x1 > − d dkgγ  h2

λ ahμ(m + γ ) − d dkgγ

< h1 (x1 ) = 0.

Therefore, there is no equilibrium when R1w < 1. λ ahμ(m + γ ) If R1w > 1, then x1 < − and d dkgγ  h2

λ ahμ(m + γ ) − d dkgγ

> h1 (x1 ) = 0.

w So, an equilibrium E2 (x2 , l2 , y2 , v2 , w2 , 0) satisfying x2 ∈  if R1 > 1, there exists λ − dx2 γ (λ − dx2 ) h λ ahμ(m + γ ) , y2 = , v2 = , and w2 = , l2 = 0, − d dkgγ m+γ a(m + γ ) g kγ g(λ − dx1 ) μ − . When R1w = 1, we get E1 = E2 . aqh(m + γ ) q kb b and from Eq. (6), we have v = . Using If w = 0 and z = 0, then y = c μc λ − dx cγ (λ − dx) a Eqs. (3)–(5), we get l = ,z= − , and m+γ pb(m + γ ) p

 f

kb x, μc

Since z ≥ 0, we have x ≤ λ ab(m + γ ) − . d dcγ

=

μc(m + ρ + γ ) (λ − dx). kb(m + γ )

λ ab(m + γ ) − . Then, there is no equilibrium if x > d dcγ

 λ ab(m + γ ) We define the function h3 on 0, − by d dcγ h3 (x) = f

(11)

 μc(m + ρ + γ ) kb (λ − dx). − x, μc kb(m + γ )

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dcμ(m + ρ + γ ) cμλ(m + ρ + γ ) ∂f  < 0, h3 (x) = + > 0, We have h3 (0) = − kb(m + γ ) ∂x kb(m + γ )   λ ab(m + γ ) λ ab(m + γ ) and h3 − − = h1 . d dcγ d dcγ Now, we introduce the response reproduction number for cellular immunity as follows: R1z =

cy1 , b

which represents the average number of the activated CTL cells by the productive infected cells during the period of infection when the humoral immunity has not been started. λ ab(m + γ ) If R1z < 1, then x1 > − and d dcγ  h3

λ ab(m + γ ) − d dcγ

< h1 (x1 ) = 0.

Thus, there is no equilibrium when R1z < 1. λ ab(m + γ ) and If R1z > 1, then x1 < − d dcγ  h3

λ ab(m + γ ) − d dcγ

> h1 (x1 ) = 0.

z Hence,  when R1 > 1, there exists an equilibrium E3 (x3 , l3 , y3 , v3 , 0, z3 ) satisfying λ ab(m + γ ) λ − dx3 b kb , y3 = , v3 = , and , l3 = x3 ∈ 0, − d dcγ m+γ c cμ a cγ (λ − dx3 ) z3 = − . If R1z = 1, we get E3 = E1 . bp(m + γ ) p h b If w = 0 and z = 0, we have y = , v = . From Eqs. (3)–(5), we obtain c g cγ (λ − dx) a λ − dx ,z= − , and l= m+γ pb(m + γ ) p

f

 h g(m + ρ + γ ) x, (λ − dx). = g h(m + γ )

Since z ≥ 0, we have x ≤ x>

λ ab(m + γ ) − . d dcγ

(12)

λ ab(m + γ ) − . Hence, there is no equilibrium if d dcγ

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 λ ab(m + γ ) by We introduce the function h4 on 0, − d dcγ  h4 (x) = f

h x, g

−

g(m + ρ + γ ) (λ − dx). h(m + γ )

gλ(m + ρ + γ ) ∂f dg(m + ρ + γ )  We have h4 (0) = − < 0, h4 (x) = + > 0, and h(m + γ )  h(m +γ ) ∂x λ ab(m + γ ) λ ab(m + γ ) h4 − − = h2 . d dcγ d dcγ Now, in addition to R1z , we define the reproduction number for cellular immunity in competition by R2z =

cy2 , b

which represents the average number of the activated CTL cells by productive infected cells during the period of infection in the presence of humoral immunity. λ ab(m + γ ) and If R2z < 1, then x2 > − d dcγ  h4

λ ab(m + γ ) − d dcγ

< h2 (x2 ) = 0.

Therefore, there is no equilibrium when R2z < 1. λ ab(m + γ ) and If R2z > 1, then x2 < − d dcγ  h4

λ ab(m + γ ) − d dcγ

> h2 (x2 ) = 0.

 λ ab(m + γ ) . Hence, Eq. (12) has a unique solution x4 ∈ 0, − d dcγ gkb μ By Eq. (6), we get w = − . As w ≥ 0, in addition to R1w , we define the qch q reproduction number for humoral immunity in competition as R2w =

gv3 , h

which represents the average number of the activated antibodies by virus during the period of infection when the cellular immunity is established.

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w Then we conclude that when R2z > 1 and an equilibrium  R2 > 1, there exists λ ab(m + γ ) λ − dx4 , l3 = E4 (x4 , l4 , y4 , v4 , w4 , z4 ) satisfying x4 ∈ 0, − , d dcγ m+γ b h cγ (λ − dx4 ) u a y3 = , v3 = , z4 = − , and w4 = (R2w − 1). c g bp(m + γ ) p q Summarizing the above discussions, we get the following theorem.

Theorem 3.1 (i) If R0 ≤ 1, then system (1) has an infection-free equilibrium of the form λ E0 (x0 , 0, 0, 0, 0, 0), where x0 = . d (ii) If R0 > 1, then system (1) has an immune-free infection equilibrium of the form  λ λ − dx1 γ (λ − dx1 ) , y1 = , , l1 = E1 (x1 , l1 , y1 , v1 , 0, 0), where x1 ∈ 0, d m+γ a(m + γ ) kγ (λ − dx1 ) and v1 = . aμ(m + γ ) (iii) If R1w > 1, then system (1) has an infection equilibrium with only humoral immunity  of the form E2 (x2 , l2 , y2 , v2 , w2 , 0), where λ ahμ(m + γ ) λ − dx2 γ (λ − dx2 ) h , y2 = , v2 = , and , l2 = x2 ∈ 0, − d dkgγ m+γ a(m + γ ) g kγ g(λ − dx1 ) μ w2 = − . aqh(m + γ ) q z (iv) If R1 > 1, then system (1) has an infection equilibrium with only cellular immunity of the form E3 (x3 , l3 , y3 , v3 , 0, z3 ), where x3 ∈  λ ab(m + γ ) λ − dx3 b kb 0, − , y3 = , v3 = , and z3 = , l3 = d dcγ m+γ c cμ a cγ (λ − dx3 ) − . bp(m + γ ) p (v) If R2z > 1 and R2w > 1, then system (1) has an infection equilibrium with both humoral and cellular immunity of the form E4 (x4 , l4 , y4 , v4 , w4 , z4 ), λ ab(m + γ ) λ − dx4 b h where x4 ∈ 0, − , y3 = , v3 = , , l3 = d dcγ m+γ c g a cγ (λ − dx4 ) u w − , and w4 = (R2 − 1). z4 = bp(m + γ ) p q

4 Global Stability of Equilibria In this section, we study the global stability of the equilibria. Theorem 4.1 If R0 ≤ 1, then the infection-free equilibrium E0 is globally asymptotically stable and it becomes unstable if R0 > 1.

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Proof Define the Lyapunov functional L0 (t) as follows: L0 (t) =

 ρα0 x α0 + x0  (x − x0 + l)2 α0 + α1 x0 x0 2(d + m + γ )(α0 + α1 x0 )x0 +l+

a(m + ρ + γ ) aq(m + ρ + γ ) m+ρ+γ y+ v+ w γ kγ kgγ

p(m + ρ + γ ) z, cγ

+

where (x) = x − 1 − ln(x) for x > 0. Calculating the fractional derivative of L0 (t) along solutions of system (1) and using the results in [19], we get D α L0 (t) ≤

 x0 α α0 D x 1− α0 + α1 x0 x ρα0 + (x − x0 + l)(D α x + D α l) + D α l (d + m + γ )(α0 + α1 x0 )x0 +

m+ρ+γ α a(m + ρ + γ ) α aq(m + ρ + γ ) α D y+ D v+ D w γ kγ kgγ

+

p(m + ρ + γ ) α D z. cγ

Using λ = dx0 , we obtain  α0 x0 dα0 (x − x0 )2 f (x, v)v − 1− (α0 + α1 x0 )x α0 + α1 x0 x  ρα0 ρα0 (x − x0 + l) x0 + l 1− (d(x0 − x) α0 + α1 x0 x (d + m + γ )(α0 + α1 x0 )x0

D α L0 (t) ≤ −

aqh(m + ρ + γ ) aμ(m + ρ + γ ) v− w kγ kgγ  bp(m + ρ + γ ) dα0 (x − x0 )2 1 ρ − z≤− + cγ x (d + m + γ )x0 (α0 + α1 x0 ) −(m + γ )l) + f (x, v)v −

−

ρα0 (m + γ )l 2 ρα0 (x − x0 )2 l aμ(m + ρ + γ ) − + (d + m + γ )(α0 + α1 x0 )x0 (α0 + α1 x0 )xx0 kγ

× (R0 − 1)v −

bp(m + ρ + γ ) aqh(m + ρ + γ ) w− z. kgγ cγ

Hence if R0 ≤ 1, then D α L0 (t) ≤ 0. In addition, the equality holds if and only if x = x0 , l = 0, y = 0, z = 0, w = 0, and (R0 − 1)v = 0. If R0 < 1, then v = 0. If R0 = 1, from (1), we get f (x0 , v)v = 0 which implies that v = 0. Consequently, the largest invariant set of {(x, l, y, v, w, z) ∈ R6+ : D α L0 (t) = 0}

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is the singleton {E0 }. Therefore, by the LaSalle’s invariance principle [20], E0 is globally asymptotically stable. The Jacobian matrix of system (1) at any equilibrium E(x, l, y, v, w, z) is given by ⎛ ⎞ ∂f ∂f −d − v ρ 0 − v − f (x, v) 0 0 ⎜ ⎟ ∂x ∂v ⎜ ∂f ⎟ ∂f ⎜ ⎟ v −(m + ρ + γ ) 0 v + f (x, v) 0 0 ⎜ ∂x ⎟ ∂v ⎜ ⎟ ⎜ ⎟. 0 γ −a − pz 0 0 −py ⎜ ⎟ ⎜ 0 0 k −μ − qw −qv 0 ⎟ ⎜ ⎟ ⎝ 0 0 0 gw gv − h 0 ⎠ 0 0 cz 0 0 cy − b (13) We recall that E is locally asymptotically stable if all the eigenvalues ξi of (13) satisfy the following condition: |arg(ξi )| >

απ . 2

From (13), the characteristic equation at E0 is given as follows: (d + ξ )(h + ξ )(b + ξ )g0 (ξ ) = 0,

(14)

where g0 (ξ ) = (m + ρ + γ + ξ )(a + ξ )(μ + ξ ) −

kγβλ . dα0 + α1 λ

Obviously, Eq. (14) has the roots ξ1 = −d, ξ2 = −h, and ξ3 = −b. If R0 > 1, we have g0 (0) = aμ(m+ρ+γ )(1−R0 ) < 0 and lim g0 (ξ ) = +∞. ξ →+∞

Then, there exists ξ ∗ > 0 satisfying g0 (ξ ∗ ) = 0. In addition, we have |arg(ξ ∗ )| = απ 0< . Consequently, when R0 > 1, E0 is unstable.  2 Theorem 4.2 (i) The immune-free infection equilibrium E1 is globally asymptotically stable if R0 > 1, R1w ≤ 1, R1z ≤ 1, and R0 ≤ 1 +

(m + ρ + γ )[α0 adμ(m + ρ) + dkλγ α2 ] + kργ α3 λ2 . aρμ(m + ρ + γ )(α0 d + λα1 )

(ii) When R1w > 1 or R1z > 1, E1 is unstable.

(15)

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Proof Define the Lyapunov functional L1 (t) as follows: L1 (t) =

  x l α0 + α2 v1 + l1  x1  α0 + α1 x1 + α2 v1 + α3 x1 v1 x1 l1 ρ(α0 + α2 v1 ) (x − x1 + l − l1 )2 2(d + m + γ )(α0 + α1 x1 + α2 v1 + α3 x1 v1 )x1   a(m + ρ + γ ) y v m+ρ+γ + y1  v1  + γ y1 kγ v1 +

+

p(m + ρ + γ ) aq(m + ρ + γ ) w+ z. kgγ cγ

Calculating the fractional derivative of L1 (t) and using λ = dx1 + (m + γ )l1 , f (xi , vi ) = f (x1 , v1 )v1 = (m + ρ + γ )l1 , γ l1 = ay1 , ky1 = μv1 , and 1 − f (x, vi )  α0 + α2 vi xi ∀1 ≤ i ≤ 4, we get 1− α0 + α1 xi + α2 vi + α3 xi vi x  f (x1 , v1 ) D α L1 (t) ≤ d 1 − (x1 − x) f (x, v1 )  f (x1 , v1 ) v f (x, v) + + (m + ρ + γ )l1 1 − f (x, v1 ) v1 f (x, v1 )   l1 f (x, v)v ly1 + (m + ρ + γ )l1 1 − + (m + ρ + γ )l1 1 − lf (x1 , v1 )v1 l1 y   v yv1 f (x1 , v1 ) + ρ(l − l1 ) 1 − + (m + ρ + γ )l1 1 − − v1 y1 v f (x, v1 )   ρ(α0 + α2 v1 ) d(x − x1 )2 + (m + γ )(l − l1 )2 + (d + m + γ )(x − x1 )(l − l1 ) − (d + m + γ )(α0 + α1 x1 + α2 v1 + α3 x1 v1 )x1 bp(m + ρ + γ )  cy1 aqh(m + ρ + γ )  gv1 −1 w+ −1 z + kgγ h cγ b  2 (α0 + α2 v1 )(x − x1 ) dρx ≤− (dx1 − ρl1 ) + ρl + xx1 (α0 + α1 x1 + α2 v1 + α3 x1 v1 ) d +m+γ ρ(α0 + α2 v1 )(m + γ )(l − l1 )2 (m + ρ + γ )(α0 + α1 x1 + α2 v1 + α3 x1 v1 )x1  f (x1 , v1 ) l1 f (x, v)v yv1 f (x, v1 ) ly1 − − − + (m + ρ + γ )l1 5 − − f (x, v1 ) lf (x1 , v1 )v1 l1 l y1 v f (x, v)  v f (x, v)v f (x, v1 ) + (m + ρ + γ )l1 −1 − + + v1 f (x, v1 )v1 f (x, v) −

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  aqh(m + ρ + γ )  w bp(m + ρ + γ )  z R1 − 1 w + R1 − 1 z kgγ cγ  (α0 + α2 v1 )(x − x1 )2 dρx ≤− (dx1 − ρl1 ) + ρl + xx1 (α0 + α1 x1 + α2 v1 + α3 x1 v1 ) d +m+γ

+

ρ(α0 + α2 v1 )(m + γ )(l − l1 )2 (m + ρ + γ )(α0 + α1 x1 + α2 v1 + α3 x1 v1 )x1  f (x1 , v1 ) l1 f (x, v)v yv1 f (x, v1 ) ly1 − − − + (m + ρ + γ )l1 5 − − f (x, v1 ) lf (x1 , v1 )v1 l1 l y1 v f (x, v) −

(α0 + α1 x)(α2 + α3 x)(v − v1 )2 v1 (α0 + α1 x + α2 v + α3 xv)(α0 + α1 x + α2 v1 + α3 xv1 )   bp(m + ρ + γ )  z aqh(m + ρ + γ )  w R1 − 1 w + R1 − 1 z. + kgγ cγ − (m + ρ + γ )l1

By the arithmetic–geometric inequality, we have 5−

li f (x, v)v yvi f (x, vi ) lyi f (xi , vi ) − − − ≤ 0 ∀1 ≤ i ≤ 4. − f (x, vi ) lf (xi , vi )vi li y yi v f (x, v)

Since R1w ≤ 1 and R1z ≤ 1, consequently D α L1 (t) ≤ 0 if dx1 ≥ ρl1 . It is easy to α see that this condition is equivalent to  (15). Furthermore,   D L1 (t) = 0 if and only if x = x1 , l = l1 , y = y1 , v = v1 , R1w − 1 w = 0, R1z − 1 z = 0. We discuss two cases: If R1w < 1 and R1z < 1, then w = 0 and z = 0. If R1w = 1 or R1z = 1. We get from (1) that D α y1 = 0 = γ l1 − ay1 − py1 z or D α v1 = 0 = ky1 − μv1 − qv1 w. Then w = 0 or z = 0. Hence, the largest invariant set of {(x, l, y, v, w, z) ∈ R6+ : D α L1 (t) = 0} is the singleton {E1 }. By the LaSalle’s invariance principle, E1 is globally asymptotically stable. At E1 , the characteristic equation of (13) is given as follows: (cy1 − b − ξ )(gv1 − h − ξ )g1 (ξ ) = 0,

(16)

where   ∂f ∂f   −d − v1 − ξ ρ 0 − v1 − f (x1 , v1 )   ∂x ∂v   ∂f ∂f  v1 v1 + f (x1 , v1 )  . −(m + ρ + γ ) − ξ 0 g1 (ξ ) =  ∂x ∂v     0 γ −a − ξ 0     0 0 k −μ − ξ We can easily see that Eq. (16) has the roots ξ1 = cy1 − b and ξ2 = gv1 − h. Then, when R1w > 1 or R1z > 1, we have ξ1 > 0 or ξ1 > 0. In this case, E1 is unstable. 

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Theorem 4.3 (i) The infection equilibrium with only humoral immunity E2 is globally asymptotically stable if R1w > 1, R2z ≤ 1, and ρβh ≤ d(m + ρ + γ )(α0 g + α2 h) + ρλ(α1 g + α3 h).

(17)

(ii) If R2z > 1, then E2 is unstable. Proof Consider the following Lyapunov functional: L2 (t) =

  x l α0 + α2 v2 + l2  x2  α0 + α1 x2 + α2 v2 + α3 x2 v2 x2 l2 ρ(α0 + α2 v2 ) (x − x2 + l − l2 )2 2(d + m + γ )(α0 + α1 x2 + α2 v2 + α3 x2 v2 )x2   a(m + ρ + γ ) y v m+ρ+γ + y2  v2  + γ y2 kγ v2  p(m + ρ + γ ) aq(m + ρ + γ ) w + w2  z. + kgγ w2 cγ +

Computing the fractional derivative of L2 (t) and using λ = dx2 + (m + γ )l2 , h f (x2 , v2 )v2 = (m + ρ + γ )l2 , γ l2 = ay2 , ky2 = (μ + qw2 )v2 , and v2 = , we get g  f (x2 , v2 ) (x2 − x) D L2 (t) ≤ d 1 − f (x, v2 )  f (x2 , v2 ) f (x, v)v + + (m + ρ + γ )l2 1 − f (x, v2 ) f (x, v2 )v2   l2 f (x, v)v ly2 + (m + ρ + γ )l2 1 − + (m + ρ + γ )l2 1 − lf (x2 , v2 )v2 l2 y   v yv2 f (x2 , v2 ) + ρ(l − l2 ) 1 − + (m + ρ + γ )l2 1 − − v2 y2 v f (x, v2 )   2 2 ρ(α0 + α2 v2 ) d(x − x2 ) + (m + γ )(l − l2 ) + (d + m + γ )(x − x2 )(l − l2 ) − (d + m + γ )(α0 + α1 x2 + α2 v2 + α3 x2 v2 )x2  bp(m + ρ + γ )  z R2 − 1 z + cγ  (α0 + α2 v2 )(x − x2 )2 dρx (dx2 − ρl2 ) + ρl + ≤− xx2 (α0 + α1 x2 + α2 v2 + α3 x2 v2 ) d +m+γ α

−

ρ(α0 + α2 v2 )(m + γ )(l − l2 )2 (m + ρ + γ )(α0 + α1 x2 + α2 v2 + α3 x2 v2 )x2

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 f (x2 , v2 ) l2 f (x, v)v yv2 f (x, v2 ) ly2 − − − + (m + ρ + γ )l2 5 − − f (x, v2 ) lf (x2 , v2 )v2 l2 l y2 v f (x, v) (α0 + α1 x)(α2 + α3 x)(v − v2 )2 v2 (α0 + α1 x + α2 v + α3 xv)(α0 + α1 x + α2 v2 + α3 xv2 )  bp(m + ρ + γ )  z R2 − 1 z. + cγ − (m + ρ + γ )l2

Hence, D α L2 (t) ≤ 0 when R2z ≤ 1 and dx2 ≥ ρl2 . This last condition is equivalent to (17). In addition, D α L2 (t) = 0 if x = x2 , l = l2 , y = y2 , v = v2 , and (R2z − 1)z = 0. When R2z < 1, then z = 0. If R2z = 1, from (1), we have D α y2 = 0 = γ l2 − ay2 − py2 z, then z = 0. Further, D α v2 = 0 = ky2 − μv2 − qv2 w, then w = w2 . Consequently, the largest invariant set of {(x, l, y, v, w, z) ∈ R6+ : D α L2 (t) = 0} is the singleton {E2 }. By the LaSalle’s invariance principle, E2 is globally asymptotically stable. Evaluating the characteristic equation of (13) at E2 , we get (cy2 − b − ξ )g2 (ξ ) = 0,

(18)

where   ∂f   ∂f −d −  v v − ξ ρ 0 − − f (x , v ) 0 2 2 2 2   ∂x ∂v   ∂f ∂f   −(m + ρ + γ ) − ξ 0 0 v2 v2 + f (x2 , v2 )   ∂x ∂v . g2 (ξ ) =   0 γ −a − ξ 0 0      0 0 k −μ − qw2 − ξ −qv2     0 0 0 gw2 gv2 − h − ξ 

Equation (18) has the root ξ1 = cy2 − b. Then, when R2z > 1, we have ξ1 > 0. Consequently, E2 is unstable.  Theorem 4.4 (i) The infection equilibrium with only cellular immunity E3 is globally asymptotically stable if R1z > 1, R2w ≤ 1, and kbρβ ≤ d(m + ρ + γ )(α0 μc + α2 kb) + λρ(α1 μc + α3 kb). (ii) If R2w > 1, E3 is unstable.

(19)

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Proof We construct the Lyapunov functional as follows:   x l α0 + α2 v3 + l3  x3  α0 + α1 x3 + α2 v3 + α3 x3 v3 x3 l3

L3 (t) =

ρ(α0 + α2 v3 ) (x − x3 + l − l3 )2 2(d + m + γ )(α0 + α1 x3 + α2 v3 + α3 x3 v3 )x3   (a + pz3 )(m + ρ + γ ) y v m+ρ+γ + y3  v3  + γ y3 kγ v3  q(a + pz3 )(m + ρ + γ ) p(m + ρ + γ ) z . + w+ z3  kgγ cγ z3 +

Then, we have D α L3 (t) ≤

  x3 α α0 + α2 v3 l3 D x+ 1− 1− Dα l α0 + α1 x3 + α2 v3 + α3 x3 v3 x l ρ(α0 + α2 v3 )(x − x3 + l − l3 )(D α x + D α l) (d + m + γ )(α0 + α1 x3 + α2 v3 + α3 x3 v3 )x3  v3 α y3 m+ρ+γ (a + pz3 )(m + ρ + γ )  1− D v 1− Dα y + + γ y kγ v  z3 p(m + ρ + γ ) q(a + pz3 )(m + ρ + γ ) α D w+ 1− D α z. + kgγ cγ z +

Using λ = dx3 + (m + γ )l3 , f (x3 , v3 )v3 = (m + ρ + γ )l3 , γ l3 = (a + pz3 )y3 , b ky3 = μv3 , and y3 = , we get c  f (x3 , v3 ) α (x3 − x) D L3 (t) ≤ d 1 − f (x, v3 )  f (x3 , v3 ) f (x, v)v + + (m + ρ + γ )l3 1 − f (x, v3 ) f (x, v3 )v3   l3 f (x, v)v ly3 + (m + ρ + γ )l3 1 − + (m + ρ + γ )l3 1 − lf (x3 , v3 )v3 l3 y   v yv3 f (x3 , v3 ) + ρ(l − l3 ) 1 − + (m + ρ + γ )l3 1 − − v3 y3 v f (x, v3 )   ρ(α0 + α2 v3 ) d(x − x3 )2 + (m + γ )(l − l3 )2 + (d + m + γ )(x − x3 )(l − l3 ) − (d + m + γ )(α0 + α1 x3 + α2 v3 + α3 x3 v3 )x3  qh(a + pz3 )(m + ρ + γ )  w R2 − 1 w + gkγ

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 (α0 + α2 v3 )(x − x3 )2 dρx (dx3 − ρl3 ) + ρl + ≤− xx3 (α0 + α1 x3 + α2 v3 + α3 x3 v3 ) d +m+γ ρ(α0 + α2 v3 )(m + γ )(l − l3 )2 (m + ρ + γ )(α0 + α1 x3 + α2 v3 + α3 x3 v3 )x3  f (x3 , v3 ) l3 f (x, v)v yv3 f (x, v3 ) ly3 − − − + (m + ρ + γ )l3 5 − − f (x, v3 ) lf (x3 , v3 )v3 l3 l y3 v f (x, v) −

(α0 + α1 x)(α2 + α3 x)(v − v3 )2 v3 (α0 + α1 x + α2 v + α3 xv)(α0 + α1 x + α2 v3 + α3 xv3 )  qh(a + pz3 )(m + ρ + γ )  w R2 − 1 w. + gkγ − (m + ρ + γ )l3

Therefore D α L3 (t) ≤ 0 if R2w ≤ 1 and dx3 ≥ ρl3 . It is not hard to see that this condition is equivalent to (19). Furthermore, D α L3 (t) = 0 if x = x3 , l = l3 , y = y3 , v = v3 , and (R2w − 1)w = 0. When R2w < 1, then w = 0. If R2w = 1, from (1), we have D α v3 = 0 = ky3 − μv3 − qv3 w, then w = 0. In addition, D α y3 = 0 = γ l3 − ay3 − py3 z, then z = z2 . Hence, the largest invariant set of {(x, l, y, v, w, z) ∈ R6+ : D α L3 (t) = 0} is the singleton {E3 }. By the LaSalle’s invariance principle, E3 is globally asymptotically stable. We compute the characteristic equation of (13) at E3 and obtain (qv3 − h − ξ )g3 (ξ ) = 0,

(20)

where   ∂f ∂f   ρ 0 − v3 − f (x3 , v3 ) 0 v3 − ξ  −d −   ∂x ∂v   ∂f ∂f   −(m + ρ + γ ) − ξ 0 + f (x , v ) 0 v v 3 3 3 3   ∂x ∂v . g3 (ξ ) =  0 −py3  0 γ −a − pz3 − ξ      0 0 k −μ − ξ 0    cy3 − b − ξ  0 0 0 cz3

Equation (20) has the root ξ1 = qv3 − h. Then, when R2w > 1, we have ξ1 > 0. So, E3 is unstable.  Theorem 4.5 The infection equilibrium with both humoral and cellular immunity E4 is globally asymptotically stable if R2z > 1, R2w > 1 and condition (17) is satisfied.

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Proof We define the Lyapunov functional as follows: L4 (t) =

  x l α0 + α2 v4 + l4  x4  α0 + α1 x4 + α2 v4 + α3 x4 v4 x4 l4 ρ(α0 + α2 v4 ) (x − x4 + l − l4 )2 2(d + m + γ )(α0 + α1 x4 + α2 v4 + α3 x4 v4 )x4   (a + pz4 )(m + ρ + γ ) y v m+ρ+γ + y4  v4  + γ y4 kγ v4   p(m + ρ + γ ) q(a + pz4 )(m + ρ + γ ) w z + . +  z4  kgγ w4 cγ z4 +

Then, we have   x4 α α0 + α2 v4 l4 D L4 (t) ≤ D x+ 1− Dα l 1− α0 + α1 x4 + α2 v4 + α3 x4 v4 x l α

ρ(α0 + α2 v4 )(x − x4 + l − l4 )(D α x + D α l) (d + m + γ )(α0 + α1 x4 + α2 v4 + α3 x4 v4 )x4  v4 y4 m+ρ+γ (a + pz4 )(m + ρ + γ )  1− 1− Dα y + + γ y kγ v  w4 q(a + pz4 )(m + ρ + γ ) p(m + ρ + γ ) 1− Dα w + × Dα v + kgγ w cγ  z4 D α z. × 1− z +

Using λ = dx4 + (m + γ )l4 , f (x4 , v4 )v4 = (m + ρ + γ )l4 , γ l4 = (a + pz4 )y4 , b h ky4 = (μ + qw4 )v4 , y4 = , and v4 = , we get c g  f (x4 , v4 ) (x4 − x) D L4 (t) ≤ d 1 − f (x, v4 )  f (x4 , v4 ) f (x, v)v + + (m + ρ + γ )l4 1 − f (x, v4 ) f (x, v4 )v4   l4 f (x, v)v ly4 + (m + ρ + γ )l4 1 − + (m + ρ + γ )l4 1 − lf (x4 , v4 )v4 l4 y   v yv4 f (x4 , v4 ) + ρ(l − l4 ) 1 − + (m + ρ + γ )l4 1 − − v4 y4 v f (x, v4 )   2 2 ρ(α0 + α2 v4 ) d(x − x4 ) + (m + γ )(l − l4 ) + (d + m + γ )(x − x4 )(l − l4 ) − (d + m + γ )(α0 + α1 x4 + α2 v4 + α3 x4 v4 )x4 α

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 (α0 + α2 v4 )(x − x4 )2 dρx (dx4 − ρl4 ) + ρl + ≤− xx4 (α0 + α1 x4 + α2 v4 + α3 x4 v4 ) d +m+γ ρ(α0 + α2 v4 )(m + γ )(l − l4 )2 (m + ρ + γ )(α0 + α1 x4 + α2 v4 + α3 x4 v4 )x4  f (x4 , v4 ) l4 f (x, v)v yv4 f (x, v4 ) ly4 − − − + (m + ρ + γ )l4 5 − − f (x, v4 ) lf (x4 , v4 )v4 l4 l y4 v f (x, v) −

− (m + ρ + γ )l4

(α0 + α1 x)(α2 + α3 x)(v − v4 )2 . v4 (α0 + α1 x + α2 v + α3 xv)(α0 + α1 x + α2 v4 + α3 xv4 )

Then, D α L4 (t) ≤ 0 when dx4 ≥ ρl4 . In addition, it is easy to see that this condition is equivalent to (17). Further, the largest invariant set of {(x, l, y, v, w, z) ∈ R6+ : D α L4 (t) = 0} is the singleton {E4 }. By the LaSalle’s invariance principle, E4 is globally asymptotically stable.  We note that, when ρ is sufficiently small or γ is sufficiently large, conditions (15), (17), and (19) are verified. Hence, the global stability of E1 , E2 , E3 , and E4 is fully determined by R0 , R1w , R1z , R2w , and R2z . According to Theorems (4.3–4.5), we have the following results. Corollary 4.6 If ρ is sufficiently small or γ is sufficiently large, then we have • The immune-free infection equilibrium E1 is globally asymptotically stable if R0 > 1, R1w ≤ 1, and R1z ≤ 1. • The infection equilibrium E2 is globally asymptotically stable if R1w > 1 and R2z ≤ 1. • The infection equilibrium E3 is globally asymptotically stable if R1z > 1 and R2w ≤ 1. • The infection equilibrium E4 is globally asymptotically stable if R2z > 1 and R2w > 1.

5 Numerical Simulations In this section, to illustrate our theoretical results, we present some numerical simulations. Using the fractional Euler’s method presented in [21], we discretize system (1). Firstly, we take the parameter values as shown in Table 1. By calculation, we have R0 = 0.0427 ≤ 1. Then system (1) has an infectionfree equilibrium E0 (719.4245, 0, 0, 0, 0, 0). By Theorem 4.1, the solution of (1) converges to E0 (see Fig. 1). Consequently, the virus is cleared and the infection dies out.

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Table 1 Parameter values of system (1) Parameters a p k μ q g

800 700 600 500 400

0

50

100

150

200

250

300

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0

20

40

60

80

100

500

20

40

60

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α = 0.5

0

0

20

80

100

40 20 0

40

60

80

100

80

100

30

CTL cells

Antibodies

1000

0

5

Time (days)

60

1500

Values 0.2 0.002 0.1 0.1 0.01 0.00001 1

10

Time (days)

2000

Free virus

Parameters h c b α1 α2 α3 α0

15

Time (days)

0

Values 0.27 0.001 800 3 0.01 0.0001

Infected cells

Values 10 0.0139 0.00024 0.01 0.0347 0.01

Latently cells

Uninfected cells

Parameters λ d β ρ m γ

0

20

40

60

80

100

20 10 0

0

Time (days)

α = 0.7

20

40

60

Time (days)

α = 0.9

α=1

Fig. 1 Stability of the infection-free equilibrium E0

For the rest, we choose β = 0.0012. Next, we keep the other parameter values. Hence, we obtain R0 = 2.1370, R1w = 0.8334, R1z = 0.125, and 1+

(m + ρ + γ )[α0 adμ(m + ρ) + dkλγ α2 ] + kργ α3 λ2 = 2.5934. aρμ(m + ρ + γ )(α0 d + λα1 )

Consequently, condition (15) is satisfied. Therefore, the immune-free infection equilibrium E1 (176.6853, 168.7712, 6.2508, 1666.9, 0, 0) is globally asymptotically stable. Figure 2 demonstrates this result. We see that in the absence of the immune response, the infection becomes chronic. Next, we take g = 0.0003 and do not change the other parameter values. In this case, we have R1w = 2.5003 and R2z = 0.0844. Consequently, system (1) has an infection equilibrium with only antibody response E2 (353.0748, 113.9208, 4.2193, 666.6667, 206.3147, 0) which is globally asymptotically stable. Figure 3 illustrates this result. We can observe that the presence of antibodies decreases the concentration of infected cells to a lower level which does not allow the activation of cellular immunity. Now, we change only the value of parameter c = 0.04. Therefore, we get R1z = 2.5003 and R2w = 0.3333. Thus, the infection equilibrium with only

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Infected cells

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Fig. 2 Stability of the immune-free infection equilibrium E1

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α = 0.7

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α=1

Fig. 3 Stability of the infection equilibrium with only humoral immunity E2

cellular response E3 (353.0748, 113.9208, 2.5, 666.6667, 0, 185.6832) is globally asymptotically stable. Figure 4 proves this result. In this case, the presence of CTL cells decreases the viral load to a lower level which is small to activate the humoral immunity. Finally, we change only the parameter values g = 0.0008 and c = 0.065. By calculation, we have R2w = 1.3333 and R2z = 1.4939. As a result, system (1) has an infection equilibrium with both humoral and cellular immunity E4 (557.2897, 50.4177, 1.25, 250, 100, 133.3418) which is globally asymptotically stable (see Fig. 5). we show that the presence of both cellular and humoral immunity increases the concentration of host cells, decreases the concentration of infected cells and viral load to a lower level but the infection remains chronic. In all Figs. 1, 2, 3, 4 and 5, we can see that the solution of system (1) converges to the equilibrium points with different values of α. However, the value of α affects the time to reach the equilibrium points. Precisely, a small value of α (long

400

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100

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Fig. 4 Stability of the infection equilibrium with only cellular immunity E3

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Fig. 5 Stability of the infection equilibrium with both humoral and cellular immunity E4

memory) entails a fast convergence to the equilibrium points. Also the presence of the parameter α reduces the oscillation slightly (see Figs. 3, 4 and 5). This behavior can be explained by the memory effect presented in the fractional derivative in terms (t − u)−α which represents the time needed for the interaction of the time kernel (1 − α) between host cells and virus and the time needed for activation of the adaptive immunity.

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6 Conclusion In this study, we have proposed a fractional order model for viral infection which includes the adaptive immunity and the cure of infected cells in latent state. The infection transmission is modeled by Hattaf–Yousfi functional response which covers the most functional response existing in the literature such as the saturated incidence rate, the Beddington–DeAngelis functional response, the Crowley–Martin functional response, and the Hattaf’s functional response used in [6]. In addition, the fractional derivative is adopted in the sense of Caputo which describes the time needed for the interactions between host cells and viral particles and the time needed for the activation of the immune response. We have shown that the solutions are bounded and non-negative which ensures the well-posedness of the model. We have derived five threshold parameters for viral infection that are the basic reproduction number R0 , the reproduction number for humoral immunity R1w , the reproduction number for cellular immunity R1z , the humoral immunity competition reproduction number R2w , and the cellular immunity competition reproduction number R2z which determine the global dynamics of the model. By constructing a suitable Lyapunov functionals and LaSalle’s invariance principle, we have proved that if the basic reproduction number R0 ≤ 1, the infection-free equilibrium E0 is globally asymptotically stable, which means that the virus is cleared and the infection dies out. However, when R0 > 1, the infection-free equilibrium E0 becomes unstable and there exists another equilibrium, namely the immune-free infection equilibrium E1 that is globally asymptotically stable when condition (15) is satisfied. In this case, the infection becomes chronic. In the presence of the adaptive immunity, three other equilibrium appeared that are: (1) the infection equilibrium with only humoral immunity E2 which is globally asymptotically stable when R1w > 1, R2z ≤ 1, and condition (17) holds; (2) the infection equilibrium with only cellular immunity E3 that is globally asymptotically stable when R1z > 1, R2w ≤ 1, and condition (19) is verified; (3) the infection equilibrium with both humoral and cellular immunity E4 which is globally asymptotically stable when R2z > 1, R2w > 1, and condition (19) is satisfied. Further, we have remarked that if the cure rate ρ is sufficiently small or γ is sufficiently large, conditions (15), (17), (19) are satisfied and the global stability of the five equilibrium are only characterized by R0 , R1w , R1z , R2w , R2z . According to the numerical simulations, we show that the fractional order parameter α has no effect on the global dynamics of our model, but it can affect the time for arriving to the steady states and reduces the oscillations (see Figs. 1, 2, 3, 4 and 5). The presence of the fractional derivative enriches the dynamical behavior of the model and gives more information about the interactions between host cells, the viral particles, and the immune response. Moreover, the main results presented in [2, 3, 6, 18] are generalized and improved.

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References ´ D. Kirschner, Remarks on modeling host-viral dynamics and 1. J. Velasco-Hernandez, ´ J. Garcia, treatment, in Mathematical Approaches for Emerging and Reemerging Infectious Diseases: An Introduction to Models, Methods, and Theory, vol. 125 (2001), pp. 287–308 2. D. Wodarz, Hepatitis C virus dynamics and pathology: the role of CTL and antibody responses. J. Gen. Virol. 84, 1743–1750 (2003) 3. N. Yousfi, K. Hattaf, A. Tridane, Modeling the adaptive immune response in HBV infection. J. Math. Biol. 63, 933–957 (2011) 4. K. Hattaf, M. Khabouze, N. Yousfi, Dynamics of a generalized viral infection model with adaptive immune response. Int. J. Dynam. Control. 3(3), 253–261 (2014) 5. A.M. Elaiw, Global stability analysis of humoral immunity virus dynamics model including latently infected cells. J. Bio. Dynam. 9(1), 215–228 (2015) 6. M. Maziane, K. Hattaf, N. Yousfi, Global stability for a class of HIV infection models with cure of infected cells in eclipse stage and CTL immune response. Int. J. Dynam. Control. 5(4), 1035–1045 (2016) 7. Y.A. Rossikhin, M.V. Shitikova, Application of fractional calculus to dynamic problems of linear and nonlinear hereditary mechanics of solids. Appl. Mech. Rev. 50, 15–67 (1997) 8. R.J. Marks II, M.W. Hall, Differintegral interpolation from a bandlimited signal’s samples. IEEE Trans. Acoust. Speech Signal Process. 29, 872–877 (1981) 9. G.L. Jia, Y.X. Ming, Study on the viscoelasticity of cancellous bone based on higher-order fractional models, in Proceeding of the 2nd International Conference on Bioinformatics and biomedical Engineering (ICBBE’08) (2006), pp. 1733–1736 10. R. Magin, Fractional calculus in bioengineering. Crit. Rev. Biomed. Eng. 32, 13–77 (2004) 11. E. Scalas, R. Gorenflo, F. mainardi, Fractional calculus and continuous-time finance. Physica A 284, 376–384 (2000) 12. L. Song, S.Y. Xu, J.Y. Yang, Dynamical models of happiness with fractional order. Commun. Nonlinear Sci. Numer. Simul. 15, 616–628 (2010) 13. R. Capponetto, G. Dongola, L. Fortuna, I. Petras, Fractional order systems: modelling and control applications. World Sci. Ser. Nonlinear Sci. Ser. A 72 (2010) 14. K.S. Cole, Electric conductance of biological systems. Cold Spring Harb. Quant. Biol., 107– 116 (1993) 15. C.M.A. Pinto, A.R.M. Carvalho, A latency fractional order model for HIV dynamics. J. Comput. Appl. Math. 312, 240–256 (2016) 16. F.A. Rihan, M. Sheek-Hussein, A. Tridane, R. Yafia, Dynamics of hepatitis C virus infection: mathematical modeling and parameter estimation. Math. Model. Nat. Phenom. 12(5), 33–47 (2017) 17. A.A.M. Arafa, S.Z. Rida, M. Khalil, A fractional-order model of HIV infection: numerical solution and comparisons with data of patients. Int. J. Biochem. 7(4), 1–11 (2014) 18. A. Boukhouima, K. Hattaf, N. Yousfi, Dynamics of a fractional order HIV infection model with specific functional response and cure rate. Int. J. Differ. Equ. 2017, 1–8 (2017) 19. C.V. De-Leon, Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24, 75–85 (2015) 20. J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal. Real World Appl. 26, 289–305 (2015) 21. Z. Odibat, S. Momani, An algorithm for the numerical solution of differential equations of fractional order. J. Appl. Math. Inform. 26, 15–27 (2008)

Optimal Temporary Vaccination Strategies for Epidemic Outbreaks K. Muqbel, A. Dénes, and G. Röst

1 Introduction Mathematical models of the transmission dynamics of infectious diseases are useful in gaining insights into the mechanisms of disease spread, in estimating key epidemiological parameters, in making predictions about the expected outcomes, and also in devising, evaluating, and comparing intervention strategies. Vaccination is the most successful and cost-effective preventive measure against many infectious diseases [2]. However, for some emerging diseases, the delay in identification of the pathogen (such as the particular strain), the time needed to develop novel vaccinations, and the limited capacity in production, distribution, and administration of vaccines may lead to a situation where vaccination programs run parallel in time with the disease outbreak. During the recent West African Ebola virus epidemic (2013–2016), at the beginning no licensed vaccines for the disease were available. The rVSV-ZEBOV vaccine was developed during the course of the epidemic [5]. Until the vaccine became available, other coordinated public health measures have been implemented [1]. A similar situation occurred also in many developed countries during the 2009 influenza H1N1 outbreak [4]. For instance, in Canada, due to the limited availability of the vaccine at the outset of the outbreak, and the inability to vaccinate the entire population simultaneously, a sequencing strategy has been developed that identified groups of different levels of priority [3]. In some countries a significant portion of the influenza vaccines were administered in the later phase of the epidemics [4], when the number of prevented cases per a unit of

K. Muqbel () · A. Dénes Bolyai Institute, University of Szeged, Szeged, Hungary G. Röst Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Oxford, UK © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_19

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administered vaccine drops sharply. This raises the question of cost-effectiveness, and also suggests that the vaccination program should stop at some well defined point of the epidemics. Motivated by this problem, in this study we propose a family of temporary vaccination strategies. For the sake of simplicity, we work in the basic SIR-framework. An intervention strategy will be defined by two parameters which determine the time interval it is applied as well as the intensity of vaccine administration. Our goal is to find out which strategy is the most cost-efficient, where costs are assigned to cases of infections and units of administered vaccines.

2 Specification of the VUHIA Strategy and Its Total Cost We consider a constant population divided into susceptible (S(t)), infected (I (t)), and removed (R(t)) compartments. New infections occur with transmission coefficient β and infected individuals recover with rate α. Upon recovery, full immunity is assumed. Vaccination of susceptibles is included in the model with time dependent vaccination rate v(t), to be specified later. Vaccination is assumed to be fully protective, thus vaccinated individuals are placed in the R-compartment as well. Hence, we consider the following system of differential equations: S  (t) = −βS(t)I (t) − v(t)S(t), I  (t) = βS(t)I (t) − αI (t),

(1)

R  (t) = αI (t) + v(t)S(t). We are interested in the situation when a small number of infected hosts are introduced into a fully susceptible population, hence we consider initial data S(0) = S0 , I (0) = I0 , R(0) = 0, where I0 is relatively small compared to the total population size N = S + I + R. The basic reproduction number is given by R0 =

βS0 , α

however by normalizing the population size at N = 1 and with I0 1. The total cost (T C) of an outbreak will be assessed by considering two components, the disease burden and the cost of vaccination. Disease burden is calculated as the total number of infections during the course of the outbreak (denoted by I˜) multiplied by the cost C1 of a single infection. Vaccination cost is calculated as the total number of administered vaccines (denoted by V˜ ) multiplied by the cost C2 of a single vaccination. This way, for the total cost we have

Optimal Temporary Vaccination Strategies for Epidemic Outbreaks

TC := C1 I˜ + C2 V˜ ,

301

(2)

where I˜ := V˜ :=



∞

βS(t)I (t)dt = α

0

∞

I (t)dt,

(3)

0 ∞

v(t)S(t)dt.

(4)

0

There has been a number of studies using optimal control theory to find the control function v(t) that minimizes some (typically quadratic) cost function (see [7] for an example). However, a continuously changing v(t), which is the common output from that approach, is not feasible to be realized as a public health policy. Hence, we aim to define a strategy v(t) in a simpler way, and we assume that v(t) is a piecewise constant function, taking values of either 0 (control is off), or some p > 0 (control is on). This means that we propose to apply vaccination with a given rate on some time interval. It remained to determine when to start and when to finish the intervention. We cannot expect in general that the intervention can start immediately, as the epidemic may not have been detected or the resources are not in place at the beginning of the outbreak. A reasonable assumption is that the starting point of interventions is when the number of infected individuals reaches a threshold value k, as it has been in [6]. However, in outbreak models using the same threshold to define the end of intervention may not be adequate, given that if k is too large then we finish vaccination too early, while when k is too small then vaccination may go on even when it does not have any significant impact on the epidemic any more. Instead, we propose to stop the vaccination when the number of infections starts decreasing, which is the same point when the number of susceptibles becomes so low that herd immunity has reached in the population. We call such an intervention a VUHIA-strategy of (k, p)-type, referring to vaccinate until herd immunity achieved with parameters (k, p). In mathematical terms, the VUHIA-strategy of (k, p)-type is defined as follows. Let  0, t ∈ / J, v(t) = p, t ∈ J, where J is the intervention interval J = [Tstart , Tend ] with Tstart = min{t ≥ 0 : I (t) ≥ k} and Tend = min{t ≥ 0 : βS(t) − α ≤ 0}.

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The time Tstart is well defined as long as k ∈ [I0 , Imax ], where Imax denotes the peak of the SIR-epidemic in the absence of any intervention. It is well known for the SIR model (with N = 1) that Imax = 1 − R−1 0 (1 + ln R0 ). Clearly we have I  (t∗ ) = 0 when S(t∗ ) = α/β, and I  (t) < 0 for any t > t∗ regardless we vaccinate or not at some t > t∗ . Since the epidemic eventually always dies out, Tend is well defined, and (4) becomes V˜ := p

Tend

(5)

S(t)dt. Tstart

Figure 1 depicts how the epidemic plays out with two different strategies. In one, we start vaccinating early with a low rate; in the other we start vaccinate later but with a higher rate. As Fig. 1 shows, it is unclear which of these two strategies is better, hence we will systematically explore this in the forthcoming sections by computer simulations.

3 The Relation Between the Total Cost and the Vaccination Rate To see how the total cost depends on the vaccination rate, we shall consider various fixed k-s and vary p. The change in the total cost then depends on d d d T C(p, k) = C1 I˜ + C2 V˜ . dp dp dp

p=0

p=1, k=0.01

p=10, k=0.2

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Fig. 1 The total number of infected (left) and vaccinated (right) people during the epidemic for different strategies. The epidemic parameters are R0 = 4, α = 6, β = 24. On the left, the red curve is the epidemic curve in the absence of intervention. On the right, we can clearly see when the vaccination starts and stops

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From Fig. 2 we can see that I˜ decreasing while V˜ decreasing in p, thus the sign of the rate of change of the total cost is determined by the ratio of C1 and C2 relative to the rates of change in I˜ and V˜ . In the sequel we always normalize the cost of disease burden C1 = 100, and we will vary C2 to compare different scenarios. What we can see in Fig. 3 is that when C2 0 C1 , the total cost is decreasing in p, meaning that when vaccination is relatively cheap, we should vaccinate as high rate as possible. On the other hand, when C2 $ C1 , the total is increasing in p, meaning that when vaccination is very expensive relative to the disease burden, the strategy that give minimal cost is to not vaccinate at all. We can also see that the total cost is more sensitive to p when the vaccination rate is small. These results are what one would expect; however, there is a curious situation when C1 and C2 are of similar magnitudes: there is a possibility that the total cost is not monotone in p. This scenario is highlighted in Fig. 3, right. In this case, vaccination with a small rate yields a higher cost than no vaccination (see the red line); however, vaccination with a high rate yields a smaller cost. Let p∗ be the value where the cost curve intersects the straight red line corresponding to cost of no vaccination. This means that if we are capable to vaccinate with a sufficiently high rate p > p∗ , then we should do it, but if with our capacities and resources only a smaller rate p < p∗ can be achieved, it is better to not vaccinate at all.

4

The Relation Between the Total Cost and the Threshold Level

Next we consider how the total cost changes when we vary k for fixed values of p. Figure 4 shows that by increasing k, that is we start vaccinating later, the total number of infections increases while the total number of vaccinations decreases. ˜ ˜ Again, the change in total depends on how C1 :C2 relates to ddkI : ddkV . This is depicted in Fig. 5 (left) for various values of C2 . Similarly as before, we see that if vaccination k=0.002 ∼ I

k=0.01

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Fig. 2 The total number of infected (left), and vaccinated (right) people during the epidemic as a function of vaccination rate p. Parameters are R0 = 1.5, α = 6, β = 9

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Fig. 3 The total cost as a function of p, for five different vaccination costs (left). In the right, the case C2 = 155 is highlighted by zooming in. Parameters are R0 = 1.5, α = 6, β = 9, k = 0.002 p=0.1 ∼

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Fig. 4 The total number of infected (left) and vaccinated (right) people during the epidemic as a function of k. Parameters are R0 = 1.5, α = 6, β = 9

is relatively cheap, it is better to start early, and when it is very expensive, it is better not to start at all. The graphs of all five curves meet at the right when k → Imax . When similar costs are assigned to disease burden and vaccination, we can see a non-monotone behavior, which is highlighted in the right of Fig. 5. The interpretation of this figure is that in the scenario of the blue dotted curve there is a k∗ , such that if we are capable to start the intervention earlier that k∗ , then we should as soon as possible. But, if for any reason we could not start the intervention before I (t) reached k∗ , then it is better not to vaccinate at all. This k∗ is given by the intersection of the blue dotted curve with the horizontal red line.

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Fig. 5 The total cost as a function of k for five different vaccination costs (left). The cases C2 = 155 and C2 = 200 are highlighted by zooming in (right). Parameters are R0 = 1.5, α = 6, β = 9, p = 1.5

5 Conclusions and Summary We have proposed a family of temporary vaccination strategies in the framework of the SIR model. These strategies are characterized by parameters (k, p), where vaccination starts when the number of infected hosts reaches the threshold value k, and with rate p we continue vaccination until herd immunity is achieved (VUHIA). The advantages of the VUHIA-strategy are the following. First, it has a clear and meaningful definition: we start the vaccination with rate p when a threshold k is reached in the level of infection, and we stop the vaccination when the number of susceptibles drops below R−1 0 , that is herd immunity achieved the number of infected will decrease anyway. Second, it is determined only by the parameters (k, p), hence all strategies from this family can be explored in a two dimensional parameter space. We have assigned a total cost to each strategy composed of cost of disease burden and cost of vaccination, and systematically investigated the dependence of the total cost on the parameters. Essentially, we have found three types of behaviors: (i) vaccination cost is very low compared to the cost associated to disease burden: in this case increasing the vaccination rate and start vaccination earlier reduce the total cost; (ii) vaccination cost is very high compared to the cost associated to disease burden: in this case the optimal strategy is to not vaccinate at all; (iii) vaccination cost and disease burden cost are of similar magnitudes: there may be non-monotone relationships between the vaccination rate, the starting threshold and the total cost. These three typical behaviors are plotted into a heatmap in Fig. 6. In case (iii), it may happen that a better strategy is to start earlier but only if we can start sufficiently early, or, it is better to increase vaccination rate but only if we can increase it to a

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Fig. 6 Dependence of the total cost on (k, p) in three typical situations: C2 = 50 0 C1 (bottom left), C2 = 500 $ C1 (bottom right), and C2 = 155 (top). Parameters are R0 = 1.5, α = 6, β = 9, and C2 = 50,155,500, respectively. The bottom plots show monotone cases, while in the top plot we can find non-monotonicity in both k and p

sufficiently high level. If we cannot meet those criteria, then the best decision is to not vaccinate. The top plot of Fig. 6 illustrates these intricate non-monotonicity properties. Depending on the available resources and public health capacities, there may be constraints on the parameters, such as k ≥ kmin and p ≤ pmax . The optimal strategy with such constraints can be found even in these cases from the graphs in Figs. 3, 5, and 6. It is very easy when the total cost depends monotonically on the parameters, for example with an upper bound on p and in the situation of C2 = 50 in Fig. 3, the optimal strategy is always p = pmax . In contrast, for C2 = 155 (see Fig. 3 right),

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Fig. 7 Effect of R0 on the monotonicity of the cost curve. Parameters are p = 0.25, C2 = 115, α = 6 and β = 7.2, resp. β = 24. For R0 = 1.2, optimal strategy is achieved by vaccinating early, while for R0 = 4 it is better to not vaccinate at all

p = pmax is the optimal strategy only if p∗ < pmax , otherwise the optimal strategy is p = 0. Another interesting phenomenon is depicted in Fig. 7, showing that for a fixed p, the monotonicity of the total cost in k can reverse varying the reproduction number. In that particular situation of Fig. 7, for a less contagious disease (R0 = 1.2), to minimize the cost vaccination should start as early as possible (k → 0), while for a more contagious disease (R0 = 4) the lowest cost comes from not vaccinating at all (k → Imax ). Although this SIR vaccination model is certainly too simplistic to apply to any real outbreak, this simple epidemiological model already exhibits some surprising and counter-intuitive features, highlighting that in real applications with more complex models, a comprehensive mathematical investigation of the possible intervention strategies is really necessary. Acknowledgements AD was supported by the Hungarian National Research, Development and Innovation Office grant NKFIH PD 128363 and the Bolyai Scholarship of HAS. KM was supported by NKFIH FK 124016. GR was supported by EFOP-3.6.1-16-2016-00008 and Marie SkłodowskaCurie Grant No. 748193.

References 1. M.V. Barbarossa, A. Dénes, G. Kiss, Y. Nakata, G. Röst, Zs. Vizi, PLoS One 10(7), 21 (2015). Paper e0131398 2. M. Drolet, É. Bénard, M. Jit, R. Hutubessy, M. Brisson, Value Health 21, 1250–1258 (2018) 3. A. Durbin, A.N. Corallo, T.G. Wibisono, D.M. Aleman, B. Schwartz, P.C. Coyte, J. Infect. Dis. Immun. 3(3), 40–49 (2011) 4. D. Knipl, G. Röst, Math. Biosci. Eng. 8(1), 123–139 (2011) 5. S. Merler, M. Ajelli, L. Fumanelli, S. Parlamento, A. Pastore y Piontti, N.E. Dean, et al., PLoS Negl. Trop. Dis. 10(11), e0005093 (2016) 6. Y. Xiao, X. Xu, S. Tang, Bull. Math. Biol. 74(10), 2403–2422 (2012) 7. G. Zaman, Y.H. Kang, I.H. Jung, BioSystems 93(3), 240–249 (2008)

On the Reproduction Number of Epidemics with Sub-exponential Growth D. Champredon and Seyed M. Moghadas

1 Introduction Compartmental epidemic models have been used for decades to gain insights into the transmission dynamics and control of infectious diseases in human populations [1–10]. Fundamentally, these models assume an early exponential growth for epidemic dynamics in the absence of interventions or population behavioural changes. However, evidence is accumulating that some epidemics unfold with a slower rate than exponential. Previous studies have demonstrated the presence of sub-exponential (i.e., polynomial) growth patterns during the early phase of disease epidemics including HIV/AIDS [11, 12], Ebola [13–15], and foot-and-mouth disease [15]. Understanding the dynamics of spread and control of epidemics that exhibit early non-exponential growth requires the development of a fundamental theoretical basis, which is currently lacking. Characterizing the early epidemic growth and its effect on disease spread, often quantified by the basic reproduction number, is the first step towards developing such theory [15]. In order to account for a range of epidemic growth rates slower than exponential, a recent study has introduced a generalized-growth model incorporating a “deceleration of growth” parameter (p) that characterizes epidemics with a constant incidence rate (p = 0); sub-exponential growth (0 < p < 1); and exponential growth (p = 1) [14]. This model is given by the following simple differential equation: dC = C  (t) = rC(t)p dt

(1)

D. Champredon · S. M. Moghadas () Agent-Based Modelling Laboratory, York University, Toronto, ON, Canada e-mail: [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_20

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where C  (t) describes the incidence of disease at time t; the solution C(t) represents the cumulative number of cases; r is a positive parameter denoting the intrinsic growth rate per unit time, and p ∈ [0, 1] is the deceleration of growth parameter. When p = 0, the cumulative number of cases grows linearly, whereas p = 1 describes exponential growth dynamics (i.e., the Malthus equation). An explicit solution of equation (1) for t > 0 and 0 < p < 1 takes the form [16]: C(t) = [(1 − p)rt + A]m

(2)

where m = 1/(1 − p) is the degree of polynomial growth in the cumulative number 1−p is a constant that depends on the initial number of of infections, and A = C0 cases, C0 . Unlike the case of early exponential growth dynamics (p = 1), there is no theoretical framework to explicitly calculate the reproduction number for epidemics with sub-exponential growth (p < 1). In fact, as we have shown in a recent study [15], this quantity varies over time even without control measures, depletion of susceptible individuals, or other factors limiting disease spread. Using real epidemic data and simulations, we found that the reproduction number for epidemics with early sub-exponential growth exhibits a declining trend that approaches unity over time [15]. This is in contrast with the case of exponential growth, where the reproduction number remains invariant in the absence of interventions or any other mechanisms diminishing the risk of disease transmission. Because the reproduction number is a key parameter that quantifies the transmission potential of an infectious disease, we sought to investigate the effect of initial epidemic growth rate on this quantity that has important implications for science and policy. Here, we build the first step in expanding the theory of epidemics with sub-exponential growth, and establish the relationship between the reproduction number and the deceleration of growth parameter [14]. This relationship provides a theoretical basis for the declining trend of the reproduction number observed in previous work [15]. We discuss our results and their relevance to the prevention and control of disease spread.

2 Reproduction Numbers When the growth rate of an epidemic is assumed to be exponential, there is a well-established theory on the calculation of the reproduction number [17, 18]. This calculation depends on the intrinsic growth rate r and the distribution of generation interval [19], which is defined as the time period between the time when an individual is infected by an infector and the time when this infector was infected. Assuming that the distribution of the generation interval is known with the probability density function G(a), one can use the moment generating method [20] to calculate:

∞ M(z) = exp(za)G(a)da, (3) 0

On the Reproduction Number of Epidemics with Sub-exponential Growth

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which is the Laplace transform of the function G(a). If M(z) converges, then the reproduction number is defined by R0 = 1/M(−r) [17–19]. For example, if the generation interval is exponentially distributed with mean Tg , then from (3) one can obtain the formulae R0 = (1 + rTg ). However, when the early epidemic growth rate is sub-exponential, there is no established theory for calculation of the reproduction number based on the distribution of the generation interval. Here, we contrast the reproduction numbers in a discrete space for exponential and sub-exponential epidemic growth dynamics. Suppose Tg is the mean generation interval, and let Ji represent the incidence of infection observed during generation i with the fixed duration Tg . The expected incidence of infection in generation i + 1 can be obtained by Ji+1 = Ri Ji , where Ri is the reproduction number for generation i (i = 1, 2, . . .) with the same probability of generating new cases.

2.1 Continuous Time with Fixed Generation Interval When p = 1, corresponding to the exponential growth model, the average number of secondary cases generated by the initial cases during the first generation interval (assumed to be fixed) is given by: R0e =

C  (Tg ) rC0 erTg J1 = =  = erTg . J0 C (0) rC0

(4)

Assuming that there is continuous supply of susceptible individuals so that the initial phase of the epidemic can be sustained over time, it can be seen that R0e remains invariant at erTg over all generations. This can be shown by: R0e =

C  ((i + 1)Tg ) rC0 er(i+1)Tg Ji+1 = = = erTg . Ji C  (iTg ) rC0 eriTg

(5)

When p < 1, corresponding to the sub-exponential growth, the reproduction number changes over generations. This can be seen by using the solution of (1) expressed by (2). Taking the derivative of (2), we obtain the reproduction number for the sub-exponential growth model in generation i by: Ris

 p 1−p r(1 − p)Tg C  ((i + 1)Tg ) Ji+1 = 1 + = = . Ji C  (iTg ) r(1 − p)iTg + A

(6)

As we observed in previous work [15], Ris decreases asymptotically to 1 as i → ∞.

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2.2 Discrete Time with Fixed Generation Interval Discretizing the exponential growth model C  = rC over the same time interval as Tg , we obtain for i ≥ 1: Ri Ji = Ji+1 = Ci+1 − Ci = rTg Ci = rTg (J0 + · · · + Ji ).

(7)

Substituting for the incidence in each generation from the reproduction number of the previous generation, we obtain Ri Ji = rTg (J0 + R∗ J0 + R∗ R1 J0 + · · · + R∗ R1 · · · Ri−1 J0 ) = rTg J0 (1 + R∗ + R∗ R1 + · · · + R∗ R1 · · · Ri−1 ),

i ≥ 1,

(8)

where R∗ denotes the reproduction number for the index generation (i = 0), with J1 = R∗ J0 . Considering that J0 = C0 at time t = 0, and therefore J1 = rTg J0 , we obtain R1 = 1 + R∗ . We claim that Ri = 1 + R∗ for all i ≥ 1. A simple mathematical induction yields the proof. Suppose Rk = 1 + R∗ , we show that Rk+1 = 1 + R∗ . From (8) we have: Rk+1 Rk · · · R1 J1 = rTg J0 (1 + R∗ + R∗ R1 + · · · + R∗ R1 · · · Rk ),

(9)

which implies: Rk+1 =

1 + R∗ + R∗ R1 + · · · + R∗ R1 · · · Rk = 1 + R∗ . R 1 · · · Rk

(10)

We now consider the case of sub-exponential growth modelled given by C  = -i to represent the reproduction number in generation i for rC p for p < 1. Using R -1 = (1 + R -∗ )p , and this model, similar calculations (in discretized system) give R - - - p -k+1 = (1 + R∗ + R∗ R1 + · · · + R∗ R1 · · · Rk ) R -k -1 · · · R R

for k ≥ 1.

(11)

-k approaches 1 for large disease Similar to the continuous case, we claim that R -k ≥ 1 for all k ≥ 1. This is generation k. In order to show this, we note that R -k into R -k+1 , which gives obtained by substituting R -k+1 = R



-∗ R -∗ R -k p -∗ + R -1 + · · · + R -1 · · · R 1+R -∗ + R -∗ R -∗ R -k−1 -1 + · · · + R -1 · · · R 1+R

 = 1+

p -k -∗ R -1 · · · R R ≥1 -∗ + R -∗ R -∗ R -k−1 -1 + · · · + R -1 · · · R 1+R

(12)

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-∗ = 0, we obtain R -k = 1 for all k. Suppose R -∗ > 0. From (11), it follows If R that: - p -∗p - R -k+1 ≤ 1 + (R∗ R1 · · · Rk ) = 1 + . R -k -1 · · · R -k )1−p -1 · · · R (R R

(13)

-k ≥ 1 + 1/nk . Thus, Let nk be the smallest positive integer such that R -∗p R -k+1 ≤ 1 +  . R  1 1 1−p 1+ ··· 1 + n1 nk

(14)

We define a new sequence of positive numbers by: k  5 1 ak = 1+ . nj

(15)

j =1

From the monotone convergence theorem [21, 22], we have the following inequalities: 1+

k k   1 1 . ≤ ak ≤ exp nj nj j =1

(16)

j =1

) If k1 1/nj diverges, then ak → ∞ as k → ∞, from (14) that  )which implies ) ∞ -k → 1. If k1 1/nj converges, we let L = exp . In this case, using R 1/n j j =1 -k ≥ 1 and rewriting (12) gives: R  -k+1 ≤ 1 + R

-∗ L p R →1 -∗ 1 + kR

as k → ∞,

which proves our claim.

2.2.1

Alternative Approach in Discrete Time

As defined above, the reproduction number for generation i is given by: Ris =

Ji+1 . Ji

(17)

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Discretizing (1) with the same time interval as Tg gives: p

Ci+1 − Ci = rTg Ci .

(18)

Thus, for i ≥ 1, we have: p

rTg Ci Ci+1 − Ci = = p Ci − Ci−1 rTg Ci−1  Ji p = 1+ . Ci−1



Ris =

Ci Ci−1

p

 =

Ji + Ci−1 Ci−1

p (19)

Iterating Eq. (18), we have p

Ci+1 = Ci + rTg Ci

p−1

)Ci

p−1

)(1 + rTg Ci−1 )Ci−1

p−1

)(1 + rTg Ci−1 ) · · · (1 + rTg C0

= (1 + rTg Ci = (1 + rTg Ci = (1 + rTg Ci =

p−1 p−1

p−1

(20) )C0

i  5 p−1 1 + rTg Ck C0 . k=0

Because Ji = Ci − Ci−1 , using Eq. (20) we have for i ≥ 1: Ji =

i−1 5

p−1

(1 + rTg Ck

p−1

)rTg Ci

C0 .

k=0

Hence, Ji = Ci−1

6i−1

p−1 p−1 )rTg Ci C0 k=0 (1 + rTg Ck 6i−1 p−1 )C0 k=0 (1 + rTg Ck

p−1

= rTg Ci

.

(21)

Finally, substituting this into (19) we obtain:  p−1 p . Ris = 1 + rTg Ci

(22)

We note that in theory, limi→∞ Ci = +∞ and Ci is an increasing sequence. Therefore, for p < 1, Ris decreases and approaches 1 as i → +∞.

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3 Simulations We applied the theoretical framework to incidence data for three epidemics. First, we considered a simulated data set because it provides a controlled environment to check the numerical methods. Then, we estimated the reproduction numbers on real-world epidemics to illustrate their initial sub-exponential growth.

3.1 Simulated Data We simulated an epidemic that is driven by the generalized growth model (GGM) given by Eq. (1) with known values for p and r. The stochastic GGM assumes that incidence Jt has a Poisson distribution: Jt = Ct ∼ Poisson(rCt ). p

(23)

We choose r = 0.3 and p = 0.7 and generated 100 Monte Carlo replicates of incidence curves for a fixed time horizon set at 50 generations. For convenience, we assumed that the mean generation interval is equal to the unit of simulation timestep. For each simulated incidence trajectory, we fitted r and p, considering that their “true” values are unknown. The fit minimizes the squared error of the simulated cumulative incidence. The confidence intervals are calculated with a parametric bootstrap based on the Poisson distribution [15]. The fitted values are shown in Fig. 1a. Using the estimates for (r, p), one can calculate the sub-exponential reproduction numbers at the kth generation, Rk , for each simulated trajectory by employing Eq. (6) or (11) (respectively, for continuous and discrete time formulation). Figure 1b shows that the empirical reproduction number calculated as the ratio of incidences in generations k + 1 and k (i.e., Ik+1 /Ik ) decreases to 1, as predicted by Eqs. (6) and (11).

3.2 Real-World Epidemic Data The analyses on simulated data in the previous sections verify that the estimation methods for (r, p) and sub-exponential reproduction numbers Rk are correctly implemented. We now consider real-world epidemic data for Measles in London, England 1948; Ebola in Bomi, Liberia 2014; and Ebola in Kenema, Sierra Leone, 2014. The incidence data for these epidemics were provided as supplementary material in Chowell et al. [15]. The weekly cumulative incidence for each epidemic

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Fig. 1 (a) Grey points (respectively, horizontal and vertical segments) represent the mean (respectively, 95% CI) estimated values for r and p from simulated incidence curves. The black diamond shows the “true” values r = 0.3 and p = 0.7 used to simulate the epidemics. (b) The dashed line (respectively, dark grey area) is the mean (respectively, 95% quantile) of the theoretical reproduction numbers based on the estimated (r, p). The solid line (respectively, light grey shaded area) is the mean (respectively, 95% quantile) of the reproduction number calculated directly from the data as the ratio of incidences in generations k + 1 and k (i.e., Ik+1 /Ik )

is plotted in Fig. 2. We fitted the model with parameters (r, p) based on incidence data during the first five generation intervals. We assumed that the generation interval for measles is gamma-distributed, with mean of 12 days and coefficient of variation cv = 0.3 [23]. For Ebola, we also assumed a gamma distribution for the generation intervals with mean of 15 days and cv = 0.66 [24]. Table 1 provides the estimated values for r and p, suggesting that these epidemics exhibited an initial sub-exponential growth. Using these estimates, one can calculate the theoretical sub-exponential reproduction numbers given by Eq. (6) or (11). Since we do not have incidence data matching the generation interval frequency, to calculate the empirical reproduction number, we use the following formula that explicitly takes into account the generation interval distribution (g): Rk = )k

Ik

j =1 Ik−j

g(j )

.

(24)

On the Reproduction Number of Epidemics with Sub-exponential Growth

(a)

(b) EbolaEbola (Bomi; Ebola (Bomi; 2014) (Bomi; 2014) 2014)

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time (in units of mean generation interval) Fig. 2 Cumulative incidence for the 2014 Ebola epidemic in the county of Bomi, Liberia (a); Kenema in Sierra Leone (b), and the measles epidemic in London in 1948 (c). Data frequency is weekly for all three epidemics, and the x-axis is represented in units of the mean generation interval. Only the data earlier than the vertical dashed line (representing five times mean generation interval) were used to fit the model with parameters r and p. Note that y-axis is represented in logscale Table 1 Mean and 95% confidence interval for fitted values of r and p based on the first five mean generation intervals Epidemic Ebola (Bomi, 2014) Ebola (Kenema, 2014) Measles (London, 1948)

r (95% CI) 4.29 (1.83–6.84) 2.74 (1.58–3.74) 5.72 (3.78–8.11)

p (95% CI) 0.290 (0.121–0.462) 0.549 (0.458–0.653) 0.601 (0.557–0.650)

The estimates of the reproduction numbers for each epidemic using (24) are shown in Fig. 3. These results highlight the difficulty to extract meaningful information about the reproduction number early in the epidemic because of the relatively large stochasticity for small incidence counts. The estimates for the Ebola epidemic in Bomi fluctuate significantly, which make a faithful comparison with the theoretical reproduction difficult. The empirical reproduction numbers for the Ebola epidemic in Kenema are generally in agreement with the theoretical ones obtained from the third generation. Unlike the Ebola epidemics, the empirical reproduction numbers for the measles epidemic do not indicate a clear decreasing trend during the first five generations. To conclude, this section with examples from real-world epidemics highlights the difficulty to obtain reliable estimates for the empirical reproduction numbers early in an epidemic. This point has also been raised by previous studies (for

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Fig. 3 Estimates of the reproduction numbers for the 2014 Ebola epidemic in the country of Bomi, Liberia (a); Kenema in Sierra Leone (b), and the measles epidemic in London in 1948 (c). The theoretical reproduction number given by Eq. (6) is represented by the dash line and its 95% confidence interval by the shaded area. The solid line shows the estimates of the empirical reproduction number given by Eq. (24)

example, [25, 26]). Although the theoretical estimates of the reproduction number in a sub-exponential framework (Eqs. (6) or (11)) have a declining secular trends, the stochastic effects (noise) in data are still present when estimating the parameters r and p [27].

4 Implications One of the most prominent application of compartmental epidemic models has been the determination of the population fraction that must be vaccinated in order to prevent the occurrence of an epidemic. In the simplest susceptible-infectedrecovered (SIR) type model, and when the early epidemic growth is exponential, the critical vaccination fraction is given by ρc = 1−1/R1 , where R1 is the reproduction number, remaining invariant over disease generations [28]. In the context of subexponential growth, the reproduction number changes over disease generations. However, we could apply the same argument and determine the critical fraction -1 = 1. The fraction ρ for vaccination by solving the equation (1 − ρ -c )R -c depends on the parameter p < 1 characterizing sub-exponential growth, and cannot exceed the corresponding fraction in the exponential epidemic growth. We note that the fraction ρ -c relates to the disease control at the onset of epidemic during the first disease generation.

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Our analysis could also provide a compelling argument for understanding the reasons that some epidemics, especially those that exhibit initial sub-exponential growth dynamics, undergo early extinction even in the absence of intervention measures. For a sufficiently small reproduction number of index generation R∗ > 0, or a sufficiently small p < 1, the reproduction number of the first disease generation -1 may be marginally greater than one, and therefore stochastic effects during the R early epidemic phase could lead to disease extinction given the declining trend in -k over disease generations. the reproduction number R

5 Discussion Since the pioneering work of Hamer [6], Sir Ross [7], and Kermack and McKendrick [3–5] that laid the ground for modern mathematical epidemiology between 1900 and 1937, a wealth of theory has been developed to study disease transmission dynamics, prevention, and control. This theory, however, relies on a fundamental assumption that the early phase of an epidemic exhibits exponential growth in the absence of any measures limiting disease spread. While this assumption holds for many epidemics, recent studies have provided data-driven evidence to show that sub-exponential growth has governed the dynamics of early epidemic phase for several infectious diseases, including HIV, foot-and-mouth disease, plague, and Ebola [11–15]. As we have shown in this study, early sub-exponential growth fundamentally affects disease dynamics, and therefore the existing theory of epidemics has limited application to study such dynamics. First and foremost is the fact that, unlike the case of exponential growth, the reproduction number varies over disease generations, even in the absence of any intervention measures. Second, due to this variation in the reproduction numbers, the cumulative number of infections does not satisfy the final size relation derived from the SIR compartmental models with exponential growth. Without further theoretical knowledge for sub-exponential growth, this effectively means that the type and intensity of measures to control the spread of a disease may not be readily determined as compared with the epidemics that exhibit early exponential growth dynamics. Comparatively, while disease spreads in a slower rate for epidemics with sub-exponential growth, this should not be interpreted as a higher degree of controllability which is affected by biological, epidemiological, and clinical characteristics of the disease [29], in addition to geographic and demographic variables of the population [30]. As noted earlier, the reproduction number for epidemics with exponential growth remains invariant over disease generations. However, our results show that there is a recursion dependence in the sequence of reproduction numbers for sub-exponential growth, which is a function of the deceleration of growth parameter as described in our analysis. This suggests that these reproduction numbers cannot be explicitly expressed based on disease-specific or epidemiological parameters. Our results here rely on the assumption of a fixed generation interval. When the shape of

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generation interval is taken into account [17], the relation between these quantities and parameters determining the distribution of generation interval remains elusive. Acknowledgements This work is supported by NSERC (Canada) and Mitacs (Canada).

References 1. R.M. Anderson, R.M. May, Infectious Diseases of Humans: Dynamics and Control (Oxford University Press, Oxford, 1991) 2. O. Diekmann, J. Heesterbeek, Mathematical Epidemiology of Infectious Diseases: Model Building, Analysis, and Interpretations (Wiley, Hoboken, 2000) 3. A.G. McKendrick, Proc. Edinb. Math. Soc. 14, 98–130 (1926) 4. W.O. Kermak, A.G. McKendrick, Proc. R. Soc. Lond. B 115, 700–721 (1927) 5. W.O. Kermack, A.G. McKendrick, J. Hyg. (Lond.) 37, 172–187 (1937) 6. W.H. Hamer, Lancet 1, 733–739 (1906) 7. R. Ross, The Prevention of Malaria (John Murray, London, 1911) 8. D. Mollison, Epidemic Models: Their Structure and Relation to Data (Cambridge University Press, Cambridge, 1995) 9. N.T.J. Bailey, The Mathematical Theory of Infectious Diseases and Its Applications (Hafner, New York, 1975) 10. S.M. Moghadas, Eur. J. Epidemiol. 21, 337–342 (2006) 11. S.A. Colgate, E.A. Stanley, J.M. Hyman, S.P. Layne, C. Qualls, Proc. Natl. Acad. Sci. U. S. A. 86, 4793–4797 (1989) 12. B. Szendroi, G. Csányi, Proc. R. Soc. Lond. B 271, S364–S366 (2004) 13. G. Chowell, C. Viboud, J.M. Hyman, L. Simonsen, PLoS Curr. 7 (2015). https://doi.org/10. 1371/currents.outbreaks.8b55f4bad99ac5c5db3663e916803261 14. C. Viboud, L. Simonsen, G. Chowell, Epidemics 15, 27–37 (2016) 15. G. Chowell, C. Viboud, L. Simonsen, S.M. Moghadas, J. R. Soc. Interface 13(123), 20160659 (2016) 16. J. Tolle, Math Gazette 87, 522–525 (2003) 17. J. Wallinga, M. Lipsitch, Proc. R. Soc. B 274(1609), 599–604 (2007) 18. M.G. Roberts, H. Nishiura, PLoS One 6(5), p.e17835 (2011) 19. J. Müller, C. Kuttler, Methods and Models in Mathematical Biology - Deterministic and Stochastic Approaches (Springer, Berlin, 2015) 20. A.M. Mood, F.A. Graybill, D.C. Boes, Introduction to the Theory of Statistics (McGraw-Hill, Singapore, 1974) 21. H. Jeffreys, B.S. Jeffreys, Methods of Mathematical Physics, 3rd edn. (Cambridge University Press, Cambridge, 1999) 22. J. Bibby, Glasgow Math. J. 15(01), 63–65 (1974) 23. M.A. Vink, M.C.J. Bootsma, J. Wallinga, Am. J. Epidemiol. 180(9), 865–875 (2014) 24. WHO Ebola Response Team, New Engl. J. Med. 371(16), 1481–1495 (2014) 25. A. Cori, N.M. Ferguson, C. Fraser, S. Cauchemez, Am. J. Epidemiol. 178(9), 1505–1512 (2013) 26. T. Obadia, R. Haneef, P.Y. Boelle, BMC Med. Inform. Decis. Mak. 12(1) (2012) 27. D. Champredon, D.J.D. Earn, Phys. Life Rev. 18, 105–108 (2016) 28. A. Scherer, A. McLean, Br. Med. Bull. 62(1), 187–199 (2002) 29. C. Fraser, S. Riley, R.M. Anderson, N.M. Ferguson, Proc. Natl. Acad. Sci. U. S. A. 101(16), 6146–6151 (2004) 30. M. Laskowski, L.C. Mostaço-Guidolin, A.L. Greer, J.Wu, S.M. Moghadas, Sci. Rep. 1(105), 1–7 (2011)

Numerical Simulations for Cardiac Electrophysiology Problems Alexey Y. Chernyshenko, A. A. Danilov, and Y. V. Vassilevski

1 Introduction In present times there has been a growing interest in developing numerical methods for models of cardiac electrophysiology. They can be used in clinical applications, such as the prediction of the arrhythmias, defibrillation therapy optimization, study of the effects of drugs and others. Efficient computational frameworks for the treatment of heart diseases are of great social interest and benefits. The systems of monodomain and bidomain equations are considered to be the most appropriate descriptions of the electrical activity in cardiac tissues. The bidomain equation is a system of two partial differential equations describing the evolution of the intracellular and extracellular potentials coupled at each point in space to a system of ordinary differential equations which describe transmembrane ionic current density. The electrophysiology model may be used to predict the severity and the duration of different types of arrhythmias under possible conditions (e.g., after radiofrequency ablation treatment of ventricular extrasystole causing premature ventricular contraction). Personalized models may be used to select the best treatment scenario and to minimize the possibility of required reoperation. The solution of electrophysiology problem may be split into two or three steps: the solution of coupled parabolic and elliptic PDEs (sequentially or fully implicitly) and the solution of a nonlinear system of ODEs. The finite element method is used conventionally for the spatial discretization of the monodomain and bidomain PDEs. There exist several computational frameworks for numerical solving of bidomain equations, including Chaste (Cancer, Heart and Soft Tissue Environment) [1],

A. Y. Chernyshenko () · A. A. Danilov · Y. V. Vassilevski Marchuk Institute of Numerical Mathematics, Moscow, Russia Sechenov University, Moscow, Russia © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_21

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CMISS/openCMISS (Continuum Mechanics, Image analysis, Signal processing and System Identification),1 CARP (Cardiac Arrhythmia Research Package),2 and FEniCS.3 Ani3D [2] framework benefits tools for usage of anisotropic grids and tensors, which are crucial in real heart modeling due to the essential anisotropic features of the myocardial tissue.

2 Models and Methods 2.1 Governing Equations Let  represent a 3D domain with piecewise smooth boundary ∂. We start with bidomain equations since monodomain equations are derived from the bidomain equations. The basic bidomain equations associate the intracellular potential φi and the extracellular potential φe defined in , through the transmembrane current density. The transmembrane voltage v = φi − φe and φe satisfy the equations  ∂v + Iion (u, v) − ∇ · (σi ∇(v + φe )) = Ii χ Cm ∂t

(1)

∇ · ((σi + σe )∇φe + σi ∇v) = −Itotal

(2)

∂u = f(u, v) ∂t

(3)

where Cm is the cell membrane capacitance, χ is the membrane surface-to-volume ratio, Ii is an intracellular volume current density stimulus, Ie is an extracellular volume current density stimulus, Itotal = Ii + Ie , and σi and σe are the intracellular and extracellular conductivity tensors, respectively. We use anisotropic conductivity tensors with their principal axes aligned with fiber orientation to account the role of electrical properties of myocyte fibers. The current density in the ionic channels Iion is defined by a function f of a vector of state variables u ≡ u(t, x) defined by a system of nonlinear ODEs. The boundary conditions for (1)–(2) represent zero flux across the boundary ∂: n · (σi ∇(v + φe )) = 0

(4)

n · (σe ∇φe ) = 0

(5)

where n is the outward unit normal vector to ∂. Initial conditions define the starting transmembrane voltage v and extracellular potential φe , as well as the initial vector of state variables of the ionic current model 1 http://opencmiss.org. 2 https://carp.medunigraz.at/. 3 https://fenicsproject.org/.

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u. Note that both intracellular and extracellular potentials are defined by Eqs. (1)– (5) only up to some constant. To ensure the uniqueness of the solution, one should enforce additional conditions: either a fixed value of φe in a selected ground node, or the zero mean potential across the domain. Equations (1) and (2) can be solved sequentially as a parabolic problem for unknown v and an elliptic problem for unknown φe , or simultaneously as a coupled problem. If the extracellular electric field can be ignored and the intracellular and extracellular conductivities are proportional, the bidomain equations can be replaced with the monodomain equation:  ∂v + Iion (u, v) − ∇ · (σ ∇v) = Istim χ Cm ∂t

(6)

and the corresponding system of ODEs (3). Here σ is the effective conductivity and Istim is the stimulus current. Typical boundary conditions are zero-flux as well. The accuracy of the monodomain model was investigated by comparing its results with those from the bidomain model, see [3]. The numerical tests demonstrated that even if differences between monodomain and bidomain results can be found, they are small enough to be ignored for most of practical applications. The bidomain-with-bath equations are used to model the case of cardiac tissue contained in an electrically conductive bath (e.g., the human body). Let b represent the bath domain. The transmembrane voltage v is defined only in the cardiac tissue , while φe is defined everywhere in  ∪ b , and inside the bath b φe satisfies ∇ · (σb ∇φe ) = 0

(7)

where σb is the conductivity of the bath. In addition to (4) one needs the continuity of φe across ∂. The remaining boundary/interface conditions are the following: n · σe ∇φe = n · σb ∇φe on ∂ (surf )

n · σb ∇φe = IE

on ∂b \ ∂

(surf )

is the stimulus current applied to the surface of the bath domain (e.g., where IE representing defibrillation electrodes).

2.2 Numerical Methods The bidomain system (1)–(2) can be solved sequentially as a parabolic problem for unknown v and an elliptic problem for unknown φe (decoupled formulation), or simultaneously as a coupled problem (coupled formulation), followed by the solution of the nonlinear system of ODEs. The advantage of the decoupled formulation is that the equations can be solved with different time steps and resulting

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linear systems are smaller than in that of the coupled formulation. On the other hand, the coupled formulation provides better accuracy for the numerical solution. We implemented both methods and decided to use the coupled formulation in our experiments. The finite element method (FEM) is used conventionally for the spatial discretization of the monodomain and bidomain PDEs. Complicated geometry of the human heart can be recovered by an unstructured tetrahedral mesh. The FEM on tetrahedral meshes is used in Chaste, FEniCS, CARP, and other frameworks. We use the Ani3D (Advanced Numerical Instruments) framework [2]. It can produce tetrahedral meshes adapted to geometric features and/or solution features through aniAFT and aniMBA libraries. The aniFEM library provides FEM discretizations of various types. Discretization in time is performed by the implicit schemes with the first (Euler) and the second (BDF) order of accuracy. Let t denote a time step χ Cm and κ = . In matrix notation the FEM coupled formulation of (1)–(2) is as t follows:

 n+1 

 v M(κvn − χ iion + ii ) κM + Ki Ki = , (8) Ki Ki+e φen+1 Mitotal where M is the FEM mass matrix, Ki and Ki+e are the FEM stiffness matrices with σi and σi + σe tensors, respectively, iion , ii , itotal are the vectors of currents and stimulus. To solve the linear system, we use the BSGStab iterative linear solver with the ILU(0) preconditioner. The solver is provided by Ani3D framework. For the numerical solution of ODE system (3) we use CVODE solver [4]. Ionic cell models are provided by biologically relevant cell models available from the CellML model repository [5].

2.3 Code Acceleration Preliminary profiling of the solution steps of the sequential code showed that ODE integration takes up to 90% of the total CPU time. We used the OpenMP technology in order to parallelize the computing of the ionic currents in each node of the computational grid. Table 1 represents the speedup of the ODEs integration step on one 24-cores node of INM RAS supercomputer. Table 1 The CPU time for ODEs integration on mesh with 500,000 nodes

CPUs 1 8 24

Time (s) 2064 290 146

Speedup 1× 7.11× 14.14×

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3 Numerical Results For verification of the code we consider several benchmarks. Verification of singlecell models (3) was based on comparison of numerical results with reference results computed with OpenCOR4 software.

3.1 Model Problems with Exact Solutions The first step for code verification is to solve model problems with given nonphysiological exact solution. We used 3D problems from [6] with known exact solutions v, u, φe and corresponding forcing terms (Iion , f). Zero stimulus currents are used for all problems in this section, Ii = Itotal = 0. The results of the convergence study on the series of hierarchically refined meshes with N nodes are presented in Tables 2, 3, 4. The numerical method shows the second order accuracy in L2 -norm. Table 2 Numerical errors for monodomain problem with exact solution

N 555 3202 21,476 152,351

||err||L2 7.045e−1 2.777e−1 7.271e−2 2.038e−3

Order – 1.73 2.24 1.96

Table 3 Numerical errors for bidomain problem with exact solution

N 555 3202 21,476 152,351

||err||L2 2.737e−1 9.534e−2 2.501e−2 6.824e−3

Order – 1.96 2.24 2.00

Table 4 Numerical errors for bidomain-with-bath problem with exact solution

N 325 2025 14,161 105,633

||err||L2 6.444e−2 2.496e−2 8.680e−3 2.759e−3

Order – 1.59 1.76 1.77

4 http://opencor.ws/.

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3.2 N -Version Benchmark The second step for verification of our framework is the N -version benchmark [7]. The domain  is a cuboid [0, 20] × [0, 7] × [0, 3] mm3 . The stimulus current (strength 50 mA/mm3 , duration 2 ms) was delivered to the volume of [1.5 × 1.5 × 1.5] mm3 located at one corner of the cuboid. The cell model is the ten Tusscher and Panfilov [8] model (epicardium variant). The monodomain formulation is used with following parameters: v0 = −85.23 mV, χ = 140 mm−1 , Cm = 0.01 µF/mm2 , and an anisotropic conduction tensor. More details of the benchmark can be found in the original paper. Comparison of the numerical result to the benchmark reference solution requires evaluation of the activation time, which is the time that the membrane potential passes through 0 mV. Figure 1 shows the activation time along the diagonal of the cuboid (from (0, 0, 0) to (20, 7, 3)) for two grids with the mesh sizes h = 0.5 mm and h = 0.2 mm and equal time steps t = 0.005 ms. The solutions demonstrate slight deviation. Figure 2 shows the activation time on the cross-section plane (details can be found in [7]). The results fall within the range of 11 simulations presented in the benchmark paper [7].

3.3 3D Rabbit Heart Finally we verify the code on a cardiac electrophysiological simulation. We address the 3D rabbit heart model [6]. We exploit the “Oxford Rabbit Heart” [9], a fine-resolution MR-based mesh of the rabbit heart with average element 45

Activation time (ms)

40 35 30 25 20 15 10

h = 0.2 h = 0.5

5 0

0

5

10 15 Distance (mm)

20

25

Fig. 1 Activation time along the cuboid diagonal for numerical solutions with t = 0.005 ms, h = 0.2 mm (solid), and h = 0.5 mm (dashed)

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Fig. 2 Activation time on the cross-section plane for numerical solutions with t = 0.005 ms, h = 0.5 mm

edge-length 125 µm. We also use two coarsened versions of this mesh, a mediumresolution mesh, and a coarse-resolution mesh with average edge length 251 and 482 µm, respectively. The reentry is induced by the S1–S2 protocol, consisting of an initial apical stimulus followed by a second stimulus applied to a large left-ventricular region 170 ms later. We solve the monodomain problem with the following parameters: v0 = −85.4485 mV, χ = 140 mm−1 , Cm = 0.01 µF/mm2 , σ = 0.093 mS/mm. The ionic cell model is the Mahajan et al. rabbit cell model [10]. Figure 3 shows several snapshots of the epicardial voltage at different time instances on the coarsest mesh. The transmembrane voltage is compared with the reference solution computed by the Chaste software. One can observe similar activation fronts for the solutions. Results on the finer meshes are similar.

4 Discussion We presented the new electrophysiological code based on the open-source Ani3D package. The code targets anisotropic conductivity tensors and adaptive unstructured meshes. The numerical solutions produced by the code were compared with the reference solutions computed by other codes on a series of benchmarks proposed in the literature. The use of HPC systems for solver acceleration requires further parallelization on distributed memory systems and highly parallel computing devices such as GPUs. The Ani3D framework allows us to extend our electrophysiological solver to electromechanical problems and other multiphysics cardiac problems.

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7.5e+00 0 –10 –30 –40 –50

solution

–20

–60 –70 –80 –8.5e+01

1.4e+01

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0 –10 –20 –30 –40 –50 –60 –70 –80 –8.9e+01

1.0e+01 0 –10 –30 –40 –50

solution

–20

–60 –70 –80 –8.6e+01

Fig. 3 Epicardial voltage at different time instances: 300 ms (top), 400 ms (middle), 500 ms (bottom). Ani3D solution (left), Chaste solution (right)

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Acknowledgement The work has been supported by the Russian Science Foundation grant 1431-00024.

References 1. G. Mirams et al., Chaste: an open source C++ library for computational physiology and biology. PLOS Comput. Biol. 9(3), e1002970 (2013) 2. Advanced Numerical Instruments 3D. https://sourceforge.net/p/ani3d 3. M. Potse et al., A comparison of monodomain and bidomain reaction-diffusion models for action potential propagation in the human heart. IEEE Trans. Biomed. Eng. 53(12), 2425–2435 (2006) 4. A.C. Hindmarsh et al., SUNDIALS: suite of nonlinear and differential/algebraic equation solvers. ACM Trans. Math. Softw. 31, 363–396 (2005) 5. T. Yu et al., The Physiome Model Repository 2. Bioinformatics 27, 743–744 (2011) 6. P. Pathmanathan, R.A. Gray, Verification of computational models of cardiac electrophysiology. Int. J. Numer. Methods Biomed. Eng. 30, 525–544 (2014) 7. S.A. Niederer et al., Verification of cardiac tissue electrophysiology simulators using an Nversion benchmark. Philos. Trans. R. Soc. 369, 4331–4351 (2011) 8. K.H.W.J. ten Tusscher, A.V. Panfilov, Alternans and spiral breakup in a human ventricular tissue model. Am. J. Physiol. Heart Circ. Physiol. 291, H1088–H1100 (2006) 9. M. Bishop et al., Development of an anatomically detailed MRI-derived rabbit ventricular model and assessment of its impact on simulations of electrophysiological function. Am. J. Physiol. Heart Circ. Physiol. 298(2), H699–H718 (2010) 10. A. Mahajan et al., A rabbit ventricular action potential model replicating cardiac dynamics at rapid heart rates. Biophys. J. 94(2), 392–410 (2008)

Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory Optimization Hypothesis Abdoulaye Mendy, Mountaga Lam, and Jean Jules Tewa

1 Introduction In Sahelian countries, pastoralism appears to be the main sustainable development activity of the fragile and irregular natural vegetation. The vast majority of fodder consumed by ruminants in the Sahelian region is still formed by natural pasture. An important issue in this part of Africa is the management of herds by pastoralists with respect to the available resource. Herd, according to mobility, has long been the main functional response of livestock keepers in the face of interannual variability in the feedstocks and watering difficulties during the dry season. Nowadays, in addition to the effect of climate events, herd community also experiences several disturbances including harvesting. Indeed, in this region herd is harvested for survival needs (food, sale, etc.). It is therefore necessary to take into account herd harvesting in order to study herd dynamics in Sahelian regions. On the other hand, there are many work on predator–prey models which consider harvesting [7, 15]. In addition, it is well-known that after consumption of resource herds need some time so that the consumed resource is beneficial to them. Taking this hypothesis into account leads to delay differential systems. In this paper, we consider a system of differential equations modeling the resource–herd dynamics with the piecewise threshold policy harvesting and a Holling response function of type III.

A. Mendy () · M. Lam Cheikh Anta Diop University, Department of Mathematics and Informatics, Dakar, Senegal J. J. Tewa University of Yaounde I, National Advanced School of Engineering, Department of Mathematics and Physics, Yaounde, Cameroon © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_22

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A common feature of plant–herbivore interactions is that as herbivory reduces the plant biomass of its maximum crop, plant productivity will increases [14]. To characterize the impact of herbivory on the dynamics of the resource, several theories have been proposed including the herbivores optimization hypothesis (HOH) [6, 11]. The HOH indicates that as the intensity of pasture increases, net primary productivity increases to optimum grazing intensity, then decreases to a lower level than the non-fattened resource [14]. The hypothesis of optimization of the grazing of herbivores was used by Lebon et al. [9]. This resource growth function is defined by (see Lebon et al. [9]):   H H 1− R − δR 2 , ϕ(R) = r 1 + β γ where γ > β and R denotes the forage resources density. The term δR 2 is the density dependent death rate of the resource as a consequence of local competition for water and nutrients. The particularity of this model, which is the difference with classic predator–prey models in the literature, is the fact that the intrinsic production rate can be negative (r < 0) [13]. This situation is frequently observed in SubSaharan Africa, particularly in Sahelian region when precipitations are scarce [13]. Intensification of this situation can conduct pastoralists to migrate with their herds: this is the transhumance. Primary production is heavily dependent on precipitation [3], which means that the intrinsic growth rate may be negative. The intrinsic growth rate is defined by Tewa et al. [13] as r = r01 p + r02 − r2 , where p denotes the mean annual precipitation, r02 is the growth rate of the resource without precipitation, r2 is the natural death. In the study of interactions between herd and resource, it is important to determine which specific form of functional response, describing the amount of resource consumed by an animal in the herd per unit of time, is ecologically meaningful. We can use the Holling response function of type III aR 2 , reflecting very small consumption when the amount defined by p(R) = b + R2  of resource is small (p (0) = 0), and a group of advantages for resource when the amount of resource is high p(R) tends to a when R tends to infinity , where a is the maximum consumption rate of herd on the resource, b is the half-saturation constant. Here, a time delay τ is also considered on the predator response term p(x(t)) in the herd equation. The piecewise threshold policy harvesting G(H ) is given by Bohn et al. [2] and Tankam et al. [12]: ⎧ ⎪ 0 if ⎪ ⎨ h(H − T ) 1 G(H ) = if ⎪ T − T1 ⎪ ⎩ 2 h if

H < T1 , T 1 ≤ H ≤ T2 ,

(1)

H ≥ T2 .

For this harvesting function, T1 and T2 are the threshold values. In this way, once the resource biomass reaches the size H = T1 , then harvesting starts and increases

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0.9 0.8 0.7 0.6 0.5 G 0.4 0.3 0.2

T1

T2

0.1 0

0

1

2

3

4

5

H

Fig. 1 Harvesting function (T1 = 0.8, T2 = 1.5, and h = 0.9) given by Eq. (1) (reproduced from Tankam et al. [12])

linearly to a limit value h that will be the constant value of the harvest from the threshold T2 (see Tankam et al. [12]) (Fig. 1). The novelties in this work are the introduction of the HOH in a resource– herbivore model with delay and the harvest with two thresholds of crops in the herbivore population. The following differential equations model interactions between forage resources and herds: ⎧   dR(t) H (t) H (t) aR 2 (t) ⎪ ⎪ = r 1+ 1− R(t) − δR 2 (t) − H (t), ⎪ ⎪ ⎨ dt β γ b + R 2 (t) ⎪ ⎪ ⎪ dH (t) aR 2 (t − τ ) ⎪ ⎩ =e H (t) − μH (t) − G(H (t)), dt b + R 2 (t − τ )

(2)

where e is the conversing rate of forage resources consumed into herd biomass. The model system (2) is considered with the following initial values: R(θ ) = ϕ1 (θ ) ≥ 0, H (θ ) = ϕ2 (θ ) ≥ 0, θ ∈ [−τ, 0], where ϕ1 , ϕ2 ∈ C([−τ, 0], R2+0 ) is the Banach space of continuous functions mapping the interval [−τ, 0] into R2+0 , where R2+0 = {(R, H )/ R ≥ 0, H ≥ 0}.

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2 Basic Results 2.1 Boundedness of Solutions We start by showing that solutions of system (2) that start in R2+ will remain there and are uniformly bounded. The following Theorem 2.1 is valid. Its proof can be done as in [12]. Theorem 2.1 Every solution of system (2) that starts in R2+ will remain there and is uniformly bounded.

2.2 Equilibria of the Model Concerning equilibria of the model, we have the following definitions. 4βγ ηa r β aR ∗ (1 − ), σR ∗ = , Let N0 = , ηs = 1 2 βσR ∗ γ b + R ∗2 2 (γ − β) (1 − ) + 4βγ ηs ∗ eaR3∗2 (T2 − T1 ) δR ea h ηa = , S01 = , S0 = , S0∗ = , and r μ μ(T2 − T1 ) + h h(b + R3∗2 ) eaT2 R4∗2 S0∗∗ = . (h + μT2 )(b + R4∗2 ) Proposition 2.1 The following results hold for system (2). (1) Assume that H < T1

r , 0 of system (2) exist without any (a) Equilibria E0 = (0, 0) and E1 = δ condition. (b) We have two cases for the coexistence equilibrium of system (2). Case 1: If N0 = 1, we have a unique equilibrium E0∗ = (R ∗ , H0∗ ) with 7  1 ∗ bμ 1 ∗ γ −β 1− . E0 is ecologically and H0∗ = R = eb a − μb 2 ηs   1 1 γ −β 1− meaningful whenever S01 > 1, ηs > 1, and < T1 . 2 ηs ∗ Case 2: If N0 < 1, we have two equilibria E1 = (R ∗ , H1∗ )  1 1  γ − β 1 − and E2∗ = (R ∗ , H2∗ ) with H1∗ = − 2 ηs 8  2    1 2 γ −β 1− − 4βγ ηa − 1 and ηs 8   2    1 1 2 1  ∗ γ −β 1− + γ −β 1− − 4βγ ηa − 1 . H2 = 2 ηs ηs ∗ E1 is ecologically meaningful whenever N0 < 1, S01 > 1, ηa > 1, and

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ηs > 1. E2∗ is ecologically meaningful whenever (S01 > 1, ηa < 1, H1∗ < T1 , and H2∗ < T1 ) or (S01 > 1, ηa < 1, ηs < 1, H1∗ < T1 , and H2∗ < T1 ). (2) Assume that T1 ≤ H ≤ T2 E3∗ = (R3∗ , H3∗ ), hT1 (b + R3∗2 ) where H3∗ =  is an ecologμ(T2 − T1 ) + h (b + R3∗2 ) − eaR3∗2 (T2 − T1 ) ically acceptable equilibrium if R3∗ is a nonnegative value which satisfies equation: A5 R3∗5 + A4 R3∗4 + A3 R3∗3 + A2 R3∗2 + A1 R3∗ + A0 = 0,

(3)

  S 0 T1 and the coefficients Ai given in the with S0 1 − S0 < S0∗ < S0 1 − T2 proof. (3) Assume that H ≥ T2 h(b + R4∗2 ) is an ecologically E4∗ = (R4∗ , H4∗ ), where H4∗ = eaR4∗2 − μ(b + R4∗2 ) ∗ acceptable equilibrium if R4 is a nonnegative value which satisfies equation: B5 R4∗5 + B4 R4∗4 + B3 R4∗3 + B2 R4∗2 + B1 R4∗ + B0 = 0,

(4)

R∗

with S0 4 > 1 and S0∗∗ < 1 and the coefficients Bi given in the proof. Proof See Appendix 1.

 

In the sequel, we deal with the problem of stability characterization of various equilibria of model (2). For that we first consider the case where there are no delays and second focus on the case where delays are considered. This process is motivated by the fact that if an equilibrium is unstable without delays, it will remain unstable when delays are considered (see Culshaw and Ruan [4] and Martin and Ruan [10]).

3 Stability Analysis and Hopf Bifurcation When τ = 0 3.1 Stability Analysis When τ = 0 Theorem 3.1 Let S0 =

aR0∗2 H0∗ ear 2 R∗ H ∗ = ; S . 1 μ(bδ 2 + r 2 ) abH0∗ + δ(b + R0∗2 )2

(1) If r < 0, then E0 is locally asymptotically stable (LAS).

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(2) If S0 < 1, then E1 is LAS. ∗ ∗ (3) Suppose that S1R H < 1. We have two cases. (a) The equilibrium E0∗ = (R ∗ , H0∗ ) is LAS if it is unique. (b) If there exist two equilibria, then one is LAS and the other is unstable. (4) The coexistence equilibrium E3∗ = (R3∗ , H3∗ ) when it exists, is LAS whenever tr(J3 ) < 0 and det (J3 ) > 0, where tr(J3 ) and det (J3 ) are the trace and determinant of the Jacobian matrix at E3∗ . (5) The coexistence equilibrium E4∗ = (R4∗ , H4∗ ) when it exists, is LAS whenever tr(J4 ) < 0 and det (J4 ) > 0, where tr(J4 ) and det (J4 ) are the trace and determinant of the Jacobian matrix at E4∗ . Proof See Appendix 2.

 

Remark 3.1 The stability of equilibrium E0 means that there will be no more forage resources and no animal of the herd in the considered area for a long moment. There can be many explanations to this situation. Droughts can cause animals mortality through starvation, emergency slaughtering, and sales, or definitive herds migrations (transhumance), which can create severe drops in the herd sizes. When this removal concerns only animals of the herbivores, depending on some climatic changes, the forage resources growth towards the maximal quantity needed for livestock and the equilibrium E1 is stable. Theorem 3.2 (Hopf Bifurcation) There is a stable limit cycle that surrounds E3∗ (b + R ∗2 )2  hT1 and δ+ ∗ ∗ when a passes through the value ah = ∗ ∗2 3 H3 R3 (T2 − T1 ) H3 (R3 − b) a stable limit cycle that surrounds E4∗ when δ passes through the value δh2 = aH4∗ (R4∗2 − b) h + ∗ ∗ . This phenomenon is known as Hopf bifurcation. Since the H4 R4 (b + R4∗2 )2 limit cycle is stable, it is a supercritical Hopf bifurcation. The pastoral interpretation of Hopf bifurcation as we said is that animals in the herd will coexist with the forage resources, exhibiting oscillatory balance behavior. Proof See Appendix 3.

 

Remark 3.2 (Non-existence of the Hopf Bifurcation) According to expressions of ah and δh in Theorem 3.2, one deduces that the Hopf bifurcation cannot occur in system 2 whenever R3∗2 ≤ b and R4∗2 ≤ b, respectively. Indeed, in both cases the trace (tr) of the Jacobian matrix at the corresponding equilibrium is such that tr < 0. Remark 3.3 The importance of this section is due to the fact that if an equilibrium of model (2) is unstable for τ = 0 it remains unstable for τ > 0 (see Culshaw and Ruan [4] and Martin and Ruan [10]).

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4 Stability Analysis with Positive Delay In this section we study the stability of equilibria in the presence of discrete delays τ > 0. Let E ∗ = (R ∗ , H ∗ ) be an equilibrium of the model, and let us define u(t) = R(t) − R ∗ and v(t) = H (t) − H ∗ . Then the linearized system of (2) at E ∗ is given, in matrix form, by d dt



u(t) v(t)



= A1

u(t) v(t)







+ A2

u(t − τ ) , v(t − τ )

where A1 =

a1 a2 0 a4

 , A2 =

0 0 a3 0



  H∗ H∗ 2abR ∗ H ∗ 1− − 2δR ∗ − , and a1 = r 1 + β γ (b + R ∗2 )2 aR ∗2 2eabR ∗ H ∗ eaR ∗2 ∗  ∗ a2 = rR βγ (γ −β −2H )− b + R ∗2 , a3 = (b + R ∗2 )2 , a4 = b + R ∗2 −μ−G (H ). The characteristic equation at equilibrium E ∗ is given by λ2 − (a1 + a4 )λ + a1 a4 − a2 a3 e−λτ = 0.

(5)

 r , 0 are When τ > 0, the characteristic equations at E0 = 0, 0 and E1 = δ independent of the delays τ , equilibria E0 and E1 keep their stability when τ > 0. We now analyze the local stability of equilibrium E ∗ when τ > 0. The characteristic equations at E ∗ when τ > 0 are given by λ2 − (a1 + a4 )λ + a1 a4 − a2 a3 e−λτ = 0.

(6)

When substituting λ = iω in (6), the real and the imaginary part satisfy − ω2 + a1 a4 = a2 a3 cos ωτ,

(7)

(a1 + a4 )ω = a2 a3 sin ωτ.

(8)

Eliminating τ by squaring and adding Eqs. (7) and (8), we get the algebraic equation ω4 + (a12 + a42 )ω2 + a12 a42 − a22 a32 = 0.

(9)

Substituting ω2 = η in the above equation, we obtain a cubic equation in η in the form

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η2 + (a12 + a42 )η + a12 a42 − a22 a32 = 0.

(10)

Case 1: Suppose H < T1 . In this case the characteristic equation (10) becomes η2 + a12 η − a22 a32 = 0.

(11)

By Descartes rule of sign, Eq. (11) has at last one positive root. The following theorem gives a criterion for the switching in the stability of E ∗ in terms of the delay parameter τ . Theorem 4.1 Suppose that E ∗ exists and is LAS for (2) with τ = 0. Also let ω02 = η0 be a positive root of (11). Then there exists a τ = τ 0 such that E ∗ is LAS for τ ∈ (0, τ 0 ] and unstable for τ > τ 0 . Furthermore, the system undergoes a Hopf bifurcation at E ∗ when τ = τ 0 . Proof See Appendix 4.

 

˜ Direction and Stability of the Hopf 4.1 Stability Analysis of E: Bifurcation In the previous section, we studied mainly the stability of the positive equilibrium E ∗ = (R ∗ , H ∗ ) of system (2) and the existence of Hopf bifurcations at this equilibrium. In this section, we shall study the properties of the Hopf bifurcations obtained in Theorem 4.1 and the stability of bifurcated periodic solutions occurring through Hopf bifurcations by using the normal form theory and the center manifold reduction for retarded functional differential equations (RFDEs) due to Hassard et al. [5]. Throughout this section, we always assume that system (2) undergoes Hopf bifurcation at the positive equilibrium E ∗ = (R ∗ , H ∗ ) for τ = τj0 , (j = 0, 1, 2....) and then ±iω0 is the corresponding purely imaginary roots of the characteristic equation (11). Theorem 4.2 There exist three real numbers μ2 , β2 , and T2 which determine the qualities of bifurcating periodic solution in the center manifold at the critical value τ 0 such that: (a) When μ2 > 0, then the Hopf bifurcation is supercritical and the bifurcating periodic solution exists for τ > τ 0 and when μ2 < 0, then the Hopf bifurcation is subcritical and the bifurcating periodic solution exists for τ < τ 0 . (b) When β2 < 0, the bifurcating periodic solutions are stable and when β2 > 0, the bifurcating periodic solutions are unstable. (c) When T2 > 0, the period increases and it decreases when T2 < 0. Proof See Appendix 5.

 

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Case 2: Suppose T1 ≤ H ≤ T2 or H ≥ T2 . In this case we have η2 + (a12 + a42 )η + a12 a42 − a22 a32 = 0.

(12)

We consider only the case when (12) possesses exactly one positive root. This occurs if a1 a4 + a3 a2 < 0. Under this condition, Eq. (12) possesses a positive root η0 = ω02 and thus (5) has the pair of purely imaginary roots ±iω0 . Theorem 4.3 Suppose that E ∗ exists and is LAS for (2) with τ = 0. Also let ω02 = η0 be a positive root of (12). Then there exists a τ = τ 0 such that E ∗ is LAS for τ ∈ (0, τ 0 ] and unstable for τ > τ 0 . Furthermore, the system undergoes a Hopf bifurcation at E ∗ when τ = τ 0 . Proof The proof is done in the same way as the proof of Theorem 4.1.

 

Using a nonstandard numerical scheme as in [1], we provide the following numerical simulations where parameter values have been chosen in such a way that they obey the conditions for stability or bifurcation.

5 Numerical Simulations In this section, we present some numerical simulations to illustrate the analytical results observed in the previous sections. We consider the following set of parameter values: r = 2; β = 0.4; γ = 0.9; δ = 0.5; a = 0.7; b = 0.6; e = 0.8; μ = 0.3; h = 0.8; T1 = 0.9; T2 = 1.5 (Figs. 2 and 3).

6 Conclusion In this paper, we developed a delayed resource–herd model taking into account the well-mastered herbivory optimization hypothesis. The aim is to study a plant– herbivore interactions model in Sahelian regions. Moreover, we take into account the harvest of the herbivore population which is motivated by empirical evidences in Sahelian regions. Indeed, in these parts of Africa, herd is harvested mainly for survival needs (food, sales, etc.). Theoretical and numerical analyses of our model reveal various equilibria and very rich dynamics. We showed that the time delay can induce instability of our equilibria. We also proved that the time delay can induce oscillations in the system outcomes via Hopf bifurcations. In a forthcoming paper, we will move towards a more realistic modeling of herd harvesting. Indeed, it is well-known that harvest is not a time-continuous event. It occurs very often at specific times (i.e., impulsive events). Thus our next model will deal with the theory of impulsive differential equations (see Lashmikantan et al. [8]).

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Fig. 3 (a) Illustration of locally asymptotically stability of coexistence equilibrium when τ = 0. (b) Phase portrait resource–herd when τ = 0

Acknowledgement Abdoulaye Mendy partially supported by grant 2-4570.5 of the Swiss National Science Foundation.

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Appendix 1: Proof of Proposition 2.1 Looking for the equilibria in system (2) is equivalent to solving the following system: ⎧   2 H H ⎪ 2 − aR ⎪ r 1 + Hb = 0; (i) 1 − R − δR ⎪ ⎪ ⎨ β γ b + R2 ⎪ ⎪ 2 ⎪ ⎪ ⎩ e aR H − μH − G(H ) = 0. (ii) b + R2

(13)

(1) If H < T1 , i.e., G(H ) = 0. (a) The trivial equilibria of system (13) are given by E0 = (0, 0) and E1 = r ( , 0). δ 7 bμ ∗ with S01 > 1 and (b) If R = 0 and Hb = 0, we have from (ii) R = ea − μ ea . Substituting R ∗ in (i) gives S01 = μ    1 H + βγ ηa − 1 = 0, H2 − γ − β 1 − ηs where ηs =

r βσR ∗

(14)

 β δR ∗ 1− . The discriminant of Eq. (14) and ηa = γ r

is given by  1 2  = (γ − β) 1 − − 4βγ (ηa − 1) . ηs 2

Let us set N0 =

4βγ ηa .  1 2 (γ − β)2 1 − + 4βγ ηs

(i) If N0 > 1, i.e.,  < 0, then (14) does not admit a real solution. ∗ (ii) If N0 = 1, i.e.,  = 0, admits a unique solution H0 defined  then (14)  1 1 by H0∗ = . γ −β 1− 2 ηs (iii) If N0 < 1, i.e.,  > 0, then (14) admits two solutions H1∗ and H2∗ given by 8 + *   1 1 2 γ −β 1− − (γ −β)2 (1 − ) − 4βγ (ηa − 1) 2 ηs ηs

1 H1∗ =

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and 8 + *   1 1 1 2 ∗ 2 H2 = γ −β 1− + (γ − β) (1 − ) − 4βγ (ηa − 1) . 2 ηs ηs Since R ∗ > 0 and 0 < Hi∗ < T1 , i = 0, 1, 2, we have E0∗ = (R ∗ , H0∗ ) is ecologically meaningful whenever S01 > 1, ηs > 1, and  1 1 γ −β 1− < T1 . 2 ηs ∗ E1 is ecologically meaningful whenever N0 < 1, S01 > 1, ηa > 1, and ηs > 1. E2∗ is ecologically meaningful whenever (S01 > 1, ηa < 1, H1∗ < T1 , and H2∗ < T1 ) or (S01 > 1, ηa < 1, ηs < 1, H1∗ < T1 , and H2∗ < T1 ). h(H − T1 ) (2) If T1 ≤ H ≤ T2 , i.e., G(H ) = . T2 − T1  hT1 b + R3∗2  We have from (ii) H3∗ =  . μ(T2 − T1 ) + h b + R3∗2 − eaR3∗2 (T2 − T1 ) Substituting H3∗ in (i) gives A5 R3∗5 + A4 R3∗4 + A3 R3∗3 + A2 R3∗2 + A1 R3∗ + A0 = 0,

(15)

 2 where A5 = −δβγ (T2 − T1 )(μ − ea) + h ,  2  A4 = δβγ (T2 − T1 )(μ − ea) + h + rhT1 (γ − β) (T2 − T1 )(μ − ea) + h − rh2 T12 ,   A3 = − (T2 − T1 )(μ − ea) + h 2δβγ (μ(T2 − T1 ) + h) + βγ ahT1 ,   A2 = 2βγ rb(μ(T2 − T1 ) + h) (T2 − T1 )(μ − ea) + h + rbhT1 (γ − β) (T2 − T1 )(2μ − ea) + 2h − 2rh2 T12 b,   A1 = −βγ b μ(T2 − T1 ) + h δb(μ(T2 − T1 ) + h) + ahT1 ,  2    A0 = rβγ b2 μ(T2 − T1 ) + h + rb2 hT1 γ − β μ(T2 − T1 ) + h − rh2 b2 T12 .   S0 T1 , where Since T1 ≤ H3∗ ≤ T2 , we have S0 1 − S0 < S0∗ < S0 1 − T2 ∗2 eaR3 (T2 − T1 ) h and S0∗ = . S0 = μ(T2 − T1 ) + h h(b + R3∗2 ) Then, E3∗ = (R3∗ , H3∗ ) is an ecologically acceptable equilibrium if R3∗ is a 

nonnegative value which satisfies Eq. (15) and S0 1 − S0 S0 T1 . T2 (3) If H ≥ T2 , i.e., G(H ) = h.

< S0∗ < S0 1 −

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  h b + R4∗2 . H4∗ > 0 ⇔ eaR4∗2 − μ b +  We have from (ii) H4∗ = eaR4∗2 − μ b + R4∗2 eaR4∗2 R∗ R∗ . Substituting H4∗ in (i) R4∗2 > 0, i.e., S0 4 > 1, where S0 4 =  ∗2 μ b + R4 gives B5 R4∗5 + B4 R4∗4 + B3 R4∗3 + B2 R4∗2 + B1 R4∗ + B0 = 0,

(16)

where B4 = rβγ (ea − μ)2 + rh(γ − β)(ea − μ) − rh2 , B5 = −δβγ (ea − μ)2 , B3 = (ea − μ)(2δβγ μb − ah), B2 = −2rβγ μb(ea − μ) + rhb(γ − β)(ea − 2μ) − 2rbh2 , B1 = δβγ μ2 b2 + ahbμ, B0 = rβγ μ2 b2 − rh(γ − β)μb2 − rh2 b2 . eaT2 R4∗2 . Since, H4∗ ≥ T2 we have S0∗∗ < 1, where S0∗∗ = (h + μT2 )(b + R4∗2 ) Then, E4∗ = (R4∗ , H4∗ ) is an ecologically acceptable equilibrium if R4∗ is a R∗

nonnegative value which satisfies Eq. (16), S0 4 > 1 and S0∗∗ < 1.

Appendix 2: Proof of Theorem 3.1 (1) By computing the Jacobian matrix at equilibrium E0 , E1 , E3∗ , or E4∗ , we deduce their stability. (2) Local stability of the equilibrium point E ∗ . The Jacobian matrix at the point E ∗ is given by ⎛

  ∗ aR ∗ H ∗ (R ∗2 − b) aR ∗2 ∗ r ∗ γ − β − 2H R − δR − ⎜ βγ (b + R ∗2 )2 b + R ∗2 ⎜ J (E ∗ ) = ⎜ ⎜ ⎝ 2abR ∗ H ∗ 0 (b + R ∗2 )2

⎞ ⎟ ⎟ ⎟, ⎟ ⎠

The characteristic polynomial: P (λ) = λ2 − m∗11 λ − m∗12 m∗21 , admits negative roots if m∗11 < 0 and m∗12 m∗21 < 0. aR ∗ H ∗ (R ∗2 −b) aR ∗2 H ∗ ∗ ∗ m∗11 = −δR ∗ < 0, i.e., S1R H = < 1. ∗2 2 ∗ (b+R ) abH + δ(b + R ∗2 )2 ∂ R˙ = ψ  (H ) − G (H ) < 0 with ψ(H ) = m∗12 m∗21 < 0, i.e., m∗12 = ∂H   H dR H = 0 is 1− and G(H ) = δR ∗ + σR ∗ H. Moreover r 1+ β γ dt equivalent to

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( )

(

)

°

°

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°

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Fig. 4 We present three different configurations. Case (i): there is no equilibria. Case (ii): the unique equilibrium is LAS. Case (iii): there exist two equilibria: one is LAS and the other is unstable

ψ(H ) − G(H ) = 0;

(17)

then, m∗12 < 0 is equivalent to ψ  (H ) − G (H ) < 0. Furthermore, by using relation (17) we deduce graphically the stability of equilibrium E ∗ (Fig. 4).

Appendix 3: Proof of Theorem 3.2 The characteristic polynomial of the Jacobian matrix at the point E3∗ is given by P (λ) = λ2 − tr(J3 )λ + det (J3 ). aR3∗ H3∗ (R3∗2 − b)

(18)

hT1 , E ∗ is the − T1 ) 3 only candidate for a Hopf bifurcation. It follows from expression of tr(J3 ) that if a Hopf bifurcation occurs, there exists ah that tr(J3 ) = 0, i.e.,

From expression of tr(J3 ) = −δR3∗ +

−δR3∗ +

(R3∗2

aR3∗ H3∗ (R3∗2 − b) (R3∗2

+ b)2

−

+ b)2

−

H3∗ (T2

hT1 = 0, H3∗ (T2 − T1 )

i.e., ah =

(b + R3∗2 )2 

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δ+

hT1 . H3∗ R3∗ (T2 − T1 )

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The discriminant of (18) is given by  = tr(J3 )2 − 4det (J3 ). (1) If a = ah and  < 0, then the eigenvalues of P (λ) are given by λ1,2 (a) = α1,2 (a) + iβ1,2 (a). (2) If a = ah and  < 0, then λ1,2 (ah ) = iβ1,2 (ah ). aR3∗ H3∗ (R3∗2 − b) ∂α1,2 (ah ) = (3) Moreover, = 0. Then we can conclude: ∂a (R3∗2 + b)2 (a) a = ah is a bifurcation value. (b) There exists ah1 > ah such that for all a ∈ [ah , ah1 ], the equilibrium point E3∗ is a stable focus. (c) For every neighborhood U of E3∗ , there exists ah2 < ah such that for every a ∈ [ah2 , ah [, the equilibrium E3∗ is an unstable focus surrounded by a stable limit √ cycle contained in U whose amplitude increases and is of the order of ah − a. Therefore, in Theorem 3.2, we have a supercritical Hopf bifurcation with the appearance of a stable limit cycle. Proof of Hopf bifurcation at point E4∗ is done in the same way.

Appendix 4: Proof of Theorem 4.1 Since ω0 is a solution of Eq. (9) the characteristic equation (5) has the pair of purely imaginary roots ±iω0 . From Eqs. (7) and (8), we have cos ω0 τ =

−ω02 . a2 a3

Hence, τn0 for n = 0, 1, . . . as a function of ω is given by τn0

 9 −ω02 1 2π n = arccos . + ω0 a2 a3 ω0

(19)

For τ = 0, Theorem 3.1 ensures that E ∗ is LAS; hence, by Butlers lemma (in [12]) E ∗ remains stable up to the minimum value of τn0 , here obtained for n = 0, i.e., for τ < τ00 , so that τ 0 = min τn0 = τ00 . n≥0

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The theorem will be completely proven if we can show that  sign

 d(Reλ(τ ))  > 0.  λ=iω0 dτ

Differentiating Eq. (5) with respect to τ yields  dλ  2λ − a1 + τ a2 a3 e−λτ = −λa2 a3 e−λτ , dτ which implies that 

dλ dτ

−1

=−

τ 2λ − a1 + τ a2 a3 e−λτ 1 1 − . =− 2 − 2 λa2 a3 e−λτ λ λ − a1 λ λ

Thus,   9  d(Reλ(τ ))  dλ −1  sign = sign Re   λ=iω0 λ=iω0 dτ dτ  9 ω02 1 = sign + 4 . ω02 ω0 + a12 ω02 

 d  (Reλ(τ )) > 0. Hence, the transversality condition is satisfied λ=iω0 dτ and a Hopf bifurcation occurs at τ = τ 0 . This achieves the proof.

We have

Appendix 5: Proof of Theorem 4.2 Let x1 (t) = R(t) − R ∗ and x2 (t) = H (t) − H ∗ ; then system (2) is equivalent to the following two dimensional system: ⎧ dx1 (t)  ⎪ ∗ )−2δR ∗ −P  (R ∗ )H ∗ x (t) ⎪ = ψ(H 1 ⎪ ⎪ dt   ⎪ ⎪ ⎨  ∗ ∗ ∗ + ψ (H )R −P (R ) x2 (t)+f1 x1 (t), x2 (t) ; ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎩ dx2 (t) = eP  (R ∗ )H ∗ x1 (t − τ ) + f2 x1 (t), x2 (t), x1 (t − τ ) , dt

(20)

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  H aR 2 H 1− , P (R) = where ψ(H ) = r 1 + , β γ b + R2    2   f1 x1 (t), x2 (t) = ψ x2 (t) + H ∗ x1 (t) + R ∗ − δ x1 (t) + R ∗ − P x1 (t) +    R ∗ x2 (t) + H ∗ − ψ(H ∗ ) − 2δR ∗ − P  (R ∗ )H ∗ x1 (t) + ψ  (H ∗ )R ∗ −     P (R ∗ ) x2 (t) and f2 x1 (t), x2 (t), x1 (t − τ ) = eP x1 (t) + R ∗ x2 (t) + H ∗ −  μ x2 (t) + H ∗ − eP  (R ∗ )H ∗ x1 (t − τ ). Let τ = τ 0 + ν, then ν = 0 is the Hopf bifurcation value of system (2) at the positive equilibrium E ∗ = (R ∗ , H ∗ ). Since system (2) is equivalent to system (20), in the following discussion we shall consider mainly system (20). In system (20), let ui (t) = xi (τ t). Then system (20) can be rewritten as follows: ⎧  dx1 (t) ⎪ 0 + ν) ψ(H ∗ ) − 2δR ∗ − P  (R ∗ )H ∗ x (t) ⎪ = (τ 1 ⎪ ⎪ dt    ⎪ ⎪ ⎨  ∗ ∗ ∗ + ψ (H )R − P (R ) x2 (t) + f1 x1 (t), x2 (t) ; ⎪ ⎪ ⎪ ⎪    ⎪ ⎪ ⎩ dx2 (t) = (τ 0 +ν) eP  (R ∗ )H ∗ x1 (t−τ )+f2 x1 (t), x2 (t), x1 (t−τ ) . dt

(21)

system (21) can be written as functional differential equation (FDE) in C([−1, 0], R2 ) of the form: u(t) ˙ = Lν (ut ) + f (ν, ut ),

(22)

where u(t) = (u1 (t), u2 (t))T , Lν : C −→ R is the linear operator, and f : (R × C) −→ R is the nonlinear operator are given, respectively, by Lν (φ) = ⎛ (τ10 + μ) ⎝

ψ(H ∗ ) − 2δR ∗ − P  (R ∗ )H ∗ ψ  (H ∗ )R ∗ − P (R ∗ ) 0

0 ⎛

+ (τ10 + μ) ⎝

and

0 eP  (R ∗ )H ∗

0 0

⎞

 ⎠ φ1 (0) (23) φ2 (0)

⎞

 ⎠ φ1 (−1) , φ2 (−1)

(24)

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⎛

f1 (φ1 (0), φ2 (0))

f (φ, ν) = (τ 0 + ν) ⎝

⎞ ⎠,

(25)

f2 (φ1 (0), φ2 (0), φ1 (−1)) where φ = (φ1 , φ2 )T ∈ C. By the Riesz representation theorem, there exists a (2 × 2) matrix, η(θ, ν), (1 ≤ θ ≤ 0) whose elements are bounded variation functions such that

Lν (φ) =

0

−1

dη(θ, ν)φ(θ ) f or φ ∈ C([−1, 0], R2 ).

(26)

where δ is the Dirac function defined by

δ(θ ) =

⎧ ⎨ 0, ⎩

θ = 0; (27)

1,

θ = 0.

For φ ∈ C([−1, 0], R2 ), define

A(ν)φ(θ ) =

⎧ dφ(θ ) ⎪ ⎪ , θ ∈ [−1; 0); ⎪ ⎪ ⎨ dθ

⎪ ⎪ ⎪ ⎪ ⎩

0 −1

(28) dη(ν, s)φ(s), θ = 0

and R(ν)φ(θ ) =

⎧ ⎨ 0, ⎩

θ ∈ [−1; 0); (29)

f (ν, θ ),

θ = 0.

Then, system (21) is equivalent to u(t) ˙ = A(ν)ut + R(ν)ut , where ut (θ ) = u(t + θ ) for θ ∈ [−1, 0]. For ϕ ∈ C([−1, 0], R2 ), define ⎧ dϕ(s) ⎪ ⎪ − , s ∈ (0, 1]; ⎪ ⎪ ⎨ ds ∗ A ϕ(s) = ⎪ 0 ⎪ ⎪ ⎪ dηT (t, 0)ϕ(−t), s = 0 ⎩ −1

(30)

(31)

Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory. . .

349

and a bilinear inner product

ϕ(s), φ(θ ) = ϕ(0)φ(0) ¯ −

0

θ

−1 ξ =0

ϕ(ξ ¯ − θ )dη(θ )φ(ξ )dξ,

(32)

where η(θ ) = η(θ, 0). Then A(0) and A∗ are adjoint operators. By Theorem 4.1, we know that ±iω0 τ 0 are eigenvalues of A(0). Thus, they are also eigenvalues of A∗ . Let q(θ ) is the eigenvector of A(0) corresponding to +iω0 τ 0 and q ∗ (θ ) is the eigenvector of A∗ corresponding to −iω0 τ 0 . T   T 0 0 Let q(θ ) = 1, α eiθω0 τ and q ∗ (s) = D 1, α ∗ eiω0 τ s . From the above discussion, it is easy to know that A(0)q(0) = iω0 τ 0 q(0) and A∗ (0)q ∗ (0) = −iω0 τ 0 q ∗ (0). That is, ⎛ τ0 ⎝

ψ(H ∗ ) − 2δR ∗ − P  (R ∗ )H ∗ ψ  (H ∗ )R ∗ − P (R ∗ ) 0 ⎛

+ τ0 ⎝

⎞ ⎠ q(0)

0

0

0

eP  (R ∗ )H ∗

0

⎞ ⎠ q(−1) = iω0 τ 0 q(0),

and ⎛ τ0 ⎝

ψ(H ∗ ) − 2δR ∗ − P  (R ∗ )H ∗ 0 ψ  (H ∗ )R ∗

⎛ + τ

0⎝

− P (R ∗ )

0 eP  (R ∗ )H ∗ 0

⎞ ⎠ q ∗ (0)

0

⎞ ⎠ q ∗ (−1) = −iω0 τ 0 q ∗ (0),

0

Thus, we can easily obtain  eP  (R ∗ )H ∗ e−iω0 τ 0 T 0 eiθω0 τ q(θ ) = 1, iω0

(33)

 ψ  (H ∗ )R ∗ − P (R ∗ ) T 0 eiω0 τ s . q ∗ (s) = D1 1, iω0

(34)

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In order to assure q ∗ (s), q(θ ) = 1, we need to determine the value of D1 . From (32), we have

0 θ q¯ ∗ (s), q(θ ) = q¯ ∗ (0)q(0) − q¯ ∗ (ξ − θ )dη(θ )φ(ξ )dξ −1 ξ =0

= q¯ ∗ (0)q(0) −



0

θ

−1 ξ =0

= q¯ ∗ (0)q(0) − q¯ ∗

0

−1

= q¯ ∗ (0)q(0) − q¯ ∗ τ 0

T   0 0 D¯ 1, α ∗ e−iω0 τ (ξ −θ ) dη(θ ) 1, α ∗ eiω0 τ ξ dξ 0

θ eiω0 τ θ dη(θ )q(0)



0 0 0 (−e−iω0 τ )q(0)  ∗ ∗ eP (R )H 0

 0 = D¯ 1 + α α¯ ∗ + τ 0 α¯ ∗ eP  (R ∗ )H ∗ e−iω0 τ .

So, we have D¯ = D=

1 1 + α α¯ ∗

0 + τ 0 α¯ ∗ eP  (R ∗ )H ∗ e−iω0 τ

1 + αα ¯ ∗

0 + τ 0 α ∗ eP  (R ∗ )H ∗ eiω0 τ

1

.

(35) (36)

Using the same notations as in [5], we first compute the coordinates to describe the center manifold C2 (0) at ν = 0. Let ut be the solution of Eq. (30) when ν = 0. Define z0 (t) = q ∗ , ut ,

W (t, θ ) = ut (θ ) − 2Re(z0 (t)q(θ ))  ¯ ) . = ut (θ ) − z0 (t)q(θ ) + z¯ 0 (t)q(θ

(37)

On the center manifold C2 (0), we have W (t, θ ) = W (z0 , z¯ 0 , θ ),

(38)

where W (z0 , z¯ 0 , θ ) = W20 (θ )

z02 z¯ 2 z3 + W11 (θ )z0 z¯ 0 + W02 (θ ) 0 + W30 (θ ) 0 + . . . , 2 2 6

(39)

where z0 and z¯ 0 are the local coordinates for center manifold C2 (0) in the direction of q ∗ and q¯ ∗ . Note that W is real if ut is real. We only consider real solutions. For solution ut ∈ C2 (0) of (21), since ν = 0, we have  z˙ 0 (t) = iω0 τ10 z0 + q¯ ∗ (0)f 0, W (z0 , z¯ 0 , 0) + 2Re(z0 q(θ )) def

= iω0 τ10 z0 + q¯ ∗ (0)f0 (z0 , z¯ 0 ).

(40)

Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory. . .

351

We rewrite this equation as z˙ 0 (t) = iω0 τ10 z0 + p(z0 , z¯ 0 ),

(41)

where p(z0 , z¯ 0 ) = q¯ ∗ (0)f0 (z0 , z¯ 0 ) = p20 (θ )

z02 z¯ 2 + p11 (θ )z0 z¯ 0 + p02 (θ ) 0 2 2

+ p21 (θ )

z02 z¯ 0 + .... 2

(42)

Remark 1 In what follows, our objective is to compute coefficients p20 , p11 , p02 , and p21 of p(z0 , z¯ 0 ). These coefficients will be used to find the direction of Hopf bifurcation.   T 0 We have ut (θ ) = u1t (θ ), u2t (θ ) and q(θ ) = 1, α eiω0 τ θ . So from (37) and (39), it follows that ut (θ ) = W (t, θ ) + 2Re(z0 (t)q(θ )), = W (t, θ ) + z0 (t)q(θ ) + z¯ 0 (t)q(θ ¯ ), = W20 (θ )

z02 z¯ 2 + W11 (θ )z0 z¯ 0 + W02 (θ ) 0 2 2

(43)

 T  T 0 0 + 1, α eiω0 τ θ z0 + 1, α¯ e−iω0 τ θ z¯ 0 + . . . , and then, we have (1)

z02 z¯ 2 (1) (1) + W11 (0)z0 z¯ 0 + W02 (0) 0 + . . . , 2 2

u1t (0)

= z0 + z¯ 0 + W20 (0)

u2t (0)

= αz0 + α¯ z¯ 0 + W20 (0)

(2)

z02 z¯ 2 (2) (2) + W11 (0)z0 z¯ 0 + W02 (0) 0 + . . . , 2 2

(1) u1t (−1) = e−iω0 τ z0 + eiω0 τ z¯ 0 + W20 (−1) 0

0

(1)

+W02 (−1)

z¯ 02 2

z02 (1) + W11 (−1)z0 z¯ 0 2

+ ...,

u2t (−1) = αe−iω0 τ z0 + αe ¯ iω0 τ z¯ 0 + W20 (−1) 0

(2)

+W02 (−1)

0

z¯ 02 + .... 2

(2)

z02 (2) + W11 (−1)z0 z¯ 0 2

(44)

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It follows together with Eq. (25) that p(z0 , z¯ 0 ) = q¯ ∗ f0 (z0 , z¯ 0 ), = q¯ ∗ f (0,  ut ), 0 ¯ = τ D 1, α¯ f (0, ut ),  m 1 , = τ 0 D¯ 1, α¯ × m2

(45)

where m1 =

 r  2 2 2 (γ − β − 2H ∗ )x2t (0)x1t (0) − x2t (0)x1t (0) − R ∗ x2t (0)) − δx1t (0) βγ

1 1 2 2 − P  (R ∗ )x2t (0)x1t (0) − P  (R ∗ )x2t (0)x1t (0) − P  (R ∗ )H ∗ x1t (0) + . . . 2 2 1 2 m2 = eP  (R ∗ )x2t (0)x1t (−1) + eP  (R ∗ )x2t (0)x1t (−1) 2 1 2 + eP  (R ∗ )H ∗ x1t (−1) + . . . . 2 So, we have p(z0 , z¯ 0 ) =

z02  0  r  2τ D¯ α(γ − β − 2H ∗ ) − α 2 R ∗ − δ − αP  (R ∗ ) 2 βγ −

  P  (R ∗ )H ∗ 1 0 0 + α¯ eP  (R ∗ )αe−iω0 τ + eH ∗ P  (R ∗ )e−2iω0 τ 2 2

+

z¯ 02  0  r  2τ D¯ α(γ ¯ − β − 2H ∗ ) − α 2 R ∗ − δ − αP ¯  (R ∗ ) 2 βγ

  P  (R ∗ )H ∗ 1 0 0 + α¯ eP  (R ∗ )αe ¯ iω0 τ + eH ∗ P  (R ∗ )e2iω0 τ 2 2   r  (α + α)(γ ¯ − β − 2H ∗ ) − 2α αR ¯ ∗ − 2δ +z0 z¯ 0 τ 0 D¯ βγ −

 −(α + α)P ¯  (R ∗ ) − P  (R ∗ )H ∗ + α¯ eP  (R ∗ )   0 0 ×Re αe−iω0 τ + αe ¯ iω0 τ + eH ∗ P  (R ∗ )

Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory. . .

353

 z02 z¯ 0  0  r  (2) (1) τ D¯ (γ − β − 2H ∗ ) 2W11 (0) + 2αW11 (0) 2 βγ (2) (1) (2) +W20 (0) + αW20 (0) − 2α 2 − 4α α¯ − 4αR ∗ W11 (0)

+

(2) (1) (1) −2R ∗ W20 (0) − 4δW11 (0) − 2δW20 (0)  (2) (1) (2) (1) −P  (R ∗ ) 2W11 (0) + 2αW11 (0) + W20 (0) + αW ¯ 20 (0) 

 (1) (1) −P  (R ∗ ) α¯ + 2α) − H ∗ P  (R ∗ ) 2W11 (0) + W20 (0)

(46)

  0 (2) (1) +α¯ eP  (R ∗ ) 2W11 (0)e−iω0 τ + 2αW11 (−1)  0 0 (2) (1) +W20 (0)eiω0 τ + αW20 (−1) + eP  (R ∗ ) 2α + αe−2iω0 τ   0 0 (1) (1) + 12 eH ∗ P  (R ∗ ) W20 (−1)eiω0 τ + 2W11 (−1)e−iω0 τ . Comparing the coefficient with (42), we obtain p20 = 2τ 0 D¯

 r  P  (R ∗ )H ∗ α(γ − β − 2H ∗ ) − α 2 R ∗ − δ − αP  (R ∗ ) − βγ 2

  1 0 0 +α¯ eP  (R ∗ )αe−iω0 τ + eH ∗ P  (R ∗ )e−2iω0 τ , 2

p02 = 2τ 0 D¯

 r  P  (R ∗ )H ∗ α(γ ¯ − β − 2H ∗ ) − α 2 R ∗ − δ − αP ¯  (R ∗ ) − βγ 2

  1 0 0 +α¯ eP  (R ∗ )αe , ¯ iω0 τ + eH ∗ P  (R ∗ )e2iω0 τ 2 p11 = τ 0 D¯

 r  (α + α)(γ ¯ − β − 2H ∗ ) − 2α αR ¯  (R ∗ ) ¯ ∗ − 2δ − (α + α)P βγ

   0 0 ¯ iω0 τ + eH ∗ P  (R ∗ ) , −P  (R ∗ )H ∗ + α¯ eP  (R ∗ )Re αe−iω0 τ + αe

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p21 = τ 0 D¯

  r  (2) (1) (2) (γ − β − 2H ∗ ) 2W11 (0) + 2αW11 (0) + W20 (0) βγ

(1) (2) (2) (1) +αW20 (0) − 2α 2 − 4α α¯ − 4αR ∗ W11 (0) − 2R ∗ W20 (0) − 4δW11 (0)  (1) (2) (1) (2) (1) −2δW20 (0) − P  (R ∗ ) 2W11 (0) + 2αW11 (0) + W20 (0) + αW ¯ 20 (0)   (1) (1) −P  (R ∗ ) α¯ + 2α) − H ∗ P  (R ∗ ) 2W11 (0) + W20 (0)   0 (2) (1) +α¯ eP  (R ∗ ) 2W11 (0)e−iω0 τ + 2αW11 (−1)  0 0 (2) (1) +W20 (0)eiω0 τ + αW20 (−1) + eP  (R ∗ ) 2α + αe−2iω0 τ   0 0 (1) (1) + 12 eH ∗ P  (R ∗ ) W20 (−1)eiω0 τ + 2W11 (−1)e−iω0 τ . (47) Since there are W20 and W11 in p21 , we still need to compute them. From (30) and (37), we have ¯ W˙ = u˙ t − z˙ 0 q − ¯˙z0 q,

=

⎧ ⎨ AW − 2Re(q¯∗ (0)f0 q(θ )), ⎩

θ ∈ [−1, 0),

AW − 2Re(q¯∗ (0)f0 q(θ )) + f0 , θ = 0,

(48)

def

= AW + H (z0 , z¯ 0 , θ ),

where H (z0 , z¯ 0 , θ ) = H20 (θ )

z02 z¯ 2 + H11 (θ )z0 z¯ 0 + H02 (θ ) 0 + .... 2 2

(49)

Substituting the corresponding series into Eq. (48) and comparing the coefficients, we obtain (A − 2iω0 τ 0 )W20 (θ ) = −H20 (θ ), (50) AW11 (θ )

= −H11 (θ ).

From Eq. (48) we know that for θ ∈ [−1; 0), ¯ ) = −p(z0 , z¯ 0 )q(θ ) − p(z ¯ 0 , z¯ 0 )q(θ ¯ ). H (z0 , z¯ 0 , θ ) = −q ∗ f0 q(θ ) − q ∗ f¯0 q(θ (51)

Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory. . .

355

Comparing the coefficients with Eq. (49), we get ¯ ), H20 (θ ) = −p20 q(θ ) − p¯ 02 q(θ

(52)

¯ ). H11 (θ ) = −p11 q(θ ) − p¯ 11 q(θ

(53)

From Eqs. (49) and (52) and the definition of A, it follows that ¯ ). W˙ 20 (θ ) = 2iω0 τ 0 W20 (θ ) + p20 q(θ ) + p¯ 02 q(θ

(54)

 T 0 Notice that q(θ ) = 1, α eiω0 τ θ . Hence W20 (θ ) =

ip20 i p¯ 02 0 0 −iω0 τ 0 θ q(0)eiω0 τ θ + q(0)e ¯ + E1 e2iω0 τ θ , 0 0 ω0 τ 3ω0 τ

(55)

 (1) (2) ∈ R2 is a constant vector. Similarly, from (49) and (53), where E1 = E1 , E1 we obtain W11 (θ ) = −

ip11 i p¯ 11 0 −iω0 τ 0 θ q(0)eiω0 τ θ + q(0)e ¯ + E2 , ω0 τ 0 ω0 τ 0

(56)

 (1) (2) where E2 = E2 , E2 ∈ R2 is also a constant vector. In what follows, we will seek appropriate E1 and E2 . From the definition of A and (49), we obtain

0 −1

dη(θ )W20 (θ ) = 2iω0 τ 0 W20 (0) − H20 (0),

0

−1

(57)

dη(θ )W11 (θ ) = −H11 (0),

(58)

where η(θ ) = η(0, θ ). By (48), we have H20 (0) = −p20 q(0) − p¯ 02 q(0) ¯ + 2τ 0 ⎞ r  P  (R ∗ )H ∗ ∗ 2 ∗  ∗ α(γ − β − 2H − δ − αP ) − α R (R ) − ⎟ ⎜ βγ 2 ⎟ ×⎜ ⎠ ⎝ ⎛

eP  (R ∗ )αe−iω0 τ + 12 eH ∗ P  (R ∗ )e−2iω0 τ 0

0

(59)

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A. Mendy et al.

H11 (0) = −p11 q(0) − p¯ 11 q(0) ¯ + 2τ 0 ⎛ r  ⎞ (α+¯α)(γ−β−2H ∗ )−2α αR ¯ ∗ −2δ−(α+¯α)P  (R ∗ )−P  (R ∗ )H ∗ ⎜ βγ ⎟ ⎟. ×⎜ ⎝ ⎠  0 0 eP  (R ∗ )Re αe−iω0 τ + αe ¯ iω0 τ + eH ∗ P  (R ∗ ) (60) Substituting Eqs. (55) and (59) into Eq. (57) and noticing that 



iω0 τ 0 I −

0

−1

0 eiω0 τ dη(θ ) q(0)

− iω0 τ 0 I −

0 −1

e−iω0

τ0

= 0,

dη(θ ) q(0) ¯ = 0.

(61)

We obtain  0 a1 , e2iω0 τ dη(θ ) E1 = 2τ 0 × a2 −1

 2iω0 τ 0 I −

0

r  P  (R ∗ )H ∗ α(γ − β − 2H ∗ ) − α 2 R ∗ − δ − αP  (R ∗ ) − and βγ 2 0 0 a2 = eP  (R ∗ )αe−iω0 τ + 12 eH ∗ P  (R ∗ )e−2iω0 τ . This leads to ⎞ ⎛  2iω0 − ψ(H ∗ ) + 2δR ∗ + P  (R ∗ )H ∗ −ψ  (H ∗ )R ∗ + P  (R ∗ ) a1 ⎟ ⎜ . ⎠ E1 = 2 ⎝ a2 0 −eP  (R ∗ )H ∗ e−2iω0 τ 2iω0 where a1 =

Solving this system for E1 , we obtain (1)

E1

(2)

E1

   a1 −ψ  (H ∗ )R ∗ + P  (R ∗ )        a  2iω0 2    2iω − ψ(H ∗ ) + 2δR ∗ + P  (R ∗ )H ∗ a  0 1  2   =  ,  1  0  −eP  (R ∗ )H ∗ e−2iω0 τ a2  2 = 1

Hopf Bifurcation in a Delayed Herd Harvesting Model and Herbivory. . .

357

where    2iω − ψ(H ∗ ) + 2δR ∗ + P  (R ∗ )H ∗ −ψ  (H ∗ )R ∗ + P  (R ∗ )  0     1 =  .   0  ∗ ∗ −2iω τ 0   −eP (R )H e 2iω0 Similarly, substituting (56) and (60) into (58), we get ⎛ ⎝

ψ(H ∗ ) − 2δR ∗ − P  (R ∗ )H ∗ ψ  (H ∗ )R ∗ − P  (R ∗ ) eP  (R ∗ )H ∗

0

⎞

 ⎠ E2 = 2 b1 , b2

where  r (α + α)(γ ¯ − β − 2H ∗ ) − 2α αR ¯  (R ∗ ) − P  (R ∗ )H ∗ b1 = ¯ ∗ − 2δ − (α + α)P βγ  0 0 and b2 = eP  (R ∗ )Re αe−iω0 τ + αe ¯ iω0 τ + eH ∗ P  (R ∗ ). Hence    b1 ψ  (H ∗ )R ∗ − P  (R ∗ )    2   E2(1) =   2   b2 0    ψ(H ∗ ) − 2δR ∗ − P  (R ∗ )H ∗ b1    2  , E2(2) =   2  eP  (R ∗ )H ∗ b2  where    ψ(H ∗ ) − 2δR ∗ − P  (R ∗ )H ∗ ψ  (H ∗ )R ∗ − P  (R ∗ )    . 2 =      ∗ ∗ eP (R )H 0 Thus, we can determine W20 and W11 from (55) and (56). Furthermore, g21 in (47) can be expressed by the parameters and delay. Thus, we can compute the following values:

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A. Mendy et al.

C2 (0) =

i  |p02 |2 p21 2 + , p p − 2|p | − 20 11 11 3 2 2ω0 τ 0

μ2 = β2

=

T2

=

−

% $ Re C2 (0) $ %, Re λ (τ 0 )

(62)

$

% 2Re C2 (0) , −

I m(C2 (0)) + μ2 I m(λ (τ 0 )) , ω0 τ 0

which determine the qualities of bifurcating periodic solution in the center manifold at the critical value τ 0 .

References 1. R. Anguelov, Y. Dumont, J.M. Lubuma, On nonstandard finite difference schemes in biosciences. AIP Conf. Proc. 1487, 212–223 (2012) 2. J. Bohn, J. Rebaza, K. Speer, Continuous threshold prey harvesting in predator-prey models. WASET Int. J. Math. Comput. Sci. 5, 777–784 (2011) 3. M. Carriere, Impact des systèmes d’élevage pastoraux sur l’environnement en Afrique et en Asie tropicale et sub-tropicale aride et sub-aride. Livestock and the Environment Finding a Balance, Scientific Environmental Monitoring Group, 1996 4. R.V. Culshaw, S. Ruan, A delay-differential equation model of HIV infection of CD4 T-cells. Math. Biosci. 165, 27–39 (2000) 5. B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Applications of Hopf Bifurcation (Cambridge University, Cambridge, 1981) 6. D.W. Hilbert, D.M. Swift, J.K. Detling, M.I. Dyer, Relative growth rates and the grazing optimization hypothesis. Oecologia 51, 14–18 (1981) 7. T.K. Kar, Modelling and analysis of a harvested prey-predator system incorporating a prey refuge. J. Comput. Appl. Math. 185, 19–33 (2006) 8. V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations (World Scientific, Singapore,1989) 9. A. Lebon, L. Mailleret, Y. Dumont, F. Grognard, Direct and apparent compensation in plantherbivore interactions. Ecol. Model. 290, 192–203 (2014) 10. A. Martin, S. Ruan, Predator-prey models with delay and prey harvesting. J. Math. Biol. 43, 247–267 (2001) 11. S.J. McNaughton, Grazing lawns: animals in herds, plants form and coevolution. Am. Nat. 124(6), 863–886 (1984) 12. I. Tankam, P.T. Mouofo, A. Mendy, M. Lam, J.J. Tewa, S. Bowong, Local bifurcations and optimal theory in a delayed predator-prey model with threshold prey harvesting. Int. J. Bifurcation Chaos 25(7), 1540015 (2015) 13. J.J. Tewa, A. Bah, S.C.O. Noutchie, Dynamical models of interactions between herds forage and water resources in Sahelian region. Abstr. Appl. Anal. 2014, 138179 (2014) 14. S.C. Williamson, J.K. Detling, J.L. Dodd, M.I. Dyer, Experimental evaluation of the grazing optimization hypothesis. J. Range Manag. Archives 42(2), 149–152 (1989) 15. D. Xiao, W. Li, M. Han, Dynamics in a ratio-dependent predator-prey model with predator harvesting. J. Math Anal. Appl. 324, 14–29 (2006)

A Fractional Order Model for HBV Infection with Capsids and Cure Rate Moussa Bachraou, Khalid Hattaf, and Noura Yousfi

1 Introduction Hepatitis B is a viral infection caused by the hepatitis B virus (HBV) that affects the liver cells (hepatocytes). It can cause chronic infection and puts people at high risk of death from cirrhosis and liver cancer. The World Health Organization (WHO) estimates that 257 million people are living with HBV infection, and about 887,000 people died mostly from HBV complications (including cirrhosis and hepatocellular carcinoma) in 2015 [1]. On the other hand, modeling and analysis using fractional differential equations (FDEs) have helped to better understand the dynamics of HBV infection [2–4]. In addition, infected hepatocytes may also revert to the uninfected state by loss of all covalently closed circular DNA (cccDNA) from their nucleus [5–7]. Motivated by these mathematical and biological reasons, we propose the following fractional order model for HBV infection with cure of infected hepatocytes: Dα H Dα I Dα D Dα V

= λ − dH − f (H, V )V + ρI, = f (H, V )V − (ρ + δ)I, = kI − (γ + δ)D, = γ D − μV ,

(1)

M. Bachraou () · N. Yousfi Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Science Ben M’sik, Hassan II University, Casablanca, Morocco K. Hattaf Laboratory of Analysis, Modeling and Simulation (LAMS), Faculty of Science Ben M’sik, Hassan II University, Casablanca, Morocco Centre Régional des Métiers de l’Education et de la Formation (CRMEF), Casablanca, Morocco © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_23

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where D α is the fractional derivative in the Caputo sense and α is a parameter that describes the order of the fractional time-derivative with α ∈ (0, 1]. The variables H (t), I (t), D(t), and V (t) are the concentrations of uninfected hepatocytes, infected hepatocytes, HBV DNA-containing capsids and virions at time t, respectively. The uninfected hepatocytes are produced from a source at a constant rate λ with a natural death rate d and become infected at rate f (H, V )V . The infected hepatocytes revert to the uninfected state by loss of all cccDNA from their nucleus at a rate ρ per infected cell. The parameter δ is the death rate for infected hepatocytes and capsids. The parameters k, γ , and μ are, respectively, the production rate of capsids from infected hepatocytes, the rate at which the capsids are transmitted to blood which gets converted to virions, and the clearance rate of virions. Here, the infection transmission is modeled by Hattaf–Yousfi functional response [8] of βH the form f (H, V ) = , where α0 , α1 , α2 , α3 ≥ 0 are α0 + α1 H + α2 V + α3 H V the saturation factors measuring the inhibitory or psychological effect, and β is a positive constant rate describing the infection process. This Hattaf–Yousfi functional response was recently used in [9, 10] and it includes various forms of incidence rate existing in the literature such as the bilinear incidence, the saturation incidence, the Beddington–DeAngelis functional response, the Crowley–Martin functional response, and the specific functional response proposed by Hattaf et al. [11]. The novelty of our model presented by system (1) is that it improves and generalizes the mathematical model formulated by ordinary differential equations (ODEs) and presented in [12], and also FDE models [2–4]. Moreover, the importance of this study is that it incorporates the effect of memory and the cure of infected hepatocytes which are neglected in ODE models [12, 13]. Our main purpose in this work is to investigate the dynamical behavior of the FDE model (1). To do this, the next section deals with the existence, uniqueness, non-negativity, and boundedness of the solutions of (1) and discusses the existence equilibria. The global stability of equilibria is established in Sect. 3. The illustrative numerical simulations are presented in Sect. 4. Finally, we end this paper with concluding remarks in Sect. 5.

2 Well-Posedness and Equilibria In this section, we will show that our model is well-posed and discuss the existence of equilibria. For these reasons, we assume that the initial data for system (1) satisfy: H (0) = H0 ≥ 0, I (0) = I0 ≥ 0, D(0) = D0 ≥ 0 and V (0) = V0 ≥ 0.

(2)

First, we investigate the non-negativity and boundedness of the solution of our system (1).

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Theorem 2.1 For any initial data satisfying ( 2), system ( 1) has a unique solution on [0, +∞). Furthermore, this solution remains non-negative and bounded for all t ≥ 0. Proof System (1) can be written as follows: D α X(t) = F (X), where ⎛

⎞ ⎛ ⎞ H (t) λ − dH − f (H, V )V + ρI ⎜ I (t) ⎟ ⎜ f (H, V )V − (ρ + δ)I ⎟ ⎟ ⎜ ⎟. X(t) = ⎜ ⎝ D(t) ⎠ and F (X) = ⎝ ⎠ kI − (γ + δ)D γ D − μV

V (t) Let

⎛ ⎞ ⎛ ⎞ −d ρ 0 0 λ ⎜ 0 −(ρ + δ) ⎜0⎟ 0 0 ⎟ ⎜ ⎟. ⎟ η=⎜ ⎝ 0 ⎠ and A1 = ⎝ 0 k −(γ + δ) 0 ⎠ 0 0 γ −μ 0 So, we discuss four cases: • If α0 = 0, then system (1) can be written as follows: D α X(t) = η + A1 X +

α0 V A2 X, α0 + α1 H + α2 V + α3 H V

where ⎞ 000 ⎟ 0 0 0⎟ ⎟. 0 0 0 0⎠ 0 000

⎛ −β α0 β α0

(3)

: α : :D X(t): ≤ η + ( A1 + V A2 ) X .

(4)

⎜ ⎜ A2 = ⎜ ⎝

Then

• If α1 = 0, we have D α X(t) = η + A1 X +

α1 H A3 X, α0 + α1 H + α2 V + α3 H V

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where ⎛

0 0 0 −β α1 ⎜ β ⎜ 0 0 0 α1 A3 = ⎜ ⎝0 0 0 0 000 0

⎞ ⎟ ⎟ ⎟. ⎠

Then : α : :D X(t): ≤ η + ( A1 + A3 ) X . • If α2 = 0, we have D α X(t) = η + A1 X +

α2 V A4 X, α0 + α1 H + α2 V + α3 H V

where ⎛ −β ⎜ ⎜ A4 = ⎜ ⎝

⎞ 0 0 0 α2 ⎟ β α2 0 0 0 ⎟ ⎟. 0 0 0 0⎠ 0 000

Then : α : :D X(t): ≤ η + ( A1 + A4 ) X . • If α3 = 0, we have D α X(t) = η + A1 X +

α3 H V A5 , α0 + α1 H + α2 V + α3 H V

where ⎛ −β ⎞ α

⎜ β3 ⎜ A5 = ⎜ α3 ⎝ 0 0

⎟ ⎟ ⎟, ⎠

Then D α X(t) ≤ η + A5 + A1 X .

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Therefore, the second condition of Lemma 4 in [14] is satisfied and system (1) has a unique solution on [0, +∞). Now, we show the non-negativity of solutions. By (1), we have Dα H |

H =0

D I |

I =0

Dα D |

D=0

= kI ≥ 0,

D V |

V =0

= γ D ≥ 0.

α

α

= λ ≥ 0, = f (H, V )V ≥ 0

Based on Lemmas 2.5 and 2.6 in [14], it is not hard to deduce that the solution of (1) remains non-negative. Finally, we prove that this solution is bounded. Define the function T (t) as T (t) = H (t) + I (t) +

δ(γ + δ) δ D(t) + V (t). 2k 4kγ

So, we have D α T (t) = D α H (t) + D α I (t) +

δ α δ(γ + δ) α D D(t) + D V (t). 2k 4kγ

Then δ δ(γ + δ) δμ(γ + δ) D α T (t) = λ − dH (t) − I (t) − D(t) − V (t) 2 4k 4kγ ≤ λ − θ T (t), where θ = min{d, 2δ , γ +δ 2 , μ}. Therefore, T (t) ≤ T (0)Eα (−θ t α ) +

 λ 1 − Eα (−θ t α ) , θ

∞ 

zα is the Mittag-Leffler function of parameter α. (αk + 1) k=0 Since 0 ≤ Eα (−θ t α ) ≤ 1 , we obtain

where Eα (z) =

T (t) ≤ T (0) + This completes the proof.

λ . θ 

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We begin the analysis of the equilibria by observing that system (1) has one λ infection-free equilibrium E0 ( , 0, 0, 0) corresponding to the extinction of HBV d infection. Then we define the basic reproduction number of (1) as follows: R0 =

λβkγ . μ(α0 d + α1 λ)(γ + δ)(ρ + δ)

(5)

To find the other equilibria, we solve the following system: λ − dH − f (H, V )V + ρI = 0,

(6)

f (H, V )V − (ρ + δ) I = 0,

(7)

kI − (γ + δ)D = 0,

(8)

γ D − μV = 0.

(9)

By (6)–(9), we get I =

λ − dH k(λ − dH ) γ k(λ − dH ) ,D= ,V = , and δ δ(γ + δ) μδ(γ + δ)

 γ k(λ − dH ) μ(γ + δ) (δ + ρ) f H, . = μδ(γ + δ) kγ

(10)

λ − dH λ  0, which implies that H  . Then there is no positive δ d λ equilibrium point when H > . So, we consider the function g defined on interval d λ [0, ] by d  μ(γ + δ) (δ + ρ) γ k(λ − dH ) . (11) − g(H ) = f H, μδ(γ + δ) kγ

We have I =

We have g(0) =

− μ(γ +δ)(ρ+δ) kγ

 λ = < 0 and g d

g  (H ) = Hence, if R0

μ(γ +δ)(ρ+δ) kγ

∂f kγ d ∂f − > 0. ∂H μδ(γ + δ) ∂V

(R0 − 1) and

(12)

1, system admits a unique endemic equilibrium  (1) λ , I1 > 0, D1 > 0, and V1 > 0. E1 (H1 , I1 , D1 , V1 ) with H1 ∈ 0, d >

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Summarizing the above discussions in the following theorem. Theorem 2.2 (i) If R0 ≤ 1, then system (1) has a unique infection-free equilibrium of the form λ E0 (H0 , 0, 0, 0), where H0 = . d (ii) If R0 > 1, the infection-free equilibrium is still present and system (1) has a unique infection equilibrium of the form E1 (H1 , I1 , D1 , V1 ), where  chronic λ λ − dH1 k(λ − dH1 ) γ k(λ − dH1 ) , D1 = and V1 = . , I1 = H1 ∈ 0, d δ δ(γ + δ) μδ(γ + δ)

3 Global Stability In this section, we focus on the global stability of the two equilibria of system (1). At first, we have Theorem 3.1 The infection-free equilibrium E0 is globally asymptotically stable when R0 ≤ 1 and it becomes unstable if R0 > 1. Proof In order to prove the first part of this theorem, we consider the following Lyapunov functional:  α0 ρ (H − H0 + I )2 H α0 L0 (t) = + H0  α0 + α1 H0 H0 2 (δ + d) (α0 + α1 H0 ) H0 +I +

ρ+δ (ρ + δ) (γ + δ) D+ V, k kγ

where (x) = x − 1 − ln(x), x > 0. By using the property of fractional derivatives given in [15], we have D α L0 (t) ≤

α0 α0 + α1 H0 +D α I +

 H0 α0 ρ (H − H0 + I ) (D α H + D α I ) 1− Dα H + H (δ + d) (α0 + α1 H0 ) H0

ρ+δ α (ρ + δ) (γ + δ) α D D+ D V. k kγ

Using λ = dH0 , we get  dα0 (H − H0 )2 ρ 1 ρα0 δI 2 + − D L0 (t) ≤ − H (δ + d)H0 α0 + α1 H0 (δ + d) (α0 + α1 H0 ) H0  μ(ρ + δ)(γ + δ) f (H, V ) ρα0 I (H − H0 )2 + −1 V R0 − (α0 + α1 H0 )H0 H kγ f (H, 0) α

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 dα0 (H − H0 )2 ρ 1 ρα0 δI 2 + ≤− − H (δ + d)H0 α0 + α1 H0 (δ + d) (α0 + α1 H0 ) H0 −

 μ(ρ + δ)(γ + δ)  ρα0 I (H − H0 )2 + R0 − 1 V . (α0 + α1 H0 )H0 H kγ

Since R0 ≤ 1, we have that D α L0 (t) ≤ 0. In addition, the largest invariant set in {(H, I, C, V , Z) | D α L0 (t) = 0} is the singleton {E0 }. It follows from LaSalle’s invariance principle [16] that the infection-free equilibrium E0 is globally asymptotically stable when R0 ≤ 1. In order to show the remaining part, we determine the characteristic equation at the infection-free equilibrium E0 . By a simple computation, the characteristic equation at E0 is given by (d + ξ ) P (ξ ) = 0, where P (ξ ) = ξ 3 + a1 ξ 2 + a2 ξ + a3 and a1 = μ + γ + ρ + 2δ a2 = μ (γ + δ) + (μ + γ + δ) (ρ + δ) a3 = μ (γ + δ) (ρ + δ)(1 − R0 ). We have lim P (ξ ) = +∞ and P (0) = μ (γ + δ) (ρ +δ)(1−R0 ). If R0 > 1, then ξ →+∞

P (0) < 0. So, there exists a ξ0 ∈ (0, +∞) such that P (ξ0 ) = 0, which implies that the characteristic equation at E0 has a positive root when R0 > 1. Consequently E0 is unstable if R0 > 1. This completes the proof.  Theorem 3.2 The chronic infection equilibrium E1 is globally asymptotically stable if R0 > 1 and R0 ≤ 1 +

[dδμ(γ + δ) + α2 dλkγ ] (ρ + δ) + ρα3 γ kλ2 . ρμδ(ρ + δ)(dα0 + λα1 )

(13)

Proof Consider the following Lyapunov functional: L1 (t) =

 H α0 + α2 V1 H1  α0 + α1 H1 + α2 V1 + α3 H1 V1 H1 ρ (α0 + α2 V1 ) (H − H1 + I − I1 )2 2 (δ + d) (α0 + α1 H1 + α2 V1 + α3 H1 V1 ) H1    ρ+δ I D V (ρ + δ) (γ + δ) + + . D1  V1  +I1  I1 k D1 kγ V1 +

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The derivative of L1 (t) along the positive solutions of (1) satisfies: D α L1 (t) ≤

α0 + α2 V1 α0 + α1 H1 + α2 V1 + α3 H1 V1

 H1 1− Dα H H

 I1 ρ (α0 + α2 V1 ) (H − H1 + I − I1 ) (D α H + D α I ) Dα I + 1− + I (δ + d) (α0 + α1 H1 + α2 V1 + α3 H1 V1 ) H1   D1 V1 ρ+δ (ρ + δ) (γ + δ) 1− Dα D + 1− Dα V . + k D kγ V

By applying λ = dH1 + δI1 , f (H1 , V1 )V1 = (ρ+ δ)I1 , kI1 = (γ + δ)D1 , and H1 α0 + α2 V1 f (H1 , V1 ) 1− = , we get 1− f (H, V1 ) α0 + α1 H1 + α2 V1 + α3 H1 V1 H D α L1 (t) ≤

−d (α0 + α2 V1 ) (H − H1 )2 (α0 + α1 H1 + α2 V1 + α3 H1 V1 ) H ρ (α0 + α2 V1 ) (H − H1 ) (I − I1 ) (α0 + α1 H1 + α2 V1 + α3 H1 V1 ) H  f (H1 , V1 ) f (H, V )V1 +f (H1 , V1 )V1 1 − + f (H, V1 ) f (H, V1 )V1  I D1 +f (H1 , V1 )V1 1 − I1 D  I1 f (H, V )V +f (H1 , V1 )V1 1 − If (H1 , V1 )V1  DV1 V − +f (H1 , V1 )V1 1 − V1 D1 V   ρ (α0 + α2 V1 ) d (H − H1 )2 + δ(I − I1 )2 + (d + δ) (H − H1 ) (I − I1 ) − . (α0 + α1 H1 + α2 V1 + α3 H1 V1 ) (d + δ)H1 +

Hence, D α L1 (t) ≤

− (α0 + α2 V1 ) (H − H1 )2 (α0 + α1 H1 + α2 V1 + α3 H1 V1 ) H H1

 ρdH + ρI (dH1 − ρI1 ) + δ+d

ρδ (α0 + α2 V1 ) (I − I1 )2 + f (H1 , V1 )V1 (α0 + α1 H1 + α2 V1 + α3 H1 V1 ) (δ + d) H1

 DV1 f (H, V1 ) f (H1 , V1 ) D1 I f (H, V ) V I1 − − − × 5− − f (H, V1 ) DI1 f (H1 , V1 ) V1 I D1 V f (H, V ) −

−

f (H1 , V1 )V1 (α0 + α1 H )(α2 + α3 H )(V − V1 )2 . (α0 + α1 H + α2 V1 + α3 H V1 )(α0 + α1 H + α2 V + α3 H V )V1

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Since the arithmetic mean is greater than or equal to the geometric mean, we have 5−

f (H, V ) V I1 f (H1 , V1 ) D1 I DV1 f (H, V1 ) − − − − ≤ 0, f (H, V1 ) DI1 f (H1 , V1 ) V1 I D1 V f (H, V )

and the equality holds only for H = H1 , I = I1 , D = D1 , and V = V1 . Clearly, D α L1 (t) ≤ 0 if R0 > 1 and ρI1 ≤ dH1 . In addition, it is not hard to show that the condition ρI1 ≤ dH1 is equivalent to R0 ≤ 1 +

[dδμ(γ + δ) + α2 dλkγ ] (ρ + δ) + ρα3 γ kλ2 . ρμδ(ρ + δ)(dα0 + λα1 )

Further, the largest compact invariant set in {(H, I, D, V ) | D α L1 (t) = 0} is the singleton {E1 }. By the LaSalle’s invariance principle, we conclude that E1 is globally asymptotically stable.  It is important to see that [dδμ(γ + δ) + α2 dλkγ ] (ρ + δ) + ρα3 γ kλ2 = ∞, ρ→0 ρμδ(ρ + δ)(dα0 + λα1 ) lim

and [dδμ(γ + δ) + α2 dλkγ ] (ρ + δ) + ρα3 γ kλ2 = ∞. γ →∞ ρμδ(ρ + δ)(dα0 + λα1 ) lim

According to Theorem 3.2, we get the following result. Corollary 3.1 Assume that ρ is sufficiently small or γ sufficiently large. The chronic infection equilibrium E1 is globally asymptotically stable when R0 > 1.

4 Numerical Simulation In this section, we give some numerical simulations in order to illustrate our theoretical results. Firstly, we choose λ = 468,500, β = 3.6 × 10−5 , d = 0.0139, δ = 0.00693, k = 800, γ = 0.87, ρ = 0.01, α0 = 1, α1 = 0.1, α2 = 0.01, α3 = 0.0001, and μ = 0.67. By calculation, the value of R0 is R0 = 0.46026 < 1. In this case, the infection-free equilibrium E0 (1.2 × 108 , 0, 0, 0) is globally asymptotically stable, which can be explained by the elimination of the virus (see Fig. 1).

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A Fractional Order Model for HBV Infection with Capsids and Cure Rate

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Fig. 2 Stability of the chronic infection equilibrium E1

Secondly, we choose β = 0.0018 and we keep the other parameter values. We have R0 = 1.2595 > 1 and 1+

[dδμ(γ + δ) + α2 dλkγ ] (ρ + δ) + ρα3 γ kλ2 = 3.5934. ρμδ(ρ + δ)(dα0 + λα1 )

Hence, condition (13) is satisfied. Therefore, the solution of system (1) converges to the chronic infection equilibrium E1 (3.36 × 107 , 1181, 43.75, 1.66 × 104 ) which is globally asymptotically stable. Figure 2 illustrates this result.

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5 Conclusion In this study, we have proposed and investigated a fractional order model for HBV infection with capsids and cure of infected cells. We have proved that the proposed model has two infection equilibria, namely the infection-free equilibrium E0 and the chronic infection equilibrium E1 . By using the direct Lyapunov method, we have proved that E0 is globally asymptotically stable when R0 ≤ 1, which means that the virions are cleared and eventually the HBV infection dies out. When R0 > 1 and the condition (13) holds, the equilibrium E1 is globally asymptotically stable, which indicates that the HBV infection becomes chronic. In addition, we have remarked that if the cure rate ρ is sufficiently small or the value of γ is sufficiently large, condition (13) is satisfied. From the above theoretical and numerical results, we deduce that the order of the fractional derivative α has no effect on the global dynamical behaviors of the model. However, when the value of α decreases (long memory), the solutions of the model converge rapidly to the steady states (see Figs. 1 and 2).

References 1. WHO, Hepatitis B (2017). http://www.who.int/news-room/fact-sheets/detail/hepatitis-b 2. X. Zhou, Q. Sun, Stability analysis of a fractional-order HBV infection model. Int. J. Adv. Appl. Math. Mech. 2(2), 1–6 (2014) 3. S.M. Salman, A.M. Yousef, On a fractional-order model for HBV infection with cure of infected cells. J. Egypt. Math. Soc. 25, 445–451 (2017) 4. L.C. Cardoso, F.L.P. Dos Santos, R.F. Camargo, Analysis of fractional-order models for hepatitis B. Comput. Appl. Math. 37, 4570–4586 (2018). https://doi.org/10.1007/s40314-0180588-4 5. L.G. Guidotti, R. Rochford, J. Chung, M. Shapiro, R. Purcell, F.V. Chisari, Viral clearance without destruction of infected cells during acute HBV infection. Science 284(5415), 825–829 (1999) 6. S.R. Lewin, R.M. Ribeiro, T. Walters, G.K. Lau, S. Bowden, S. Locarnini, A.S. Perelson, Analysis of hepatitis B viral load decline under potent therapy: complex decay profiles observed. Hepatology 34, 101–1020 (2001) 7. K. Hattaf, N. Yousfi, Hepatitis B virus infection model with logistic hepatocyte growth and cure rate. Appl. Math. Sci. 5, 2327–2335 (2011) 8. K. Hattaf, N. Yousfi, A class of delayed viral infection models with general incidence rate and adaptive immune response. Int. J. Dynam. Control 4, 254–265 (2016) 9. M. Mahrouf, E.M. Lotfi, M. Maziane, K. Hattaf, N. Yousfi, A stochastic viral infection model with general functional response. Nonlinear Anal. Differ. Equ. 4, 435–445 (2016) 10. D. Riad, K. Hattaf, N. Yousfi, Dynamics of capital-labour model with Hattaf-Yousfi functional response. Br. J. Math. Comput. Sci. 18(5), 1–7 (2016) 11. K. Hattaf, N. Yousfi, A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays. Appl. Math. Comput. 221, 514–521 (2013) 12. K. Manna, S.P. Chakrabarty, Chronic hepatitis B infection and HBV DNA-containing capsids: modeling and analysis. Commun. Nonlinear Sci. Numer. Simul. 22, 383–395 (2015) 13. K. Manna, Global properties of a HBV infection model with HBV DNA-containing capsids and CTL immune response. Int. J. Appl. Comput. Math. 3(3), 2323–2338 (2017)

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14. A. Boukhouima, K. Hattaf, N. Yousfi, Dynamics of a fractional order HIV infection model with specific functional response and cure rate. Int. J. Differ. Equ. 2017, 1–8 (2017) 15. C.V. De-Leon, Volterra-type Lyapunov functions for fractional-order epidemic systems. Commun. Nonlinear Sci. Numer. Simul. 24, 75–85 (2015) 16. J. Huo, H. Zhao, L. Zhu, The effect of vaccines on backward bifurcation in a fractional order HIV model. Nonlinear Anal. Real World Appl. 26, 289–305 (2015)

Modeling Anaerobic Digestion Using Stochastic Approaches Oussama Hadj Abdelkader and A. Hadj Abdelkader

1 Introduction Anaerobic digestion is a biological decomposition that takes place without oxygen in closed bioreactors and generates biogas and solid valuable residuals. It is mostly used for waste-water treatment as a natural, low cost, and energy efficient procedure. More specifically, it consists of the decomposition of organic substance by microorganisms such as bacteria in two stages: the substance digestion by a first bacteria population that yields to the production of volatile fatty acids (VFAs) and the VFAs digestion by a second bacteria population that generates biogas. Anaerobic digestion models can be classified according to different criteria, but we mostly distinguish two types of models: the descriptive high-order models which represent all the chemical and biological reactions of the system and loworder models that are useful for control and supervision operations. The first Anaerobic Digestion Model (ADM1) introduced by Batstone et al. [1] is the most known descriptive model. It was used to simulate a two-stage anaerobic digestion laboratory process [2] and also plant-wide benchmarks [3]. A review of its extensions, applications, and analysis could be found in [4]. Many authors proposed simplifications to this model [5] but without leading to any concrete use of the obtained results. The most famous low-order model of anaerobic digestion is the two-reaction AM2 model introduced first by Bernard et al. [6], describing the digestion of the organic substrate by a population of acidogenic bacteria to produce CO2 and volatile fatty acids (VFAs), and then the digestion of the VFAs by a population of

O. H. Abdelkader () · A. H. Abdelkader LAT, Laboratoire d’Automatique de Tlemcen, Université de Tlemcen, Tlemcen, Algeria © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_24

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methanogenic bacteria to produce finally CO2 and CH4 . This model represents then the key variables of the process with the lowest possible level of complexity. The AM2 was widely used in digestion process control and supervision, thanks mainly to its simple structure and the relatively reduced number of parameters and also to its ability to simulate the operational behavior of the system up to a satisfying level [7–9]. Its stability analysis for monitoring purposes was initially performed by Hess and Bernard [10] and followed by Benyahia et al. [11] for an additional bifurcation analysis and more generic stability and operating conditions. Some extensions to the AM2 model were also performed by a number of authors, such as [12] who introduced the AM2b, which includes a biological model of the soluble microbial products specific to membrane bioreactors and also [13] who built the AM2HN, which is an extension of the AM2 including a subsequent hydrolysis step. All the previous work on the AM2 aimed only to add further descriptions of the digestion dynamics to the initial model, without being interested in the accuracy of the model itself in different operating conditions. Indeed, the AM2 represents well the process at the macroscopic level where the bacteria populations are large and the substrate concentration is relatively high. However, at lower concentrations or for small bacteria populations, the model response does not fit anymore the concentrations dynamics. For a better representation of the process, some authors proposed stochastic versions of some simpler waste-water treatment models. The stochastic modeling provides a deeper description of the process by adding a quantification of the uncertain and noisy dynamics. For instance, the authors in [14] developed a stochastic version of the simple chemostat model. Also, the logistic deterministic and stochastic models were introduced in [15]. In this paper, we propose some stochastic versions of the AM2 model. We begin in Sect. 2 by reviewing the original deterministic model. Then, in Sect. 3 we derive a pure jump Markov model for the AM2 starting from the same mass balance principle that led to the deterministic one. Next, we propose two different methods of deriving a continuoustime diffusion approximation to this Markov model: in the first method, we use a Taylor series development, whereas in the second method, we use discrete-time Poisson and Normal approximations. In Sect. 4 we show how to obtain other similar models using the same method. In Sect. 5 we present the simulation algorithms for the presented models. The simulation results as well as a comparison with the deterministic model are given in Sect. 6.

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2 Deterministic AM2 Model with ODE We consider the following two biological reactions which describe the anaerobic digestion process in the bioreactor: r1

k1 s1 −→ b1 + k2 s2 + k4 CO2 r2 k3 s2 −→ b2 + k5 CO2 + k6 CH4

(1)

where r1 = μ1 b1 and r2 = μ2 b2 are the velocities of the reactions, s1 represents the organic matter concentration, s2 represents the volatile fatty acids concentration, b1 is the acidogenesis bacteria, b2 is the methanization bacteria, CH4 and CO2 are biogases (methane and carbon dioxide), ki are stoichiometric coefficients. We also consider the following mass balance system describing the variation of a component ξt denoting either b1 , b2 , s1 , or s2 at time t: dξt = ξt,in − ξt,out + ξt,produced − ξt,consumed/dead dt

(2)

Based on the reaction (1) and the mass balance principle (2), the AM2 model was derived by Bernard et al. [6]. This model represents the variation of the bacteria and the substrates concentrations x = (b1 , b2 , s1 , s2 )∗ in time, and it is described by the following ordinary differential equations (ODE) system: b˙1 (t) = (μ1 (s1 (t)) − αD)b1 (t) s˙1 (t) = D (s1in − s1 (t)) − k1 μ1 (s1 (t))b1 (t) b˙2 (t) = (μ2 (s2 (t)) − αD)b2 (t) s˙2 (t) = D (s2in − s2 (t)) + k2 μ1 (s1 (t))b1 (t) − k3 μ2 (s2 (t))b2 (t)

(3)

where each function of time denotes the concentration of the respective species or compounds as already defined in (1). μ1 (s1 (t)) and μ2 (s2 (t)) are the specific growth rate functions, s1in and s2in are the input concentrations of the substrates s1 (t) and s2 (t), respectively, D is the dilution rate, α is the part of b1 (t) and b2 (t) that has left the bioreactor, k1 , k2 , k3 are yield coefficients. We suppose that μ1 (s1 (t)) is an uninhibited Monod type growth function [16] and μ2 (s2 (t)) is an inhibited Haldane type growth function [17] given, respectively, by: μ1 (s1 (t)) = μ1max μ2 (s2 (t)) = μ2max

s1 (t) s1 (t) + K1 s2 (t)

s22 (t) Ki

(4)

(5)

+ s2 (t) + K2

The model (3) is valid only for large population sizes. In this situation the stochastic effects can be neglected according to the law of large numbers. Indeed, this model

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is efficient for some applications, it may give good results in suitable environment conditions and a perfectly stirred and homogeneous bioreactor which is the case in the well-controlled laboratory conditions. However, it is not the case in the real conditions. In this situation, the perturbations appear because of imprecise operating conditions. Therefore, the stochastic phenomena cannot be neglected. The same problem appears at small population sizes, especially in the case of multi-species such as our system here (two types of bacteria). Hence, the stochastic effects should be taken into account. In the next section, we aim to build a model that describes the variation of the bacteria and the substrates and also takes account of the stochastic information in the system.

3 Stochastic AM2 Models First we recall the same mass balance principle (2) applied on the interval [t, t + t), with b1t , s1t , b2t , s2t denoting the real (with stochastic effects) concentrations at time t: b1t+t − b1t s1t+t − s1t b2t+t − b2t s2t+t − s2t

= b1t,prod + b1t,out = s1t,cons + s1t,in + s1t,out = b2t,prod + b2t,out = s2t,cons + s2t,prod + s2t,in + s2t,out

(6)

where, within the time interval [t, t + t), we have b1t,prod and b2t,prod represent the increase of the species b1 and b2 due to their growth; b1t,out , b2t,out , s1t,out , s2t,out are, respectively, the proportion of the bacteria and the substrates which has left the bioreactor; s1t,in and s2t,in are the proportion of the substrates which has entered the bioreactor; s1t,cons and s2t,cons represent the decrease of the substrates s1 and s2 due to their consumption by the bacteria; s2t,prod is the part of s2 which has been produced from the first reaction (degradation of s1 ), note that it depends on s1 and b1 . Assuming that none of the previous concentrations change significantly during the interval [t, t + t). We denote by b¯1 (t), s¯1 (t), b¯2 (t), s¯2 (t) the sequence of the means of the concentrations b1t , s1t , b2t , s2t , which are assumed to stay constant all over the interval [t, t + t). Then, by taking only these means, system (6) becomes   b¯1 (t + t) − b¯1 (t) = E b1t,prod + b1t,out   s¯1 (t + t) − s¯1 (t) = E s1t,cons + s1t,in + s1t,out   b¯2 (t + t) − b¯2 (t) = E b2t,prod + b2t,out   s¯2 (t + t) − s¯2 (t) = E s2t,cons + s2t,prod + s2t,in + s2t,out

(7)

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377

Then, by applying the mass action law [18] and assuming a perfect stirring and homogeneity of the medium, we get   E b1t,prod = μ1 (s¯1 (t))b¯1 (t)t ;   E s1t,cons = −k1 μ1 (s¯1 (t))b¯1 (t)t ;  1  E st,out =  −D s¯1 (t)t ; E b2t,prod = μ2 (s¯2 (t))b¯2 (t)t ;   E s2t,cons = −k3 μ2 (s¯2 (t))b¯2 (t)t ;   E s2t,out = −D s¯2 (t)t ;

  E b1t,out = −αD b¯1 (t)t ;   E s1t,in = Ds1in t ;   E b2t,out = −αD b¯2 (t)t ;   E s2t,in = Ds2in t ;   E s2t,prod = k2 μ1 (s¯1 (t))b¯1 (t)t

By replacing these last results in (7), we get the following system of Eq. (8): b¯1 (t + t) − b¯1 (t) = (μ1 (s¯1 (t)) − αD)b¯1 (t)t   s¯1 (t + t) − s¯1 (t) = D (s1in − s¯1 (t)) − k1 μ1 (s¯1 (t))b¯1 (t) t b¯2 (t + t) − b¯2 (t) =  (μ2 (s¯2 (t)) − αD)b¯2 (t)t s¯2 (t + t) − s¯2 (t) = D (s2in − s¯2 (t)) + k2 μ1 (s¯1 (t))b¯1 (t)

(8)

6cm − k3 μ2 (s¯2 (t))b¯2 (t) t

System (8) represents a discrete version of model (3) and is obtained by using an Euler scheme with a time step t. Recursively, the model (3) is the limit of system (8) as t −→ 0. Yet, some attention with the experimental conditions should be made in this case since t has to be large enough to avoid the stochastic variations, this means that model (3) does not describe the bioreactor at its natural scale. If the time step t is too small, then the conditions under which (3) is obtained are not satisfied. Although the stochastic effects should be taken into account in this case.

3.1 Pure Jump Markov Model Now, introduce a pure jump model for the bioreactor denoted as Xt =  1 we ∗ Bt , St1 , Bt2 , St2 . Unlike the model (3), the pure jump process does respect the natural scale of the system. This stochastic model provides a microscopic representation of the bioreactor, which contains a description of the discrete events in the dynamics. As described earlier, the stochastic effects are now considered as we are taking a small t. Yet, taking a very small time step t is unnecessary since we do not aim to describe the dynamics of one unit of substrates or bacteria but the events resulting from adding many jumps of the same type.

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From the mass balance system (6), we remark we can have 11 jumps: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

b1 biological increase of size ν1 (x) at rate λ1 (x). s1 biological decrease of size ν2 (x) at rate λ2 (x). s1 inflow of size ν3 (x) at rate λ3 (x). b1 outflow of size ν4 (x) at rate λ4 (x). s1 outflow of size ν5 (x) at rate λ5 (x). b2 biological increase of size ν6 (x) at rate λ6 (x). s2 biological decrease of size ν7 (x) at rate λ7 (x). s2 biological increase of size ν8 (x) at rate λ8 (x). s2 inflow of size ν9 (x) at rate λ9 (x). b2 outflow of size ν10 (x) at rate λ10 (x). s2 outflow of size ν11 (x) at rate λ11 (x).

For a small time step t, let Bt1,bio , St1,bio , St1,in ,Bt1,out , St1,out , Bt2,bio , 2,prod , St2,in ,Bt2,out , St2,out denote the cumulated jumps of type St2,bio , St 1, . . . , 11, respectively, within the time interval [t, t + t). The probability of having each one of these cumulated jumps, given the concentrations x = (b1 , s1 , b2 , s2 )∗ at time t, is

P (Xt+t

⎧ ⎪ λi (x) t + o(t) if α = νi (x) , ⎪ ⎪ ⎪ ⎪ ⎪ for i = 1, . . . , 11 ⎨ 11 = x + α | Xt = x) ) ⎪ 1− λi (x)t +o(t) if α = 0 ⎪ ⎪ ⎪ i=1 ⎪ ⎪ ⎩ o(t) otherwise (9)

Supposing that the holding times follow an exponential distribution, the number of jumps of type i in the interval [t, t + t) is approximately P (λi (x) t) [18]. So, the expectations of each jump are given by: E [Xt+t = x + νi (x) | Xt = x] = λi (x) νi (x) t , for i = 1, . . . , 11

(10)

So, supposing that the concentrations at time t are b1 , s1 , b2 , s2 , the jumps νi (x) and their rates λi (x) satisfy ⎛ ⎜ ⎜ λ1 (x) ν1 (x) = ⎜ ⎝

⎞ μ1 (s1 )b1 ⎟ 0 ⎟ ⎟; ⎠ 0

0 ⎞ 0 ⎜ ⎟ ⎜ Ds ⎟ λ3 (x) ν3 (x) = ⎜ 1in ⎟ ; ⎝ 0 ⎠ ⎛

0

⎛

⎞ 0 ⎜ ⎟ ⎜ −k μ (s )b ⎟ λ2 (x) ν2 (x) = ⎜ 1 1 1 1 ⎟ ; ⎝ ⎠ 0 0 ⎞ −αDb1 ⎟ ⎜ 0 ⎟ ⎜ λ4 (x) ν4 (x) = ⎜ ⎟; ⎠ ⎝ 0 ⎛

0

Modeling Anaerobic Digestion Using Stochastic Approaches ⎛

⎞ 0 ⎜ ⎟ ⎜ −Ds1 ⎟ λ5 (x) ν5 (x) = ⎜ ⎟; ⎝ 0 ⎠ 0

⎞ 0 ⎟ ⎜ 0 ⎟ ⎜ λ7 (x) ν7 (x) = ⎜ ⎟; ⎠ ⎝ 0 −k3 μ2 (s2 )b2 ⎛ ⎞ 0 ⎜ ⎟ ⎜ 0 ⎟ λ9 (x) ν9 (x) = ⎜ ⎟; ⎝ 0 ⎠ ⎛

Ds ⎛ 2in ⎞ 0 ⎜ ⎟ ⎜ 0 ⎟ λ11 (x) ν11 (x) = ⎜ ⎟ ⎝ 0 ⎠ −Ds2

379 ⎛

⎞ 0 ⎜ ⎟ 0 ⎜ ⎟ λ6 (x) ν6 (x) = ⎜ ⎟; ⎝ μ2 (s2 )b2 ⎠ 0 ⎛ ⎞ 0 ⎜ ⎟ ⎜ k μ (s )b ⎟ λ8 (x) ν8 (x) = ⎜ 2 1 1 1 ⎟ ; ⎝ ⎠ 0 0 ⎞ 0 ⎜ ⎟ 0 ⎜ ⎟ λ10 (x) ν10 (x) = ⎜ ⎟; ⎝ −αDb2 ⎠ 0 ⎛

To determine the values of νi and λi we introduce some scale parameters Ki , i = 1, . . . , 11 and we choose the values given by definition in Table 1. These values mean that jumps of type i will have a size K1i and the corresponding rate will be of size Ki . Large Ki produce small and frequent jumps. Only the jumps in the positive orthant are considered. The choice of these values of jump sizes and rates is not unique. We see that the scale parameters Ki do not affect the mean values of the increments but affect their variances (large Ki will correspond to small variances) according to the Poisson argument mentioned above. The Ki can thus be considered as tuning parameters quantifying the uncertainty or regularity of the corresponding source of variation [14]. The pure jump model can be simulated using the Gillespie algorithm [19]. However, this model is only useful in the case of small population sizes (microscopic scale), its simulation can have a very large computational cost in all other scales which makes it impractical in real situations. The solution is to use approximations of this pure jump process. This can be done in two ways: in continuous time using a Taylor series development of the infinitesimal generator of the process or using a discrete-time Poisson and normal approximations to finally get a diffusion process described by stochastic differential equations. These two methods are presented in the next subsections.

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Table 1 Pure jump process and its 11 jumps and rates Jump i 1

2

3

4

5

6

Size νi ⎛ 1 ⎞ K1

⎟ ⎜ ⎜ 0 ⎟ ⎟ ⎜ ⎝ 0 ⎠ 0 ⎛ ⎞ 0 ⎜ 1 ⎟ ⎜ ⎟ − ⎜ K2 ⎟ ⎝ 0 ⎠ 0 ⎛ ⎞ 0 ⎜ 1 ⎟ ⎜ K3 ⎟ ⎜ ⎟ ⎝ 0 ⎠ ⎛

0 1

⎟ ⎜ ⎜ 0 ⎟ −⎜ ⎟ ⎝ 0 ⎠ 0 ⎞ ⎛ 0 ⎜ 1 ⎟ ⎟ ⎜ − ⎜ K5 ⎟ ⎝ 0 ⎠ 0

Jump i

K1 μ1 (s1 )b1

7

Size νi ⎛ 0 ⎜ 0 ⎜ −⎜ ⎝ 0

⎞ ⎟ ⎟ ⎟ ⎠

Rate λi K7 k3 μ2 (s2 )b2

1

K2 k1 μ1 (s1 )b1

8

K3 Ds1in

9

⎛

K7

0

⎜ 1 ⎜ K8 ⎜ ⎝ 0 0 ⎛ 0 ⎜ 0 ⎜ ⎜ ⎝ 0

⎞ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎠

K8 k2 μ1 (s1 )b1

K9 Ds2in

1

⎞

K4

⎛

Rate λi

K4 αDb1

10

K9

⎞ 0 ⎜ 0 ⎟ ⎟ ⎜ −⎜ 1 ⎟ ⎝K ⎠ ⎛

K10 αDb2

10

K5 Ds1

11

⎛ ⎜ ⎜ −⎜ ⎝

0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎠

K11 Ds2

1

⎞

0 ⎜ 0 ⎟ ⎜ ⎟ ⎜ 1 ⎟ ⎝K ⎠

K11

K6 μ2 (s2 )b2

6

0

3.2 Continuous-Time Approximation We propose an approximation of the pure jump process with the help of the infinitesimal generator [20] of the pure jump process which is given by: Aφ (x) =

11 

λi (x) (φ (x + νi (x)) − φ (x))

(11)

i=1

for all function φ (x) Borel and bounded. By using the second order Taylor series development (12): 1 φ (x + νi (x)) − φ (x) - G∗ νi (x) + νi (x)∗ H νi (x) 2 with G and H are the gradient and the Hessian of φ (x), respectively.

(12)

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381

We get the following infinitesimal generator: ˜ (x) = Aφ

11  i=1

 1 λi (x) G∗ νi (x) + νi (x)∗ H νi (x) 2

(13)

which corresponds to the following diffusion process: 11   !     Xtn+1 = f Xtn dt + νi Xtn λi Xtn dWti

(14)

i=1

where f (·) is exactly the right-hand side of system (3) and Wti are standard Brownian motions. The explicit formula of (14) is given by Eq. (28) in Appendix 1.

3.3 Discrete-Time Approximations 3.3.1

Poisson Approximation

∗  We propose a discrete-time Poisson approximation Xtn = Bt1n , St1n , Bt2n , St2n of  1 1 2 2 ∗ the pure jump process Xt = Bt , St , Bt , St with tn = nt given a small time step t. We suppose the jump sizes νi (x) and the rates λi (x) are constant on an interval [tn , tn+1 ) and they are equal to νi Xtn and λi Xtn , respectively. So that the number of the jumps on the interval [tn , tn+1 ) is approximated by a Poisson distribution. This approximation is given as: Xtn+1 = Xtn +

11 

      νi Xtn Pin λi Xtn t

(15)

i=1

    where Pin λi Xtn t ,i = 1, . . . , 11 and n ∈ N are independent Poisson variables with parameters λi Xtn t. We have 



E Xtn+1 | Xtn = x = x +

11 

      νi (x) E Pin λi Xtn t | Xtn = x

i=1

= x + t

11  i=1

νi (x) λi (x)

(16)

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Let fn (x) 

11 

νi (x) λi (x)

(17)

i=1

fn (x) is exactly the right-hand side of system (3). We also have 11            cov Xtn+1 | Xtn = x = 0 + cov νi Xtn Pin λi Xtn t | Xtn = x

⎛

i=1

σ12

0 ⎜ 0 σ2 2 =⎜ ⎝ 0 0 0 0

0 0 σ32 0

⎞ 0 0 ⎟ ⎟ 0 ⎠ σ42

(18)

with σ12 = σ22 =

(K1 )

σ42 =

  cov P1n (λ1 (x) t) +

2

  cov P2n (λ2 (x) t) +

1 (K2 ) +

σ32 =

2

1

1 (K5 )

  cov P6n (λ6 (x) t) +

2

  cov P7n (λ7 (x) t) +

1 (K7 ) +

1 (K9 )

2

  cov P4n (λ4 (x) t)

(K4 )

2

  cov P3n (λ3 (x) t)

1 (K3 )

  cov P5n (λ5 (x) t)

2

1 (K6 )

2

2

1

1 2

(K10 ) 1 (K8 )

  cov P9n (λ9 (x) t) +

2

 n  cov P10 (λ10 (x) t)

  cov P8n (λ8 (x) t)

1 2

(K11 )

 n  cov P11 (λ11 (x) t)

(19)

that is σ12 = K11 μ1 (s1 ) b1 t + K14 αDb1 t σ22 = K12 k1 μ1 (s1 ) b1 t + K13 Ds1in t + K15 Ds1 t σ32 = K16 μ2 (s2 ) b2 t + K110 αDb2 t σ42 = K17 k3 μ2 (s2 ) b2 t + K18 k2 μ1 (s1 ) b1 t + K19 Ds2in t + K111 Ds2 t

(20)

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3.3.2

383

Normal Approximation

In (15), the variable Pin (λi (x) t) is Poisson distributed with parameter λi (x) t. When this parameter is large (greater than 10), this last distribution is very close to a normal distribution of mean λi (x) t and variance λi (x) t. Hence, we  ∗ 1 1 ˜ ˜ ˜ of Xt = get a discrete-time normal approximation Xtn = Btn , Stn , B˜t2n , S˜t2n  1 1 2 2 ∗ Bt , St , Bt , St : X˜tn+1 = X˜tn +

11 

 νi X˜tn Nin

(21)

i=1

where x = X˜tn , X˜t0 = Xt0 = X0 and Nin are independent Gaussian random variables: Nin ∼ N (λi (x) t, λi (x) t) So, X˜tn+1 is normally distributed with mean (16) and covariance matrix (18) ∗  conditionally on X˜tn = x, i.e., B˜t1 , S˜t1 , B˜t2 , S˜t2 = (b1 , s1 , b2 , s2 )∗ . n

n

n

n

Equation (21) can be written as: X˜tn+1 = x + f (x) t +

11 

νi (x) λi (x) twti

(22)

i=1

where wni ∼ N (0, 1) , i = 1, . . . , 11. 3.3.3

Diffusion Process (Back to Continuous Time)

∗  System (22) is the discretization of a diffusion process Xt = Bt1 , St1 , Bt2 , St2  1 1 2 2 ∗ using an Euler–Maruyama scheme. The process Xt = Bt , St , Bt , St is the solution of the following stochastic differential equation: 11   !     dXtn+1 = f Xtn dt + νi Xtn λi Xtn dWti

(23)

i=1

We attained the same approximation obtained using the continuous method. Also, f (·) is exactly the right-hand side of system (3) and Wti , i = 1, . . . , 11 are standard Brownian motions. The explicit formula of (23) is given by Eq. (28) in Appendix 1. In the pure jump process section, we mentioned that the scale parameters Ki do not affect the mean values and affect their variances. This argument appears clearly

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Table 2 Pure jump process and its jumps and rates Jump νi

First reaction ⎛ ⎞ 1 ⎜ ⎟ 1 ⎜ −k1 ⎟ ⎟ K1 ⎜ ⎝ 0 ⎠

Second reaction ⎛ ⎞ 0 ⎜ ⎟ 1 ⎜ 0 ⎟ ⎟ K2 ⎜ ⎝ 1 ⎠

K1 μ1 (s1 )b1

K2 μ2 (s2 )b2

Inflow ⎛ 1

K4

K3 D

0

⎞

⎜ ⎟ ⎜ s1in ⎟ ⎜ ⎟ ⎝ 0 ⎠

−s2

−k3

k2 Rate λi

Outflow ⎛ ⎞ −αb1 ⎜ ⎟ 1 ⎜ −s1 ⎟ ⎟ K3 ⎜ ⎝ −αb2 ⎠

s2in K4 D

in the normal approximation (48) and the diffusion process (28), (32). Also, in these approximations, it is easy to remark that the large values of Ki will correspond to small variances. Moreover, it is clearly noticeable that system (32) converges almost sure (a.s.) to system (3) when Ki −→ ∞.

4 Another Model As mentioned in the first part of the previous section, the choice of the jump sizes and rates is not unique. In the previous jump process, we have chosen jump sizes values that do not depend on x. We present here another way to describe the jumps along with their rates to have another model as described in Table 2. In this case, the explicit formulas of the model change. The discrete time Poisson approximation is given by (44) in Appendix 1. Also, the discrete-time normal approximation is given by (52) and the continuous-time diffusion model is given by (36) in Appendix 1.

5 Model Simulation The simulation of the pure jump model can be performed using Algorithm 1 in Appendix 2, this algorithm is adapted from the standard Gillespie algorithm. Also, the discrete-time Poisson model can be simulated using the tau-leap method as shown in Algorithm 2 in Appendix 2. Recently many papers have addressed the numerical analysis of this approximation scheme [21–23]. In this method the time step should be small enough so that it fulfills the following “leap condition”:   & '      n (24) νi  (x) Pi  (λi  (x) t) − λi (x) ≤ ελ (x) λ i x +    i

for i = 1 . . . 11, where 0 < ε 0 1.

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To apply this method, the authors in [24] proposed an automatic and simple way of determining the largest time step t compatible with the leap condition: mi (x) 

11 

λi  (x) (∇λi (x) νi  (x))

(25)

λi  (x) (∇λi (x) νi  (x))2

(26)

i  =1

vi (x) 

11  i  =1

for i  = 1 . . . 11. Then t is given by: 

ελ ε 2 λ2 , t = min i |mi (x)| |vi (x)|

 (27)

Finally, to simulate the diffusion process or its discrete-time normal approximation, we use Algorithm 3 in Appendix 2 which is based on the Euler–Maruyama method.

6 Results and Discussion After implementing the previous algorithms in MATLAB we obtained the following results: Fig. 1 contains a comparison between the Poisson approximation, the diffusion approximation, and the deterministic AM2 model for t = [1, 100] and Ki = 103 . By changing the scale parameters Ki we can change the variance of these approximations, for instance: Fig. 2 shows the same simulation but using Ki = 104 instead of Ki = 103 , the variance decreases significantly in this case. Also, considering a different choice of jump sizes and rates would change the variance of the proposed stochastic models. This appears in Fig. 3 which contains the simulation of the second model proposed in Sect. 4 for t = [1, 100] and Ki = 103 . Figures 4 and 5 show the evolution of the mean and standard deviation of the diffusion model for t = [1, 100] and using Ki = 103 in Fig. 4 and Ki = 104 in Fig. 5. The decrease of the variance produced by the change of the scale parameters Ki appears clearly in this simulation. Figure 6 is an approximation of the concentrations densities at time t = 10 h and with Ki = 104 . This approximation was obtained by simulating the system until t = 10 using 10,000 independent Monte Carlo trials and “ksdensity” MATLAB function.

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Fig. 1 Evolution of the biomass and substrate concentrations of the first model in time with Ki = 103 . In Blue: Deterministic model, Red: Diffusion approximation, Green: Poisson approximation

Fig. 2 Evolution of the biomass and substrate concentrations of the first model in time with Ki = 104 . In Blue: Deterministic model, Red: Diffusion approximation, Green: Poisson approximation

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Fig. 3 Evolution of the biomass and substrate concentrations of the second model in time with Ki = 103 . In Blue: Deterministic model, Red: Diffusion approximation, Green: Poisson approximation

Fig. 4 Evolution of the biomass and substrate concentrations of the first model in time with Ki = 103 . In Blue: Deterministic model, Red: Mean of the diffusion approximation, Black: Mean ± Standard deviation of the diffusion approximation

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Fig. 5 Evolution of the biomass and substrate concentrations of the first model in time with Ki = 104 . In Blue: Deterministic model, Red: Mean of the diffusion approximation, Black: Mean ± Standard deviation of the diffusion approximation

Fig. 6 Approximated densities for the biomass and substrate concentrations of the first model at time t = 10 with Ki = 104 . In Black: Deterministic model, Red: Diffusion approximation, Blue: Poisson approximation. These densities were obtained by computing the concentrations at t = 10 using 10,000 independent Monte Carlo trials and ksdensity MATLAB function

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7 Conclusion In this paper, we presented a pure jump Markov model for the two species (acidogenic and methanogenic bacteria) anaerobic digestion process. Among the proposed models in this paper, this model is the only one allowing to describe the dynamics of small-size biomass populations since it takes account of all the events occurring in the process. Different approximations such as the Gaussian, the Poisson, and the diffusion models were proposed. These approximations led us to the final stochastic formulation of the model in two different ways to obtain the SDEs. We simulated the models using specific algorithms such as the Gillespie and the tau-leap algorithms. The simulations showed a total accordance between all the presented models and the initial deterministic model, demonstrating then the validity of our approach. Further contribution would be the stochastic modeling of additional dynamics or phenomena in the process such as the hydrolysis phase or the fouling phenomenon.

Appendix 1: Explicit Formulas Diffusion Process Model 1 The diffusion approximation of the first model is given by the following system of equations: 8 dBt1

=

(μ1 (St1 ) − αD)Bt1 dt

+

8   μ1 St1 Bt1 αDBt1 1 dWt + dWt4 K1 K4

 dSt1 = D s1in − St1 − k1 μ1 (St1 )Bt1 dt + 

8

8 +

Ds1in dWt3 + K3

  k1 μ1 St1 Bt1 dWt2 K2

DSt1 dWt5 K5

8 dBt2 = (μ2 (St2 ) − αD)Bt2 dt +

8

(28)

8   μ2 St2 Bt2 αDBt2 dWt6 + dWt10 K6 K10

(29)

(30)

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 dSt2 = D s2in − St2 + k2 μ1 (St1 )Bt1 − k3 μ2 (St2 )Bt2 dt + 

8 +

8

8 8   k2 μ1 St1 Bt1 Ds2in DSt2 8 9 dWt + dWt + dWt11 K8 K9 K11

  k3 μ2 St2 Bt2 dWt7 K7 (31)

The Brownian motions in (28) can be grouped to have the following system: 8 dBt1

=

(μ1 (St1 ) − αD)Bt1 dt

+

  μ1 St1 Bt1 αDBt1 + dWtB1 K1 K4

  dSt1 = D s1in − St1 − k1 μ1 (St1 )Bt1 dt 8   k1 μ1 St1 Bt1 Ds1in DSt1 + + + dWtS1 K2 K3 K5 8 dBt2

=

(μ2 (St2 ) − αD)Bt2 dt

+

  μ2 St2 Bt2 αDBt2 + dWnB2 K6 K10

  dSt2 = D s2in − St2 + k2 μ1 (St1 )Bt1 − k3 μ2 (St2 )Bt2 dt 8     k3 μ2 St2 Bt2 k2 μ1 St1 Bt1 Ds2in DSt2 + + + + dWnS2 K7 K8 K9 K11

(32)

(33)

(34)

(35)

where WtB1 , WtS1 , WtB2 , WtS2 are independent standard Brownian motions.

Model 2 The diffusion approximation of the second model is given by the following system of equations: 8 dBt1

=

(μ1 (St1 ) − αD)Bt1 dt

+

  2  μ1 St1 Bt1 −αBt1 D + dWt1 K1 K3

(36)

Modeling Anaerobic Digestion Using Stochastic Approaches

  dSt1 = D s1in − St1 − k1 μ1 (St1 )Bt1 dt 8    1 2 −St D (−k1 )2 μ1 St1 Bt1 (s1in )2 D + + + dWt2 K4 K1 K3 8 dBt2 = (μ2 (St2 ) − αD)Bt2 dt +

  2  μ2 St2 Bt2 −αBt2 D + dWt3 K2 K3

391

(37)

(38)

  dSt2 = D s2in − St2 + k2 μ1 (St1 )Bt1 − k3 μ2 (St2 )Bt2 dt 8  2 2     −St D (k2 )2 μ1 St1 Bt1 (−k3 )2 μ2 St2 Bt2 (s2in )2 D + + + + dWt4 K4 K1 K2 K3 (39)

Poisson Approximation Model 1 ∗  Given Bt1n , St1n , Bt2n , St2n = (b1 , s1 , b2 , s2 )∗ , the Poisson approximation (15) can be written as follows: Bt1n+1 = b1 +

1 n 1 n P1 (K1 μ1 (s1 )b1 t) − P (K4 αDb1 t) K1 K4 4

St1n+1 = s1 − − Bt2n+1 = b2 +

St2n+1 = s2 − +

(40)

1 n 1 n P2 (K2 k1 μ1 (s1 )b1 t) + P (K3 Ds1in t) K2 K3 3

1 n P (K5 Ds1 t) K5 5

(41)

1 n 1 n P (K6 μ2 (s2 )b2 t) − P (K10 αDb2 t) K6 6 K10 10

(42)

1 n 1 n P (K7 k3 μ2 (s2 )b2 t) + P (K8 k2 μ1 (s1 )b1 t) K7 7 K8 8

1 n 1 n P9 (K9 Ds2in t) − P (K11 Ds2 t) K9 K11 11

(43)

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Model 2 The Poisson approximation of the second model is given by the following system of equations: Bt1n+1 = b1 +

1 n αb1 n P1 (K1 μ1 (s1 )b1 t) − P (K3 Dt) K1 K3 3

St1n+1 = s1 + − Bt2n+1 = b2 +

s1in n k1 n P (K4 Dt) − P (K1 μ1 (s1 )b1 t) K4 4 K1 1

s1 n P (K3 Dt) K3 3

(45)

1 n αb2 n P2 (K2 μ2 (s2 )b2 t) − P (K3 Dt) K2 K3 3

(46)

St2n+1 = s2 + −

(44)

s2in n k2 n P4 (K4 Dt) + P (K1 μ1 (s1 )b1 t) K4 K1 1

k3 n s2 n P (K2 μ2 (s2 )b2 t) − P (K3 Dt) K2 2 K3 3

(47)

Normal Approximation Model 1 X˜tn+1 is normally distributed with mean (16) and covariance matrix (18) condition ∗  ally on X˜tn = x, i.e., B˜t1 , S˜t1 , B˜t2 , S˜t2 = (b1 , s1 , b2 , s2 )∗ . n

n

n

n

Equation (21) can be written as: 8 B˜t1n+1 = b1 + (μ1 (s1 ) − αD)b1 t +

8 μ1 (s1 ) b1 t 1 wn + K1

αDb1 t 4 wn K4

(48)

8 S˜t1n+1 = s1 + (D (s1in − s1 ) − k1 μ1 (s1 )b1 ) t + 8 +

8 Ds1in t 3 wn + K3

Ds1 t 5 wn K5

k1 μ1 (s1 ) b1 t 2 wn K2 (49)

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8

8 μ2 (s2 ) b2 t 6 wn + K6

B˜t2n+1 = b2 + (μ2 (s2 ) − αD)b2 t +

αDb2 t 10 wn K10 (50)

8 k3 μ2 (s2 ) b2 t

S˜t2n+1 = s2 + (D (s2in − s2 ) + k2 μ1 (s1 )b1 − k3 μ2 (s2 )b2 ) t +

8 +

8

8 k2 μ1 (s1 ) b1 t K8

wn8 +

K7

Ds2in t K9

wn9 +

Ds2 t K11

wn11

wn7

(51)

where wni ∼ N (0, 1) , i = 1, . . . , 11. Model 2 The discrete-time normal approximation of the second model can be written as: 8 B˜t1n+1 = b1 + (μ1 (s1 ) − αD)b1 t +

μ1 (s1 ) b1 (−αb1 )2 D √ + twn1 K1 K3

S˜t1n+1 = s1 + (D (s1in − s1 ) − k1 μ1 (s1 )b1 ) t 8 (−k1 )2 μ1 (s1 ) b1 (−s1 )2 D √ (s1in )2 D + + twn2 + K4 K1 K3 8 B˜t2n+1 = b2 + (μ2 (s2 ) − αD)b2 t +

μ2 (s2 ) b2 (−αb2 )2 D √ + twn3 K2 K3

(52)

(53)

(54)

S˜t2n+1 = s2 + (D (s2in − s2 ) + k2 μ1 (s1 )b1 − k3 μ2 (s2 )b2 ) t

8

+

(s2in )2 D K4

+

(k2 )2 μ1 (s1 ) b1 K1

+

(−k3 )2 μ2 (s2 ) b2 K2

+

(−s2 )2 D √ twn4 K3

(55)

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Appendix 2: Simulation Algorithms

Algorithm 1 Gillespie algorithm adaptation for the pure jump model defined in Table 1 t =0 x (t) = x0 while t ≤ T do compute λi (x) # see Table 1 ) λ = 11 i=1 λi (x) t ∼ Exp (λ) # exponential distribution u ∼ U [0, 1] # uniform distribution t = t + t if u ≤ λ1 (x) /λ then x (t) = x (t − t) + ν1 # b1 increase else if u ≤ (λ1 (x) + λ2 (x)) /λ then x (t) = x (t − t) + ν2 # s1 decrease .. . ) else if u ≤ 10 i=1 λi (x) /λ then x (t) = x (t − t) + ν10 # b2 outflow else x (t) = x (t − t) + ν11 # s2 outflow end if end while

Algorithm 2 Tau-leap algorithm adaptation for system (15) t =0 x (t) = x0 while t ≤ T do compute λi (x) # see Table 1 ) λ = 11 i=1 λi (x) compute mi$(x) , vi (x) % t ∼ mini ελ/ |mi (x)| , ε2 λ2 / |vi (x)| t = t + t Pi ∼ P (λi (x) t) , i = 1, . . . , 11 # Poisson distribution x (t) = max ( 0 , x (t − t) + ν1 P1 + ν2 P2 + . . . + ν11 P11 ) end while

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Algorithm 3 Euler–Maruyama method adaptation for systems (22) and (23) t =0 x (t) = x0 while t ≤ T do t = t + t wj ∼ N (0, 1)

, j = 1, . . . , 4 # normal distribution 7   μ S1

8 +

1 1 (μ1 (St−t ) − αD)Bt−t t

B1

1 t−t t−t = + + K1   1 1 1 1 St = D s1in − St−t − k1 μ1 (St−t )Bt−t t

Bt1

 1  1 k1 μ1 St−t Bt−t K2

+

Ds1in K3

+

1 αDBt−t

K4

√ tw1

1 √ DSt−t tw2 K5 7  

2 2 2 √ μ2 St−t αDBt−t Bt−t 2 2 Bt2 = (μ2 (St−t ) − αD)Bt−t t + + K10 tw3 K6   2 2 1 1 2 2 St = D s2in − St−t + k2 μ1 (St−t )Bt−t − k3 μ2 (St−t )Bt−t t

8 +

 2  2 k3 μ2 St−t Bt−t

 K7  Bt1 = max 0 , Bt1   St1 = max 0 , St1   Bt2 = max 0 , Bt2   2 2 St = max 0 , St end while

+

 1  1 k2 μ1 St−t Bt−t K8

+

Ds2in K9

+

2 √ DSt−t tw4

K11

References 1. D.J. Batstone, J. Keller, I. Angelidaki, S.V. Kalyuzhnyi, S.G. Pavlostathis, A. Rozzi, W.T.M. Sanders, H. Siegrist, V. A. Vavilin, The IWA anaerobic digestion model no. 1 (ADM1). Water Sci. Technol. 45(10), 65–73 (2002) 2. F. Blumensaat, J. Keller, Modelling of two-stage anaerobic digestion using the IWA Anaerobic Digestion Model No. 1 (ADM1). Water Res. 39(1), 171–183 (2005) 3. C. Rosen, D. Vrecko, K.V. Gernaey, M-N. Pons, U. Jeppsson, Implementing ADM1 for plantwide benchmark simulations in Matlab/Simulink. Water Sci. Technol. 54(4), 11–19 (2006) 4. D.J. Batstone, J. Keller, J. P. Steyer, A review of ADM1 extensions, applications, and analysis: 2002–2005. Water Sci. Technol. 54(4), 1–10 (2006) 5. S. Hassam, B. Cherki, E. Ficara, J. Harmand, Towards a systematic approach to reduce complex bioprocess models—Application to the ADM1, in 2012 20th Mediterranean Conference on Control & Automation (MED) (IEEE, Piscataway, 2012), pp. 573–578 6. O. Bernard, Z. Hadj-Sadok, D. Dochain, A. Genovesi, J-P. Steyer, Dynamical model development and parameter identification for an anaerobic wastewater treatment process. Biotechnol. Bioeng. 75(4), 424–438 (2001) 7. J. Hess, O. Bernard, Advanced dynamical risk analysis for monitoring anaerobic digestion process. Biotechnol. Prog. 25(3), 643–653 (2009) 8. A. Rincon, F. Angulo, G. Olivar, Control of an anaerobic digester through normal form of fold bifurcation. J. Process Control 19(8), 1355–1367 (2009) 9. J.P. Steyer, O. Bernard, D. Batstone, I. Angelidaki, Lessons learnt from 15 years of ICA in anaerobic digesters. Water Sci. Technol. 53(4–5), 25–33 (2006)

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10. J. Hess, O. Bernard, Design and study of a risk management criterion for an unstable anaerobic wastewater treatment process. J. Process Control 18(1), 71–79 (2008) 11. B. Benyahia, T. Sari, B. Cherki, J. Harmand, Bifurcation and stability analysis of a two step model for monitoring anaerobic digestion processes. J. Process Control 22(6), 1008–1019 (2012) 12. B. Benyahia, T. Sari, B. Cherki, J. Harmand, Anaerobic membrane bioreactor modeling in the presence of Soluble Microbial Products (SMP)–the Anaerobic Model AM2b. Chem. Eng. J. 228, 1011–1022 (2013) 13. S. Hassam, E. Ficara, A. Leva, J. Harmand, A generic and systematic procedure to derive a simplified model from the anaerobic digestion model No. 1 (ADM1). Biochem. Eng. J. 99, 193–203 (2015) 14. F. Campillo, M. Joannides, I. Larramendy-Valverde, Stochastic modeling of the chemostat. Ecol. Model. 222(15), 2676–2689 (2011) 15. F. Campillo, M. Joannides, Modeles logistiques deterministes et stochastiques, in CARI 2010 (2010), pp. 110–118 16. J. Monod, The growth of bacterial cultures. Annu. Rev. Microbiol. 3(1), 371–394 (1949) 17. H.L. Smith, P.E. Waltman, The Theory of the Chemostat: Dynamics of Microbial Competition (Cambridge University Press, Cambridge, 1995) 18. D.J. Wilkinson, Stochastic Modelling for Systems Biology (Chapman and Hall/CRC, Milton, 2006) 19. D.T. Gillespie, Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115(4), 1716–1733 (2001) 20. S.N. Ethier, T.G. Kurtz, Markov Processes: Characterization and Convergence, vol. 282 (Wiley, New York, 2009) 21. D.F. Anderson, A. Ganguly, T.G. Kurtz et al., Error analysis of tau-leap simulation methods. Ann. Appl. Probab. 21(6), 2226–2262 (2011) 22. T. Li, Analysis of explicit tau-leaping schemes for simulating chemically reacting systems. Multiscale Model. Simul. 6(2), 417–436 (2007) 23. M. Rathinam, L.R. Petzold, Y. Cao, D.T. Gillespie, Consistency and stability of tau-leaping schemes for chemical reaction systems. Multiscale Model. Simul. 4(3), 867–895 (2005) 24. D.T. Gillespie, L.R. Petzold, Improved leap-size selection for accelerated stochastic simulation. J. Chem. Phys. 119(16), 8229–8234 (2003)

Alzheimer Disease: Convergence Result from a Discrete Model Towards a Continuous One M. Caléro, I. S. Ciuperca, L. Pujo-Menjouet, and L. M. Tine

1 Introduction 1.1 Biological Context Alzheimer disease (AD) is referred as one of the most common form of dementia with up to 75% of dementia cases [26]. Predictions show an increase in Alzheimer over time, especially those over 65 years old [11]. Although AD arises most of the time in the elderly people, the disease is not considered as a normal consequence of ageing. Indeed the precise causes of AD are not yet known and several tracks are followed to explain the genesis of the disease. Francis et al. support the hypothesis of cholinergic receptors [7], while the hypotheses of amyloid plaques accumulation [18] and Tau proteins [22] are widely investigated. These two hypotheses could evolve simultaneously [25] and the investigation of the AD patient cerebrospinal fluid shows abnormal concentrations of amyloid Aβ-40 and Aβ-42 [14]. The emergence of Aβ monomers is due to the cleavage of the amyloid precursor protein APP [24] in major isoforms composed of 39–43 amino acids. The Aβ40 is the most common type, while the Aβ-42 is indexed to be a plausible evidence of Alzheimer disease [2, 16]. Two distinct polymerization pathways are identified in the literature. The first one is the fibrillation pathway [2] where the obtained Aβ fibrils are reported to be able to depolymerize [4]. The second one is the Aβ oligomerization pathway which follows the same nucleation elongation

M. Caléro · I. S. Ciuperca · L. Pujo-Menjouet · L. M. Tine () CNRS UMR 5208 Institut Camille Jordan, Université Claude Bernard Lyon 1, Université de Lyon, Villeurbanne Cedex, France Inria Team Dracula, Inria Grenoble Rhône-Alpes Center, Villeurbanne, France e-mail: [email protected] © Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1_25

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mechanism as for the fibrillation one except the fact that oligomerization can lead to the formation of highly stable Aβ assemblies [1, 23]. Thanks to the works done in [15, 17, 19] the Aβ oligomers are involved in several neurotoxic processes, particularly the binding of Aβ oligomers to normal prion PrPC is reported to be involved in a death-signal transduction inside the neurons through the catalysis of the PrPC to PrPol (oligomeric prions) that are misfolded [5, 10, 19, 20]. This catalysis process can be interpreted in physical-chemistry point of view as the succession of two steps: the first one leads to the formation of PrPC /Aβ complex [8, 9] and the second one where, after a delay , the obtained complex evolves to generate PrPol . The observed delay corresponds to a structural rearrangement [13]. For biologists, it is not easy to explain the observed kinetic behaviours in experiments, thereby collaboration with mathematicians allows to establish new mathematical model where all considerations are discussed and validated by biological hypothesis and measures. To the best of our knowledge, only two mathematical models have been proposed to investigate the interaction between Aβ and PrPC [6, 12]. In [12], the whole polymerization process is not taken into account but the authors investigate a plausible in vivo interaction between Aβ oligomers and PrPC . In [6], a complex Aβ polymerization and interaction with PrPC model is proposed. For this last model, the size of polymers is assumed to be continuous so the corresponding model is based on PDEs (partial differential equations) inspired by Lifshitz–Slyozov modelling approach [21].

1.2 Purpose of this Study The aim of this work is to show a convergence result from a modified version of the discrete model [12] towards a continuous one. For the considered discrete model no fragmentation event is taken into account for oligomers but a polymerization rate for maximal size fibrils is added. This paper is organized as follows: in Sect. 2 we present the considered discrete model and its scaling version. Section 3 is devoted to the analysis of the convergence of the model. In Sect. 4, we present some numerical simulations.

2 The Mathematical Model Our model of study is based on the one established in ref. [12]. It describes several processes involved in the polymerization mechanism of Aβ and the interaction with prion PrPC . During the polymerization mechanism, monomers Aβ can either form proto-oligomers or fibrils by gain or loss of monomers. Proto-oligomers and fibrils of a given size i can in turn evolve by gain of monomers at rate ri depending

Alzheimer Disease: Convergence Result from a Discrete Model Towards a. . .

399

on the size i and loss of monomers at a constant rate b. When proto-oligomers reach the size i0 they become stable and can either polymerize or depolymerize and they are called oligomers. There are two behaviours for Aβ oligomers: they can be transported to the amyloid plaque at rate γ or bind to PrPC and form a complex at rate δ. These complexes release after a delay τ some misfolded PrPC which are called PrPol and correspond to the catalysis of the normal prion to an abnormal one. Of course the released oligomer is intact and can interact with another prion or be transported to the plaque. Fibrils can reach a maximum size if . At each size i lower than if , they can still polymerize at rate rf,i and depolymerize at constant rate bf . They can also be transported to amyloid plaque at constant rate γf or escape from amyloid plaque with rate ba . In order to manage very well the analytical study of our convergence result we assume a mass decreasing hypothesis of our model by allowing fibrils of maximal size if to be able to polymerize with the rate rf,if . The schematic whole considered model is given in Fig. 1. All the parameters are described in Table 1. The considered discrete model can be splitted into two sub-models: one for the polymerization of Aβ monomers and another one for the interaction between PrPC and Aβ oligomers.

Fig. 1 Schematic representation of Aβ polymerization processes and interactions with PrPC prions. Orange part (top left) corresponds to monomer evolution. Blue part (left) is related to fibrillation process, while green part (top) is related to oligomerization part. Finally pink box (top right corner) corresponds to Aβ/PrPC interaction. All parameters, quantities and interactions are described in the main text

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Table 1 Description of the parameters Parameters Monomers c1 (t) = cf,1 (t) Oligomerization i0 ci (t), i ∈ [[2, i0 − 1]] ci0 (t), ca (t) ri , i ∈ [[2, i0 − 1]] b γ Fibrillation if cf,j (t), ca,j (t), j ∈ [[2, if ]] rf,j , j ∈ [[1, if ]] bf , ba γf Interactions of Aβ/PrPC pc (t), pol (t), C(t) δ τ ι

Definitions Aβ monomers concentration Maximal size of proto-oligomers Aβ Aβ proto-oligomers of size i Aβ oligomers and Aβ oligomers inside plaque Polymerization rate for proto-oligomers of size i Depolymerization rate of proto-oligomers Aβ Transportation rate of oligomers Aβ towards plaque Maximal size of Aβ fibrils Aβ fibrils of size j inside and outside plaque Polymerization rate for Aβ fibrils of size j Depolymerization rate for Aβ fibrils inside and outside plaque Transportation rate for fibrils Aβ towards plaque Concentration of PrPC , PrPol and complex Aβ/PrPC Reaction rate between oligomers Aβ and PrPC Duration of interaction between Aβ and PrPC The direct transformation rate of PrPC to PrPol

2.1 Sub-models for Aβ Polymerization The equations corresponding to the first sub-model are given by Monomers : c˙1 (t) = −2(r1 c1 (t)2 − bc2 (t)) −

i 0 −2

(ri ci (t)c1 (t) − bci+1 (t)) − ri0 −1 ci0 −1 (t)c1 (t)

i=2 if −1

− 2(rf,1 c1 (t)2 − bf cf,2 (t)) −



(rf,j cf,j (t)c1 (t) − bf cf,j +1 (t)) + 2ba ca,2 (t)

j =2 if −1

+



ba ca,j +1 (t)

(1)

j =2

Proto-oligomers (i ∈ [[2, i0 − 2]]): c˙i (t) = ri−1 ci−1 (t)c1 (t) + bci+1 (t) − ri ci (t)c1 (t) − bci (t), c˙i0 −1 (t) = ri0 −2 ci0 −2 (t)c1 (t) − ri0 −1 ci0 −1 (t)c1 (t) − bci0 −1 (t)

(2)

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Fibrils (j ∈ [[2, if − 1]]): c˙f,j (t) = rf,j −1 cf,j −1 (t)c1 (t) + bf cf,j +1 (t) − rf,j cf,j (t)c1 (t) − bf cf,j (t) −γf cf,j (t),

(3)

c˙f,if (t) = rf,if −1 cf,if −1 (t)c1 (t) − bf cf,if (t) − rf,if cf,if (t)c1 (t) − γf cf,if (t) Fibrils inside plaque (j ∈ [[2, if − 1]]) : c˙a,j (t) = ba ca,j +1 (t) − ba ca,j (t) + γf cf,j (t), c˙a,if (t) = −ba ca,if (t) + γf cf,if (t)

(4)

where t ∈ R+ for Eqs. (1)–(4) with the initial data satisfying ⎧ ⎪ c1 (t = 0) := c10 > 0, ⎪ ⎪ ⎨ c (t = 0) := c0  0, i ∈ [[2, i0 − 1]], i i 0  0, := (t = 0) c j ∈ [[2, if ]], c ⎪ f,j f,j ⎪ ⎪ ⎩ c (t = 0) := c0  0, j ∈ [[2, i ]]. a,j f a,j

(5)

Equation (1) describes the dynamic of Aβ monomers by evaluating the gain and loss of monomers from each proto-oligomer and each fibril. Equation (2) is for proto-oligomers evolution. More precisely, a proto-oligomer of size i can either come from a monomer which is attached to a proto-oligomer of size i − 1 or from proto-oligomer of i + 1 size which loses one monomer. The fibrils, given in (4), follow the same mechanism as the proto-oligomers except the loss of fibrils that are transported into plaque which is described by (4).

2.2 Sub-Model for Prion Catalysis This sub-model gives the interaction between oligomers and prions. The equations can be written as: Oligomers: c˙i0 (t) = ri0 −1 ci0 −1 (t)c1 (t) − γ ci0 (t) − δci0 (t)pc (t) + δci0 (t − τ )pc (t − τ )

(6)

Oligomers inside plaque: c˙a (t) = γ ci0 (t)

(7)

Prions PrPC : p˙ c (t) = −δci0 (t)pc (t) − ιpc (t)

(8)

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Prions PrPol :

p˙ ol (t) = δci0 (t − τ )pc (t − τ ) + ιpc (t)

˙ = δci0 (t)pc (t)−δci0 (t −τ )pc (t −τ ) Complex Oligomers Aβ/PrPC : C(t)

(9) (10)

where t ∈ R+ for Eqs. (7)–(8) and t ∈ [τ, +∞[ for Eqs. (6)–(9)–(10). On t ∈ [0, τ [, Eqs. (6)–(9)–(10) become ⎧ ⎨ c˙i0 (t) = ri0 −1 ci0 −1 (t)c1 (t) − γ ci0 (t) − δci0 (t)pc (t), (11) p˙ (t) = ι pc (t), ⎩ ˙ol C(t) = δci0 (t)pc (t), where the initial data fulfil

⎧ ⎪ ci0 (t = 0) := ci00  0, ⎪ ⎪ ⎪ ⎪ ⎨ ca (t = 0) := ca0  0, pc (t = 0) := pc0  0, ⎪ ⎪ 0 = 0, ⎪ pol (t = 0) := pol ⎪ ⎪ ⎩ 0 C(t = 0) := C = 0.

(12)

In this sub-model, Eq. (6) describes the oligomers behaviour and the delay τ represents the constant binding time that are needed for oligomers to misfold normal prions. Equations (7) and (8) represent, respectively, the evolution of oligomers inside amyloid plaques and the evolution of PrPC . Finally, Eqs. (9) and (10) model, respectively, the evolution of misfolded prions, PrPol , and the evolution of the complex Aβ/PrPC . In order to simplify the mathematical expression, we use in what follows the notations: c˙1 (t) := c˙1,1 (t) + c˙1,2 (t), where c˙1,1 (t) := −2(r1 c1 (t)2 − bc2 (t)) −

i 0 −2

(ri ci (t)c1 (t) − bci+1 (t))

i=2

− ri0 −1 ci0 −1 (t)c1 (t),

(13) if −1

c˙1,2 (t) := −2(rf,1 c1

(t)2

− bf cf,2 (t)) − if −1

+2ba ca,2 (t) +





(rf,j cf,j (t)c1 (t) − bf cf,j +1 (t))

j =2

ba ca,j +1 (t).

j =2

(14)

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From Fig. 1, one can guess that the total mass of the system is decreasing. Indeed, the fact that fibrils of maximal size if are allowed to polymerize implies a kind of leak for those fibrils. So, if we denote by Q(t) the total mass of the system and mp the characteristic mass of prion one can deduce easily the property ˙ Q(t) ≤ 0 ∀t ∈ [0, ∞[. We leave the proof to the reader. The general analytical results including existence, uniqueness, positivity and stability of equilibrium are already done in [12].

2.3 Scaling of the Aβ Polymerization Sub-Model Let define the following parameters: • • • •

T : characteristic time, U : characteristic value of proto-oligomers, fibrils and fibrils inside plaque, M : characteristic value of Aβ monomers, G : characteristic gain rate.

So, the new dimensionless variables are written as follows: ⎧ t ⎪ ⎪ ⎪s = T , ⎪ ⎪ ⎪ ci (t) ⎪ ⎪ ui (s) = , i ∈ [[2, i0 − 1]], ⎪ ⎪ U ⎪ ⎪ ⎪ c (t) f,j ⎪ ⎪ fj (s) = , j ∈ [[2, if ]], ⎪ ⎪ U ⎨ ca,j (t) fa,j (s) = , j ∈ [[2, if ]], ⎪ ⎪ U ⎪ r ⎪ i ⎪ i ∈ [[2, i0 − 1]], g = , ⎪ ⎪ ⎪ i Gr ⎪ ⎪ f,j ⎪ ⎪ , j ∈ [[2, if ]], gf,j = ⎪ ⎪ G ⎪ ⎪ (t) c ⎪ ⎩ m(s) = 1 = u1 = f1 , M M rf,1 M r1 = . with g1 = gf,1 = GU GU For simplicity, we use the notations ⎧ ⎪ α = T GM, α¯ = T G, ⎪ ⎨ U T bf UT b U T ba β= , βf = , βa = , ⎪ M M M ⎪ ⎩ ν = T b, νf = T bf , νγ = T γf , νa = T ba .

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Thanks to these previous considerations, the Aβ polymerization sub-model can be recasted in this following dimensionless model: Monomers: i f 0 −1       m(s) ˙ = β u2 (s) + ui (s) + βf f2 (s) + fj (s) + βa fa,2 (s) i

j =2

i=2 if

i 0 −1

if −1

i=2

j =2

   + fa,j (s) − αU ¯ m(s) j =2

gi ui (s) +



fj (s)gf,j + 4m(s)g1

 (15)

Proto-oligomers (i ∈ [[2, i0 − 2]]):     u˙ i (s) = αm(s) gi−1 ui−1 (s) − gi ui (s) + ν ui+1 (s) − ui (s) ,   u˙ i0 −1 (s) = αm(s) gi0 −2 ui0 −2 (s) − gi0 −1 ui0 −1 (s) − νui0 −1 (s)

(16)

Fibrils (j ∈ [[2, if − 1]]):     f˙j (s) = αm(s) gf,j −1 fj −1 (s) − gf,j fj (s) + νf fj +1 (s) − fj (s) − νγ fj (s),   f˙if (s) = αm(s) gf,if −1 fif −1 (s) − gf,if fif (s) − νf fif (s) − νγ fif (s) (17) Fibrils inside plaque (j ∈ [[2, if − 1]]):   f˙a,j (s) = νa fa,j +1 (s) − fa,j (s) + νγ fj (s), f˙a,if (s) = −νa fa,if (s) + νγ fif (s)

(18)

Mass decreasing relation: 0 β β d  m(s) + iui (s) + j (fj (s) + fa,j (s)  0, ds ν ν

i −1

if

i=2

j =2

(19)

where s ∈ [0, S], S > 0 arbitrary, for Eqs. (21)–(24) with initial conditions satisfying ⎧ ⎪ m(s = 0) := m0 > 0, ⎪ ⎪ ⎨ u (s = 0) := u0  0, i ∈ [[2, i0 − 1]], i i 0  0, := (s = 0) f j ∈ [[2, if ]], f ⎪ j j ⎪ ⎪ ⎩ f (s = 0) := f 0  0, j ∈ [[2, i ]]. a,j f a,j

(20)

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The proof of the obtained dimensionless model for Aβ polymerization process is left in exercise for the interested reader.

3 Convergence of the Dimensionless Model From the discrete model we want to deduce a continuous one based on PDEs equations. For this we let our maximal sizes tend to infinity carefully because protooligomers are known to have a limited size. So, let denote N := i0 − 2, N := if − 2, and x :=

1 . N

We then assume the following hypothesis Hypothesis 1 We assume N = N 2 , such that N −→ +∞. N x→0 With hypothesis 1 we can discretize the size with respect to x for fibrils, fibrils inside the plaque and proto-oligomers. For proto-oligomers, the size is now between 0 and 1, while for fibrils the size is between 0 and +∞. Of course, a maximal size greater than 1 can be obtained as in [6] by discretizing with respect to x0 x with x0 the maximal size. We also consider the following hypothesis for the parameters Hypothesis 2 There exist α0 , β0 , βf,0 , νγ ,0 , βa,0 positive constants not depending on N such that ⎧ α ⎨ α = 0 , αU ¯ = α0 x, β = β0 x, βf = βf,0 x, βa = βa,0 x, x β ⎩ ν = 0 , ν = βf,0 , ν = ν , and ν = βa,0 . f γ γ ,0 a x x x Thanks to hypothesis 2, it is easy to deduce the relation U = (x)2 , M

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and we can rewrite our dimensionless model in order to show the explicit dependence on N as follows: Monomers: N +1 N +2       m(s) ˙ = β0 x u2 (s) + ui (s) + βf,0 x f2 (s) + fj (s) j =2

i=2

N +2 +1    N + βa,0 x fa,2 (s) + fa,j (s) − α0 xm(s) gi ui (s) j =2

+

N +1 

i=2

fj (s)gf,j + 4m(s)g1



(21)

j =2

Proto-oligomers (i ∈ [[2, N]]):    α0 β0  m(s) gi−1 ui−1 (s) − gi ui (s) + ui+1 (s) − ui (s) , x x   α0 β0 m(s) gN uN (s) − gN +1 uN +1 (s) − ui −1 (s) u˙ N +1 (s) = x x 0

u˙ i (s)

=

(22)

Fibrils (j ∈ [[2, N + 1]]):   βf,0   α0 m(s) gf,j −1 fj −1 (s) − gf,j fj (s) + fj +1 (s) − fj (s) x x −νγ ,0 fj (s),   βf,0 α0 m(s) gf,N +1 fN +1 (s) − gf,N +2 fN +2 (s) − fN +2 (s) f˙N +2 (s) = x x −νγ ,0 fif (s) (23) f˙j (s)

=

Fibrils inside plaque (j ∈ [[2, N + 1]]):  βa,0  fa,j +1 (s) − fa,j (s) + νγ ,0 fj (s), x βa,0 fa,N +2 (s) + νγ ,0 fN +2 (s) f˙a,N +2 (s) = − x f˙a,j (s)

=

(24)

Mass decreasing relation:   d  m(s) + (x)2 iui (s) + (x)2 j (fj (s) + fa,j (s)  0, ds N +1

N +2

i=2

j =2

where s ∈ [0, S] and initial data remain the same as in (20).

(25)

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For all the polymerization rates of the model, we set the following hypotheses: Hypothesis 3 There exists an increasing function g : ] − ∞, 1[−→ R+ belonging to C 1 (R∗+ ) and a positive constant Mg independent to N such that   g (i − 2)x ∀i ∈ [[2, N + 1]], with g(x) = 0 if x  0, gi = α0 if x > 0. g(x)  x Mg , Hypothesis 4 There exists an increasing function gf : R −→ R+ belonging to C 1 (R∗+ ) and a positive constant Mgf independent to N such that   gf (j − 2)x gf,j = ∀j ∈ [[2, N + 1]], with gf (x) = 0 if x  0, α0 if x > 0. gf (x)  x Mgf , Some hypotheses are needed for the initial data of our model and we assume Hypothesis 5 There exists a function uin : ] − ∞, 1[−→ R+ such that   u0i = uin (i − 2)x ∀i ∈ [[2, N + 1]], with uin (x) = 0 if x  0,   uin ∈ L∞ [0, 1[ . Hypothesis 6 There exists f in : R −→ R+ such that   fj0 = f in (j − 2)x ∀j ∈ [[2, N + 2]], with f in (x) = 0 if x  0,     f in ∈ L1 R+ , (1 + x)dx ∩ L∞ R+ . Hypothesis 7 There exists a function f a,in : R −→ R+ such that   0 = f a,in (j − 2)x ∀j ∈ [[2, N + 2]], with f a,in (x) = 0 if x  0, fa,j     f a,in ∈ L1 R+ , (1 + x)dx ∩ L∞ R+ . Let remark that on the one hand m(s) depends either on N or N and on the other hand, thanks to relation (25), the equation on the mass of the system can be recasted as m(s) + (x)2 +(x)2

+1 N i=2

+1 N i=2

i ui (s) +

N +2 

  j fj (s) + fa,j (s)  m0

j =2

+2    N      i uin (i − 2)x + j f in (j − 2)x + f a,in (j − 2)x . j =2

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3.1 Estimates on Solutions 3.1.1

L∞ Estimates

We investigate here some L∞ estimates of the solutions for proto-oligomers, fibrils and fibrils inside plaque. Proposition 3.1 Let Ku = maxi∈[[2,N +1]] u0i and Kf = maxj ∈[[2,N +2]] fj0 . Denoting by R := max(Ku , Kf ), one deduces (i) ui L∞ ([0,S])  R, ∀i ∈ [[2, N + 1]]. (ii) fj L∞ ([0,S])  R, ∀j ∈ [[2, N + 2]]. (iii) fa,j L∞ ([0,S])  R(1 + S), ∀j ∈ [[2, N + 2]]. Proof The proof of (i) and (ii) is obvious and the argument is based on quasipositivity technique that we recall as follows Lemma 3.1 Let F : Rn −→ Rn a continuous and locally Lipschitz function and X = (X1 , X2 , . . . , Xn ) ∈ Rn . Let consider the Cauchy problem: 

˙ X(s) = F (X(s)), s ∈ [0, S],  n X(0) = X0 ∈ [0, R] ,

with R and S two positive constants. If the two below relations are satisfied for an arbitrary i ∈ [[1, n]] : Xi = 0 and Xj ∈ [0, R] ∀j ∈ [[1, n]] ⇒ Fi (X)  0, Xi = R and Xj ∈ [0, R] ∀j ∈ [[1, n]] ⇒ Fi (X)  0, then Xi ∈ [0, R]. In particular, the unique solution of the Cauchy problem fulfils Xi ∈ [0, R] ∀s ∈ [0, S]. For the proof of (iii), the quasi-positivity technique is unusable, so we write the differential equations for fa,j , j ∈ [[2, N + 2]] in matrix form 

U˙ (s) = A U (s) + B(s), U (0) = U 0 ,

with ⎛ ⎛

fa,2 (s) ⎜. U (s) = ⎜ ⎝ .. fa,N +2 (s)

⎞

⎛ f2 (s) ⎟ ⎜. ⎟ , B(s) = νγ ,0 ⎜ . ⎠ ⎝. fN +2 (s)

⎞ ⎟ ⎟ , and A = βa,0 ⎠ x

−1 1

⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎜ ⎝0

0

0

...

0

⎞

⎟ −1 1 0 . . . 0 ⎟ ⎟ ⎟ .. .. . . . 0 ⎟ 0 ⎟ ⎟ 0 0 −1 1 ⎠ ... ... 0

−1

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Duhamel formula gives the unique solution:

U (s) = exp(A s) U 0 +

s

  exp (s − t) A B(t) dt.

0

In order to compute exp(A s), we then split the matrix as follows: As = ⎞ ⎛ 0 1 0 ... 0 ⎜0 0 1 0... 0⎟ ⎟ ⎜ βa,0 ⎟ ⎜ s(−Id + C), where the matrix C = ⎜ 0 0 . . . . . . 0 ⎟ . So on the one ⎟ ⎜ x ⎝0 0 0 0 1⎠ 0 ... ... 0

0

hand, we have exp(−

βa,0 βa,0 sId) = exp(− s) Id. x x

On the other hand, one remarks that C is nilpotent because C N +2 vanishes. Then the ith component of U (s) denoted by Ui (s) satisfies for all i ∈ [[1, N + 1]] : N βa,0  (βa,0 s)k Ui (s) = exp(− s) fa,i+2 (0) x (x)k k! k=i−1

s N  (νa (s − t))k βa,0 (s − t)) fi+2 (s) ds + exp(− x k! 0 k=i−1

N βa,0  (βa,0 (s − t))k s)  R exp(− x (x)k k! k=i−1

s N  (βa,0 (s − t))k βa,0 (s − t)) ds + R exp(− x (x)k k! 0 k=i−1 βa,0 βa,0 s)exp( s)  R exp(− x x

s βa,0 βa,0 (s − t))exp( (s − t)) ds + R exp(− x x 0  R(1 + S),

which achieves the proof.

3.1.2

 

L1 Estimates

For the study of the convergence of our model, we need some L1 estimates for fibrils and fibrils inside plaque.

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Proposition 3.2 There exist MF and MFa two constants not dependent on N such that (a) x

N +2 

fj (s)  MF .

j =2

(b) x

N +2 

fa,j (s)  MFa .

j =2

Proof From the fundamental analysis theorem, one has N +2 

xfj (s) = x

j =2

N +2 

  f in (j − 2)x + x

sN +2  f˙j (t) dt. 0 j =2

j =2

By Riemann integrals and thanks to hypothesis 6

x

N +2 

  f in (j − 2)x −→

N →+∞

j =2

+∞

f in (x) dx < +∞,

0

By (23), one deduces N +2 

N +2 



x f˙j (t) = α0 m(t)

j =2

+βf,0

gf,j −1 fj −1 (t) − gf,j fj (t)



j =2 N +1 

N +1    fj +1 (t) − fj (t) − νγ ,0 x fj (t)

j =2

j =2

−βf,0 fN +2 (t) − νγ ,0 xfN +2 (t) N +1     α0 m(t) gf,j −1 fj −1 (t) − gf,j fj (t) +βf,0

j =2 N +1 

  fj +1 (t) − fj (t)

j =2

+α0 m(t)gf,N +1 fN +1 (t) − βf,0 fN +2 (t).

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N +1 

  gf,j −1 fj −1 (t) − gf,j fj (t) .

• Step 1: Computation of α0 m(t)

j =2

Thanks to hypothesis 4, N +1 

N    gf,j −1 fj −1 (t) − gf,j fj (t) = α0 m(t) gf,j fj (t)

α0 m(t)

j =2

j =1

N +1 

−α0 m(t)

gf,j fj (t)

j =2

= −α0 m(t)gf,N +1 fN +1 (t) +αm(t)xgf,1 f1 (t) = −α0 m(t)gf,N +1 fN +1 (t)

• Step 2: Computation of βf,0

N +1 

  fj +1 (t) − fj (t) .

j =2

βf,0

N +1 

N +2 N +1     fj +1 (t) − fj (t) = βf,0 fj (t) − βf,0 fj (t),

j =2

j =3

j =2

= βf,0 (fN +2 (t) − f2 (t))  βf,0 fN +2 (t).

So, N +2 

x f˙j (t)  0,

j =2

which implies N +2 

N +2 

j =2

j =2

xfj (s)  x

  f in (j − 2)x  Cste,

where Cste is a constant, which proves item (a). For the proof of item (b) we use again the fundamental analysis theorem and obtain N +2 

xfa,j (s) = x

j =2

N +2  j =2

f

a,in

  (j − 2)x + x

sN +2  0 j =2

f˙a,j (t) dt.

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By Riemann and thanks to hypothesis 7 one has x

N +2 

f

a,in

  (j − 2)x −→

N →+∞

j =2

+∞

f a,in (x) dx < +∞,

0

So, from (24) : N +2 

N +1 

j =2

j =2

x f˙a,j (t) = βa,0

  fa,j +1 (t) − fa,j (t)

+νγ ,0 x

N +2 

fj (t) − βa,0 fa,N +2 (t),

j =2

= −βa,0 fa,2 (t) + νγ x  νγ ,0 x

N +2 

N +2 

fj (t),

j =2

fj (s),

j =2

 

which achieves the proof in regard to hypothesis 2 and item a).

3.1.3

Mass Control for Proto-Oligomers, Fibrils and Fibrils Inside Plaque

We give here some estimates that control the mass of the system in order to prove strong convergence result for monomers. Proposition 3.3 There exist two positive constants MF˜ and MF˜a not dependent on N such that (x)

2

N +2 

j fj (s)  MF˜ .

(26)

j fa,j (s)  MF˜a .

(27)

j =2

2

(x)

N +2  j =2

Proof By the mass equation relation, we deduce N +2

N +1

j =2

i=2

   in   β0 β0 (x)2 j fj (s)  m0 + (x)2 i u (i − 2)x ν0 ν0 +

N +2 

     j f in (j − 2)x + f a,in (j − 2)x .

j =2

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On the one hand, we have N +2       β0 (x)2 j f in (j − 2)x + f a,in (j − 2)x ν0 j =2

=

N +2

      β0 (x)2 (j − 2) f in (j − 2)x + f a,in (j − 2)x ν0 j =2

N +2

     2β0 + f in (j − 2)x + f a,in (j − 2)x . (x)2 ν0 j =2

On the other hand, we have N +1

N +1

i=2

i=2

      β0 β0 (x)2 i uin (i − 2)x = (x)2 (i − 2) uin (i − 2)x ν0 ν0 N +1

+

   2β0 (x)2 uin (i − 2)x . ν0 i=2

By Riemann sums one deduces N +2       β0 (x)2 (j − 2) f in (j − 2)x + f a,in (j − 2)x ν0 j =2

β0 ∞ −→ x(f in (x) + f a,in (x)) dx, x→0 ν0 0 N +2   in     2β0 f (j − 2)x + f a,in (j − 2)x −→ 0, (x)2 x→0 ν0 j =2

N +1    β0 β0 1 in (x)2 (i − 2) uin (i − 2)x −→ xu (x) dx, x→0 ν0 0 ν0 i=2 N +1

   2β0 (x)2 uin (i − 2)x −→ 0. x→0 ν0 i=2

  Knowing that f in ,f a,in ∈ L1 R+ , (1 + x)dx and uin ∈ L∞ ([0, 1[) ⊆   L1 [0, 1[, (1 + x)dx , then all the terms are finite and not dependent on N .

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The proof of relation (27) follows the same argument as above and the beginning point is the inequality: N +2

N +1

   in   β0 β0 (x)2 j fa,j (s)  m0 + (x)2 i u (i − 2)x ν0 ν0 j =2

i=2

N +2 

+

     j f in (j − 2)x + f a,in (j − 2)x .

j =2

3.1.4

 

Strong Convergence for Monomers m(s)

For monomers dynamics, we are looking for a strong convergence result. So we prove the following proposition Proposition 3.4 There exist Mm , Mm˙ positive constants not depending on N such that (i) m L∞ ([0,1])  Mm , (ii) m ˙ L∞ ([0,1])  Mm˙ . Proof The proposed estimates in (i) and (ii) are obvious and we let it as an exercise to the interested reader. They are based on the mass decreasing relation for (i) and on Eq. (21) for (ii).   From the estimates in Proposition 3.4 we deduce, thanks to compact injection theorem, the existence of a subsequence of m denoted also by m that converges towards an m∗ in L∞ ([0, 1]).

3.1.5

Construction of Size Dependence of the Scaling Model

To introduce size dependence on the solutions of the scaling model, we define the following notations:   u (s) if x ∈ [(i − 2)x, (i − 1)x[, ∀i ∈ [[2, N + 1]], ∀s ∈ [0, S],  UN (s, x) =  i 0 if x < 0,   f (s) if x ∈ [(j − 2)x, (j − 1)x[, ∀j ∈ [[2, N + 2]], ∀s ∈ [0, S],  FN (s, x) =  j 0 if x > N or x < 0,   f (s) if x ∈ [(j − 2)x, (j − 1)x[, ∀j ∈ [[2, N + 2]], ∀s ∈ [0, S],  Fa,N (s, x) =  a,j 0 if x > N or x < 0.

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  We will also use the notation UN s, (i−2)x to designate ui (s). The same notation is used for fibrils and fibrils inside plaque. Let state the following proposition Proposition 3.5

  (1) UN is bounded in L∞ [0, S] × [0, 1] . Furthermore FN and Fa,N are bounded in L∞ [0, S] × R+ .    (2) FN is bounded in L∞ [0, S]; L1 R+ , (1 + x)dx .    (3) Fa,N is bounded in L∞ [0, S]; L1 R+ , (1 + x)dx . Proof The proof of item (1) is already done, one has to adapt it in regard to the previous notations. For item (2), we split the integral as follows:

0

∞

FN (s, x)(1 + x)dx =

1

On [0, 1] one has 0

1

0

FN (s, x)(1 + x)dx +

1

FN (s, x)(1 + x)dx.

1 FN (s, x)(1 + x)dx  2 FN (s, x)dx < ∞. 0

On [1, ∞[ we have

1

∞

∞

FN (s, x)(1 + x)dx  2

∞ 0

x FN (s, x)dx.

Knowing that FN is a piecewise constant function, we deduce, thanks to the mass control relation,

2

∞ 0

N +1 (j −1)x 

x FN (s, x) dx = 2

  x FN s, (j − 2)x dx,

=

j =2 (j −2)x N +1   2 (x) F s, (j



j =2 N +1  2(x)2 j

N

 − 2)x (2j − 3),

  FN s, (j − 2)x < ∞.

j =2

The proof of item (3) follows the same reasoning as for item (2).

 

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Weak∗ Convergence for Proto-Oligomers, Fibrils and Fibrils Inside Plaque

We denote     H1 := L1 [0, S] × [0, 1] and H2 := L1 [0, S], X , with % $ X := f : R+ −→ R+ measurable : ∃C > 0 : f (x)  C (1 + x) p.p x ∈ R+ , : f (x) : :   : . and f (x) X = : : 1 + x L∞ R+ We then state the following result Proposition 3.6 There exist subsequences that we will note again UN , FN and Fa,N which converge in weak∗ topology, respectively, towards U ∗ in H1 , towards F ∗ in H2 and towards Fa∗ in H2 . The proof is essentially based on classical analysis results that one can find, for instance, in [3]. So we already know that proto-oligomers are bounded in H1 , thanks to Proposition 3.5.   The continuous embedding result of L1 R+ , (1 + x)dx in X shows that fibrils and fibrils inside plaque are bounded in H2 . So, we achieve the proof by applying to L1 space the following lemma from [3] Lemma 3.2 Let E be a Banach separable space and let (fn ) a bounded sequence in E  . Then there exists a subsequence (fnk ) which converges for weak∗ topology in E.

3.2 Weak Formulation of the Scaling Model In this part we propose a weak formulation of our model. Before that let use the following notations for spaces:  • ω :=]0, S[, 1 :=]0, S[×]0, 1[, 2 :=]0, S[×]0, +∞[, V := φ(s)  0 ∈    C ∞ ω¯ : φ(S) = 0 ,    ¯ 1 : ψ(S, x) = ψ(s, 0) = ψ(s, 1) = 0 ∀(s, x) ∈ • W1 := ψ(s, x)  0 ∈ Cc∞   1 ⊆ H1 ,    ¯ 2 ∩ H2 : ∂s ψ, ∂x ψ ∈ H2 : ψ(S, x) = ψ(s, 0) = • W2 := ψ(s, x)  0 ∈ Cc∞   0 ∀(s, x) ∈ 2 .

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    Theorem 3.1 There exists m(s), UN (s, x), F(s, x), Fa (s, x) ∈ L∞ (ω)×H ¯ 1×   H2 × H2 such that ∀ (φ, ψ1 , ψ2 ) ∈ (V × W1 × W2 ) we have the following weak formulation for our model: Monomers:

S ˙ ds − m0 φ(0) − m(s)φ(s) 0

+1 SN 

=− 0

−

i=2

SN +2  0

+

  φ(s)xUN s, (i − 2)x v(s, (i − 2)x) ds

j =2

SN +2  0

  φ(s)xFa,N s, (j − 2)x βa,0 ds

j =2 S

+

  φ(s)xFN s, (j − 2)x vf (s, (j − 2)x) ds

0

  φ(s)x UN (s, 0)β0 + βf,0 FN (s, 0) + βa,0 Fa,N (s, 0) ds

(28)

where v(s, x) := m(s)g(x) − β0 , vf (s, x) := m(s)gf (x) − βf,0 .

(29)

Proto-oligomers:

−

+1 SN  0

    ∂s ψ1 s, (i − 2)x UN s, (i − 2)x x ds

i=2 N +1 

−x

    ψ1 0, (i − 2)x uin (i − 2)x

i=2

=

S N

   m(s) ψ1 s, (i − 1)x

0 i=2 

     −ψ1 s, (i − 2)x g (i − 2)x UN s, (i − 2)x ds

S N +1       UN s, (i − 2)x ψ1 s, (i − 2)x − ψ1 s, (i − 3)x ds −ν0 0

S i=3       − m(s)ψ1 s, (N − 1)x g (N − 1)x UN s, (N − 1)x ds 0

(30)

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Fibrils:

SN +2      x∂s ψ2 s, (j − 2)x FN s, (j − 2)x ds − 0

j =2

− x

N +2 

    ψ2 0, (j − 2)x f in (j − 2)x

j =2

= −νγ ,0 x

SN +2      ψ2 s, (j − 2)x FN s, (j − 2)x ds 0

− βf,0

SN +2  0

+

j =2

      FN s, (j − 2)x ψ2 s, (j − 2)x − ψ2 s, (j − 3)x ds

j =3

SN +1        m(s)gf (j − 2)x FN s, (j − 2)x ψ2 s, (j − 1)x 0

j =2

  − ψ2 s, (j − 2)x ds −

S

0

m(s)ψ2 (s, N )gf (N )FN (s, N ) ds

(31)

Fibrils in plaque: −

SN +2  0

    xFa,N s, (j − 2)x ∂s ψ2 s, (j − 2)x ds

j =2

− x

N +2 

    f a,in (j − 2)x ψ2 0, (j − 2)x

j =2

=

SN +2  0

−

    νγ ,0 xFN s, (j − 2)x ψ2 s, (j − 2)x ds

j =2

SN +2  0

    βa,0 Fa,N s, (j − 2)x ψ2 s, (j − 2)x

j =3



− ψ2 s, (j − 3)x



ds

(32)

Mass decreasing:

S 0

N +2 N +1    ˙ φ(s) m(s) + (x)2 iui (s) + (x)2 j (fj (s) + fa,j (s)) ds i=2

j =2

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N +1 N +2         φ(0) m0 + (x)2 iuin (i − 2)x + (x)2 j (f in (j − 2)x j =2

i=2

  + f a,in (j − 2)x )

(33)

Proof The proof of Theorem 3.1 consists to multiply the considered equation by test function and integrate it on [0, S]. So for monomers we use the previous notations and rewrite Eq. (21) as follows: N +1 

m(s) ˙ =

xUN (s, (i − 2)x)(β0 − m(s)g((i − 2)x))

i=2

N +2 

+

   xFN s, (j − 2)x βf,0 − m(s)gf ((j − 2)x)

j =2

+βa,0 x

N +2 

fa,j (s) + xβ0 UN (s, 0) + xβf,0 FN (s, 0)

j =2

+xβa,0 Fa,N (s, 0), then we multiply this relation by a test function φ ∈ V and integrate it on [0, S]. For proto-oligomers we consider a test function ψ1 ∈ W1 . We then multiply Eq. (22) by ψ1 (s, (i − 2)x ), take the sum on all proto-oligomers and obtain

+1 SN 

  u˙ i (s)ψ1 s, (i − 2)x ds

0

i=2

=

+1 SN 

   α0 m(s)ψ1 s, (i − 2)x gi−1 ui−1 (s) − gi ui (s) ds 0 i=2 x

S  β0  ψ1 s, (N − 1)x uN +1 (s) − 0 x N    β0  ψ1 s, (i − 2)x ui+1 (s) − ui (s) ds. − x i=2

• Step 1 : Computation of the left term.

S N +1   u˙ i (s)ψ1 s, (i − 2)x ds = − 0

i=2 N +1 

+

i=2

+1 SN 

  ui (s)∂s ψ1 s, (i − 2)x ds

0

i=2

      ψ1 S, (i − 2)x ui (S) − uin (i − 2)x ψ1 0, (i − 2)x .

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Knowing ψ1 ∈ W1 , then

  u˙ i (s)ψ1 s, (i − 2)x ds = −

0

+1 SN 

+1 SN 

i=2

0 i=2 N +1   in

−

u

  ui (s)∂s ψ1 s, (i − 2)x ds    (i − 2)x ψ1 0, (i − 2)x .

i=2

Using the piecewise function notation introduced above, we obtain

+1 SN 

− 0

−

    ∂s ψ1 s, (i − 2)x UN s, (i − 2)x ds

i=2 N +1 

    ψ1 0, (i − 2)x uin (i − 2)x .

i=2

• Step 2: Computation of

N +1  i=2

N +1  i=2

   α0 m(s)ψ1 s, (i − 2)x gi−1 ui−1 (s) − gi ui (s) . x

   α0 m(s)ψ1 s, (i − 2)x gi−1 ui−1 (s) − gi ui (s) x

=

N +1 

  α0 m(s)ψ1 s, (i − 2)x gi−1 ui−1 (s) x

i=2 N +1 

−

i=2

  α0 m(s)ψ1 s, (i − 2)x gi ui (s) x

N    α0 m(s)ψ1 s, (i − 1)x gi ui (s) = x i=1 N +1 

−

i=2

  α0 m(s)ψ1 s, (i − 2)x gi ui (s) x

N       α0 m(s)gi ui (s) ψ1 s, (i − 1)x − ψ1 s, (i − 2)x = x i=2 α0 m(s)ψ1 (s, 0)g1 u1 (s) + x   α0 m(s)ψ1 s, (N − 1)x gN +1 uN +1 (s). − x

Alzheimer Disease: Convergence Result from a Discrete Model Towards a. . .

Thanks to ψ1 ∈ W1 , we deduce N +1  i=2

   α0 m(s)ψ1 s, (i − 2)x gi−1 ui−1 (s) − gi ui (s) x

N       α0 m(s)gi ui (s) ψ1 s, (i − 1)x − ψ1 s, (i − 2)x = x i=2   α0 − m(s)ψ1 s, (N − 1)x gN +1 uN +1 (s). x

Integrating from 0 to S and using the hypotheses of the model, we obtain 1 x

S N

      m(s)g (i − 2)x UN s, (i − 2)x ψ1 s, (i − 1)x

0

i=2  −ψ1 s, (i − 2)x ds

S       1 − m(s)ψ1 s, (N − 1)x g (N − 1)x UN s, (N − 1)x ds. x 0

• Step 3 : Computation of

N    β0  ψ1 s, (i − 2)x ui+1 (s) − ui (s) . x i=2

N  i=2

= =

  β0  ψ1 s, (i − 2)x ui+1 (s) − ui (s) x

N N     β0  β0  ψ1 s, (i − 2)x ui+1 (s) − ψ1 s, (i − 2)x ui (s), x x i=2 N +1 

i=2

N    β0  β0  ψ1 s, (i − 3)x ui (s) − ψ1 s, (i − 2)x ui (s), x x

i=3 N 

i=2

     β0 ui (s) ψ1 s, (i − 2)x − ψ1 s, (i − 3)x =− x i=3  β0  β0 u2 (s)ψ1 (s, 0) + ψ1 s, (N − 2)x uN +1 (s). − x x The fact that ψ1 ∈ W1 implies N    β0  ψ1 s, (i − 2)x ui+1 (s) − ui (s) x i=2

N       β0 ui (s) ψ1 s, (i − 2)x − ψ1 s, (i − 3)x =− x i=3  β0  ψ1 s, (N − 2)x uN +1 (s). + x

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If we add the term − N +1 

−

i=3

 β0  ψ1 s, (N − 1)x uN +1 (s), we obtain x

     β0 ui (s) ψ1 s, (i − 2)x − ψ1 s, (i − 3)x . x

Integrating on [0, S] and using hypotheses of the model one deduces −

1 x

+1 SN 

0

      β0 UN s, (i − 2)x ψ1 s, (i − 2)x − ψ1 s, (i − 3)x ds.

i=3

Finally we multiply by x in the equation and obtain the expected weak formulation for proto-oligomers. For fibrils and fibrils inside plaque, the proof of the weak formulation works the same as for proto-oligomers. For the mass decreasing weak formulation we have   d  m(s) + (x)2 iui (s) + (x)2 j (fj (s) + fa,j (s)) ds  0. ds N +1

N +2

i=2

j =2

Multiplying this inequality by φ ∈ V and integrating on [0, S] one obtains

S

φ(s) 0

  d  m(s) + (x)2 iui (s) + (x)2 j (fj (s) + fa,j (s)) ds  0. ds N +1

N +2

i=2

j =2

Knowing that φ(S) = 0, we get

S 0

N +2 N +1    ˙ φ(s) m(s) + (x)2 iui (s) + (x)2 j (fj (s) + fa,j (s)) ds i=2

j =2

N +2 N +1         φ(0) m0 + (x)2 iuin (i − 2)x + (x)2 j (f in (j − 2)x

  +f a,in (j − 2)x ) , i=2

which achieves the global proof of the theorem.

j =2

 

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3.3 Weak Convergence Towards a Continuous Model Let define the following continuous model: Monomers:

1

m(s) ˙ = − u(s, x)v(s, x) dx − 0

∞

∞

f (s, x)vf (s, x) dx + βa,0

0

fa (s, x) dx 0

v(s, x) := m(s)g(x) − β0 , vf (s, x) := m(s)gf (x) − βf,0

(34)

  Proto-oligomers (x ∈ [0, 1[): ∂s u(s, x) + ∂x u(s, x)v(s, x) = 0

(35)

  Fibrils (x ∈ [0, ∞[): ∂s f (s, x) + ∂x vf (s, x)f (s, x) = −νγ ,0 f (s, x)

(36)

Fibrils inside plaque (x ∈ [0, ∞[):∂s fa (s, x) − βa,0 ∂x fa (s, x) = νγ ,0 f (s, x) (37) Mass decreasing: ˙ E(s)  0,

1

E(s) := m(s) +

∞

x u(s, x) dx +

0

x(f (s, x) + fa (s, x)) dx,

(38)

0

with s ∈ [0, S] and initial data satisfying ⎧ ⎪ m(0) = m0 , ⎪ ⎪ ⎪ ⎪ ⎨ u(0, x) = uin (x), f (0, x) = f in (x), ⎪ ⎪ a,in (x), ⎪ ⎪ fa (0, x) = f ⎪ ⎩ E(0) = E 0 .

(39)

Let us define what we mean by strong solution and weak solution for the continuous model. Definition 3.1 (m, u, f, fa ) is called strong solution of the continuous model if it fulfils the equations of the model and belongs at least to the space of C 1 functions.

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Definition 3.2 Let m ∈ W 1, ∞ ([0, S]), u ∈ H1 , f ∈ H2 , fa ∈ H2 . (m, u, f, fa ) is called weak solution of the continuous model if ∀(φ, ψ1 , ψ2 ) ∈ (V, W1 , W2 ) we have Monomers:

S

S 1 ˙ ds − m0 φ(0) = − − m(s)φ(s) φ(s)u(s, x)v(s, x) dxds 0

0 S 0 ∞ φ(s)f (s, x)vf (s, x) dxds −

0 S 0 ∞ φ(s)fa (s, x)βa,0 dxds + 0

0

Proto-oligomers:

S 1

1 − ∂s ψ1 (s, x)u(s, x) dxds − ψ1 (0, x)uin (x)dx 0 0 0 S 1 v(s, x)u(s, x)∂x ψ1 (s, x) dxds = 0

0

Fibrils:

S −

∞ ∂s ψ2 (s, x)f (s, x) dxds − ψ2 (0, x)f in (x) dx 0 0 0

S ∞

S ∞ ψ2 (s, x)f (s, x)dxds + vf (s, x)f (s, x)∂x ψ2 (s, x)dxds = −νγ ,0 ∞

0

0

0

0

Fibrils in plaque:

S ∞

∞ − fa (s, x)∂s ψ2 (s, x) dxds − f a,in (x)ψ2 (0, x) dx 0 0 0 S ∞ S ∞ νγ ,0 f (s, x)ψ2 (s, x) dxds − βa,0 fa (s, x)∂x ψ2 (s, x) dxds = 0

0

0

S

Mass decreasing:

0

˙ φ(s)E(s) ds + E 0 φ(0)  0

0

Thanks to the previous definitions we can state the main result of this paper Theorem 3.2 The limit of the discrete model (m∗ , U ∗ , F ∗ , Fa∗ ) is a weak solution of the continuous model.

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Proof The proof of this main result is quite long and we will give its sketch. For monomers: Let φ ∈ V, and let recall the monomers weak formulation

S ˙ ds − m0 φ(0) − m(s)φ(s) 0

S N +1   =− φ(s)xUN s, (i − 2)x v(s, (i − 2)x) ds 0

i=2

0

j =2

SN +2    φ(s)xFN s, (j − 2)x vf (s, (j − 2)x) ds −

SN +2    + φ(s)xFa,N s, (j − 2)x βa,0 ds 0

S j =2   + φ(s)x UN (s, 0)β0 + βf,0 FN (s, 0) + βa,0 Fa (s, 0) ds. 0

So the goal is to prove that previous equation converges towards the formulation

S

S 1 ˙ ds − m0 φ(0) = − − m∗ (s)φ(s) φ(s)U ∗ (s, x)v ∗ (s, x) dxds 0

0 S 0 ∞ φ(s)F ∗ (s, x)vf∗ (s, x) dxds −

0 S 0 ∞ φ(s)Fa∗ (s, x)βa,0 dxds. + 0

0

It remains to investigate the convergence of each term involved in the equations. For proto-oligomers: We already have the following formulation:

−

+1 SN  0

    ∂s ψ1 s, (i − 2)x UN s, (i − 2)x x ds

i=2 N +1 

−x

    ψ1 0, (i − 2)x uin (i − 2)x

i=2

S N        = m(s) ψ1 s, (i − 1)x − ψ1 s, (i − 2)x g (i − 2)x 0 i=2 

 · UN s, (i − 2)x ds

S N +1       −β0 UN s, (i − 2)x ψ1 s, (i − 2)x − ψ1 s, (i − 3)x ds 0

S i=3       − m(s)ψ1 s, (N − 1)x g (N − 1)x UN s, (N − 1)x ds. 0

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So, for ψ1 ∈ W1 the goal is to prove the convergence towards the relation:

1

S 1 ∗ ∂s ψ1 (s, x)U (s, x) dxds − ψ1 (0, x)uin (x)dx − 0 0 0 S 1 m∗ (s)g(x)U ∗ (s, x)∂x ψ1 (s, x) dxds = 0 0 S 1 −β0 U ∗ (s, x)∂x ψ1 (s, x) dxds. 0

0

Here again, the idea is to investigate the convergence of each term involved in relation. For fibrils: We recall the weak formulation for fibrils −

SN +2      x∂s ψ2 s, (j − 2)x FN s, (j − 2)x ds 0

j =2

− x

N +2 

    ψ2 0, (j − 2)x f in (j − 2)x

j =2

= −νγ ,0 x

SN +2      ψ2 s, (j − 2)x FN s, (j − 2)x ds 0

− βf,0

0

+

j =2

SN +2 

      FN s, (j − 2)x ψ2 s, (j − 2)x − ψ2 s, (j − 3)x ds

j =3

SN +1        m(s)gf (j − 2)x FN s, (j − 2)x ψ2 s, (j − 1)x 0

j =2

  − ψ2 s, (j − 2)x ds −

0

S

m(s)ψ2 (s, N )gf (N )FN (s, N ) ds.

For ψ2 ∈ W2 , we aim to prove the convergence of this previous equation towards the formulation:

∞

S ∞ ∂s ψ2 (s, x)F ∗ (s, x) dxds − ψ2 (0, x)f in (x) dx − 0 0 0

S ∞ ψ2 (s, x)F ∗ (s, x) dxds = −νγ ,0

0 S 0 ∞ F ∗ (s, x)∂x ψ2 (s, x) dxds −βf,0

S 0∞ 0 m∗ (s)gf (x)F ∗ (s, x)∂x ψ2 (s, x) dxds. + 0

0

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As previously, the goal is to pass to the limit at each term involved in the formulation. For fibrils inside plaque: Here again, from the formulation of fibrils inside plaque, we aim to prove for ψ2 ∈ W2 the convergence towards the relation

∞

S ∞ ∗ Fa (s, x)∂s ψ2 (s, x) dxds − f a,in (x)ψ2 (0, x) dx − 0 0 0 S ∞ S ∞ ∗ νγ ,0 F (s, x)ψ2 (s, x) dxds − νa,0 Fa∗ (s, x)∂x ψ2 (s, x) dxds. = 0

0

0

0

The global idea is to investigate the convergence of each term of the formulation. For mass decreasing relation: From the mass decreasing formulation that we recall as follows:

S

N +1   ˙ φ(s) m(s) + (x)2 iUN (s, (i − 2)x)

0

+(x)2

N +2 

i=2

    j (FN s, (j − 2)x + Fa,N s, (j − 2)x ) ds

j =2

N +2 N +1         φ(0) m0 + (x)2 iuin (i − 2)x + (x)2 j (f in (j − 2)x

+f a,in



(j − 2)x

j =2

i=2



,

we set φ ∈ V and aim to prove the convergence towards the formulation:

1

∞   ˙ xU ∗ (s, x) dx + x F ∗ (s, x) + Fa∗ (s, x) dx ds φ(s) m∗ (s) + 0

10

∞0  in  φ(0) m0 + xu (x) dx + x(f in (x) + f a,in (x) dx .

S

0

0

Once again, we investigate the convergence of each term involved in the formulation.  

4 Numerical Simulations In this section, as said in Sect. 3.2, we add in the equation for monomers dynamics, the term if c1 (t)cf,if (t)rf,if in order to reconstruct the mass preservation relation. We will illustrate two cases: the first case which consists to consider g1 = gf,1 = 0 (Figs. 2 and 3), and the second case which consists to take g1 = gf,1 = 0 (Figs. 4 and 5).

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This second case is very close to the case with the parameters used in Ref. [6], and model the fact that monomers cannot polymerize.

4.1 Case g1 = gf,1 = 0

Fig. 2 Monomers dynamics for the discrete model in the case g1 = gf,1 = 0. In left we observe an oscillation dynamics with a decreasing intensity. In right we plot a zoom

Fig. 3 Oligomers concentration inside the plaque with g1 = gf,1 = 0. In left we observe an increase of the concentration until it reaches a threshold. In right we make a zoom

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4.2 Case g1 = gf,1 = 0

Fig. 4 Monomers dynamics in the case g1 = gf,1 = 0. In left we plot result from the discrete model and in right we have the result obtained from the continuous model. For the continuous model we use the same code as in Ref. [6] with the same parameters. We observe the same dynamic for the monomers according to the two models

Fig. 5 Oligomers concentration inside the plaque with g1 = gf,1 = 0. In left we plot result from the discrete model and in right we have the result obtained from the continuous model. For the continuous model we use the same code as in Ref. [6] and we remark that the numerical results are the same according to the two models

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5 Conclusion In this study, we have shown a convergence result from a discrete model inspired by Helal et al. [12] towards a continuous one very close to the model studied in [6]. To get these results we first do a scaling of the discrete model; then, with some hypotheses in consideration, we established prior estimates in order to find adequate spaces for functions and finally we proved that the weak formulation of a certain continuous model is the weak limit of our discrete model. Numerical simulations are performed in order to compare the discrete model and the continuous one.

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Index

A Aβ oligomerization pathway, 397 Adaptive immunity, 271–296 Adjoint operator, 349 Adjoint semigroup, 27, 28 Aedes aegypti, 107 Akaike information criterion (AIC), 244, 246–250, 252, 253 Alignment of domains, 139, 140 Allee effect, 49 Alphavirus, 107 Alzheimer disease, 397–430 Amyloid Aβ–40 and Aβ–42, 397 Amyloid plaques accumulation, 397 Anaerobic digestion, 373–395 Ani3D package, 322, 324, 327, 328 Antigenic stimulation, 60, 166 Aphids, 209–229 Aphid-symbiont interactions, 229 Arbitrary Lagrangian Eulerian approach, 256, 259–260 Arrhythmias, 321, 322 Asymptotically stable endemic equilibrium, 39–41, 44, 45, 62, 87, 88, 111, 112, 115, 284, 296, 368 Asynchronous exponential growth, 80 Atherosclerosis, 1 Autowave systems, 8

B Bacteria-infected aphids, 210–212 Bacterial endosymbionts, 210–229 Banach separable space, 416

Banach space, 24, 333 Bang–bang control solution, 119 Basic reproduction number, 37, 38, 45, 63, 87, 109, 116, 277, 300 Basic SIR framework, 300 Beddington–DeAngelis functional response, 271, 273, 296, 360 Bellman principle, 125 Bidomain equations, 321–323 Bifurcating periodic solution, 338, 358 Bifurcation parameter, 53–55 Blood coagulation, 1–14, 18 Blood flow, 2, 4, 12, 238, 256, 258, 260 Blood loss, 1 Boundedness of solutions, 60, 109, 115, 166, 273, 334 Brownian motion, 88, 192, 381, 383, 390

C Cancer drug paclitaxel, 232, 233 Capsids load, 29 Cardiac Arrhythmia Research Package (CARP), 322, 324 Cardiac electrophysiology, 321–328 Carreau’s law, 256 Catchability rate of the fleet, 121 Cauchy problem, 127, 128, 408 CD4+ T cells, 63 Cell population evolution, 60 Cellular immunity, 271, 279–282, 288, 290, 293–296 Center manifold reduction, 338, 350 Chemostat model, 374

© Springer Nature Switzerland AG 2019 R. P. Mondaini (ed.), Trends in Biomathematics: Mathematical Modeling for Health, Harvesting, and Population Dynamics, https://doi.org/10.1007/978-3-030-23433-1

433

434 Chikungunya virus (CHIKV), 107–116 Cholinergic receptors, 397 Chronic inflammatory response Chronic infection equilibrium, 110–112, 114–116, 365, 366, 368–370 Closed greenhouse systems, 228 Clotting, 2, 5, 10, 11, 13 Coagulation activation, 4, 8, 13 Coagulation cascade, 3, 4, 6, 8 Coagulation system, 1–4, 8, 9, 13, 18 Coexistence equilibrium, 211, 215, 220, 227, 334, 336, 340 Combined therapy protocol, 154, 159 Compact subset, 37 Compartmental epidemic models, 309, 318 Concavity, 145, 146, 149, 169, 179–184 Conditional entropy measure, 179 Conditional probability of occurrence, 179 Conservation of forest resources, 48 Crowley–Martin functional response, 360 Cytotoxic and cytostatic drugs, 153 Cytotoxic T Lymphocytes (CTL), 59–69, 153, 165–175, 271–273, 280, 281, 293–295

D de Moivre law, 191 Debrillation therapy optimization, 321 Deceleration of growth parameter, 310, 319 Delay differential systems, 331 Delayed resource-herd model, 339 Deterministic AM2 models, 375–376, 385 Differential equation of Itô type, 90 Differential games theory, 119 Diffusion models, 384, 385, 389 Diffusion process, 379, 381, 383–385, 389–390 Discrete-time Poisson model, 384 Disease extinction, 96–99, 319 Disease-free equilibrium, 38, 39, 41, 43–45, 60–62, 64, 65, 67, 87 Disease outbreak, 88, 89, 299 Disease persistence, 99–101 DNA Polymerase, 71 Drug miberfradil, 233, 241 Dynamics of HBV infection, 359

E Early phase of an epidemic, 319 Ebola virus, 35–45 Effect of memory, 360 Effector cells, 153–163 Endemic disease, 41, 43–45 Endemic equilibrium point, 62, 68

Index Enzymatic reactions, 3, 9, 13, 234 Epidemic outbreaks, 299–307 Epithelial membranes, 234 Euler finite-difference scheme method, 63 Exclusive Economic Zone, 119 Extrinsic pathway of coagulation, 4

F Feasibility conditions, 213, 214, 221 Feasible coexistence equilibrium, 227 Fibrillation pathway, 397 Fibrils, 397–401, 403–406, 408, 409, 412–416, 418, 422–424, 426, 427 Fibrils inside plaque, 401, 403, 404, 406, 408, 409, 412, 415, 416, 422, 423, 427 Fibrinogen, 4, 8 Fibrin polymerization, 12 Finite element method, 265, 321, 324 First anaerobic digestion model, 373 Fishery resource management problems, 120 Fish population, 51, 52 Fleet exploitation strategy, 120 Fluid structure interaction problems (FSI), 255–258, 260, 261, 268 Fokker–Planck equation, 142 Foot-and-mouth disease, 319 Forest exploitation, 49, 50, 54, 56 Forest resources, 48, 52 Fractal calculus, 233 Fractal pharmacokinetics, 233 Fractional derivative, 272, 273, 283, 285, 287, 295, 296, 360, 365, 370 Fractional derivative in the Caputo sense, 273, 360 Fractional order differential equations (FDE), 347, 360 Fractional order model for virus infection, 272, 296, 359–369, 373 Free-flowing leukocytes, 256, 262–267 Fundamental fishery model, 119

G Game theory, 119 Generalized Bayes law, 189 Gibbs–Shannon entropy, 143–144, 147, 149, 181, 187–188 Gillespie algorithm, 379, 384, 394 Globally asymptotic stable, 41, 43–45, 88, 112, 114–116, 284, 286, 288, 290, 292, 294, 296, 365, 366, 368–370 Global populations, 47 Greatest water consumption, 50

Index Greenhouse effect, 48 Growth of the effector cells, 154, 155 Growth rate, 50, 51, 56, 155, 309–311, 332, 375 H Haldane type growth factor, 375 Hamilton–Jacobi–Bellman equation, 125–126 Havrda–Charvat entropy, 144–149, 151, 187–188 Hayick limit, 72 Heat capacity, 142 Helmholtz free energy, 143, 144, 148, 150, 151 Hepatitis B virus (HBV), 23–32, 271, 359–370 Hepatitis B virus DNA-containing capsids, 23–32, 360 Hepatitis B virus virion, 31 Hepatitis C virus (HCV), 271, 272 Hepatocytes, 24, 32, 359, 360, 369 Herbivory optimization hypothesis, 331–358 Hessian matrix, 26, 169 Hi-Lo algorithm, 206–208 Holling response function of type III, 331 Hopf bifurcation, 54, 331–358 Hopf-type bifurcation points, 53 Host cells, 24, 59, 166, 273, 294–296 Human immunodeficiency virus (HIV), 59–69, 165–175, 271, 272, 309, 319 Humoral immunity, 115, 271, 279–281, 287, 294, 296 Hypothermia, 11 I Immune-free infection equilibrium, 282, 284, 292, 294, 296 Immunotherapy, 153, 159 Impulsive differential equations theory, 339 Independent Brownian motions, 88, 192 Inelastic strain tensor, 258 Infected hepatocytes and capsids, 32, 359, 360 Infection equilibrium points, 60 Infection-free equilibrium, 61, 109, 111, 115, 116, 277, 282, 293, 296, 364–366, 368–370 Infestation control, 216 Infinitesimal generator, 379–381 Inflammatory processes, 255–268 Intrinsic growth rate, 310, 332 Itô formula, 91, 94, 98, 194

J Jaccard entropy measure, 185–186

435 K Kaviar’s approximation, 162 Kearns model, 247–250 Kernel approximation for viability and resilience (Kaviar), 14 Khinchin-Shannon inequalities, 177–190 Klausmeier’s framework, 51 Kuznetsov’s model, 13

L Lagrange multipliers, 122, 143 Lamé coefficients, 259 Landsberg–Vedral entropy measure, 187 Laplace transform, 311 Large number theorem for martin gales, 98 LaSalle invariance principle, 41, 43 Latently infected cells, 172, 173, 271, 273 Lee–Carter model, 199 Lévy jumps, 87–104 Lifshitz–Slyozov modelling approach, 398 Lipschitz condition, 93 Livestock production, 48 Local and global stability, 37 Locally asymptotically stable, 39, 40, 62, 67, 68, 111, 284, 335 Local stability of equilibrium, 337 Local thrombin concentration, 4 Logistic growth, 165–175 Lyapunov functional, 41, 112, 283, 285, 287, 289, 291, 365, 366

M Malthusian coefficient, 310 Markov chains, 191–208 Mass action law, 377 Mass decreasing relation, 404, 406, 414, 427 Master equation, 142 Mathematical epidemiology, 319 Maximization of entropy, 143–144 Meyers angle bracket process, 92 Michaelis–Menten kinetics, 11, 233–244, 250 Michaelis–Menten mechanism, 9 Milevsky–Promislov model, 191–208 Misfolded prions, 402 Mittag-Leffler function, 51 Model of Wang and Liu, 108 Moment generating method, 310 Monte Carlo simulations, 238 Mooney–Rivlin model, 259 Multiphysics cardiac problems, 327 Mutual information, 185

436 N Nash equilibrium, 120–122, 124, 126, 136 Non-additive entropy measures, 177–190 Non-cooperative differential game, 10 Non-Gaussian process, 192 Non-Gaussian scalar linear filter, 194 Nonlinear model of hyperelasticity, 258 Non-monotonicity properties, 306 Non-Newtonian fluid, 256, 262 Nonstandard numerical scheme, 339 N-version benchmark, 326 O Objective functional, 25–27, 168, 169 Olovnikov assumption, 72 Optimal control problem, 27, 119, 166, 168, 170 Optimality proof, 127–136 Optimality system, 27–29, 170–171 P Panteleev’s model, 12, 13 Parametric bootstrap, 315 Parasitoid wasps, 209, 228, 229 Parental-offspring vertical transmission, 210 Partition function, 144, 146–148, 150, 151 Pesticide resistance, 209 Phases G1-S-G2-M, 71 Phenotypic heterogeneity, 153 Piola–Kirchhoff stress and strain tensors, 258 Plant-herbivore interactions model, 332 Platelet aggregation, 2 Poisson and Normal approximations, 374, 379 Poisson distribution, 315, 394 Poissonization process, 12 Pontryagin’s maximum Principle, 32, 175 Population growth of CHIKV, 107–108 Positiveness and concavity requirements, 179 Postulate of Gompertz, 191 Predator-prey models, 48, 331, 332 Prion catalysis, 401–403 Protease inhibitor, 166 Protein family database (PFAM), 139, 140 Prothrombinase, 4, 9, 10, 12, 14 Proto-oligomers evolution, 401 Public health policy, 301 Pure jump Markov model, 377–380, 389 R Reaction-diffusion-convection equations, 2 Renyi entropy measure, 187 Replicative senescence, 71

Index Residence time in the body, 234 Retarded functional differential equations (RFDEs), 338 Reverse transcriptase inhibitor, 166 Riemann integrals, 410 Risk of disease transmission, 310 Risk of parasitism, 210 Routh-Hurwitz condition, 214, 224 Routh-Hurwitz theorem, 40, 69 rVSV-ZEBOV vaccine, 299

S Sample space, 139–142 Scaling hypothesis, 238 S-escort probabilities, 145 Sharma–Mittal entropy, 149, 179, 182, 187–188 Susceptible-infected recovered (SIR)epidemic, 89, 302 Susceptible-infected recovered (SIR) vaccination model, 307 Small-size biomass populations, 389 Stability analysis, 62, 111–114, 335–339, 374 Stability analysis of equilibria, 111–114 Steady states, 6–8, 15, 16, 61–62, 165, 239–241, 247, 296, 370 Stochastic AM2 models, 376–384 Stochastic epidemic models, 88 Stochastic hybrid systems, 192 Stochastic Lotka–Volterra population, 89 Stochastic SIRS epidemic models, 89 Strictly monotonic increasing function, 108 Strongly continuous semigroup, 255 Sub-exponential growth patterns, 309–320

T Tau-leap algorithms, 384, 389, 394 Tau proteins, 397 Therapy control, 31, 173 Threshold level, 306 Thrombin generation, 1–18 Thrombomodulin, 4, 16 Thrombosis, 2 Transhumance, 332, 336 Transition probability, 193, 198 Transmission dynamics of infectious diseases, 299 Transversality conditions, 28, 171, 346 Tusscher & Panlov model, 326 Two-level optimization, 120 Two-reaction AM2 model, 373

Index U Uninfected aphids, 210–212, 215–219, 224, 228, 229 Uninfected cells, 25, 29, 30, 32, 64, 65, 108, 166, 168, 171, 175, 293–295 Unique endemic equilibrium, 38, 39, 43, 44, 364 Unstructured tetrahedral mesh, 324 V Vaccination, 45, 89, 299–307, 318 Vaccination until herd immunity is achieved (VUHIA) strategy, 300–302, 305 Ventricular fibrillation, 233 Viability approach, 120 Viability kernel, 154, 156–160, 162, 163 Viability theory, 154, 163 Viable trajectories, 156

437 Viral antigen, 60, 166 Viral infection model, 23–32, 165, 166 Viral load, 25, 32, 168, 175, 294 Virions, 23, 24, 31, 63, 271, 273, 360, 370 Volatile fatty acids (VFA), 373, 375 Volterra–Lotka model, 48

W Wagenvoord model, 10 Wasp population, 228 Weighted residual sum square (WRSS), 243, 244, 246–250, 252, 253 Wiener process, 194 Within-host CHIKV infection model, 115

Y Yew tree, 233