Age Structured Epidemic Modeling [1st ed.] 9783030424954, 9783030424961

Table of contents :
Front Matter ....Pages i-xii
Linear Age-Structured Population Models as a Base of Age-Structured Epidemic Models (Xue-Zhi Li, Junyuan Yang, Maia Martcheva)....Pages 1-21
Age-Structured Epidemic Models (Xue-Zhi Li, Junyuan Yang, Maia Martcheva)....Pages 23-67
Nested Immuno-Epidemiological Models (Xue-Zhi Li, Junyuan Yang, Maia Martcheva)....Pages 69-103
Age-Since-Infection Structured Models Based on Game Theory (Xue-Zhi Li, Junyuan Yang, Maia Martcheva)....Pages 105-151
Age Structured Models on Complex Networks (Xue-Zhi Li, Junyuan Yang, Maia Martcheva)....Pages 153-209
Vector-Borne Age-Structured Models (Xue-Zhi Li, Junyuan Yang, Maia Martcheva)....Pages 211-257
Metapopulation and Multigroup Age-Structured Models (Xue-Zhi Li, Junyuan Yang, Maia Martcheva)....Pages 259-299
Class Age-Structured Epidemic Models (Xue-Zhi Li, Junyuan Yang, Maia Martcheva)....Pages 301-359
Back Matter ....Pages 361-383

Citation preview

Interdisciplinary Applied Mathematics 52

Xue-Zhi Li Junyuan Yang Maia Martcheva

Age Structured Epidemic Modeling

Interdisciplinary Applied Mathematics Volume 52 Editors Anthony Bloch, University of Michigan, Ann Arbor, MI, USA Charles L. Epstein, University of Pennsylvania, Philadelphia, PA, USA Alain Goriely, University of Oxford, Oxford, UK Leslie Greengard, New York University, New York, USA Advisors L. Glass, McGill University, Montreal, QC, Canada R. Kohn, New York University, New York, NY, USA P. S. Krishnaprasad, University of Maryland, College Park, MD, USA Andrew Fowler, University of Oxford, Oxford, UK C. Peskin, New York University, New York, NY, USA S. S. Sastry, University of California Berkeley, CA, USA J. Sneyd, University of Auckland, Auckland, New Zealand Rick Durrett, Duke University, Durham, NC, USA

More information about this series at http://www.springer.com/series/1390

Xue-Zhi Li • Junyuan Yang • Maia Martcheva

Age Structured Epidemic Modeling

Xue-Zhi Li College of Mathematics and Information Sciences Henan Normal University Xinxiang, Xinxiang, China

Junyuan Yang Complex Systems Research Center Shanxi University Taiyuan, Shanxi, China

Maia Martcheva Dept of Math, Little Hall 358 University of Florida Gainesville, FL, USA

ISSN 0939-6047 ISSN 2196-9973 (electronic) Interdisciplinary Applied Mathematics ISBN 978-3-030-42495-4 ISBN 978-3-030-42496-1 (eBook) https://doi.org/10.1007/978-3-030-42496-1 Mathematics Subject Classification (2010): 34D05, 34D20, 34D23, 34D45, 34F10, 35A09, 35B09, 35B10, 35B30, 35D30, 35B32, 35B40, 35D35, 35E15, 37C10, 37C75, 37G15, 45A05, 45D05, 45E10, 45J05, 45P05, 46E15, 46E30, 47A10, 47A11, 47A75, 47B07, 47B33, 47B34, 47B38, 47H05, 47H09, 47H10, 49K40, 49G50, 62P10, 62P25, 65F50, 65L11, 65R10, 74G30, 74G35, 91A05, 91A22, 92B05, 92D25, 92D30, 93C20, 93D20, 93C15 © Springer Nature Switzerland AG 2020 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, expressed 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

Age-structured epidemic models date back to the early twentieth century when Kermack and McKendrick created their well-known SIR model to explain disease outbreaks such as the plague or cholera. Even one of the original Kermack-McKendrick models is, in fact, discrete age-since-infection structured to capture better the potential changes in infectivity and recovery, depending on time-since-infection. Since Kermack and McKendrick, age-structured epidemic models have had a steady presence among the infectious disease literature. Their major contribution to mathematics was a novel type of partial differential equations with nonlocal boundary conditions, whose analysis has challenged mathematicians in the last 70 years. The mathematical setup for these first order PDEs is completely different than the “traditional second order PDEs”—it requires working in L1 spaces which are not Hilbert spaces and are not reflexive. Interest in age-structured models was renewed at the beginning of the twenty-first century with the appearance of several types of multi-scale immuno-epidemiological models. All these newly developed models, some of which may become a standard of modeling in infectious diseases, will continue challenging mathematicians for years to come. There are a lot of very nice introduction books to age-structured modeling. Some of them involve the edited book “Mathematical Epidemiology” (Springer), the textbook “Introduction to Mathematical Epidemiology” by M. Martcheva (Springer, 2015), the “Mathematical Models in Population Biology and Epidemiology” by F. Brauer and C. Castillo-Chavez (Springer, 2002), “The Basic Approach to AgeStructured Population Dynamics” by M. Iannelli and F. Milner (Springer, 2017), and others. The present book is not meant as an introduction to age-structured modeling—it is meant as a tool to develop the skills of graduate students and researchers who wish to make age-structured epidemic modeling their area of expertise. The mathematical tools that currently serve age-structured epidemic modeling are presented in the book through examples. Many of these tools already exist in the literature—published in papers. However, the book demonstrates them on different examples and at a slower pace, with more explanations compared to regular articles. The presentation of the tools in the book is also graded in difficulty, from reguv

vi

Preface

lar functional-analytic tools to C0 -semigroup tools to integrated semigroup tools. The only other book that the authors know of that comes this close in developing the mathematical tools is a recent book by H. Inaba “Age-Structured Population Dynamics in Demography and Epidemiology” (Springer, 2017). Our goal with the book is that this book serves to train a whole new generation of mathematical epidemiologists, ready to tackle the analytical and numerical challenges of the novel age-structured population and epidemic modeling. The book starts with an introduction to chronological age-structured population models which is only meant to prepare the reader to tackle chronological agestructured epidemic models, which are discussed in Chap. 2. Along with the models, we introduce here a general theorem for proving well posedness of age-structured models, we discuss, basic analysis, some early tools for detecting backward bifurcation in age-structured models as well as setting up and simulating optimal control problems. All these techniques are routinely used with ODE epidemic models but are rarely in the arsenal of mathematical epidemiologists when it comes to agestructured modeling. Chapter 3 focuses on immuno-epidemiological modeling. We believe that age-structured population modeling, analysis, and simulations will be driven in the years to come primarily by developments in immuno-epidemiological or multi-scale modeling. Chapter 3 describes some well-known and some novel (bidirectionally linked) immuno-epidemiological models as well as some of the existing analysis tools but much of the tools necessary for the analysis and simulation of these types of models are still yet to come. Some of the open questions here are: How do we find oscillations in unidirectionally linked immuno-epidemiological models, if the tools developed for regular age-since-infection models do not apply here? How do we analyze bidirectionally linked immuno-epidemiological models given that their complexity is higher and presents challenges to existing tools? Chapter 4 introduces age-since-infection structured epidemic models coupled with game theory. Game theory has been used to analyze behavioral implications on infectious disease epidemiology. It is often used to study the decisions people make to comply (or not to comply) with some control strategies, such as vaccination. Most of the models in this direction are ODE models. In this chapter, we introduce a number of age-since-infection structured PDE models together with the appropriate functionalanalytic tools for their analysis. Chapter 5 discusses network age-structured models. Complex networks are the standard of modeling of complex systems but the models are typically simulational and do not lend themselves to closed-form equations that can be analyzed. Here we present, under some assumptions, closed-form agestructured epidemic models built on complex, typically scale-free, networks. We develop functional-analytic tools for their analysis. Chapter 6 introduces a number of chronological age and age-since-infection structured models of vector-borne diseases. Although models like these can be found readily in the literature, we use them as examples to illustrate a number of functional-analytic techniques, as well as techniques based on C0 -semigroup theory. C0 -semigroups are analytical tools that have had long existence but have been applied to age-structured epidemic models only since the late twentieth century. One nice introductory textbook to C0 -semigroups is the one by A. Pazy “Semigroups of Linear Operators and Applications to Partial Dif-

Preface

vii

ferential Equations” (Springer, 1983) and a more recent reference by K. Engel and R. Nagel “One-Parameter Semigroups for Linear Evolution Equations” (Springer, 2000). C0 -semigroups are further used as the main tool of analysis also in Chap. 7 which introduces and tackles metapopulation epidemic models and multi-group epidemic models with chronological age. C0 -semigroups are a powerful tool to study age-structured models but they often fall short in the analysis of age-since-infection or other class-age structured models. The main reason is that class-age structured models often have as a boundary condition of the PDE a term that depends in a nonlinear way on the dependent variables. That, in general, precludes the use of C0 semigroups, unless an appropriate change of variables is made to transform the problem into one with a zero boundary condition. Since this change of variables is nonobvious or impossible to make, this type of problems are best treated by the socalled integrated semigroups or semigroups with non-densely defined generators. The best introduction here may be an article by H. Thieme “Integrated Semigroups and Integrated Solutions to Abstract Cauchy Problems” in JMAA, 1990. We use integrated semigroups to analyze a number of example class-age models in Chap. 8. We include some frequently used mathematical tools in the appendix. This book emerged from the long-term friendship of its authors and it is a result of collaboration of scientists from across the world, often supported and encouraged by the National Funding Agencies. The authors would like to thank the National Science Foundation of the USA and the National Science Foundation of China for bringing them together. Further, the authors thank the warm hospitality of University of Florida that made their face collaboration on the book possible. The authors would like to thank their Springer editors Donna Chernyak and Danielle Walker who believed in the book and encouraged them to work. The authors also thank their families for their patience while the authors were working on the book. We believe that this book is unique in the sense that it is perhaps the first attempt to merge the best of the mathematics education and training in Asia, North America, and Europe. The differences in these educational systems made the writing of the book somewhat challenging. We hope we have risen to the challenge and the book that has emerged will contribute seriously to the training of the next generation experts in age-structured population and epidemic modeling. Xinxiang, China Taiyuan, China Gainesville, FL, USA

Xue-Zhi Li Junyuan Yang Maia Martcheva

Contents

1

2

Linear Age-Structured Population Models as a Base of Age-Structured Epidemic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 The Lotka-McKendrick-von Foerster Model . . . . . . . . . . . . . . . . . . . . 1.1.1 The McKendrick Age-Structured PDE Model . . . . . . . . . . . . 1.1.2 The Lotka Integral Equation Model . . . . . . . . . . . . . . . . . . . . . 1.2 Properties of the Solutions of Lotka-McKendrick Model . . . . . . . . . . 1.2.1 Existence and Uniqueness of Solutions of the Lotka-McKendrick Model . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Long-Term Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Persistent Solutions of the Lotka-McKendrick Model . . . . . . 1.2.4 The Sharpe-Lotka Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Combining the Age-Structured Population with Epidemic Models . . 1.3.1 Homogeneous Age-Structured Epidemic Models . . . . . . . . . . 1.3.2 Age-Structured Epidemic Models with Stationary Population 1.3.3 Age-Structured Models with Disease-Induced Mortality . . . . 1.3.4 Age-Structured Epidemic Models with Constant Total Birth Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Early Age-Structured Epidemic Models . . . . . . . . . . . . . . . . . . . . . . . . Age-Structured Epidemic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Analysis of Age-Structured Epidemic Models . . . . . . . . . . . . . . . . . . . 2.1.1 Well Posedness of Age-Structured Epidemic Models . . . . . . 2.1.2 Computing R0 for Age-Structured Epidemic Models . . . . . . 2.1.3 Backward Bifurcation in Age-Structured Models . . . . . . . . . . 2.2 An Age-Structured SIR Model with Reinfection . . . . . . . . . . . . . . . . . 2.2.1 Basic Analysis of the SIR Model with Reinfection . . . . . . . . 2.2.2 Stability of the Disease-Free Equilibrium . . . . . . . . . . . . . . . . 2.3 Numerical Methods for Age-Structured Epidemic Models . . . . . . . . . 2.4 Optimal Control of Age-Structured Models . . . . . . . . . . . . . . . . . . . . . 2.4.1 The Optimal Control Problem . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 1 3 4 5 6 8 9 10 12 15 16 18 19 23 23 23 27 30 32 33 38 40 42 43 ix

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2.4.2

Deriving the Adjoint System and the Characterization of the Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Numerical Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Two-Strain Age-Structured Epidemic Models . . . . . . . . . . . . . . . . . . . 2.5.1 Disease-Free Equilibrium and Reproduction Numbers . . . . . 2.5.2 Strain One and Strain Two Equilibria and Invasion Numbers Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

4

Nested Immuno-Epidemiological Models . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Nested Immuno-Epidemiological Modeling . . . . . . . . . . . . . . . . . . . . 3.1.1 Within-Host Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Composing Immuno-Epidemiological Models . . . . . . . . . . . . 3.1.3 An Immuno-Epidemiological Model of Disease with Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analysis of Immuno-Epidemiological Models . . . . . . . . . . . . . . . . . . . 3.2.1 Analysis of the SIR Immuno-Epidemiological Model . . . . . . 3.2.2 Stability of the SI Immuno-Epidemiological Model of HIV . 3.2.3 Impact of Within-Host Parameters on the Between-Host Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Bidirectionally Linked Immuno-Epidemiological Models . . . . . . . . . 3.3.1 A Bidirectionally Linked Model of HIV . . . . . . . . . . . . . . . . . 3.3.2 Bidirectionally Linked Immuno-Epidemiological Model of Cholera . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Immuno-Epidemiological Multi-Strain Models . . . . . . . . . . . . . . . . . . 3.4.1 An n-Strain Immuno-Epidemiological Competitive Exclusion Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.2 A Two Strain Model Structured by Inoculum Fraction . . . . . 3.4.3 Multi-Strain Models with Trade-Off Mechanisms on the Between-Host Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . .

44 48 50 53 54 58 60 69 69 69 72 74 76 77 80 83 84 85 89 94 95 96 99

Age-Since-Infection Structured Models Based on Game Theory . . . . . 105 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 4.2.1 Existence of Equilibria and Their Local Stability . . . . . . . . . . 115 4.2.2 The Attractivity of Boundary Equilibria and Disease Persistence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.2.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles . . . . . . . 134 4.3.1 Stability of the Disease-Free Equilibrium . . . . . . . . . . . . . . . . 137 4.3.2 Boundary Equilibrium and the Endemic Equilibrium . . . . . . 140 4.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 4.4 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151

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5

Age Structured Models on Complex Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.1.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.1.2 Basic ODE Epidemic Models on Networks . . . . . . . . . . . . . . . 155 5.2 An Age-Structured SIS Model on Complex Networks . . . . . . . . . . . . 157 5.2.1 Global Asymptotic Stability of the Disease-Free Equilibrium 162 5.2.2 Existence and Stability of the Endemic Equilibrium . . . . . . . 165 5.2.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.3 An SIR Model with Demography on Complex Networks . . . . . . . . . . 174 5.3.1 Baseline Analysis of the SIR Model . . . . . . . . . . . . . . . . . . . . . 176 5.3.2 Stability of Disease-Free Equilibrium . . . . . . . . . . . . . . . . . . . 178 5.3.3 Existence and Stability of the Endemic Equilibrium . . . . . . . 181 5.3.4 Global Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.3.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 5.4 Two Strain Models with Age Structure on Complex Networks . . . . . 191 5.4.1 A SIS Age-Structured Model on Complex Networks . . . . . . . 191 5.4.2 A Two Strain Model with Age Structure and Mutation on Complex Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 5.4.3 A Two Strain Model with Age Structure and Superinfection on Complex Networks . . . . . . . . . . . . . . . . . . . 203

6

Vector-Borne Age-Structured Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.2.1 The Model Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.2.2 Existence and Uniqueness of Solution . . . . . . . . . . . . . . . . . . . 214 6.2.3 Existence of Steady States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 6.2.4 Local Stability of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection . . 231 6.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 6.3.2 The Reproduction Numbers and Threshold Dynamics . . . . . . 233 6.3.3 Global Stability of the Disease-Free Equilibrium . . . . . . . . . . 237 6.3.4 Existence and Stability of Boundary Equilibria . . . . . . . . . . . 240 6.3.5 Preliminary Results and Uniform Persistence . . . . . . . . . . . . . 244 6.3.6 Principle of Competitive Exclusion . . . . . . . . . . . . . . . . . . . . . 252

7

Metapopulation and Multigroup Age-Structured Models . . . . . . . . . . . 259 7.1 Metapopulation Age-Structure Model . . . . . . . . . . . . . . . . . . . . . . . . . . 259 7.1.1 Monotone Dynamical System Approach . . . . . . . . . . . . . . . . . 261 7.1.2 SIS Epidemic Model with Age and Patch Structures . . . . . . . 264 7.1.3 Definition of the Semiflow . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 7.1.4 Existence and Uniqueness of the Endemic Equilibrium . . . . . 271 7.1.5 Global Attractivity of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . 277

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7.2

8

Multigroup Epidemic Model with Age Structure . . . . . . . . . . . . . . . . . 279 7.2.1 Multigroup SIR Vector-Borne Epidemic Model . . . . . . . . . . . 280 7.2.2 The Basic Reproduction Number of System (7.50) . . . . . . . . 286 7.2.3 Existence of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 288 7.2.4 Global Stability of the Disease-Free Equilibria . . . . . . . . . . . . 295

Class Age-Structured Epidemic Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.2 SIRS Model with Age of Infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.2.1 Volterra Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 308 8.2.2 Asymptotical Smoothness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309 8.2.3 Existence of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311 8.2.4 Stability of Equilibria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 8.3 SIRS Model with Age of Recovery . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324 8.3.1 Preliminary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 8.3.2 Stability of the Disease-Free Equilibrium . . . . . . . . . . . . . . . . 329 8.3.3 Local Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334 8.3.4 Global Hopf Bifurcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 8.3.5 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 346 8.4 SVIR Model with Age of Vaccination . . . . . . . . . . . . . . . . . . . . . . . . . 347 8.4.1 Introduction and Model Formulation . . . . . . . . . . . . . . . . . . . . 347 8.4.2 Local Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 8.4.3 Global Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355

Appendix A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 A.1 Hille-Yosida Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362 A.2 Volterra Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 A.3 Positive Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365 A.4 Some Features and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 369 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381

Chapter 1

Linear Age-Structured Population Models as a Base of Age-Structured Epidemic Models

1.1 The Lotka-McKendrick-von Foerster Model Age-structured epidemic models are based on age-structured population models. Typically the linear age-structured population model, called the Lotka-McKendrick model, is used as a baseline population model in epidemic systems. The Lotka-McKendrick model is an age-structured PDE analogue of the ODE Malthus model of population growth. Malthus model assumes that the population changes in an unlimited environment. If P(t) is the total population size at time t, it is assumed that the population reproduces at a per capita rate β and dies at a per capita rate μ . Hence, the model takes the form: P (t) = β P − μ P. Malthus’ model is not hard to solve and the solutions are exponential. The Lotka-McKendrick model takes its name from Alfred James Lotka (March 2, 1880–December 5, 1949) who derived the model in an integral form and Anderson Gray McKendrick (8 September 1876 to 30 May 1943) who first derived it in a PDE form. The equation was later rediscovered by biophysics professor Heinz von Foerster. For those interested in the history of mathematical population dynamics and epidemiology we refer to the excellent book of Nicolas Bacaër [9]. We introduce the McKendrick and the Volterra versions of the model in the next two subsections.

1.1.1 The McKendrick Age-Structured PDE Model The McKendrick age-structured PDE model that we introduce in this section is an age-structured extension of Malthus model—it structures individuals by their chronological age. All other assumptions for the derivation of the model are the same as in Malthus’ model: the model assumes that the population is isolated and lives in an invariant environment with unlimited resources. All individuals are iden© Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1_1

1

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1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

tical, except for their age a. The population of age a at time t is given by the density function p(a,t). The function p(a,t) is a density in the sense that  a2

p(a,t)da

a1

gives the number of individuals in the age interval [a1 , a2 ]. If A is the maximal age, which may be assumed finite or infinite, then P(t) =

 A

p(a,t)da

(1.1)

0

is the total population at time t. If β (a) and μ (a) are the age-specific fertility and mortality rates, the McKendrick model takes the form ⎧ pt (a,t) + pa (a,t) = −μ (a)p(a,t), ⎪ ⎪  A ⎨ (1.2) β (a)p(a,t)da, p(0,t) = ⎪ 0 ⎪ ⎩ p(a, 0) = p0 (a). In this system B(t) =

 A 0

β (a)p(a,t)da

is the total number of newborns in the time unit, also called total birth rate, and p0 (a) is an initial age distribution, assumed known. Model (1.2) is the basic model that describes the evolution of a single one-sex population. The model in this form was first proposed by McKendrick [130] in 1926. Because of its importance in population and epidemic problems, it has been extensively studied. Some excellent references on the topic are [38, 85, 88, 96, 179]. We discuss here its main properties which are relevant for its application to epidemic models. The natural setting to study model (1.2) is the space L1 (0, A), equipped with the integral norm || f ||1 =

 A 0

| f (a)|da,

since the integral in (1.39) must converge for the total population size to be defined. The vital rates β (a) and μ (a) must be nonnegative. The total birth rate satisfies

β (.) ∈ L∞ (0, A),

supa β (a) = β¯ .

To understand the assumptions on the mortality rate, we first introduce an important quantity in the age-dependent population dynamics—the probability of survival to age a, which is given by a π (a) = e− 0 μ (s)ds . (1.3) We require that the probability of survival till maximal age A is zero; that is, π (A) = 0 which guarantees that no individuals survive past maximal age. That requires that

1.1 The Lotka-McKendrick-von Foerster Model

3

the age-specific death rate satisfies  A 0

μ (s)ds = ∞.

Hence, we require that 1 μ (a) ∈ Lloc [0, A),

which means that μ (a) is integrable on every compact subset of [0, A). Further, essinfa∈[0,A) μ (a) = μ∗ > 0. If A = ∞, then typically μ (a) is required to be essentially bounded above. Finally, the initial population distribution is also nonnegative and satisfies p0 (a) ∈ L1 (0, A). To prove well posedness of age-structured models, one typically transforms the PDE model into an integral form and studies the solutions of the corresponding integral equation model. We do that in the next subsection.

1.1.2 The Lotka Integral Equation Model Model (1.2) can be recast in integral form. This is done through a procedure called integration along the characteristic lines. Model (1.2) is a first order PDE model whose characteristic lines are lines with slope one in the first quadrant of (a,t) plane. Since age and time progress simultaneously, t − a = c, where c is a constant independent of time and age. Parameterized along the characteristic lines, the PDE becomes a first order linear ODE and can be integrated. To see that, let (a0 ,t0 ) be a point in the first quadrant of the (a,t) plane. Then the characteristic line through that point is given by a = a0 + s,t = t0 + s where s is a variable. Then p(a,t) = ˆ Similarly μ (a) = μ (a0 + s) = μˆ (s). The PDE becomes p(a0 + s,t0 + s) = p(s). d p(s) ˆ = −μˆ (s) p(s). ˆ ds Solving the ODE, we obtain − p(s) ˆ = p(0)e ˆ

s 0

μˆ (σ )d σ

.

There are two options depending on which boundary the characteristic line through the point (a,t) will hit. In the case a > t, the characteristic line hits the a-axis. We set t0 = 0, a0 = a − t, s = t and we obtain p(a,t) = p0 (a − t)e−

t

0 μ (a−t+σ )d σ

= p0 (a − t)

π (a) . π (a − t)

In the other case a < t, the characteristic line through the point (a,t) hits the t-axis. We set t0 = t − a, a0 = 0, s = a and p(0) ˆ = B(t − a). Hence, for the solution we

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1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

obtain p(a,t) = B(t − a)π (a). Thus, the integrated solution of (1.2) is given by  (a) p0 (a − t) ππ(a−t) p(a,t) = B(t − a)π (a)

a > t,

(1.4)

a < t.

This is not an explicit solution since B(t) depends on p(a,t). From here we can obtain an integral equation for the birth rate B(t) by substituting p(a,t) from (1.4) in B(t). If A < ∞, we again have two cases. First, if t < A, we have B(t) =

 t 0

B(t − a)β (a)π (a)da +

 A t

β (a)p0 (a − t)

π (a) da. π (a − t)

(1.5)

As p0 is given, the second integral is a known function, often denoted by F(t). We have  t B(t) = B(t − a)β (a)π (a)da + F(t). (1.6) 0

If t > A, we have B(t) =

 A 0

B(t − a)β (a)π (a)da.

(1.7)

Equation (1.6) is known as Lotka one-sex population model or the renewal equation. It was discussed by Lotka in 1939 [114]. Model (1.6) is a Volterra integral equation in the case A = ∞ or t < A, if A < ∞. The case A < ∞ can be extended to the case A = ∞ if β (a), π (a), and p0 (a) are defined as zero for a > A. In this case the Lotka model becomes  t B(t − a)K(a)da + F(t), (1.8) B(t) = 0

where K(a) = β (a)π (a) is the convolution kernel and F(t) =

 A t

β (a)p0 (a − t)

π (a) da. π (a − t)

It is not hard to show that F(t) is a nonnegative continuous function of t and F(t) ≤ β¯ ||p0 ||1 .

1.2 Properties of the Solutions of Lotka-McKendrick Model In this section we review some of the properties of the Lotka-McKendrick model. Those are well known (see [85, 88, 179]) but are necessary to understand how we build age-structured epidemic models on the foundation of Lotka-McKendrick agestructured population model.

1.2 Properties of the Solutions of Lotka-McKendrick Model

5

1.2.1 Existence and Uniqueness of Solutions of the Lotka-McKendrick Model The existence and uniqueness of solutions of the Lotka-McKendrick model are not hard to prove. Multiple approaches exist. We will prove the result based on the following general lemma: Lemma 1.1. For T < ∞, suppose fk (t) ∈ C[0, T ] is a sequence of functions satisfying | fn+1 − fn |(t) ≤ K

 t 0

| fn − fn−1 |(s)ds

for all

t ∈ [0, T ]

(1.9)

for some constant K > 0. Then, fn is Cauchy in C[0, T ], and fn → f ∈ C[0, T ]. Proof. Since all functions are continuous, max | f1 − f0 | ≤ M. t

Then, | f2 − f1 | ≤ KMt. Using induction, one can show that | fn+1 − fn | ≤ M

(Kt)n . n!

Next, we prove that { fn } is Cauchy. For n > m with m large enough so that KT /(m+ 1) < 1, we apply the triangle inequality:

n−1 (Kt)m + · · · + | fn − fm | ≤ M (Kt) m! (n−1)! (1.10) (KT )m 1 − (KT /(m + 1))n−m → 0 as n, m → ∞. ≤M m! 1 − KT /(m + 1) 

This completes the proof. Next, we formulate the existence and uniqueness of the renewal equation.

Theorem 1.1. The renewal equation (1.6) has a unique solution B(t) that is continuous on R+ and such that ¯ 0 ≤ B(t) ≤ β¯ eβ t ||p0 ||1 .

(1.11)

Proof. To see the existence and uniqueness of solution of (1.6), we build a sequence Bn (t) defined as follows: B0 (0) = F(t) Bn+1 =

 t 0

K(a)Bn (t − a)da + F(t).

(1.12)

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1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

Clearly, B0 (t) is continuous and nonnegative, since F(t) is continuous and nonnegative. Then, by induction, Bn (t) is continuous and nonnegative for all n. Since, supa K(a) = β¯ , we have |Bn+1 − Bn | ≤ β¯

 t 0

|Bn (s) − Bn−1 (s)|ds.

Then Lemma 1.1 implies existence with the limit B(t) ≥ 0. To see uniqueness, assume B and B¯ are two solutions corresponding to the same initial condition. Then, ¯ ≤ β¯ |B − B|

 t 0

¯ |B(s) − B(s)|ds.

¯ Gronwall inequality then implies B ≡ B. To see the estimate, notice B(t) ≤ β¯

 t 0

B(s)ds + β¯ ||p0 ||1 .

Gronwall inequality then implies estimate (1.11). This completes the proof.

.

The continuity of B(t) implies that p(a,t) is continuous for t > a. From population perspective, we would like to know when the total birth rate B(t) ≡ 0. The Volterra equation (1.8) has a zero solution if and only if F(t) ≡ 0, that is, if and only if  ∞ 0

β (a + t)p0 (a)

π (a + t) da ≡ 0 for all t ≥ 0. π (a)

Thus, we need β (a+t)p0 (a) ≡ 0 almost everywhere on [0, A] for all t. Since β (a+t) is a shift to the left t units, β (a +t)p0 (a) ≡ 0 almost everywhere on [0, A] if and only if the support of β (a) is to the left of the support of p0 (a). This last requirement means that the initial population density consists only of post-reproductive ages. The initial population only ages and dies and no new individuals are born into the population. In this case p0 (a) is called trivial datum.

1.2.2 Long-Term Behavior The Lotka-McKendrick model (1.2) (or (1.6)) is a linear model, so we expect to have exponential solutions. With A finite, for t > A we have from (1.6) B(t) =

 A 0

B(t − a)β (a)π (a)da

t > A.

(1.13)

We look for an exponential solution of this equation, namely, a solution of the form B(t) = beλ t . Substituting in (1.13) we obtain the following equation in λ :

1.2 Properties of the Solutions of Lotka-McKendrick Model

1=

 A 0

e−λ a β (a)π (a)da.

7

(1.14)

This equation is called Lotka characteristic equation. We denote by G (λ ) =

 A 0

e−λ a β (a)π (a)da.

We define by R = G (0), that is, R=

 A 0

β (a)π (a)da.

R is called net reproduction rate. Regarding the solutions of Eq. (1.14) we have the following result. Proposition 1.1. The solutions λ of Eq. (1.14) satisfy: • Equation (1.14) has a unique real solution λ ∗ , which is a simple root; • The following are equivalent: – λ ∗ < 0 iff R < 1; – λ ∗ = 0 iff R = 1; – λ ∗ > 0 iff R > 1 • All complex solutions of (1.14) occur in conjugate pairs and satisfy ℜ(λ ) < λ ∗ Proof. Consider G (λ ) for λ real. G (λ ) is a strictly decreasing function of λ , since β (a)π (a) ≥ 0. Hence, if a real solution exists, it is unique. Since G (λ ) → ∞ if λ → −∞ and G (λ ) → 0 if λ → ∞, then the equation has a unique real solution λ ∗ . From the definition of R, we have that R = 1 iff λ ∗ = 0. Furthermore, if λ ∗ < 0, then R < 1 and vice versa. To see the last claim, let λ = x + yi be a complex root. Separating the real and the imaginary parts of (1.14) we have  A 0 A 0

e−xa cos yaβ (a)π (a)da = 1 e−xa sin yaβ (a)π (a)da = 0

(1.15)

Clearly, the above equations are satisfied by the complex conjugate as well. In addition:  G (λ ∗ ) = 1 = |

A

0

e−xa cos yaβ (a)π (a)da| < G (x).

Since G is a decreasing function, that implies that x < λ ∗ . This completes the proof. 

λ ∗ is called Malthusian parameter or intrinsic growth rate. The reason for that name is the following theorem giving the long-term behavior of B(t).

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1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

Theorem 1.2. Let λ ∗ be the Malthusian parameter. Then, ∗

B(t) = b0 eλ t (1 + Ω (t)), where b0 ≥ 0 and is zero if and only if B(t) ≡ 0. b0 is given by  A

b0 = 

∗

eλ t F(t)dt

0 A 0

∗

eλ t t β (t)π (t)dt

.

Furthermore, the function Ω (t) is continuous and there exists a constant K > 0 such that Ω (t) ≤ K for all t ≥ 0. Furthermore, we have lim Ω (t) = 0.

t→∞

The proof of this theorem can be found in [85]. From the expression for b0 and the estimate on F(t) we easily obtain that b0 ≤ M||p0 ||1 , where M is a constant independent of p0 . For t > A we have p(a,t) = b0 eλ and hence,

∗ (t−a)

(1 + Ω (t − a))π (a),

(1.16)

ˆ λ ∗ t ||p0 ||1 , ||p(.,t)||1 ≤ Me

where Mˆ is a constant independent of p0 . The main implication of Theorem 1.2 is that the solutions of Lotka-McKendrick model grow or decline asymptotically exponentially with growth rate λ ∗ . If λ ∗ = 0, the population is called stationary.

1.2.3 Persistent Solutions of the Lotka-McKendrick Model Separable solutions are a special type of solutions of PDE models which are a product of a function of t and a function of a, that is, they are of the form f (t)φ (a). It is not hard to see that solutions of this form of the McKendrick model (1.6) must have f (t) as an exponential function. Thus, we look for solutions of (1.6) in the form p(a,t) = eλ t φ (a), where λ is an unknown parameter and φ (a) is an unknown function. Substituting in the (1.6), we obtain φ  (a) + λ φ (a) = −μ (a)φ (a),  A (1.17) φ (0) = β (a)φ (a)da. 0

1.2 Properties of the Solutions of Lotka-McKendrick Model

9

Solving the differential equation and substituting in the initial condition, we obtain Lotka’s characteristic equation (1.14) for the unknown λ . We know that the unique real solution of this equation is λ ∗ . Hence, ∗

φ (a) = φ (0)e−λ a π (a). Therefore, for the separable solution we have ∗

∗

p(a,t) = φ (0)eλ t e−λ a π (a).

(1.18)

The unknown constant φ (0) can be determined from the initial condition. From (1.6), ∗ p(a, 0) = p0 (a) = φ (0)e−λ a π (a). Integrating, we have



A p0 (a)da . φ (0) =  A0 ∗ −λ a π (a) 0 e

(1.19)

Thus, the separable solution is obtained from (1.18) with the constant evaluated from (1.19). Solutions of this form are called persistent solutions. They play the role of equilibria in linear and homogeneous of degree one models (see Sect. 1.3.1).

1.2.4 The Sharpe-Lotka Theorem One of the most important results in mathematical population theory is the theorem of F.R. Sharpe and A.J. Lotka describing the long-term behavior of an age-structured population. This celebrated result first appeared in 1911 [158]. A rigorous proof was published in 1941 by W. Feller [51]. Sharpe-Lotka Theorem is listed below without proof: Theorem 1.3 (Sharpe-Lotka [158, 178]). Let μ (a) and β (a) be continuous from [0, ∞) to [0, ∞). Let μ ≤ μ (a) ≤ μ¯ , β (a) ≤ β¯ for a ≥ 0, where μ , μ¯ , β¯ are positive constants. Let λ = λ ∗ be the necessarily unique real root of the equation 1=

 ∞ 0

e−λ a β (a)π (a)da.

Then, the following hold: 1. Let λ ∗ < 0. There exists M > 0 and ω < 0 such that if p0 (a) ∈ L1 (0, ∞), then ||p(·,t)||L1 ≤ Meω t ||p0 ||L1

t ≥ 0.

2. Let λ ∗ ≥ 0. There exists M > 0 and ω < 0 such that if p0 (a) ∈ L1 (0, ∞), then ||p(·,t) − Π p0 ||L1 ≤ Meω t ||p0 ||L1

t ≥ 0,

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1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

where −λ ∗ a

Π p0 (a) = e

π (a)

∞ 0

∗x  x

∗

x

eλ y e− y μ (ξ )d ξ p0 (y)dydx ∞ −λ ∗ x π (x)dx 0 β (x)e

β (x)e−λ

0

a ≥ 0.

As a fundamental classical result in the age-structured population theory, SharpeLotka theorem states two things: 1. A population with stable fertility and mortality rates has a growth rate that is the same for all ages and for the total population. 2. That a population subject to stable fertility and mortality schedules will tend asymptotically toward an age distribution dependent only on the fertility and mortality rates and not on the initial age distribution, and will do so with the growth rate of the population. For systems where the role of equilibria is played by exponential solutions, it is typical to consider an associate system for the proportions of the age classes into the total population size. The age distribution is the fraction of the population age density into the total population size. Thus, following [88], we define the age distribution as

ω (a,t) =

p(a,t) , P(t)

where P(t) is the total population defined by (1.39). For the age distribution to be defined we need P(t) > 0. Long-term behavior of solutions can then be established by considering the long-term behavior of the age-distribution (see [88] for discussion along these lines). Linear age-structured population models played an important role in the development of the age-structured epidemic models, because they are the population model on the basis of which an epidemic model was built. In the next section we consider several ways of developing age-structured epidemic models.

1.3 Combining the Age-Structured Population with Epidemic Models Epidemic models date back to Sir Ronald Ross and his model of malaria at the beginning of the twentieth century as well as Kermack-McKendrick [102] and their susceptible-infected-removed (SIR) model, proposed to explain the observed epidemic curve of the number of infected individuals which emerged in epidemics such as the plague (London 1665–1666, Bombay 1906) and cholera (London 1865). Epidemics models the population, stratified in nonoverlapping classes, named by their epidemic status. For instance, a population may consist of susceptible and infected individuals and that would lead to an SI or SIS model, which the last one meaning

1.3 Combining the Age-Structured Population with Epidemic Models

11

that infected individuals can recover and return to susceptible state. Further structures may include susceptible, exposed, infected, recovered individuals, leading to an SEIR model. The original Kermack-McKendrick model was an SIR model, thus stratifying the population by susceptible individuals at time t – S(t), infected individuals at time t – I(t), recovered individuals at time t – R(t). Susceptible individuals become infected through a contact with an infectious individual at the bilinear rate β S(t)I(t), called incidence. β in this case is called transmission coefficient. The infected individuals recover at a per capita rate γ . The model takes the form: S (t) = −β SI, I  (t) = β SI − γ I, R (t) =

(1.20)

γ I.

The epidemic in Kermack-McKendrick model is assumed to develop on a very short scale, so the births and deaths of the individuals in the population are neglected. If demographic variables are added, and the simplest demographic model is used, that is Malthus model, then the Kermack-McKendrick model with demography will take the form: S (t) = bN(t) − β SI − μ S, I  (t) = β SI − (γ + μ )I, R (t)

(1.21)

= γ I − μ R,

where b is the per capita birth rate and μ is the natural mortality rate, and N(t) = S(t) + I(t) + R(t) is the total population size. In what follows in this chapter, we derive age-structured analogues of epidemic models with demography. To this end, in this section we discuss options for combining the Lotka-McKendrick model with epidemic models. As we saw, the solutions of the Lotka-McKendrick model grow or decline exponentially, while epidemic models have nonlinear terms, most commonly in the incidence, which typically suggest existence of a steady state. This creates a conflict between the behavior of the epidemiological classes and the behavior of the total population size. In this section we discuss cases in which the combination will be well defined, although, besides the cases listed below, other cases may be possible. Most of the models discussed below are analogues of an SI or SIS ODE models of the form: S (t) = bN(t) − f (S, I) − μ S + γ I, I  (t) = f (S, I) − (γ + μ )I,

(1.22)

where f (S, I) is the incidence term. The incidence can be mass action if f (S, I) = β SI as above or standard if f (S, I) = βNSI , but it can have different forms as well.

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1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

1.3.1 Homogeneous Age-Structured Epidemic Models A function (or functional) g(x) of n variables is called homogeneous of degree one if for a constant α , we have g(α x) = α g(x). Standard incidence is a homogeneous functional of degree one. We can combine the age-structured Lotka-McKendrick population model with an epidemic model with standard incidence. To illustrate this, consider an SI age-structured epidemic model with s(a,t) and i(a,t) being the densities of the susceptible and infected individuals, respectively. Standard incidence, which in the ODE case has the form f (S, I) =

β SI , N

in age-structured epidemic models can be given in a number of ways. The following variants are possible as well as combinations of those: k0 (a)s(a,t)i(a,t) P(t) k1 (a)s(a,t)

 A 0

k2 (a)i(a,t)da

P(t)

(1.23) ,

where ki (a), i = 0, 1, 2 are age-structured transmission rates and p(a,t) = s(a,t) + i(a,t). We can also have a more general form as  A

s(a,t) 0

k(a, a )i(a ,t)da P(t)

.

(1.24)

P(t) as before is the total population size. It is not hard to see that these functionals of s and i are homogeneous of degree one. They are well defined because their denominators are strictly positive, if the initial population size is a nontrivial datum. The first incidence is a standard intra-cohort incidence where the individuals mix only with their own age groups. This type will be appropriate for childhood diseases. The second incidence is standard inter-cohort incidence, where an individual of age a mixes with all other age groups. This one will be appropriate for example for modeling spread of HIV. Age-structured epidemic models have been used to study age-related infectious disease questions and vaccination policies since the 1980s, starting with the works of Anderson and May (e.g., [2]). The importance of age-structure in infectious disease modeling cannot be underestimated, particularly for diseases which are strongly age-related such as childhood diseases or malaria, where immunity is age-related. A simple SIS epidemic model with the second incidence will take the form

1.3 Combining the Age-Structured Population with Epidemic Models

13

⎧ st + sa = −Λ (a,t)s(a,t) + γ (a)i(a,t) − μ (a)s(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ (a,t)s(a,t) − (μ (a) + γ (a))i(a,t), ⎪ ⎪it + ia = Λ  A ⎪ ⎪ ⎨ β (a)p(a,t)da, s(0,t) = 0 ⎪ ⎪ i(0,t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ s(a, 0) = s0 (a), ⎪ ⎪ ⎩ i(a, 0) = i0 (a), where

Λ (a,t) =

k1 (a)

 A 0

(1.25)

k2 (a)i(a,t)da P(t)

are the force of infection and γ (a) is the recovery rate. Here, p(a,t) = s(a,t) + i(a,t)

and

P(t) =

 A

p(a,t)da 0

are the total population density and the total population size, respectively. The above model assumes that all newborns are born susceptible (no vertical transmission). Summing the equations, we obtain the McKendrick age-structured population model (1.2). The above model is of SIS type. As before, the natural space to consider the solutions is in L1 (0, A). The assumptions on μ are the same as for the McKendrick model. The remaining rates all belong to L∞ [0, A]. We will discuss well posedness of age-structured epidemic models in Chap. 2. Here we will only address the various ways of combining population and epidemic models. Since (1.2) is linear and therefore homogeneous of degree one, and the nonlinear term in (1.25) is homogeneous of degree one, it is not hard to see that the full model (1.25) is homogeneous of degree one. As such, (1.25) has exponentially increasing or decreasing solutions, just like model (1.2). The role of equilibria is played by the persistent solutions. The presence of the persistent solutions is given by the following proposition: Proposition 1.2. Assume p0 (a) is a nontrivial initial condition. Then, system (1.25) has a persistent solution of the form ∗

s(a,t) = eλ t ψ (a),

∗

i(a,t) = eλ t η (a),

where λ ∗ is the Malthusian parameter. Proof. It is not hard to see that if we look for separable solutions, the timedependent function needs to be an exponential. Hence, we look for solutions of the form s(a,t) = eλ t s(a), i(a,t) = eλ t i(a), where λ , s(a), and i(a) are to be determined. Substituting in system (1.25), we obtain:

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1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

⎧ ⎪ λ s + sa = −Λ (a)s(a) + γ (a)i(a) − μ (a)s(a), ⎪ ⎪ ⎪ ⎪ ⎨λ i + ia = Λ (a)s(a) − (μ (a) + γ (a))i(a),  A

(1.26)

⎪ β (a)p(a)da, s(0) = ⎪ ⎪ ⎪ 0 ⎪ ⎩i(0) = 0. where

Λ (a) = k1 (a)

 A 0

k2 (ξ )i(ξ )d ξ

p(a) = s(a) + i(a).

and

Adding the two equations, we obtain ⎧ ⎨λ p + pa = −μ (a)p(a), ⎩ p(0) =

 A 0

(1.27)

β (a)p(a)da.

Solving the differential equation and substituting into the initial condition we obtain the Lotka characteristic equation 1=

 A 0

β (a)e−λ a π (a)da. ∗

We know that the unique real solution is λ = λ ∗ . Hence p(a) = p(0)e−λ a π (a). It  is not hard to see that 0A p(a)da = P0 , see (1.19). We write

Λ (a) = k1 (a)Λ , where Λ is an unknown parameter, such that A

Λ=

0

k2 (a)i(a)da . P0

Replacing s(a) = p(a) − i(a) and solving the differential equation for i we have i=Λ

 a 0

k1 (σ )p(σ )e−λ

∗ (a−σ )

e−Λ

a

σ k1 (τ )d τ

e−

a σ

μ (τ )+γ (τ )d τ

dσ .

Substituting in the equation for Λ we obtain that Λ is the unique positive solution, independent of P0 (or p(0)) of the equation 1=

1 P0

 A 0

k2 (a)

 a 0 ∗

k1 (σ )p(σ )e−λ

∗ (a−σ )

e−Λ

a

σ k1 (τ )d τ

e−

a σ

μ (τ )+γ (τ )d τ

d σ da.

Since p(σ ) = p(0)e−λ σ π (σ ), substituting in P0 and the above expression, we obtain the following simplified equation for Λ :

1.3 Combining the Age-Structured Population with Epidemic Models

 A

1

1= ∞

−λ ∗ a π (a)da 0 0 e

∗

k2 (a)e−λ a π (a)

 a 0

k1 (σ )e−Λ

a

σ k1 (τ )d τ

15

e−

a σ

γ (τ )d τ

d σ da.

A positive solution to the above equation exists iff ∞

1

 A

−λ ∗ a π (a)da 0 0 e

∗

k2 (a)e−λ a π (a)

 a 0

k1 (σ )e−

a σ

γ (τ )d τ

d σ da > 1.

This completes the proof.  One can show that every solution approaches a separable solution. Further information about homogeneous age-structured epidemic models can be obtained from [87, 94].

1.3.2 Age-Structured Epidemic Models with Stationary Population If the incidence of the epidemic model is of mass action type, then it is not homogeneous of degree one. Thus separable solution of an age-structured models with mass action incidence cannot be expected. The only way to reconcile the exponential growth (or decay) of the population model with the mass action incidence is to assume that the intrinsic growth rate of the population is equal to zero, that is, the population is stationary. Mass action incidence can also have several forms: k1 (a)S(a,t)

 A 0

k2 (a)i(a,t)da or

k0 (a)s(a,t)i(a,t),

where the first one is of inter-cohort type and assumes mixing of an individual of age a with individuals of all other ages. This one will be appropriate for influenza. The second one is of intra-cohort type and assumes that individuals of age a mix only with individuals of the same age. This one will again be appropriate for childhood diseases. In this case with the notation in the previous subsection, the SIS model with mass action incidence becomes: ⎧ st + sa = −Λ (a,t)s(a,t) + γ (a)i(a,t) − μ (a)s(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ it + ia = Λ (a,t)s(a,t) − (μ (a) + γ (a))i(a,t), ⎪ ⎪ ⎪  A ⎪ ⎪ ⎨ β (a)p(a,t)da, s(0,t) = (1.28) 0 ⎪ ⎪ i(0,t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s(a, 0) = s0 (a), ⎪ ⎪ ⎪ ⎪ ⎩ i(a, 0) = i0 (a),

16

1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

where we take the inter-cohort incidence

Λ (a,t) = k1 (a)

 A 0

k2 (a)i(a,t)da

and p(a,t) = s(a,t) + i(a,t)

and

P(t) =

 A

p(a,t)da 0

are the total population density and the total population size respectively. The population model of model (1.28) is again the McKendrick model (1.2). To avoid the exponential growth (or decay) so that the population model is consistent with the nonlinearity in the epidemic model, we need to assume that the Malthusian parameter in the population model is zero, that is, we assume that R = 1 or that  A 0

β (a)π (a)da = 1.

(1.29)

In this case model (1.28) would have a steady state. The state could be derived in the same way as the persistent solutions were derived in the previous subsection, but with λ ∗ = 0. The SIR version of the model is studied in a number of references (e.g., see [32]).

1.3.3 Age-Structured Models with Disease-Induced Mortality When the epidemic model has mass action incidence and disease-induced mortality, then in the presence of the disease, the population model is no longer the McKendrick model (1.2). In this case assumption (1.29) is not necessary and we need to assume R > 1. As a result, in the absence of disease, the population grows exponentially; however, in the presence of disease the disease regulates the population growth and that may lead to a stable steady state solution. To illustrate the scenario, with the notation in the previous section, we consider the model: ⎧ s + sa = −Λ (a,t)s(a,t) − μ (a)s(a,t), ⎪ ⎪ ⎪t ⎪ ⎪ ⎪ ⎪it + ia = Λ (a,t)s(a,t) − (μ (a) + α (a))i(a,t), ⎪ ⎪ ⎪  ⎪ ⎪ ⎨s(0,t) = A β (a)p(a,t)da, (1.30) 0 ⎪ ⎪ i(0,t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ s(a, 0) = s0 (a), ⎪ ⎪ ⎪ ⎪ ⎩ i(a, 0) = i0 (a), where α (a) is the age-specific disease-induced mortality and

1.3 Combining the Age-Structured Population with Epidemic Models

Λ (a,t) = k1 (a)

 A 0

17

k2 (a)i(a,t)da.

With p(a,t) = s(a,t) + i(a,t), the equation for the total population size becomes: ⎧ ⎨ pt + pa = −μ (a)p(a,t) − α (a)i(a,t),  A (1.31) ⎩ p(0,t) = β (a)p(a,t)da. 0

This is an SI model with disease-induced mortality. In the absence of disease (1.31) is the same as (1.2) and since R > 1, the solutions grow exponentially at an intrinsic growth rate λ ∗ > 0. Proposition 1.3. Assume R > 1. Then system (1.30) has a unique endemic equilibrium. Proof. We recall that λ ∗ > 0 satisfies:  A 0

∗

β e−λ a e−

a 0

μ (x)dx

da = 1.

The system for the equilibria takes the form: ⎧ ⎪ λ ∗ s + sa = −Λ (a)s(a) − μ (a)s(a), ⎪ ⎪ ⎪ ⎪ ⎨λ ∗ i + ia = Λ (a)s(a) − (μ (a) + α (a))i(a),  A

(1.32)

⎪ β (a)p(a)da, s(0) = ⎪ ⎪ ⎪ 0 ⎪ ⎩ i(0,t) = 0.

We again write Λ (a) = Λ k1 (a), where Λ is a real positive number to be determined. Solving the equation for s we have: s(a) = s(0)e−Λ

a

0 k1 (τ )d τ

π (a),

where as before π (a) is the probability of survival till age a

π (a) = e−

a 0

μ (σ )d σ −λ ∗ a

e

.

In the expression for s(a), s(0) is an unknown constant, to be determined. Substituting s(a) in the equation for i(a) and solving we obtain i(a) = s(0)Λ π (a)

 a 0

k1 (σ )e−Λ

σ 0



k1 (τ )d τ − σa α (τ )d τ e dσ

= s(0)Λ π (a) f (a, Λ ), (1.33)

where we have used the notation f (a, Λ ) =

 a 0

k1 (σ )e−Λ

σ 0



k1 (τ )d τ − σa α (τ )d τ e dσ .

18

1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

Substituting in the equation for s(0) and canceling s(0) we obtain the following equation for Λ : 1=

 A 0

β (a)π (a)(e−Λ

a

0 k1 (τ )d τ

+ Λ f (a, Λ ))da.

(1.34)

Let the right-hand side of the above equality be denoted by a function F (Λ ). Integration by parts shows that e−Λ

a

0 k1 (τ )d τ

+ Λ f (a, Λ ) = e−

a σ

α (τ )d τ

+

 a 0

α (σ )e−Λ

σ 0

 k1 (τ )d τ − σa α (τ )d τ

e

dσ .

Therefore F (Λ ) is a decreasing function of Λ with  A

lim F (Λ ) =

Λ →∞

0

β (a)π (a)e−

a 0

α (x)dx

.

Since F (0) = R > 1, and  A 0

β (a)π (a)e−

a 0

α (x)dx


t, a < t.

(1.36)

The population model (1.35) does not exhibit exponential growth and can be coupled with any nonlinear incidence rate to build an epidemic model. We remark that constant birth rate occurs in the particular case λ ∗ = 0, i.e., when R = 1. In this sense, this is a special case of Sect. 1.3.2. We suggest readers use the approach of Sect. 1.3.2.

1.4 Early Age-Structured Epidemic Models Age-structure is an important demographic characteristics with significant impact on infectious disease dynamics and spread. There are two main reasons to incorporate age-structure: (1) Age-structure is necessary where susceptibility and infectivity are strongly age-related. This is the case, for instance for all childhood diseases, or for many sexually transmitted diseases. (2) Age structure is necessary in models of chronic infections, such as HIV, Hepatitis C, and TB where the demography of the population, and particularly survivorship need more realistic distributions, in place of exponential distributions modeled by ODEs [34]. We saw in the previous section that age-structured population models can be recast into age-structured epidemic models. The baseline ODE epidemic models of SIS, SI, and SIR types have also been considered as age-structured models. The inclusion of age-structure enriches the dynamics of the model. One of the best wellunderstood dynamically age-structured epidemic model is the SIS model without vertical transmission. To introduce the model, let s(a,t) be the density of susceptibles and i(a,t) be the density of infecteds. We have p(a,t) = s(a,t) + i(a,t). The SIS model is given by the following system: ⎧ st + sa = −Λ (a,t)s(a,t) + γ (a)i(a,t) − μ (a)s(a,t), ⎪ ⎪ ⎪ ⎪ ⎨ it + ia = Λ (a,t)s(a,t) − (γ (a) + μ (a))i(a,t), (1.37) ∞ ⎪ ⎪ s(0,t) = 0 β (a)p(a,t)da, ⎪ ⎪ ⎩ i(0,t) = 0, where A(a,t) is the force of infection, γ (a) is the recovery rate, β (a) is the birth rate, and μ (a) is the natural death rate. There are several constitutive formulas for the force of infection. Busenberg et al. [21] consider the so-called mixed inter-cohort, intra-cohort case where the force of infection is given by K0 (a)i(a,t) +

 ∞ 0

K(a, ξ )i(ξ ,t)d ξ ,

(1.38)

where K0 (a) and K(a, ξ ) are two given functions. The force of infection is called intra-cohort if K(a, ξ ) = 0. The force of infection is called inter-cohort if K0 (a) = 0.

20

1 Linear Age-Structured Population Models as a Base of Age-Structured Epidemic. . .

The force of infection is called separable if K(a, ξ ) = K1 (a)K2 (ξ ). This model assumes that all individuals are born susceptible. Adding the two equations, we obtain the equation of the total population size  pt + pa = −μ (a)p(a,t),  (1.39) p(0,t) = 0∞ β (a)p(a,t)da. The equation of the total population size is a linear age-structured population equation and its solutions are exponential. For consistency with the epidemic model, which is nonlinear, we assume that the growth rate of the population is zero; that is, the population is stationary. Under the above conditions, Busenberg et al. [21] show that the model has a unique disease-free equilibrium which is globally asymptotically stable if the reproduction number is smaller than one. If the reproduction number is larger than one, the system has a unique endemic equilibrium, which is also globally asymptotically stable. In a continuation of article [21], Busenberg et al. consider in [22] an SIS model with vertical transmission. With the same notation as before, an SIS model with vertical transmission takes the form ⎧ st + sa = −Λ (a,t)s(a,t) + γ (a)i(a,t) − μ (a)s(a,t), ⎪ ⎪ ⎪ ⎪ ⎨ it + ia = Λ (a,t)s(a,t) − (γ (a) + μ (a))i(a,t), (1.40)  ⎪ s(0,t) = 0∞ β (a)(s(a,t) + qi(a,t))da, ⎪ ⎪ ⎪  ⎩ i(0,t) = (1 − q) 0∞ β (a)i(a,t)da, where q is the proportion of newborns to infected individuals who are susceptibles. Under some additional assumptions Busenberg et al. [22] ruled out oscillations and proved global stability. The dynamics of the age-structured SIR model with vertical transmission does not appear to be that simple. Inaba [92] and Cha et al. considered the question of multiple endemic equilibria and derived conditions for existence of a unique equilibrium [31, 32, 92]. We introduce the SIR model considered, which involved vertical transmission. With the same notation as before, we have ⎧ st + sa = −Λ (a,t)s(a,t) − μ (a)s(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ it + ia = Λ (a,t)s(a,t) − (γ (a) + μ (a))i(a,t), ⎪ ⎪ ⎪ ⎪ ⎨ rt + ra = γ (a)i(a,t) − μ (a)r(a,t) (1.41) ∞ ⎪ ⎪ s(0,t) = 0 β (a)(s(a,t) + qi(a,t) + r(a,t))da, ⎪ ⎪  ⎪ ⎪ i(0,t) = (1 − q) ∞ β (a)i(a,t)da ⎪ 0 ⎪ ⎪ ⎩ r(0,t) = 0

1.4 Early Age-Structured Epidemic Models

21

The force of infection is of mixed inter-cohort, intra-cohort case and is given by

Λ (a,t) = K0 (a)i(a,t) +

 ∞ 0

K(a, ξ )i(ξ ,t)d ξ ,

(1.42)

where K0 (a) and K(a, ξ ) are two given functions. The problem of multiple equilibria of the age-structured SIR model was finally resolved by Franceschetti et al. [60]. The special case example in [60] shows nonuniqueness of positive equilibria which arise not from a backward transcritical bifurcation at the disease free equilibrium. The multiple equilibria of the SIR model with vertical transmission arise through two saddle-node bifurcations of the positive equilibrium. The authors analyze numerically the dynamical behavior of the model; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasi-periodic and chaotic attractors. This article shows complex dynamical behavior in a somewhat simple age-structured epidemic model. While the mathematical tools for analysis and simulations of ODE epidemic models, from proving well posedness and global stability of the disease-free equilibrium, to backward bifurcation, oscillations, chaos, and optimal control are well developed and widely used even in very complex epidemic models, that is not the case with age-structured and age-since-infection structured models. It is our goal in this book to develop and illustrate on multiple examples various techniques for the analysis and simulation of age and age-since-infection structured epidemic models. In the next section we develop tools for establishing well posedness of age-structured models and illustrate methods for deriving necessary and sufficient conditions for backward bifurcation in age-structured epidemic models. It is difficult to acknowledge all authors who have contributed to age-structured population modeling. The first age-dependent population modeling is credited to Euler [78]. Nonlinear age-structured models that we do not discuss here were introduced and analyzed by Gurtin and MacCamy [74–76]. The analytical tools for these models were developed in [179]. A more realistic versions of age-structured population models can be found in the works of Greenhalgh [71], Anderson and May [1, 126] in the late 1980s and 1990s, and Adreasen et al. [27]. At the same time the age-structured analytical theory was developed as we mentioned before by Iannelli [85], Thieme, Busenberg [21, 22], and Inaba [92]. A generalization of the age-structured models is the so-called physiologically structured models that were extensively explored in the 1970s and 1980s by J.A.J. Metz and O. Diekmann [133, 134] as well as other authors. In Chap. 2 we discuss more advanced techniques to study age-structured epidemic models. These techniques are very common for ODE epidemic models but have been developed and used little in age-structured epidemic models.

Chapter 2

Age-Structured Epidemic Models

2.1 Analysis of Age-Structured Epidemic Models In this section we present some basic and advanced analysis of age-structured epidemic models. In the next subsection, we discuss well posedness.

2.1.1 Well Posedness of Age-Structured Epidemic Models General theorems for existence, uniqueness, and nonnegativity of solutions of agestructured models are rare. Typically the well posedness of age-structured epidemic models is approached through recasting the problem in semigroup formulation (see [96] for illustration). Another approach to well posedness is to use the results in [179]. We present here a general theorem and use functional analytical tools to prove the result. Let X be a Banach space with norm ||.||X and A > 0 be the maximal age. We set X = Rn × Y where Y = (L1 (0, A))m . Let v ∈ C([0, T ], Rn ) and u ∈ C([0, T ],Y ). Let D ≥ 0 be a Rm × Rm diagonal matrix of functions which may not be bounded on [0, A), that is, we may have lima→A d j (a) = ∞ for some or all js and if A is finite. Consider the system v = G(t; v, u) ut + ua = F(t, a; v, u) − Du, v(0) = v0 ,

(2.1)

u(0,t) = B(t; v, u), u(a, 0) = u0 (a), where G, B : [0, T ] × [0, A) × X −→ Rn , F : [0, T ] × [0, A) × X −→ Y are generally nonlinear. Let X+ denote the positive cone in X. Let (v0 , u0 ) ∈ X+ . Let Ω ⊂ X+ be a bounded invariant set for system (2.1). © Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1_2

23

24

2 Age-Structured Epidemic Models

Theorem 2.1. Assume 1. G, B, F are Lipschitz continuous, that is, for all v, v¯ ∈ Rn and all u, u¯ ∈ Y (or ¯ ∈ Ω ), and all t ∈ [0, T ] we have (v, u) ∈ Ω , (¯v, u) ¯ Rn ≤ KG (|v − v¯ |Rn + ||u − u|| ¯ Y) |G(t; v, u) − G(t; v¯ , u)| ¯ Rm ≤ KB (|v − v¯ |Rn + ||u − u|| ¯ Y) |B(t; v, u) − B(t; v¯ , u)| ¯ Y ≤ KF (|v − v¯ |Rn + ||u − u|| ¯ Y ). ||F(t; v, u) − F(t; v¯ , u)|| 2. F(t, a; v, u) = F1 (t, a; v, u) − F2 (t, a; v, u)u where F1 , F2 ≥ 0 for all t ∈ [0, T ], a ∈ [0, A), v ∈ Rn+ and u ∈ Y+ and |F2 (t, a; v, u)|Rm ≤ K2 for all t ∈ [0, T ], a ∈ [0, A], v ∈ Rn+ and u ∈ Y+ or all (v, u) ∈ Ω . 3. G(t; v, u) = G1 (t; v, u) − G2 (t; v, u)v where G1 , G2 ≥ 0 for all t ∈ [0, T ], v ∈ Rn+ and u ∈ Y+ and |G2 (t; v, u)|Rn ≤ K2 for all t ∈ [0, T ], v ∈ Rn+ and u ∈ Y+ or all (v, u) ∈ Ω . 4. B(t; v, u) ≥ 0 for all t ∈ [0, T ], v ∈ Rn+ and u ∈ Y+ . 5. |G(t; 0, 0)|Rn , ||F(t; 0, 0||Y , B(t; 0, 0) are bounded. Then, for every (v0 , u0 ) ∈ X+ or in Ω , the system (2.1) has a unique nonnegative solution. Proof. Let λ > 0 a real number. Adding λ v to both sides of the first equation and λ u to both sides of the second equation in (2.1) and integrating we have the following system of integral equations: 

v(t) = e−λ t v0 + 0t e−λ (t−s) (λ v + G(s; v, u))ds ⎧  e−λ t π (a,t;t)u0 (a − t) + 0t (λ u + F(v, u))(a − τ ,t − τ )e−λ τ ⎪ ⎪ ⎪ ⎪ ⎨ π (a,t; τ )d τ a > t u(a,t) =  ⎪ ⎪ e−λ a π (a,t; a)B(v, u)(t − a) + 0a (λ u + F(v, u))(a − τ ,t − τ )e−λ τ ⎪ ⎪ ⎩ π (a,t; τ )d τ a < t, (2.2) where π is the diagonal matrix

π (a,t; τ ) = e−

τ 0

D(a−s,t−s)ds

.

The equations above define a fixed point problem (v, u)T = O(v, u). First we show that for λ large enough O maps the positive cone into itself. Alternatively, we may show that O maps Ω into itself. To show that O maps the positive cone into itself, we use assumptions 2., 3., and 4., that is, we assume that |G2 (t, v, u)|Rn ≤ K2 and |F2 (t, a; v, u)|Rm ≤ K2 for all t ∈ [0, T ], a ∈ [0, A], v ∈ Rn+ and u ∈ Y+ . If we take λ > K2 , it is not hard to see that λ v − G2 (t, v, u)v ≥ 0 and λ u − F2 (t, a; v, u)u ≥ 0. Hence, O maps the positive cone into itself.

2.1 Analysis of Age-Structured Epidemic Models

25

Next, we define an iteration (vn+1 , un+1 )T = O(vn , un ), where (v0 , u0 ) = (0, 0). We note that if the initial conditions are not identically zero, (0, 0) is not a fixed point of O. Consider |vn+1 − vn | + ||un+1 − un ||Y ≤ I1 + I2 + I3 + I4 , where I1 =

 t 0

e−λ (t−s) (λ |vn − vn−1 | + |G(s; vn , un ) − G(s; vn−1 , un−1 )|)ds

and I2 =

 t 0

|B(t − a; vn , un ) − B(t − a; vn−1 , un−1 |da.

For I3 and I4 we have after the change of order of integration I3 =

I4 =

 t t τ

0

 t A 0

t

(λ |un − un−1 | + |F(vn , un ) − F(vn−1 , un−1 )|)(a − τ ,t − τ )dad τ

(λ |un − un−1 | + |F(vn , un ) − F(vn−1 , un−1 )|)(a − τ ,t − τ )dad τ .

Hence, I3 + I 4 =

 t A 0

τ

(λ |un − un−1 | + |F(vn , un ) − F(vn−1 , un−1 )|)(a − τ ,t − τ )dad τ .

Applying the Lipschitz property for each of the nonlinear terms (assumption 1.), we have the following estimate: |vn+1 − vn |Rn + ||un+1 − un ||Y ≤ κ

 t 0

(|vn − vn−1 | + ||un − un−1 ||Y )(t − τ )d τ .

Let Nn+1 (t) = |vn+1 − vn |Rn + ||un+1 − un ||Y . Then the above inequality becomes: Nn+1 (t) ≤ κ

 t 0

Nn (η )d η

for some constant κ which depends on λ as well as KG , KF , KB . Assumption 5 implies that N1 ≤ Q for some constant Q. Since N2 (t) ≤ κ

 t 0

N1 (η )d η ≤ κ Qt

26

2 Age-Structured Epidemic Models

by induction we can show that Nn+1 (t) ≤ Q

(κ t)n . n!

Hence,

(κ T )n → 0 as n → ∞. n! We notice that in the above argument, we do not assume that T is small, just that T is fixed but arbitrary. To show that the sequence is Cauchy, let m, n be large enough, such that n + 1 > κ T and m > n. Then, Nn+1 (t) ≤ Q

|vm+1 − vn |Rn + ||um+1 − un ||Y ≤ Nm+1 (t) + · · · + Nn+1 (t) ≤κ

 t

Nm (s) + · · · + Nn (s)ds

0

≤ κQ

 t m (κ s)( j−1)

∑

0 j=n

( j − 1)!

ds

(2.3) (κ t) j j=n j!  (κ t)m−n (κ t)n +···+1 =Q n! m . . . (n + 1) 1 (κ t)n ≤Q →0 as n → ∞. κT n! 1 − n+1 m

=Q∑

Thus, the sequence (vn , un ) is Cauchy in X+ and therefore convergent to a fixed point of O. To show that the fixed point is unique, we assume that O has two fixed points: ¯ Y . Similar computations ¯ We denote by N (t) = |v − v¯ |Rn + ||u − u|| (v, u) and (¯v, u). lead to the inequality  N (t) ≤ κ

t

0

N (s)ds.

Gronwall inequality then implies that N (t) ≡ 0. Hence, the fixed point is unique. This completes the proof.  We illustrate the techniques for the analysis of age-structured epidemic models, techniques such as computing R0 , showing local stability of the disease-free equilibrium on an example in the next subsection.

2.1 Analysis of Age-Structured Epidemic Models

27

2.1.2 Computing R0 for Age-Structured Epidemic Models In this section we demonstrate how to compute R0 for age-structured models. As an example we consider an age-structured SIR model. The density of the susceptibles is denoted by s(a,t), of the infecteds—by i(a,t) and by recovered—by r(a,t). ⎧ st + sa = −Λ (a,t)s(a,t) − μ (a)s(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ it + ia = Λ (a,t)s(a,t) − (γ (a) + μ (a))i(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ rt + ra = γ (a)i(a,t) − μ (a)r(a,t) s(0,t) = 0∞ β (a)(s(a,t) + i(a,t) + r(a,t))da, (2.4) ⎪ ⎪ ⎪ ⎪ i(0,t) = 0 ⎪ ⎪ ⎪ ⎪ ⎪ r(0,t) = 0 ⎪ ⎪ ⎪ ⎩ s(a, 0) = s0 (a), i(a, 0) = i0 (a), r(a, 0) = r0 (a), where s0 (a), i0 (a), r0 (0) are given nonnegative functions, μ (a) is the natural mortality and γ (a) is the recovery rate, β (a) is the birth rate, and the force of infection is of inter-cohort type:

Λ (a,t) = k1 (a)

 ∞ 0

k2 (ξ )i(ξ ,t)d ξ ,

(2.5)

where k1 (a), k2 (a) are given functions. The total population size p(a,t) = a(a,t) + i(a,t) + r(a,t) is assumed constant, that is  ∞ 0

β (a)π (a)da = 1

and where

a π (a) = e− 0 μ (ξ )d ξ .

p(a,t) = p∞ (a) = b0 π (a), The constant b0 is determined from the initial conditions

b0 =

∞ 0

(s0 (a) + i0 (a) + r0 (a))da ∞ . 0 π (a)da

We begin by computing the equilibria. These satisfy the system: ⎧ sa = −Λ (a)s(a) − μ (a)s(a), ⎪ ⎪ ⎪ ⎪ ⎪ ia = Λ (a)s(a) − (γ (a) + μ (a))i(a), ⎪ ⎪ ⎪ ⎪ ⎨ ra = γ (a)i(a,t) − μ (a)r(a,t) ∞ ⎪ ⎪ s(0) = 0 β (a)(s(a) + i(a) + r(a))da, ⎪ ⎪ ⎪ ⎪ i(0) = 0 ⎪ ⎪ ⎪ ⎩ r(0) = 0.

(2.6)

28

2 Age-Structured Epidemic Models

The system has the disease-free equilibrium, where i(a) = 0 and r(a) = 0. The susceptibles satisfy sa = −μ (a)s(a), which gives s(a) = s(0)π (a) where we can see that the boundary condition for s(0) is trivially satisfied and s(0) = b0 . Thus, the disease-free equilibrium is given by E0 = (b0 π (a), 0, 0). To compute the reproduction number R0 , we need to study the stability of the disease-free equilibrium. To this end we look at the perturbations s(a,t) = s(a) + x(a,t),

i(a,t) = y(a,t)

r(a,t) = z(a,t).

The nonlinear term given by the incidence is linearized as follows:  ∞

s(a,t) 0

K(a, ξ )i(ξ ,t)d ξ = (s(a) + x(a,t)) ≈ s(a)

 ∞ 0

 ∞ 0

K(a, ξ )y(ξ ,t)d ξ

K(a, ξ )y(ξ ,t)d ξ .

Then the equations for the perturbations x(a,t), y(a,t)z(a,t) take the form: ⎧ xt + xa = −Λ p (a,t)s(a) − μ (a)x(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ yt + ya = Λ p (a,t)s(a) − (γ (a) + μ (a))y(a,t), ⎪ ⎪ ⎪ ⎪ ⎨ zt + za = γ (a)y(a,t) − μ (a)z(a,t) ∞ ⎪ ⎪ x(0,t) = 0 β (a)(x(a,t) + y(a,t) + z(a,t))da, ⎪ ⎪ ⎪ ⎪ y(0,t) = 0 ⎪ ⎪ ⎪ ⎩ z(0,t) = 0, where

Λ p (a,t) = k1 (a)

 ∞ 0

(2.7)

k2 (ξ )y(a,t)da.

Next, we look for solution of the form x(a,t) = eλ t x(a)

y(a,t) = eλ t y(a)

z(a,t) = eλ t z(a).

We obtain from the system above, the following eigenvalue problem: ⎧ ⎪ ⎪ λ x + xa = −Λ p (a)s(a) − μ (a)x(a), ⎪ ⎪ ⎪ λ y + ya = Λ p (a)s(a) − (γ (a) + μ (a))y(a), ⎪ ⎪ ⎪ ⎪ ⎨ λ z + za = γ (a)y(a) − μ (a)z(a) 

⎪ x(0,t) = 0∞ β (a)(x(a,t) + y(a,t) + z(a,t))da, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y(0,t) = 0 ⎪ ⎪ ⎩ z(0,t) = 0.

(2.8)

We need to determine the λ s that allow the system above to have a nonzero solution. We notice that the equation for y has separated and is independent of the other

2.1 Analysis of Age-Structured Epidemic Models

29

equations. Also notice that Λ p (a) = k1 (a)Λ , where Λ is an unknown constant. We integrate the equation for y: y(a) = Λ

 a 0

 − ξa (λ +μ (σ )+γ (σ ))d σ

k1 (ξ )s(ξ )e

dξ .

Substituting in the expression for Λ we get

Λ =Λ

 ∞ 0

k2 (a)

 a 0

 − ξa (λ +μ (σ )+γ (σ ))d σ

k1 (ξ )s(ξ )e

d ξ da.

Since we are looking for nonzero solutions, Λ = 0 and we can cancel it, this way obtaining the following characteristic equation: 1=

 ∞ 0

k2 (a)

 a 0

 − ξa (λ +μ (σ )+γ (σ ))d σ

k1 (ξ )s(ξ )e

d ξ da

or 1 = G (λ ), where G (λ ) denotes the right-hand side. We define R0 =

 ∞ 0

k2 (a)

 a 0

 − ξa (μ (σ )+γ (σ ))d σ

k1 (ξ )s(ξ )e

d ξ da.

Epidemiologically, R0 is the number of secondary cases produced by one infectious individual in an entirely susceptible population during the lifespan as infectious. Mathematically R0 is a reproduction number if it serves as threshold for the stability of the disease-free equilibrium. The following theorem justifies this: Theorem 2.2. If R0 < 1, then E0 is locally asymptotically stable; if R0 > 1, then E0 is unstable. Proof. Assume first R0 > 1. Then G (λ ) is a decreasing function of λ , if λ is viewed as a real variable. Since G (0) = R0 > 1 and limλ →∞ G (λ ) = 0, then there exists a real λ ∗ > 0 such that G (λ ∗ ) = 1. Hence the characteristic equation has a real positive eigenvalue and E0 is unstable. Assume now R0 < 1. Then for λ with ℜλ ≥ 0 we have: |G (λ )| ≤ G (ℜλ ) ≤ G (0) = R0 < 1. Hence the equation does not have roots with nonnegative real part. Thus, E0 is locally asymptotically stable. Model (2.4) has a unique endemic equilibrium if R0 > 1. To see that we solve system (2.6). From the first equation we have s(a) = b0 e−Λ where Λ = we have

∞ 0

a

0 k1 (σ )d σ

π (a),

k2 (a)i(a)da is an unknown constant. Solving the equation for i(a) i(a) = Λ

 a 0

 − ξa (μ (σ )+γ (σ ))d σ

k1 (ξ )s(ξ )e

dξ .

30

2 Age-Structured Epidemic Models

Replacing i(a) in the formula for Λ and substituting the formula for s(a) we have

Λ = Λ bo

 ∞ 0

k2 (a)π (a)

 a 0

k1 (ξ )e−Λ

ξ 0



a k1 (σ )d σ − ξ γ (σ )d σ e d ξ da.

Canceling Λ , we obtain the following equation for Λ : 1 = bo

 ∞ 0

k2 (a)π (a)

 a 0

k1 (ξ )e−Λ

ξ 0



a k1 (σ )d σ − ξ γ (σ )d σ e d ξ da.

Denote by H (Λ ) the right-hand side of the equation above. Since H (0) = R0 > 1 and limΛ →∞ H (Λ ) = 0 and the function H (Λ ) is decreasing, then the equation H (Λ ) = 1 has a unique positive solution Λ ∗ . With that value we can obtain the components of E ∗ = (s∗ (a), i∗ (a), r∗ (a)).

2.1.3 Backward Bifurcation in Age-Structured Models Backward bifurcation plays an important role in epidemic models. Forward bifurcation with respect to R0 occurs when near the critical value R0 = 1, for R0 < 1 only the disease-free equilibrium exists and it is locally asymptotically stable, while for R0 > 1 besides the disease-free equilibrium, which is unstable, there exists a locally asymptotically stable endemic equilibrium. Backward bifurcation with respect to R0 occurs when near the critical value R0 = 1, for R0 < 1 besides the diseasefree equilibrium, which is locally asymptotically stable, there exists an endemic equilibrium which is unstable, while for R0 = 1 the endemic equilibrium collides with the disease-free equilibrium and disappears while the disease-free equilibrium becomes unstable for R0 > 1. The forward bifurcation and the backward bifurcation are illustrated in Fig. 2.1. Backward bifurcation is important because when it occurs, even if control measures reduce R0 slightly below one, the disease will not disappear by itself, as in the case with forward bifurcation. Additional effort is necessary for the elimination of the disease. Having necessary and sufficient conditions for backward bifurcation I*

I*

0.6

1.4

0.5

1.2 1.0

0.4

0.8

0.3

0.6

0.2

0.4

0.1

0.2 0.2

0.4

0.6

0.8

1.0

1.2

1.4

0

0.2

0.4

Fig. 2.1 Forward bifurcation (left) and backward bifurcation (right)

0.6

0.8

1.0

1.2

1.4

0

2.1 Analysis of Age-Structured Epidemic Models

31

is critical to understand the roots of this phenomenon, and more importantly, what control measures will be most effective to eliminate it. The importance of backward bifurcation to epidemic modeling has led to a number of methods that give necessary and sufficient conditions in ODE models. Most important of those is a result based on the Center Manifold Theory [25] that provides easy criteria to derive conditions for backward bifurcation. Unfortunately, the development of methods for deriving conditions for backward bifurcation in age-structured PDE models has not kept pace with the ODE models. Some of the existing methods work both with ODEs and PDEs after minor modifications. We illustrate the most used one below. This method is based on the observation that if we think of some real valued epidemiological quantity, such as the total number of infectives at endemic equilibrium I ∗ , as a function of R0 , that is I ∗ (R0 ), and we plot the curve in the (R0 , I ∗ ) plane, the bifurcation is forward, if the slope of the tangent to the curve I ∗ (R0 ) at the critical point (1, 0) is positive, that is dI ∗ |R =1,I ∗ =0 > 0 dR0 0 (see Fig. 2.1 (left)). The bifurcation is backward, if the slope of the tangent to the curve I ∗ (R0 ) at the critical point (1, 0) is negative, that is, dI ∗ |R =1,I ∗ =0 < 0 dR0 0 (see Fig. 2.1 (right)). In practice we do not have to consider R0 as a bifurcation parameter. We can consider any of the parameters composing R0 as a bifurcation parameter. Let p be a parameter in R0 and p∗ be the value of p that makes R0 = 1. Then by the chain rule we have ∂ R0 dI ∗ dI ∗ | p=p∗ ,I ∗ =0 = . |R =1,I ∗ =0 dp dR0 0 ∂p Hence, if ∂∂Rp0 > 0, then the bifurcation is forward, if the slope of the tangent to the curve I ∗ (p) at the critical point (p∗ , 0) is positive, that is dI ∗ | p=p∗ ,I ∗ =0 > 0 dp (see Fig. 2.1 (left)). The bifurcation is backward, if the slope of the tangent to the curve I ∗ (p) at the critical point (p∗ , 0) is negative, that is, dI ∗ | p=p∗ ,I ∗ =0 < 0 dp (see Fig. 2.1 (right)). If, however, ∂∂Rp0 < 0, then the bifurcation is forward, if the slope of the tangent to the curve I ∗ (p) at the critical point (p∗ , 0) is negative, that is

32

2 Age-Structured Epidemic Models

dI ∗ | p=p∗ ,I ∗ =0 < 0. dp The bifurcation is backward, if the slope of the tangent to the curve I ∗ (p) at the critical point (p∗ , 0) is positive, that is, dI ∗ | p=p∗ ,I ∗ =0 > 0. dp To simplify things and avoid confusion, we will always take the application of this criterium in the next section.

∂ R0 ∂p

> 0. We illustrate

2.2 An Age-Structured SIR Model with Reinfection Although age-structured SIR models with and without vertical transmission have been previously widely considered, little attention has been paid to the age-structured SIR model with reinfection. SIR models with reinfection are suitable for all diseases, more notably influenza and tuberculosis, where recovered individuals can continue to be infected, at a potentially lower rate. ODE models with reinfection have been found to exhibit backward bifurcation as well as the phenomenon, called “reinfection threshold” [68]. The ODE SIR model with reinfection exhibits backward bifurcation; however, that happens only at the assumption that reinfection occurs at a higher rate than the original infection [26, 54]. We consider an SIR model without vertical transmission, as the presence of vertical transmission complicates the dynamics significantly. With the notations in the previous chapter, we have: ⎧ st + sa = −Λ (a,t)s(a,t) − μ (a)s(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ it + ia = Λ (a,t)s(a,t) + ρΛ (a,t)r(a,t) − (γ (a) + μ (a))i(a,t), ⎪ ⎪ ⎨ rt + ra = γ (a)i(a,t) − ρΛ (a,t)r(a,t) − μ (a)r(a), (2.9) ⎪ ⎪ ∞ ⎪ ⎪ s(0,t) = 0 β (a)p(a,t)da, ⎪ ⎪ ⎪ ⎪ ⎩ i(0,t) = 0, r(0,t) = 0, where ρ is the coefficient of enhancement or reduction at reinfection. This model satisfies the following initial conditions: s(a, 0) = s0 (a)

i(a, 0) = i0 (a)

r(a, 0) = r0 (a),

where we assume (s0 (a), i0 (a), r0 (a)) ∈ (L1 (0, ∞))3 are given functions. We will take the force of infection of the inter-cohort separable form; that is, we assume

Λ (a,t) = k1 (a)

 ∞ 0

k2 (a )i(a ,t)da .

(2.10)

2.2 An Age-Structured SIR Model with Reinfection

33

For the parameter functions, we assume: k1 (a), k2 (a), γ (a), β (a) ∈ L∞ (0, ∞). We further assume that all initial conditions are nonnegative and belong to the class L1 (0, ∞). The function μ (a) may or may not be bounded on (0, ∞). We recall that for model (2.9) we assume that the total population size remains constant; that is, we assume that  ∞ β (a)π (a)da = 1 0

and the total growth rate of the population is zero. Compared to the basic SIR model (2.4), model (2.9) includes reinfection of recovered individuals. Since both susceptible and recovered individuals can get infected (at different rates), then the model has multiple groups of individuals who can be infected and that is a typical mechanism that leads to backward bifurcation. As we see below model (2.9) exhibits backward bifurcation.

2.2.1 Basic Analysis of the SIR Model with Reinfection In this section we illustrate some basic analysis of age-structured models using primarily functional analytic techniques. We begin by applying Theorem 2.1 to show well posedness. We take Y = (L1 (0, ∞))3 . Here X = Y . D = diag(μ (a), γ (a) + μ (a), μ (a)). Furthermore, B(t) = ( 0∞ β (a)p(a,t)da, 0, 0)T . We set u = (s, i, r) and F(u) = (−Λ s, Λ s + ρΛ r, γ i − ρΛ r)T . Since B is linear and β (a) is bounded, it is not hard to see that it is also Lipschitz. To see that F is Lipschitz, we consider u = (s, i, r) and u¯ = (s, ¯ i¯, r¯) in Y+ and ||F(u) − F(u)|| ¯ Y ≤ 2||Λ (i)s − Λ (i¯)s|| ¯ Y + 2ρ ||Λ (i)r − Λ (i¯)¯r||Y + γ¯||i − i¯||Y , where γ¯ = ess supa γ (a). We consider the first term in a more detailed way; the rest are similar. ||Λ (i)s − Λ (i¯)s|| ¯ Y ≤ ||Λ (i)s − Λ (i)s|| ¯ Y + ||Λ (i)s¯ − Λ (i¯)s|| ¯ Y ≤ κ0 ||i||L1 ||s − s|| ¯ L1 + κ0 ||i − i¯||L1 ||s|| ¯ L1 , where κ0 = ess supa k1 (a) ess supa k2 (a). We recall that the growth rate of the population is zero. As a result from (1.15) it follows that ||i||L1 ≤ M||u0 ||Y where u0 = (s0 (a), i0 (a), r0 (a)). Similarly for ||s||L1 and ||r||L1 . Consequently, ||F(u) − F(u)|| ¯ Y ≤ KF ||u − u|| ¯ Y.

34

2 Age-Structured Epidemic Models

Furthermore, F2 = (Λ , 0, ρΛ )T . We have |Λ (i)| ≤ κ0 ||i||L1 ≤ MΛ ||u0 ||Y = K2 . The remaining conditions are trivially satisfied. Hence, by Theorem 2.1, model (2.9) is well posed. We next consider the analysis of equilibria. We expect the presence of backward bifurcation. Backward bifurcation has rarely been found in age-structured models (but see Inaba [93]). Equilibria of model (2.9) satisfy the system of ODEs: ⎧ sa = −Λ k1 (a)s(a) − μ (a)s(a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i = Λ k1 (a)s(a) + ρΛ k1 (a)r(a) − (γ (a) + μ (a))i(a), ⎪ ⎨ a ra = γ (a)i(a) − ρΛ k1 (a)r(a) − μ (a)r(a), (2.11) ⎪ ∞ ⎪ ⎪ ⎪ s(0) = 0 β (a)p∞ (a)da, ⎪ ⎪ ⎪ ⎩ i(0) = 0, r(0) = 0, where

Λ=

 ∞ 0

k2 (a)i(a)da.

Λ is a real number which is unknown as it depends on i(a). To find λ , one usually solves the differential equations and substitutes i(a) in the expression for λ , thus obtaining an equation for λ . Because of the reinfection term, this idea does not work here in a completely straightforward way. Instead of solving each equation separately, we first solve for i(a) + r(a). Adding the second and third equations, we have (2.12) (i(a) + r(a))a = Λ k1 (a)s(a) − μ (a)(i(a) + r(a)). Solving this first order equation with the zero initial condition, we obtain: i(a) + r(a) = Λ

 a 0

k1 (η )s(η )e−

a η

μ (ξ )d ξ

dη .

(2.13)

We note that s(a) is not unknown but it can be obtained from the first equation in system (2.11): a (2.14) s(a) = s(0)e− 0 Λ k1 (ξ )+μ (ξ )d ξ , where s(0) is a given constant. We can express r from (2.13) and substitute in the equation for i. We obtain the following differential equation for i which can be solved in terms of the parameters: ia = Λ k1 (a)s(a) + ρΛ 2 k1 (a)

 a 0

−(ρΛ k1 (a) + γ (a) + μ (a))i(a).

k1 (η )s(η )e−

a η

μ (ξ )d ξ

dη

(2.15)

2.2 An Age-Structured SIR Model with Reinfection

35

Solving this differential equation, we obtain i(a) = Λ

 a

 a

+Λ 2 ρ

0

k1 (τ )s(τ )e−

k1 (τ )e−

0

a τ

a

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ

τ

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ

 τ 0

dτ

k1 (η )s(η )e−

τ η

μ (ξ )d ξ

(2.16)

dη dτ .

Multiplying both sides of the above expression with k2 (a) and integrating, we have the following equation for λ :

Λ =Λ +Λ 2 ρ

 ∞ 0  ∞ 0

k2 (a)

k2 (a)

 a



0 a

0

k1 (τ )s(τ )e−

k1 (τ )e−

a τ

a τ

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ

 τ 0

d τ da

k1 (η )s(η )e−

τ η

μ (ξ )d ξ

d η d τ da.

(2.17) Clearly Λ1 = 0 is one solution to the above equation. That solution corresponds to the disease-free equilibrium E0 whose remaining components are given by: ⎧ a ⎪ s0 (a) = s(a) = s(0)e− 0 μ (ξ )d ξ ⎪ ⎨ (2.18) i(a) = 0 ⎪ ⎪ ⎩ r(a) = 0. The remaining, nonzero solutions of (2.17) are also solutions of the equation 1=

 ∞ 0

k2 (a)

+Λ ρ

 ∞ 0

 a 0

k1 (τ )s(τ )e−

k2 (a)

 a 0

a

k1 (τ )e−

τ

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ

a τ

d τ da

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ

 τ 0

k1 (η )s(η )e−

τ η

μ (ξ )d ξ

d η d τ da. (2.19)

Each nonzero solution of the above equation gives one endemic equilibrium of the SIR model with reinfection. The components of that equilibrium are given by (2.14), (2.16), and (2.13). We denote the right-hand side of the above equation by H (Λ ). We define the reproduction number to be R0 = H (0). Hence, we define the reproduction number as R0 =

 ∞ 0

k2 (a)

 a 0

k1 (τ )s0 (τ )e−

a τ

γ (ξ )+μ (ξ )d ξ

d τ da.

(2.20)

Later we will show that with this definition, R0 plays the usual threshold role. Proposition 2.1. Assume R0 > 1. Then Eq. (2.19) has at least one positive solution. Proof. Since R0 > 1, then H (0) > 1. On the other hand, it is not hard to see that for Λ real and positive limΛ →∞ H (Λ ) = 0. Hence, there is at least one solution to the equation H (Λ ) = 1.

36

2 Age-Structured Epidemic Models

To consider the backward bifurcation, we will take the parameter k2 to be independent of a, that is, a constant, and we will think of λ as a function of k2 . The function Λ (k2 ) exhibits backward bifurcation at the critical value of k2∗ that turns R0 = 1, if and only if dΛ ∗ < 0. | dk2 Λ =0,k2 =k2 We derive conditions on the parameters so that this condition is satisfied. To do that, we rewrite Eq. (2.19) in the form 1 = k2 A0 (λ ) + k2Λ B0 (λ ), where A0 (λ ) and B0 (λ ) denote the integrals A0 (Λ ) = B0 (λ ) = ρ

 ∞ a 0

0

 ∞ a 0

0

k1 (τ )e−

a τ

k1 (τ )s(τ )e−

a τ

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ

 τ 0

d τ da

k1 (η )s(η )e−

τ η

μ (ξ )d ξ

d η d τ da.

Substituting s(a) in A0 (λ ) and B0 (λ ), we obtain A0 (Λ ) =

 ∞ a 0

B0 (Λ ) = ρ

0

k1 (τ )s(0)e−

 ∞ a 0

 τ 0

0

k1 (τ )e−

k1 (η )s(0)e−

τ

a τ

η 0

0



Λ k1 (ξ )+μ (ξ )d ξ − τa ρλ k1 (ξ )+γ (ξ )+μ (ξ )d ξ e d τ da

ρΛ k1 (ξ )+γ (ξ )+μ (ξ )d ξ 

Λ k1 (ξ )+μ (ξ )d ξ − ητ μ (ξ )d ξ e d η d τ da.

We differentiate with respect to k2 . We obtain: 0 = A0 (Λ ) + Λ B0 (Λ ) + k2 (A0 (Λ ) + B0 (Λ ) + Λ B0 (Λ ))

dΛ . dk2

Setting Λ = 0 and k2 = k2∗ , and solving for the derivative, we obtain: dΛ −A0 (0) . |Λ =0,k2 =k2∗ = ∗  dk2 k2 (A0 (0) + B0 (0)) We conclude that the necessary and sufficient condition for backward bifurcation is A0 (0) + B0 (0) > 0, where A0 (Λ ) = s(0)

 ∞

e−

a 0

μ (ξ )d ξ

0

 a 0

e−

τ 0

(2.21)

 τ  a k1 (τ ) − k1 (ξ )d ξ − ρ k1 (ξ )d ξ 0



Λ k1 (ξ )d ξ − τa ρλ k1 (ξ )+γ (ξ )d ξ e d τ da.

τ

2.2 An Age-Structured SIR Model with Reinfection

37

Hence,  τ  ∞   a  a ⎧  − 0a μ (ξ )d ξ ⎪ ⎪ A0 (0) = s(0) e k1 (τ ) − k1 (ξ )d ξ − ρ k1 (ξ )d ξ ⎪ ⎪ 0 0 τ ⎨ 0 − τa γ (ξ )d ξ d τ da, e ⎪  ∞   a  τ ⎪ a ⎪ a ⎪ ⎩ B0 (0) = s(0)ρ e− 0 μ (ξ )d ξ k1 (τ ) k1 (ξ )d ξ e− τ γ (ξ )d ξ d τ da. 0

0

0

(2.22) From these expressions it is clear that backward bifurcation occurs only in ρ > 1. This is consistent with the ODE case. Example 2.1. We would exhibit a specific example so that inequality (2.21) is satisfied and backward bifurcation occurs. Assume μ and γ are constants and

k1 , 0 < a < A, k1 (a) = (2.23) 0, a > A, where A is an arbitrary cut-off value, assumed finite and positive. With these assumptions, A0 (0) and B0 (0) take the form: ⎧  A  a  −μ a ⎪ ⎪ A (0) = s(0) e k1 (−k1 τ − ρ k1 (a − τ ))e−γ (a−τ ) d τ da ⎪ 0 ⎪ 0 0 ⎪ ⎪  ∞  A ⎪ ⎪ ⎪ ⎪ ⎨ +s(0) e−μ a k1 (−k1 τ − ρ k1 (A − τ ))e−γ (a−τ ) d τ da A

0

 A  a ⎪ ⎪ −μ a ⎪ B (0) = s(0) ρ e k1 k1 τ e−γ (a−τ ) d τ da ⎪ 0 ⎪ ⎪ 0 0 ⎪  ∞  a ⎪ ⎪ ⎪ ⎩ +s(0)ρ e−μ a k1 τ e−γ (a−τ ) d τ da. A

(2.24)

0

Using Mathematica, one can compute the following expression for the numerator of A0 (0) + B0 (0): eAμ (γ + μ )(−1 + eAμ − Aμ )

1 − e−A(γ +μ ) μ 2 ρ + eA(γ +μ ) γ (γ + μ + μρ ) − eAγ (γ + μ )(γ + Aγ μ + μρ ) , γ (2.25) where s(0) has been omitted and assumed as one. With γ = 1 and μ = 1/70, the area where this expression is positive is given in Fig. 2.2.

38

2 Age-Structured Epidemic Models

Fig. 2.2 Region of backward bifurcation. One can clearly see that ρ > 1 is necessary for backward bifurcation. The inset shows the backward bifurcation

2.2.2 Stability of the Disease-Free Equilibrium In this section we investigate the local stability of the disease-free equilibrium and we show that R0 as defined by (8.13) is the reproduction number of model (2.9). In particular we have the following classical result. Proposition 2.2. The disease-free equilibrium E0 is locally asymptotically stable if R0 < 1 and unstable if R0 > 1. Proof. As in the ordinary differential equation case we need to linearize the system around the disease-free equilibrium. In age-structured models this is done directly by looking at the perturbations. Let s(a,t) = s0 (a) + x(a,t), i(a,t) = y(a,t), and r(a,t) = z(a,t). Then the linearized system obtained by neglecting the higher order terms takes the form ⎧ xt + xa = −Λ (a,t)s0 (a) − μ (a)x(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ y + ya = Λ (a,t)s0 (a) − (γ (a) + μ (a))y(a,t), ⎪ ⎨ t (2.26) zt + za = γ (a)y(a,t) − μ (a)z(a), ⎪ ⎪ ⎪ ⎪ x(0,t) = 0, ⎪ ⎪ ⎪ ⎩ y(0,t) = 0, z(0,t) = 0, where the force is of infection of the inter-cohort separable form:

Λ (a,t) = k1 (a)

 ∞ 0

k2 (a)y(a,t)da.

(2.27)

The term containing ρ has disappeared because it is a second order term in the perturbations. Looking for exponential solutions, that is x(a,t) = x(a)eλ t , y(a,t) = y(a)eλ t , z(a,t) = z(a)eλ t , we obtain the following linear eigenvalue problem.

2.2 An Age-Structured SIR Model with Reinfection

39

⎧ λ x + xa = −Λ k1 (a)s0 (a) − μ (a)x(a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ λ y + ya = Λ k1 (a)s0 (a) − (γ (a) + μ (a))y(a), ⎪ ⎪ ⎪ ⎪ ⎨ λ z + za = γ (a)y(a) − μ (a)z(a), ⎪ x(0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ y(0) = 0, ⎪ ⎪ ⎪ ⎪ ⎩ z(0) = 0,

(2.28)

where the unknown constant Λ is given by

Λ=

 ∞ 0

k2 (a)y(a)da.

(2.29)

As with the approach to proving existence of equilibria, we solve the differential equations and substitute y(a) in the formula for Λ . Since the equation for y(a) is independent from the remaining equations, we solve only that one and we substitute in Λ :  a a y(a) = Λ k1 (τ )s0 (τ )e− τ λ +γ (ξ )+μ (ξ )d ξ d τ . (2.30) 0

Replacing y(a) in the expression for Λ , canceling Λ since we are looking for a nonzero value, we obtain the following characteristic equation of the disease-free equilibrium: 1=

 ∞ 0

k2 (a)

 a 0

k1 (τ )s0 (τ )e−

a τ

λ +γ (ξ )+μ (ξ )d ξ

d τ da.

(2.31)

Let H (λ ) denote the right-hand side of the characteristic equation. If R0 > 1, then for λ real and positive, we have H (0) = R0 > 1. At the same time lim H (λ ) = 0.

λ →∞

Hence, the characteristic equation has a real, positive root and the disease-free equilibrium is unstable. Now assume that R0 < 1 and consider λ = ξ1 + iξ2 such that ξ1 ≥ 0. In this case: |H (λ )| ≤ ≤

 ∞ 0

 ∞

k2 (a)

0

k2 (a)

 a 0

 a 0

k1 (τ )s0 (τ )|e−λ (a−τ ) |e−

k1 (τ )s0 (τ )e−ξ1 (a−τ ) e−

a τ

a τ

γ (ξ )+μ (ξ )d ξ

γ (ξ )+μ (ξ )d ξ

d τ da

d τ da ≤ H (0) = R0 < 1. (2.32)

Hence, in this case the characteristic equation does not have complex roots with nonnegative real parts. Since H (λ ) is a strictly decreasing function of a real λ , the characteristic equation does not have real positive roots. We conclude that the disease-free equilibrium is locally asymptotically stable.

40

2 Age-Structured Epidemic Models

2.3 Numerical Methods for Age-Structured Epidemic Models Most age-structured models cannot be solved analytically, so we need to solve them numerically and investigate their solutions through simulations. Research on numerical methods for age-structured models has been performed in the past, predominantly on models of population dynamics but also some for epidemic models [4, 5, 89, 138]. A very good introduction to numerical methods for age-structured models is [88]. Mathematically age-structured models are systems of first order integro-differential equations. As with age-structured population models, characteristic lines of the system are lines with slope 45◦ , so it is best if our mesh in the (a,t) plane is square. Let T be the maximal time to which we will compute. Maximal time depends on our interests. Let A be the maximal age to which we will compute. How A do we choose A? A should be a large number, big enough so that e− 0 μ (ξ )d ξ is so small that from the perspective of the computer is equal to zero. We compose a square mesh with step Δ t in the rectangle [0, T ] × [0, A]. We have: tn = nΔ t

a j = jΔ t,

where j = 1, . . . , M and n = 1, . . . , N. For a dependent variable u, subject to the differential equation ut + ua = F(t, u) − G(t, u)u, where F and G are given functions, we discretize the derivative along the characteristic lines using backward Euler method: ut + ua ≈

unj − un−1 j−1

Δt

,

where unj ≈ u(a j ,tn ). We evaluate the right-hand side of the equation at (a j ,tn ). We obtain: unj − un−1 j−1 = F(tn , unj ) − G(tn , unj )unj . Δt This is a nonlinear system and needs Newton’s method to be solved. It is faster and still adequate to “linearize” nonlinear terms by evaluating them at the previous time level: Thus, instead, we want: unj − un−1 j−1

Δt

n−1 n = F(tn , un−1 j ) − G(tn , u j )u j .

We illustrate this methodology on system (2.9). In particular, by discretizing and evaluating at (a j ,tn ), we obtain the following numerical method.

2.3 Numerical Methods for Age-Structured Epidemic Models

41

⎧ n n−1 ⎪ ⎪ s j − s j−1 = −Λ (a ,t )s(a ,t ) − μ (a )s(a ,t ), ⎪ ⎪ j n j n j j n ⎪ Δt ⎪ ⎪ ⎪ n−1 ⎪ ⎪ inj − i j−1 ⎪ ⎪ = Λ (a j ,tn )s(a j ,tn ) + ρΛ (a j ,tn )r(a j ,tn ) − (γ (a j ) + μ (a j ))i(a j ,tn ), ⎪ ⎨ Δt n−1 n r j − r j−1 ⎪ ⎪ = γ (a j )i(a j ,tn ) − ρΛ (a j ,tn )r(a j ,tn ) − μ (a j )r(a j ,tn ), ⎪ ⎪ Δt ⎪  ⎪ ∞ ⎪ ⎪ ⎪ ⎪ s(0,t ) = β (a)p(a,tn )da, n ⎪ ⎪ 0 ⎪ ⎩ i(0,tn ) = 0, r(0,tn ) = 0. (2.33) This system is highly nonlinear and difficult to solve. To simplify things, we “linearize” the system by evaluating some terms not at (a j ,tn ) but at the previous time level. We replace the integral with right endpoint sum. We obtain the system ⎧ n n−1 ⎪ ⎪ s j − s j−1 = −λ n−1 sn − μ sn , ⎪ ⎪ j j j j ⎪ Δt ⎪ ⎪ ⎪ n−1 ⎪ inj − i j−1 ⎪ ⎪ ⎪ = Λ jn−1 snj + ρΛ jn−1 rnj − (γ j + μ j )inj , ⎪ ⎪ ⎨ Δt (2.34) rnj − rn−1 j−1 ⎪ = γ j inj − ρΛ jn−1 rnj − μ j rnj , ⎪ ⎪ ⎪ Δt ⎪ ⎪ M ⎪ ⎪ ⎪ n ⎪ s = β j pnj Δ t, ⎪ ∑ 0 ⎪ ⎪ 1 ⎪ ⎩ n r0n = 0. i0 = 0, This system is very easy to solve. This could be done by hand. Solving the first equation for snj , the third for rnj , and substituting rnj in the second equation and solving the second equation for inj we obtain: ⎧ sn−1 ⎪ j−1 n ⎪ ⎪ sj = , ⎪ n−1 ⎪ 1 + Λ j Δ t + μ jΔ t ⎪ ⎪ ⎪ ⎪ n−1 ⎪ in−1 Δ tsnj ρΛ jn−1 Δ trn−1 ⎪ j−1 + Λ j j−1 n ⎪ ⎪ i , + = ⎪ j ⎪ D D(1 + ρΛ n−1 Δ t + μ j Δ t) ⎨

γ j Δ tinj + rn−1 j−1

j

⎪ ⎪ rnj = , ⎪ ⎪ 1 + ρΛ jn−1 Δ t + μ j Δ t ⎪ ⎪ ⎪ ⎪ M ⎪ ⎪ n ⎪ s = β j pnj Δ t, ⎪ ∑ 0 ⎪ ⎪ ⎪ 1 ⎩ n r0n = 0, i0 = 0, where D = 1 + (γ j + μ j )Δ t − ρΛ jn−1 Δ t

γ jΔ t . n−1 1 + ρΛ j Δ t + μ j Δ t

Computation by these formulas is easy. For j = 1, . . . , M, we compute snj , inj and rnj for level n from level n − 1. Then we compute sn0 , in0 , r0n . Besides the easiness

42

2 Age-Structured Epidemic Models

of the computation, this numerical scheme has several very strong features. First, it is not hard to see that D > 0 and the solutions remain positive for any time step. This property of the numerical scheme is very important in biological applications. Second, it could be shown that the scheme is convergent O(Δ t) for any size of the step. Third, if one does not need to plot the surfaces, one can only keep time level n − 1 and n which will allow to compute to larger values of T .

2.4 Optimal Control of Age-Structured Models In epidemic models we are often interested in the impact of control measures on the spread of a disease. Common control measures include vaccination, treatment, isolation, quarantine, and others. Control measures are explicitly incorporated in the model and to each control measure there corresponds a rate which measures the rate of vaccination, treatment, isolation, and others. In age-structured models these rates are typically age dependent. To illustrate, we introduce a simple SIR age-structured model with vaccination. We will assume that vaccinated individuals, upon vaccination, move into the recovered class. Such a model will be adequate for childhood diseases. This model is similar to model (2.9) but to simplify matters, we omit the reinfection (ρ = 0). The age-structured model with vaccination takes the form. ⎧ st + sa = −Λˆ (a,t)s(a,t) − (μ (a) + ψ (a))s(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ it + ia = Λˆ (a,t)s(a,t) − (γ (a) + μ (a))i(a,t), ⎪ ⎪ ⎨ rt + ra = ψ (a)s(a,t) + γ (a)i(a,t) − μ (a)r(a), (2.35) ⎪ ⎪  ⎪ ⎪ ⎪ s(0,t) = 0A β (a)p(a,t)da, ⎪ ⎪ ⎪ ⎩ i(0,t) = 0, r(0,t) = 0, where ψ (a) is the vaccination rate, and A is the maximal age, which at this point may be infinite. As before, will take the force of infection of the inter-cohort separable form; that is, we assume

Λˆ (a,t) = k1 (a) Further, we denote by

Λ (t) =

 A 0

 A 0

k2 (x)i(x,t)dx.

k2 (x)i(x,t)dx.

(2.36)

(2.37)

The total population again satisfies the linear McKendrick-von Foerster age-structured model and we assume that the growth rate is zero, that is  A 0

β (a)e−

a 0

μ (σ )d σ

da = 1.

2.4 Optimal Control of Age-Structured Models

43

We can investigate the impact on the solutions of ψ (a) through simulations, or we can search for a ψ (a), possibly also dependent on time, such that the solutions corresponding to that ψ minimize some critical epidemiological quantities such as total prevalence and/or the total number of vaccines used. In this case the resulting problem is an optimal control problem.

2.4.1 The Optimal Control Problem Optimal control problems are often formulated for ODE models to investigate optimal control strategies and their impact on the epidemic. Ideally one wants to be able to do that in the context of age-structured and age-since-infection structured models. Optimal control problems for age-structured models are distinct from both the ODE and the diffusion PDE models, as the underlying space for age-structured models is L1 which is not Hilbert. In this section we illustrate on a specific example how to obtain the equations for the adjoint variables. The system of the state variables and the adjoint variables can then be solved numerically simultaneously to give the optimal control. In investigation of the optimal control, an important problem is the existence and uniqueness of the control. However, this is rather technical, and we leave it out of the discussion. Instead we present simulations and code that can help with simulations. We will recast model (2.35) as an optimal control problem replacing ψ (a) with the control variable u1 (a,t) and adding a control variable u2 (a,t) to represent treatment. With these two control variables the model becomes: ⎧ st + sa = −k1 (a)Λ (t)s(a,t) − (μ (a) + u1 (a,t))s(a,t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ it + ia = k1 (a)Λ (t)s(a,t) − (γ (a) + u2 (a,t) + μ (a))i(a,t) ⎪ ⎪ ⎨ rt + ra = u1 (a,t)s(a,t) + (γ (a) + u2 (a,t))i(a,t) − μ (a)r(a) (2.38) ⎪  A ⎪ ⎪ ⎪ ⎪ s(0,t) = β (a)p(a,t)da ⎪ ⎪ ⎪ 0 ⎪ ⎩ i(0,t) = 0, r(0,t) = 0. The problem becomes: Find u∗1 , u∗2 ∈ U where U = {(u1 , u2 ) ∈ (L∞ ((0, A)×(0, T )), L∞ ((0, A)×(0, T ))|0 ≤ u1 ≤ U1 , 0 ≤ u2 ≤ U2 }, and U1 and U2 are positive constants so that J (u∗1 , u∗2 ) = min J (u1 , u2 ) u1 ,u2 ∈U

and J (u1 , u2 ) = +

 T A

 0 1 T A

2

0

0

0

A0 i(a,t) + A1 u1 (a,t)s(a,t) + A2 u2 (a,t)i(a,t)dadt

(B1 u21 (a,t) + B2 u22 (a,t))dadt.

(2.39)

44

2 Age-Structured Epidemic Models

The constants A0 , A1 , A2 are positive weights and B1 , B2 are the costs of the controls. T < ∞ is the final time of application of control, usually assumed small enough so that existence and uniqueness of the optimal control can be established. Age-structured optimal control epidemic problems are considered rarely in the literature (but see [59]). The methodology we use here was proposed by Barbu and Iannelli [11] and then further developed by Fister and Lenhart [57, 58]. See also Numfor et al. for detailed analysis [142]. The methodology for age-structured models is quite different than the one for ODEs and diffusion models which can be found in [110]. Instead of composing a Hamiltonian and looking at its derivatives to obtain the adjoint system as in the ODE case, in PDE models, the adjoint system is derived from the so-called sensitivities, which are derivatives of the state variables with respect to a small parameter giving a perturbation of the control variables. The existence of these derivatives has to be rigorously established; however, here we will not be concerned with the technical details, but only with the steps to produce the adjoined system and solve the problem for the optimal control numerically.

2.4.2 Deriving the Adjoint System and the Characterization of the Control In order to characterize the optimal control pair, we need to differentiate the objective functional with respect to the controls. Since the objective functional depends on the state variables, and the state variables depend on the controls, we need to differentiate the state variables with respect to the controls. Since the controls are functions of both a and t, we take Gateaux derivatives. To obtain the derivative with respect to u1 and u2 , we replace them with u1 (a,t) + ε l1 (a,t) and u2 (a,t) + ε l2 (a,t) where ε is a scalar and l1 (a,t) and l2 (a,t) are arbitrary functions (unknown). With these substitutions system (2.38) becomes: ⎧ st + sa = −k1 (a)Λ (t)s(a,t) − (μ (a) + u1 (a,t) + ε l1 (a,t))s(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ it + ia = k1 (a)Λ (t)s(a,t) − (γ (a) + u2 (a,t) + ε l2 (a,t) + μ (a))i(a,t), ⎪ ⎪ ⎪ ⎨ rt + ra = (u1 (a,t)+ε l1 (a,t))s(a,t)+(γ (a)+u2 (a,t)+ε l2 (a,t))i(a,t) − μ (a)r(a), ⎪  A ⎪ ⎪ ⎪ ⎪ s(0,t) = β (a)p(a,t)da, ⎪ ⎪ ⎪ 0 ⎪ ⎩ i(0,t) = 0, r(0,t) = 0. (2.40) The solutions of system (2.40) are functions of ε . We denote the derivatives of s, i, and r with respect to ε , evaluated at ε = 0 by ψ (a,t), φ (a,t), and θ (a,t) respectively. These are the sensitivities. Differentiating formally system (2.40) with respect to ε and setting ε = 0 we get the system for the sensitivities.

2.4 Optimal Control of Age-Structured Models

⎧  A ⎪ ⎪ ψ + ψ = −k (a) Λ (t) ψ (a,t) − k (a) k2 (x)φ (x,t)dxs(a,t), ⎪ t a 1 1 ⎪ ⎪ 0 ⎪ ⎪ ⎪ −(μ (a) + u1 (a,t))ψ (a,t) − l1 (a,t)s(a,t) ⎪ ⎪ ⎪ ⎪  A ⎪ ⎪ ⎪ ⎪ k2 (x)φ (x,t)dxs(a,t), ⎪ φt + φa = k1 (a)Λ (t)ψ (a,t) + k1 (a) ⎪ ⎪ 0 ⎪ ⎪ ⎪ −(γ (a) + u2 (a,t) + μ (a))φ (a,t) − l2 (a,t)i(a,t), ⎪ ⎨ θt + θa = u1 (a,t)ψ (a,t) + l1 (a,t)s(a,t) + (γ (a) + u2 (a,t))φ (a,t), ⎪ ⎪ ⎪ ⎪ ⎪ +l2 (a,t)i(a,t) − μ (a)θ (a,t), ⎪ ⎪ ⎪ ⎪  ⎪ A ⎪ ⎪ ⎪ ψ (0,t) = β (a)(ψ (a,t) + φ (a,t) + θ (a,t))da, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ φ (0,t) = 0, θ (0,t) = 0, ⎪ ⎪ ⎪ ⎪ ⎩ ψ (a, 0) = φ (a, 0) = θ (a, 0) = 0.

45

(2.41)

To derive the adjoint equations we write the system of sensitivities above in vector form. Define ⎛ ⎞ ⎞ ⎛ ⎞ ⎛ ψ ψ Lψ (2.42) L ⎝ φ ⎠ = ⎝ Lφ ⎠ + M ⎝ φ ⎠ Lθ θ θ, d where L = da + dtd and M is defined as follows: ⎛ ⎞ ψ M⎝ φ ⎠ θ ⎛ ⎞  A k (a) Λ (t) ψ (a,t)+k (a) k (x) φ (x,t)dx+( μ (a)+u (a,t)) ψ (a,t) 1 1 2 1 ⎜ ⎟  A0 ⎜ ⎟ =⎜ ⎟. k2 (x)φ (x,t)dx+(γ (a)+u2 (a,t)+ μ (a))φ (a,t)⎠ ⎝−k1 (a)Λ (t)ψ (a,t)−k1 (a) 0

−u1 (a,t)ψ (a,t)−(γ (a)+u2 (a,t))φ (a,t)+ μ (a)θ (a,t). (2.43) Then the system for the sensitivities takes the form: ⎛ ⎞ ⎛ ⎞ ψ −l1 (a,t)s(a,t) ⎜ ⎟ ⎜ ⎟ ⎟ ⎜ ⎟. −l2 (a,t)i(a,t) L⎜ ⎝φ ⎠=⎝ ⎠ θ l1 (a,t)s(a,t) + l2 (a,t)i(a,t)

(2.44)

Let q, ξ , η be the adjoint variables corresponding to s, i, r respectively. To derive the adjoint equations, we formally need to multiply the first equation in the system for sensitivities (2.41) with q, the second equation with ξ , and the third equation with η and integrate with respect to a and t. Then, we need to integrate by parts the terms with the derivatives, so that the derivatives are transferred onto the adjoint variables. In the remaining integrals we need to change the order of integration so that the sensitivity variables are in the out-most integral. All terms multiplied by ψ

46

2 Age-Structured Epidemic Models

give the equation for q; similarly, all terms multiplied by φ give the equation for ξ and finally all terms multiplied by θ give the equation for η . We demonstrate the procedure for the first equation. We consider: ⎞ ⎛ ⎞ ⎛  T A  T A ψ ψa + ψt (q, ξ , η )L ⎝ φ ⎠ dadt = (q, ξ , η ) ⎝ φa + φt ⎠ dadt 0 0 0 0 θa + θt θ ⎛ ⎞  T A ψ (2.45) (q, ξ , η )M ⎝ φ ⎠ dadt. + 0 0 θ We begin by considering  T A 0

0

 T

q(ψa +ψt )dadt = −

0

 T A

q(0,t)ψ (0,t)dt−

0

0

 T A

qa ψ dadt−

0

0

qt ψ dadt.

(2.46) Here a number of terms are zero because we assume q(A,t) = q(a, T ) = 0 and we also have ψ (a, 0) = 0. We consider more closely the first term in the right-hand side above. We have  T

 A

q(0,t) 0

0

β (a)(ψ + φ + θ )dadt =

 T A 0

0

β (a)q(0,t)(ψ + φ + θ )dadt. (2.47)

Thus, ⎞ ⎞ ⎛ ⎛ ψa + ψt −(qa + qt )− β (a)q(0,t)   T A ⎟ ⎟ ⎜ ⎜ −(ξa + ξt )− β (a)q(0,t) ⎟ φa + φt ⎟ dadt = (q, ξ , η )⎜ (ψ , φ , θ )⎜ ⎠ ⎠dadt. ⎝ ⎝ 0 0 0 0 −(θa + θt )− β (a)q(0,t) θa + θt (2.48) Next, we consider the second term in (2.45). We illustrate again with the first equation for the sensitivities.  T A

 T A 0

0

qk1 (a)Λ (t)ψ (a,t) + (μ (a) + u1 (a,t))ψ (a,t)q

+k1 (a)q(a,t)s(a,t)

=

 T A

0

0

k2 (σ )φ (σ ,t)d σ dadt

qk1 (a)Λ (t)ψ (a,t) + (μ (a) + u1 (a,t))ψ (a,t)qdadt

0 0 T A

+

A

0

k2 (σ )φ (σ ,t)

 A 0

(2.49)

k1 (a)q(a,t)s(a,t)dad σ dt,

where in the triple integral we have changed the order of integration. Since the sensitivity involved in this integral is φ , the expression will go into the equation for ξ . The right-hand side of the adjoint equations is given by the derivative of the objective functional. We write the objective functional in terms of ε

2.4 Optimal Control of Age-Structured Models

J (u1 , u2 , ε ) =

 T A 0

0

47

[A0 i(a,t) + A1 (u1 (a,t) + ε l1 (a,t))s(a,t) + A2 (u2 (a,t) 



1 T A (B1 (u1 + ε l1 (a,t))2 (a,t) 2 0 0 +B2 (u2 + ε l2 (a,t))2 (a,t))dadt. (2.50) Differentiating the objective functional with respect to ε and setting ε = 0 we obtain +ε l2 (a,t))i(a,t)]dadt +

dJ |ε =0 = dε

 T A 0

+

A0 φ (a,t) + A1 u1 (a,t)ψ (a,t) + A2 u2 (a,t)φ (a,t)dadt

0 T  A 0

0

A1 l1 (a,t)s(a,t) + A2 l2 (a,t)i(a,t)

(2.51)

+B1 u1 (a,t)l1 (a,t) + B2 u2 (a,t)l2 (a,t)dadt. Thus, in vector form, the adjoint system has the form ⎞ ⎛ ⎞ ⎛ A1 u1 (a,t) q L ∗ ⎝ ξ ⎠ = ⎝ A0 + A2 u2 (a,t) ⎠ . η 0

(2.52)

In expanded form the system for the adjoint variables becomes: ⎧ −(qa + qt ) = β (a)q(0,t) − k1 (a)Λ (t)q(a,t) − (μ (a) + u1 (a,t))q(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +k1 (a)Λ (t)ξ (a,t) + u1 (a,t)η (a,t) + A1 u1 (a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −(ξa + ξt ) = β (a)q(0,t) − (μ (a) + γ (a) + u2 (a,t))ξ (a,t), ⎪ ⎪ ⎪ ⎪  A ⎪ ⎪ ⎨ −k2 (a) k1 (x)(q(x,t) + ξ (x,t))s(x,t)dx + (γ (a) + u2 (a,t))η (a,t), 0 ⎪ ⎪ +A0 + A2 u2 (a,t) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −(ηa + ηt ) = β (a)q(0,t) − μ (a)η (a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ q(A,t) = ξ (A,t) = η (A,t) = 0, ⎪ ⎪ ⎪ ⎪ ⎩ q(a, T ) = ξ (a, T ) = η (a, T ) = 0. (2.53) The last step is to characterize the controls. This is done from the derivative of J with respect to ε . We have to “transfer” the first part of the double integral from dependent on sensitivities to dependent on adjoint functions. We write that first part in vector form. We obtain ⎞ ⎛  T A A1 u1 (a,t) dJ = (ψ , φ , θ ) ⎝ A0 + A2 u2 (a,t) ⎠ dadt + other terms dε 0 0 ⎛ ⎞0 (2.54)  T A ψ = (q, ξ , η )L ⎝ φ ⎠ dadt + other terms. 0 0 θ

48

2 Age-Structured Epidemic Models

Thus, we obtain ⎛ dJ = dε =

 T A 0

0

 T A 0

0

−l1 (a,t)s(a,t)

⎜ (q, ξ , η ) ⎜ ⎝

−l2 (a,t)i(a,t)

⎞ ⎟ ⎟ dadt + other terms ⎠

l1 (a,t)s(a,t) + l2 (a,t)i(a,t) −l1 (a,t)s(a,t)q(a,t) − l2 (a,t)i(a,t)ξ (a,t)

+(l1 (a,t)s(a,t) + l2 (a,t)i(a,t))η dadt + other terms = +

 T A 0  T

0  A

0

0

l1 (a,t)(s(a,t)(η − q + A1 ) + B1 u1 (a,t))dadt l2 (a,t)(i(a,t)(η − ξ + A2 ) + B2 u2 (a,t))dadt.

(2.55) From here we obtain the following characterization of the optimal control. Set s(a,t)(q(a,t) − η (a,t) − A1 ) , B1 i(a,t)(ξ (a,t) − η (a,t) − A2 ) . u¯2 (a,t) = B2 u¯1 (a,t) =

(2.56)

Hence, u∗1 (a,t) = min {max {0, u¯1 } ,U1 } ,

u∗2 (a,t) = min {max {0, u¯2 } ,U2 } , (2.57)

where U1 and U2 are the upper bounds for the controls. The next step is to solve numerically the state system with the adjoint system (2.45) and the optimal control above. Next section gives a generalization of the forward-backward sweep for agestructured PDEs.

2.4.3 Numerical Methods For ODE models, Lenhart and Workman [110] propose a forward-backward sweep method which consists of application of a fourth order Runge-Kutta method for ODEs for the forward systems and then for the adjoint system. The entire method is an iterative procedure, which starts with zero controls, solves the forward and the backward systems, then updates the controls and repeats the procedure until relative errors become small. Although application of fourth order Runge-Kutta methods to age-structured PDEs is possible to achieve higher convergence rate [138], to simplify matters we use here a first order method along the characteristic lines, the same as the one used in the previous section.

2.4 Optimal Control of Age-Structured Models

49

For the forward sweep, we use the same discretization method as in the previous section. We compose a square mesh with step Δ t in the rectangle [0, T ] × [0, A]. We have: a j = jΔ t, tn = nΔ t where j = 1, . . . , M and n = 1, . . . , N. We discretize the derivative along the characteristic lines using backward Euler method: st + sa ≈

snj − sn−1 j−1

Δt

.

We evaluate the right-hand side of the equation at (a j ,tn ). We replace the integral with right endpoint sum. We obtain the system ⎧ n n−1 s j −s j−1 ⎪ ⎪ = −k1 j Λ n−1 snj − (μ j + un1, j )snj , ⎪ Δt ⎪ ⎪ ⎪ ⎪ inj −in−1 ⎪ j−1 ⎪ ⎪ = k1 j Λ n−1 snj − (γ j + un2, j + μ j )inj , ⎪ Δt ⎪ ⎨ rnj −rn−1 j−1 (2.58) = un1, j snj + (γ j + un2, j )inj − μ j rnj , ⎪ Δt ⎪ ⎪ ⎪ ⎪ M ⎪ ⎪ n n n n ⎪ ⎪ s0 = ∑ βk (sk + ik + rk )Δ t, ⎪ ⎪ ⎪ ⎩ n k=1 i0 = 0, r0n = 0. We also initialize time level n = 0 from the given initial conditions. For the backward sweep we move backwards for j = M − 1, . . . , 0 and n = N − 1, . . . , 0. The “current” time level is n + 1, the “next” time level is “n.” We discretize the adjoint system, evaluating the right-hand side at the next level n. However, as before, in order to avoid a highly nonlinear system, we need to “linearize” some quantities at level n + 1. We obtain the following “linearized” system: ⎧ n+1 q − qnj ⎪ ⎪ n+1 n ⎪− j+1 =β j qn+1 q j −(μ j + un1, j )qnj +k1 j Λ n+1 ξ jn+1 +un1, j η nj +A1 un1, j , ⎪ 0 −k1 j Λ ⎪ ⎪ Δ t ⎪ n+1 ⎪ ⎪ M ⎪ ξ j+1 − ξ jn ⎪ n+1 n n ⎪ − = β q − ( μ + γ + u ) ξ − k + ξln+1 )snl Δ t ⎪ j j j 2 j ∑ k1l (qn+1 2, j j 0 l ⎪ ⎪ Δt ⎪ l=1 ⎨ +(γ j + un2, j )η nj + A0 + A2 un2, j , ⎪ ⎪ n ⎪ η n+1 ⎪ j+1 − η j ⎪ ⎪ − = β j qn+1 − μ j η nj , ⎪ 0 ⎪ Δt ⎪ ⎪ ⎪ n = 0, ⎪ qnM = ξMn = ηM n = N, . . . , 0, ⎪ ⎪ ⎪ ⎪ ⎩qN = ξ N = η N = 0, j = M, . . . , 0. j j j (2.59) As before the forward and the backward systems can be solved. Thus, we compute the forward system by

50

2 Age-Structured Epidemic Models

⎧ sn−1 ⎪ j−1 ⎪ n ⎪ s , = ⎪ j n−1 ⎪ 1 + (k1, j Λ + μ j + un1, j )Δ t ⎪ ⎪ ⎪ ⎪ n−1 sn Δ t ⎪ in−1 ⎪ j j−1 + k1, j Λ n ⎪ ⎪ i , = ⎪ j ⎪ 1 + (μ j + γ j + un2, j )Δ t ⎨ n n n n rn−1 j−1 + u1, j s j Δ t + (γ j + u2, j )i j Δ t n ⎪ ⎪ r , = j ⎪ ⎪ 1 + μ jΔ t ⎪ ⎪ ⎪ M ⎪ ⎪ n ⎪ ⎪ s = βk (snk + ink + rkn )Δ t, ⎪ ∑ 0 ⎪ ⎪ ⎪ ⎩ n k=1 r0n = 0. i0 = 0,

(2.60)

We can solve also the backward sweep equations. There we start the computing from η as the simplest equation, then we compute q, and finally ξ . ⎧ n+1 η n+1 ⎪ j+1 + β j q0 Δ t n ⎪ ⎪ , η = j ⎪ ⎪ 1 + μ jΔ t ⎪ ⎪ ⎪ n+1 ⎪ + k1 j Λ n+1 ξ jn+1 + un1, j η nj + A1 un1, j )Δ t qn+1 ⎪ ⎪ j+1 + (β j q0 n ⎪ qj = , ⎪ ⎪ ⎪ 1 + (k1 j Λ n+1 + un1, j + μ j )Δ t ⎪ ⎪ ⎪ M ⎨ I n+1 = k2, j ∑ k1l (qn+1 + ξln+1 )sn+1 Δ t, j l l ⎪ l=1 ⎪ ⎪ n+1 ⎪ ⎪ ξ j+1 + (β j qn+1 − I n+1 + (γ j + un2, j )η nj + A0 + A2 un2, j )Δ t ⎪ j 0 n ⎪ , ξ = ⎪ j ⎪ ⎪ 1 + (γ j + un2, j + μ j )Δ t ⎪ ⎪ ⎪ qn = ξ n = η n = 0, ⎪ n = N, . . . , 0, ⎪ M M M ⎪ ⎪ ⎪ ⎩ N j = M, . . . , 0. q j = ξ jN = η Nj = 0,

(2.61)

The iteration starts with controls u1 = 0 and u2 = 0. Then, after each forward and backward sweep, the controls are updated by the formula: u¯n1, j =

snj (qnj −η nj −A1 ) , B1

u¯n2, j =

inj (ξ jn −η nj −A2 ) . B2

(2.62)

Finally, we set the new values of the controls by n u∗n 1, j = min{max{u¯1, j , 0},U1 },

n u∗n 2, j = min{max{u¯2, j , 0},U2 }.

(2.63)

Then, we repeat the forward-backward sweep.

2.4.4 Numerical Simulations We write a MATLAB code to implement the numerical method. The MATLAB code is provided in the appendix. Simulations suggest that vaccination has to be applied early in the time period, while treatment has to be applied late in the time period. Thus the two control measures complement each other. For low limits on the controls vaccination has to be applied throughout the entire duration with boost at the beginning of the period, see Fig. 2.3.

2.4 Optimal Control of Age-Structured Models

51 10

Treatment

Vaccination

3

2

1

0

5

0

0

5

10

0

1 with control without control

0.8 0.6 0.4 0.2 0

5

Time

5

10

Time Infected Individuals

Susceptible Individuals

Time

10

15

with control without control

10

5 0

5

10

Time

Fig. 2.3 First row: control profiles. Left: vaccination. Right: treatment. Second row: Solution numbers: Left: susceptibles. Right: infectives. Third row: age and time distribution of controls. Left: vaccination. Right: treatment. Upper limit for both controls is 1

Treatment is applied at the end of time interval, see Fig. 2.3. With small limit on the controls, which is what is physically reasonable, the number of susceptibles in time is slightly decreased for most of the duration, see Fig. 2.3. The number of infected individuals is increased for most of the duration, except for the end of the time interval, see Fig. 2.3. The age-time structure of the vaccination control is concentrated along the a-axis, while age-time structure of the treatment control is concentrated along the t-axis. As the bound of the control increases, the integral of the vaccination control over all ages is applied for a shorter and shorter period. For unrealistically large bounds of the control, vaccination is applied for a very

52

2 Age-Structured Epidemic Models 105

0.8

Treatment

Vaccination

10

5

0 0

5

0.6 0.4 0.2 0

10

0

5

10

Time

4

15 with control without control

3

Infected Individuals

Susceptible Individuals

Time

2 1 0 0

5

Time

10

with control without control 10

5

0 0

5

10

Time

Fig. 2.4 First row: control profiles. Left: vaccination. Right: treatment. Second row: Solution numbers: Left: susceptibles. Right: infectives. Third row: age and time distribution of controls. Left: vaccination. Right: treatment. Upper limit for both controls is 10,000

short duration at very high level and then steeply drops, see Fig. 2.4. The integral of the treatment control in age continues to be supported primarily at the end of the time interval (Fig. 2.4). With very high bounds, the number of susceptibles decreases substantially, see Fig. 2.4 as more of them are now vaccinated. Controls with high bounds are now able to control and reduce the number of infected individuals (Fig. 2.4). The age-structured profile of the vaccination control is concentrated again along the axes, more so along the t-axis and to a lesser extent along the a-axis. The treatment control has a complex age-profile but it is again concentrated near the end of the time period.

2.5 Two-Strain Age-Structured Epidemic Models

53

As the upper bound increases, the vaccination profile for a given fixed t seems to approximate a Dirac delta function, concentrated at zero. This is not accidental or parameter dependent. In a sequence of papers J. Müller [79] showed that in a time independent situation, the optimal age profile, while minimizing the cost of vaccination, is a delta function, concentrated at one age A. Thus in a static situation the optimal age-dependent vaccination strategy is vaccinate at one age. Our simulations here show that this result extends to the dynamic situation. Moreover, the age-integral of the vaccination also appears to be a delta function in time, in our case concentrated at zero. We hypothesize that the long-term optimal vaccination strategy is also a Dirac delta function in age, in our case concentrated at zero. Thus the optimal vaccination strategy is to vaccinate newborns and to vaccinate them early in the time period.

2.5 Two-Strain Age-Structured Epidemic Models Pathogen genetic diversity, modeled as multi-strain models, impacts the distribution and control of diseases. Pathogen genetic diversity has been found to change with age. For instance, article [70] reports that Escherichia coli, isolated from 266 individuals, belonged to four groups (A, B1, B2, or D). In males, the probability of isolating A or D strains increased with host age, whilst the probability of detecting a group B2 strain declined. In females, the probability of recovering A or B2 strains increased with increasing host age and the likelihood of isolating B1 or D strains declined. Furthermore, evidence exists that different serotypes of Streptococcus pneumoniae cause infection in children and adults [152]. The fact that different serotypes can colonize different age groups suggests that serotypes may coexist even if no other trade-off mechanisms exist. This property is captured by mathematical models and article [125] suggests that age structure alone can generate coexistence. In this section we study an extension of model (2.35) which includes two strains; that is, we study an age-structured two-strain model with vaccination. With the notations before, we denote by i(a,t) the density of individuals infected with strain one and by j(a,t) the density of individuals infected with strain two. We further assume that the vaccine is perfect with respect to both strains. ⎧ st + sa = −k1 (a)λ1 (t)s(a,t) − k2 (a)λ2 (t)s(a,t) − (μ (a) + ψ (a))s(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ it + ia = k1 (a)λ1 (t)s(a,t) − (γ1 (a) + μ (a))i(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ jt + ja = k2 (a)λ2 (t)s(a,t) − (γ2 (a) + μ (a)) j(a,t), (2.64) ⎪ + r = ψ (a)s(a,t) + γ (a)i(a,t) + γ (a) j(a,t) − μ (a)r(a,t), r t a 1 2 ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎪ β (a)p(a,t)da, s(0,t) = ⎪ ⎪ ⎪ 0 ⎪ ⎩ i(0,t) = 0, j(0,t) = 0, r(0,t) = 0.

54

2 Age-Structured Epidemic Models

As before, we will take the force of infection of the inter-cohort separable form; that is, we assume

λˆ 1 (a,t) = k1 (a)

 ∞ 0

λˆ 2 (a,t) = k2 (a)

ρ1 (x)i(x,t)dx,

 ∞ 0

ρ2 (x) j(x,t)dx, (2.65)

and we will denote by

λ1 (t) =

 ∞ 0

ρ1 (a)i(a,t)da,

λ2 (t) =

 ∞ 0

ρ2 (a) j(a,t)da.

(2.66)

The total population size satisfies the linear age-structured model. We assume again that the growth rate of the total population is zero so that p(a,t) → p∞ (a),

as

t → ∞.

Model (2.64) is a model with perfect vaccination, that is, once vaccinated individuals are fully protected with respect to both strains.

2.5.1 Disease-Free Equilibrium and Reproduction Numbers Equilibria are solutions to the system where the time derivatives are equal to zero: ⎧ sa = −k1 (a)λ1 s(a) − k2 (a)λ2 s(a) − (μ (a) + ψ (a))s(a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ia = k1 (a)λ1 s(a) − (γ1 (a) + μ (a))i(a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ja = k2 (a)λ2 s(a) − (γ2 (a) + μ (a)) j(a), ⎪ ra = ψ (a)s(a) + γ1 (a)i(a) + γ2 (a) j(a) − μ (a)r(a), ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎪ s(0) = β (a)p∞ (a)da, ⎪ ⎪ ⎪ 0 ⎪ ⎩ i(0) = 0, j(0) = 0, r(0) = 0. The system has a unique disease-free equilibrium, which is a solution to the system sa = −(μ (a) + ψ (a))s(a), − μ (a)r(a), ra = ψ (a)s(a)  ∞

s(0) =

0

β (a)p∞ (a)da,

r(0) = 0.

Solving the first equation, we have s0 (a) = s(0)π (a)πψ (a) where

π (a) = e−

a 0

μ (σ )d σ

,

πψ (a) = e−

a 0

ψ (σ )d σ

.

2.5 Two-Strain Age-Structured Epidemic Models

55

Solving the equation for r(a) we have  a

r(a) = s(0)

0

ψ (η )π (η )πψ (η )e−

a η

μ (s)ds

dη .

After some simplification, we obtain r0 (a) = s(0)π (a)(1 − πψ (a)). The disease-free equilibrium is given by E 0 = (s0 (a), 0, 0, r0 (a)). There exists a unique equilibrium corresponding to strain one and strain two E1 = (s∗ (a), i∗ (a), 0, ˆ 0, jˆ(a), rˆ(a)). To show the existence of E1 we consider the r∗ (a)) and E2 = (s(a), system sa = −k1 (a)λ1 s(a) − (μ (a) + ψ (a))s(a), ia = k1 (a)λ1 s(a) − (γ1 (a) + μ (a))i(a), ra = ψ (a)s(a) + γ1 (a)i(a) − μ (a)r(a), s(0) =

 ∞ 0

β (a)p∞ (a)da,

i(0) = 0,

r(0) = 0.

Integrating the equation for s we obtain s∗ (a) = s(0)e− grating the equation for i(a) we obtain i∗ (a) = λ1

 a 0

k1 (η )s(η )e−

a

a

η k1 ( σ ) λ1 d σ

η ( μ (s)+γ1 (s))ds

π (a)πψ (a). Inte-

dη .

To find i∗ (a) we need to find λ1 . Substituting in the expression for λ1 we obtain the following equation for λ1 : 1=

 ∞ 0

ρ1 (a)

 a 0

k1 (η )s∗ (η )e−

a

η ( μ (s)+γ1 (s))ds

d η da.

(2.67)

Since s∗ (a) is a function of λ1 , we consider the right-hand side of the above equation to be a decreasing function of λ1 , denoted by F1 (λ1 ). We define the reproduction number of strain one to be F1 (0); that is, R1 (ψ ) = s(0)

 ∞ 0

ρ1 (a)

 a 0

k1 (η )π (η )πψ (η )e−

a

η ( μ (s)+γ1 (s))ds

d η da.

(2.68)

Clearly, if R1 > 1, then Eq. (2.67) has a unique positive solution λ1∗ and the endemic equilibrium of strain one exists. If R1 < 1, there is no endemic equilibrium corresponding to strain one only. The reproduction number of strain two similarly is defined as R2 (ψ ) = s(0)

 ∞ 0

ρ2 (a)

 a 0

k2 (η )π (η )πψ (η )e−

a

η ( μ (s)+γ2 (s))ds

d η da.

(2.69)

56

2 Age-Structured Epidemic Models

The force of infection of strain two satisfies the equation 1=

 ∞ 0

ρ2 (a)

 a 0

k2 (η )s( ˆ η )e−

a

η ( μ (s)+γ2 (s))ds

d η da,

(2.70)

a

where s(a) ˆ = s(0)e− η k2 (σ )λ2 d σ π (a)πψ (a). Since s(a) ˆ is a function of λ2 , we consider the right-hand side of Eq. (2.70) to be a decreasing function of λ2 , denoted by F2 (λ2 ). We define the reproduction number of strain two to be F2 (0) (see (2.69)). Clearly, if R2 > 1, then Eq. (2.70) has a unique positive solution λ2∗ and the endemic equilibrium of strain two exists. If R2 < 1, there is no endemic equilibrium corresponding to strain two only. To confirm the reproduction numbers and derive the invasion reproduction numbers, we have to look at the stability of these equilibria. Stability of the disease-free equilibrium will give conditions on the reproduction numbers. Stability of equilibrium E1 will give the invasion reproduction number of strain two, and stability of E2 will give the invasion reproduction number of strain one. To consider linear stability, we denote the perturbations of the equilibrium E0 with x(a,t), y(a,t), z(y,t) and w(a,t) respectively. The linearized equations, where we have omitted all quadratic terms in the perturbations, become: ⎧ xt + xa = −k1 (a)λ1y (t)s0 (a) − k2 (a)λ2z (t)s0 (a) − (μ (a) + ψ (a))x(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ yt + ya = k1 (a)λ1y (t)s0 (a) − (γ1 (a) + μ (a))y(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ z 0 ⎪ ⎨ zt + za = k2 (a)λ2 (t)s (a) − (γ2 (a) + μ (a))z(a,t), ⎪ wt + wa = ψ (a)x(a,t) + γ1 (a)y(a,t) + γ2 (a)z(a,t) − μ (a)w(a,t), ⎪ ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎪ x(0,t) = β (a)u(a,t)da, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎩ y(0,t) = 0, z(0,t) = 0, w(0,t) = 0, where u(a,t) = x(a,t) + y(a,t) + z(a,t) + w(a,t) and

λ1y (t) = λ2z (t) =

 ∞ 0

 ∞ 0

ρ1 (a)y(a,t)da, (2.71)

ρ2 (a)z(a,t)da.

rt , y(a,t) = y(a)e rt , z(a,t) = z¯(a)ert , ¯ Looking for exponential solutions x(a,t) = x(a)e ¯ rt , we obtain the following linear eigenvalue problem: and w(a,t) = w(a)e ¯

2.5 Two-Strain Age-Structured Epidemic Models

57

⎧ rx + xa = −k1 (a)λ1y s0 (a) − k2 (a)λ2z s0 (a) − (μ (a) + ψ (a))x(a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ry + ya = k1 (a)λ1y s0 (a) − (γ1 (a) + μ (a))y(a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ rz + za = k2 (a)λ2z s0 (a) − (γ2 (a) + μ (a))z(a), (2.72)

⎪ rw + wa = ψ (a)x(a) + γ1 (a)y(a) + γ2 (a)z(a) − μ (a)w(a), ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎪ β (a)u(a)da, x(0,t) = ⎪ ⎪ ⎪ 0 ⎪ ⎩ y(0,t) = 0, z(0,t) = 0, w(0,t) = 0, where we have omitted the bars, u(a) = x(a) + y(a) + z(a) + w(a), and

λ1y = λ2z =

 ∞  0∞ 0

ρ1 (a)y(a)da, (2.73)

ρ2 (a)z(a)da.

To solve the system (2.72), we notice that the equations for y and z split from the rest of the system. Solving the equation for y (the solution of z is similar) we obtain: y(a)

= λ1y

 a 0

k1 (η )s0 (η )e−r(a−η ) e−

a

η (γ1 (s)+ μ (a))ds

dη .

Substituting in the formula for λ1y and canceling λ1y we obtain the first characteristic equation: 1 = G1 (r) where G1 (r) =

 ∞ 0

ρ1 (a)

 a 0

k1 (η )s0 (η )e−r(a−η ) e−

a

η (γ1 (s)+ μ (a))ds

d η da.

Notice that G1 (0) = R1 . Thus, if R1 > 1, as a decreasing function of the real variable r, G1 (r) has a positive real root and the disease- free equilibrium is unstable. If R1 < 1, then G1 (0) < 1. Consider complex solutions r with nonnegative real part. Then |G1 (r)| ≤ G1 (ℜr) < G1 (0) = R1 < 1. Hence, this characteristic equation does not have complex roots with nonnegative real parts. Similar considerations lead to the second characteristic equation: 1 = G2 (r) where G2 (r) =

 ∞ 0

ρ2 (a)

 a 0

k2 (η )s0 (η )e−r(a−η ) e−

a

η (γ2 (s)+ μ (a))ds

d η da.

We have again that G2 (0) = R2 . Hence, if R2 > 1, then the disease-free equilibrium is unstable. If R1 < 1 and R2 < 1, then neither characteristic equation has roots with nonnegative real parts and the disease-free equilibrium is locally asymptotically stable.

58

2 Age-Structured Epidemic Models

2.5.2 Strain One and Strain Two Equilibria and Invasion Numbers To derive the invasion reproduction numbers we look at the stability of the equilibria E1 and E2 . As with the disease-free equilibrium we look at the perturbations of equilibrium E1 . We denote the perturbations as before. The linearized system for the perturbations takes the form: ⎧ xt + xa = −k1 (a)λ1y (t)s∗ (a) − k1 (a)λ1∗ x(a,t) − k2 (a)λ2z (t)s∗ (a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ −(μ (a) + ψ (a))x(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ yt + ya = k1 (a)λ1y (t)s∗ (a) + k1 (a)λ1∗ x(a,t) − (γ1 (a) + μ (a))y(a,t), ⎪ ⎪ ⎪ ⎨ zt + za = k2 (a)λ2z (t)s∗ (a) − (γ2 (a) + μ (a))z(a,t), ⎪ ⎪ ⎪ ⎪ wt + wa = ψ (a)x(a,t) + γ1 (a)y(a,t) + γ2 (a)z(a,t) − μ (a)w(a,t), ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎪ x(0,t) = β (a)u(a,t)da, ⎪ ⎪ ⎪ 0 ⎪ ⎩ y(0,t) = 0, z(0,t) = 0, w(0,t) = 0, where u(a,t) = x(a,t) + y(a,t) + z(a,t) + w(a,t) and

λ1∗ =

 ∞ 0

λ1y (t) = λ2z (t) =

ρ1 (a)i∗ (a)da,

 ∞  0∞ 0

ρ1 (a)y(a,t)da,

(2.74)

ρ2 (a)z(a,t)da.

rt , y(a,t) = y(a)e rt , z(a,t) = z¯(a)ert , Looking for exponential solutions x(a,t) = x(a)e ¯ ¯ rt , we obtain the following linear eigenvalue problem: and w(a,t) = w(a)e ¯ ⎧ rx + xa = −k1 (a)λ1y s∗ (a) − k1 (a)λ1∗ x(a) − k2 (a)λ2z s∗ (a) − (μ (a) + ψ (a))x(a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ry + ya = k1 (a)λ1y s∗ (a) + k1 (a)λ1∗ x(a,t) − (γ1 (a) + μ (a))y(a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ rz + za = k2 (a)λ2z s∗ (a) − (γ2 (a) + μ (a))z(a),

⎪ rw + wa = ψ (a)x(a) + γ1 (a)y(a) + γ2 (a)z(a) − μ (a)w(a), ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎪ x(0,t) = β (a)u(a)da, ⎪ ⎪ ⎪ 0 ⎪ ⎩ y(0,t) = 0, z(0,t) = 0, w(0,t) = 0. The equation for z splits from the remaining system above. We solve the differential equation for z but the solution contains the unknown λ2z :

2.5 Two-Strain Age-Structured Epidemic Models

z(a) = λ2z

 a 0

59

k2 (η )s∗ (η )e−r(a−η )−

a

η (γ2 (σ )+ μ (σ ))d σ

dη .

(2.75)

Substituting in the formula for λ2z , and canceling λ2z from both sides of the equation, we obtain the following characteristic equation 1 = J2 (r) where J2 (r) =

 ∞ 0

ρ2 (a)

 a 0

k2 (η )s∗ (η )e−r(a−η )−

a

η (γ2 (σ )+ μ (σ ))d σ

d η da.

(2.76)

We define the invasion reproduction number of strain two at the equilibrium of strain one as R21 = J2 (0), that is as R21 =

 ∞ 0

ρ2 (a)

 a 0

k2 (η )s∗ (η )e−

a

η (γ2 (σ )+ μ (σ ))d σ

d η da.

(2.77)

Clearly, if R21 > 1, then the equilibrium E1 is unstable because the characteristic equation has a real positive root. If R21 < 1 similar consideration as before can show that the characteristic equation 1 = J2 (r) does not have roots with nonnegative real parts and the stability of the equilibrium E1 depends on the system ⎧ xt + xa = −k1 (a)λ1y (t)s∗ (a) − k1 (a)λ1∗ x(a,t) − (μ (a) + ψ (a))x(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ y ⎪ ∗ ∗ ⎪ ⎪ yt + ya = k1 (a)λ1 (t)s (a) + k1 (a)λ1 x(a,t) − (γ1 (a) + μ (a))y(a,t), ⎪ ⎪ ⎨ wt + wa = ψ (a)x(a,t) + γ1 (a)y(a,t) − μ (a)w(a,t), ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎪ β (a)u(a,t)da, x(0,t) = ⎪ ⎪ 0 ⎪ ⎪ ⎩ y(0,t) = 0, w(0,t) = 0, which is the system governing the stability of the endemic equilibrium of a strain one model where strain two is not present (single-strain model). This system, however, may not be stable for all parameter values and oscillation may occur. This will compromise the value of the invasion number as a threshold for stability. Regardless, we will consider the invasion number as our measure for stability. Stability of E2 is governed by an invasion reproduction number of strain one at the equilibrium of strain two, similarly defined and derived: R12 =

 ∞ 0

ρ1 (a)

 a 0

k1 (η )s( ˆ η )e−

a

η (γ1 (σ )+ μ (σ ))d σ

d η da.

(2.78)

Often it can be shown that if both invasion reproduction numbers are greater than one, coexistence of the two strains occurs. This is difficult to prove in the general context of model (2.64) but see [125]. Table 2.1 summarizes the competitive outcomes in a two-strain model with or without age structure.

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2 Age-Structured Epidemic Models

Table 2.1 Competitive outcomes in a two-strain model Reproduction numbers

Invasion numbers

Outcome

Interpretation

R1 < 1, R2 < 1 R1 > 1, R2 < 1 R1 < 1, R2 > 1 R1 > 1, R2 > 1 R1 > 1, R2 > 1 R1 > 1, R2 > 1 R1 > 1, R2 > 1

– – – R12 < 1, R21 > 1 R12 > 1, R21 < 1 R12 > 1, R21 > 1 R12 < 1, R21 < 1

I1 → 0, I2 → 0 I1 persists, I2 → 0 I2 persists, I1 → 0 I2 persists, I1 → 0 I1 persists, I2 → 0 I1 persists, I2 persists I1 may persist, I2 may persist

Extinctiona Competitive exclusiona Competitive exclusiona Competitive exclusion Competitive exclusion Coexistence Bistable dominance

a May

not hold if backward bifurcation occurs

Acknowledgements The authors thank Suzanne Lenhart for valuable comments and Eric Numfor and Necibe Tuncer for help with the code.

Appendix In this appendix we include the MATLAB code that executes the numerical method in the section.

% This function computes the optimal control problem % of an eage-structured sir model clc; clear all; close all; format long test = -1; tol = 0.05; % set tolerance for the iteration N = 100; % number of subdevisions in time M = 100; % number of subdevisions in age FT = 10 ;% final time h = FT/N; % step size t = 0:h:FT; % t-variable mesh aa = 0:h:FT; %a-variable mesh s = zeros(length(aa),length(t)); ix = zeros(length(aa),length(t)); r = zeros(length(aa),length(t)); eta = zeros(length(aa),length(t)); q = zeros(length(aa),length(t)); xi = zeros(length(aa),length(t));

% initialization

2.5 Two-Strain Age-Structured Epidemic Models lam=zeros(1,length(t)); IntSS=zeros(1,length(t)); IntI=zeros(1,length(t)); IntR=zeros(1,length(t)); IntU1=zeros(1,length(t)); IntU2=zeros(1,length(t)); IntSS2=zeros(1,length(t)); IntI2=zeros(1,length(t)); IntR2=zeros(1,length(t)); IntB=0; u1 = zeros(length(aa),length(t)); u2 = zeros(length(aa),length(t)); k2 = @(x) 1*x*exp(-0.005*x); % defining parameters k1 = 0.5; beta = 0.5; % As constants beta and mu must be equal mu = 0.5; gamma = 0.1; s0 = @(x) exp(-x); ix0 = @(x) 1.5; %initial conditions for the forward system r0 = @(x) 0.01; A0 = 10; A1 = 0.01; A2 = 0.01; B1 = 0.1; B2 = 0.1; U1max =1; U2max = 1;

while (test 1, the infection equilibrium T∗ =

δw cw βw ρw

Ti∗ =

cw dw (ℜ0 − 1) ρw βw

V∗ =

dw (ℜ0 − 1) βw

is locally and globally stable. To obtain an outbreak model of this type, we may set the recruitment rate and the clearance rate of the healthy cells to zero. The

3.1 Nested Immuno-Epidemiological Modeling

71

outbreak model takes the form: T  (τ ) = −βw TV Ti (τ ) = βw TV − δw Ti V  (τ ) =

(3.4)

ρw Ti − cwV.

• Within-host models that involve pathogen replication cycle and the cellular immune response. Cellular immune response consists of various kinds of T-cells which attack the infected target cells. Infected target cells stimulate the proliferation of the T -cells. A modification of model (3.3) including cellular immune response is given as T  (τ ) = Λw − βw TV − dw T Ti (τ ) = βw TV − δw Ti − ε CTi

(3.5)

V  (τ ) = ρw Ti − cwV C (τ ) = aCTi − mwC,

where mw is the clearance of the cellular immune response and a and ε have the same meaning as in (3.2). This model again models chronic infection; if we want an outbreak model of this kind, we may take mw = 0 or set the recruitment rate and clearance rate equal to zero. Model (3.5) has three equilibria: infection-free equilibrium, immune response-free equilibrium, and coexistence equilibrium. The immune-response-free equilibrium is the same as with model (3.3) as well as the reproduction number ℜ0 . This system, however, has a second reproduction number, an invasion number of the immune response ℜi =

βw ρwΛw a . δw (βw ρw mw + dw acw )

If ℜi < 1, then the immune response does not get activated to persist; if ℜi > 1, then the immune response does get activated and persists with the pathogen. The coexistence equilibrium of the pathogen and the immune response is given by T∗ =

cwΛw a βw ρw mw + dw acw

Ti∗ =

mw a

V∗ =

ρw mw acw

C∗ =

δw (ℜi − 1). ε

• Within-host models that involve the pathogen replication cycle and the humoral immune response, B-cells, and antibodies. The humoral immune system components attack the free pathogen. Thus, a within-host model with humoral immune response takes the form: T  (τ ) = Λw − βw TV − dw T Ti (τ ) = βw TV − δw Ti V  (τ ) = ρw Ti − cwV − ε BV B (τ ) = aBV − mw B.

(3.6)

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3 Nested Immuno-Epidemiological Models

This model again describes the chronic case scenario but the model can be modified as described in the previous part to be an outbreak model. As a chronic model, it again had three equilibria: an infection-free equilibrium, an immuneresponse-free equilibrium, and a coexistence equilibrium. The within-host reproduction number ℜ0 is defined as before as are the infection-free equilibrium and the immune-response-free equilibrium. The coexistence equilibrium and the immune invasion number are different. We define the immune response invasion number as ρw βw Λw a . ℜi = cw δw βw mw + dw a Then the coexistence equilibrium is given by

Λw a Λw a mw βw Ti∗ = βw mw + dw a aδw βw mw + dw a m c w w B∗ = (ℜi − 1). V∗ = a ε T∗ =

(3.7)

• The most complex type of models includes both the humoral and the cellular immune response. Such a model was introduced by Wodarz [181]. This model has five equilibria: an infection-free equilibrium, immune responses-free equilibrium, cellular-immune-response-free equilibrium, humoral immune responsefree equilibrium, and coexistence equilibrium. The cellular and the humoral immune responses “compete” for control over the infection. The different outcomes are controlled by three reproduction numbers. Because of its complexity these types of models are rarely used in immuno-epidemiological modeling. There are various reasons why one within-host model should be preferred over another. One reason is that the within-host model should be properly selected to match the between-host model and the disease being modeled. For instance an immunoepidemiological model of HIV should link a chronic within-host model with an SI-type, no recovery, between-host model. Another important reason that helps select one within-host model over another is the questions that are to be addressed with the immuno-epidemiological model. Typical questions involve the trade-off between within-host and between-host strain competition, the impact of within-host dynamics on the population level prevalence and reproduction numbers, evolution of virulence or coevolution of pathogens and hosts. All within-host models satisfy initial conditions. If fitted to within-host data, the initial conditions for the viral load can be determined from data.

3.1.2 Composing Immuno-Epidemiological Models Immuno-epidemiological models consist of three components: a within-host model, a between-host model, and linking functions. After within-host model is selected, a between-host model needs to be specified. The between-host model is an agesince-infection structured model whose infected class is structured with age-since-

3.1 Nested Immuno-Epidemiological Modeling

73

infection. If the population level model has multiple infected classes (such as exposed, asymptomatic, and symptomatic) and we are interested in the within-host dynamics of each of these classes, they all need to be age-since-infection structured and linked to a generally different within-host model. The immuno-epidemiological models that we discuss in this chapter are of nested type. The first nested immuno-epidemiological model was proposed by Gilchrist and Sasaki [66]. As a starting point we introduce here an immuno-epidemiological model of a chronic disease without recovery, such as HIV. The between-host model is an SI model where the infected class is structured with time since infection. Agesince-infection models are PDE models of mixed type, meaning that some classes are modeled with ODEs and others with PDEs. To introduce the age-since-infection model, let S(t) be the number of susceptibles at time t and i(τ ,t) be the density of infected individuals of time-since-infection τ at time t. The equation of the density of infected individuals is very much like a Kermack-McKendrick equation. The boundary condition, however, is an integral which gives all new cases per unit of time (incidence). The model takes the form:  ⎧ S ∞  ⎪ S = Λ − β (τ )i(τ ,t)d τ − μ S, ⎪ ⎪ ⎪ N 0 ⎪ ⎨ iτ + it = −(α (τ ) + μ )i, ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎩ i(0,t) = S β (τ )i(τ ,t)d τ , N 0

(3.8)

where α (τ ) is the disease-induced mortality, and β (τ ) is the transmission coefficient. The within-host models will give shape of these two functions. Model (3.8) is a standard SI age-since-infection structured model. One can obtain background information about these models from many books [85, 88, 119]. We develop the mathematical techniques to analyze these models in Chap. 8. To become a nested immuno-epidemiological model, we need to link the parameters β (τ ) and α (τ ) in model (3.8) to the within-host system. For the within-host system we may use (3.5) as it has both replication and immune response. The linking between the withinhost and between-host systems can be done in multiple ways. We list some of the possibilities, together with some references in Table 3.1. The choice of the linking functions in each specific case depends on the disease as well as the choice of classes in the two scales. Ideally the linking function should be derived from data as we demonstrate for the transmission coefficient in Sect. 3.4.3. However, in many cases data is not available. Then the linking functions are derived from parsimony and logical considerations. For instance, the simplest linking function for the transmission coefficient is β (τ ) = β0V . However, one can surmise that the probability of transmission that is a part of the linking function is bounded by one, while the contact rate is a constant or a decreasing function of V . Hence, even though very simple the linking function β (τ ) = β0V is not particularly realistic. Other linking functions are given in Table 3.1. Parameters β0 , β1 , K, η are called linking parameters. Models (3.5)–(3.8) together with the choices of β (τ ) and α (τ ) in terms of within-host variables

74

3 Nested Immuno-Epidemiological Models

Table 3.1 Linking functions Case

β (τ )

α (τ )

Reference

1

β0 V β0 V V +K

η aCV

[66]

ηV  1 1 a − η  T T0 a 1 1 − η T T0

[119]

2 3

β0V + β1 Ti∗

4

β0 V z

aT 0

=

[65] [65]

Λw dw

comprise a nested immuno-epidemiological model. This type of nested immunoepidemiological models assume that all individuals exhibit the same within-host dynamics. If one has to have classes of infected individuals exhibiting different within-host dynamics, one has to consider a multi-group model [143]. In the next subsection we introduce an immuno-epidemiological model of a disease with recovery.

3.1.3 An Immuno-Epidemiological Model of Disease with Recovery Immuno-epidemiological model of diseases with recovery differ from the immunoepidemiological model in the previous section in two ways: (1) the within-host model is an outbreak model; (2) the between-host model is an SIR type model. Actually recovery can be incorporated in an immuno-epidemiological model in two ways. The first one was first suggested by Gilchrist and Sasaki [66] and consists of all individual who reach a “recovery time” T moving to the recovered class. Thus recovery occurs at a single time-since-infection. We refer to [66] for further information on modeling recovery through that approach. The second way recovery may be incorporated is through recovery rate which is linked to the within-host dynamical variables. In this case individuals recover more gradually as the pathogen gets cleared. For the within-host model, we use an outbreak modification of model (3.6). For instance, the within-host model can be given by: T  (τ ) = −βw TV, Ti (τ ) = βw TV − δw Ti , V  (τ ) = ρw Ti − cwV − ε BV,

(3.9)

B (τ ) = aBV. It is easy to see that the only equilibrium of this model is (T ∗ , 0, 0, B∗ ), where T ∗ and B∗ are nonzero quantities, which may differ depending on the initial conditions.

3.1 Nested Immuno-Epidemiological Modeling

75

Fig. 3.1 Fit of the within-host model (3.9) with ε = 1 and immune response to data in [61]. Inset figure shows the scaled immune response

To see that limτ V = 0 we argue by contradiction. Assume that limτ T = T ∞ > 0 and such that βw T ∞ cw ≥ ρw δw . That implies that the dynamical system Tˆi (τ ) = βw T ∞Vˆ − δw Tˆi , (3.10)

Vˆ  (τ ) = ρw Tˆi − cwVˆ .

has a solution Tˆi → ∞ and Vˆ → ∞, which implies that T (τ ) → 0. This is a contradiction. Thus limτ T = T ∞ > 0 satisfying βw T ∞ cw < ρw δw . Hence, we may shift the dynamical system so that T (0) = T0 is such that βw T0 cw < ρw δw . In this case the following dynamical system Tˆi (τ ) = βw T0Vˆ − δw Tˆi , (3.11)

Vˆ  (τ ) = ρw Tˆi − cwVˆ

has a solution that satisfies Ti ≤ Tˆi → 0, Ti ≤ Tˆi → 0, which completes the proof. This implies that model (3.9) is an outbreak model. We compare model (3.9) with within-host influenza data. Various within-host models of influenza have been fitted to data. We fit our model (3.9) to data in [61]. The fit is shown in Fig. 3.1. The within-host parameters for this fit are listed in

Table 3.2 Parameter values for the within-host model Parameter

Value

Units

Comment

βw cw ρw δw T0 V0 Ti (0) ε a B(0)

0.001 10 5 2 35,000 1 0.001 1 0.001 0.0001

Per cell per day Per day Per day Per day Per ml Per ml Per ml Per cell per day Per cell per day Per ml

Fitted [147] Fitted [147] Fitted Data Assumeda See text Fitted Assumeda

a Small

number

76

3 Nested Immuno-Epidemiological Models

Table 3.2. Prior investigation of identifiability suggests that ε may not be identifiable ˆ τ ) = ε B. Scaling and from data on the viral load. We may scale it out by setting B( dropping the hats amounts to taking ε = 1 (see [170] for further details). We augment the within-host model (3.9) with an SIR between-host model: ⎧  ∞  ⎪ ⎪ S = Λ − S β (τ )i(τ ,t)d τ − μ S, ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎨ iτ + it = −(α (τ ) + γ (τ ) + μ )i,  ∞ (3.12) ⎪ β ( τ )i( τ ,t)d τ , i(0,t) = S ⎪ ⎪ ⎪  ∞ 0 ⎪ ⎪ ⎪ R = ⎩ γ (τ )i(τ ,t)d τ − μ R. 0

Notice that the epidemiological component is not an outbreak model because even though the disease disappears in each individual, it may be endemic on population level. Such a disease, for instance, is influenza. The transmission rate and diseaseinduced mortality can be taken again from Table 3.1. We need to specify a recovery rate. The recovery rate γ (τ ) must be inversely proportionate to the pathogen load. Thus, we may have: κ γ (τ ) = , V + ε0 where κ is a linking parameter and ε0 is a small number. In the case of an immune model that involves memory cells such as model (3.9), we may have γ (τ ) proportionate to the memory cells as well:

γ (τ ) =

κB . V + ε0

We notice that in the classical nested models the within-host model is linked to the between-host model, but the between-host dynamical variables do not affect the within-host dynamical variables. This type of models are called unidirectionally linked immuno-epidemiological models. Unidirectionally linked models are the main kind considered in the literature as, particularly for human diseases, it is not clear how the population-level prevalence should affect the within-host dynamics.

3.2 Analysis of Immuno-Epidemiological Models Since the models introduced in the previous section are unidirectionally linked, their analysis is often performed with tools for analysis of age-since-infection structured epidemic models. This is the case with establishing well posedness. The well posedness of the within-host model is established with methods for ODEs applied to the within-host model. The well posedness of the immuno-epidemiological model is established with methods for age-structured PDEs, which we introduced in Chap. 2 (see also [142]). We use semigroup approach to establish well posedness of agesince-infection models in Chap. 8.

3.2 Analysis of Immuno-Epidemiological Models

77

3.2.1 Analysis of the SIR Immuno-Epidemiological Model In this section we derive the equilibria, the reproduction number, and and the endemic equilibrium of the nested immuno-epidemiological models (3.9)–(3.12). In analyzing immuno-epidemiological models, one keeps the within-host model as a dynamical model and studies the between-host model. The between-host model (3.12) has a disease-free equilibrium E0 = ( Λμ , 0, 0) and an endemic equilibrium which solve the following system: ⎧  ∞ ⎪ ⎪ 0 = Λ − S β (τ )i(τ )d τ − μ S, ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ iτ = −(α (τ ) + γ (τ ) + μ )i, ⎨  ∞ (3.13) ⎪ i(0) = S β (τ )i(τ )d τ , ⎪ ⎪ ⎪ ⎪  ∞ 0 ⎪ ⎪ ⎪ ⎩0 = γ (τ )i(τ ,t)d τ − μ R. 0

Solving the differential equation, we obtain i(τ ) = i(0)π (τ ) where

π (τ ) = e−μτ −

τ 0

(α (s)+γ (s))ds

is the probability of survival in the infected class. Substituting the expression for i(τ ) in the initial condition, we obtain a value for S S = ∞ 0

1 . β (τ )π (τ )d τ

(3.14)

From the first equation, we obtain i(0) and R i(0) = μ S(R0 − 1),

R=

i(0)

∞ 0

γ (τ )π (τ )d τ , μ

(3.15)

where R0 is the basic reproduction number, defined as follows: R0 =

Λ μ

 ∞ 0

β (τ )π (τ )d τ .

(3.16)

We have the following result: Proposition 3.1. The system (3.12) has a unique endemic equilibrium, which exists if and only if R0 > 1. Although the reproduction number and the components of the endemic equilibrium look exactly the same as obtained from an age-since-infection model, here they are fundamentally different, because β (τ ), γ (τ ), α (τ ) depend on the within-host dynamical variables and their dynamical behavior. We explore that dependence more carefully in the next subsection. Here we continue with local analysis of the equilibria.

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3 Nested Immuno-Epidemiological Models

Proposition 3.2. The disease-free equilibrium is locally asymptotically stable if R0 < 1 and unstable if the opposite inequality holds. Proof. To see this result, we linearize around disease-free equilibrium. Let S = Λμ + x, i(x,t) = y(x,t), R(t) = z(t). Just like in Chap. 2 we linearize and then look for exponential solutions. That leads to the following linear eigenvalue problem: ⎧  Λ ∞ ⎪ ⎪ λ x = − β (τ )y(τ )d τ − μ x, ⎪ ⎪ μ 0 ⎪ ⎪ ⎪ ⎪ ⎨ yτ + λ y = −(α (τ ) + γ (τ ) + μ )y,  (3.17) Λ ∞ ⎪ ⎪ β ( τ )y( τ )d τ , y(0) = ⎪ ⎪ 0 ⎪ μ ⎪ ∞ ⎪ ⎪ ⎩λz = γ (τ )y(τ )d τ − μ z. 0

Solving the differential equation, we have y(τ ) = y(0)π (τ )e−λ τ . Substituting in the initial condition, we obtain the following characteristic equation:

Λ μ

 ∞ 0

β (τ )π (τ )e−λ τ d τ = 1.

Denote the left-hand side by G (λ ). Assume R0 < 1. Then, we have for λ with ℜλ ≥ 0: |G (λ )| ≤ G (0) = R0 < 1, and hence the characteristic equation has no roots with nonnegative real part. Thus the disease-free equilibrium is locally asymptotically stable. If R0 > 1 and we consider λ as real variable, we have G (0) = R0 > 1. In addition limλ →∞ G (λ ) = 0. Hence, there is a positive λ ∗ > 0 such that G (λ ∗ ) = 1. Thus the disease-free equilibrium is unstable. This completes the proof. Next we establish the local stability of the endemic equilibrium. Proposition 3.3. Assume R0 > 1. Then the unique endemic equilibrium is locally asymptotically stable. Proof. We linearize the between-host model around the endemic equilibrium. We set S = S∗ + x, i = i∗ + y, and R = R∗ + z. The linearized between-host system becomes: ⎧  ∞  ∗ ⎪ ⎪ x = −S β (τ )y(τ ,t)d τ − xB − μ x, ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ yτ + yt = −(α (τ ) + γ (τ ) + μ )y, ⎨  ∞ (3.18) ∗ ⎪ y(0,t) = S β (τ )y(τ ,t)d τ + Bx, ⎪ ⎪ ⎪ 0 ⎪  ∞ ⎪ ⎪  ⎪ ⎩z = γ (τ )y(τ ,t)d τ − μ z, ∞

0

β (τ )i∗ (τ )d τ .

where B = 0 Looking for exponential solutions we get the following linear eigenvalue problem.

3.2 Analysis of Immuno-Epidemiological Models

79

⎧  ∞ ∗ ⎪ ⎪ λ x = −S β (τ )y(τ )d τ − xB − μ x, ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ yτ + λ y = −(α (τ ) + γ (τ ) + μ )y,  ∞

⎪ y(0) = S∗ β (τ )y(τ )d τ + Bx, ⎪ ⎪ ⎪ ⎪  ∞ 0 ⎪ ⎪ ⎪ ⎩λz = γ (τ )y(τ )d τ − μ z.

(3.19)

0

Solving the differential equation we have y(τ ) = y(0)e−λ τ π (τ ). The first and the third equations in system (3.19) above imply x=−

y(0) . λ +μ

Substituting x and y(τ ) in the equation for y(0) and canceling y(0) we obtain the following characteristic equation:

λ +μ +B = S∗ λ +μ

 ∞ 0

β (τ )e−λ τ π (τ )d τ .

(3.20)

Consider λ with ℜλ ≥ 0. Then, |S∗ On the other side

 ∞ 0

β (τ )e−λ τ π (τ )d τ | ≤ S∗

 ∞ 0

β (τ )π (τ )d τ = 1.

   λ + μ +B (ℜλ + μ + B)2 + (ℑλ )2    λ + μ  = (ℜλ + μ )2 + (ℑλ )2 > 1.

Hence, the characteristic equation does not have roots with nonnegative real part. That completes the proof. Among other things Propositions 3.2 and 3.3 show that performing local analysis with unidirectionally linked immuno-epidemiological models uses the same techniques as analysis of age-since-infection structured epidemiological models. This is often the case, particularly when the linking is not necessary to obtain the result, but not always. The endemic equilibrium in nested immuno-epidemiological models tends to be more stable than the endemic equilibrium in the corresponding age-since-infection structured epidemiological model. For instance Castillo-Chavez and Thieme [169] show that an SI age-since-infection structured model with standard incidence exhibits oscillations. Those were shown through simulations by Milner and Pugliese [137]. Milner and Pugliese used a step function for β (τ ). That cannot be done with immuno-epidemiological models since their epidemiological parameters β (τ ), α (τ ), and γ (τ ) are not arbitrary functions but solutions of a system of ODEs. As such they are smooth and often related to each other.

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3 Nested Immuno-Epidemiological Models

In the next section we consider a specific immuno-epidemiological model to show an example of this situation.

3.2.2 Stability of the SI Immuno-Epidemiological Model of HIV We consider a specific immuno-epidemiological model of HIV. What is different for this model compared to the model in Sect. 3.8 is that the between-host model has standard incidence and no recovery. This model is the one considered in [169] with C(S + I) = C, where C is a constant. The results in [169] and [137] imply that the endemic equilibrium of the age-since-infection epidemiological model can be destabilized and sustained oscillations are possible. In contrast, the specific immunoepidemiological model we consider here with standard incidence and β (τ ) that is proportional to the viral load has a locally stable endemic equilibrium [121]. Hence, unidirectionally linked immuno-epidemiological models are more stable than their age-since-infection structured epidemiological counterparts and the reason for that is that β (τ ), α (τ ) and γ (τ ) are not arbitrary functions but solutions of a system of ODEs. We consider models (3.6)–(3.8) with the following linking functions: β (τ ) = β0V (τ ) and α (τ ) = η V (τ ). The model has a disease-free equilibrium E0 = ( Λμ , 0). We define the basic reproduction number R0 as follows: R0 =

 ∞ 0

β (τ )π (τ )d τ ,

(3.21)

where π (τ ) is the probability of survival in the infected class, defined as follows:

π (τ ) = e−μτ −

τ 0

α (s)ds

.

Concerning the endemic equilibrium, we have Proposition 3.4. Assume R0 > 1. Then, the system has a unique endemic equilibrium, given by Λ Π R0 ΛΠ , S∗ = , N∗ = R0 − 1 + μΠ R0 − 1 + μΠ (3.22) Λ (R0 − 1) i∗ (τ ) = i∗ (0)π (τ ), i∗ (0) = , R0 − 1 + μΠ where Π =

 ∞ 0

π (τ )d τ .

Proof. The endemic equilibrium is a solution to the following system:  ⎧ S ∞ ⎪ ⎪ 0 = Λ − β (τ )i(τ )d τ − μ S, ⎪ ⎪ N 0 ⎨ iτ = −(α (τ ) + μ )i, ⎪  ∞ ⎪ ⎪ ⎪ ⎩ i(0) = S β (τ )i(τ )d τ . N 0

(3.23)

3.2 Analysis of Immuno-Epidemiological Models

81

Solving the differential equation, we have i(τ ) = i(0)π (τ ). Substituting that in the equation for i(0) we have S 1 = . N R0 Thus, we rewrite the equations for the equilibrium in the following form: ⎧ 0 = Λ − i(0) − μ S, ⎪ ⎪ ⎪ ⎨ N = S + i(0)Π , ⎪ ⎪ ⎪ ⎩S = 1 . N R0

(3.24)

Solving this system for S, N, and i(0) we obtain the endemic equilibrium. Proposition 3.5. The disease-free equilibrium is locally asymptotically stable if R0 < 1. It is unstable if R0 > 1. Proof is similar to the proof of Proposition 3.2 and is omitted. Next we show that the endemic equilibrium is locally asymptotically stable. Proposition 3.6. Assume R0 > 1. Then the endemic equilibrium is locally asymptotically stable. Proof. From [169] we know that the endemic equilibrium of the age-since-infection structure epidemiological model is not locally stable for all parameter values. So we need to use the specific properties of the immuno-epidemiological model. We linearize (3.8) with S = S∗ + x, i(τ ,t) = i∗ (τ ) + y(τ ,t), N = N ∗ + n to obtain:  ⎧ S∗ ∞ xB S∗ Bn  ⎪ x = − β (τ )y(τ ,t)d τ − ∗ + ∗ 2 − μ x, ⎪ ⎪ ∗ ⎪ N 0 N (N ) ⎨ yτ + yt = −(α (τ ) + μ )y, ⎪ ⎪ ∗  ∞ ⎪ xB S∗ Bn ⎪ ⎩ y(0,t) = S β (τ )y(τ ,t)d τ + ∗ − ∗ 2 . ∗ N 0 N (N )

(3.25)

Looking for exponential solutions we obtain the following linear eigenvalue problem:  ⎧ S∗ ∞ xB S∗ Bn ⎪ λ x = − β ( τ )y( τ )d τ − + − μ x, ⎪ ⎪ ⎪ N∗ 0 N ∗ (N ∗ )2 ⎨ yτ + λ y = −(α (τ ) + μ )y, (3.26) ⎪  ∞ ⎪ ∗ ∗ ⎪ xB S Bn ⎪ ⎩ y(0) = S β (τ )y(τ )d τ + ∗ − ∗ 2 . N∗ 0 N (N ) We solve the differential equation to get y(τ ) = y(0)e−λ τ π (τ ). We introduce the following notation:

ρ (λ ) =

 ∞ 0

e−λ τ π (τ )d τ ,

B(λ ) =

 ∞ 0

β (τ )e−λ τ π (τ )d τ .

(3.27)

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3 Nested Immuno-Epidemiological Models

We note that ρ (0) = Π and B(0) = R0 . We solve system (3.26). Using the first and last equations we have y(0) x=− . (3.28) λ +μ Since n = x +

∞ 0

y(τ )d τ = x + y(0)ρ (λ ), we have  1 n = y(0) ρ (λ ) − . λ +μ

(3.29)

Substituting in the equation for y(0) and canceling y(0) we obtain the characteristic equation  S∗ B B 1 S∗ − ∗ 2 ρ (λ ) − 1 = ∗ B(λ ) − ∗ . (3.30) N N (λ + μ ) (N ) λ +μ The key observation here is a specific property of the immuno-epidemiological model that we are considering. It is based on the form of β (τ ) and α (τ ), We have B(λ ) =

c (1 − (λ + μ )ρ (λ )). η

Therefore the above characteristic equation becomes:  S∗ B B η 1 = 1+ ∗ B(λ ) 1 + ∗ . N (λ + μ ) N ∗ N c λ +μ

(3.31)

(3.32)

Simplifying further we have

λ + μ + NB∗ S∗ = B(λ ). η N∗ λ + μ + NB∗ c

(3.33)

From Eq. (3.31) we also have R0 = B(0) =

c (1 − μρ (0)). η

Since R0 > 1, that implies that c > η . Hence, for λ with ℜλ ≥ 0 we have    λ +μ + B   N∗    > 1,  λ + μ + NB∗ ηc  while

(3.34)

(3.35)

 ∗  ∗ S   B(λ ) ≤ S R0 = 1.  N∗  N∗

Therefore, the characteristic equation cannot have roots with nonnegative real part. This completes the proof.

3.2 Analysis of Immuno-Epidemiological Models

83

Proposition 3.6 says that the endemic equilibrium of the immuno-epidemiological model is locally asymptotically stable, even though the corresponding age-sinceinfection model may lose stability and exhibit oscillations. In that respect immunoepidemiological models are more stable than age-since-infection models. In fact to date there are no results showing oscillations in a unidirectionally linked immunoepidemiological model whose all between-host rates are linked to the within-host model variables.

3.2.3 Impact of Within-Host Parameters on the Between-Host Dynamics One of the questions that we can address with immuno-epidemiological models is about the impact of the within-host dynamics on the between-host dynamics. There are many approaches to address this question but the simplest one can be applied to models of diseases without recovery, such as our SI model of HIV. In this simplest approach, we assume that the within-host system has reached an equilibrium. In this case the time-since-infection coefficients in the between-host system are constants and the between-host system becomes a system of ODEs: ⎧ β SI ⎨  − μ S, S =Λ− (3.36) N ⎩ I  = β SI − (α + μ )I. N

We couple this system with the within-host system (3.3) and the following linking functions:  1 β0 V ∗ dw β= ∗ α =η − . (3.37) V +K T ∗ ΛW Substituting the equilibrial values for V ∗ and T ∗ , we obtain the following dependence of the between-host parameters β and α on the within-host parameters:

Λw βw ρw . dw cw δw (3.38) To explore the impact of the within-host parameters on the between-host dynamics, we start by considering the immuno-epidemiological reproduction number β=

β0 dw (ℜ0 − 1) dw (ℜ0 − 1) = βw K

R0 =

α =η

dw (ℜ0 − 1) Λw

where

ℜ0 =

β β0Λw dw (ℜ0 − 1) . = α +μ (dw (ℜ0 − 1) + βw K)(η dw (ℜ0 − 1) + μ )

(3.39)

From this expression, we see that R0 is actually a non-monotone function of the within-host reproduction number; first increasing and then a decreasing function. That is a rather surprising result. The increase happens over a small interval of ℜ0 over one, and for most values R0 is a decreasing function of ℜ0 . The expression suggests that the reason for the decrease is that the disease-induced mortality in-

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3 Nested Immuno-Epidemiological Models

Fig. 3.2 Dependence of the immuno-epidemiological reproduction number R0 on the within-host reproduction number ℜ0

Fig. 3.3 Dependence of the immuno-epidemiological prevalence I ∗ on the withinhost reproduction number ℜ0

creases faster with the within-host reproduction number than the transmission rate. Our investigations so far suggest that there is a critical value ℜ∗0 of the within-host reproduction number that maximizes the between-host reproduction number. This result is important in the context of evolution (see Fig. 3.2). Next, we consider the dependence of the endemic equilibrium on the withinhost parameters, and in particular the dependence of the endemic prevalence on the within-host ℜ0 . The endemic equilibrium has components  Λ Λ R0 1 ∗ ∗ S = I = . 1− μ R0 + α (R0 − 1) μ R0 + α (R0 − 1) R0 The equilibrium prevalence I ∗ depends on R0 and R0 depends in non-monotone way on the within-host reproduction number ℜ0 . However, simulations suggest that the dependence is mainly decreasing in terms of ℜ0 (see Fig. 3.3). The parameters used in Figs. 3.2 and 3.3 are Λw = 1000, dw = 0.5, βw = 1, K = 100, Λ = 1, μ = 0.01, β0 = 0.005, α = 1. This result is also somewhat counter-intuitive, but it is understandable under the scenario when disease-induced mortality grows faster than the transmission. In the next section we introduce bidirectionally linked models.

3.3 Bidirectionally Linked Immuno-Epidemiological Models Classical immuno-epidemiological models as the ones considered in the previous section, link the between-host model to the within-host model but do not link the within-host model to the between-host dynamical variables. The main reason for

3.3 Bidirectionally Linked Immuno-Epidemiological Models

85

that is that it is hard to imagine how the between-host variables affect the withinhost viral load or immune response. One natural linking in that direction occurs with environmentally driven diseases where the host may be continuously ingesting the pathogen from the environment [52]. Another natural linking occurs in animal diseases where a competitor or a predator may adversely affect the immune response [17].

3.3.1 A Bidirectionally Linked Model of HIV We introduce a different type of linking of the within-host model to between-host variables where we assume that the population-level prevalence affects the efficacy of the medications used on within-host level. The simplifying assumption is that the higher the population level prevalence of infectives, the better the efficacy of the medications. The dependence of the within-host model on the proportion infected I(t) leads to dependence of the within-host variables individuals in the population N(t) on chronological time t, thus transforming the within-host model from an ODE into a PDE. To introduce the within-host model, we begin with HIV within-host model (3.3). Although HIV medications have different modes of operation, we will consider a medication that obstructs the production of new virons. We assume that the more individuals in the population are infected, the more efficient the medication. This assumption is reasonable because more resources are concentrated into developing better medications. Such an assumption works well with diseases like HIV, where the prevalence I(t) increases in time and medications improve in time. To capture this dependence, we transform the term ρw Ti into ρw (1 − ε NI )Ti . That leads to the within-host dependent variables being variables of both τ and t. Thus, we have T (τ ,t), Ti (τ ,t),V (τ ,t). We have the following within-host PDE model. Tt + Tτ = Λw − βw TV − dw T, (Ti )t + (Ti )τ = βw TV − δw Ti ,

(3.40)

Vt +Vτ = ρw (1 − ε NI )Ti − cwV. This model is equipped with initial and boundary conditions: T (t, 0) = T0 , Ti (t, 0) = Ti,0 ,V (t, 0) = V0 . T (0, τ ) = φ1 (τ ), Ti (0, τ ) = φ2 (τ ),V (0, τ ) = φ3 (τ ),

(3.41)

functions where T0 , Ti,0 , and V0 are given constant, while φi (τ ), i = 1, . . . , 3 are given  of τ , specifying the initial distribution of T , Ti and V . In addition, I(t) = 0∞ i(τ ,t)d τ . This model is coupled with the between-host model (3.8):

86

3 Nested Immuno-Epidemiological Models

 ⎧ S ∞ ⎪ ⎪ S = Λ − β (τ ,t)i(τ ,t)d τ − μ S, ⎪ ⎪ N 0 ⎨ iτ + it = −(α (τ ,t) + μ )i, ⎪  ∞ ⎪ ⎪ ⎪ ⎩ i(0,t) = S β (τ ,t)i(τ ,t)d τ . N 0

(3.42)

We may specify some general linking functions but other choices are also possible:

β (τ ,t) =

βcV (τ ,t) K +V (τ ,t)

α = η V (τ ,t).

This is a novel, bidirectionally linked immuno-epidemiological model. These types of models have been studied little. We investigate here the equilibria of models (3.40)–(3.42). Those satisfy the system Tτ = Λw − βw TV − dw T, (Ti )τ = βw TV − δw Ti , Vτ = ρw (1 − ε NI )Ti − cwV, 0 = Λ − NS

∞ 0

β (τ )i(τ )d τ − μ S,

(3.43)

iτ = −(α (τ ) + μ )i, i(0) =

S N

 ∞ 0

β (τ )i(τ )d τ .

We begin with the disease-free equilibrium. We denote the solution of the withinhost system with I = 0 as (T 0 (τ ), Ti0 (τ ),V 0 (τ )) and the corresponding betweenhost rates as β0 (τ ) and α0 (τ ). We solve the between-host equation for i and obtain i(τ ) = i(0)π0 (τ ) where τ π0 (τ ) = e−μτ − 0 α0 (s)ds . 

Since I = i(0) 0∞ π0 (τ )d τ = 0 that implies that i(0) = 0. Hence, the disease-free equilibrium is given by E0 = (T 0 (τ ), Ti0 (τ ),V 0 (τ ), Λμ , 0). We define the reproduction number as  ∞ R0 = β0 (τ )π0 (τ )d τ . 0

We notice that the within-host components of the disease-free equilibrium are not zero. That is because, the within-host dynamics is described, given infection, so that as long as I moves a little from zero, the within-host dynamics is not zero. Now, we look for endemic equilibria. Assume I = 0. We denote the solution of the immune system with generic I/N as (T, Ti ,V ) and the corresponding between-host rates as β (τ ) and α (τ ). We solve the between-host equation for i and we obtain i(τ ) = i(0)π (τ ) where τ π (τ ) = e−μτ − 0 α (s)ds .

3.3 Bidirectionally Linked Immuno-Epidemiological Models

87

We substitute the solution in the initial condition, and we obtain: 1 S 1 = ∞ =: I . N β ( τ ) π ( τ )d τ B( 0 N) Furthermore, for the equation of the total population size N = S + I we have S I = 1− . N N Thus, we have the following equation for

I N:

1 I = 1− . I N B( N )

(3.44)

Proposition 3.7. Assume R0 > 1. Then the system (3.43) has at least one endemic equilibrium. Proof. Assume I = 0. Then, B(0) = R0 > 1. Hence, the left-hand side of (3.44) is smaller than one, while the right-hand side is exactly one. Next, take I = N. Then the left-hand side is positive, while the right-hand side is zero. Since both sides are continuous, Eq. (3.44) has at least one solution on (0, N). Once we know NI , we also know NS . The remaining components are computed as in the case of unidirectionally linked model. We have that N∗ = I∗ =

I N

ΛΠ , + μΠ NS

Λ Π NI I N

+ μΠ NS

,

S∗ =

Λ Π NS I N

+ μΠ NS

,

I∗ i∗ (0) = Π

(3.45)



where Π = 0∞ π (τ )d τ . To gain more insight into the behavior of system we make the assumption that the within-host dynamical variables are subject to a faster dynamics, and given I(t), they equilibrate sooner. That allows us to set their derivatives to zero, and express each one of these variables in terms of I/N. Denote by i f = I/N. We have δw cw , T (t) = βw ρw (1 − ε i f (t)) V (t) =

dw βw



Λ βw ρw (1 − ε i f (t)) −1 , δw cw dw

cw V (t). ρw (1 − ε i f (t))     We define β NI = cV and α NI = η V . The between-host model becomes: Ti (t) =

(3.46)

88

3 Nested Immuno-Epidemiological Models

 I SI − μ S, S = Λ − β N N   I SI I − (μ + α I = β )I. N N N

(3.47)

In analyzing this simple ODE model, we begin with equilibria. The system for equilibria is obtained from (3.47) by setting the derivatives to zero. The system has the disease-free equilibrium E0 = ( Λμ , 0). As before, we define the basic reproduction number cV 0 R0 = , (3.48) ηV 0 + μ where V0 =

dw βw



Λ βw ρw −1 . δw cw dw

We see that the basic reproductive number depends on the within-host parameters. To obtain the endemic equilibria, we cancel I in the second equation and solve for S/N. As before, we obtain   μ + α NI S   . = N β NI It is not hard to see that both α and β are decreasing functions of I/N. Furthermore, one can show that NS is an increasing function of I/N. Indeed, differentiating S/N with respect to i f , we have

α  β − (α + μ )β  η cVV  − (η V + μ )cV  −μ cV  = = >0 β2 β2 β2 since V  < 0, that is V is a decreasing function of i f . The fraction I/N satisfies the following equation:   μ + α NI I  I  = 1− . N β N

(3.49)

Since the left-hand side is an increasing function of I/N while the right-hand side is a decreasing function of I/N, this equation may have at most one solution.     Proposition 3.8. Assume R0 > 1 and β NI = cV and α NI = η V . Then Eq. (3.49) has at exactly one solution. Assume R0 < 1. Then Eq. (3.49) has no solutions. Proof is similar to the one of Proposition 3.7 and is omitted.     Proposition 3.9. Assume R0 > 1 and β NI = cV and α NI = η V . Then the unique endemic equilibrium is locally asymptotically stable Proof. The proof is based on the observation that if we rewrite Eq. (3.7) in the form μ + α (i f ) = β (i f )(1 − i f ), then at the equilibrium the derivatives of the two sides satisfy

3.3 Bidirectionally Linked Immuno-Epidemiological Models

α  > −β + β  s f ,

89

(3.50)

where s f is the fraction of susceptibles. Recall that we have s f + i f = 1. Computing 2,2 the Jacobian we have J = ( jik )i=1,k=1 where j11 = β  s f i2f − β i f + β s f i f − μ < 0, j12 = −β  s f i f + β  s f i2f − β s f + β s f i f , j21 = −β  s f i2f + β i f − β s f i f + α  i2f > 0,

(3.51)

j22 = β  s f i f − β  s f i2f − β s f i f − α  s f i f < 0. The third and the fourth inequalities follow from (3.50). We also recall that β  < 0 and α  < 0. Given the inequalities above, it is easy to see that Tr J < 0. To see that Det J > 0, we consider, (β  s f i2f − β i f + β s f i f − μ )(β  s f i f − β  s f i2f − β s f i f − α  s f i f ) −(−β  s f i f + β  s f i2f − β s f + β s f i f )(−β  s f i2f + β i f − β s f i f + α  i2f ) = (β  s f i2f − β i f + β s f i f − μ )(−s f i f (α  + β − β  s f ))) −(−β  s f i f + β  s f i2f − β s f + β s f i f )(i2f (α  + β − β  s f )) = (α  + β − β  s f )i f (−β  s2f i2f + β s f i f − β s2f i f − β  s f i2f − β  s f i3f + β s f i f − β s f i2f ) (α  + β − β  s f )s f i2f (−β  s f i f + β − β s f − β  i f − β  i2f + β − β i f ) > 0. This completes the proof.

3.3.2 Bidirectionally Linked Immuno-Epidemiological Model of Cholera Cholera is an environmentally driven disease which can be naturally modeled with bidirectional linking since infected individuals can continuously ingest the pathogen Vibrio cholerae from the environment. A recent immuno-epidemiological model of cholera with bidirectional linking of ODE type was considered in [174]. We begin by introducing the within-host model which is given by a simple logistic equation. We denote the amount of the pathogen within a host with P(τ ,t) where τ is the time-since-infection and t is the chronological time. With B(t) denoting the bacteria in the environment, we consider the following within-host model: ⎧ Pτ + Pt = rP(1 − qP) + η B, ⎪ ⎪ ⎨ P(τ , 0) = φ (τ ), (3.52) ⎪ ⎪ ⎩ P(0,t) = P0 ,

90

3 Nested Immuno-Epidemiological Models

where φ (τ ) and P0 are given. φ can be taken as the solution of the differential equation with B(0). Furthermore, r = (ˆr − d) with rˆ being the growth rate within a host ˆ − d/ˆr) and d being death rate of the bacteria within a host. q = 1/K where K = K(1 ˆ where K is the carrying capacity. In addition η is the rate of ingesting the pathogen from the environment. We define the within-host reproduction number ℜ0 =

rˆ . d

We will assume ℜ0 > 1. The between-host model consists of susceptible individuals S(t), infected i(τ ,t), and recovered R(t). The model takes the form: ⎧ β BS ⎪ ⎪ S = Λ − − μ S, ⎪ ⎪ B +D ⎪ ⎪ ⎪ ⎪ iτ + it = −(μ + α (τ ,t) + γ )i, ⎪ ⎪ ⎪ ⎨ β BS (3.53) , i(0,t) = ⎪ B+D ⎪ ⎪ ⎪ R = γ I − μ R, ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎩ B = ξ (τ ,t)i(τ ,t)d τ − δ B, 0

where I(t) =

 ∞ 0

i(τ ,t)d τ

is the total infected population size. We assume only transmission through the environment which is the main mode for cholera. The parameters are listed in Table 3.3. To link the between-host model to the within-host model, we specify the linking Table 3.3 Parameters for model (3.53) Parameter

Meaning

Units

Λ μ β γ α (τ ,t) ξ (τ ,t) δ D

Recruitment rate Natural death rate Transmission rate Recovery rate Disease-induced death rate Shedding rate Clearance rate of bacteria from environment Half saturation constant for bacteria

People/unit of time 1/time 1/time 1/time 1/time Bacteria/(individual*time) 1/time Number of bacteria

functions. We take the simplest form:

α (τ ,t) = α0 P(τ ,t),

ξ (τ ,t) = ξ0 P(τ ,t).

(3.54)

3.3 Bidirectionally Linked Immuno-Epidemiological Models

91

Analytical tools for these type of models will have to be developed in the future. Here we discuss the disease-free equilibrium and derive the reproduction number. In the disease-free equilibrium B = 0. Then Eq. (3.52) is a logistic equation and ˆ τ ). Since B = 0, we have from the can be solved. We denote the solution by P( between-host model (3.53) that i(0) = 0. Hence i(τ ) = 0. That gives R = 0 and ˆ 0, 0, 0) where Sˆ = Λ . In ˆ τ ), S, S = Λμ . Hence, the disease-free equilibrium is E0 = (P( μ deriving the reproduction number, we consider the local stability of the disease-free ˆ ˆ τ )+u(τ ,t), S(t) = S+x(t), i(τ ,t) = y(τ ,t), B(t) = z(t). equilibrium. Let P(τ ,t) = P( We linearize systems (3.52) and (3.53). Then we look for exponential solutions. We obtain a linear eigenvalue problem: ⎧ ˆ + η z, uτ + λ u = ru(1 − 2qP) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u(0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ β zSˆ ⎪ ⎪ ⎨ λ x = D − μ x, (3.55) ˆ τ ) + γ )y, yτ + λ y = −(μ + α0 P( ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ β zSˆ ⎪ ⎪ y(0) = , ⎪ ⎪ ⎪ ⎪  ∞D ⎪ ⎪ ⎪ ˆ τ )y(τ )d τ − δ z. ⎩λz = ξ0 P( 0

The system for y and z separates from the rest of the equations. We solve the differential equation to obtain β zSˆ −λ τ y(τ ) = e π0 (τ ), (3.56) D τ

where π0 (τ ) = e−(μ +γ )τ − 0 α0 P(s)ds . Substituting y(τ ) in the equation for z and canceling z we obtain the following characteristic equation: ˆ

βΛ ξ0 μ D(λ + δ )

 ∞ 0

ˆ τ )e−λ τ π0 (τ )d τ = 1. P(

(3.57)

We define the reproduction number R0 =

βΛ ξ0 μ Dδ

 ∞ 0

ˆ τ )π0 (τ )d τ . P(

(3.58)

Proposition 3.10. The disease-free equilibrium is locally asymptotically stable if R0 < 1. The disease-free equilibrium is unstable if R0 > 1. Proof. If R0 > 1, then if we treat the left-hand side of the characteristic equation as a function of the real variable λ and denote it by G (λ ). Then we have G (0) = R0 > 1. In addition, limλ →∞ G (λ ) = 0. Hence, the characteristic equation has a positive solution and the disease-free equilibrium is unstable. Now, assume R0 < 1 and λ is such that ℜλ ≥ 0. Then |G (λ )| ≤ G (0) = R0 < 1.

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3 Nested Immuno-Epidemiological Models

Hence, the characteristic equation has no roots with nonnegative real part. That gives z = 0 and y(0) = 0. From the equation for x we have λ = −μ . The remaining eigenvalues may come from the equation for u but since z = 0 and u(0) = 0, this equation has no nonzero solutions for any λ . To gain further understanding, we consider the associated ODE model. Assuming that the within-host dynamics develops on a much faster scale, we set the derivatives with respect to τ and t equal to zero in (3.52). We express the internal pathogen load through the pathogen in the environment  r + r2 + 4rqη B(t) . P(t) = 2rq We will use the notation P(t) = ν (B(t)). The ODE immuno-epidemiological model becomes ⎧ β BS ⎪ ⎪ S = Λ − − μ S, ⎪ ⎪ B+D ⎪ ⎨ β BS (3.59) − (μ + α0 ν (B) + γ )I, I = ⎪ ⎪ ⎪ B + D ⎪ ⎪ ⎩  B = ξ0 ν (B)I − δ B, where we have omitted the equation for R. We investigate model (3.59). The model has a unique disease-free equilibrium ( Λμ , 0, 0, 0). We define the reproduction number β ξ0 ν (0)Λ , (3.60) R0 = μ Dδ (μ + α0 ν (0) + γ ) where ν (0) = K. To study the endemic equilibria, we express I in terms of B: I=

δB . ξ0 ν (B)

Furthermore, we express S from the first equation: S=

Λ βB B+D

+μ

.

We replace I and S in the second equation of (3.59) to obtain the following equation in B: β ξ0 ν (B)Λ = (μ + α0 ν (B) + γ )δ . (3.61) β B + μ (B + D) Proposition 3.11. Assume R0 > 1. Then Eq. (3.61) has at least one solution. Proof. Denote the left-hand side of (3.61) by f (B) and the right-hand side as g(B). Since R0 > 1, we have that f (0) > g(0). On the other hand, limB→∞ f (B) = 0 while limB→∞ g(B) = ∞. Hence, the equality of the two functions has at least one solution. That completes the proof.

3.3 Bidirectionally Linked Immuno-Epidemiological Models

93

Fig. 3.4 Backward bifurcation in model (3.59)

Proposition 3.12. Equation (3.61) exhibits backward bifurcation at the critical value of R0 = 1 if and only if the parameters satisfy (μ + α0 ν (0) + γ )(β ∗ + μ ) ν  (0) − (μ + γ ), μD ν (0) where

β∗ =

μ Dδ (μ + α0 ν (0) + γ ) . ξ0 ν (0)Λ

Proof. Equation (3.61) exhibits backward bifurcation at the critical value of R0 = 1 if and only if ∂B ∗ | < 0. ∂ β β =β ,B=0 We differentiate Eq. (3.61) with respect to β

ξ0 ν (0)Λ β ∗ ξ0 ν  (0)Λ μ D − β ∗ ξ0 ν (0)Λ (β ∗ + μ ) ∂ B ∂B + = α0 ν  (0)δ . (3.62) μD (μ D)2 ∂β ∂β Solving for

∂B ∂β

and posing the condition

α0 ν  (0)δ −

∂B ∂β

< 0 we obtain the following inequality:

β ∗ ξ0 ν  (0)Λ μ D − β ∗ ξ0 ν (0)Λ (β ∗ + μ ) < 0. (μ D)2

Replacing β ∗ with its equal and simplifying, we get (μ + α0 ν (0) + γ )(β ∗ + μ ) − This completes the proof.

ν  (0) μ D < 0. ν (0)

94

3 Nested Immuno-Epidemiological Models

Since ν  (0) = η /r that suggests that large enough η will lead to backward bifurcation. On the other hand, if η = 0 backward bifurcation does not occur. Hence the mechanism of the backward bifurcation is the continual ingestion of bacteria by the host. Figure 3.4 shows the backward bifurcation for several values for η . The ingestion rate η controls the depth of the backward bifurcation; the larger the η , the more pronounced the backward bifurcation. This suggests that one critical control measure necessary for Cholera control is to prevent infected individuals from ingesting further the pathogen. Not much can be proved about the stability of endemic equilibria. The lower endemic equilibrium is typically unstable. For some parameter values the upper endemic equilibrium when R0 < 1 is locally asymptotically stable. Figure 3.5 shows bistability in model (3.59) when R0 < 1. If R0 > 1, the endemic equilibrium, even if unique, is not always stable. Hopf bifurcation may occur and sustained oscillations are possible. We illustrate those in Fig. 3.6. The oscillations in B(t) and I(t) are of similar amplitude and within phase.

3.4 Immuno-Epidemiological Multi-Strain Models One of the main problems in modeling multi-strain interactions is composing a tractable model [180]. The simplest scenario with immuno-epidemiological models is the case when each individual is infected by a unique strain and, when recovered, is protected from infection with any other strain or recovery is not an option. This last case potentially mimics highly pathogenic avian influenza, where poultry rarely recover since they either die from the disease or are destroyed. More complex scenarios can involve any type of interaction between the strains both within-host and between-host. In the next subsection we introduce the simplest type of unidirectionally linked multi-strain immuno-epidemiological models. Fig. 3.5 Bistability when R0 < 1 in model (3.59)

3.4 Immuno-Epidemiological Multi-Strain Models

95

Fig. 3.6 Sustained oscillations when R0 > 1 in model (3.59)

3.4.1 An n-Strain Immuno-Epidemiological Competitive Exclusion Model An n-strain nested immuno-epidemiological model where all n strains affect distinct individuals is a n-strain age-since-infection structured model whose transmission, recovery, and disease-induced death rate are related to a within-host dynamics with strain j. To introduce the model, we begin with the within-host model of strain j which is a variant of model (3.3). ⎧  T (τ ) = Λw − βwj TV j − dw T, ⎪ ⎪ ⎨ Ti, j (τ ) = βwj TV j − δwj Ti, j , ⎪ ⎪ ⎩  V j (τ ) = ρwj Ti, j − cwj V j ,

(3.63)

for j = 1, . . . , n. Here V j are the virons of strain j and Ti, j are the infected cells with strain j. On the between-host level, we have an n-strain age-since-infection structured SI epidemic model with mass action incidence.  ∞ ⎧  ⎪ S = Λ − S β j (τ )i j (τ ,t)d τ − μ S, ⎪ ⎪ ⎪ 0 ⎨ i j,τ + i j,t = −(α j (τ ) + μ )i j , (3.64) ⎪  ∞ ⎪ ⎪ ⎪ ⎩ i j (0,t) = S β j (τ )i j (τ ,t)d τ , 0

where i j (τ ,t) are the infected individuals with strain j. The within-host model is linked to the between-host model through the transmission rate and disease-induced mortality: β j (τ ) = β0, jV j (τ ), α j (τ ) = α0, jV j (τ ). Defining the probability of survival in the infected class j as

π j (τ ) = e−μτ −

τ 0

α j (s)ds

,

96

3 Nested Immuno-Epidemiological Models

we obtain the basic reproduction number Rj =

Λ μ

 ∞ 0

β j (τ )π j (τ )d τ .

For this model it can be proved (see [39]) that the system has a unique diseasefree equilibrium which is locally and globally stable if R j < 1 for j = 1, . . . , n and unstable if there exists R j0 > 1 for some j0 . Furthermore, if R j > 1, there exists a unique endemic equilibrium corresponding to strain j: E j . The single strain equilibrium E j is locally and globally stable if R j is the maximal reproduction number, that is R j = max{R1 , . . . , Rn }.

3.4.2 A Two Strain Model Structured by Inoculum Fraction Multi-strain immuno-epidemiological models require dealing with significant complexity and different frameworks have been proposed to handle the complexity. Coombs et al. [35] investigate a framework where the infected individuals are structured not only by age-since-infection but also by inoculum fraction. Thus, it is assumed that any infected individual becomes infected with either one or both strains simultaneously. The within-host model is built on bases of model (3.3) but consists of two strains that mutate in each other. With V1 , V2 denoting the two strains and Ti,1 and Ti,2 denoting the target cells infected with strain one and two respectively, the model (3.3) becomes: ⎧  T (τ ) = Λw − βw,1 TV1 − βw,2 TV2 − dw T, ⎪ ⎪ ⎪ ⎪ ⎪  ⎪ ⎪ ⎪ Ti,1 (τ ) = (1 − ε )βw,1 TV1 + εβw,2 TV2 − δw,1 Ti , ⎨ V1 (τ ) = ρw,1 Ti,1 − cw,1V1 , (3.65) ⎪ ⎪ ⎪  (τ ) = (1 − ε )β TV + εβ TV − δ T , ⎪ Ti,2 ⎪ w,2 2 w,1 1 w,2 i,2 ⎪ ⎪ ⎪ ⎩  V2 (τ ) = ρw,2 Ti,2 − cw,2V2 . The parameter meanings are as before, and ε is mutation rate, assumed small. The model is equipped with the following initial conditions T (0) = T 0 , Ti, j (0) = 0,V j (0) = V j0 where j = 1, 2. The between-host model is an SI model structured by initial inoculum x0 where V0 x0 = 0 1 0 . V1 +V2 Clearly 0 ≤ x0 ≤ 1. Assuming that V10 + V20 = ξ is a constant, with ξ being the inoculum concentration, the initial conditions for the immune system become T (0) = T 0 , Ti, j (0) = 0,V10 = ξ x0 ,V20 = ξ (1 − x0 ).

3.4 Immuno-Epidemiological Multi-Strain Models

97

In the between-host model β (τ , x0 , x0 ) is the transmission rate from individuals first infected with strain mix x0 to individuals first infected with strain mix x0 . We will assume the transmission rate in separable form, that is

β (τ , x0 , x0 ) = β1 (τ , x0 )β2 (x0 ). If S(t) represents the number of susceptible, and i(τ ,t, x0 ) represents the density of infected individuals, infected with strain mix x0 the between-host model becomes: ⎧  ∞ 1 1 ⎪  ⎪ S = Λ − S β (τ , x0 , x0 )i(τ ,t, x0 )dx0 dx0 d τ − μ S, ⎪ ⎪ ⎪ 0 0 0 ⎨ it + iτ = −(μ + α (τ , x0 ))i(τ ,t, x0 ), (3.66) ⎪ ⎪   ⎪ ∞ 1 ⎪ ⎪ ⎩ i(0,t, x0 ) = S β (τ , x0 , x0 )i(τ ,t, x0 )dx0 d τ . 0

0

For the linking function we take

β1 (τ , x0 ) = b1 (V1 +V2 ),

α (τ , x0 ) = a1 μ (T 0 − T ).

For now we leave β2 (x0 ) not specified. Besides the initial inoculum, the strain mix as a function of τ is given by V1 x= . V1 +V2 The model has a disease-free equilibrium E0 = ( Λμ , 0). We denote by

π (τ , x0 ) = e−μτ

τ 0

α (s,x0 )ds

the probability of survival in the infectious class and the between-host reproduction number as   Λ ∞ 1 β1 (τ , x0 )β2 (x0 )π (τ , x0 )dx0 d τ . R0 = μ 0 0 Proposition 3.13. If R0 > 1, then the system has a unique endemic equilibrium. Proof. The endemic equilibrium satisfies the equations: ⎧  ∞ 1 1 ⎪ ⎪0 = Λ −S β (τ , x0 , x0 )i(τ , x0 )dx0 dx d τ − μ S, ⎪ ⎪ ⎪ 0 0 0 ⎨ iτ = −(μ + α (τ , x0 ))i(τ , x0 ), ⎪ ⎪  ∞ 1 ⎪ ⎪ ⎪ ⎩ i(0, x0 ) = S β (τ , x0 , x0 )i(τ , x0 )dx0 d τ . 0

Denote by I=

0

 ∞ 1 0

0

β1 (τ , x0 )i(τ , x0 )dx0 d τ .

(3.67)

98

3 Nested Immuno-Epidemiological Models

Then, i(0, x0 ) = β2 (x0 )IS. Multiplying both sides by β1 (τ , x0 )π (τ , x0 ) and integrating by x0 and τ we obtain the following formula for S: S = ∞1 0

0

1

β1 (τ , x0 )β2 (x0 )π (τ , x0 )dx0 d τ

.

With S known, from the first equation we have 0 = Λ −S Solving for I we have I = 1 0

 1 0

β2 (x0 )dx0 I − μ S.

μ β2 (x0 )dx0

(R0 − 1) .

Finally, we have i(τ , x0 ) = i(0, x0 )π (τ , x0 )

i(0, x0 ) = β2 (x0 )IS.

(3.68)

In what follows we investigate the stability of equilibria. We begin with the diseasefree equilibrium. Proposition 3.14. Assume R0 < 1. Then the disease-free equilibrium is locally asymptotically stable. Assume R0 > 1. Then the disease-free equilibrium is unstable. Proof. We linearize system (3.69) around the disease-free equilibrium. With S = S0 + x and i(τ ,t, x0 ) = y(τ ,t, x0 ) and looking for exponential solutions we have the following eigenvalue problem:  ∞ 1 1 ⎧ 0 ⎪ ⎪ λ x = −S β (τ , x0 , x0 )y(τ , x0 )dx0 dx d τ − μ x, ⎪ ⎪ 0 0 0 ⎨ λ y + yτ = −(μ + α (τ , x0 ))y(τ , x0 ), ⎪  ∞ 1 ⎪ ⎪ ⎪ ⎩ y(0, x0 ) = S0 β (τ , x0 , x0 )y(τ , x0 )dx0 d τ . 0

(3.69)

0

Solving the differential equation for y we have y(τ , x0 ) = y(0, x0 )e−λ τ π (τ , x0 ). Multiplying the last equation of (3.69) by β1 (τ , x0 )e−λ τ π (τ , x0 ) we have the following characteristic equation G (λ ) = 1 where G (λ ) = S 0

 ∞ 1 0

0

β1 (τ , x0 )β2 (x0 )e−λ τ π (τ , x0 )dx0 d τ .

Techniques from before result in the claim.

3.4 Immuno-Epidemiological Multi-Strain Models

99

The endemic equilibrium does not have to be stable. We omit the derivation of the characteristic equation.

3.4.3 Multi-Strain Models with Trade-Off Mechanisms on the Between-Host Scale Complexity of the epidemiological models grows very fast when they involve various trade-off mechanisms on the between-host scale. Such trade-off mechanisms involve: mutation, cross-immunity, super-infection, co-infection, and others. Each of these models has to be linked with appropriate within-host model. The resulting immuno-epidemiological model may be very complex. On the other hand, the importance of studying these models cannot be underestimated. Multi-strain immunoepidemiological models can shed light on how strains compete both within-host and on population level and how the outcomes of the two scale competition affect each other. For instance, it is interesting to know how antibody enhancement in dengue on the within-host level affects the disease transmission and virulence evolution. We introduce a model of dengue but to simplify the system, we define susceptible individuals S, i(τ ,t) infected with primary infection individuals, j(τ ,t) infected with secondary infection individuals, where τ is time since infection and t is chronological time, R recovered from primary infection individuals and W are the recovered from both infections. The vector population consists of susceptible vectors Sv and infected vectors Iv . The vector component of the between-host model takes the form: Sv

= Λv − Sv

Iv = Sv

 ∞ 0

 ∞ 0

(βi (τ )i(τ ,t) + β j (τ ) j(τ ,t))d τ ) − μv Sv ,

(βi (τ )i(τ ,t) + β j (τ ) j(τ ,t)) d τ − μv Iv ,

(3.70)

where N is the total human population size N = S+

 ∞ 0

i(τ ,t)d τ + R +

 ∞ 0

j(τ ,t)d τ +W.

The equation for the humans takes the form: ⎧  S = Λ − βS SIv − μ S, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ iτ + it = −(μ + αi (τ ) + γi (τ ))i(τ ,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ i(0,t) = βS SIv , ⎪ ⎪ ⎪  ∞ ⎪ ⎨ R = γi (τ )i(τ ,t)d τ − βR RIv − μ R, 0 ⎪ ⎪ ⎪ jτ + jt = −(μ + α j (τ ) + γ j (τ )) j(τ ,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ j(0,t) = βR RIv , ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎩W = γ j (τ ) j(τ ,t)d τ − μ W. 0

(3.71)

100

3 Nested Immuno-Epidemiological Models

Fig. 3.7 Fit of the function −2x y = f (x) = e−κ e to data in [141]. Estimated parameter is κ = 69671.8

This formulation, although much simpler, is derived under the assumption that at least two serotypes persist. This assumption is consistent with reality but is also necessary to study the antibody enhancement. Models (3.70) and (3.71) contain two classes of infected individuals: those with primary infection and those with secondary infection. We need to specify a within-host model for each class. We can capture the cross-immunity with the within-host models. A recent article discusses simple within-host models of dengue that can reproduce the primary and the secondary infections [15]. Article concludes that the model with natural killer (NK) cells reproduces best the primary infection. Thus, we use that model to model infection in i individuals. The model is an outbreak model with target cells T , infected target cells Ti , and NK cells NK and takes the form T  = −βw Ti T, Ti = βw Ti T − κ Ti Nk , Nk

(3.72)

= qN Ti − dN N.

Article [15] also derives a model that can mimic secondary infection by augmenting the primary infection model (3.72) with the T-cell population C(t). Thus, the secondary infection model that will be liked to class j is given by T  = −βw Ti T, Ti = βw Ti T − κ Ti Nk − δCC, Nk = qN Ti − dN N,

(3.73)

C = qc TiC − dCC. In determining the linking parameters we utilize data in transmission of dengue from [141]. With x = log10 V and y being the probability of transmission, given a bite, we fit a function of Gomperz type y = f (x) = e−κ e

−2x

.

3.4 Immuno-Epidemiological Multi-Strain Models

101

We illustrate the fit in Fig. 3.7. Hence, the transmission functions βi and β j are best approximated by a function of the form

βi (τ ) = ρi e−κiV ,

β j (τ ) = ρ j e−κ jV ,

p

p

(3.74)

where p = −2/ ln 10. To see the value of p, we set x = log10 V in e−2x = e−2 log10 V = elog10 V

−2

=e

lnV −2 ln 10

2

= V − ln 10 = V p .

Acute illness in dengue occurs while the virus is present [77] so we take the diseaseinduced mortality proportional to the viral load.

αi (τ ) = ξiV,

α j (τ ) = ξ jV,

(3.75)

and the recovery inversely proportional to the viral load:

γi ( τ ) =

ηi , V + ε0

γ j (τ ) =

ηj . V + ε0

(3.76)

To understand the dynamics, we study models (3.70) and (3.71). The model has a disease-free equilibrium E0 = ( Λμvv , 0, Λμ , 0, 0, 0). To compute the reproduction number, we look at the stability of the disease-free equilibrium. We linearize around the disease-free equilibrium with perturbation variables Sv = Sv0 + u, Iv = v, S = S0 + x, i = y, R = r, j = z, where Sv0 = Λμvv and S0 = Λμ . Looking at exponential solutions, we obtain the following eigenvalue problem.  ∞ ⎧ 0 ⎪ λ u = −S (βi (τ )y(τ ) + β j (τ )z(τ ))d τ ) − μv u, ⎨ v  ∞0 (3.77) ⎪ ⎩ λ v = Sv0 (βi (τ )y(τ ) + β j (τ )z(τ ))d τ ) − μv v. 0

For the host system, we obtain: ⎧ ⎪ λ x = −βS S0 v − μ x, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ yτ + λ y = −(μ + αi (τ ) + γi (τ ))y(τ ), ⎪ ⎪ ⎪ ⎪ ⎪ y(0) = β S0 v, ⎨ S

 ∞

⎪ ⎪ λr = γi (τ )y(τ )d τ − μ r, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ zτ + λ z = −(μ + α j (τ ) + γ j (τ ))z(τ ), ⎪ ⎪ ⎪ ⎪ ⎩ z(0) = 0.

(3.78)

Solving the equation for z in (3.78), we obtain that z(τ ) = 0. This simplifies the system of the perturbation significantly. The following system for y and v separates from the remaining equations:

102

3 Nested Immuno-Epidemiological Models

 ∞

⎧ ⎪ λ v = Sv0 βi (τ )y(τ )d τ − μv v, ⎪ ⎪ ⎨ 0 yτ + λ y = −(μ + αi (τ ) + γi (τ ))y(τ ), ⎪ ⎪ ⎪ y(0) = β S0 v. ⎩

(3.79)

s

We solve for y(τ ) and substitute in the equation for v, thus obtaining the following characteristic equation G (λ ) = 1 where G (λ ) = and πi (τ ) = e−μτ − ber as follows:

τ 0

βs Sv0 S0

 ∞ 0

βi (τ )e−λ τ πi (τ )d τ λ + μv

,

αi (s)+γi (s)ds . That suggests that we define the reproduction num-

R0 =

βs Sv0 S0

 ∞ 0

βi (τ )πi (τ )d τ μv

.

(3.80)

Standard analysis for age-since-infection structured models now implies the following result: Proposition 3.15. Assume R0 < 1. Then the disease-free equilibrium is locally asymptotically stable. If R0 > 1, then the disease-free equilibrium is unstable. Next we consider endemic equilibria. Solving the ordinary differential equations in the system for the equilibria we get i(τ ) = βS SIv πi (τ ),

j(τ ) = βR RIv π j (τ ),

(3.81)

where π j is defined similar to πi . We compute S from the equation for S: S=

Λ =: g1 (Iv ). βS Iv + μ

(3.82)

We define the following quantities:

Γ=

 ∞ 0

γi (τ )πi (τ )d τ ,

Bi =

 ∞ 0

βi (τ )πi (τ )d τ ,

Bj =

 ∞ 0

β j (τ )π j (τ )d τ . (3.83)

From the equation for R we have

βS SIvΓ =: g2 (Iv ). βR Iv + μ

(3.84)

Λv . Bi g1 (Iv )Iv + B j g2 (Iv )Iv + μv

(3.85)

R= From the equation for Sv we have Sv =

3.4 Immuno-Epidemiological Multi-Strain Models

103

Fig. 3.8 Backward bifurcation in models (3.70) and (3.71). Parameters as follows: Λ = 1, Λv = 0.5, μ = 0.01, μv = 0.3, βS = 0.001, βR = 0.1, Γ = 1.5

Substituting in the equation for Iv we have the following equation for Iv : Sv0 [Bi g1 (Iv ) + B j g2 (Iv )] = 1. Bi g1 (Iv )Iv + B j g2 (Iv )Iv + μv

(3.86)

We have the following result: Proposition 3.16. Assume R0 > 1. Then systems (3.70) and (3.71) have at least one endemic equilibrium. Assume R0 < 1. Then systems (3.70) and (3.71) may have zero or a positive number of endemic equilibria occurring through backward bifurcation. Proof. Denoting by H (Iv ) the left-hand side of Eq. (3.86), we have H (0) = R0 > 1. On the other hand, limIv →∞ H (Iv ) = 0. Hence, there is at least one solution to the equation. In the case R0 < 1 we illustrate the backward bifurcation in Fig. 3.8. The backward bifurcation occurs as a result of reinfection of recovered individuals. Figure 3.8 shows that the larger the B j , the bigger the depth of the bifurcation. B j is larger in case of antibody enhancement during secondary infection with another dengue serotype. Antibody enhancement has been found before to generate backward bifurcation. It was used to explain why the dengue serotypes are quite different from each other [100]. Nested immuno-epidemiological models build upon the properties of age-sinceinfection structured models. Since their transmission, recovery and other rates are defined in terms of the dependent variables of the within-host model they are more stable and much harder to destabilize. Unidirectional nested models are also easier to fit to data since data on both scales exist [170]. Bidirectionally linked nested immuno-epidemiological models are just beginning to be investigated. Future research will illuminate better how the linking functions need to be selected.

Chapter 4

Age-Since-Infection Structured Models Based on Game Theory

4.1 Introduction

Game theory is the process of modeling strategic interactions between two or more players in a situation containing set rules and outcomes. Game theory is used in many disciplines, but we are interested in introducing here its application to infectious diseases. Vaccination against all childhood diseases poses an interesting dilemma to the parents: if enough children in the population are vaccinated, then their child may be protected and taking the risk and the potential side effects of vaccination might be unnecessary. Thus, every parent must answer the question whether to vaccinate or not their child. Thus, situation can be studied and analyzed with game theory. Because the decision depends on the number of infected/recovered and vaccinated children in the population, the decision depends on time. This leads to application of evolutionary game theory. By definition game is any set of circumstances that has an outcome dependent on the actions of two or more “players” (e.g., vaccination can be considered as a game and the players are the parents who take the decision to vaccinate or not). A strategy is a complete plan of action a player will take given the set of circumstances that might arise in the game. Payoff is the payout a player receives from arriving at a particular outcome. The payout can be in any quantifiable outcome. A replicator is a central notion in evolutionary game theory. A replicator is a player producing its own strategy. A replicator system is a set of replicators in a particular environment with a structured pattern of interaction among agents. Talor and Jonker [164] are the pioneers who cast game theory in terms of differential equations. They assumed that the game consists of n strategies. The elements of the payoff matrix ai j give the payoff for strategy i interacting with strategy j. xi denotes the frequencies of players playing strategy i and xi is the frequency of strategy i, that is, the ratio of the players playing strategy i in total population of players. The

© Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1_4

105

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4 Age-Since-Infection Structured Models Based on Game Theory

expected payoff of strategy i is given by fi =

n

∑ ai j x j

j=1

called fitness of strategy i and the average payoff (average fitness) is given by Φ = ∑ xi fi . xi are called the replicators. Then the replicator dynamics with fitness is i=1

described by

dxi (t) = xi ( fi − Φ ), i = 1, 2, · · · , n. dt For instance, if there are only two strategies A and B, the payoff matrix is as follows: AB A a b. B c d

The expected payoff fA and fB satisfy the following equations: fA = xA a + xB b, fB = xA c + xB d. The average payoff is given by the following equation:

Φ = xA fA + xB fB . The replicator dynamics with fitness is described by xA = xA ( fA − xA fA − xB fB ) = xA (xB fA − xB fB ) = xA (1 − xA )( fA − fB ) with

fA − fB = xA (a − c) + xB (b − d) = xA (a − c) + (1 − xA )(b − d) = xA (a − c − b + d) + (b − d).

Then the fitness (expected payoff) for population A satisfies the following differential equation: dxA (t) = xA (1 − xA )[(a − b − c + d)xA + b − d], dt here we have used the fact that xA + xB = 1. From this equation, one can conclude five situations that characterize the dynamics:

4.1 Introduction

107

A dominates B: When a > c and b > d, xA = 1 is the stable equilibrium such that EA = (1, 0). This implies that the strategy A always eliminates B, and selection will favor population playing A no matter what the size of the population playing B is. B dominates A: When a < c and b < d, xB = 1 is the unique stable equilibrium such that EB = (0, 1). By the symmetric characters, selection will result in the state B where all the population plays the strategy B. d−b is an unstable interior A and B are bistable: When a > c and b < d, x∗ = a−c−b+d d−b a−c ∗ equilibrium such that E = ( a−c−b+d , a−c−b+d ). The stability of the boundary equilibria depends on the initial conditions. If xA (0) < x∗ , then EA is stable; if xA (0) > x∗ , then EB is stable. Outcome will change with respect to the initial population playing A. A and B stable coexistence: When a < c and b > d, the interior equilibrium E ∗ = (xA∗ , xB∗ ) is stable. Everyone chooses the opposite strategy of what the opponent does. This indicates that strategy A is the best response to strategy B, and strategy B is the best response to strategy A. This behavior results in a positive fraction playing A and another positive fraction playing B. This means that E ∗ is a Nash equilibrium. A and B are neutral: When a = c and b = d (that is, player 1 plays A and player 2 plays B, then any fraction playing A and another fraction playing B remains constant in time), any constant between 0 and 1 is the equilibrium. It is easy to see the payoff for any population stays the same and has no change. This means that any mixture of A and B is an equilibrium for selection dynamics. Therefore, the natural selection is completely described by the replicator dynamics. In this chapter, based on the replicator dynamics, we focus on the vaccine-making decisions and analyze population behaviors for some diseases. This is helpful for us to quantify how payoff perception impacts the vaccine uptake and coverage levels. Vaccination has been widely considered as one of the effective methods to reducing the morbidity and mortality from infectious diseases. There are two key vaccination policies: voluntary vaccination and mandatory vaccination. There has been a vigorous debate about voluntary vaccination policy failing to protect population adequately. A rational vaccine decision-making is determined by various factors, such as perceived infection risk, potential side effects from vaccination, and vaccinating behaviors of other individuals. Because of the declining familiarity with the interplay between perceived infection risk and potential side effects from vaccination, parents have various reasons for avoiding potential side effects for their children, relying instead on enough other children being vaccinated to provide herd immunity (that is, a form of indirect protection from infectious disease that occurs when a large percentage of a population has become immune to an infection, thereby providing a measure of protection for individuals who are not immune). Therefore, rational decision may lead to a reduced number of vaccination intakes. This is a free-riding dilemma, that is a dilemma to benefit from everybody contributing to the common good without contributing ourselves, between individuals and the public good. Game theory builds a bridge connecting the epidemic models with individual behaviors.

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4 Age-Since-Infection Structured Models Based on Game Theory

Vaccine decision-making based on game theory has been extensively investigated (see, for example, [37, 154]). In these works, it is usually assumed that individuals have full information for perceived infection risk and potential vaccine health risk. Under this assumption, rational vaccine decision-making will get the highest personal utilities, i.e., there exists a Nash equilibrium where no individuals could be better off by unilaterally changing to a different strategy [14, 56]. In [183], Xia and Liu employed a computational approach to characterize the impact of social influence on individuals’ vaccination decision-making while in [185] they also investigated the impact of the two factors, information of the disease prevalence and the perceived vaccination risk, coupled with fading coefficient of awareness spread. Recently, Xu and Cresmman [187] built a nonlinear epidemic model with the smoothed best response to study the game theory based decisions on vaccination. Their results showed that if there is a perceived cost for vaccination, the smoothed best response is very effective in controlling the disease spread, but if vaccination is free, the best response is a good control. Zhang et al. [194] investigated the “double-edged sword” effect on public health conditions for rational decision-making. Shim et al. used game dynamic models to gain insight into the decision-making between vaccine skeptics and vaccine believers [160] and the decision-making with regard to antiviral intervention during an influenza pandemic [159], respectively. The above-mentioned references focus on vaccine uptake based on the perfect information, indicating that parents mastered on perceived infection risk and potential vaccine side effects. In fact, the vaccine uptake behaviors evolve with respect to time. Individuals’ decision to vaccination or not is conducted by imitating others who appear to have adopted more successful strategies. The process is called “imitation dynamics” and it has been studied by some researchers. Bauch [13] investigated parents’ vaccinating decisions for their children with the assumption that the susceptibles behave strategically in accordance with imitation dynamics and studied the dependence of epidemic prevalence and coverage of vaccination on these strategic decisions. Fu et al. [63] proposed an agent based model on a social network with game theory to shed light on how imitation of peers shapes individual vaccination choices. d’Onofrio et al. [44] assumed that the perceived risk of vaccination is a function with respect to the incidence and studied an SIR transmission model with dynamic vaccine demand based on an imitation mechanism. A common assumption in these existing vaccination models on imitation dynamics is that individuals are homogeneously mixed or heterogeneously mixed on social networks [36, 43, 186]. This cannot capture the feature of different transmission rates for childhood diseases which we mentioned earlier. In order to address the imitation behaviors on vaccination decision-making, we develop an age-since-infection epidemic model based on the game theory. We assume that the transmission incidence rate varies according to the infection age and individuals act their behaviors with imperfect information about perceived infection risk and potential vaccine side effects. The payoff of the perceived infection risk is also related with the infection age. Parents imitate the vaccination decision-making of their neighbors and then adopt a vaccination policy.

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

109

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game Theory In [13], Bauch proposed an epidemic model to predict vaccinating behavior based on game theory. The total population is divided into three classes: the Susceptible S(t), the Recovered R(t), and the Infected I(t) at time t. It is assumed that all newborns are susceptible; All the groups decrease due to natural death; Susceptible individuals are infected by infectious individuals and enter the infected group; The infectious individuals exit the group due to natural death and recovery; The recovered group increases due to recovered infectious individuals and the vaccinated newborns decrease due to natural death; The recovered individuals acquire permanent immunity from the disease or vaccination and never leave this group. The incidence has a bilinear form. In order to understand a strategic interaction between individuals when they are deciding whether or not to vaccinate, a replicator equation for an evolutionary population game is introduced, and the payoff function varies according to time. If x denotes the proportion of children that are vaccinated at birth, the equation for x formally reads as dx = kx(1 − x)[−rv + ri mI], dt where the vaccination cost is −rv and the non-vaccination cost is −ri mI. Here ri denotes the perceived probability of suffering significant morbidity upon infection, m quantifies the sensitivity of vaccinating behavior to change in prevalence, k denotes the imitation rate. Under these assumptions, Bauch [13] obtained the following model based on game theory, ⎧ dS(t) ⎪ ⎪ = Λ (1 − x) − β S(t)I(t) − μ S(t), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ dI(t) ⎪ ⎪ = β S(t)I(t) − (μ + γ )I(t), ⎨ dt (4.1) ⎪ dR(t) ⎪ ⎪ = Λ x + γ I(t) − μ R(t), ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dx(t) = kx(1 − x)[−rv + ri mI], dt where Λ is the recruitment rate (note that we modified the recruitment rate to be Λ instead of μ as in [13]), β is the transmission rate, μ is the natural death rate, γ is the recovery rate. It is well known that most childhood diseases have incubation period (the length of time from when a child is exposed to the illness to when the first symptoms appear in that child). For example, the ranges for the incubation period of measles is 7–21 days [30], of mumps is 16–25 days [29], of pertussis is 4–21 days [29]. During the incubation period, whether the exposed child is infectious or not depends

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4 Age-Since-Infection Structured Models Based on Game Theory

on the disease. During the infectious period (the time during which a sick child can transmit the disease to others), the infectivity of infectious individuals may vary. Moreover, inadequate vaccination coverage may increase congenital rubella syndrome numbers by increasing the average age at infection [132]. To describe these phenomena, we introduce the infection age. Let i(t, a) denote the density of infected individuals of infection age a at time t. Similar to the bilinear form, we assume that the incidence rate depends on the infectious age a in the following way,

λ (t) = S(t)

 ∞ 0

β (a)i(t, a)da, 

where β (a) is the transmission rate with infection age a. In epidemiology, 0∞ β (a) i(t, a)da is called the force of infection at time t. For convenience, the perceived vaccination cost is −r , while the potential risk cost for  ∞ non-vaccinator mainly depends on the epidemic transmission, so its cost  i(t,a)da is − 0 N(t) , where N(t) = S(t) + 0∞ i(t, a)da + R(t), or the payoff benefit is ∞ 0

i(t,a)da

 ∞ N(t)

. Then the payoff gain for a vaccinator compared to a non-vaccinator is

− r . This is formally analogous with [187]. Therefore, similar to [13], we obtain the replicator equation for the proportion x of vaccinated children at birth as follows:  ∞ i(t, a)da dx = δ  x(1 − x) 0 − r . dt N(t) 0

i(t,a)da N(t)

Based on the above assumptions, the model with game theory is given by the following system of ordinary and partial differential equations, ⎧  ∞ dS(t) ⎪ ⎪ = Λ (1 − x) − S(t) β (a)i(t, a)da − μ S(t), ⎪ ⎪ ⎪ dt 0 ⎪ ⎪ ⎪ ⎪ ∂ i(t, a) ∂ i(t, a) ⎪ ⎪ ⎪ ⎨ ∂ t + ∂ a = −(μ + γ (a))i(t, a), (4.2)  ∞ ⎪ dR(t) ⎪ ⎪ = Λx+ γ (a)i(t, a)da − μ R(t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪  ∞ ⎪ ⎪ dx(t) ⎪   0 i(t, a)da ⎪ = δ x(1 − x) −r , ⎩ dt N(t) with the boundary condition i(t, 0) = S(t)

 ∞ 0

β (a)i(t, a)da,

t > 0,

and initial conditions S(0) = S0 ≥ 0,

1 i(0, a) = i0 (a) ∈ L+ ,

R(0) = R0 ≥ 0,

x(0) = x0 ∈ [0, 1],

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

111

1 is the set of all integrable functions from R = [0, ∞) into R . γ (a) is the where L+ + + recovery rate with infection age a, δ  is the imitation rate which has the biological meaning of k in (4.1). Due to the biological background, Λ , μ , δ  , and r are all positive, and β = 0 belongs to CBU (R+ , R+ ) with β being not identically zero, where CBU (R+ , R+ ) denotes the set of all bounded and uniformly continuous functions from R+ into R+ . Note that the third equation of (4.2) is decoupled from the others and N  (t) = Λ − μ N(t). So we can assume that N(t) = Λμ . By the last equation of (4.2), we have

∞ dx(t) i(t,a)da = δ  x(1 − x) 0 N(t) − r dt

∞ i(t,a)da = δ  x(1 − x) 0 Λ μ − r

  ∞ Λ r = μδ x(1 − x) i(t, a)da − 0 Λ μ ∞

=: δ x(1 − x) ( 

0

(4.3)

i(t, a)da − r)



Λr where δ = μδ Λ and r = μ . Then we can ignore the third equation and only consider the following system: ⎧  ∞ dS(t) ⎪ ⎪ = Λ (1 − x) − S(t) β (a)i(t, a)da − μ S(t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎨ ∂ i(t, a) ∂ i(t, a) + = −(μ + γ (a))i(t, a), (4.4) ⎪ ∂t ∂a ⎪ ⎪   ⎪ ∞ ⎪ dx(t) ⎪ ⎪ ⎩ = δ x(1 − x) i(t, a)da − r dt 0

with the boundary condition i(t, 0) = S(t)

 ∞ 0

β (a)i(t, a)da,

t >0

and initial condition S(0) = S0 ≥ 0,

1 i(0, a) = i0 (a) ∈ L+ ,

x(0) = x0 ∈ [0, 1].

Thus it follows from the above section, together with [84, 115], (4.4) is well posed, that is, (4.4) has a unique nonnegative solution on [0, ρ ). In the sequel, whenever we say a solution of (4.4), the above assumptions on the boundary and initial conditions are satisfied. Proposition 4.1. Let (S(t), i(t, a), x(t)) be a solution of (4.4) with the maximal interval of existence [0, ρ ) (ρ is allowed to be ∞). Then S(t) ≥ 0, and x(t) ∈ [0, 1] for t ∈ [0, ρ ).

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4 Age-Since-Infection Structured Models Based on Game Theory

Proof. Firstly, from the third equation of (4.4), we have x(t) =

x0 eδ

t ∞ 0 ( 0 i(s,a)da−r)ds

1 − x0 + x0 eδ

for t ∈ [0, ρ ).

t ∞ 0 ( 0 i(s,a)da−r)ds

(4.5)

Since x0 ∈ [0, 1], it follows that x(t) ∈ [0, 1] for t ∈ [0, ρ ). Secondly, by the first equation of (4.4), we see that S(t) = S0 e−μ t−

t ∞ 0 0

β (a)i(s,a)dads

+Λ

 t 0

(1 − x(s))e−μ (t−u)−

t ∞ u 0

β (a)i(s,a)dads

du

for t ∈ [0, ρ ). Clearly, S(t) ≥ 0 for t ∈ [0, ρ ) as S0 ≥ 0 and x(t) ∈ [0, 1] for t ∈ [0, ρ ). Lemma 4.1. Let i0 (a) ≥ 0 hold. Then for t ∈ [0, ρ ], B(t) := i(t, 0) = S(t)

 ∞ 0

β (a)i(t, a)da ≥ 0.

(4.6)

Proof. Integrating the second equation of (4.4) with the boundary condition yields that, for t ∈ [0, ξ ] and a ∈ R+ , we have  t ≥ a, B(t − a)π (a), i(t, a) = (4.7) (a) , t < a, i0 (a − t) ππ(a−t) where B(t) is defined in (4.6) for t ∈ R+ and a ∈ R+ and π (a) = e− Note that, with the help of (4.7), for t ∈ [0, ρ ] B(t) = S(t)

 t 0

a

0 ( μ +γ (s))ds

.

β (a)B(t − a)π (a)da + F(t)



a

(a) where F(t) = S(t) t∞ β (a)i0 (a − t) ππ(a−t) da and π (a) = e− 0 (μ +γ (s))ds . Note that, with the help of (4.7) and Proposition 4.1, S(t) ≥ 0 implies that F(t) =  S(t) 0∞ β (a)i0 (a) ππ(a+t) (a) da ≥ 0. We take constructive form of Picard sequences as follows: ⎧  t ⎨ B (t) =S(t) β (a)B (t − a)π (a)da + F(t), n n−1 , n ∈ N. (4.8) 0 ⎩ B0 =F(t),

First, we show that for any n ∈ N, Bn (t) ∈ C+ [0, ρ ]. By induction method, we readily check that B0 = F(t) ≥ 0. We assume that Bn (t) ≥ 0 holds for n ∈ N. Then Bn+1 (t) = S(t)

 t 0

β (a)Bn (t − a)π (a)da + F(t) = S(t)

 t 0

β (t − a)Bn (a)π (t − a)da + F(t),

so that it follows from the nonnegativity of β and π that Bn+1 (t) is nonnegative for t ∈ [0, ρ ].

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

113

Next, we need to show that Bn (t) converges to B(t) for any t ∈ [0, ρ ]. This implies that {Bk (t)}k∈N is a Cauchy sequence. In order to apply the contraction mapping principle, we define Bˆ k (t) = e−λ t Bk (t), λ > 0, t ∈ [0, ρ ]. In view of the definition of Bk (t), multiplying by e−λ t both sides of Bk (t) yields Bˆ k (t) = S(t)

 t 0

ˆ − a)da + F(t), ˆ β (a)π (a)e−λ a B(t

(4.9)

ˆ = e−λ t F(t). We note that if we prove Bˆ k (t) converges to B(t) ˆ for any where F(t) t ∈ [0, ρ ] then Bk (t) do well B(t). For any n ∈ N,

Bˆ n+1 (t) − Bˆ n (t) ∞ =S(t) ≤S(t) ≤S(t)

 t 0

 t 0

  β (a)π (a)e−λ a Bˆ n (t − a) − Bˆ n−1 (t − a) da

β (a)π (a)e−λ a |Bˆ n (t − a) − Bˆ n−1 |da

 ∞ 0

(4.10)

β (a)π (a)e−λ a da Bˆ n (t − a) − Bˆ n−1 ∞ . 

By the virtue of S(t), we have that S(t) 0∞ β (a)π (a)e−λ a da ≤ R+ . Plugging this estimation into (4.11), we obtain

Λ β¯ μλ

:= Mλ for λ ∈

Bˆ n+1 (t) − Bˆ n (t) ∞ ≤ Mλ Bˆ n (t) − Bˆ n−1 (t) ∞ ≤ Mλn Bˆ 1 (t) − Bˆ 0 (t) ∞ . Therefore, for any m, n ∈ N, and λ large enough

Bˆ m (t) − Bˆ n (t) ∞ ≤ Bˆ m (t) − Bˆ m−1 (t) ∞ + · · · + Bˆ n+1 (t) − Bˆ n (t) ∞ ≤(Mλm−1 + · · · + Mλn ) Bˆ 1 (t) − Bˆ 0 (t) ∞ ≤

(4.11)

Mλn

Bˆ 1 (t) − Bˆ 0 (t) ∞ , 1 − Mλ

hence, Bˆ m (t)− Bˆ n (t) ∞ → 0 as n → ∞ since Mλ ≤ 1 if we choose λ is large enough. ˆ as n → ∞. So does B(t). This implies that Bˆ n (t) → B(t) 1 for t ∈ [0, ρ ). The total population Lemma 4.1 and Eq. (4.7) imply that i(t, ·) ∈ L+ size in system (4.4) 

N1 (t) = S(t) +

∞

i(t, a)da 0

for t ∈ [0, ρ ). With the help of Proposition 4.1, we can obtain dN1 (t) ≤ Λ − μ N1 (t) dt

(4.12)

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4 Age-Since-Infection Structured Models Based on Game Theory

for t ∈ [0, ρ ), which implies that lim sup N1 (t) < ∞. Therefore, the maximal interval t→ρ −

1 × [0, 1]) → of existence for every solution of (4.4) is R+ . Let Φ : R+ × (R+ × L+ 1 R+ × L+ × [0, 1] be the solution semiflow associated with (4.4), that is,

Φ (t, (S0 , i0 , x0 )) = (S(t), i(t, ·), x(t)) Note that, by (4.12), lim sup N1 (t) ≤ R+ , then N1 (t) ≤

Λ μ

t→∞

1 × [0, 1]. for (S0 , i0 , x0 ) ∈ R+ × L+

Λ μ.

Moreover, if N1 (t0 ) ≤

Λ μ

for some t0 ∈

for all t ≥ t0 . Define

1 × [0, 1] : S(t) + Γ = {(S, i, x) ∈ R+ × L+

∞ 0

i(t, a)da ≤

Λ μ }.

Then we have proved the following result. Proposition 4.2. Γ is an attractive and positively invariant set for (4.4). To end this section, we mention that, by (4.5), both sets {(S, i, x) ∈ Γ : x = 0} and {(S, i, x) ∈ Γ : x = 1} are also positively invariant subsets of (4.4). It is clear that for solutions in {(S, i, x) ∈ Γ : x = 1} we have lim (S(t), i(t, a), x(t)) = (0, 0, 1) for t→∞

a ∈ R+ . On the other invariant set {(S, i, x) ∈ Γ : x = 0}, (4.4) reduces to the case without vaccination, that is, ⎧  ∞ dS(t) ⎪ ⎪ = Λ − S(t) β (a)i(t, a)da − μ S(t), ⎨ dt 0 (4.13) ⎪ ⎪ ⎩ ∂ i(t, a) + ∂ i(t, a) = −(μ + γ (a))i(t, a). ∂t ∂a This model has been studied by Magal et al. [118] and their main results are as follows. Theorem 4.1 ([118, Theorems 1.2 and 1.3]). Λ

∞

β (a)π (a)da

(i) If 0 μ ≤ 1, then the disease-free equilibrium ( Λμ , 0) is globally asymptotically stable for the semiflow generated by (4.13). (ii) Let a¯ = sup{a ≥ 0 : β (a) > 0} and 1 : M0 = R+ × {i ∈ L+

Λ

∞

β (a)π (a)da

 a¯ 0

i(a)da > 0}

and

1 )\M . ∂ M0 = (R+ × L+ 0

> 1. Then every solution of (4.13) with the initial value Assume 0 μ in ∂ M0 (respectively, in M0 ) stays in ∂ M0 (respectively, in M0 ). Moreover, each solution with initial value in ∂ M0 converges to ( Λμ , 0). Furthermore, every solution with an initial value in M0 converges to the endemic equilibrium (  ∞ β (a)1π (a)da , (Λ −  ∞ β (a)μπ (a)da )π (a)), which is globally asymptotically stable. 0

0

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

115

The attractivity of the endemic equilibrium in Theorem 4.1(ii) is established by the approach of Lyapunov functionals. We also mention that Theorem 4.1 will play an important role in dealing with the attractivity of the boundary equilibria of (4.4).

4.2.1 Existence of Equilibria and Their Local Stability Recall that π (a) = e−

a

0 ( μ +γ (s))ds

for a ∈ R+ . Define

K=

 ∞ 0

β (a)π (a)da

and R0 =

ΛK . μ

(4.14)

Note that π (a) is the survival probability at infection age a and hence K is the mean value of infectiousness of an infectious child. As the total population size is Λ /μ , R0 is the average number of cases that one typical infected individual can produce during his/her infectious period. In epidemiology, R0 is called the basic reproduction number. As we will see soon, the structure of equilibria of (4.4) depends on the values of R0 . 1 × [0, 1] is an equilibrium of (4.4) if it satisfies (S∗ , i∗ , x∗ ) ∈ R+ × L+  ⎧ 0 = Λ (1 − x∗ ) − S∗ 0∞ β (a)i∗ (a)da − μ S∗ , ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ di∗ (a) = −(μ + γ (a))i∗ (a), ⎨ da

(4.15)

 ⎪ ⎪ i∗ (0) = S∗ 0∞ β (a)i∗ (a)da, ⎪ ⎪ ⎪ ⎪ ⎪  ⎩ 0 = δ x∗ (1 − x∗ )( 0∞ i∗ (a)da − r).

By a direct computation, we see that (4.15) may have at most four equilibria, E1 = (0, 0, 1), E2 = ( Λμ , 0, 0), E3 = (S∗ , i∗1 (0)π (a), 0), E4 = (S∗ , i∗ (0)π (a), x∗ ), where S∗ =

1 , K

where K1 =

i∗1 (0) = ∞ 0

μ (R0 − 1), K

i∗ (0) =

r , K1

x∗ =

μ (R0 − 1) − rK K1 ΛK

.

π (a)da. The existence of equilibria of (4.4) is summarized below.

Theorem 4.2. Let R0 be defined in (4.14). The following statements hold: (i) If R0 ≤ 1, then (4.4) has two equilibria E1 and E2 . , then (4.4) has three equilibria E1 , E2 , and E3 . (ii) If 1 < R0 ≤ 1 + KrK 1μ rK (iii) If R0 > 1 + K1 μ , then (4.4) has four equilibria E1 , E2 , E3 , and E4 .

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4 Age-Since-Infection Structured Models Based on Game Theory

From the biological viewpoint, E1 is the disease-free and pure vaccinator strategy equilibrium, E2 is the disease-free and non-vaccinator strategy equilibrium, E3 is the endemic and non-vaccinator strategy equilibrium, and E4 is the endemic and vaccinator strategy equilibrium. Let E¯ ∗ = (S¯∗ , i¯∗ , x¯∗ ) be an equilibrium of (4.4). Then the linearized system at ∗ ¯ E is ⎧  ∞ dS(t) ¯∗ ∞ ¯∗ ⎪ ⎪ dt = −Λ x − S 0 β (a)i(t, a)da − S 0 β (a)i (a)da − μ S, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ i(t,a) + ∂ i(t,a) = −(μ + γ (a))i(t, a), ∂t ∂a   ⎪ ⎪ i(t, 0) = S(t) 0∞ β (a)i¯∗ (a)da + S¯∗ 0∞ β (a)i(t, a)da, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∞ ∗ ∞ ⎩ dx(t) ∗ ∗ ∗ ¯ dt = δ (1 − 2x¯ )x( 0 i (a)da − r) + δ x¯ (1 − x¯ ) 0 i(t, a)da.

Letting S(t) = s0 eλ t , i(t, a) = y(a)eλ t , and x(t) = x0 eλ t leads to the equation at E¯ ∗ , which is   ˆ λ)  λ + μ + 0∞ β (a)i¯∗ (a)da S¯∗ K( Λ     ∞ ˆ λ)−1 ¯∗ 0= S¯∗ K( 0 0 β (a)i (a)da    0 −δ x¯∗ (1 − x¯∗ )Kˆ (λ ) λ −C 1



1

characteristic      ,   



¯ ∗) ˆ λ ) = 0∞ β (a)π (a)e−λ a da, Kˆ1 (λ ) = 0∞ π (a)e−λ a da and C1 = δ (1 − 2x where K( ∞ ∗ ¯ ( 0 i (a)da − r). Then we can get the local stability of E1 , E2 , and E3 as follows. Theorem 4.3. Let R0 be defined in (4.14). The following statements hold. (i) The disease-free and pure vaccinator equilibrium E1 is always unstable. (ii) The disease-free and non-vaccinator equilibrium E2 is locally asymptotically stable if R0 < 1 and it is unstable if R0 > 1. (iii) The endemic and non-vaccinator equilibrium E3 is locally asymptotically stable and it is unstable if R0 > 1 + KrK . if 1 < R0 < 1 + KrK 1μ 1μ Proof. (i) The characteristic equation for E1 is   λ +μ 0 Λ    0 −1 0  = (λ + μ )(λ − δ r) = 0,    0 0 λ −δr  which has a positive root δ r. This implies that E1 is unstable. (ii) The characteristic equation at E2 is

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

 λ +μ     0   0

117

       = (λ + μ )(λ + δ r) Λ K( Λ  λ ) − 1 = 0. 0  μ K(λ ) − 1 μ  λ +δr  0 Λ  μ K(λ )

Λ

Besides the two negative roots −μ and −δ r, the other roots are given by the equation Λ K(λ ) = 1. (4.16) μ Since

Λ  μ K(0) = R0

 and lim K( λ ) = 0, it follows that (4.16) has a positive root λ →∞

if R0 > 1. If R0 < 1, we claim that all roots of (4.16) have negative real parts. Otherwise, if λ0 is a root of (4.16) with nonnegative real part, then    ∞  Λ  Λ  Λ∞ Λ  1 =  K( λ ) =  β (a)π (a)e−λ0 a da ≤ β (a)π (a)da = K = R0 < 1, μ μ 0 μ 0 μ a contradiction. This proves the claim. Therefore, E2 is stable if R0 < 1 and it is unstable if R0 > 1. (iii) The characteristic equation at E3 is      λ) Λ  λ + μ + i∗1 (0)K S∗ K(      ∗ (0)K ∗ K(  0=  i S λ ) − 1 0 1     ∗  0 0 λ − δ i1 (0)K + δ r   = (λ − δ i∗1 (0)K1 + δ r)[(λ + μ )(S∗ K( λ ) − 1) − i∗1 (0)K]. First, the root of λ − δ i∗1 (0)K1 + δ r = 0 is δ i∗1 (0)K1 − δ r = δ μKK1 (R0 − 1 − rK rK K  μ ), which is positive (respectively, negative) if R0 > 1 + K  μ (respectively,  λ ) − 1) − i∗1 (0)K has R0 < 1 + KrK μ ). Second, we claim that 0 = (λ + μ )(S∗ K( no root with nonnegative real part. By way of contradiction, suppose the characteristic equation has a root λ1 with nonnegative real part. Then  λ1 ) − 1) − i∗1 (0)K. 0 = (λ1 + μ )(S∗ K( Hence,

λ 1 + μ R0 = It follows that

λ1 + μ  K(λ1 ). K

   λ1 + μ    | λ 1 + μ R0 | =  K(λ1 ) ≤ |λ1 + μ |, K

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4 Age-Since-Infection Structured Models Based on Game Theory

which is a contradiction since |λ + μ R0 |2 − |λ1 + μ |2 = 2Re(λ1 )μ (R0 − 1) + μ 2 (R02 − 1) is positive if Re(λ1 ) ≥ 0 and R0 > 1. This proves the claim. To summarize, we have proved (iii). Theorem 4.3 tells us that the disease-free and pure vaccinator equilibrium E1 is always unstable. This means that if the level of the vaccinated children is very high then the unvaccinated children have no incentives to vaccinate since the herd immunity can protect them and they do not care about the potential risk from vaccination. For some small level of disease prevalence, E3 is stable, that is the disease persists at low level but the parents have no incentive to vaccinate. If the disease prevalence increases beyond this threshold, then a proportion of parents do vaccinate their children. The analysis for the stability of E4 is not so easy. In fact, the characteristic equation at E4 is   ˆ λ)  λ + μ + i∗ (0)K  S∗ K( Λ     ∗ ∗ ˆ λ)−1 0= i (0)K S K( 0     ∗ ∗ ∗ ∗ 0 −δ x (1 − x )Kˆ1 (λ ) λ − δ (1 − 2x )(i (0)K1 − r)    λ +μ 1 Λ      λ) K( (4.17) =  rK 0  K1 K −1    0 −δ x∗ (1 − x∗ )K   1 (λ ) λ    ˆ λ) K( rK  = λ (λ + μ ) −1 − −CK 1 (λ ), K K1 where C = Λ rδ x∗ (1−x∗ ) KK1 . Obviously, 0 is not a root to (4.17) when R0 > 1+ KrK . 1μ Then (7.48) is equivalent to ˆ λ) K( r K C Kˆ1 (λ ). = 1+ + K λ + μ K1 λ (λ + μ )

(4.18)

It follows easily that (4.17) has no nonnegative real roots. However, it is hard to see whether (4.17) has roots with nonnegative real parts or not. In order to have a clear picture of the dynamics of (4.4), we make the following further assumption. Assumption 1 Assume that the transmission rate is homogeneous, that is,

β (a) = β , for any a ∈ R+ ,

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

and the transfer rate is



γ (a) =

119

0, a ≤ τ ,

γ, a > τ.

Based on Assumption 1, we observe that K=

 ∞

K1 =

0

β (a)π (a)da =

 ∞ 0

1 − x∗ =

π (a)da =

β β  eγτ , K( λ) = eγτ , λ +μ +γ λ +μ +γ

1 γτ  1 e , K1 (λ ) = eγτ , μ +γ λ +μ +γ

(4.19)

(β r + μ )(μ + γ ) −γτ μ (R0 − 1) − β r1 = 1− e . ΛK βΛ

Theorem 4.4. Let the basic reproduction number R0 > 1 + KrK and Assumption 1 1μ hold. If β > δ x∗ , then the endemic and vaccinator equilibrium E ∗ is locally asymptotically stable. Proof. Plugging (4.19) in (4.18), we have

μ +γ βr rδ x∗ (β r + μ )(μ + γ ) . = 1+ + λ +μ +γ λ + μ λ (λ + μ )(λ + μ + γ ) We simplify this equation and obtain

λ 3 + a1 λ 2 + a2 λ + a3 = 0,

(4.20)

where a1 = β r + μ , a2 = β r(μ + γ ), a3 = rδ x∗ (β r + μ )(μ + r). By Routh-Hurwitz criteria, we have to show that a1 a2 − a3 =β r(μ + γ )(β r + μ ) − rδ x∗ (β r + μ )(μ + r) =r(μ + γ )(β r + μ )(β − δ x∗ )

(4.21)

is positive. Hence, if β > δ x∗ , then the endemic and vaccinator equilibrium E ∗ is locally asymptotically stable. Theorem 4.4 states that if the imitated rate is not large enough, then system (4.4) remains stable under Assumption 1. This means that some parents do not like to imitate others vaccinating their children or this imitative ability is not strong enough, the disease will develop an endemic disease and do not oscillate with respect to time t. From mathematical points of view, there exists a Hopf bifurcation when β = δ x∗ . Theorem 4.5. If R0 > 1 and β = δ x∗ , system (4.4) exhibits a Hopf bifurcation. Proof. In order to prove the existence of Hopf bifurcation for system (4.4), we need to show (4.21) has a pair purely imaginary roots. We assume that λ = iw (w > 0) is a purely imaginary root of (4.21), after a substitution we obtain

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4 Age-Since-Infection Structured Models Based on Game Theory

− iw3 − (β r + μ )w2 + iwβ r1 (μ + γ ) + rδ x∗ (μ + γ )(β r + μ ) = 0.

(4.22)

Separating the real parts and imaginary parts of (4.22), we obtain rδ x∗ (μ + γ )(β r + μ ) = (β r + μ )w2 ,

(4.23)

wβ r(μ + γ ) = w3 .

Solving (4.23)yields w2 = rδ x∗ (μ + γ ) = β r(μ + γ ). Hence, there exists a positive solution w = β r(μ + γ ) if β = δ x∗ . In what follows, we will verify the transversal condition. Directly deviating both sides of (4.21) with respect to δ , we admit d λ (δ ) rδ x∗ (μ + γ )(β r + μ ) . =− 2 dδ 3λ + 2(β r + μ )λ + β r(μ + γ ) Assume that λ (δ ) = v(δ ) + iw(δ ) with v(δ ) = 0 and w(δ ) = Therefore, 

dRe(λ (δ )) −1  −3w2 + β r(μ + γ ) =−  dδ rδ x∗ (μ + γ )(β r + μ ) λ =λ ( δ ) =2 =2

(4.24) 

β r(μ + γ ) rδ x∗ (μ + γ )(β r + μ ) β δ x ∗ (β r + μ )

rδ x∗ (μ + γ ).

(4.25)

> 0.

Hence, the transversal condition readily satisfies and there exists a limit circle around the endemic and vaccinator equilibrium E ∗ . For another specific case, we assume that Assumption 2 Assume that there exists a constant τ ∈ R+ such that

γ (a) = γ , for any a ∈ R, 

and

β (a) =

β ∗ ,t ≥ τ , 0,t < τ .

In what follows, we suppose that β ∗ = β e(μ +γ )τ . Then we can calculate the following values

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

K1 = K=

 ∞ 0

 ∞

1 , μ +γ

β (a)π (a)da =

0

 K 1 (λ ) =  K( λ) =

π (a)da =

 ∞ 0

 ∞ 0

121

β , μ +γ

π (a)e−λ a da =

1 , λ +μ +γ

β (a)π (a)e−λ a da =

β , λ +μ +γ

(4.26)

 λ) μ + γ −λ τ K K( = e , = β, K λ +μ +γ K1 1 − x∗ =

(μ + γ )(μ + β r) , Λβ

C = Λ rδ x∗ (1 − x∗ )

K = rδ x∗ (μ + γ )(μ + β r). K1

With a substitution (4.26) into Eq. (4.18), we have

λ 3 + a1 λ 2 + a2 λ + a3 = (μ + γ )(λ 2 + μλ )e−λ τ where

(4.27)

a1 =2μ + γ + β r, a2 =(μ + γ )(μ + β r), a3 =rδ x∗ (μ + γ )(μ + β r).

If τ = 0, Eq. (4.27) changes into

λ 3 + b1 λ 2 + b2 λ + b3 = 0, with

(4.28)

b1 = μ + β r, b2 = β r(μ + γ ), b3 = rδ x∗ (μ + γ )(μ + β r).

We readily verify that b1 b2 − b3 = r(μ + γ )(μ + β r)(β − δ x∗ ). From what has been discussed, we obtain the following lemma. Lemma 4.2. If τ = 0, and β > δ x∗ , then the endemic equilibrium E ∗ of system (4.4) is locally asymptotically stable. Next, we will verify that λ = iw(w > 0) is a solution of Eq. (4.27). Lemma 4.3. Suppose that β > δ x∗ > β (μ + γ )/(2(μ + β r)). Equation (4.27) has two pairs purely imaginary solutions.

122

4 Age-Since-Infection Structured Models Based on Game Theory

Proof. We separate the real and imaginary parts and obtain −w3 + a2 w =(μ + γ )(w2 sin wτ + μ w cos wτ ), −a1 w2 + a3 =(μ + γ )(−w2 cos wτ + μ w sin wτ ).

(4.29)

Squaring (4.29) and summing together yields w6 + (a21 − 2a2 − (μ + γ )2 )w4 + (a22 − 2a1 a3 − μ 2 (μ + γ )2 )w2 + a23 = 0.

(4.30)

Let w2 = z. Then (4.30) can be rewritten as z3 + (a21 − 2a2 − (μ + γ )2 )z2 + (a22 − 2a1 a3 − μ 2 (μ + γ )2 )z + a23 = 0. If the above equation has a positive solution, system (4.30) has a positive solution √ w = z. By the Descartes’ rule of signs, we need to check the coefficients of (4.30). Note that a21 − 2a2 − (μ + γ )2 = (μ + β r)2 , and a22 − 2a1 a3 − μ 2 (μ + γ )2 = (μ + γ )2 (μ + β r)2 − 2(2μ + γ + β r)(μ + γ )(μ + β r)rδ x∗ − μ 2 (μ + γ )2 = β r(μ + γ )2 (2μ + β r) − 2(2μ + γ + β r)(μ + γ )(μ + β r)rδ x∗ = r(μ + γ )[β (2μ + β r)(μ + γ ) − 2δ x∗ (2μ + γ + β r)(μ + β r)] = r(μ + γ )[(2μ + β r)(β (μ + γ ) − 2δ x∗ (μ + β r) − 2δ x∗ (μ + β r)]. If δ x∗ > β (μ + γ )/(2(μ + β r)), then a22 − 2a1 a3 − μ 2 (μ + γ )2 < 0. Hence, system (4.29) has two positive solutions denoted by w+ and w− , respectively. Now, we are going to show the transversal condition holds when λ = iw± . First, we give a general lemma to show the transversal condition. Proposition 4.3. Suppose equation

λ 3 + a1 λ 2 + a2 λ + a3 = e−λ τ λ (b1 λ + b2 )

(4.31)

has one or two different purely imaginary root λ = iw, w > 0. Then (4.31) satisfies the transversal condition, i.e. dReλ  = 0.  d τ λ =iw Proof. First, we rewrite the form of Eq. (4.31) as follows:

λ + d1 +

d2 a3 = b1 e−λ τ , + c + λ λ (λ + c)

(4.32)

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

123

where c = bb21 , d1 = a1 − c, and d2 = (a1 − c)c. We assume that λ = iw is the solution of (4.32) in the term of iw + d1 +

a3 d2 + = b1 e−iwτ . c + iw iw(c + iw)

(4.33)

Separating the real and imaginary parts of (4.33), we have d2 w 2 c + w2

w−

−

a3 c 2 w(c + w2 )

= −b1 sin wτ , (4.34)

d2 c a3 − = b1 cos wτ . d1 + 2 c + w2 c2 + w2 Squaring (4.34) and summing together, we obtain H(w) := (w −

d2 w a3 c d2 c a3 )2 + (d1 + 2 − − )2 − b21 = 0. c2 + w2 w(c2 + w2 ) c + w2 c2 + w2

It follows from the assumption that H(w) = 0 has one or two different roots. Observe that

  d2 w a3 c d(c2 − w2 ) a3 c(c2 + 3w2 ) H  (w) =2 w− 2 − + 1 − c + w2 w(c2 + w2 ) c2 + w2 w(c2 + w2 )    d2 c a3 2a3 w 2d2 cw − + + d1 + 2 − . c + w2 c2 + w2 c2 + w2 (c2 + w2 )2 Second, we deviate Eq. (4.32) with respect to τ to obtain 

dλ dτ

−1

=

a3 (c+2λ ) d2 1 − (λ +c) 2 − λ 2 (λ +c)2

−λ (λ + d1 +

d2 λ +c

+

a3 λ (λ +c) )

−

τ , λ

(4.35)

With a substitution λ = iw of (4.35), we have 

dλ dτ

−1   

λ =iw

=

a3 (c+2iw) d2 1 − (iw+c) 2 − −w2 (iw+c)2 a3 d2 −iw(iw + d1 + iw+c + iw(iw+c) )

τ iw

a3 c(c −3w ) 2 (c −w ) 1 − d(w 2 +c2 )2 + w2 (w2 +c2 )2 2

=

−

2

2

2

 a3 a3 c d2 w 2c −iw d1 + w2d+c 2 − w2 +c2 + i(w − c2 +w2 − w(w2 +c2 ) ) +i

2d2 cw (c2 +w2 )2

3w − (c22a+w 2 )2

 a3 a3 c d2 w 2c −iw d1 + w2d+c 2 − w2 +c2 + i(w − c2 +w2 − w(w2 +c2 ) )

τ iw f1 f3 + f2 f4 + i( f1 f4 − f2 f3 ) = , −iw( f12 + f22 ) −

(4.36)

124

where

4 Age-Since-Infection Structured Models Based on Game Theory

d2 c a3 − , w2 + c2 w2 + c2 d2 w a3 c , − f2 =w − 2 c + w2 w(w2 + c2 )

f1 =d1 +

f3 =1 −

2d2 cw 2 (c + w2 )2

f4 = Therefore,

d2 (c2 − w2 ) a3 c(c2 − 3w2 ) + 2 2 , (w2 + c2 )2 w (w + c2 )2



dReλ dτ

−1

−

2a3 w . 2 (c + w2 )2

=−

f1 f4 − f2 f3 , w( f12 + f22 )

(4.37)

Recalling H  (w), we have ! Sign

dReλ dτ

−1 "

  = Sign H  (w) .

By the virtue of H(w) with respect to w, we obtain H(w− ) < 0, and H(w+ ) > 0,

−1 λ = 0. where w± is the solution of H(w). So that dRe dτ Solving Eq. (4.29) for τ j , j = 1, 2, · · · , we have ⎧ (a −μ )w2 +(a2 μ −a3 ) ⎪ ), ⎨2 jπ + w1 arccos( 1 (μ +γ )(μ 2 −w 2) τj = ⎪ ⎩2( j + 1)π − 1 arccos( (a1 −μ )w2 +(a2 μ −a3 ) ), w (μ +γ )(μ 2 −w2 )

(a1 −μ )w2 +(a2 μ −a3 ) (μ +γ )(μ 2 −w2 )

≥ 0,

(a1 −μ )w2 +(a2 μ −a3 ) (μ +γ )(μ 2 −w2 )

< 0,

j=1, 2, · · · . (4.38)

Theorem 4.6. Let the condition of Lemma 4.3 hold. Then system (4.4) admits the oscillation in form of Hopf bifurcation at τ = iw± .

4.2.2 The Attractivity of Boundary Equilibria and Disease Persistence In this section, we first study the attractivity of the boundary equilibria E2 and E3 by applying Theorem 4.1 and the comparison principle. To establish the attractivity of E2 , we need Fluctuation Lemma A.12 in Appendix A. Theorem 4.7. Suppose that R0 ≤ 1. Then the equilibrium E2 attracts all solutions of (4.4) with (S0 , i0 , x0 ) ∈ {(S, i, x) ∈ Γ : x ∈ [0, 1)}.

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

125

Proof. Let (S(t), i(t, a), x(t)) be any solution of (4.4) with initial condition and boundary conditions from {(S, i, x) ∈ Γ : x ∈ [0, 1)}. Then x(t) ∈ [0, 1) for t ∈ R+ and hence (S(t), i(t, x)) satisfies ⎧ ∞ ⎨ dS(t) dt ≤ Λ − S(t) 0 β (a)i(t, a)da − μ S(t), ⎩ ∂ i(t,a)

+ ∂ i(t,a) ∂ a = −( μ + γ (a))i(t, a).

∂t

¯ i¯(t, a)) be the solution of the auxiliary system Let (S(t), ⎧ ∞ ⎨ dS(t) dt = Λ − S(t) 0 β (a)i(t, a)da − μ S(t) ⎩ ∂ i(t,a) ∂t

(4.39)

+ ∂ i(t,a) ∂ a = −( μ + γ (a))i(t, a) 

∞ ¯ ¯ with (S¯0 , i¯0 ) = (S0 , i0 ) and i¯(t, 0) = S(t) 0 β (a)i(t, a)da. With the help of the com¯ parison principle, it is easy to see that 0 ≤ S(t) ≤ S(t) and i(t, a) ≤ i¯(t, a) for 2 ¯ i¯(t, a)) converges (t, a) ∈ R+ . Since R0 ≤ 1, it follows from Theorem 4.1 that (S(t), Λ to ( μ , 0). Then

0 ≤ lim sup t→∞

 ∞ 0

i(t, a)da ≤ lim sup t→∞

which implies that

 ∞

lim

t→∞ 0

 ∞ 0

i¯(t, a)da = 0,

i(t, a)da = 0.

Next, we show lim x(t) = 0. Choose ε ∈ (0, r). Then there exists a t0 ∈ R+ such that

∞ 0

t→∞

β (a)i(t, a)da ≤ ε for t ≥ t0 . By the third equation of (4.4), we get dx ≤ δ x(1 − x)(ε − r) dt

for t ≥ t0 .

Remembering x(t) ∈ [0, 1) for t ∈ R+ and ε < r, we easily see that lim x(t) = 0. Finally, we show lim S(t) = S(sn ) → S∞ , and

t→∞ dS(sn ) dt →

Λ μ.

t→∞

By Lemma A.14, choose {sn } such that sn → ∞,

0 as n → ∞. Taking limitation in

dS(sn ) = Λ (1 − x(sn )) − S(sn ) dt

 ∞ 0

β (a)i(sn , a)da − μ S(sn )

produces 0 = Λ − μ S∞ or S∞ =

Λ μ.

This, combined with

S∞

≤

Λ μ,

yields lim S(t) = t→∞

Λ μ.

To summarize, we have shown that (S(t), i(t, a), x(a)) converges to E2 . This completes the proof.

126

4 Age-Since-Infection Structured Models Based on Game Theory

For the attractivity of E3 , we define

Γ0 = {(S, i, x) ∈ Γ :

 a¯ 0

i(a)da > 0, x ∈ [0, 1)},

where a¯ = sup{a ∈ R+ : β (a) > 0}. With the integrated semigroup approach, one can show (similar to Magal et al. [118], for example) that Γ0 is a positively invariant subset for (4.4). Theorem 4.8. Suppose that 1 < R0 < 1 + μrK K1 . Then the equilibrium E3 attracts all solutions of (4.4) with (S0 , i0 , x0 ) ∈ Γ0 . Proof. Let (S(t), i(t, a), x(t)) be a solution of (4.4) with (S0 , i0 , x0 ) ∈ Γ0 . We first show lim x(t) = 0. From the proof of Theorem 4.7, we know that t→∞

¯ 0 ≤ S(t) ≤ S(t)

and

0 ≤ i(t, a) ≤ i¯(t, a)

(4.40)

for (t, a) ∈ R2+ . Since R0 > 1, it follows from Theorem 4.1 that ¯ → S∗ S(t)

1 as t → ∞. i¯(t, ·) → i∗1 (0)π (·) in L+

and

(4.41)

Then we can obtain  ∞

lim sup t→∞

0

i(t, a)da ≤ lim sup t→∞

≤ lim sup t→∞

=

 ∞ 0

 0

i¯(t, a)da

∞

i∗1 (0)π (a)da +

 ∞ 0

 |i¯(t, a) − i∗1 (0)π (a)|da

K1 μ (R0 − 1). K

. Choose ε > 0 such that KK1 μ (R0 − 1) + ε < r, which is possible since R0 < 1 + KrK 1μ So there exists t0 ∈ R+ such that  ∞ 0

i(t, a)da ≤

K1 μ (R0 − 1) + ε K

for t ≥ t0 .

By the third equation of (4.4), we have dx(t) K1 μ ≤ δ x(t)(1 − x(t))( (R0 − 1) + ε − r) dt K This, together with lim x(t) = 0.

K1 μ K (R0

for t ≥ t0 .

− 1) + ε − r < 0 and x(t) ∈ [0, 1) for t ∈ R+ , gives

t→∞

1 as t → ∞. For any Next, we show that S(t) → S∗ and i(t, ·) → i∗1 (0)π (·) in L+ 1 η ∈ (0, 1 − R0 ), it follows from lim x(t) = 0 that there exists t1 ∈ R+ such that t→∞

0 ≤ x(t) ≤ η

for t ≥ t1 .

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

127

Then, for t ≥ t1 , (S(t), i(t, a)) satisfies ⎧ ∞ ⎨ dS(t) dt ≥ Λ (1 − η ) − S(t) 0 β (a)i(t, a)da − μ S(t), ⎩ ∂ i(t,a) ∂t

+ ∂ i(t,a) ∂ a = −( μ + γ (a))i(t, a).

ˆ iˆ(t, a)) be the solution of the auxiliary system Let (S(t), ⎧ ∞ ⎨ dS(t) dt = Λ (1 − η ) − S(t) 0 β (a)i(t, a)da − μ S(t) ⎩ ∂ i(t,a) ∂t

+ ∂ i(t,a) ∂ a = −( μ + γ (a))i(t, a)

ˆ with (Sˆ0 , iˆ0 ) = (S(t1 ), i(t1 , ·)) and iˆ(t, 0) = S(t) parison principle yields ˆ ≤ S(t + t1 ) 0 ≤ S(t)

∞ 0

iˆ(t, a)da. Then applying the com-

0 ≤ iˆ(t, a) ≤ i(t + t1 , a)

and

(4.42)

for (t, a) ∈ R2+ . Since (1 − η )R0 > 1, applying Theorem 4.1 again, we see that ˆ → S∗ S(t)

and

μ iˆ(t, ·) → (R0 (1 − η ) − 1)π (·) in L1+ as t → ∞. K

(4.43)

Notice that, for (t, a) ∈ R2+ , |i(t, a) − i∗1 (0)π (a)| ≤ |i¯(t, a) − i∗1 (0)π (a)| + |iˆ(t, a) − i∗1 (0)π (a)| ≤ |i¯(t, a) − i∗1 (0)π (a)| + |iˆ(t, a) −

μ (R0 (1 − η ) − 1)π (a)| K

+Λ ηπ (a). By (4.40)–(4.43), we have S∗ = Sˆ∞ ≤ S∞ ≤ S∞ ≤ S¯∞ = S∗ and  ∞

lim sup t→∞

0

|i(t, a) − i∗1 (0)π (a)|da ≤ lim sup t→∞

+ + ≤



∞ 0

|i¯(t, a) − i∗1 (0)π (a)|da

 ∞  μ ˆ  i(t, a) − (R0 (1 − η ) − 1)π (a) da 0

 ∞ 0

Λ η. μ

K  Λ ηπ (a)da

128

4 Age-Since-Infection Structured Models Based on Game Theory



By the arbitrariness of η , we have lim supt→∞ 0∞ |i(t, a) − i∗1 (0)π (a)|da = 0. There1 as t → ∞. This completes the proof. fore, S(t) → S∗ and i(t, ·) → i∗1 (0)π (·) in L+  ∞ Finally, we study the disease persistence of (4.4). Define ρ : (S, i, x) ∈ Γ → 0

i(a)da. Let

D0 = {(S0 , i0 , x0 ) ∈ Γ : there exists a t ∈ R+ such that ρ (Φ (t, (S0 , i0 , x0 )) > 0} and x ∈ [0, 1)} . We distinguish two kinds of persistence. • The disease in (4.4) is uniformly weakly ρ -persistent if there exists an η > 0, independent of the initial conditions, such that if (S0 , i0 , x0 ) ∈ D0 then lim sup ρ (Φ (t, (S0 , i0 , x0 )) > η . t→∞

• The disease in (4.4) is uniformly strongly ρ -persistent if there exists an η > 0, independent of the initial conditions, such that if (S0 , i0 , x0 ) ∈ D0 then lim inf ρ (Φ (t, (S0 , i0 , x0 )) > η . t→∞

Lemma 4.4. Assume R0 > 1. Then (4.4) is uniformly weakly ρ -persistent. Proof. Since R0 > 1, there exists an ε such that 0 < ε < min(1, r) and  Λ (1 − ε ) ˆ ε ) > 1. K( − ε μ + β¯ ε

(4.44)

By way of contradiction, we assume that there exists (S0 , i0 , x0 ) ∈ D0 such that

ε lim sup ρ (Φ (t, (S0 , i0 , x0 )) < . 2 t→∞ Then there is a t1 ∈ R+ such that

ρ (Φ (t, (S0 , i0 , x0 )) < ε

for t ≥ t1 .

As before, it follows from x(t) ∈ [0, 1) for t ∈ R+ , ε < r, and dx ≤ δ x(1 − x)(ε − r) dt

for t ≥ t1

that lim x(t) = 0. Therefore, there is a t2 ≥ t1 such that t→∞

0 ≤ x(t) < ε

for t ≥ t2 .

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

Note that

∞ 0

129

 β (a)i(t, a)da ≤ β¯ 0∞ i(t, a)da < β¯ ε for all t ≥ t2 . As a result, we have

dS(t) ≥ Λ (1 − ε ) − (μ + β¯ ε )S(t) dt which implies that lim inf S(t) ≥ t→∞

for t ≥ t2 ,

Λ (1 − ε ) . μ + β¯ ε

Then there exists t3 ≥ t2 such that S(t) ≥

Λ (1 − ε ) −ε μ + β¯ ε

for t ≥ t3 .

With the help of (4.7), we easily see that   t Λ (1 − ε ) B(t) ≥ −ε β (a)B(t − a)π (a)da μ + β¯ ε 0

for t ≥ t3 .

(4.45)

By replacing the initial condition with (S(t3 ), i(t3 , ·), x(t3 )), we can assume that (4.45) holds for all t ∈ R+ . Note that both B(·) and β (·)π (·) are bounded functions on R+ and hence their Laplace transforms exist at least on (0, ∞). It follows from (4.45) (with t3 = 0) that  Λ (1 − ε ) ˆ ˆ λ )K( ˆ λ) B(λ ) ≥ − ε B( for λ > 0, (4.46) μ + β¯ ε ∧

where · denotes the Laplace transform of a function. As B(·) is not identically zero ˆ λ ) > 0 for λ ∈ (0, ∞). It follows from (4.46) that ( Λ (1−¯ ε ) − on R+ , we know that B( μ +β ε ˆ λ ) ≤ 1 for λ ∈ (0, ∞). In particular, ( Λ (1−¯ ε ) − ε )K( ˆ ε ) ≤ 1, which contradicts ε )K( μ +β ε with (4.44). This completes the proof.

Next, we establish the uniformly strongly ρ -persistence. For this purpose, it is crucial to show Φ has a global compact attractor in D0 . A global compact attractor A is a maximal compact invariant set in D0 such that for any open set that contains A , all solutions of (4.4) that start at zero from a bounded set are contained in that open set, at least for sufficiently large time. To establish the existence of global attractors, we need the two lemmas (see Lemmas A.13 and A.14) in Appendix A. The next result will be needed in the discussion of Φ being asymptotically smooth. Lemma 4.5. For any ε > 0, there exists δ > 0 such that |B(t + h) − B(t)| ≤ ε h

for all t ∈ R+ , 0 < h < δ , and (S0 , i0 , x0 ) ∈ Γ . (4.47)

The proof follows a familiar process Lemma A.10 in Appendix. This lemma helps us to look for a global attractor of the solution semiflow Φ . By Lemma 4.4, we know that Φ (t, D0 ) ⊆ D0 for t ∈ R+ . So it induces a semiflow on D0 .

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4 Age-Since-Infection Structured Models Based on Game Theory

Lemma 4.6. If R0 > 1, then there exists a global attractor A for the solution semiflow Φ of (4.4) in D0 . Proof. With the help of Proposition 4.2 and Lemma 4.4, we only need to show that the restricted semiflow on D0 is asymptotically smooth. This is done by applying Lemma 3.4.6 in [161] as follows. For t ∈ R+ and (S0 , i0 , x0 ) ∈ D0 , let

Φˆ (t, (S0 , i0 , x0 )) = (0, iˆ(t, ·), 0), Φ˜ (t, (S0 , i0 , x0 )) = (S(t), i˜(t, ·), x(t)), where



i˜(t, a) =

i(t, a)

for 0 ≤ a ≤ t

0

for t < a

and

#

 iˆ(t, a) = i(t, a) − i˜(t, a) =

 =

B(t − a)π (a)

for 0 ≤ a ≤ t

0

for t < a

0 (a) i0 (a − t) ππ(a−t)

for 0 ≤ a ≤ t, for t < a.

# (4.48)

(4.49)

Then Φ = Φˆ + Φ˜ . Clearly, both iˆ and i˜ are nonnegative. It follows from (4.49) that

Φˆ (t, (S0 , i0 , x0 )) = iˆ(t, ·) 1 = =

 ∞ t

 ∞ 0

i0 (a − t) i0 (a)

≤ e−μ t

 ∞ 0

π (a) da π (a − t)

π (a + t) da π (a)

i0 (a)da

= e−μ t i0 1 ≤ e−μ t (S0 , i0 , x0 ) and hence Φˆ satisfies the assumptions in Theorem A.3 in Appendix A. Now, we show that Φ˜ is complete continuous, that is, the set {Φ˜ (t, (S0 , i0 , x0 )) : (S0 , i0 , x0 ) ∈ B} is precompact for any fixed t ∈ R+ and any bounded set B ⊆ D0 . This is done by applying the Fréchet-Kolmogorov Theorem [193]. First, it follows easily from the definitions of Φ˜ , D0 , and Γ that {Φ˜ (t, (S0 , i0 , x0 )) : (S0 , i0 , x0 ) ∈ B} is bounded and this verifies the first condition of the Fréchet-Kolmogrov Theorem. Second, by (4.48), the third condition of Fréchet-Kolmogorov Theorem is satisfied. Finally, we verify the second condition of the Fréchet-Kolmogrov Theorem. It suffices to show that

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . .

131

lim i˜(t, ·) − i˜(t, · + h) 1 = 0 uniformly in i˜(t, ·) ∈ {Φ˜ (t, (S0 , i0 , x0 )) : (S0 , i0 , x0 ) ∈ B}.

h→0+

(4.50)

If t = 0, then (4.50) holds automatically since i˜(0, ·) = 0 by (4.48). Without loss of generality, we assume that t > 0. Since we concern with the limit as h tends to 0+ , we only consider h ∈ (0,t). Then

i˜(t, ·) − i˜(t, · + h) 1 = =

 ∞ 0

|i˜(t, a) − i˜(t, a + h)|da

 t−h 0

|B(t − a − h)π (a + h) − B(t − a)π (a)|da

 t

B(t − a)π (a)da  2  t−h Λ h+ |B(t − a − h)π (a + h) − B(t − a)π (a)|da ≤ β¯ μ 0  2  t−h Λ ¯ ≤β h+ B(t − a − h)|π (a + h) − π (a)|da μ 0 +

t−h

 t−h

|B(t − a − h) − B(t − a)|π (a)da 2  t−h Λ ¯ + h(1 + t k) |B(t − a − h) − B(t − a)|π (a)da ≤ β¯ μ 0   a+h as we know B(t) ≤ β¯ ( Λμ )2 for t ∈ R+ and |π (a+h)− π (a)| = π (a) 1 − e− a k(s)ds ¯ This, combined with Lemma 8.6, immediately gives (4.50) and hence we have ≤ kh. completed the proof. +

0



Now, with the assistance of Lemmas 4.4, 4.6, and [167, Theorem 2.3], we can obtain the following result. Theorem 4.9. If R0 > 1, then (4.4) is uniformly strongly ρ -persistent.

4.2.3 Numerical Simulations In this section, assume that Assumption 1 holds. Numerical results are used to illustrate the theoretical results obtained in the previous sections. We fix the transmission rate as follows:  0, 0 ≤ a ≤ τ , β (a) = β , a ≥ τ. Then we choose the other parameters as Λ = 4, β = 0.01, μ = 1/(50 × 365), and γ (a) = 0.01(1 + sin(((a − 5)π )/10)) + 1/50. Theorem 4.3, Proposition 7.9, and Theorem 4.8 imply that δ , τ , and r play an important role in the evolution of (4.4). Recall that δ represents the imitation rate

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4 Age-Since-Infection Structured Models Based on Game Theory

describing the imitation behaviors and the vaccination behaviors for children are determined by the imitation rate. For some larger imitation rate δ , system (4.4) exhibits limit cycles and hence it can be destabilized by the imitation rate δ . Figure 4.1a shows the bifurcation diagram by taking δ as a bifurcation parame0.35

3.5

0.3

3

0.25

x(t)

I(t)

2.5

2

0.2

0.15 1.5 0.1 1

0.05

0.5 0.5

0

1

1.5

δ

(a)

2

0

50

100

150

200

250

300

Time t

(b)

Fig. 4.1 With r = 1.8 and τ = 0.01, the imitation rate δ can destabilize (4.4). (a) The bifurcation diagram showing I(t) with respect to the imitation rate δ with δ varying from 0.5 to 2; (b) Representative phase diagram in the Ix-plane with δ being 0.5(blue), 1.0(red), and 1.5(orange), respectively, and the initial condition (S0 , i0 , x0 ) = (0.5, 5e−0.4a , 0.1)

ter, which indicates that the amplitude of the oscillations increases as the imitation rate does, while Fig. 4.1b shows time series of x(t) with δ = 0.5, 1.0, and 1.5, and the initial condition (S0 , i0 , x0 ) = (0.5, 5e−0.4a , 0.1). The perceived vaccine risk r impacts the prevalence of the disease spread. If the cost of the vaccination is high, such as a vaccine scare or a strong side effect, the frequency of the vaccination will gradually decrease and the prevalence of the disease will increase. Figure 4.2a shows bifurcation diagram of the solution with 1 . We can see respect to r with Λ = 4, δ = 0.1, τ = 0.01, β = 0.01, and μ = 50×365 that the vaccination proportion decreases as the cost of vaccination increases. The latent period τ is also a key parameter for controlling the disease spread. For a childhood disease, if some medicine can extend the latent period of the disease, the prevalence of the disease may be slowed down. Figure 4.2b shows that τ can destabilize (4.4) and oscillations occur when τ is from 0.1 to 0.5. Here R = 1.8 and the other parameters except τ have the same values as above for the impact of r. Also, the prevalence of the disease oscillates with respect to τ when τ > 0.2.

4.2 Imitation Dynamics of Vaccine Decision-Making Behaviors Based on the Game. . . 0.75

133

5 4.5

0.7

4 3.5

I(t)

x(t)

0.65 0.6

3 2.5 2

0.55

1.5 1

0.5

0.5 0.45 0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95

1

0 0.1

r

(a)

0.15

0.2

0.25

0.3

τ

0.35

0.4

0.45

0.5

(b)

Fig. 4.2 With Λ = 4, δ = 0.1, β = 0.01, μ = (a) bifurcation diagram with payoff of the vaccination rate r varying from 0.5 to 1; (b) bifurcation diagram with respect to the latent period τ as bifurcation parameter and r = 1.8 1 50×365 ,

4.2.4 Discussion From the theoretical analysis in the above sections, the pure vaccinator equilibrium E1 is always unstable. Parents have no incentive to vaccinate if the vaccination coverage is high. The disease-free and non-vaccinator equilibrium E2 is globally asymptotically stable if and only if R0 < 1. If 1 < R0 , then there is an endemic and non-vaccinator equilibrium E3 , which is locally asymptotically stable if . The behavior of the endemic and vaccinator equilibrium E4 is 1 < R0 < 1 + KrK 1μ determined by the relationship between the transmission rate β and imitation proportion δ x∗ . From the biological perspective, the childhood diseases are controlled if the transmission rates are small enough; Otherwise, the diseases will spread and become endemic diseases in some region. Moreover, if vaccination cost is high, parents have a heavy burden from vaccinating their children, then they would avoid vaccinating their children and this leads to the spread of childhood diseases. If the vaccination cost is not too expensive and the total sampling rate for non-vaccinators imitating vaccinators is small enough, even if some of the parents would vaccinate their children, childhood diseases still spread in these areas. If, however, the sampling rate for non-vaccinators imitating vaccinators is large enough, this will lead to imitating phenomena switching, i.e., first, non-vaccinators imitating vaccinators decreases, and then the vaccination side effects will increase. Hence, vaccinators imitating non-vaccinators increases, and then the diseases will spread. Then the phenomenon will continue to cycle. The differences in infectivity, involved in childhood diseases, is incorporated through β (a), the different transmission ability for the different values of the infection age a. For special cases, the different transmissibility can be described by the parameters τ and β . Suitable higher value of τ will produce a limit cycle.

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4 Age-Since-Infection Structured Models Based on Game Theory

When there is no cost for vaccination, i.e., r = 0, the payoff benefit is larger  than the cost of vaccination as long as 0∞ i(t, a)da is not zero. From the third equation of (4.4), the frequency of vaccination monotonously increases. It follows from the first two equations combined with the third equation of (4.4) that system (4.4) other hand, if the cost of vaccination is not free, and the initial evolves to E2 . On the  conditions satisfy 0∞ i(t0 , a)da > r, the frequency of the vaccination monotonously increases. It follows from  the first and second equation of (4.4) that infected individuals decrease until 0∞ i∗ (a)da = r. System (4.4) evolves to the endemic and non-vaccinator equilibrium E3 or a stable endemic and vaccinator equilibrium E4 which depends on whether the initial vaccinated proportion is equal or larger than  the propor0. When 0∞ i(t, a)da < r, it follows from the third equation of (4.4) that  tion vaccinated x decreases. Prevalence of the disease increases until 0∞ i∗ (a)da = r. This implies that (4.4) also evolves to the endemic and non-vaccinator equilibrium E3 or the endemic and vaccinator equilibrium E4 which depends on R0 . In summary, parents can make rational decisions in favor of disease control if they understand the interplay between the perceived vaccination risk, prevalence of the disease, and the variable transmission abilities of the children. Our investigation implies that high vaccination coverage can’t guarantee the elimination of the disease for a voluntary vaccination policy. On the other hand, limited vaccination may be harmful for control of the disease spread. The amount of up-to-date information for the vaccine use has two opposite effects: knowledge about the disease prevalence encourages parents to take the vaccine for their children, on the other hand the potential side effects discourage them to take vaccine for their children. Rational decision-making individuals, depending on updated information about the perceived vaccine risk compared with the prevalence of the disease, decide whether or not to vaccinate their children. Based on the game theory, we conclude that there should be sufficient information for the parents to make rational decision whether or not to vaccinate their children.

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles In the last section of this chapter, we present the voluntary strategy implementing vaccination of newborns in the childhood diseases. In this section, we mainly focus on the vaccination strategies applied to the susceptible individuals (of any age). Making a decision on a vaccine uptake depends on the potential payoff and perceived cost of the vaccine. Benefit-cost compels the individuals to make rational decisions. Game theory has been used to predict the evolution of the individuals behavior. Generally, the vaccination rate is a constant which is not changed with respect to the individuals behavior. To our best knowledge, individuals making vaccine decision based on interplay between the benefit and the cost of side-effects for the vaccination has not been considered. Motivated by the above assumptions, we use the variable vaccination rate instead of the constant vaccination rate based on the game theory. The total population is divided into four classes: susceptible,

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles

135

infectious, recovered, vaccinated classed, denoted by S(t), i(t, a), R(t), and V (t), respectively. x(t) denotes the probability of choosing vaccination. The model is given in the following differential equations: ⎧  ∞ dS(t) ⎪ ⎪ = Λ − S(t) β (a)i(t, a)da − μ S(t) − φ x(t)S(t) + ε V (t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎪ ⎪ ∂ i(t, a) ∂ i(t, a) ⎪ ⎪ + = −(μ + γ (a))i(t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪ ⎪  ⎪ ⎪ ⎪ i(t, 0) = S(t) 0∞ β (a)i(t, a)da, ⎪ ⎨  ∞ (4.51) dR(t) ⎪ ⎪ = γ (a)i(t, a)da − μ R(t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎪ ⎪ dV (t) ⎪ ⎪ = φ x(t)S(t) − (μ + ε )V (t), ⎪ ⎪ ⎪ dt ⎪ ⎪   ⎪ ⎪ σ 0∞ i(t, a)da ∞ − 0a γ (s)ds dx(t) ⎪  ⎪ ⎩ = θ x(t)(1 − x(t))( e da + P − r ), dt N(t) 0 where N(t) = S(t) + I(t) + R(t) +V (t). An individual’s vaccine uptake depends on his or her benefit-cost analysis. φ is the rate that susceptible individuals get vaccinated when it brings the maximum advantages and the cost is low. ε is the rate that vaccinated individuals lose their immunity and then move to susceptible class. Based on the game theory, the payoff or the fitness difference can be separated as two opposite parts (positive and negative). For positive payoff, susceptible individuals risk to get infected, which depends on the fraction infectious individuals of the  total individuals 0∞ i(t, a)da/N(t); the length of the infectious period has the in a verse effect on the positive payoff, this can be denoted as 0∞ e− 0 γ (s)ds da; the pain of the suffering denoted by σ . An individual getting the expected positive payoff is formulated as   σ 0∞ i(t, a)da ∞ − 0a γ (s)ds e da. N(t) 0 Another positive payoff contribution is rooted in the community. It has been verified that social community is a successful method to resist an epidemic outbreak. This contribution is assumed to be a positive constant P . For negative payoff, the sideeffects and potential risks of the vaccine are main contributions, denoted by r as discussed in last section. Individuals imitate other behaviors due to the benefit-cost analysis at rate θ  . The total payoff for an individual is described as

σ

∞ 0

i(t, a)da N(t)

 ∞ 0

e−

a 0

γ (s)ds

da + P − r .

Observe that the total population satisfies the following equation: dN(t) = Λ − μ N(t). dt

136

4 Age-Since-Infection Structured Models Based on Game Theory

If we assume that N0 = Λμ , the total population remains a constant N(t) = stituting this value into the last equation of (4.51), we obtain

Λ μ.

  dx(t) σ ∞ i(t,a)da  ∞ − a γ (s)ds 0 = θ  x(t)(1 − x(t))( 0 N(t) da + P − r ) 0 e dt

= θ x(t)(1 − x(t))(σ 



∞ 0

Sub-

(4.52)

a  i(t, a)da ∞ e− 0 γ (s)ds da + P − r)

0



PΛ rΛ where θ = μθ Λ , P = μ and r = μ . All the other parameters have the same biological meaning as in the above section. Note that the variable R(t) does not appear in the other equations, so we can ignore it and then consider the following system: ⎧  ∞ dS(t) ⎪ ⎪ = Λ − S(t) β (a)i(t, a)da − μ S(t) − φ xS(t) + ε V (t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎪ ⎪ ∂ i(t, a) ∂ i(t, a) ⎪ ⎪ + = −(μ + γ (a))i(t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪ ⎪  ⎨ i(t, 0) = S(t) 0∞ β (a)i(t, a)da, (4.53) ⎪ ⎪ ⎪ ⎪ dV (t) ⎪ ⎪ = φ xS(t) − (μ + ε )V (t), ⎪ ⎪ dt ⎪ ⎪ ⎪  ∞  ∞  ⎪ dx(t) ⎪ a ⎪ γ (s)ds ⎪ 0 = θ x(t)(1 − x(t)) σ i(t, a)da e da + P − r , ⎪ ⎩ dt 0 0

with the following initial values 1 S(0) = S0 ∈ R+ , i(0, a) = i0 (a) ∈ L+ ,V (0) = V0 ∈ R+ , x(0) = x0 ∈ [0, 1].

As discussed in the above section, system (4.53) has a positively invariant set 1 Γ = {(S, i,V, x) ∈ R+ × L+ × R+ × [0, 1]|S(t) +

 ∞ 0

i(t, a)da +V (t) ≤

Λ }. μ

Next, we just focus on the initial values of system (4.53) chosen from Γ . The existence, uniqueness, and positivity of solution can be proved as before, assuming β (a) ∈ L∞ (0, +∞) is a uniformly bounded continuous function. 1 × R × [0, 1]) → R × L1 × R × [0, 1] be the solution Let Φ : R+ × (R+ × L+ + + + + semiflow associated with (4.55), that is, 1 Φ (t, X0 ) = (S(t), i(t, ·),V (t), x(t)) for X0 = (S0 , i0 (a),V0 , x0 ) ∈ R+ ×L+ ×R+ ×[0, 1].

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles

137

4.3.1 Stability of the Disease-Free Equilibrium For investigating the dynamics of system (4.53), we should define the survival function as a π (a) = e− 0 (μ +γ (s))ds , the survival function and the transmission ability K=

 ∞ 0

β (a)π (a)da,

K1 =

 ∞ 0

π (a)da.

They are two important values to determine the dynamics of system (4.53). If i∗ (a) = 0, there are three cases as follows: When P > r, then x0 = 1, S0 = When P < r, then x0 = 0, S0 =

Λ μ +ε 0 μ μ +ε +φ ,V Λ 0 μ ,V = 0;

When P = r, then x0 ∈ [0, 1], S0 =

=

φ Λ μ μ +ε +φ ;

μ +ε Λ 0 μ μ +ε +φ x0 ,V

=

Λ φ x0 μ μ +ε +φ .

All these three cases give a disease-free equilibrium E0 = (S0 , 0,V 0 , x0 ). The basic reproduction number can be defined as R(φ ) = S0 K. Here S0 is the susceptible element of the disease-free equilibrium E0 . As we well know, this threshold is a key quantity to determine whether or not the disease persists. For the local stability of the disease-free equilibrium E0 , we linearize system (4.53) at the disease-free equilibrium E0 and obtain ⎧  ∞ dS(t) ⎪ 0 ⎪ = −S β (a)i(t, a)da − μ S(t) − φ x0 S(t) − φ S0 x(t) + ε V (t), ⎪ ⎪ ⎪ dt 0 ⎪ ⎪ ⎪ ∂ i(t, a) ∂ i(t, a) ⎪ ⎪ + = −(μ + γ (a))i(t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎨  ∞ i(t, 0) = S0 0 β (a)i(t, a)da, (4.54) ⎪ ⎪ ⎪ dV (t) ⎪ ⎪ = φ x0 S(t) + φ S0 x(t) − (μ + ε )V (t), ⎪ ⎪ ⎪ dt ⎪  ∞ ⎪ dx(t) ⎪ ⎪ 0 0 0 0 ⎪ = θ (1 − x )(P − r)x − θ (P − r)x x + θ (1 − x )x B i(t, a)da, ⎩ dt 0 

a

here B = σ 0∞ e− 0 γ (s)ds da. Let S = s0 eλ t , i(t, a) = i0 (a)eλ t ,V = v0 eλ t and x = x0 eλ t be a solution of system (4.54). Substituting them into (4.54), we get the characteristic equation of (4.53) as follows:

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4 Age-Since-Infection Structured Models Based on Game Theory

   λ + μ + φ x0 ˆ λ) S0 K( −ε φ S0    0 D1 (λ ) 0 0  =0 D(λ ) =  0 0 λ + μ + ε −φ S0   −φ x  0 −θ (1 − x0 )x0 BKˆ 1 (λ ) 0 D2  where D1 (λ ) = 1 − S0 K(ˆλ ), D2 (λ ) = λ − θ (1 − x0 )(P − r) − θ (P − r)x0 show the local stability of the disease-free equilibrium E0 , we analyze the characteristic roots of D(λ ) and get the following theorem. Theorem 4.10. When R(φ ) < 1. Then the following three cases hold: (i) if P > r, then the disease-free equilibrium E0 = (S0 , 0,V 0 , 1) is locally asymptotically stable. (ii) if P < r, then the disease-free equilibrium E0 = (S0 , 0, 0, 0) is locally asymptotically stable. (iii) if P = r, we do not know whether or not the disease-free equilibrium E0 = (S0 , 0,V 0 , x0 ) is locally asymptotically stable. Proof. In view of the expressions of the disease-free equilibrium E0 , the characteristic equation of system (4.53) satisfied D(λ ) = D1 (λ )D2 (λ )D3 (λ ), where D1 and D2 are defined as before. D3 is defined as follows D3 (λ ) = λ 2 + (2μ + ε + φ x0 )λ + μ 2 + (φ x0 + ε )μ = 0. Obviously, it follows from D j (λ ), j = 2, 3 that they have three characteristic roots with negative real parts when P = r. Except for the case P = r, the local stability of the disease-free equilibrium is finally determined by ˆ λ) = 1 S0 K(

(4.55)

By the similar argument as in [192], (4.55) just has the characteristic roots with negative real parts. For P = r, we observe D2 (λ ) = 0, we can’t obtain the local asymptotically stable because there exists a characteristic root λ = 0. This finishes the proof. Theorem 4.10 implies that E0 is the basin of attraction in the neighborhood of the disease-free equilibrium E0 when R(φ ) < 1 except P = r. Theorem 4.11. If R(0) < 1 and P = r, the disease-free equilibrium E0 is globally asymptotically stable.

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles

139

Proof. Note that S(t) ≤ Λμ . Integrating the second equation along t − a = C, together with the boundary equation and initial condition, yields ⎧ ⎨ Z(t − a)π (a),t ≥ a, i(t, a) = ⎩ i0 (a − t) π (a) ,t < a π (a−t) here Z(t) = i(t, 0). It follows from the boundary equation of (4.53) that Z(t) = S(t) here F(t) =

∞ t

 t 0

β (a)Z(t − a)π (a)da + F(t),

(a) β (a)i0 (a − t) ππ(a−t) da and lim F(t) = 0. Then

Z(t) = S(t)

t→∞

 ∞ 0

β (a)Z(t − a)π (a)da ≤

Λ μ

 ∞ 0

β (a)Z(t − a)π (a)da.

Employing the Lemma A.12 yields Z ∞ ≤ R(0)Z ∞ . 

Thistells us that Z ∞ = 0. For t large enough, it follows theexpression of 0∞ i(t, a)da < ε0 for t > t0 that 0∞ i(t, a)da → 0 as t → ∞. There exists a t0 such that 0∞ i(t, a)da  and for any small ε0 . Combining β (a) ∈ CBU [0, +∞) indicates that 0∞ β (a)i(t, a)da < β¯ ε0 for t > t0 .  For P > r, it follows from the fact of i(t, a) small enough that B 0∞ i(t, a)da + P − r > P−r > 0. In view of the equation x, we know that x is an increasing function with respect to t. Therefore, x tends to 1 as t goes to infinity. We again use Lemma A.12 to obtain lim S(t) = S0 and lim V (t) = V 0 . t→∞



t→∞

For P < r, then B 0∞ i(t, a)da + P − r < Bε0 + P − r < 0 for t > t0 if we choose ε0 < r−P B . It is easy to see that x is a decreasing function with respect to t. Therefore, x goes to 0 when t tends to infinity. This, together with Lemma A.12, means that S(t) → S0 ,V (t) → V 0 as t → ∞. For P = r, it follows from the last equation of (4.53) that dx(t) = θ Bx(1 − x) dt

 ∞

i(t, a)da.

(4.56)

x0 θ B 0t 0∞ i(s,a)dads 1−x0 e   . x0 θ B 0t 0∞ i(s,a)dads 1 + 1−x e 0

(4.57)

0

We solve it and obtain  

x(t) =

If R(0) < 1, then it follows from the above discussion that there exists a t0 such that for any t > t0 and any small ε0

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4 Age-Since-Infection Structured Models Based on Game Theory

 ∞ 0

i(t, a)da < ε0 .

This, combining with (4.57) and the definition of the limitation, we obtain x(t) apμ +ε ) proaches 1. Similarly to the case of P > r, we have lim S(t) = S0 = μΛ(μ(+ ε +φ ) and t→∞

μ +ε ) lim V (t) = V 0 = μΛ(μ(+ ε +φ ) . Theorem 4.11 tells us that decreasing the basic reprot→∞ duction number less than 1 can control the disease spread. All the solutions are absorbed to the disease-free equilibrium E0 .

4.3.2 Boundary Equilibrium and the Endemic Equilibrium Let S, i(a),V and x be a solution of (4.54), which is not a function of t. Then the endemic equilibrium satisfies the following system ⎧  0 = Λ − S 0∞ β (a)i(a)da − μ S − φ xS + ε V, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ di(a) ⎪ ⎪ = −(μ + γ (a))i(a), ⎪ ⎪ ⎨ da  i(0) = S 0∞ β (a)i(a)da, (4.58) ⎪ ⎪ ⎪ ⎪ 0 = φ xS − (μ + ε )V, ⎪ ⎪ ⎪ ⎪ ∞ ⎪ ⎪ ⎩ 0 = δ x(1 − x)(B 0 i(a)da + P − r). By the second equation, i(a) = i(0)π (a). Substituting i(a) into the third equation yields S = K1 . In view of the fourth equation, we have V=

φ xS . μ +ε

For the sake of convenience, define R(φ ) =

μ +ε Λ K. μ μ + ε + φ x0

(4.59)

Case 1: If x = 0, then V = 0. In view of the first equation of (4.58), we have i1 (0) =

μ (R(0) − 1). K

Case 2: If x = 1, then the variable V satisfies V2 = first equation of (4.58) leads to i2 (0) =

φS μ +ε .

Substituting V2 into the

μ (μ + ε + φ ) (R(φ ) − 1). (μ + ε )K

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles

141

Case 3: If x ∈ (0, 1), it is easy to see that i∗ (0) = r−P BK1 by the last equation of (4.58). Substituting S,V and i(0) into the first equation of (4.58) yields x∗ =

μ +ε B1 K(r − P) (R(0) − − 1), B1 = . φ μ BK1

In summary, we have the following theorem: Theorem 4.12. Let R(φ ) be defined in (4.59), the following results hold: (i) E1 = (S1 , i1 (0)π (a), 0, 0) is a semi-boundary equilibrium if R(0) > 1. (ii) E2 = (S2 , i2 (0)π (a),V2 , 1) is a semi-boundary equilibrium if R(φ ) > 1. (iii) E3 = (S∗ , i∗ (0)π (a),V ∗ , x∗ ) is an endemic equilibrium when 1 + Bμ1 < R(0) < φ μ +ε

+ 1 + Bμ1 and P < r.

As discussed in Theorem 4.2, E1 is the endemic and non-vaccinator equilibrium, E2 is the endemic and pure-vaccinator equilibrium, E3 is the endemic and vaccinator strategy equilibrium from biological points of view. Linearizing system (4.53) around the endemic equilibrium E ∗ and letting S(t) = s(t) + S∗ , i(t, a) = y(t, a) + i∗ (a),V (t) = z(t) +V ∗ and x(t) = w(t) + x∗ to obtain ⎧  ∞  ∞ ds(t) ⎪ ∗ ⎪ = −S β (a)y(t, a)da − ( β (a)i∗ (a)da + μ + φ x∗ )s(t) ⎪ ⎪ dt ⎪ 0 0 ⎪ ⎪ ⎪ ⎪ − φ S∗ w(t) + ε z(t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ i(y, a) ∂ i(y, a) ⎪ ⎪ + = −(μ + γ (a))y(t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪   ⎨ y(t, 0) = S∗ 0∞ β (a)y(t, a)da + s(t) 0∞ β (a)i∗ (a)da, (4.60) ⎪ ⎪ ⎪ dz(t) ⎪ ⎪ ⎪ = φ x∗ s(t) + φ S∗ w(t) − (μ + ε )z(t), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ dw(t) ⎪ = θ (Bi(0)K1 + P − r)w(t)(1 − 2x∗ ) ⎪ ⎪ ⎪ dt ⎪  ⎪ ⎪ ⎪ + θ (1 − x∗ )x∗ B 0∞ y(t, a)da. ⎩ Similar to the method in the above subsection, substituting s(t) = s0 eλ t , z(t) = zλ0 t , w(t) = w0 eλ t and y(t, a) = y(a)eλ t into (4.60) yields  ∞

 λ) φ K( + w0 − ε z0 = 0, K K

(4.61)

K(ˆλ ) ) = i(0)Ks0 , K

(4.62)

−φ x∗ s0 − φ S∗ w0 + (λ + μ + ε )z0 = 0,

(4.63)

−θ (1 − x∗ )x∗ BKy(0) + (λ − θ (Bi(0)K1 + P − r)(1 − 2x∗ ))w0 = 0.

(4.64)

(λ +

0

β (a)i∗ (a)da + μ + φ x∗ )s0 +

y(0)(1 −

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4 Age-Since-Infection Structured Models Based on Game Theory

Hence, if x∗ = 0 then we know from (4.64) that there exists a characteristic root

λ=

B1 θ μ BK1 (R(0) − − 1). K μ

On the other hand, if x∗ = 1, (4.64) has a characteristic root

λ =−

θ BK1 μ (μ + ε + φ ) B1 (R(φ ) − 1 − ). K(μ + ε ) μ

Otherwise, w0 = 0. Clearly, if x∗ = 0, according to (4.63), we obtain λ = −(μ + ε ) or z0 = 0. For this case, the remaining characteristic roots are equivalent to an SI model with infection age which is considered as examples in [84]; if x∗ = 1, the remaining characteristic roots are equivalent to an SIV S model with infection age which is a special case in [188]. Important thing is to determine the local stability when x∗ ∈ (0, 1). In fact, the characteristic equation can be changed as ˆ λ) K( φ S∗ Δ λ +μ +ε = 1+ B1 Kˆ 1 (λ ) + ∗ K λ (λ + μ + ε + φ x ) (λ + μ )(λ + μ + ε + φ x∗ )

(4.65)

here Δ = θ (1 − x∗ )x∗ BB1 . By a standard argument, we make sure that the following theorem holds. Theorem 4.13. If 1 < R(0) < 1 + Bμ1 , then the semi-boundary equilibria E1 are locally asymptotically stable. If R(φ ) > 1 + Bμ1 E2 is locally asymptotically stable.

The stability of the endemic equilibrium E ∗ is so complex and determined by the characteristic equation (4.65). For clarifying the details of the stability for endemic equilibrium, we make the following assumptions. Assumption 3 Let ε = 0, γ (a) = γ and

∗ β , a≥τ β (a) = 0, a < τ , where β ∗ = β e(μ +γ )τ . Based on Assumption 3, we readily derive the following equalities: K(ˆλ ) =

 ∞

 ∞

0

β (a)π (a)e−λ a da =

β , K= λ +μ +γ 

 ∞ 0

β (a)π (a)da =

β , μ +γ

∞ 1 1 K1 = π (a)da = , K1 (ˆλ ) = π (a)da = , μ +γ λ +μ +γ 0 0  ∞ β (r − P)γ ∗ 1 Λ β B1 = β (a)i∗ (a)da = i∗ (0)K = ,x = ( − B1 − μ ), σ φ μ +γ 0 μ +γ 1 Δ = θ (1 − x∗ )x∗ BB1 = θ (1 − x∗ )x∗ β (r − P), S∗ = = . K β

(4.66)

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles

143

Replacing all the equalities in (4.65) by the quantities of (4.66), we derive the following equation: B1 μ + γ −λ τ φ S∗ Δ + e = 1+ . λ +μ +γ λ (λ + μ + φ x∗ )(λ + μ + γ ) λ + μ + φ x∗

(4.67)

With an equivalent change for (4.67), we have

λ 3 + a1 λ 2 + a2 λ + a3 = (b1 λ 2 + b2 λ )e−λ τ ,

(4.68)

where a1 = 2μ + γ + φ x∗ , a2 = (μ + γ )(μ + B1 + φ x∗ ), a3 = φ Δ S∗ , b1 = (μ + γ ), b2 = (μ + γ )(μ + φ x∗ ). Lemma 4.7. Supposed Assumption 3 holds. For τ = 0, if βΛ > φ x∗ S∗ θ B(1 − x∗ ), then the endemic equilibrium E ∗ is locally asymptotically stable. Proof. If τ = 0, then (4.68) has been changed as

λ 3 + aˆ1 λ 2 + aˆ2 λ + aˆ3 = 0, where

aˆ1 = μ + γ + φ x∗ + B1 , aˆ2 = B1 (μ + γ ), aˆ3 = φ Δ S∗ .

Hence, aˆ j > 0, j = 1, 2, 3. From Routh-Hurwitz criterion, we need to show that aˆ1 aˆ2 − aˆ3 = (μ + γ )B1 (μ + φ x∗ + B1 ) − φ Δ S∗ = B1 (βΛ − φ x∗ S∗ θ B(1 − x∗ )). Hence, if βΛ > φ x∗ S∗ θ B(1 − x∗ ), then the endemic equilibrium E ∗ is locally asymptotically stable. Now we are in position for τ > 0. Assume that (4.68) has a purely imaginary root λ = iw, w > 0. Then this solution satisfies −iw3 − a1 w2 + a2 iw + a3 = (−b1 w2 + b2 iw)(cos wτ − i sin wτ ). Separating the real and imaginary parts, we arrive at −b1 w2 cos wτ + b2 w sin wτ = −a1 w2 + a3 , b1 w2 sin wτ + b2 w cos wτ = −w3 + a2 w.

(4.69)

Squaring the above equations and summing together, we have w6 + (a21 − 2a2 − b21 )w4 + (a22 − 2a1 a3 − b22 )w2 + a33 = 0.

(4.70)

Note that a21 − 2a2 − b21 = (μ + φ x∗ + B1 )2 > 0. Next, we will show the sign of the coefficient for w2 .

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4 Age-Since-Infection Structured Models Based on Game Theory

a22 − 2a1 a3 − b22 =(μ + γ )2 (μ + φ x∗ + B1 )2 − 2φ Δ S∗ (μ + γ )(2μ + γ + φ x∗ + B1 ) − ( μ + γ )2 ( μ + φ x ∗ )2   2Λ β =(μ + γ )B1 ( ) − 2φ θ x∗ (1 − x∗ )B − B1 . μ +γ Consequently, we have the following lemma by the Descartes’ rule of sign. Lemma 4.8. If μΛ+βγ − positive solutions.

B1 2

< φ θ x∗ (1 − x∗ )S∗ B < Λ β , Eq. (4.70) has two different

Lemma 4.8 implies that the characteristic equation (4.68) has two pair purely imaginary roots, denoted by w± and without loss of generality assume w+ > w− . In view of Proposition 4.3, we have the following corollary. Corollary 4.1. Suppose the condition of Lemma 4.8 holds, then the transversal conλ) dRe(λ ) dition has been satisfied, i.e., dRe( d τ |τ =iw− < 0, and d τ |τ =iw+ > 0,. Return to Eq. (4.69), we have ⎧ (a2 b2 −a3 b1 )+(a1 b1 −b2 )w2 1 ⎪ ), ⎪ ⎪ 2 jπ + w arccos( b22 −b21 w2 ⎨ τj = )+(a1 b1 −b2 )w2 ), 2( j + 1)π − w1 arccos( (a2 b2 −a3bb21−b ⎪ 2 w2 ⎪ ⎪ 2 1 ⎩ j = 1, 2, · · · .

(a2 b2 −a3 b1 )+(a1 b1 −b2 )w2 b22 −b21 w2

≥ 0,

(a2 b2 −a3 b1 )+(a1 b1 −b2 )w2 b22 −b21 w2

< 0, (4.71)

Theorem 4.14. Let the condition of Lemma 4.8 hold. Then system (4.53) oscillates in form of Hopf bifurcation at τ = τ j , , j = 1, 2, · · · , τ j is defined in (4.71). By the same methods in Sect. 4.2.2, we have the following familiar results. Theorem 4.15. Suppose that 1 < R0 < 1 + K(r−P) μ BK1 . Then the two cases hold: For r is large enough, the equilibrium E1 attracts all solutions of system (4.53). For P > r, the endemic and pure-vaccinator equilibrium E2 attracts all the solutions of system (4.53). 

Proof. For case one, if r is large enough, i.e., r > ΛμB + P, we obtain B 0∞ i(t, a)da + P − r is always negative. Then x(t) is a decreasing function associated with time t. So, x(t) converges to 0 when t goes to infinity. This indicates that there exists a t0 for any small ε0 and t > t0 such that x(t) < ε0 . Observing the fourth equation of (4.53), for t > t0 dV (t) φ ε0Λ ≤ − μ V (t). (4.72) dt μ So, there exists a t1 > t0 such that V (t) ≤ ε1 for all t > t1 where ε1 = φ με02Λ . It follows ¯ i(t, a) ≤ i¯(t, a) where S(t) ¯ and i¯(t, a) denote the from system (4.53) that S(t) ≤ S(t), solution of the following system.

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles

⎧  ∞ dS(t) ⎪ ⎪ = ( Λ + εε ) − S(t) β (a)i(t, a)da − μ S, 1 ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎨ ∂ i(t, a) ∂ i(t, a) + = −(μ + γ (a))i(t, a), ⎪ ∂t ∂a ⎪ ⎪ ⎪  ⎪ ∞ ⎪ ⎪ ⎩ i(t, a) = S(t) β (a)i(t, a)da.

145

(4.73)

0

¯ → S1 , and i¯(t, a) → iε1 (0)π (a) in L1 (0, +∞) where Using Theorem 4.1, we have S(t) 1 iε11 (0) =

μ Λ + εε1 ( K − 1). K μ

ˆ and iˆ(t, a) be the solution of the system as follows: On the other hand, let S(t) ⎧  ∞ dS(t) ⎪ ⎪ = Λ − S(t) β (a)i(t, a)da − μ S(t) − φ ε0 S(t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎨ ∂ i(t, a) ∂ i(t, a) (4.74) + = −(μ + γ (a))i(t, a), ⎪ ∂ t ∂ a ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎩ i(t, a) = S(t) β (a)i(t, a)da. 0

By constructing Lyapunov functional method and employing Theorem 4.1, we have ˆ → S1 , and iˆ(t, a) → iε0 (0)π (a) in L1 (0, +∞) where S(t) 1

ε

i10 (0) =

μ Λ ( K − 1). K μ + φ ε0

From the choice of ε j , j =  0, 1, it follows that E1 attracts all the solutions in Γ . For case 2: If P > r, 0∞ i(t, a)da + P − r is always positive. Then x(t) is an increasing function associated with time t and x(t) approaches 1 as t goes to infinity. Hence, given ε2 > 0, there exists a t2 so that t > t2 such that 1 − ε2 ≤ x(t) ≤ 1 + ε2 . We can construct an auxiliary system as follows: ⎧  ∞ dS(t) ⎪ ⎪ = Λ − S(t) β (a)i(t, a)da − μ S(t) − φ (1 − ε2 )S(t) + ε V (t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ i(t, a) + ∂ i(t, a) = −(μ + γ (a))i(t, a), ⎨ ∂t ∂a (4.75)  ∞ ⎪ ⎪ ⎪ i(t, a) = S(t) β (a)i(t, a)da, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dV (t) = φ (1 + ε2 )S(t) − (μ + ε )V (t) dt

146

4 Age-Since-Infection Structured Models Based on Game Theory

˜ i˜(t, a), V˜ (t)) approaches ( 1 , iε1 (0)π (a), φ (1+ε2 ) ) where iε2 (0) whose solution (S(t), 2 K 2 (μ +ε )K = (μ + φ μ −εμ2 (+με+2ε ) ) K1 (R ε2 (φ ) − 1) and R ε2 (φ ) = Λμ μ (μ +ε )+φμ(μ+−ε ε (μ +2ε )) . 2 On the other hand, we can construct another auxiliary system as follows: ⎧  ∞ dS(t) ⎪ ⎪ = Λ − S(t) β (a)i(t, a)da − μ S(t) − φ (1 + ε1 )S(t) + ε V (t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ∂ i(t, a) ∂ i(t, a) ⎪ ⎪ + = −(μ + γ (a))i(t, a), ⎨ ∂t ∂a (4.76)  ∞ ⎪ ⎪ ⎪ i(t, a) = S(t) β (a)i(t, a)da, ⎪ ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dV (t) = φ (1 − ε2 )S(t) − (μ + ε )V (t) dt

ˇ iˇ(t, a), Vˇ (t)) converges to ( 1 , iˇε2 (0)π (a), φ (1−ε2 ) ), where which solution (S(t), K 2 (μ +ε )K μ +ε ˇiε2 (0) = (μ + φ μ +ε2 (μ +2ε ) ) 1 (Rˇε2 (φ ) − 1) and Rˇε2 (φ ) = Λ 2 μ +ε K μ μ (μ +ε )+φ (μ +ε2 (μ +2ε )) . ˇ ≤ S(t) ≤ S(t), ˜ Combining system (4.75) and system (4.76), we obtain S(t) ˇiε2 (0) ≤ i2 (0) ≤ i˜ε2 (0) and Vˇ (t) ≤ V (t) ≤ V˜ (t). This, together with the arbitrary 2 2 of ε2 , indicates system (4.53) approaches E2 as t goes to infinity. In order to prove the strongly ρ -persistence of the disease, we use the familiar methods as discussed in Sect. 4.2.2. Firstly, we show the weekly ρ -persists when R(0) > 1. We define ρ (S, i,V, x) = 0∞ i(t, a)da and so that ρ ∈ Γ . Lemma 4.9. If r > P + BμΛ and R0 > 1, system (4.53) is weekly ρ -persistent. Proof. Since R0 > 1, we can pick up a ε3 > 0 such that (

Λ − ε3 )K > 1. μ + (φ + β¯ )ε3

(4.77)

By the contradiction method, we assume that there exists X0 = (S0 , i0 (a),V0 , x0 ) ∈ Γ such that lim sup ρ (t, Φ (t, X0 )) < ε3 . t→∞

There exists a t1 > 0 for all t > t1 such that

ρ (t, X0 ) < ε3 . As discussed before, if r > P+ BμΛ , then lim x(t) = 0. Therefore, there exists a t2 > t1 t→∞ for all t > t2 such that 0 ≤ x(t) < ε3 .

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles

147

By the first equation of (4.53), we obtain dS(t) ≥ Λ − μ S(t) − φ ε3 S(t) − β¯ ε3 S. dt

(4.78)

Equation (4.78) implies that lim inf S(t) ≥ μ +ε Λ(φ +β¯ ) . Then there exists a t3 > t2 for 3 all t > t3 such that Λ S(t) ≥ − ε3 . μ + ε3 (φ + β¯ ) Borrowing the third equation of (4.53), we have Z(t) ≥ (

Λ − ε3 ) μ + ε3 (φ + β¯ )

 t 0

β (a)Z(t − a)π (a)da.

(4.79)

Taking Laplace transform on both sides of (4.79), we obtain ˆ λ) ≥ ( Z(

Λ ˆ λ )K( ˆ λ ). − ε3 )Z( μ + ε3 (φ + β¯ )

(4.80)

where ˆ· denotes a Laplace transform. Equation (4.79) indicates that ( μ +ε (Λφ +β¯ ) − ˆ λ ) < 1 for any λ . This leads to a contradiction with (4.77) when we take λ = 0. ε )K( Lemma 4.10. If P > r and R(φ ) > 1, system (4.53) is weekly ρ -persistent. Proof. Since R(φ ) > 1, we can pick up a ε0 > 0 such that (

Λ (μ + ε ) − ε0 )K > 1. (μ + β¯ ε0 )(μ + ε ) + φ [μ + ε0 (μ + 2ε )]

(4.81)

By the contradiction method, we assume that there exists X0 = (S0 , i0 (a),V0 , x0 ) ∈ Γ such that lim sup ρ (t, Φ (t, X0 )) < ε0 . t→∞

There exists a t1 > 0 for all t > t1 such that

ρ (t, X0 ) < ε0 . As discussed before, if P > r, then lim x(t) = 1. Therefore, there exists a t2 > t1 for t→∞ all t > t2 such that 1 − ε0 < x(t) < 1 + ε0 . Employing Lemma A.12, we can choose a sequence {tn } such that V  (tn ) → 0 as t → ∞ and V (tn ) → V∞ as n goes to infinity where f∞ = lim inf f (t). From the last t→+∞

equation of (4.53), it follows that V∞ ≥

φ (1 − ε0 ) S∞ . μ +ε

(4.82)

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4 Age-Since-Infection Structured Models Based on Game Theory

We again use the Lemma A.12, together with (4.82) and the first equation of (4.53) to obtain Λ (μ + ε ) S∞ ≥ . (4.83) ¯ (μ + β ε0 )(μ + ε ) + φ [μ + ε0 ε ] Then there exists a t3 > t2 for all t > t3 such that S(t) ≥

Λ (μ + ε ) − ε0 . ¯ (μ + β ε0 )(μ + ε0 ) + φ [μ + ε0 ε ]

Borrowing the third equation of (4.53), we have Z(t) ≥ (

Λ (μ + ε ) − ε0 ) ¯ (μ + β ε0 )(μ + ε ) + φ [μ + ε0 ε ]

 t 0

β (a)Z(t − a)π (a)da

(4.84)

Taking Laplace transform on both sides of (4.84), we obtain ˆ λ) ≥ ( Z(

Λ (μ + ε ) ˆ λ )K( ˆ λ ). − ε0 )Z( ¯ (μ + β ε0 )(μ + ε ) + φ [μ + ε0 ε ]

(4.85)



ε) where fˆ(λ ) = 0∞ f (a)e−λ a da. Equation (4.84) indicates that ( (μ +β¯ ε )(Λμ(+με+)+ φ [ μ +ε0 ε ] 0 ˆ λ ) < 1 for any λ . This leads to a contradiction with (4.81) taking λ = 0. −ε0 )K(

Define the following invariant sets D0 = {(S, i,V, x) ∈ Γ |S(t) + and D1 = {(S, i,V, x) ∈ Γ |S(t) +

 ∞ 0

 ∞ 0

i(t, a)da ≤

Λ , x(t) = V (t) = 0}, μ

i(t, a)da +V (t) ≤

Λ , x(t) = 1}. μ

We repeat the proving processes for Lemmas 4.4 and 4.6, and obtain the following lemma. Lemma 4.11. Let R(φ ) be defined in (4.59). (1) If r > P + BμΛ and R(0) > 1, the system (4.53) has a global attractor in D0 ; (2) If P > r and R(φ ) > 1, the system (4.53) has a global attractor in D1 . With the help of Lemmas 4.9 and 4.10, together with Lemma 4.11, we obtain uniformly strongly ρ -persistent. Theorem 4.16. If R(φ ) > 1, then system (4.53) is uniformly strongly ρ -persistence.

4.3.3 Discussion From the disease control perspective, a quality of interest is the final infected level. The equilibrium value of I for system (4.53) with respect to the parameters is shown in Figs. 4.3 and 4.4. The parameter r denotes the cost of the vaccine side-effects. If

4.3 Imitation Dynamics in the Case of Vaccinating Susceptibles

149

r is large enough, individuals will have heavy burden if they take vaccination. From the game theory based cost-benefit analysis, susceptible individuals will not vaccine. Then the number of infectives will increase. The parameter P indicates the benefit from uptake vaccination. Increasing P means an individual gets much more positive benefits from vaccination. Hence, summing the benefits in the community take advantages against the disease prevalence. An individual suffering from the disease (pain severity and the distress) helps susceptible individuals increase the perceived costs and improves vaccination rates. In our models, the parameter σ is used to explain the suffering from the disease, which encourages susceptible individuals to take vaccination. As a result, the disease slows down. In order to investigate the impact of various strategies on the dynamics of system (4.53), we take the parameters as follows:  0 0 ≤ a ≤ τ, β (a) = γ (a) = γ . β exp(−(μ + γ )τ ), a ≥ τ ,

1

4

r= 7 r=7.5

3.5 0.8 3 0.6

x(t)

x*

2.5 2

0.4

1.5 1

0.2

0.5 0

3

4

5

6

r

7

0 0

8

50

100

150

200

250

300

350

400

Time t

(a)

(b)

Fig. 4.3 The frequency of vaccinated individuals varies with respect to its side-effects r. (a) The component of vaccinated ratio varies with respect to payoff of its side-effects r. (b) The vaccinated ratio varies with different payoff of its side-effects r = 7.0 and r = 7.5, respectively 9 3.5

r=7 r=7.5

8 7

3

I(t)

6

I*

2.5

5 4

2

3 2

1.5

1 1 3

4

5

r

(a)

6

7

8

0 0

50

100

150

200

250

300

350

400

Time t

(b)

Fig. 4.4 The infected individuals varies with respect to the side-effects of r. (a) The component of infectives with respect to τ . (b) The infectives varies with r = 7.0 and r = 7.5, respectively

150

4 Age-Since-Infection Structured Models Based on Game Theory

9

5

t=0.08 t=0.14

8

f=1.3 f=1.8

4

7 5

I(t)

I(t)

6 4

3 2

3 2

1

1 0 0

50

100

150

200

250

300

350

400

0 0

50

100

150

200

250

300

350

400

Time t

Time t

(a)

(b)

Fig. 4.5 (a) The infected individuals with respect to latent period τ . (b) The infectives varies with different vaccination rates

Under the above assumption on parameters, we directly obtain the basic reproduction number in terms of φ R(φ ) = S0 K =

β S0 . μ +γ

Other parameters are chosen as follows:

Λ = 2, μ = 0.01, φ = 0.8, ε = 0.01, γ = 0.5, σ = 1, ε = 0.1, β = 0.1. If the vaccine benefit from mass immunization is better than the side-effects of the vaccination, this persuades all the individuals to get vaccination and the frequency of the vaccinated population converges to 1. Otherwise, if the side-effects of the vaccination are always worse than the benefits from the vaccine, then people are scared to vaccine and decline to receive vaccination. This leads to the frequency of the vaccinated population converging to 0. At the same time, the game  evolutionary behaviors depend on the initial infectives. If the initial infectives B 0∞ i0 (a)da + P > r, it follows from the last equation of (4.53) that the frequency of the vaccinated population increases and the susceptible individuals decrease. This  leads to the infected individuals decreasing until B 0∞ i(t, a)da + P = r. The stability of the equilibria depends on the parameters with respect to the reproduction number, which is stated in Theorem 4.13. The side-effects from the vaccination reduce the complexity of system (4.53) and returns to the nature of the system with increasing r values. This results in x(t) approaching to zero. System (4.53) degenerates to an SI epidemic model and its dynamics are determined by an SI system. If the impact of the mass immunization is good enough, this gives people confidence to gain immunity from the vaccination. System (4.53) changes as an SIVS epidemic model and its dynamics is totally determined in [190]. Since the basic reproduction number is unrelated with the two parameters r and P, the side-effects of the vaccination and benefits from the mass immunizations

4.4 Comparison

151

have no effect on the persistence of the disease. However, we readily see that r and P have some contributions on the final transmission level by the expression i(0) in the endemic equilibrium. From Theorem 4.12, the stronger the side-effect of vaccination, the higher the infected level. For the latent period τ , lengthening the latent period reduces the magnitude of the infected individuals, and delays the first peak arrival time (see Fig. 4.4), which is suitable for early preventions and warning. Although the latent period τ does not have any contributions on the basic reproduction number R(φ ) and the components of the endemic equilibrium E ∗ , this parameter plays an important role in investigating the stability of the endemic equilibrium E ∗ (see Fig. 4.5a). Theorem 4.14 shows that there exists a τ ∗ such that if τ < τ ∗ then E ∗ is locally stable, otherwise, there exists a Hopf bifurcation if the condition of Lemma 4.8 holds. The vaccination rate φ plays a key role in the effect of vaccination impacting the control of the disease. It follows from the expression of the basic reproduction number R0 that there exists a critical vaccination coverage value

φ∗ =

(μ + ε )(Λ K − μ ) μ

such that if φ > φ ∗ the disease dies out, otherwise it becomes endemic when P ≥ r. This implies that improving the vaccine coverage is beneficial for controlling the disease transmission if the side-effects of vaccination are enough weak.

4.4 Comparison Comparing the two voluntary vaccination strategies, side-effects of vaccination discourage the susceptible individuals to get immunity from vaccination. Suffering from the disease prompts them to take vaccination and avoid to be infected by the disease. However, it follows from the first model for the new born vaccination scheme improving the vaccine coverage is not always beneficial to control the disease. For the second vaccination scheme, that is continuous vaccination, improving the vaccine is always good at slowing down the disease transmission. The expressions of the game equations (replicator equations) have familiar formulas. From the theoretical and numerical analysis, introducing replicator equation modifies the structure of SIR and SIV S models [118, 190]. This revision enhances the complexity of the two models and exhibits the oscillation phenomena. The game framework evolves with respect to time t and leads to a Hopf bifurcation. Hence, the payoff function has dual characteristics coupling negative with positive trade-off. In fact, the two vaccination schemes in this section evolves to the form of game for two players in the introduction discussed. The optimal vaccine-making strategy strongly depends on the behaviors of game objects and evolves with respect to time.

Chapter 5

Age Structured Models on Complex Networks

5.1 Introduction Human behaviors affect epidemic spread. One of the main components of the disease transmission is the contact rate which in all models so far has been assumed constant or varying by age (age-since-infection) only. However, the contact rate is not constant from individual to individual; in particular some individuals have high contact rate while others may have much lower. This is particularly the case in sexually transmitted diseases but it can be observed in many others. Heterogeneity is produced due to individuals heterogeneous mixing. All the individuals and contacts generate a network. How one network structure (topology) affects the disease transmission has become a hot topic in recent years. This chapter is based on the individual contact behavior to investigate the disease transmission. In order to understand the network structure, we fist introduce some basic knowledge about networks.

5.1.1 Definitions Graph theory [144] is a useful tool to generalize a network from the point of view of mathematics. A graph consists of nodes(vertices) and edges(links). If every two nodes are connected by edges, this suggests there exists some relations between the two nodes. We assume the number of the nodes in a network is N. Then an adjacency matrix A consists of N ×N elements with ai j = 1 if there exists a connection between i node and j node, otherwise, ai j = 0. In order to clearly understand the background of networks, we need to know some necessary basic definitions and facts. Definition 5.1 (Undirected Networks and Directed Networks). Connections of a network can be bidirectional, which is called undirected networks, otherwise, it is called directed networks.

© Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1_5

153

154

5 Age Structured Models on Complex Networks

Definition 5.2 (Degree and Degree Distribution). The degree k of one node on undirected networks denotes one node emanates total of k edges. The degree of distribution of the network P(k) is the probability that a randomly chosen node has degree k. Definition 5.3 (Degree Correlations). The degree correlation between two vertices is defined as the conditional probability P(k |k) that an edge departing from a vertex of degree k is connected to a vertex of degree k . The conditional probability P(k |k)  ) of an uncorrelated network is independent of the originating node, P(k |k) = k P(k k , n

where the average degree of a network is defined by k = ∑ kP(k). k=1

Definition 5.4 (Random Graph). The concept of random graph was first proposed by Erdös and Rényi [48]. A random graph is generated from a set of N nodes in which each node stochastically connects with another node with probability p. If the number of the nodes is large and the probability p is a constant, this network can be called regular network. Scale of Networks A scale of a network is define as S(G) =

s(G) , smax

where smax is the maximum value of s(H) for H in the set of all graphs with degree distribution identical to that of G, which is a graph with edge set E, and deg(v) denotes the degree of a vertex v. Besides, s(G) =

∑

deg(u) · deg(v).

(u,v)∈E

S(G) gives a metric between 0 and 1, where a graph G with small S(G) is “scalerich,” and a graph G with S(G) close to 1 is “scale-free.” Definition 5.5 (Complex Networks). Between regular networks and random graphs, there exist networks called complex networks. Two main types of complex networks are considered: one is small world networks, the other is scale-free networks. Small world network was first proposed by Watts and Strogatz [177] and is based on rewiring procedure of an ordered graph. A small network has small shorted path (a small number of edges traversed), and high clustering coefficient defined as the ratio between the number of triangles and the number of the connected triples. A scale-free network (see Fig. 5.1) is structured in a hierarchy of nodes with a few nodes having very large degree (the hub nodes), while the vast majority of nodes have much smaller degrees. The degree distribution of the scale-free networks approximates as a power-law function of the form P(k) ∼ k−r , r ∈ (2, 3]. The models in this chapter will be based on Scale-free networks. In general, there are four approaches to combine infectious disease models with a network: the bond

5.1 Introduction Fig. 5.1 A scale-free network with 100 nodes and k = 3.2 (average degree). Minimum degree of the network is 0 and Maximum degree of the network is 9

155 BA network

100

60

2 8

80 9 84 20

60

26

87 66 97 7

63

25 98

7038

40 19

20

65 78 35 47 77 100 4 24 29 50

0

20

79 93

30

54 18 21

48 92 81

85

40 17 44

90

58 45

10 37

0

74

82

62

28 89

99 12 55

31 46 22 68 88 43 67

5223 16

32 27 69 57

73 76 75 15 3 39 41 61 94 80 59 83 64 1413

95 42 49

40

72

34 96 71

5 6 36

60

1 56 11

33 51

80

86

53 91

100

percolation [140], the mean-field models [69], the pairwise models [46], the edgebased compartmental models [135]. In this chapter, we only use one of the second approach.

5.1.2 Basic ODE Epidemic Models on Networks Classical models assume that all individuals are uniformly mixed and hence have the same probability contacting others. This assumption is reasonable in modeling infections that have become established or in well-mixed homogenous populations. In a standard SIS model, the population is divided into two classes: susceptibles (S) and infected (I). A simple SIS model without demography is described by the following system of ordinary differential equations: ⎧ dS I ⎪ = −kσ β S + γ I, ⎨ dt S+I (5.1) ⎪ ⎩ dI = kσ β S I − γ I. dt S+I Here k is the average number of contacts per unit of time, σ is the effective exposure of a susceptible to the infected individuals (which varies according to behavior change, prevention strategies, and so on), β is the probability of transmission, and γ is the recovery rate of the infected individuals. In (5.1), the standard incidence rate is used. Evolution of society and behavior of humans has led to individual contacts exhibiting obvious heterogeneity. Modeling epidemics on contact networks is one of the main approaches to capture the diversity in human contacts. Significant amount of research implies that the contact infectivity networks have the scale-free property over several orders of magnitude. In 1999, Barabási and Albert [10] built an epidemic model on scale-free networks. They showed that if p(k) is the probability of a node with degree k (which links with k other nodes), that probability satisfies the power law distribution p(k) = c f (k)k−r , where r ∈ (2, 3], and f (k) is a function of k. Many researchers used the mean-field approximations to study the epidemic spread on static networks. Pastor-Satorras and Vespignani [145, 146] studied disease transmission through an SIS epidemic model on heterogeneous networks (that is networks that are not regular networks). They defined the model as follows:

156

5 Age Structured Models on Complex Networks

⎧ dSk ⎪ ⎪ = −kσ SkΘ (Ik ) + γ Ik , ⎪ ⎪ dt ⎪ ⎪ ⎨ dI k = kσ SkΘ (Ik ) − γ Ik , ⎪ dt ⎪ ⎪ n

n

⎪ ⎪ ⎪ ⎩ Θ (Ik ) = ∑ kIk (t) / ∑ kNk (t) k=1

(5.2)

k=1

here Sk and Ik denote the susceptible nodes and infected nodes with degree k. The authors showed that the epidemic transmission threshold λc (Actually, λc = 1/R0 ) is small enough when the scale of the networks is large enough. Θ (I) denotes the probability of an edge to be coming from an infected nodes. Zhou [195] modified the model and established that the threshold λc is a constant regardless of the size of the network and the degree distribution. Wang et al. [175] introduced the vector infection to an SIS epidemic model on complex networks, and investigated the stability and effects of immunization schemes. Throughout, these results mainly focus on epidemic spread on a static network (Static network is a network that has no rewiring process, that is, the structure of the network is fixed and does not change in time). If the network structure changes in time the network is called dynamic network. For large scale (dynamic network) the time behavior of the network structure and the demography of the population are impacting significantly the dynamics of the disease. Zhang and Jin in [97] constructed an SIS model with birth and death into a vacant node to investigate the dynamics of the model with several types of immunization schemes. Jin et al. in [99] investigated an SIS epidemic model on a dynamic network with birth of susceptible nodes and death of susceptible or infected nodes and obtained the global dynamics of the model. Compared with the epidemic models on static networks, the basic reproduction number is a property of models with birth and death of nodes. Actually, most of the existing epidemic models on complex networks consist of ordinary differential equations. This means that infected individuals are assumed to be equally infectious during the infectious period. However, evidence suggest that infected individuals have different infectiousness during their infectious period. This phenomenon often occurs for some chronic diseases such as HIV, TB (Tuberculosis), Hepatitis C, and Diphtheria. As for TB, most of TB cases have a long latent period until the immune system is comprised due to long term of infection. This heterogeneity in individual infectiousness has attracted some attention and research has been directed towards describing the disease transmission as it varies with infection age (classical age-since-infection can also be found in earlier chapter models as the ones considered, as well as in [33, 85, 116, 118, 176, 179]). To the best of our knowledge, not much has been done for epidemic models with infection age on complex networks [191].

5.2 An Age-Structured SIS Model on Complex Networks

157

5.2 An Age-Structured SIS Model on Complex Networks To introduce structure of a static network and infection age, we subdivide the population as follows. Let n be the maximal degree of the complex network. For k ∈ Nn  {1, 2, . . . , n}, let Sk (t) be the number of susceptible vertices of degree k at time t, and Ik (t, a) stand for the density of infected vertices of degree k at time t and with infection age a. Then an SIS model with infection age on complex networks is as follows: ⎧  ∞ dSk (t) ⎪ ⎪ = −k σ S (t) Θ (I(t, ·)) + γ (a)Ik (t, a)da, ⎪ k ⎪ ⎪ dt 0 ⎪ ⎨ ∂ Ik (t, a) ∂ Ik (t, a) (5.3) k ∈ Nn , + = −γ (a)Ik (t, a), ⎪ ⎪ ⎪ ∂ t ∂ a ⎪ ⎪ ⎪ ⎩ Ik (t, 0) = kσ Sk (t)Θ (I(t, ·)), where γ (a) is the recovery rate at infection age a from infected class to susceptible class, n ∞

n

Θ (I(t, ·)) = ∑ k β (a)Ik (t, a)da / ∑ kNk (t) (5.4) k=1

0

k=1

denotes the force of infection with Nk (t) = Sk (t) +

 ∞ 0

Ik (t, a)da being the total

number of vertices with degree k at time t, I(t, ·) = (I1 (t, ·), I2 (t, ·), . . . , In (t, ·)), and β (a) is the transmission probability at infection age a. Assumption 4

γ (a), β (a) ∈ L∞ (R+ ), this implies that

(H1)

β¯ = ess. sup β (a), γ¯ = ess. sup γ (a). a∈[0,+∞)

(H2)

a∈[0,+∞)

limt→∞ ππ(t+·) (·) ∞ = 0,

where

π (a) = e−

a 0

γ (s)ds

(5.5)

for a ∈ R+

is the probability of the infected individual still staying in the infected compartment. Condition (5.5) automatically holds if there exist a0 ≥ 0 and γ0 > 0 such that γ (a) ≥ γ0 for a ≥ a0 . The initial condition of (5.3) is 1 x0 = (S0 , I0 ) = (S10 , S20 , . . . , Sn0 , I10 , I20 , . . . , In0 ) ∈ (R+ )n × (L+ (R+ ))n ,

where 1 L+ (R+ ) = {ϕ ∈ L1 (R+ )|ϕ (a) ≥ 0a.ea ∈ R+ }.

158

5 Age Structured Models on Complex Networks

We define the following functional spaces: X = (L1 (R))n ,Y = (R)n × X n × ({0})n . with norm

φ X =

ψ Y =

 ∞ n

∑ |ϕ j (a)|da, φ ∈ X,

0

j=1

n

∑ |ψ1 j | + ψ2 X , ψ1 = (ψ11 , ψ12 , · · · , ψ1n ) ∈ (R)n , ψ2 ∈ X.

j=1

One readily see that   dNk (t) = −kσ Sk (t)Θ (I(t, ·)) + 0∞ γ (a)Ik (t, a)da + 0∞ ∂ Ik∂(t,a) t da dt ∞ = −kσ Sk (t)Θ (I(t, ·)) + 0 γ (a)Ik (t, a)da   ∞ ∂ Ik (t, a) + γ (a)Ik (t, a) da − ∂a 0 = −kσ Sk (t)Θ (I(t, ·)) − Ik (t, ∞) + Ik (t, 0)

= 0, which implies that Nk (t) is a constant for k ∈ Nn . Here we have implicitly assumed n

that Ik (t, +∞) = 0. Denote N(t) = ∑ Nk (t). Let p(k) = ik (t, a) =

Ik (t,a) Nk (t)

k=1

Nk (t) N(t) ,

sk (t) =

Sk (t) Nk (t) ,

and

for k ∈ Nn . Dividing both sides of (5.3) by Nk gives

⎧  ∞ dsk (t) ⎪ ⎪ = −kσ sk (t)Θ (i(t, ·)) + γ (a)ik (t, a)da, ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎨ ∂ ik (t, a) ∂ ik (t, a) + = −γ (a)ik (t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ ⎪ ⎩ ik (t, 0) = kσ sk (t)Θ (i(t, ·)),

k ∈ Nn .

(5.6)

Deviating N(t) on both numerator and denominator of Θ (I(t, a)) defined in (5.4), we have n N (t)  ∞ Ik (t, a) n Nk (t)

k da / ∑ k Θ (i(t, ·)) = ∑ k β (a) Nk (t) k=1 N(t) 0 k=1 N(t) (5.7)  ∞ n

= ∑ kp(k) β (a)ik (t, a)da /k, k=1

0

5.2 An Age-Structured SIS Model on Complex Networks

with a notice that k =

159

n

 ∞

k=1

0

∑ kp(k). Since sk (t) +

ik (t, a) = 1, (5.6) is equivalent to

⎧ ∂ ik (t, a) ∂ ik (t, a) ⎪ ⎪ ⎪ ⎨ ∂ t + ∂ a = −γ (a)ik (t, a),  ∞ ⎪ ⎪ ⎪ ⎩ ik (t, 0) = σ k(1 − ik (t, a)da)Θ (i(t, ·)),

k ∈ Nn

(5.8)

0



1 (R ))n such that ∞ i (a)da ≤ with the initial condition i0 = (i10 , i20 , . . . , in0 ) ∈ (L+ + 0 k0 1 for k ∈ Nn . We mention that (5.8) is also an age-structured extension of the model studied by Wang and Dai [171].

Definition 5.6. A set is called invariant under the semiflow of Φ if

Φ (K) = K for all t ∈ R+ . K is called attractor of M if Φt (K) = K and d(Φt (M), K) → 0 as t → ∞. Moreover, it is not difficult to show the following result. Proposition 5.1. The set 1 (R ) and Ω = {i = (i1 , i2 , . . . , in )|ik ∈ L+ +

 ∞ 0

ik (a)da ≤ 1 for k ∈ Nn }

is a positively invariant set of (5.8). Lemma 5.1. If Assumption 4 holds, then system (5.8) has a unique mild solution. Proof. From Theorem 2.1 in Chap. 2, we note that Bk (t, i) = k(1 −

 ∞ 0

n

 ∞

j=1

0

ik (t, a)da)(k)−1 ∑ j p( j)

β (a)i j (t, a)da.

1 , it is obvious to see that B (t, i) ≥ 0 for a.e a ∈ R+ . For all t ∈ [0, T ], ik (t, a) ∈ L+ k Moreover, Bk (t, 0) = 0 is bounded. Next, we need to show that Bk is Lipschitz continuous. For all i, i¯ ∈ L1 , k ∈ [0, n] and t ∈ [0, T ], we have n

 ∞

j=1

0

|Bk (t, i) − Bk (t, i¯)| =k ∑ j p( j) −k +k ≤kβ¯

 ∞ 0

 ∞ 0 n

β (a)[i j (t, a) − i¯j (t, a)]da n

 ∞

j=1

0

n

 ∞

j=1

0

ik (t, a)da ∑ j p( j) i¯k (t, a)da ∑ j p( j)

∑ j p( j) i(t, a) − i¯(t, a) L1

j=1

β (a)i j (t, a)da β (a)i¯j (t, a)da

160

5 Age Structured Models on Complex Networks

+ kβ¯ ik (t, a) − i¯k (t, a) L1

n

∑ j p( j) i j (t, a) L1

j=1

+ kβ¯ i¯k (t, a) L1

n

∑ j p( j) i j (t, a) − i¯j (t, a) L1

j=1 n

≤n2 β¯ max{1, k} ∑ i j (t, a) − i¯j (t, a) L1

(5.9)

k=1

here we use the fact i j (t, a) L1 ≤ 1, j ∈ Nn . Since n is a finite number, it follows from the Theorem 2.1 that system (5.8) has a unique mild solution. Moreover, it is not difficult to show that the solution exists for t ∈ R+ . By the model formulation and Proposition 5.1, we only need to consider (5.8) with initial conditions in Ω . In the following, we will show that the semiflow Φ is asymptotically smooth. In order to do this, we separate the semiflow Φ (t) = U(t) + V (t) where U(t) = (0, iU (t)), iU (t) = (i˜1 , i˜2 , · · · , i˜n ) and V (t) = (S(t), iV (t)), iv (t) = (iˆ1 , iˆ2 , · · · , iˆn ). Here 

0, t ≥ a, Bk (t − a)π (a), t ≥ a, ˆ i˜k (t, a) = (t, a) = (5.10) , i (a) k 0, t < a. ,t 0 such that γ (a) > γ for all a ∈ [a0 , +∞). The semiflow Φ is asymptotically smooth. Proof. Let K be a closed subset of X. For each x ∈ K and x K ≤ r, we have the following estimation: n

U(t) = ∑

 ∞

k=1 0 n

=∑

 ∞

k=1 t n

=∑

 ∞

k=1 0

≤e−γ t

n

∑

i˜k (t, a)da

π (a) i˜k0 (a − t) da π (a − t) π (a + t) i˜k0 (a) da π (a)  ∞

k=1 0

(5.11)

i˜k0 (a)da

=e−γ t i0 K . So that U(t) ≤ re−γ t approaches to 0 as t goes to infinity. Next, it follows from Theorem B.2 [161] that we have to check the following conditions:

5.2 An Age-Structured SIS Model on Complex Networks n

(i) The supremum of  ∞

(ii) lim

h→∞ h n

∑ h→0+

j=1 0

n



 ∞

j=1 0 h

j=1 0

iˆj (t, a)da for all initial data x0 ∈ Ω is finite.

iˆj (t, a)da = 0 uniformly in x0 ∈ Ω .

(iii) lim (iv) lim

∑

 ∞

161

|iˆj (t, a + h) − iˆj (t, a)|da uniformly in x0 ∈ Ω .

iˆj (t, a)da = 0 uniformly in x0 ∈ Ω .

By (5.10),

0 ≤ iˆk (t, a)da ≤ kβ¯ e−γ a , k = 1, 2, · · · , n.

This implies that (i), (ii), and (vi) immediately hold. In what follows, we will show that iˆj (t, a) is completely continuous in terms of infection age a. We assume that h ∈ (0,t) without loss of generality. Then, from (5.10), we have n

H =∑

 ∞

j=1 0 n

=∑

|iˆj (t, a + h) − iˆj (t, a)|da

 t−h

j=1 0

n

|iˆj (t, a + h) − iˆj (t, a)|da + ∑

 t

j=1 t−h

|0 − iˆj (t, a)|da

t−h

≤β¯ nh + ∑ |B j (t − a − h)π (a + h) − B j (t − a)π (a)|da j=1

t−h

≤β¯ nh + ∑ |B j (t − a − h) − B j (t − a)|π (a + h)|da j=1

t−h

+ ∑ B j (t − a)|π (a + h) − π (a)|da. j=1

From the first equation of (5.6), we have |sj (t)| ≤ jσ s jΘ (i) + γ¯

 ∞ 0

i j (t, a)da (5.12)

≤ jσ β¯ + γ¯  Lsj , and it follows from the second equation of (5.6) that |Θ j (t)| ≤ Lsj . Borrowing from (5.12), we have

162

5 Age Structured Models on Complex Networks n

∑ |B j (t − a − h) − B j (t − a)|

j=1

=

n

∑ |σ js j (t − a − h)Θ (t − a − h) − σ js j (t − a)Θ (t − a)|

j=1 n

≤σ

∑ j|s j (t − a − h) − s j (t − a)|Θ (t − a − h)

(5.13)

j=1 n

+σ ≤σ

∑ js j (t − a)|Θ (t − a) − Θ (t − a)|

j=1 n

n

∑ jLsj h + σ Δ ∑ jh  Hh,

j=1

where H = σ

n

∑ jLsj +

j=1

j=1

σ Δ n(n + 1) and Δ = 2 n

H =∑

 ∞

j=1 0

n β¯ k ∑ j=1

j p( j)Lsj . So,

|iˆj (t, a + h) − iˆj (t, a)|da

n(n + 1)σ β¯ γ + h ≤β¯ nh + Hh + 2 here we have used the fact that |e−x − e−y | ≤ |x − y|. Consequently, H uniformly approaches 0 as h → 0+ for any element in Ω . This indicates (iii) of Theorem B.2 [161] holds. From Lemma 5.2, it follows that the positive orbit of the semiflow Φ (t) is relative compact. Lemma 5.3. Under the Assumption 4, for any x0 ∈ Ω , {Φ (t, x0 ) : t ≥ 0} has compact closure in Y . Proof. Since sup Φ (t)x0 Y ≤ 1 t≥0

for all x0 ∈ Ω , it follows from Lemma 5.2 and Theorem A.3 or Proposition 3.3 [178] that {Φ (t)x0 : t ≥ 0} has compact closure in Y .

5.2.1 Global Asymptotic Stability of the Disease-Free Equilibrium In this section, we will investigate properties of the disease-free equilibrium E0 and also give the basic reproduction number R0 , which determines whether or not the disease persists. In epidemiology, R0 called the basic reproduction number, which is the average number of newly infected individuals produced by introducing an

5.2 An Age-Structured SIS Model on Complex Networks

163

infectious individual to a totally susceptible population. Define K = and

k2 

n

= ∑

k2 p(k).

∞ 0

β (a)π (a)da

The basic reproduction number is defined as follows:

k=1

R0 =

n σ k2  lNl K = ∑l n k l=1 ∑ lNl

 ∞ 0

β (a)π (a)da,

(5.14)

l=1

n

lNl /( ∑ lNl ) denotes the probability of choosing one infected edge with degree l. l=1 n

lNl /( ∑ lNl ) denotes the secondary infected nodes produced by the choosing edge l=1

n

with degree l during its infectious period. Then l 2 Nl /( ∑ lNl ) denotes secondary l=1

infected edges by one of infected edge during its infectious period. R0 denotes the total secondary infected edges by introducing one infected edge into a network with only susceptible nodes during its infectious period. Therefore, the definition of R0 in (5.14) agrees with the biological interpretation of the basic reproduction number. The equilibrium E0 = (0, 0, . . . , 0) ∈ Ω is a disease-free equilibrium of (5.8). Theorem 5.1. Let R0 be defined in (5.14). The disease-free equilibrium E0 of (5.8) is locally asymptotically stable if R0 < 1 and it is unstable if R0 > 1. Proof. We linearize system (5.8) around the disease-free equilibrium E0 . By setting ik (t, a) = yk (t, a), we arrive at ⎧ ⎨ ∂ yk (t, a) + ∂ yk (t, a) = −γ (a)y (t, a), k ∂t ∂a (5.15) k ∈ Nn . ⎩ yk (t, 0) = σ kΘ (y(t, ·)), where y(t, ·)  (y1 (t, ·), y2 (t, ·), · · · , yn (t, ·)). We look for solution of the form yk (t, a) = yk (a)eλ t into (5.15) to get ⎧ ⎨ dyk (a) = −(λ + γ (a))y (a), k da (5.16) k ∈ Nn . ⎩ yk (0)  yk0 = σ kΘ (y(·)), Solving the first equation of (5.16) yields yk (t) = yk0 π (a)e−λ a .

(5.17)

Substituting (5.17) into the second equation of (5.16), we have yk0 =

σ n k ∑ l p(l) k l=1

 ∞ 0

yl0 β (a)π (a)e−λ a da,

k ∈ Nn .

164

Define Θ (y(λ )) =

5 Age Structured Models on Complex Networks 1 k k

n

∑ l p(l)yl0 K(λ ). Multiplying

l=1

jp( j)K(λ ) k

and then adding to-

gether from 1 to n, we have n

Θ (y(λ )) =

σ ∑ l 2 p(l)K(λ ) l=1

k

Θ (y(λ )).

Cancelling Θ (y(λ )) on both side of the above equation yields 1−

σ k2   K(λ ) = 0. k

(5.18)

We know that E0 is locally asymptotically stable if all the roots of (5.18) have negative real parts and it is unstable if (5.18) has at least one root with positive real part. First, suppose R0 < 1. We claim that all roots of (5.18) have negative real parts. By way of contradiction, suppose that (5.18) has a root λ0 with Re(λ0 ) ≥ 0. Then it 2  λ0 ) = 0 that follows from 1 − σ k  K( k

  2   σ k2   λ0 ) ≤ σ k  K = R0 , K( 1 =   k k a contradiction. This proves the claim and hence E0 is locally asymptotically stable if R0 < 1. Now, suppose R0 > 1. In this case, we have 1−

σ k2   K(0) = 1 − R0 < 0 k 

and lim

λ →∞

1−

σ k2   K(λ ) = 1. k

By the Intermediate Value Theorem, (5.18) has at least one positive root. Therefore, E0 is unstable if R0 > 1. This completes the proof. Theorem 5.2. When R0 < 1, the disease-free equilibrium E0 of (5.8) is globally asymptotically stable. Proof. For any solution i with i0 ∈ Ω , integrating (5.8) along the characteristic line t − a = c (a constant) gives us  − ξa γ (s)ds

ik (a + c, a) = ik (a + ξ , a)e

where ξ is determined by the sign of c,  0, if c > 0, ξ= −c, otherwise.

,

5.2 An Age-Structured SIS Model on Complex Networks

165

Therefore,  ik (t, a) =

Bk (t − a)π (a),

t ≥ a,

(a) , ik0 (a − t) ππ(a−t)

t < a,

k ∈ Nn ,

(5.19)

where Bk = ik (t, 0). It follows from Proposition 5.1 that Bk is nonnegative and bounded for k ∈ Nn . Substituting (5.19) into (5.8), we obtain Bk (t) ≤

σ kk−1 [  ∞

n

 t

l=1

0

∑ l p(l)(

β (a)Bl (t − a)π (a)da

π (a) da)] π (a − t) t   for k ∈ Nn . For simplicity of notation, let B∞ = lim sup Bk (t) 1×n . Note that +

(5.20)

β (a)il0 (a − t)

t→∞

 ∞ t

β (a)il0 (t − a)



∞ π (a) π (a + t) da = da β (a + t)il0 (a) π (a − t) 0 $ $π (a) $ $ ≤ β ∞ il0 1 $ ππ(t+·) (·) $ . ∞

This, together with (5.20) and Proposition 3.1 [85], produces n

−1 B∞ ∑ l p(l)B∞l K. k ≤ σ kk

(5.21)

l=1

Multiplying kp(k)/k−1 on both sides of (5.20) and then summing up k from 1 to n yield Θ (B∞ ) ≤ R0Θ (B∞ ). (5.22) Equation (5.22) holding just needs Θ (B∞ ) = 0. Otherwise, there exists some k0 such that p(k0 ) = 0. If this case holds, we return to (5.21) and readily obtain B∞ k0 = 0. Hence B∞ = 0 by the nonnegativity of B∞ . Then the result follows directly from (5.19).

5.2.2 Existence and Stability of the Endemic Equilibrium By Theorem 5.1, if R0 > 1 then the disease-free equilibrium of (5.8) is unstable. Let E ∗ = i∗ = (i∗1 , i∗2 , . . . , i∗n ) ∈ Ω be an endemic equilibrium of (5.8). Then, for k ∈ Nn , i∗k = 0 and ⎧ ∗ ⎪ dik (a) ⎨ = −γ (a)i∗k (a), da  ∞ (5.23) ⎪ ⎩ i∗k (0)  B∗k = σ k(1 − i∗k (a)da)Θ (i∗ ). 0

166

5 Age Structured Models on Complex Networks

Solving (5.23) yields i∗k (a) = B∗k π (a). We substitute i∗k (a) into the expression of B∗k to get σ kk−1 K Bˆ ∗ , k ∈ Nn , (5.24) B∗k = 1 + σ kk−1 K1 K Bˆ ∗ where K1 =

∞ 0

n

π (a)da and Bˆ ∗ = ∑ kp(k)B∗k . It follows that k=1

σK n k2 p(k)Bˆ ∗ Bˆ ∗ = ∑ 1 + σ kk−1 K1 K Bˆ∗  f (Bˆ∗ ). k k=1

(5.25)

One can see that f  (Bˆ ∗ ) > 0 with f  (0) = R0 > 1 and f  (Bˆ ∗ ) < 0. It follows that (5.25) has a unique positive solution and this solution combined with (5.24) and (5.23) gives the existence of a unique endemic equilibrium. In summary, we have proved the following result. Theorem 5.3. Suppose R0 > 1. Then (5.8) has a unique endemic equilibrium E ∗ = (i∗1 , i∗2 , . . . , i∗n ), which is in Ω . Next, we discuss the local stability of system (5.8). First, we linearize system (5.8) around the endemic equilibrium E ∗ and let ik (t, a) = yk (t, a) + i∗k (a) obtain ⎧ ∂ yk (t, a) ∂ yk (t, a) ⎪ ⎪ + = −γ (a)yk (t, a), ⎪ ⎪ ∂t ∂ a ∞  ∞ ⎪ ⎨ i∗k (a)da)Θ (y) − σ kΘ ∗ yk (t, a)da, yk (t, 0) = σ k(1 − 0 0 ⎪ ⎪ n ∞ ⎪ 1 ⎪ ⎪ l p(l) β (a)yl (t, a)da. ⎩ Θ (y) = ∑ k l=1 0

(5.26)

Second, let (5.26) have a solution in the form of yk (t, a) = yk (a)eλ t . We substitute yk (t, a) into (5.26) and obtain ⎧ dyk (a) ⎪ ⎪ = −(λ + γ (a))yk (a), ⎪ ⎪  ∞  ∞ ⎪ ⎨ da i∗k (a)da)Θ (y) − σ kΘ ∗ yk (a)da, yk (0) = σ k(1 − (5.27) 0  0 ⎪ ⎪ n ∞ ⎪ 1 ⎪ ⎪ ⎩ Θ (y) = ∑ l p(l) 0 β (a)yl (a)da. k l=1 Theorem 5.4. Let R0 > 1 and γ (a) = γ , then the endemic equilibrium E ∗ is locally asymptotically stable. Proof. Solving the first equation of (5.27), we obtain yk (a) = yk (0)π (a)e−λ a . Then we substitute yk (a) into the second equation of (5.27) and obtain 

yk (0) =

σ k(1 − 0∞ i∗k (a)da)Θ (y(0))  . 1 + σ kΘ ∗ 0∞ π (a)e−λ a da

(5.28)

5.2 An Age-Structured SIS Model on Complex Networks

Multiplying

1 ˆ k kp(k)K(λ )

167

on both side of (5.28) yields 

σ l 2 p(l)(1 − 0∞ i∗l (a)da)  Θ (y(0)) = Θ (y(0)). k 1 + σ lΘ ∗ 0∞ π (a)e−λ a da

(5.29)

If Θ (y(0)) = 0, we cancel (5.29) and obtain the following characteristic equation: 

1=

σ l 2 p(l)(1 − 0∞ i∗l (a)da)  . k 1 + σ lΘ ∗ 0∞ π (a)e−λ a da

If γ (a) = γ , we employ the relationship 1 = tain 1=

σ k

n

∑ l 2 p(l)(1 −

l=1

(5.30) ∞ ∗ 0 il (a)da)K and ob-

 ˆ λ )(λ + γ ) σ l 2 p(l)(1 − 0∞ i∗l (a)da)K(

k

λ + γ + σ lΘ ∗ .

(5.31)

By the way of contradiction, we assume that (5.31) has a characteristic root with positive real parts, then the right side of (5.31)  σ K(λ ) n l 2 p(l)(1 −  ∞ i∗ (a)da)(ˆλ + γ )    0 l   ∑ k l=1 λ + γ + σ lΘ ∗  ∞ σ n 2 < l p(l)(1 − i∗l (a)da)Kˆ = 1, ∑ k l=1 0

(5.32)

which leads to a contradiction with (5.31). Therefore, if Θ (y(0)) = 0 the endemic equilibrium E ∗ is locally asymptotically stable. 1 If Θ (y(0)) = 0, it follows from the second equation that yk (0) = −σ kΘ ∗ yk (0) λ + γ. ∗ Hence, λ = −(σ kΘ + γ ) is negative. This indicates that system (5.8) is locally asymptotically stable. To establish the stability of the endemic equilibrium E ∗ , we need the following two results. The first result is about the uniform strong ρ -persistence of the disease. The proof is based on the theory of uniform persistence [167]. Though the arguments are tedious, they are standard (see, for example, [20]). Definition 5.7. A total trajectory of Φ in X is a function h : R → X such that Φ (t, h(r)) = h(t + r) for all r ∈ R and t ∈ R+ . Theorem 5.5 (Theorem 1.40 (Persistence Theory) [167]). A set A is invariant if and only if for every element x0 ∈ A. There exists a total trajectory through (0, x0 ) with values in A. The next result states that the compact attractor of bounded sets is the union of all bounded total trajectories. Proposition 5.2 (Proposition 2.34 [167]). Let A be compact attractor of bounded sets for Φ on X. Then A contains every bounded backward invariant set (for t ∈ (−∞, 0])

168

5 Age Structured Models on Complex Networks

A = {x ∈ X : ∃ bounded total trajectory Φ though x} x0 = {x ∈ X, ∀t ∈ R+ , ρ (Φ (t, X)) = 0}. It follows from Proposition 5.2 that the compact attractor of bounded sets consists of the orbits of total trajectories. A total trajectory of Φ in X is a function h : R → X such that Φ (t, h(r)) = h(t + r) for all r ∈ R and t ∈ R+ . Let r ∈ R be fixed and arbitrary. Define skr (t) = sk (t + r), Θr (t) = Θ (t + r), ikr = ik (t + r, a). Then (Skr (t), ikr (t, a) = Φ (t, (Sk (r), ik (r, ·))). It follows from the definition of the semiflow,  skr = −σ kskrΘ (ir (t, ·)) + 0∞ γ (a)ikr (t, a)da, ⎧ ⎨ σ kskr (t − a)Θ (ir (t − a))π (a), t > a, ikr (t, a) = ⎩ i (r, a − t) π (a) ,t < a, k0 π (a−t) skr (0) = Sk (r). Hence, for all t > 0 and k ∈ N, 

sk (t + r) = −σ ksk (t + r)Θ (i(t + r, ·)) + 0∞ γ (a)ik (t + r, a)da, ⎧ ⎨ σ ksk (t + r − a)Θ (i(t + r − a))π (a), t > a, ik (t + r, a) = ⎩ i (r, a − t) π (a) ,t < a, k0 π (a−t) skr (0) = Sk (r). For any ξ ≥ r, let t = ξ − r, 

sk (ξ ) = −σ ksk (ξ )Θ (i(ξ , ·)) + 0∞ γ (a)ik (ξ , a)da, ⎧ ⎨ σ ksk (ξ − a)Θ (i(ξ − a))π (a), ξ − r > a, ik (ξ , a) = ⎩ i (r, a − (ξ − r)) π (a) , ξ − r < a, k0 π (a−(ξ −r)) and

Θ (i(ξ − a)) =

1 k

+

n

∑ l p(l)

l=1

 ξ −r 0

βl (a)sl (ξ − a)Θ (i(ξ − a))da

∞

ξ −r βk (a)ik0 (r, a − (ξ

π (a) − r)) π (a−( ξ −r)) da.

Then we let r go to −∞ and obtain the following equations: sk (ξ ) = −σ ksk (ξ )Θ (i(s, ·)) +

∞ 0

γ (a)ik (ξ , a)da,

ik (s, a) = σ ksk (ξ − a)Θ (i(ξ − a))π (a), and

(5.33)

5.2 An Age-Structured SIS Model on Complex Networks

Θ (i(s − a)) =

1 k

n

∑ l 2 p(l)

∞

l=1

0

169

β (a)sl (ξ − a)Θ (i(ξ − a))da.

(5.34)

We define ρ : L1 (0, +∞) → R by  ∞

ρ (sk (t), ik (t, ·)) = Then ρ (Φ (t, x)) =

∞ 0

0

ik (t, a)da = Ik (t).

(5.35)

ik (t, a)da = Ik (t).

Theorem 5.6. If R0 > 1, there exists a k0 such that Ik0 (t) is uniformly weekly ρ persistent. Proof. Since R0 > 1, there exists a small ε > 0 such that

σ (1 − ε )

k2  K > 1. k

(5.36)

By the way of contradiction, we assume that there exists an x0 ∈ Ω such that for all k, lim sup ρ (Φ (t, x0 )) ≤ ε2 . Then there exists a t0 > 0 for t > t0 such that t→+∞

ρ (Φ (t, x0 )) =

 ∞ 0

ik (t, a)da < ε .

For any k ∈ N, it follows from the boundary condition that Bk (t) ≥ (1 − ε )

1 k

 t

n

∑

j p( j) 0

j=1

β (a)B j (t − a)π (a)da.

(5.37)

Taking Laplace transform on both side of (5.37), we have  λ ), ˆ λ )K( Bk (t) ≥ kσ (1 − ε )B( ˆ λ) = where B(

1 k

n

∑ j p( j)B j (λ ). Multiplying

j=1

1 k kp(k)

(5.38) on both side of (5.38) and

then summing up, we obtain  λ ) ≥ σ (1 − ε ) k  K(  λ )B(  λ ). B( k 2

(5.39)

If we take λ = 0, (5.39) implies that σ kk K < 1 which leads to a contradiction with (5.35). 2

We list two additional conditions for supporting the following theorem. Assumption 5 (H0) Φ has a compact attractor A attractor each x ∈ A . (H1) There is no trajectory h : R → A such that ρ (h(0)) = 0, ρ (h(−r)) > 0 and ρ (h(t)) > 0 for r,t ∈ R.

170

5 Age Structured Models on Complex Networks

Asymptotical smoothness of the semiflow Φ (t, x), together with Theorem 5.6 inA for all x ∈ A . To verify (H1) of dicates that Φ (t, x) has a compact attractor ∞ i (a, 0)da = 0, 0∞ ik0 (a, −r)da > 0, and Assumption 5, we need to show that k 0 0 ∞ 0 ik0 (a,t)da > 0 do not simultaneously hold. Observing the definition of the total trajectory defined in (5.76), we obtain  ∞ 0

 ∞ 0

ik0 (a, 0)da =

ik0 (a, −r)da = =

 ∞ 0

 ∞ 0 r ∞

sk0 (−a)Θ (i(−a))π (a)da = 0

sk0 (−r − a)Θ (i(−r − a))π (a)da

sk0 (−a)Θ (i(−a))π (−r − a)da

=e−γ r

(5.40)

 r ∞

(5.41)

sk0 (−a)Θ (i(−a))π (−a)da.

Both (5.40) and (5.41) holding together is impossible if we let r → 0. Hence, (H1) of Assumption 5 holds. / ρ ◦ Φ is continuous and Φ is uniTheorem 5.7 (Theorem 5.2 [167]). If x0 = 0, formly weakly ρ -persistent, then Φ is uniformly strongly ρ -persistent. 

Theorem 5.3 states that x0 = 0. / ρ ◦ Φ = 0∞ ik (t, a)da ≤ 1 implies that ρ ◦ Φ is continuous. It follows from Theorems 5.6 and 5.7 that Ik (t) is uniformly strongly persistent. Lemma 5.4. Suppose R0 > 1. Then the disease is uniformly strongly ρ -persistent, that is, there is an η > 0 such that if i0 ∈ Ω satisfies i0 > 0 then there exists a k0 such that the solution i of (5.8) satisfies lim inf Ik (t) ≥ η . t→∞

In order to prove the global stability of the endemic equilibrium, we must mention the Volterra-type function defined as g(x) = x − 1 − ln x.

(5.42)

It is easy to see that g(x) > 0 for all x ≥ 0 and it has a minimum value 0 at x = 1. Assume that f (t, a) ∈ L1 (R+ ) is a solution of the following system:

∂ f (t, a) ∂ f (t, a) + = −m(a) f (t, a), ∂t ∂ a∞ f (t, 0) = L(t) η (a) f (t, a)da,

(5.43)

0

f (0, a) = f0 (a),

where m(a), η (a) ∈ L1 (R+ ), and L(t) ∈ R. The nontrivial equilibrium E1∗ of system (5.43) satisfies the following equations: d f ∗ (a) = −m(a) f ∗ (a), da  ∞ f ∗ (0) = L∗ η (a) f ∗ (a)da. 0

(5.44)

5.2 An Age-Structured SIS Model on Complex Networks

Define a Lyapunov function as V1 (t) = f ∗ (s)ds.  ∞

Lemma 5.5. If 0

 ∞ a

0

M(a)g( ff (t,a) ∗ (a) )da, where M(a) =

∞ a

β (s)

| ln f0 (a)|da is bounded, then

dV1 (t) = dt where M(a) =

∞

171

 ∞ 0

  f (t, a) f (t, 0) ) − g( ∗ ) da, M(a) f (a) g( ∗ f (0) f (a) ∗

(5.45)

η (s) f ∗ (s)da.

Proof. Note that

∂ g( ff (t,a) ∗ (a) ) ∂a

=

f (t, a) ∂ f (t, a) ( − 1 − ln ∗ ) ∂ a f ∗ (a) f (a)

=

f (t, a) ∂ f (t, a) ∂ − ln ∗ ∂ a f ∗ (a) ∂ a f (a) 

f ∗ (a) fa (t, a) f ∗ (a) − f (t, a) fa∗ (a) f  (t, a) f ∗ (a) − f (t, a) fa∗ (a) − = a ( f ∗ (a))2 f (t, a) ( f ∗ (a))2 % &  1 1 fa∗ (a)  − ) fa (t, a) − f (t, a) ∗ =( ∗ f (a) f (t, a) f (a) =(

1 1 1 1 − ) f  (t, a) + ( ∗ − )m(a) f (t, a), f ∗ (a) f (t, a) a f (a) f (t, a)

(5.46) where we denote hl (s, l) = ∂ h(s,l) . ∂l It follows from the assumption of Lemma 5.5 that V1 (t) is well defined for all t ≥ 0. Deviating it along the solution of (5.43), we obtain  ∞ ∂ g( ff (t,a) ∗ (a) ) dV1 (t)  da = M(a)  dt (5.43) 0 ∂t ∞ f ∗ (a) ft (t, a) ) da = M(a)(1 − f (t, a) f ∗ (a) 0 ∞ 1 1 = − )(− fa (t, a) − m(a) f (t, a))da M(a)( ∗ f (a) f (t, a) 0  ∞ ∞ 1 1 − ) fa (t, a)da − =− M(a)( ∗ M(a)m(a) f (t, a)da. f (a) f (t, a) 0 0 (5.47) Observing (5.46), we have  ∞ ∂ g( ff (t,a) ∗ (a) ) dV1 (t)  da. = − M(a)  dt (5.43) ∂a 0

(5.48)

172

5 Age Structured Models on Complex Networks

With the help of integrating by parts, we obtain  ∞ dV1 (t)  f (t, a) ∞ f (t, a) ) + )da = −M(a)g( ∗ Ma (a)g( ∗  dt (5.43) f (a) 0 f (a) 0 

∞ f (t, 0) f (t, a) )− )da η (a) f ∗ (a)g( ∗ ∗ f (0) f (a) 0    ∞ f (t, a) f (t, 0) ∗ = ) − g( ∗ ) da, η (a) f (a) g( ∗ f (0) f (a) 0

= M(0)g(

here we used the fact M(0) =

∞ 0

(5.49)

η (a) f ∗ (a)da, and M  (a) = −η (a) f ∗ (a).

Theorem 5.8. If γ (a) = γ and R0 > 1, then the endemic equilibrium E ∗ is the global attractor of system (5.6). Proof. Construct the following Lyapunov function: Vk (t) = s∗k g(

n sk (t) ∗ ) + ks j p( j) ∑ k s∗k j=1

 ∞ 0

α j (a)g(

ik (t, a) )da, k ∈ Nn , i∗k (a)



where α j (a) = a∞ β (a)i∗j (a)da. It follows from Lemma 5.4 that Vk (t) is well defined for t ∈ R. Taking the derivative of Vk (t) with respect to time t, along the trajectory of (5.6) and Lemma 5.5 yields  ∞ n s∗k i∗ (a) ∂ ik (t, a) dVk (t)   ∗ )s ) da =(1 − + ks j p( j) α j (a)(1 − k  k k ∑ dt (5.6) sk (t) ik (t, a) ∂t 0 j=1

=(1 −

n s∗k )(k−1 Sk ∑ kp(k) sk (t) k=1 n

+ k−1 ∑

 ∞

k=1 0

αk (a)(1 −

 ∞ 0

β (a)ik (t, a)da + γ (1 − sk (t)))

i∗k (a) ∂ ik (t, a) )[− − γ ik (t, a)]da, ik (t, a) ∂a

n s∗ = − γ (1 − k )(sk (t) − s∗k ) + ks∗k k−1 ∑ j p( j) sk (t) j=1

× [1 −

0

β (a)i∗k (a)

s∗k sk (t) i j (t, a) i j (t, a) − ∗ ∗ + ∗ ]da sk (t) sk i j (a) i j (a) n

 ∞

j=1

0

+ ks∗k k−1 ∑ j p( j) =γ s∗k (2 − %

 ∞

β j (a)i∗j (a)[g(

ik (t, a) ik (t, 0) ) − g( ∗ )]da ∗ ik (0) ik (a)

n s∗k sk (t) ∗ −1 ) + kS − k j p( j) ∑ k s∗k sk (t) j=1

 ∞ 0

β (a)i∗j (a)

& s∗k i∗k (0) sk (t) i j (t, a) ik (t, a) × 1− + ln − ∗ ∗ + ln ∗ da. sk (t) ik (t, 0) sk i j (a) ik (a) (5.50)

5.2 An Age-Structured SIS Model on Complex Networks

173

Note that from (5.23) and (5.24) s∗

n

∑ k2 p(k) kk K = 1,

k=1

so that

n

∑ k2 p(k)

k=1

=

s∗k n ∑ l p(l) k l=1 s∗

n

=

0

β (a)i∗l (a)

il (t, 0) i∗l (0)

n

∑ k2 p(k) kk K ∑ l p(l)il (t, 0)

k=1

=

 ∞

l=1

(5.51)

n

n

l=1

k=1

∑ l p(l)il (t, 0) = ∑ kp(k)ik (t, 0) n

∑ k2 p(k)

k=1

s∗k n ∑ l p(l) k l=1

Let V (t) =

 ∞ 0

β (a)i∗l (a)

sk (t)il (t, 0) . s∗k i∗l (0)

n

∑ kp(k)Vk (t).

k=1

Combining (5.50) and (5.51) yield n s∗ n dV (t) ≤ ∑ k2 p(k) k ∑ l p(l) dt k l=1 k=1

 ∞

s∗ n = ∑ k p(k) k ∑ l p(l) k l=1 k=1

 ∞

n

2

0

0

β (a)i∗l (a)

  il (t, a)i∗l (a) s∗k + ln 1− da sk (t) il (t, 0)i∗l (0)

β (a)i∗l (a)

 sk (t)i∗ (0)ik (t, a) s∗ ) −g( k ) − g( ∗ k sk (t) sk ik (t, 0)i∗k (a)



+ ln

il (t, a)i∗l (a) da. il (t, 0)i∗l (0)

(5.52) By the virtue of g, the first term of (5.51) is negative. In the following, we need to show that the last term is also negative as well. Observing the last term of (5.51), we have  n n s∗k il (t, 0) ik (t, 0) 2 ∗ ∑ ∑ k p(k) k l p(l)Kil ln i∗ (0) − ln i∗ (0) k l k=1 l=1 =

n

∑

n

∑ k2 p(k)

k=1 l=1

=

s∗k s∗ l p(l)K m k k

 il (t, 0) ik (t, 0) ∗ − ln j p( j)i (0)K ln (5.53) ∑ j i∗k (0) i∗l (0) j=1 n

 il (t, 0) ik (t, 0) − ln ν ln , ∑ ∑ kl i∗k (0) i∗l (0) k=1 l=1 n

n

174

5 Age Structured Models on Complex Networks

where

νkl = K 2 l p(l)k2 p(k)

s∗k s∗l k k

n

∑ j p( j)i∗j (0).

j=1

By the symmetrical character of νkl with respect to k and l, Eq. (5.53) is equal to zero. This, together with Eq. (5.51) implies that dVdt(t) ≤ 0. It follows from V˙ (t) = 0 that i j (t, a) ik (t, 0) ik (t, a) = ∗ = ∗ . (5.54) sk (t) = s∗k , ∗ i j (a) ik (0) ik (a) From the first equation of system (5.6), we admit

Θ (i) =

γ (1 − s∗k ) = Θ (i∗ ), kσ s∗k

this implies that ik (t, 0) does not depend on t and then ik (t, 0) = ci∗k (0). Recalling the first equation of (5.6) ensures that c = 1. Therefore, the largest invariant set is M = {(s(t), i(t, a)) ∈ Ω |V˙ (t) = 0} = {(s∗ , i∗ (a))}. The LaSalle invariance principle implies that E ∗ is globally asymptotically stable in Ω .

5.2.3 Numerical Simulations Based on the preferential algorithm [145, 146], we can generate a Barabasi-Albert (BA) network (A few nodes have many links, but most nodes have very few links, connectivity is in power-law form [10]) with k ≈ 4.0118 (the average degree) and N = 10,000. The dynamics of the nodes is implemented based on the BA network with 10,000 nodes. In order to illustrate the theoretical results, we do the numerical simulations based on the above network. For this purpose, we first take σ = 1,

0, a ≤ τ, β (a) = β ∗, a > τ. If we take β ∗ = 0.48, then R0 = 0.96 < 1. According to Theorem 5.2, the diseasefree equilibrium E0 is globally stable (see Fig. 5.2a). Now, we enlarge the transmission probability by taking β ∗ = 0.08. Then R0 = 1.6 > 1. According to Theorem 5.8, the endemic equilibrium E ∗ is globally asymptotically stable (see Fig. 5.2b).

5.3 An SIR Model with Demography on Complex Networks Define a complex network with degree n where each node is either occupied by an individual or vacant. The states for the epidemic transmission process on the network are divided into vacant state V , susceptible state S, infected state i, and recovered state R. The vacant state can give birth to a susceptible state. Susceptible, in-

5.3 An SIR Model with Demography on Complex Networks 10-3

7

0.35

β=0.08,R0=1.6

6

0.3

0.25

β *=0.48,R0 =0.96

4

I(t)

I(t)

5

0.2

3

0.15

2

0.1

1

0.05

0

175

0

50

100

150

Time t

200

250

300

0

0

50

100

(a)

150

Time t

200

250

300

(b)

Fig. 5.2 Stability of the equilibria of (5.8). (a) The disease-free equilibrium is globally asymptotically stable when R0 < 1. Here β ∗ = 0.048, γ = 0.2, and τ = 2; (b) The endemic equilibrium is globally stable when R0 > 1. Here β = 0.08, γ = 0.2, and τ = 2

fected, and recovered states can change their state into a vacant state at natural death rate μ , a susceptible state can be infected and then change into an infected state, an infected state moves to a recovered state due to treatment or spontaneous recovery. j) For a uncorrelation network, the conditional probability p ( j|k) = jp( k denotes a node with degree k linking to a node with degree j where k = ∑ kp(k). Here k

we will assume that every infectious node infects at least one neighbor. Therefore, the edges for an infectious individual with degree j freely linking with a susceptible individual is at most j − 1 since one edge of each infected node must be pointing to j) . an infected node. Then the conditional probability is chosen as p( j|k) = ( j−1)p( k With the structure of a complex network and infection age, let Sk (t) be the fraction of susceptible vertices of degree k at time t in all nodes n, Rk denote the fraction of recovered vertices of degree k at time t, and ik (t, a) stand for the density of infected vertices of degree k at time t and with infection age a for k ∈ Nn = {1, 2, . . . , n}. Then we can formulate the following SIR model with infection age on complex networks, ⎧ dSk (t) ⎪ = bk[1 − Nk (t)]Ψ − μ Sk − kSk (t)Θ (ik (t, ·)), ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ ik (t, a) + ∂ ik (t, a) = −(μ + γ (a))ik (t, a), ∂t ∂a ⎪ ⎪ (t, 0) = kS (t) Θ (i(t, ·)), i ⎪ k k ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎩ dRk = γ (a)ik (t, a)da − μ Rk (t), dt 0

k ∈ Nn ,

(5.55)

where γ (a) is the recovery rate at infection age a from infected class to recovered n

 class, Θ (i(t, ·)) = ∑ (k − 1)p(k) 0∞ βk (a)ik (t, a)da /k denotes the force of ink=1

176

5 Age Structured Models on Complex Networks

fection, and βk (a) is the transmission rate at infection age a. The coefficient b is n

the input rate or birth rate, μ is the natural death rate. Ψ (t) = ∑ p ( j|k) 1j N j (t) j=1

[99] describes the probability of fertility contacts between nodes with degree k and their neighbors with degree j with stochastic connection. Therefore, bk(1 − Nk (t))Ψ stands for the number of newborn susceptible individuals of degree k per unit time. The initial condition of (5.55) is 1 (S0 , i0 , R0 ) = (S10 , . . . , Sn0 , i10 , . . . , in0 , R10 , . . . , Rn0 ) ∈ (R+ )n × (L+ (R+ ))n × (R+ )n .

5.3.1 Baseline Analysis of the SIR Model Define the state spaces as follows: X = X1 × X2 × X3 ,

X1 = Rk , X2 = (L1 )k ,

X3 = Rk , k = 1, 2, · · · , n

with the norm

X X = ∑ |Sk (t)| + ∑ k

k

 ∞ 0

ik (t, a)da + ∑ |Rk (t)|. k

On the other hand, Y = X1 × X2 ,

X1 = Rk , X2 = (L1 )k , k = 1, 2, · · · , n

with the norm Y = X1+ × X2+ ,

1 k X1+ = Rk+ , X2+ = (L+ ) , k = 1, 2, · · · , n.

One can easily see that dNk (t) = bk(1 − Nk )Ψ − μ Nk (t). dt

(5.56)

Obviously, (5.56) has an equilibrium that satisfies Nk∗ =

bkΨ ∗ . μ + bkΨ ∗

Note that Nk (t) is the fraction of occupied nodes with degree k. We assume n

∑ Nl (t) < 1. Plugging the expression Nk into Ψ yields

l=1

Ψ=

kp(k)bΨ 1 = f (Ψ ). k ∑ k μ + bkΨ

(5.57)

5.3 An SIR Model with Demography on Complex Networks

177

Note that f (1) < 1, f  (Ψ ) > 0, and f  (0) = bk μ > 1 if the scale of the networks is large enough or we will assume b > μ in the rest of this section. If not, all nodes ultimately become vacant. This implies that (5.57) has a unique fixed equilibrium bkΨ ∗ Ψ ∗ . Therefore, lim Nk (t) = μ +bk Ψ ∗ . Without a loss of generality, we can assume t→∞

that Nk (t) is a constant for k ∈ Nn . Since the equation for Rk has no relation with the first two equations, we just consider the following system: ⎧ dSk (t) ⎪ = bk[1 − Nk∗ ]Ψ ∗ − μ Sk − kSk (t)Θ (i(t, ·)), ⎪ ⎪ ⎪ ⎨ dt ∂ ik (t, a) ∂ ik (t, a) + = −(μ + γ (a))ik (t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪ ⎩ ik (t, 0) = kSk (t)Θ (i(t, ·))

k ∈ Nn ,

(5.58)

1 (R ))n with the initial condition (S0 , i0 ) = (S10 , · · · , S20 , i10 , i20 , . . . , in0 ) ∈ Rn × (L+ + such that Sk + 0∞ ik0 (a)da ≤ 1 for k ∈ Nn . On the other hand, we make some assumptions on the parameters which stem from their biological interpretations. As discussed Assumption 4 in Sect. 5.1.2, we want the following condition to hold: (5.59) limt→∞ ππ(t+·) (·) ∞ = 0,

where

π (a) = e−

a

0 ( μ +γ (s))ds

for a ∈ R+

is the probability of the infected individual remaining in the infected compartment at age-since-infection a. Condition (5.59) automatically holds since μ > 0. Then (5.58) is well posed by the same method mentioned in (5.55). Moreover, we can define a semiflow Φ (t, X0 ), X0 = (S0 , i0 ) for t ≥ 0 such as

Φ (t, X0 ) = (S1 (t, ), S2 (t), · · · , Sn (t), i1 (t, ·), i2 (t, ·), · · · , in (t, ·)). Note that Sk (t) +

 ∞ 0

ik (t, a)da ≤ Nk =

proposition.

(5.60)

bkΨ < 1. So we have the following μ + bkΨ

Proposition 5.3. The set 1 (R ))n and S + Ω = {(S, i)|S ∈ Rn , i ∈ (L+ + k

∞ 0

ik (a)da ≤ 1 for k ∈ Nn }

is a positively invariant set of (5.58). Lemma 5.6. If the Assumption 4 holds, then system (5.58) has a unique mild solution. Proof. It follows from the proving processes of Lemma 5.1 that Bk (t, S, i) = ik (t, 0) satisfies the condition of Theorem 2.1 in Chap. 2. We observe that G(t, S, i) = bk[1 − Nk∗ ]Ψ ∗ − μ Sk − kSk (t)Θ (i(t, ·)) = G1 (t, S, i) − G2 (t, S, i) where G1 (t, S, i) = bk[1 − Nk∗ ]Ψ ∗ and G2 = μ Sk + kSk (t)Θ (i(t, ·)). Obviously, for any Sk (t) ∈ R+ and ik (t, a) ∈ 1 (0, +∞), G (t) ≥ 0, j = 1, 2. On the other hand, G(t, 0, 0) = bk[1−N ∗ ]Ψ ∗ ≤ bkΨ ∗ L+ j k

178

5 Age Structured Models on Complex Networks

which is bounded. Therefore, it follows from Theorem 2.1 in Chap. 2 that system (5.58) has a unique mild solution for every X0 ∈ Y+ . By the model formulation and Proposition 5.3, we only need to consider (5.58) with initial conditions in Ω . This implies that Φ (t, X) is point dissipative, i.e. Φ (t, X0 ) ⊂ Ω , for all X0 ∈ Ω .

5.3.2 Stability of Disease-Free Equilibrium Equilibrium E0 = (S10 , S20 , · · · , Sn0 , 0, 0, . . . , 0) ∈ Ω is the disease-free equilibrium of (5.58) where Sk0 =

bk(1−Nk∗ )Ψ ∗ , μ

k ∈ Nn .

Theorem 5.9. The disease-free equilibrium E0 of (5.58) is locally asymptotically stable if R0 < 1 and it is unstable if R0 > 1, where n

∑ j( j − 1)S0j p( j)K j

R0 = with K j =

∞ 0

j=1

(5.61)

k

β j (a)π (a)da, j ∈ N+ .

Proof. The linearized system of (5.58) around the disease-free equilibrium E0 is ⎧ dS (t) ⎪ ⎪ k = −μ Sk − kSk0Θ (i(t, ·)), ⎪ ⎪ ⎨ dt ∂ ik (t, a) ∂ ik (t, a) + = −(μ + γ (a))ik (t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪ ⎩ ik (t, 0) = kSk0Θ (i(t, ·)),

k ∈ Nn ,

(5.62)

where i(t, ·)  (i1 (t, ·), i2 (t, ·), · · · , in (t, ·)) is the vector of perturbation. Substituting Sk (t) = Sk0 eλ t , ik (t, a) = ik (a)eλ t into (5.15) yields ⎧ 0 = −(λ + μ )Sk0 − kSk0Θ (i(·)), ⎪ ⎪ ⎪ ⎪ ⎨ dik (a) (5.63) k ∈ Nn . = −(λ + μ + γ (a))ik (a), ⎪ da ⎪ ⎪ ⎪ ⎩ ik (0) = kSk0Θ (i(·))  ik0 . Solving the second equation of (5.63) yields ik (a) = ik0 π (a)e−λ a .

(5.64)

Substituting (5.64) into the third equation of (5.63), we obtain ik0 = kSk0Θ (i(λ )),

(5.65)

5.3 An SIR Model with Demography on Complex Networks

where Θ (i(λ )) =

1 k

179

n

∑ (l − 1)p(l)il0 K j (λ ). Multiplying both sides by

l=1

1 k ( j

−

1)p( j)K j (λ ) and then adding together, we have

Θ (i(λ )) =

j( j − 1)p( j)S0j K j (λ )

n

∑

k

j=1

Θ (i(λ )).

(5.66)

Cancelling Θ (i(λ )) on both side of (5.66) yields 1=

j( j − 1)p( j)S0j K j (λ )

n

∑

k

j=1

.

(5.67)

We claim that if R0 < 1 the disease-free equilibrium E0 is locally asymptotically stable, that is all the roots of (5.67) have negative real parts. In order to show that, we proceed by way of contradiction. Suppose that (5.67) has a root λ0 with Re(λ0 ) ≥ 0. Then it follows from (5.67) that   n n  'j (λ )   ∑ j( j − 1)S0j p( j)K ∑ j( j − 1)S0j p( j)K j  j=1  j=1 ≤ = R0 . 1 =   k k     This is a contradiction. Hence E0 is locally asymptotically stable if R0 < 1. Now, suppose R0 > 1, define n

'j (λ ) ∑ j( j − 1)S0j p( j)K

G(λ ) = note that

j=1

k

lim G(λ ) = 0, G (λ ) < 0, G(0) = R0 .

λ →∞

Therefore, (5.67) has at least one positive root. Therefore, E0 is unstable if R0 > 1. This completes the proof. Theorem 5.10. When R0 < 1, the disease-free equilibrium E0 of (5.58) is globally asymptotically stable. Proof. Define a function as follows:

αk (a) =

 ∞ a

βk (s)e−

s

a ( μ +γ (ξ ))d ξ

ds,

then αk (0) = Kk and

αk (a) = (μ + γ (a))αk (a) − βk (a).

180

5 Age Structured Models on Complex Networks

Define a Lyapunov function as V (t) = VS (t) +Vi (t), n

where VS (t) = ∑ j( j − 1)p( j)K j S0j g( j=1

Vi (t) =

S j (t) ), g(x) = S0j

n

 ∞

j=1

0

∑ ( j − 1)p( j)

x − 1 − ln x and

α j (a)i j (t, a)da.

Taking the derivative of VS (t) along the solution of (5.58) yields 

dVS (t)  dt (5.58)

=

S0j

n

∑ ( j − 1)p( j)K j (1 − S j (t) )Sj (t)

j=1 n

∑ ( j − 1)p( j)K j (1 −

S0j

)(−μ (S j (t) − S0j ) − jS j (t)Θ (i(t, a))) S (t) j j=1 n μ (S j (t) − S0j )2 n = −∑( j − 1)p( j)K j − ∑ j( j − 1)p( j)K j S j (t)Θ (i(t, a)) S j (t) j=1 j=1

=

n

+ ∑ j( j − 1)p( j)K j S0j Θ (i(t, a)). j=1

(5.68) On the other hand, with an integration by parts, we have 

dVi (t)  dt (5.58)

 ∞

∂ i j (t, a) da ∂a j=1  ∞ n ∂ i j (t, a) + (μ + γ )i j (t, a))da = − ∑ ( j − 1)p( j) α j (a)( ∂a 0 j=1 =

= =

n

∑ ( j − 1)p( j)

0

n

α j (a)

∑ ( j − 1)p( j)[α j (0)i j (t, 0) −

j=1 n

 ∞ 0

(5.69)

β j (a)i j (t, a)]

∑ j( j − 1)p( j)K j S j (t)Θ (i(t, a)) − kΘ (i(t, a)).

j=1

Summing (5.68) and (5.74) yields n μ (S j (t) − S0j )2 dV (t)  = − ∑ ( j − 1)p( j)K j + k(R0 − 1)Θ (i(t, a)).  dt (5.58) Sj j=1

Therefore,

dV (t) dt

≤ 0 if R0 < 1. It is easy to show that the largest invariant set M =

{(S(t), i(t, a) ∈ Ω | dVdt(t) = 0} is the singleton {E0 }. This implies that the disease-free equilibrium E0 is globally asymptotically stable.

5.3 An SIR Model with Demography on Complex Networks

181

5.3.3 Existence and Stability of the Endemic Equilibrium Assume R0 > 1. By Theorem 5.10, if R0 > 1 then the disease-free equilibrium of (5.58) is unstable. We may expect that (5.58) has at least one positive endemic equilibrium E ∗ . In order to get the existence and uniqueness for E ∗ , assume E ∗ = (S∗ , i∗ ) = (S1∗ , . . . , Sn∗ , i∗1 , . . . , i∗n ) ∈ Ω be an endemic equilibrium of (5.58). Then, for k ∈ Nn = {1, 2, · · · , n}, i∗k = 0 and ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

0 = bk(1 − Nk∗ )Ψ ∗ − μ Sk∗ − B∗k ,

di∗k (a)

= −(μ + γ (a))i∗k (a), da i∗k (0)  B∗k = kSk∗Θ (i∗ ).

(5.70)

∗

It follows from the second and third equations of (5.70) that i∗k (a) = Bk π (a) for k ∈ Nn . Moreover, it follows from the first and second equations of (5.70) that Sk∗ = μ Sk0 μ +kΘ (i) ∗

for k ∈ Nn . Recall Sk0 =

bk (1−N ∗ )Ψ ∗ . μ

Substituting Sk∗ into the expression of

Θ (i ) gives ∑ (k − 1)p(k)

Θ (i ) =

k∈Nn

=

k∈Nn

∗

∞ 0

βk (a)B∗k π (a)da

k ∗ (k − 1)p(k)K ∑ k Bk k

k(k − 1)p(k)Kk μ Sk0Θ (i∗ ) 1 . = ∑ k k∈Nn μ + kΘ (i∗ ) Clearly, Θ (i∗ ) = 0 is a solution to the above equation, which gives the disease-free equilibrium. If Θ (i∗ ) > 0 then we have an endemic equilibrium. For an endemic equilibrium, Θ (i∗ )(> 0) is a root of f (Θ ) with f (Θ ) =

k(k − 1)p(k)Kk μ Sk0 1 − 1. ∑ k k∈Nn μ + kΘ

Note that f is decreasing on R+ , f (0) = R0 , and lim f (Θ ) = −1. Then we immediately get the following result.

Θ →∞

Theorem 5.11. Suppose R0 > 1. Then (5.58) has a unique endemic equilibrium E ∗ = (S∗ , i∗ ), which is in Ω . To establish the stability of the endemic equilibrium E ∗ , we need to linearize system (5.58) and then analyze the characteristic equation obtained by system (5.58). Theorem 5.12. Suppose R0 > 1. Then the endemic equilibrium E ∗ = (S∗ , i∗ ) of system (5.58) is locally asymptotically stable.

182

5 Age Structured Models on Complex Networks

Proof. Linearizing system (5.58) at the endemic equilibrium E ∗ , i.e. Sk (t) = xk (t) + Sk∗ , ik (t, a) = yk (t, a) + i∗k , we have ⎧ dxk ∗ ∗ ⎪ ⎪ ⎪ dt = −μ xk − kSk Θ (y) − kΘ (i )xk , ⎪ ⎨ ∂ yk (t, a) ∂ yk (t, a) + = −(μ + γ (a))yk (t, a), ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎩ yk (t, 0) = kSk∗Θ (y) + kΘ (i∗ )xk .

(5.71)

Let system (5.58) have the solution as xk = xk0 eλ t , yk (t, a) = yk (a)eλ t , one can get the characteristic equation for system (5.58) ⎧ 0 = −(λ + μ + kΘ (i∗ ))xk0 − kSk∗Θ (y), ⎪ ⎪ ⎪ ⎨ dyk (a) = −(λ + μ + γ (a))yk (a), ⎪ ⎪ da ⎪ ⎩ yk (0) = kSk∗Θ (y) + kΘ (i∗ )xk0 . Solving the first and second equations of (5.72) yields xk0 = − yk

(0)π (a)e−λ a .

(5.72)

kSk∗Θ (y) λ +μ

and yk (a) =

Replacing xk0 and yk (a) by the above solution, we have yk (0) =

(λ + μ )kSk∗Θ (y) . λ + μ + kΘ (i∗ )

With a transformation, one has

Θ (y) =

n

(λ + μ )kS∗Θ (y)

k  . λ) ∑ k(k − 1)p(k)K( λ + μ + kΘ (i∗ )

k=1

Cancelling Θ (y) on both sides of the above equation, one obtains 1=

1 k

(λ + μ )kS∗

n

∑ ( j − 1)p( j)K j (λ ) λ + μ + kΘ (ik ∗ )  G(λ ).

(5.73)

j=1

By way of contradiction, we assume (5.73) has a characteristic root with positive real part Reλ0 ≥ 0. Note that |G(λ0 )| ≤ ≤ ≤

1 k 1 k 1 k

n

∑ ( j − 1)p( j)|K j (λ )||

j=1

(λ + μ )S∗j

λ + μ + jΘ (i∗ )

n

∑ ( j − 1)p( j)|K j (Re(λ0 ))||

j=1 n

∑ ( j − 1)p( j)|K j (Re(λ0 ))|.

j=1

|

(λ + μ )S∗j

λ + μ + jΘ (i∗ )

|

5.3 An SIR Model with Demography on Complex Networks

183

This leads to a contradiction with g(λ ) = 1. Thus the endemic equilibrium E ∗ is locally asymptotically stable.

5.3.4 Global Dynamics In this section, we mainly establish the global stability of the endemic equilibrium E ∗ by constructing Lyapunov functional. In order to guarantee its existence over the solutions of system (5.58), we need to show the uniform persistence of system (5.58). First, we prove the semiflow Φ is asymptotically smooth. This can be established from Propositions 5.4 and Theorem A.3. Proposition 5.4 (Proposition 6 [128]). Let D ⊂ R. For j = 1, 2, suppose f j : D → R is a bounded Lipschitz continuous function with bound K j and Lipschitz coefficient M j . Then the product function f1 f2 is Lipschitz with coefficient K1 M2 + K2 M1 . Theorem 5.13. Let Φ be defined in (5.60). The semiflow Φ is asymptotically smooth. Proof. In order to apply Theorem A.3, we can decompose Φ (t) on any bounded set of X as follows: Φ (t) = U(t) +V (t) here U(t) = (0, iU (t)),V (t) = (S(t), iV (t)) iU (t) = (iˆ1 , iˆ2 , · · · , iˆn ), S(t) = (S1 , S2 , · · · , Sn ), iV (t) = (i˜1 , i˜2 , · · · , i˜n ) and

 iˆk (t) =

0, t ≥ a, (a) , t < a, ik0 (a − t) ππ(a−t)

i˜k (t) =

Bk (t − a)π (a), t ≥ a, 0, t < a.

(5.74)

Let D be a bounded set of Ω . For U(t), we should find a function k(t, r) → 0 such that U(t)x0 ≤ k(t, r) and k(t, r) → 0 as t → ∞ if x0 < r for all x0 ∈ D. Note that

Uk (t) D =

 ∞ 0

 ∞

iˆk (t, a)da

π (a) da π (a − t)  ∞ π (a + t) da = ik0 (a) π (a) 0 =

t

ik0 (a − t)

≤ e−μ t

 ∞ 0

ik0 (a)da

= e−μ t ik0 D . Then U(t) D ≤ kre−μ t for x0 < r. This implies that lim U(t) D = 0. t→∞

184

5 Age Structured Models on Complex Networks

Next, it is necessary to show that V (t) is completely continuous. It follows from Proposition 5.3 that Sk (t) still remains in the compact set [0, 1]. By Theorem B.2 [161] (see also Lemma 5.2 in this chapter), we should verify the condition (i) and (ii) of Theorem B.2. In order to verify the condition (ii), note that n

 ∞

j=1

0

i˜k (t, a) =kSk (k)−1 ∑ ( j − 1)p( j) ≤k

(k − 1)β¯k  π (a). k

This implies that lim | h→∞

1 (0, +∞). L+

β j (a)i j (t − a, a)daπ (a)

∞ h

ik (t, a)da| = 0 for all X0 ⊂ B. Hence V (t, B) ⊂ [0, 1] ×

This indicates condition (ii) is satisfied. By the first and last equations of (5.58), together with uniform continuity of βk (a), Sk (t) and Jk (t)  0∞ βk (a)ik (t, a)da are local Lipschitz continuous, denoted the Lipschitz coefficients by LSk and LJk , respectively. By Proposition 5.4, Bk (t) is also locally Lipschitz continuous and assume that the Lipschitz coefficient is LB . In order to verify condition (i), for small enough h ∈ (0,t), we have

i˜k (t, ·) − i˜k (t, · + h) L1

= = ≤

 ∞ 0

|ik (t, a + h) − ik (t, a)|da

 t−h 0

 t−h 0

+

|Bk (t − a − h)π (a + h) − Bk (t − a)π (a)|da +

t−h

|Bk (t − a)π (a)|da

Bk (t − a − h)|π (a + h) − π (a)|da

 t−h 0

 t

|Bk (t − a − h) − Bk (t − a)|π (a)da + k

k(k − 1)β¯k  h k

LB (k − 1)β¯k  ≤ 2k h + k k h. k μ ˜

Therefore, i˜k is completely continuous, and it remains in a pre-compact subset C1i of 1 . Condition (i) is also established. It follows from Theorem A.3 that the semiflow L+ Φ is asymptotically smooth. Next, we investigate the uniform persistence of system (5.58). Define a function ρ : X → R as ρ (X) = Θ (i(t, a)), k ∈ N+ . Let Ω0 = {x0 ∈ Ω : there exists t0 ∈ R+ such that ρ (Φ (t0 , x0 )) > 0}. Clearly, if x0 ∈ Ω \ Ω0 then lim (S(t), i(t)) = E0 . If lim sup ρ (Φ (t, x0 )) > ε , for any ε > 0, then t→∞

t→∞

system (5.58) is ρ -weekly persistent. On the other hand, if lim inf ρ (Φ (t, x0 )) > ε , t→∞ for any ε > 0, then it is ρ -strongly persistent. The following result will be used in establishing the persistence of system (5.58).

5.3 An SIR Model with Demography on Complex Networks

185

Proposition 5.5. If R0 > 1, for any k ∈ N+ , Θ (i(t)) is uniformly weekly ρ -persistent. Proof. Since R0 > 1, there exists a sufficiently small ε > 0 such that 1 k

∑

'j (ε )( j( j − 1)p( j)K

j∈Nn

μ S0j − ε ) > 1. μ + jε

By way of contradiction, assume that there exists an x0 ∈ Ω0 such that lim sup ρ (Φ (t, t→∞

x0 )) ≤ ε2 . Then there exists t0 ∈ R+ (without loss of generality, we assume t0 = 0 since we can replace x0 with Φ (t0 , x0 )) such that

ρ (Φ (t, x0 )) = Θ (i(t, ·)) ≤ ε

for t ≥ t0 = 0.

For k ∈ Nn , it follows from the first equation of (5.58) that dSk (t) ≥ μ Sk0 − ε kSk (t) − μ Sk (t) dt μ S0

for t ∈ R+ , which implies that lim inft→∞ Sk (t) ≥ μ +kkε for k ∈ Nn . Therefore, there exists t1 ≥ t0 and again we can assume that t1 = 0 such that Sk (t) ≥

μ Sk0 −ε μ + kε

for all t ≥ t1 = 0 and k ∈ Nn .

Then, for k ∈ Nk ,   μ S0 Bk (t) ≥ k μ +kkε − ε k−1 ∑ ( j − 1)p( j) 0t β j (a)B j (t − a)π (a)da. j∈Nn

Taking Laplace transform gives us  μ Sk0 'j (λ )Bj (λ ) 'k (λ ) ≥ k − ε k−1 ∑ ( j − 1)p( j)K B μ + kε j∈Nn

(5.75)

for k ∈ Nn . For convenience, define  λ) = B(

∑ (k − 1)p(k)K'k (λ )B'k (λ ).

k∈Nn

k (λ ) and then summing them up, Multiplying both sides of (5.75) by (k − 1)p(k)K one gets !  " 0 μ S −1   λ ) ≥ B(  λ )k B( λ ), ∑ k(k − 1)K'k (λ ) μ + kkε − ε > B( k∈Nn

186

5 Age Structured Models on Complex Networks

 λ ) > 0 for λ > 0. This completes which contradicts with our choice of ε since B( the proof. In view of Theorem 5.13 and Proposition 5.5, we immediately have the following theorem by Theorem 5.7. Theorem 5.14. Suppose R0 > 1. Then the disease is uniformly strongly ρ -persistent, that is, there is an η > 0 such that if i0 ∈ Ω0 satisfies Θ (i0 ) > 0 then the solution i of (5.58) satisfies lim inf Θ (t) ≥ η . t→∞

We note that Bk (t) = kSk (t)Θ (i(t, ·)). Thus, there exists a positive value η0 for all k ∈ Nn lim inf Θ (t) ≥ η0 . Theorem 5.13 implies that system (5.58) has a global t→∞

attractor A such that Φ (t, X) ∈ A . A total trajectory of Φ in X is a function h : R → X such that Φ (t, h(r)) = h(t + r) for all r ∈ R and t ∈ R+ . Let r ∈ R be fixed and arbitrary. Define Skr (t) = S(t + r), Θr (t) = Θ (t + r), ikr (t, a) = ik (t + r, a). Then (Skr (t), ikr (t, a)) = Φ (t, (Sk (r), ik (r, ·))). Repeating the familiar steps as Sect. 5.3.4, we have Sk (s) =bk[1 − Nk∗ ]Ψ ∗ − μ Sk (s) − kSk (s)Θ (i(s − a)), ik (s, a) =kSk (s − a)Θ (i(s − a)),

(5.76)

and

Θ (i(s − a)) =

1 n ∑ l p(l) k l=1

 ∞ 0

βl (a)Sl (s − a)Θ (i(s − a))da.

(5.77)

Theorem 5.15. Suppose R0 > 1, and (Sk (s), ik (s, a)) is a total trajectory in A . Then there exists an η0 > 0 such that Sk > η0 , ik (s, a) > η0 πk (a), for s > s0 , s0 ∈ R. n

Proof. Note that Sk (s) ≤ Sk0 , and Bk (s) ≤ kSk0 (k)−1 ∑ ( j − 1)p( j) βk ∞ for all j=1

s ∈ R. By the first equation of (5.76), we obtain n

Sk (s) ≥ μ Sk0 − μ Sk − kSk0 (k)−1 ∑ ( j − 1)p( j) βk ∞ j=1

then there exist a s0 > 0 such that for all s > s0 we have n

μ Sk0 − kSk0 (k)−1 ∑ ( j − 1)p( j) βk ∞ Sk (s) ≥

j=1

μ

 η1 .

In view of Theorem 5.14, there exists a η  such that Bk (s) > η  , then ik (s, a) > η  πk (a).

5.3 An SIR Model with Demography on Complex Networks

187

Therefore, η0 = max{η1 , η  } is one such number that we are looking for. The proof is complete. The following lemma is helpful in proving global stability of E ∗ of system (5.58). Lemma 5.7. Suppose (Sk∗ , i∗k (a)) is the equilibrium of system (5.58), then  ∞

Sk i j (t, a) da Sk∗ i∗j (a) 0 j=1  ∞ n ik (t, 0) da, = kSk∗ (k)−1 ∑ ( j − 1)p( j) β j (a)i∗j (a) ∗ ik (0) 0 j=1 n

kSk∗ (k)−1 ∑ ( j − 1)p( j)

n

 ∞

j=1

0

kSk∗ (k)−1 ∑ ( j − 1)p( j)

β j (a)i∗j (a)

β j (a)i∗j (a)(1 −

Sk (t)i j (t, a)i∗k (0) )da = 0. Sk∗ i∗j ik (t, 0)

(5.78)

(5.79)

Proof. Note that n

 ∞

j=1

0

i∗k (0) = kSk∗ (k)−1 ∑ ( j − 1)p( j)

β j (a)i∗j (a)da.

(5.80)

By definition n

ik (t, 0) = kSk (t) ∑ ( j − 1)p( j) j=1

 ∞

n

 ∞

j=1

0

= kSk∗ ∑ ( j − 1)p( j)

0

β j (a)i j (t, a)da

β j (a)

Sk (t) i j (t, a) ∗ i (a)da. Sk∗ i∗j (a) j

On the other hand, from (5.80) we multiply both sides with ik (t, 0) and divide by ik (0) to obtain n

 ∞

j=1

0

ik (t, 0) = kSk∗ (k)−1 ∑ ( j − 1)p( j) Therefore, (5.78) holds. Multiplying by

i∗k (0) ik (t,0)

n

 ∞

j=1

0

kSk∗ (k)−1 ∑ ( j − 1)p( j)

 ∞

j=1

0

From here, we obtain (5.79).

ik (t, 0) da. i∗k (0)

on both side of Eq. (5.78) yields

β j (a)i∗j (a)

n

= kSk∗ (k)−1 ∑ ( j − 1)p( j)

β j (a)i∗j (a)

Sk i j (t, a) i∗k (0) da Sk∗ i∗j (a) ik (t, 0)

β j (a)i∗j (a)da.

188

5 Age Structured Models on Complex Networks n

Theorem 5.16 (Theorem 3.1 [112]). Consider the function V (t, u) = ∑ ciVi (t, ui ). i=1

Assume that the following assumptions are satisfied. (1) There exist functions Vi (t, ui ), Fi j (t, ui , u j ), and constants ai j ≥ 0 such that V˙i (t, ui ) ≤

n

∑ ai j Fi j (t, ui , u j ), t > 0, ui ∈ Di , i = 1, 2, · · · , n.

j=1

(2) Along each directed cycle C of the weighted digraph (G, A), A = (ai j ),

∑

Frs (t, ur , us ) ≤ 0, t > 0, ur ∈ Dr , us ∈ Ds

(s,r)∈E(C )

or there exists a function Gm (t, um ), such that Fi j (t, ui , u j ) ≤ Gi (t, ui )−Gi (t, u j ), 1 ≤ i, j ≤ n. (3) The constants ci are given as follows: ci =

∑

T ∈Ti

w(T ), i = 1, 2, · · · , n,

where Ti is the set of all spanning trees T of (G, A) that are rooted at vertex i, and w(T ) is the weight of T . n

Then the function V (t, u) = ∑ ciVi (t, ui ) satisfies V˙ (t, u) ≤ 0 for t ≥ 0 and u ∈ D, i=1

namely, V is a Lyapunov function. Theorem 5.17. If R0 > 1, then the endemic equilibrium E ∗ is globally asymptotically stable. Proof. Employing function g(x) defined in (5.42), we define a Lyapunov functional for all k ∈ Nn Vk (t) = Sk∗ g(

n Sk (t) ) + kSk∗ ∑ ( j − 1)p( j) ∗ Sk j=1

 ∞ 0

α j (a)g(

ik (t, a) )da, i∗k (a)



here αk (a) = a∞ βk (a)i∗k (a)da and αk (a) = −βk (a)i∗k (a). By Assumption 4 and Theorem 5.17, Vk (t) is well defined. Lemma 5.5 implies that d

∞ 0

α j (a)g( iki∗(t,a) (a) )da k

dt

=

 ∞ 0

β j (a)i∗j (a)

  ik (t, a) ik (t, 0) ) − g( ∗ ) da. g( ∗ ik (0) ik (a)

Then, taking a derivative of Vk (t) along the solutions of system (5.58) yields

5.3 An SIR Model with Demography on Complex Networks n S∗ dVk (t) =(1 − k )Sk (t) + kSk∗ (k)−1 ∑ ( j − 1)p( j) dt Sk (t) j=1

=−

189

 ∞ 0

α j (a)

n μ (Sk (t) − Sk∗ )2 + kSk∗ (k)−1 ∑ ( j − 1)p( j) Sk (t) j=1

 ∞ 0

∂ g( iki∗(t,a) (a) ) k

∂t

da

β j (a)i∗j (a)

& Sk∗ Sk (t) i j (t, a) i j (t, a) − ∗ ∗ + ∗ da × 1− Sk (t) Sk i j (a) i j (a) %

n

 ∞

j=1

0

+ kSk∗ (k)−1 ∑ ( j − 1)p( j)

β j (a)i∗j (a)[g(

ik (t, a) ik (t, 0) ) − g( ∗ )]da ∗ ik (0) ik (a)

n μ =− (Sk (t) − Sk∗ )2 + kSk∗ (k)−1 ∑ ( j − 1)p( j) Sk (t) j=1

 ∞ 0

β j (a)i∗j (a)

& i∗k (0) Sk∗ Sk (t) i j (t, a) ik (t, a) + ln − ∗ ∗ + ln ∗ da. × 1− Sk (t) ik (t, 0) Sk i j (a) ik (a) %

It follows from Lemma 5.7 that n dVk (t) = ∑ kSk∗ (k)−1 ( j − 1)p( j) dt j=1

− g( n

 ∞

j=1

0

here  ∞ 0

0

β j (a)i∗j (a)[−g(

Sk∗ ) Sk (t)

Sk (t)i j (t, a)i∗k (0) )]da Sk∗ i∗j ik (t, 0)

+ ∑ ak j

Δk j =

 ∞

β j (a)i∗j (a)[Gk (ik (t, a)) − G j (i j (t, a))]da

ak j = kSk∗ (k)−1 ( j − 1)p( j),

β j (a)i∗j (a) [Gk (ik (t, a)) − G j (i j (t, a))] da, Gk (ik ) = ln

ik (t, a) . i∗k (a)

Since the network is scale-free network, the adjacent matrix is an irreducible matrix. Then, ak j satisfies the condition (1) of Theorem 5.16. Based on the definition of Gk (ik ), condition (2) readily holds. The last condition is verified as Theorem 5.8. Therefore, Δk j Gk, j and Vk satisfy conditions of Theorem 5.16 or Theorem 3.1 in [112]. This implies that V (t) = ∑ ckVk (t) can be defined as a Lyapunov functional k

for system (5.58) according to [112]. Therefore, V  (t) ≤ 0 for system (5.58). By

190

5 Age Structured Models on Complex Networks

the first equation of (5.58), the largest invariant set for {(S(t), i(t, ·))| dVdt(t) = 0} is a singleton point E ∗ . It follows from LaSalle’s invariant set principle that the endemic equilibrium E ∗ is globally asymptotically stable.

5.3.5 Numerical Results To illustrate the results, we consider an example. We take the parameters as follows: b = 0.16, μ = 0.01, p(k) =

k−2.4 , τ = 2. ∑ k−2.4 k

We choose the transmission rate and the recovery rate as the following function:  0, 0 ≤ a < τ , γ (a) = γ = 0.2, βk (a) = β ∗ , Otherwise. Then the basic reproduction number is R0 =

k(k − 1)Kk Sk0  . k

Through manipulations, the basic reproduction number R0 ≈ 0.5922 < 1 when we choose β ∗ = 0.001. Fig. 5.3a shows that the densities of infected individuals decline to zero when time goes to infinity. Otherwise, if β ∗ = 0.003 the basic reproduction number R0 ≈ 1.77 > 1 then the densities of infected individuals with different degree approach some positive levels (see Fig. 5.3b). 0.015

0.3

0.25 * =0.5922 β =0.001,R 0

b=0.16

0.2

I(t)

I(t)

0.01

0.005

0.15

0.1

0.05 0

0

10

20

30

40

50

Time t

(a)

60

70

80

90

100

0

0

20

40

60

80

100

Time t

120

140

160

180

200

(b)

Fig. 5.3 Stability of the disease-free equilibrium E0 of Eq. (5.58) with the parameters listed in the text except β ∗ . (a) Time series of Ik with respect to time t with β ∗ = 0.001. (b) Time series of Ik (t) with respect to time t for β ∗ = 0.003

5.4 Two Strain Models with Age Structure on Complex Networks

191

5.4 Two Strain Models with Age Structure on Complex Networks An important aspect of studying multi-strain epidemic models is the identification of conditions that lead to the coexistence of different strains or competitive exclusion of different strains. Various mechanisms have been related to this issue, including the effect of age [125], superinfection [28, 90, 182], cross-immunity [3, 27], co-infection [122, 127], and mutation [149]. These results have been verified for multi-strain epidemic models on regular networks. To the best of our knowledge comparatively little research has focused on multi-strain models on complex networks [150, 182]. In [150], Wu et al. proposed a two-strain SIS model on a scale-free network without any trade-off mechanism, which means that the two strains cannot co-infect or super-infect a single host at any time. The authors found the emergence of an epidemic threshold that would guarantee the persistence of the second strain. Besides this, a two-strain epidemic model with superinfection on scale-free networks was proposed in [182]. This work indicates that the superinfection is a key mechanism for the coexistence of the two strains. More specifically, superinfection can allow a strain to persist even if the strain’s reproduction number is less than one. Age structure is a one of heterogeneities caused by chronological age or classage, which is identified by the experiments and has a well-defined biological meaning. Multi-strain age-structured models on complex networks is a challenging topic not only from biological viewpoint but also mathematically. This section is devoted to the investigation of mechanisms such as competitive exclusion or coexistence and also provides some mathematical methods to prove the existence of the endemic equilibrium.

5.4.1 A SIS Age-Structured Model on Complex Networks We base the discussion in this subsection on the competition two-strain epidemic model proposed in [150], which is described as follows: 2 d ρ jk (t) = β j k(1 − ∑ ρ jk (t))Θ j (ρ jk (t, ·)) − γ j ρ jk (t), j = 1, 2. da j=1

k ∈ Nn (5.81)

here ρ jk denotes the proportion of infected nodes with strain j and degree k, β j and γ j , respectively, denote the transmission rate and recovered rate for strain j, j = 1, 2. The term Θ1 (resp. Θ2 ) stands for the probability of a link pointing to a node infected by strain 1 (resp. strain 2). The conclusions in [150] showed that the strains are subject to competitive exclusion. Actually, this model just focuses on the heterogeneity of human contact behavior, and ignores infection age and transmission heterogeneity. To incorporate heterogeneity of infectivity as the disease progresses, we incorporate infection age into the model proposed in [150]. We also assume that human contact behavior impacts the transmission during the transmission process. Then the model is modified as follows:

192

5 Age Structured Models on Complex Networks

⎧ ∂ ρ jk (t, a) ∂ ρ jk (t, a) ⎪ ⎪ + = −γ j (a)ρ jk (t, a), ⎨ ∂t ∂a j = 1, 2. 2  ⎪ ⎪ ρ jk (t, 0) = k(1 − ∑ 0∞ ρ jk (t, a)da)Θ j (ρ jk (t, ·)), ⎩

k ∈ Nn (5.82)

j=1

here γ j (a) is the recovery rate for strain j, j = 1, 2 at age a. β jk (a) is transmission rate with respect to strain j, j = 1, 2 which depends on the degree distributions of the node at infection age a. The expression Θ j j = 1, 2 is given

Θ j (ρ jk (t, ·)) =

1 n ∑ kp(k) k k=1

 ∞ 0

β jk (a)ρ jk (a)da, j = 1, 2.

For convenience, define the survive probability function as

π j (a) = e−

a 0

γ j (s)ds

, j = 1, 2

and average number of infected nodes infected by one infected node during the infectious period  K jk =

∞

0

β jk (a)π j (a)da, j = 1, 2.

Then we can give the basic reproduction number with respect to strain j, j = 1, 2, respectively, as n

Rj =

∑ k2 p(k)K jk

k=1

n

∑ kp(k)

=

k2 K jk  , j = 1, 2. k

k=1

R j gives the average number of infected nodes infected by one infected link during the infectious period on the scale-free networks with respect to strain j, j = 1, 2. Let ∗ , ρ ∗ ) be a equilibrium of system (5.82). Then E ∗ satisfies E ∗ = (ρ1k 2k ⎧ d ρ jk (a) ⎪ ⎪ = −γ j (a)ρ jk (a), ⎨ da (5.83) j = 1, 2. k ∈ Nn . 2  ∞ ⎪ ⎪ ⎩ ρ jk (0) = k(1 − ∑ 0 ρ jk (a)da)Θ j (ρ jk (·)), j=1

Solving the first equation yields

ρ jk (a) = ρ jk (0)π j (a), j = 1, 2. First we show that system (5.83) cannot have a coexistence equilibrium. To achieve this goal, define  Kj =

∞

0

Then ρ jk (0) satisfies

ρ jk (0) =

π j (a)da, j = 1, 2.

kΘ j , j = 1, 2. 1 + kK jΘ j

(5.84)

5.4 Two Strain Models with Age Structure on Complex Networks

193

Substituting (5.82) into (5.83) yields

ρ jk (0) = k(1 − ρ1k (0)K1 − ρ2k K2 )Θ j (ρ jk (0)).

(5.85)

Multiplying j p( j)K jk and then summing j from 1 to n on both sides of (5.87) yields A1 + A2 = R1 − 1,

(5.86)

A1 + A2 = R2 − 1, n

where Al = (k)−1 ∑ j2 p( j)K j Kl j ρl j (0), l = 1, 2. The coexistence equilibrium E ∗ j=1

must satisfy (5.88). However, (5.88) is inconsistent unless R1 = R2 . We assume R1 = R2 and then (5.88) does not have a coexistence equilibrium. Next, we will show the existence of the strain dominated equilibria E j . Let ρ1k (0) = 0 or ρ2k = 0 hold. Multiplying kp(k) k β jk (a)π j (a) and then integrating on both sides of (5.84), finally summing it from 1 to n yields

Θj = n

Here f (x) = ∑

k=1

k2 p(k) 1+kK j x .

k2 p(k)K jk Θ j  f (Θ j )Θ j , j = 1, 2. ∑ k=1 1 + kK jΘ j n

It follows from the underlying features of f (Θ j ), similar

to Sect. 5.2, that the existence of the boundary equlibria is shown the following theorem. Theorem 5.18. System (5.83) always has disease-free equilibrium E0 = (0, 0). If R1 > 1, then system (5.83) has only the strain one dominated equilibrium E1 = (ρ1∗j , 0). If R2 > 1, then system (5.83) has only the strain two dominated equilibrium E2 = (0, ρ2∗j ). In order to determine the local stability of the dominant strain equilibrium, we first define the invasion reproduction number with respect the strain j = 1, 2 as follows: R12 R21

= =

k2 (1 −

∞ ∗ 0 ρ1k (a)da)K2k 

k

k2 (1 −

∞ ∗ 0 ρ2k (a)da)K1k 

k

, ,

where ρ ∗jk (a) = ρ jk0 π j (a). R ij implies the ability of strain i invade strain j, i = 1, 2 and j = 1, 2, i = j, and determine the stability of the strain j. Theorem 5.19. Define R0 = max{R1 , R2 }, then (1) if R0 < 1, then the disease-free equilibrium E0 is locally stable; If R0 > 1, then the disease-free equilibrium E0 is unstable; (2) if γ1 (a) = γ1 , R1 > 1, and R12 < 1, then equilibrium E1∗ is locally stable; (3) if γ2 (a) = γ2 , R2 > 1, and R21 < 1, then equilibrium E2∗ is locally stable.

194

5 Age Structured Models on Complex Networks

∞ ∞ −λ a da, K −λ a da, S∗ =  Proof. Denote by K j (λ ) = 0 π j (a)e jk (λ ) = 0 β jk (a)π j (a)e jk ∞ ∗ 1 − 0 ρ jk (a)da, j = 1, 2, k = 1, 2, · · · , n. Linearizing system (5.83), we obtain the following the characteristic equation at E0 :    A1 0    (5.87)  0 A2  = 0.

Therefore, the characteristic roots of (5.87) are determined by the characteristic roots of A j , j = 1, 2. Note that   1 − 1 p(1)K'j1 (λ ) − 1 2p(2)K'j2 (λ ) k k    − 1 2p(1)K' (λ ) 1 − 1 4p(2)K' (λ ) j1 j2  k k  |A j | =  .. ..   . .   1  − np(1)K'j1 (λ ) − 1 2np(2)K'j2 (λ ) k k

··· ··· ..

.

···

 1 − k np(n)K'jn (λ )   1 − k 2np(n)K'jn (λ )    = 0, j = 1, 2. ..   .   1 − 1 n2 p(n)K'jn (λ )  k

As proceeded in Sect. 5.2.1, the characteristic equation is equivalent to 1=

1 n 2 ∑ k p(k)K'jk (λ )  G j (λ ), j = 1, 2. k k=1

If R0 < 1, then the disease-free equilibrium E0 is locally asymptotically stable. Assume R j > 1, j = 1, 2, then G j (0) = R j > 1. Note that G j (0) → 0 as λ → 0. Therefore, there exists a λ ∗ > 0 such that G j (λ ∗ ) = 1. This implies that E0 is unstable. Second, we linearize system (5.83) at strain one dominated equilibrium E1 , and obtain the characteristic equation as follows:    B1 ∗    (5.88)  0 B2  = 0. Here B1 = ⎛1+K (λ )Θ ∗ −(k)−1 S∗ 1

⎜ ⎜ ⎜ ⎜ ⎝



11 p(1)K11 (λ ))

1

∗ 2p(1)K −(k)−1 S12 11 (λ ))

∗ 2p(2)K −(k)−1 S11 12 (λ ))

∗ np(n)K −(k)−1 S11 1n (λ ))

···

∗ −1 ∗   1+2K 1 (λ )Θ1 −(k) S12 4p(2)K12 (λ )) ···

.. .

.. .

∗ np(1)K −(k)−1 S1n 11 (λ ))

∗ 2np(2)K −(k)−1 S1n 11 (λ ))

..

∗ 2np(n)K −(k)−1 S12 1n (λ ))

.. .

.



11 p(1)K21 (λ ))

⎟ ⎟ ⎟ ⎟ ⎠

∗ −1 ∗ 2   ··· 1+nK 1 (λ )Θ1 −(k) S1n n p(n)K1n (λ ))

and ⎛ 1−(k)−1 S∗

⎞

∗ 2p(2)K −(k)−1 S11 22 (λ )) ···

⎜ −(k)−1 S∗ 2p(1)K −1 ∗  21 (λ )) 1−(k) S12 4p(2)K12 (λ )) ··· 12 ⎜ B2 = ⎜ ⎜ .. .. .. ⎝ . . .

∗ np(n)K −(k)−1 S11 2n (λ )) ∗ 2np(n)K −(k)−1 S12 2n (λ ))

.. .

∗ np(1)K −1 ∗ −1 ∗ 2   −(k)−1 S1n 21 (λ )) −(k) S1n 2np(2)K21 (λ )) ··· 1−(k) S1n n p(n)K2n (λ ))

⎞ ⎟ ⎟ ⎟. ⎟ ⎠

5.4 Two Strain Models with Age Structure on Complex Networks

195

Obviously, the characteristic roots of (5.88) are determined by the roots of B1 and S∗ B2 . To calculate the characteristic roots of B1 , we multiply the first row by − S∗1i i 11 and then add it to the row i ⎞ ⎛  ∗ −1 ∗ −1 ∗ −1 ∗    ⎜ ⎜ ⎜ B1 = ⎜ ⎜ ⎜ ⎝

1+K1 (λ )Θ1 −(k) 2S∗ − S∗1i 11

S11 p(1)K11 (λ )) −(k)

∗  (1+K 1 (λ )Θ1 )

S11 2p(2)K12 (λ )) ··· −(k)

S11 np(n)K1n (λ ))

∗  1+2K 1 (λ )Θ1

···

0

.. .

.. .

..

.. .

nS ∗  − S∗1i (1+K 1 (λ )Θ1 ) 11

0

···

∗

Then, we take column i and multiply it by

.

∗ 1+K ∗  iS1i 1 (λ )Θ1 , ∗ S11 ∗  1+iK ( λ ) Θ 1 1

⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

∗  1+nK 1 (λ )Θ1

and then sum all the

columns to the first column to obtain % & ∗ l 2 p(l)K n  1 n S1l 1l (λ ) ∗  − 1 = 0. ∏ (1 + jK1 (λ )Θ1 ) k ∑ ∗  j=1 l=1 1 + l K 1 (λ )Θ

(5.89)

1

 ∞ −(λ +γ )a 1  1 da = As γ1 (a) = γ1 , K 1 (λ ) = 0 e λ +γ1 . Substituting them into (5.89), we have % & ∗ l 2 p(l)(λ + γ )K n  1 n S1l 1 1l (λ ) ∗ − 1 = 0. (5.90) ∏ (λ + γ1 + jΘ1 ) k ∑ λ + γ1 + lΘ1∗ j=1 l=1

Equation (5.90) has characteristic roots λ = −γ1 − jΘ1∗ , j ∈ Nn . The other roots of (5.90) are determined by ∗ l 2 p(l)(λ + γ )K  1 n S1l 1 1l (λ ) = 1. ∑ ∗ k l=1 λ + γ1 + lΘ1

(5.91)

By way of contradiction, we assume that (5.91) has roots with nonnegative real parts. From the first and second equations of (5.91), we have ∗ ρ1k (0) = kS1k Θ1∗ .

Multiplying

1 k l p(l)K1l

(5.92)

on both side of (5.92) and then summing l from 1 to n yield

Θ1∗ =

1 n 2 ∑ l p(l)S1l∗ K1lΘ1∗ . k l=1

(5.93)

Cancelling Θ1∗ on both side of (5.93), we have 1=

1 n 2 ∑ l p(l)S1l∗ K1l . k l=1

(5.94)

196

5 Age Structured Models on Complex Networks

Immediately following (5.91) and (5.94), we obtain 1=

∗ l 2 p(l)(λ + γ )K  1 n 2 1 n |S1l 1 1l (λ )| < ∑ ∑ l p(l)S1l∗ K1l (0) = 1 k l=1 |λ + γ1 + lΘ1∗ | k l=1

(5.95)

which leads to a contradiction. This implies that (5.91) has no roots with nonnegative real parts. Similarly following the above process, we obtain that the characteristic equation of B2 is 1 n ∗ 2 (5.96) 2k (λ ) = 1. ∑ S1k k p(k)K k k=1 Equation (5.95), together with (5.96) implies that if γ1 (a) = γ1 , R12 < 1 then strain 1 dominated equilibrium E1 is locally asymptotically stable. By the symmetricity of system (5.82), if γ2 (a) = γ2 , R21 < 1, then the strain 2 dominated equilibrium E2 is locally asymptotically stable. Theorem 5.19 indicates system (5.82) may exhibit competitive exclusion or coexistence in the special case R1 = R2 . This result coincides with the results of the two strain model in [150]. The main reason is that all the assumptions are based on a static network not a dynamical network. In [150], the network in [150] is a static network and the system is an ODE system. The difference in our model is the presence of age-structure, we could rule out oscillations through Hopf bifurcation in E ∗j in the case γ j (a) = γ j (const). However, oscillations through Hopf bifurcation may occur if γ j (a) is not a constant.

5.4.2 A Two Strain Model with Age Structure and Mutation on Complex Networks A common phenomenon in disease spread processes is pathogen mutation. One example was swine flu, which emerged in 2009 in Mexico and the USA in addition to the seasonal influenza. The disease then spread very rapidly and had caused about 17,000 deaths by the start of 2010. Another widely spread disease is the human immunodeficiency virus (HIV). It is well known that HIV can cause HIV infection and acquired immunodeficiency syndrome (AIDS) which mutates rapidly which for instance prevents the development of a vaccine. In the two examples, mutation causes some serious problems in treating the associated diseases. To simplify matters, we consider two strains of the same pathogen (or two distinct pathologies). Let ρ jk (t, a), j = 1, 2 to be densities of nodes infected by strain j with degree k. Strain 1 can mutate into strain 2, at rate δ (a). The other parameters have the same meanings as in Sect. 5.4.1. The model with mutation and age structure on complex networks is described as follows:

5.4 Two Strain Models with Age Structure on Complex Networks

197

⎧ ∂ ρ1k (t, a) ∂ ρ1k (t, a) ⎪ ⎪ + = −(γ1 (a) + δ (a))ρ1k (t, a), ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ 2  ⎪ ⎪ ⎪ ρ1k (t, 0) = k(1 − ∑ 0∞ ρ jk (t, a)da)Θ1 (ρ1k (t, ·)), ⎪ ⎪ ⎪ j=1 ⎪ ⎨ ∂ ρ2k (t, a) ∂ ρ2k (t, a) + = −γ2 (a)ρ2k (t, a), ⎪ ∂t ∂a ⎪ ⎪ ⎪ ⎪ 2  ⎪ ⎪ ⎪ ρ2k (t, 0) = k(1 − ∑ 0∞ ρ jk (t, a)da)Θ2 (ρ2k (t, ·)) ⎪ ⎪ ⎪ ⎪  j=1 ⎪ ⎪ + 0∞ δ (a)ρ1k (t, a)da, ⎩ where Θ j (ρ jk ) =

1 k

∑ l p(l)

l=1

∞ 0

k ∈ Nn

(5.97)

β jl (a)ρ jl (t, a)da, j = 1, 2. The equilibria of system

(5.97) satisfy the following equations: ⎧ d ρ1k (a) ⎪ ⎪ = −(γ1 (a) + δ (a))ρ1k (a), ⎪ ⎪ da ⎪ ⎪ ⎪ ⎪ 2  ⎪ ∞ ⎪ ⎪ ⎨ ρ1k (0) = k(1 − ∑ 0 ρ jk (a)da)Θ1 (ρ1k (·)), j=1

d ρ2k (a) ⎪ ⎪ = −γ2 (a)ρ2k (a), ⎪ ⎪ ⎪ da ⎪ ⎪ ⎪ 2  ⎪  ⎪ ⎪ ρ2k (0) = k(1 − ∑ 0∞ ρ jk (a)da)Θ2 (ρ2k (·)) + 0∞ δ (a)ρ1k (a)da. ⎩

k ∈ Nn

j=1

(5.98) To analyze the equilibria, and particularly the coexistence equilibrium, induced by the mutation, we define the survived probabilities as

π1 (a) = e−

a

0 (γ1 (a)+δ (a))da

, π2 (a) = e−

a

0 γ2 (a)da

and average number of infected nodes with degree k infected by one infected node with degree k during the infectious period K jk =

 ∞ 0

β jk (a)π j (a)da, j = 1, 2.

Then we can define the basic reproduction number with respect to strain j, j = 1, 2 Rj =

k2 K jk  , j = 1, 2. k

(5.99)

Therefore, the basic reproduction number for system (5.97) is defined as R0 = max{R1 , R2 }. Solving (5.98) yields

ρ jk (a) = ρ jk (0)π j (a), j = 1, 2.

(5.100)

198

5 Age Structured Models on Complex Networks

Define K j =

∞ 0

π j (a)da. Inserting (5.100) into (5.98) leads to

⎧ 2 ⎪ ⎪ ⎨ ρ1k (0) = k(1 − ∑ ρ jk (0)K j )Θ1 (ρ1k (a)), j=1

2 ⎪ ⎪ ⎩ ρ2k (0) = k(1 − ∑ ρ jk (0)K j )Θ2 (ρ2k (a)) + ρ1k (0)Δ ,

k ∈ Nn ,

(5.101)

j=1

∞

here Δ = 0 δ (a)π1 (a)da. Thinking of Θ1 as given and solving the system for ρ1k (0) and ρ2k (0), we obtain ⎧ kΘ1 ⎪ ⎪ ⎨ ρ1k (0) = 1 + k(K Θ + K Θ + K Θ Δ ) , 1 1 2 2 2 1 k ∈ Nn . (5.102) ⎪ Θ Δ + Θ ) k( 1 2 ⎪ ⎩ ρ2k (0) = , 1 + k(K1Θ1 + K2Θ2 + K2Θ1 Δ ) 1 Multiplying by k kp(k)K jk and then summing on both side of (5.102) from 1 to n yield ⎧ k2 p(k)K1kΘ1 1 n ⎪ ⎪ ⎪ ⎨ Θ1 = k ∑ 1 + k(K1Θ1 + K2Θ2 + K2Θ1 Δ ) , k=1 k ∈ Nn . (5.103) ⎪ k2 p(k)K2k (Θ1 Δ + Θ2 ) 1 n ⎪ ⎪ , Θ = ⎩ 2 ∑ 1 + k(K1Θ1 + K2Θ2 + K2Θ1 Δ ) k k=1

Θ1 = Θ2 = 0 is a solution of (5.102) and that implies ρ jk (0) = 0, ρ jk (a) = 0 for all k and j. Hence, system (5.97) has one disease-free equilibrium E0 . Assume Θ1 = 0, Θ2 = 0. Then Θ2 is a solution of Θ2 =

1 n k2 p(k)K2k ∑ 1 + kK2Θ2 Θ2  g2 (Θ2 )Θ2 . k k=1

(5.104)

g2 has similar characteristics as f in equation (5.25). Hence, (5.104) has a unique nontrivial solution. This nonzero solution gives the boundary equilibrium E1 = ∗ ). We can obtain the following theorem. (0, ρ2k Theorem 5.20. Let R j be defined in (5.99). (1) System (5.97) always has disease-free equilibrium E0 . (2) If R2 > 1, then system (5.97) has only the strain two dominated equilibrium E1 . Define the invasion reproduction number R21 =

1 n l 2 p(l)K1l ∑ 1 + lK Θ E1 . k l=1 2 2

Theorem 5.21. Define R0 = max{R1 , R2 }.

5.4 Two Strain Models with Age Structure on Complex Networks

199

(1) If R0 < 1, the disease-free equilibrium E0 is locally stable. (2) If R2 > 1, γ2 (a) = γ2 , and R21 < 1 then the strain two dominated equilibrium E1 is locally stable. Proof. Linearizing system (5.97) at the disease-free equilibrium E0 , for k ∈ Nn we obtain ⎧ ∂ ρ1k (t, a) ∂ ρ1k (t, a) ⎪ ⎪ + = − (γ1 (a) + δ (a))ρ1k (t, a), ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ρ1k (t, 0) =kΘ1 (ρ1k (t, ·)), ⎪ ⎨ (5.105) ⎪ ∂ ρ2k (t, a) ∂ ρ2k (t, a) ⎪ ⎪ + = − γ2 (a)ρ2k (t, a), ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎩ ρ2k (t, 0) =kΘ2 (ρ2k (t, ·)) + δ (a)ρ1k (t, a)da. 0

Let system (5.105) have the solution with ρ jk (t, a) = x jk (a)eλ t , j = 1, 2. We substitute these solutions into system (5.105) and obtain ⎧ dx1k (a) ⎪ ⎪ = − (λ + γ1 (a) + δ (a))x1k (a), ⎪ ⎪ da ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ x1k (0) =kΘ1 (x1 (·)), (5.106) ⎪ dx2k (a) ⎪ ⎪ = − (λ + γ2 (a))x2k (a), ⎪ ⎪ da ⎪ ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ⎪ ⎩ x2k (0) =kΘ2 (x2 (·)) + δ (a)x1k (a)da. 0

Solving the first equation, we have x1k (a) = x1k (0)π1 (λ , a), π1 (λ , a) = e− +δ (s))ds . Then plugging x (a) into the second equation, we obtain 1k x1k (0) = kΘ1 (x1 (0)). Multiplying obtain

1  k l p(l)K1l (λ )

a

0 (λ +γ1 (s)

(5.107)

on both sides of (5.107) and summing l from 1 to n, we 1=

1 n 2 1l (λ ). ∑ l p(l)K k l=1

(5.108)

It follows by similar argument as in Theorem 5.19 that (5.106) has no characteristic roots with nonnegative real parts if R10 < 1. Otherwise, Θ1 (x1 (0)) = 0 implies that x10 = 0 by the second equation of (5.106). Similar calculation with the third and fourth equations of system (5.106) results in the following characteristic equation:

200

5 Age Structured Models on Complex Networks

1=

1 n 2 2l (λ ). ∑ l p(l)K k l=1

(5.109)

Equation (5.109), together with (5.108) indicates that the disease-free equilibrium E0 is locally asymptotically stable when R0 < 1. In order to show the local stability of E1 , we linearize system (5.97) and let E1 ρ1k (t, a) = x1k (a)eλ t and ρ2k (t, a) = ρ2k (a) + x2k (a)eλ t . Then we obtain the following equivalent problem of system (5.97): ⎧ dx1k (a) ⎪ ⎪ = − (λ + γ1 (a) + δ (a))x1k (a), ⎪ ⎪ da ⎪ ⎪  ∞ ⎪ ⎪ E1 ⎪ ⎪ x (0) =k(1 − ρ2k (a)da)Θ1 (x1 (·)), ⎪ 1k ⎪ ⎪ 0 ⎪ ⎨ dx2k (a) (5.110) = − (λ + γ2 (a))x2k (a), ⎪ da ⎪ ⎪  ∞ ⎪ ⎪ E1 ⎪ ⎪ x2k (0) =k(1 − ρ2k (a)da)Θ2 (x2 (·)) ⎪ ⎪ 0 ⎪ ⎪  ∞  ∞ ⎪ ⎪ ⎪ ⎩ −k x2k (a)daΘ2E1 + δ (a)x1k (a)da. 0

0

Solving the first equation of system (5.110), we obtain x1k (a) = x1k (0)π1 (λ , a). Substituting it into the second equation of system (5.110) yields x1k (0) = k(1 − Multiplying yield

1  k l p(l)K1l (λ )

1=

 ∞ 0

E1 ρ2k (a)da)Θ1 (x1 (0)).

(5.111)

on both sides of (5.112) and summing l from 1 to n

1 n 2 ∑ l p(l)(1 − k l=1

 ∞ 0

 ρ2lE1 (a)da)K 1l (λ ),

E1 here we assumed that Θ (x1 (0)) = 0. Since ρ2k (0) =

rewritten as 1=

E

kΘ2 1

E

1+kK2Θ2 1

 1 n l 2 p(l)K 1l (λ ) . ∑ k l=1 1 + kK2Θ2E1

(5.112)

, then (5.112) can be

(5.113)

Therefore, if the invasion reproduction number R21 , then (5.113) has only characteristic roots with negative real parts. Otherwise, if Θ (x1 (0)) = 0, then the third and fourth equations satisfy

5.4 Two Strain Models with Age Structure on Complex Networks

201

dx2k (a) = − (λ + γ2 (a))x2k (a), da  x2k (0) =k(1 − −k



∞

0 ∞

E1 ρ2k (a)da)Θ2 (x2 (·))

(5.114)

x2k (a)daΘ2E1 .

0

Similarly, solving the first equation of (5.114) and substituting the solution into the second equation of (5.114), we drive 1=

1 n l 2 p(l)(1 − ∑ k l=1

 ∞ E1 0

 ρ2k (a)da)(λ + γ2 )K 2l (λ )

λ + γ2 + lΘ2E1

,

(5.115)

here we used the fact γ2 (a) = γ2 and Θ (x2 (0)) = 0. Note that 1 n 2 ∑ l p(l)(1 − k l=1

 ∞ 0

E1 ρ2k (a)da)K2l = 1.

By way of contradiction, we assume that (5.115) has a characteristic root with nonnegative real parts. From the right side of (5.115), it follows that  1 n l 2 p(l)(1 −  ∞ ρ E1 (a)da)(λ + γ )K   2 2l (λ )   0 2k 1=  ∑ k l=1 λ + γ2 + lΘ2E1 1 n 2 < ∑ l p(l)(1 − k l=1 ≤

1 n 2 ∑ l p(l)(1 − k l=1

 ∞ 0

 ∞ 0

E1  ρ2k (a)da)|K 2l (λ )|

E1 ρ2k (a)da)|K2l = 1,

which leads to a contradiction. Hence, (5.115) has no characteristic roots with nonnegative real parts. Otherwise, if Θ (x2 (0)) = 0, then it follows from the third and fourth equations of (5.115) that λ = −(γ2 + kΘ2E1 ) < 0. Therefore, the strain two dominated equilibrium E1 is locally asymptotically stable. As in Sect. 5.4.1, analyzing the characteristic equation of system (5.97) yields the local stability of the boundary equilibria. Now we look for coexistence equilibria. Rewriting system (5.103) yields ⎧ k2 p(k)K1k 1 n ⎪ ⎪ 0 = f ( Θ , Θ ) = ⎪ 1 1 2 ∑ 1 + k(K1Θ1 + K2Θ2 + K2Θ1 Δ ) − 1, ⎪ k k=1 ⎪ ⎪ ⎪ ⎨ k2 p(k)K2k Θ1 Δ 1 n 0 = f2 (Θ1 , Θ2 ) = (5.116) ∑ k 1 + k(K Θ + K Θ + K Θ Δ ) Θ2 ⎪ 1 1 2 2 2 1 ⎪ k=1 ⎪ ⎪ ⎪ k2 p(k)K2k 1 n ⎪ ⎪ − 1. + ⎩ ∑ k k=1 1 + k(K1Θ1 + K2Θ2 + K2Θ1 Δ )

202

5 Age Structured Models on Complex Networks

Assume R21 > 1, then

1 k

n

∑ l p(l)K1l 1+lK1 Θ E2 > 1. Denoting Θ ∗ = K1Θ1 + K2Θ2 + 2

l=1 < Θ ∗ . We

K2Θ1 Δ yields can solve the first equation of (5.116) for Θ ∗ . Since ∗ we know Θ , we can express Θ1 in terms of Θ2 K2Θ2E1

Θ1 =

Θ ∗ − K2Θ2 . K1 + Δ K2

(5.117)

Substituting (5.117) into the second equation of (5.103) yields

Θ2 = =

1 n l 2 p(l)K2l ΔΘ ∗ − K2Θ2 Δ ∑ 1 + lΘ ∗ ( K1 + Δ K2 + Θ2 ) k l=1 1 n l 2 p(l)K2l ΔΘ ∗ + K1Θ2 ∑ 1 + lΘ ∗ ( K1 + Δ K2 ) k l=1

(5.118)

B + B1Θ2 , where B =

1 k

n

∑

l=1

l 2 p(l)K2l ΔΘ ∗ 1+lΘ ∗ K1 +Δ K2 , and B1

leads to

From

Θ2 =

K1 K1 +Δ K2

we have that 1 k

n

1 k

n

∑

l=1

l 2 p(l)K2l K1 1+lΘ ∗ K1 +Δ K2 . Solving

B . 1 − B1

(5.118) (5.119)

< 1 and (5.116), it follows that B1 < 1. So that Θ2 > 0. By (5.116), 1 k

j2 p( j)K2 j E 1+kK 2Θ 1 j=1

∑

=

n

∑

j=1

j2 p( j)K2 j 1+kΘ ∗

< 1. On the other hand, from (5.104), we have

= 1. Hence, K2Θ E1 < Θ ∗ . Additionally, K2 B −Θ∗ 1 − B1 K2 B − Θ ∗ + B1Θ ∗ = 1 − B1

K2Θ2 − Θ ∗ =

=

1 k

n

∑

l=1

l 2 p(l)K2l ΔΘ ∗ K2 1+lΘ ∗ ( K1 +Δ K2 n

l 2 p(l)K2l K1 1+lΘ ∗ K1 +Δ K2 l=1 n 2 l p(l)K2l 1 k ∑ 1+lΘ ∗ ] l=1

1 1 − k ∑

Θ ∗ [1 − =−


0. From what has been discussed, we conclude the existence of the coexistence equilibrium E ∗ . Theorem 5.22. Suppose that R1 > 1, R2 > 1 and R21 > 1 hold. Then (5.116) has a unique coexistence equilibrium E ∗ .

5.4.3 A Two Strain Model with Age Structure and Superinfection on Complex Networks In this subsection, we consider a two strain epidemic model with superinfection (a more virulent strain can superinfect a host already infected by a less virulent strain) on complex networks. Early studies of P. falciparum malaria suggested that high rates of exposure to malaria could result in simultaneous infection with multiple parasites, termed superinfection. P. falciparum malaria is characterized with multiplicity of infection (MOI). Another example is the generation of a superinfected Ae. aegypti mosquito line simultaneously infected with two avirulent Wolbachia strains, wMel and wAlbB. Superinfection may lead to replacement of either of the single constituent infections already presenting in the mosquito population. Superinfection is one of the main mechanisms resulting in a coexistence endemic equilibrium instead of replacement of the strains or clearance. In the following, ρ jk denotes the nodes infected by strain j, j = 1, 2 with its degree k. The node infected by strain 1 contacts a node infected by strain 2 and then changes its state into a node infected by strain 2 at rate α (a). The model is given as follows: ⎧ ∂ ρ1k (t, a) ∂ ρ1k (t, a) ⎪ ⎪ + = −(γ1 (a) + ψ (k)α (a)Θ2 (ρ2k (t))ρ1k (t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪  ⎪ ⎪ ρ (t, 0) = k(1 − ∞ ρ (t, a)da − ρ )Θ (ρ (t, ·)), ⎨ 1 1k 1k 2k 0 1k k ∈ Nn  ∞ ⎪ d ρ2k (t) ⎪ ⎪ ⎪ = k(1 − ρ1k (t, a)da − ρ2k )Θ2 (ρ2k (t)) ⎪ ⎪ dt 0 ⎪ ⎪  ⎩ + ψ (k) 0∞ α (a)ρ1k (t, a)daΘ2 − γ2 ρ2k (t), (5.121)

ψ (k) denotes the effective contact in networks or a restrain habitat owing to its physical fitness, which satisfies ψ (k) ≤ k. All the other parameters have the same definitions and biological meanings as in Sect. 5.3.2. The equilibria satisfy the following equations: ⎧ d ρ1k (a) ⎪ ⎪ = −(γ1 (a) + ψ (k)α (a)Θ2 )ρ1k (a), ⎪ ⎪ ⎪ da ⎪  ⎪ ⎨ ρ1k (0) = k(1 − 0∞ ρ1k (a)da − ρ2k )Θ1 (ρ1k (·)), k ∈ Nn . (5.122) ∞ ⎪ ⎪ ⎪ ρ (a)da − ρ ) Θ ( ρ ) 0 = k(1 − 2 2k 2k ⎪ 0 1k ⎪ ⎪ ⎪ ∞ ⎩ + ψ (k) 0 α (a)ρ1k (a)daΘ2 (ρ2k ) − γ2 ρ2k ,

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5 Age Structured Models on Complex Networks

For convenience, we make the following notations:

π1 j (a, Θ2 ) = e− K1 j (Θ2 ) =

 ∞ 0

π1 j (a, Θ2 )da, K1 j =

 ∞ 0

a

0 (γ1 (s)+ψ ( j)α (s)Θ2 )ds

K1 j (α , Θ2 ) =

 ∞ 0

,

α (a)π1 j (a, Θ2 )da.

β1 j (a)π1 j (a, Θ2 )da,

Δ = γ2 + k(Θ2 + γ2 K1 j (Θ2 )Θ1 + ψ (k)K1 (α , Θ2 )Θ1Θ2 ). Solving the first and second equation of (5.122), we have

ρ1 j (a) = ρ1 j (0)π1 j (a), j = 1, 2. Substituting the above equation into the second and third equation of system (5.122), one obtains (1 + jK1 j (Θ2 )Θ1 )ρ1 j (0) + jΘ1 ρ2 j = jΘ1 , ( jK1k (Θ2 )Θ2 − ψ ( j)K1 j (α , Θ2 )Θ2 )ρ1 j (0) + ( jΘ2 + γ2 )ρ2 j = jΘ2 ,

j ∈ Nn .

(5.123) By Cramer’s Rule in algebra, we solve (5.123) and obtain the following equations: ⎧ jγ2Θ1 ⎪ ⎪ , ⎨ ρ1 j (0) = Δ (5.124) j ∈ Nn . ⎪ ⎪ ⎩ ρ2 j = jΘ2 (1 + ψ ( j)K1 (α , Θ2 )Θ1 ) , Δ By a transformation similar to the one used in Sect. 5.3.2, we have ⎧ 1 n l 2 p(l)γ2 K1l (Θ2 ) ⎪ ⎪ Θ1 , Θ = 1 ⎪ ∑ ⎨ k l=1 Δ ⎪ 1 n l 2 p(l)β2l (1 + ψ (l)K1 (α , Θ2 )Θ1 ) ⎪ ⎪ Θ2 . ⎩ Θ2 = ∑ k l=1 Δ

(5.125)

Define the basic reproduction number with respect to the strain j, j = 1, 2 R1 =

1 n 2 ∑ l p(l)K1l (0), k l=1 (5.126)

1 n l 2 p(l)β2l R2 = ∑ γ2 . k l=1 It follows from analyzing underlying features of the system (5.125) that the boundary equilibria satisfy the following theorem. Theorem 5.23. Suppose that R j j = 1, 2 are defined in (5.126)

5.4 Two Strain Models with Age Structure on Complex Networks

205

(1) System (5.121) always has disease-free equilibrium E0 . (2) If R1 > 1, then system (5.121) has only the strain 1 dominance equilibrium E1 . (3) If R2 > 1, then system (5.121) has only the strain 2 dominance equilibrium E2 . Proof. System (5.122) always has a disease-free equilibrium E ∗ = (0, 0). In order 1 to prove the existence of the dominant strain equilibrium E j , provided Θ1E = 0 and Θ2 = 0, the second equation of (5.125) is trivially satisfied. 1=

1 n l 2 p(l)K1l (0) 1 n l 2 p(l)γ2 K1l (0) = ∑ ∑ 1 + K (0)Θ E1 . E k l=1 Δ (Θ1 1 , 0) k l=1 1l 1

(5.127)

As discussed in Sect. 5.2, system (5.121) has a unique positive strain one dominant equilibrium E1 = (Θ1E1 , 0) if the basic reproduction number R1 > 1. By the symmetric of system (5.121), system (5.121) also has a strain two dominated equilibrium E2 = (0, ΘE2 ) if the basic reproduction number R2 > 1. In order to consider the local stability of strain dominant equilibrium E j , j = 1, 2, we define the invasion reproduction numbers as follows: E

R21 = and

1 n l 2 p(l)γ2 K1l (Θ2 2 ) ∑ Δ (0, Θ E2 ) > 1, k l=1 2 E

R12 =

1 n l 2 p(l)β2l (1 + ψ (l)K1 (α , 0))Θ1 1 > 1. ∑ k l=1 Δ (Θ1E1 , 0)

Theorem 5.24. Let γ1 (a) = γ1 hold. (1) If R1 > 1 and R21 < 1, then the strain one dominant equilibrium E1 is locally asymptotically stable; (2) If R2 > 1 and R12 < 1, then the strain two dominant equilibrium E2 is locally asymptotically stable. Proof. First, we study the local stability of strain one dominant equilibrium E1 . Let E1 ρ1k (t, a) = x1k (t, a) + ρ1k (a), ρ2k = y2k . We linearize system (5.121) and obtain the following system: ⎧ ∂ x1k (t, a) ∂ x1k (t, a) E1 ⎪ ⎪ + = − γ1 (a)x1k (t, a) − ψ (k)α (a)ρ1k (a)Θ (y2 (t)), ⎪ ⎪ ∂ t ∂ a ⎪ ⎪  ⎪ ∞ ⎪ E1 ⎪ ⎪ ρ1k (a)da)Θ1 (ρ1 (t, ·)) x1k (t, 0) =k(1 − ⎪ ⎪ ⎪ 0 ⎪  ⎨ ∞ −k x1k (t, a)daΘ1 (ρ1E1 (·)) − kΘ1E1 y2k (t), k ∈ Nn . ⎪ 0 ⎪ ⎪  ∞ ⎪ ⎪ dy2k (t) E1 ⎪ ⎪ =k(1 − ρ1k (a)da)Θ2 (y2 (t)) ⎪ ⎪ dt 0 ⎪ ⎪  ∞ ⎪ ⎪ E1 ⎪ ⎩ + ψ (k) α (a)ρ1k (a)daΘ2 (y2 (t)) − γ2 y2k (t), 0

(5.128)

206

5 Age Structured Models on Complex Networks

Let x1k = x1k (a)eλ t and y2k = y2k0 eλ t be a solution of system (5.127). Then (5.127) can be changed as follows: ⎧ dx1k (a) E1 ⎪ ⎪ = − (λ + γ1 (a))x1k (a) − ψ (k)α (a)ρ1k (a)Θ (y), ⎪ ⎪ da ⎪  ∞ ⎪ ⎪ ⎪ E1 ⎪ ⎪ x1k (0) =k(1 − ρ1k (a)da)Θ1 (ρ1k (·)) ⎪ ⎪ ⎨  ∞0 E1 k ∈ Nn −k x1k (a)daΘ1 (ρ1k (·)) − kΘ1E1 y2k0 , ⎪ ⎪ 0 ⎪ ⎪  ∞  ∞ ⎪ ⎪ E1 E1 ⎪ ⎪ 0 =k(1 − ρ (a)da) Θ (y) + ψ (k) α (a)ρ1k (a)daΘ2 (y) ⎪ 2 1k ⎪ ⎪ 0 0 ⎪ ⎩ − (λ + γ2 )y2k0 , (5.129) Solving the third equation of (5.129) yields (λ + γ2 )y2k0 = [k(1 −

 ∞ 0

ρ1k (a)da) + ψ (k)

Multiplying by

l p(l)β2k k

(λ + γ2 )Θ2 (y) =

1 n ∑ kp(k)β2k [k(1 − k k=1

 ∞ 0

E1 α (a)ρ1k (a)da]Θ2 (y).

(5.130)

both sides of (5.130) and summing from 1 to n, we have  ∞ 0

ρ1k (a)da) + ψ (k)

 ∞ 0

E1 α (a)ρ1k (a)da]Θ2 (y).

(5.131)

If Θ2 (y) = 0, cancelling Θ2 on both side of (5.131) yields

λ + γ2 =

1 n ∑ kp(k)β2k [k(1− k k=1

Note that ρ1k (0) =

E

kΘ1 1

E

1+kK1 (0)Θ1 1

 ∞ 0

ρ1k (a)da)+ ψ (k)

 ∞ 0

E1 α (a)ρ1k (a)da]. (5.132)

E1 . We substitute ρ1k (a) into (5.132) and obtain

%

& E 1 n l 2 p(l)β2l [1 + ψ (l)K1l (α , 0)Θ1 1 ] λ = γ2 −1 ∑ k l=1 γ2 (1 + lK1l (0)Θ1E1 )   = γ2 R12 − 1 .

(5.133)

Hence, if the invasion number R12 < 1, then the characteristic roots of (5.133) are less than 0. Otherwise, we set Θ2 (y) = 0 and substitute x1k (a) into the first and the second equations of (5.130) ⎧ ⎪ ⎨ dx1k (a) = −(λ + γ1 (a))x1k (a), da k ∈ Nn . ⎪ ⎩ x (0) = k(1 −  ∞ ρ E1 (a)da)Θ (x (·)) − k  ∞ x (a)daΘ E1 , 1 1k 1k 0 1k 0 1k 1 (5.134)

5.4 Two Strain Models with Age Structure on Complex Networks

207

Solving (5.134) yields x1k (a) = x1k (0)π1k (λ , a, 0). We plug it into the second equation of (5.134) and obtain x1k (0) = Multiplying by 1 to n yields

k(1 −

 ∞ E1 0

ρ1k (a)da)Θ1 (x1 (0))

1k (λ )Θ E1 1 + kK 1

1  k kp(k)K1k (λ , 0)

.

(5.135)

both sides of (5.135) and then summing k from

 ∞ E1 1 n l 2 p(l)K 1l (λ , 0)(1 − 0 ρ1l (a)da)Θ1 (x1 (0)) Θ1 (x1 (0)) = . ∑ 1 (λ )Θ E1 k l=1 1 + lK 1

(5.136)

Cancelling Θ1 (x1k (0)) on both sides of (5.136) if Θ1 (x1k (0)) = 0 yields 

∞ E 1 n l 2 p(l)K1l (λ , 0)(1 − 0 ρ1l1 (a)da) . 1= ∑ 1 (λ )Θ E1 k l=1 1 + lK 1

It follows from ρ1k (0) = 1=

(5.137)

E

kΘ1 1

E

1+kK1kΘ1 1

that

l 2 p(l)K1l (λ , 0) 1 n ∑ (1 + lK Θ E1 )(1 + l K (λ )Θ E1 )  H(λ ). k l=1 1 1 1 1

(5.138)

If γ1 (a) = γ1 , (5.138) can be rewritten as 1=

l 2 p(l)(λ + γ1 )K1l (λ , 0) 1 n  H(λ ). ∑ k l=1 (1 + lK1Θ1E1 )(λ + γ1 + lΘ1E1 )

(5.139)

By way of contradiction, we assume that (5.140) has one characteristic root with Reλ > 0. Then taking an absolute value, we have 1 = |H(λ )|
1. Let Θ¯ satisfy 0 = f1 (Θ¯ 2 ). There may be more than one Θ¯ 2 . Each one is a solution to F1 (0, Θ¯ 2 ) = 1. Let g(Θ¯ 2 ) = F1 (0, Θ¯ 2 ) 1 n l 2 p(l)γ2 K1l (Θ¯ 2 ) ∑ Δ (0, Θ¯ 2 ) k l=1 1 n l 2 p(l)γ2 K1l (Θ¯ 2 ) . = k ∑ γ2 + lΘ¯ 2

=

l=1

5.4 Two Strain Models with Age Structure on Complex Networks

209

Clearly, g(Θ¯ ) is a decreasing function of Θ¯ 2 . Since g(Θ2E2 ) = R12 > 1 and g(Θ¯ 2 ) is decreasing and g(Θ¯ 2 ) goes to 0 as Θ¯ 2 approaches 0. This implies that there exists a Θ¯ 2∗ such that g(Θ¯ 2∗ ) = 1. Moreover, Θ¯ 2∗ > Θ E2 . Hence, it follows from (5.143) that G(Θ¯ 2∗ ) = F2 (0, Θ¯ 2∗ ) < F2 (0, Θ2E2 ) = 1. We conclude that there exists a Θ2∗ such that G( f1 (Θ2∗ ), Θ2∗ ) = 1 and 0 < Θ2∗ < Θ¯ 2∗ . Therefore, Θ1∗ = f1 (Θ2∗ ) which is a solution of 1=

1 n l 2 p(l)γ2 K1l (Θ2∗ ) ∑ Δ (Θ1 , Θ ∗ ) . k l=1 2

(5.144)

We note that (5.144) has a unique solution because g(Θ ∗ ) =

1 n l 2 p(l)γ2 K1l (Θ2∗ ) ∑ Δ (0, Θ ∗ ) > 1. k l=1 2

Note that F1 (Θ1 , Θ2∗ ) is decreasing function of Θ1 and since F1 (0, Θ2∗ ) = g(Θ2∗ ) > 1 and F(Θ1 , Θ ∗ ) → 0 as Θ1 → 0. So there exists a Θ1 > 0 such that F1 (Θ1 , Θ2∗ ) = 1. We solve it and obtain Θ1∗ = f1 (Θ2∗ ). Theorem 5.25. Let R12 > 1 and R21 > 1 hold. System (5.122) has at least one coexistence equilibrium E ∗ = (Θ1∗ , Θ2∗ ).

Chapter 6

Vector-Borne Age-Structured Models

6.1 Introduction Chapters 1 and 3 introduce and analyze several age-structured or class age-structured epidemic models. These models describe the diseases transmitted from human to human directly. However, there are many diseases, such as malaria, Chagas disease, dengue fever, which are transmitted via a vector. In this chapter, we formulate and analyze age-structured or class age-structured vector-borne disease models.

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host In this section, we formulate a vector-host model for the disease which is transmitted via vector and through blood transfusion in host population. In order to reflect the fact that the age structure of population affects the dynamics of disease transmission, the host population is structured by the chronological age, and we assume that the instantaneous death and infection rates depend on the age. We assume that removed host population is not infected again. We use C0 -semigroups for the analysis of the method. A basic introduction to C0 -semigroup can be found in Appendix A. The basic proof idea of this chapter is based on the argument of Inaba [92].

6.2.1 The Model Formulation In this section, we assume that a is the chronological age of the host, the host population is divided into three classes: susceptible, infective, and removed. Let S(a,t), I(a,t), and R(a,t) be the age-densities of, respectively, the susceptible, infective, and removed host population at time t. We divide the vector population into two groups: © Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1_6

211

212

6 Vector-Borne Age-Structured Models

susceptible and infective. We assume the total vector population size is constant and normalized to one, and V0 (t) and V (t) represents the fractions of susceptible and infective vectors. We assume that there is no disease-induced death, and the total population size of host remains constant with respected to time. Let N(a) be the density with respect to age of the total number of the host population. N(a) satisfies N(a) = μ ∗ Ne−

a 0

μ (σ )d σ

,

N(a) = S(a,t) + I(a,t) + R(a,t), where μ (a) denotes the instantaneous death rate at age a of the host population, the constant N is the total size of the host population, and N = 0∞ N(a)da, μ ∗ is the crude death rate of the host population. We assume that μ (a) is nonnegative, locally integrable on [0, +∞), and satisfies  +∞ 0

μ (σ )d σ = +∞.

The crude death rate of the host population μ ∗ is determined such that

μ∗

 +∞ 0

f (a)da = 1,

a

where f (a) = e− 0 μ (σ )d σ is the survival function which is the proportion of individuals who survive to age a. Then we have the relation N(a) = μ ∗ N f (a). Let δ1 be the number of bites per vector per unit time and c be the proportion of infected bites on susceptible hosts that give rise to infection. Then the force of infection to susceptible hosts is given by η V (t), where η = δ1 c. On the other hand, the force of infection for the direct transmission in the host population, denoted by λ (a,t), is defined by

λ (a,t) = γ1 (a)

 +∞ 0

γ2 (a)

I(a,t) da, N(a)

where γ1 (a) is age-specific contagion rate, γ2 (a) is age-specific infectiousness, γ1 (a), γ2 (a) are continuous, nonnegative, and bounded on [0, ∞). Let δ2 (a) be the proportion of bites to infected hosts with age a that give rise to infection in vectors. Then the number of new infection of vectors per unit time from infected hosts is  ¯ given by δ1 0∞ δ2 (a) I(a,t) N(a) daV0 (t). We let constant μ be the per capita death rate of −1 vectors, α denote the average infectious period in the host population. With these assumptions, we obtain the following system of equations which describe the dynamics of the vector-host model:

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host

⎧ ∂ S(a,t) ∂ S(a,t) ⎪ ⎪ ⎪ ∂ t + ∂ a = −(μ (a) + λ (a,t) + η V (t))S(a,t), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ I(a,t) ∂ I(a,t) ⎪ ⎪ + = (λ (a,t) + η V (t))S(a,t) − (μ (a) + α )I(a,t), ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ ⎨ ∂ R(a,t) ∂ R(a,t) + = α I(a,t) − μ (a)R(a,t), ⎪ ∂t ∂a ⎪ ⎪  +∞ ⎪ ⎪ dV0 (t) I(a,t) ⎪ ⎪ ⎪ = μ¯ − δ1 daV0 (t) − μ¯ V0 (t), δ2 (a) ⎪ ⎪ N(a) 0 ⎪ ⎪ dt ⎪  +∞ ⎪ ⎪ dV (t) I(a,t) ⎪ ⎩ = δ1 daV0 (t) − μ¯ V (t), δ2 (a) dt N(a) 0

213

(6.1)

with boundary and initial conditions: S(0,t) = μ ∗ N, I(0,t) = 0, R(0,t) = 0, S(a, 0) = S0 (a), I(a, 0) = I0 (a), R(a, 0) = R0 (a),V0 (0) = V0 ,V (0) = V1 ,

(6.2)

where S0 (a) ≥ 0, I0 (a) ≥ 0, R0 (a) ≥ 0,V0 ≥ 0,V1 ≥ 0, S0 (a) + I0 (a) + R0 (a) = N(a),

(6.3)

V0 (t) +V1 (t) = 1,

(6.4)

λ (a,t) = γ1 (a)

 +∞ 0

γ2 (b)

I(b,t) db. N(b)

(6.5)

We use ODE and a chronological age-structured PDE to describe the change of the population of the vector and the host, respectively, because vector lifespan is much shorter compared with the host. Considering the fractions of susceptible, infectious, and removed population at age a and time t: s(a,t) =

S(a,t) , N(a)

i(a,t) =

I(a,t) , N(a)

r(a,t) =

R(a,t) . N(a)

Then system (6.1)–(6.5) can be written in a simpler form ⎧ ∂ s(a,t) ∂ s(a,t) ⎪ ⎪ + = −(λ (a,t) + η V (t))s(a,t), ⎪ ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ ⎪ ∂ i(a,t) ∂ i(a,t) ⎪ ⎪ + = (λ (a,t) + η V (t))s(a,t) − α i(a,t), ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎨ ∂ r(a,t) ∂ r(a,t) + = α i(a,t), ⎪ ∂t ∂a ⎪ ⎪  +∞ ⎪ ⎪ dV0 (t) ⎪ ⎪ ¯ = μ − δ δ2 (a)i(a,t)daV0 (t) − μ¯ V0 (t), ⎪ 1 ⎪ dt ⎪ 0 ⎪ ⎪  ⎪ +∞ ⎪ dV (t) ⎪ ⎩ = δ1 δ2 (a)i(a,t)daV0 (t) − μ¯ V (t). dt 0

(6.6)

214

6 Vector-Borne Age-Structured Models

λ (a,t) = γ1 (a) s(0,t) = 1,

 ∞ 0

γ2 (σ )i(σ ,t)d σ ,

i(0,t) = 0,

N(a) = μ ∗ N f (a),

r(0,t) = 0,

(6.7)

s(a,t) + i(a,t) + r(a,t) = 1. In the following, we mainly consider system (6.6)–(6.7) with the initial conditions s(a, 0) = s0 (a),

i(a, 0) = i0 (a),

V0 (a) = V0 ,

V (0) = V1 .

(6.8)

In order to make sense, we need to make the following assumption for the parameters. Assumption 6 All the parameters are satisfies (1) γ j (a) ∈ L∞ (R+ ), j = 1, 2, δ2 (a) ∈ L∞ (R+ ). (2) All the parameters are nonnegative.

6.2.2 Existence and Uniqueness of Solution In this section, we shall show that the initial-boundary value problem (6.6), (6.7), and (6.8) has a unique solution. First we note that it suffices to consider the system in terms of only s(a,t), i(a,t), and V (t) since, once these functions are known, r(a,t) and V0 (t) can be obtained by r(a,t) = 1− s(a,t) − i(a,t) and V0 (t) = 1−V (t), respectively. First we introduce a new variable s(a,t) = 1 − i(a,t) − r(a,t). Then we obtain the new system for i(a,t), r(a,t) and V (t), ⎧ ∂ i(a,t) ∂ i(a,t) ⎪ ⎪ + = (λ (a,t) + η V (t))(1 − i(a,t) − r(a,t)) − α i(a,t), ⎪ ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ r(a,t) + ∂ r(a,t) = α i(a,t), ∂t ∂a (6.9) ⎪  +∞ ⎪ ⎪ dV (t) ⎪ ⎪ = δ1 δ2 (a)i(a,t)da(1 −V (t)) − μ¯ V (t), ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ⎩ s(0,t) ˆ = 0, i(0,t) = 0. Let us consider the initial-boundary value problem of the system composed of (6.9) as an abstract Cauchy problem on the Banach Space X = L1 (0, +∞)×L1 [0, +∞)×R with the norm

x = x1 + x2 + |x3 |, xi =

 +∞ 0

|xi (a)|da, (i = 1, 2),

for x(a) = (x1 (a), x2 (a), x3 )T ∈ X, where · i is the ordinary norm of L1 (0, +∞). Let A be a linear operator defined by

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host

⎞ 1 (a) − α x1 (a) − dxda ⎟ ⎜ (Ax)(a) = ⎝ − dx2 (a) ⎠, da −μ¯ x3

215

⎛

x(a) = (x1 (a), x2 (a), x3 )T ∈ D(A),

(6.10)

where U T is the transpose of the vector U and the domain D(A) is given as D(A) = {x ∈ X|xi is absolutely continuous on[0, +∞), (x1 (0), x2 (0))T = (0, 0)T }. We define a nonlinear operator F : X → X by ⎛ ⎜ F(x) = ⎜ ⎝

(P(x1 ) + η x3 )(1 − x2 − x3 )

α x2

⎞ ⎟ ⎟, ⎠

x ∈ X,

(6.11)

δ1 H(x1 )(1 − x3 ) where P and H are bound linear operators on L1 (0, +∞) given by (Pϕ )(a) = γ1 (a)

 ∞ 0

γ2 (σ )ϕ (σ )d σ ,

(H ϕ )(a) =

 ∞ 0

δ2 (σ )ϕ (σ )d σ .

(6.12)

Note that Pϕ , H ϕ ∈ L∞ (0, +∞) for ϕ ∈ L1 (0, +∞), and hence, the nonlinear operator F is defined on the whole space X. Let u(t) = (i(·,t), r(·,t),V (t)) ∈ X. Then we can rewrite the initial-boundary value problem (6.9) as the abstract semilinear initial value problem in X: du(t) = Au(t) + F(u(t)), da

u(0) = u0 ∈ X,

(6.13)

where u0 (a) = (i0 (a), r0 (a),V0 )T . It is easily seen that the operator A is the infinitesimal generator of C0 -semigroup T (t), t ≥ 0, and F is continuously Frechet differentiable on X. Then for u0 ∈ X, there exists a maximal interval of existence [0,t0 ), and a unique continuous mild solution t → u(t, u0 ) from [0,t0 ) to X such that u(t, u0 ) = T (t)u0 +

 t 0

T (t − s)F(u(s, u0 ))ds,

(6.14)

for all t ∈ [0,t0 ) and either t0 = +∞ or limt→t − u(t, u0 ) = ∞. Moreover if u0 ∈ 0 D(A), then u(t, u0 ) ∈ D(A) for 0 ≥ t ≥ t0 and the function t → u(t, u0 ) is continuously differentiable and satisfies (6.13) on [0,t0 ) (see [179, P. 194, Proposition 4.16]). Since 1 = N(a,t) = s(a,t) + i(a,t) + r(a,t), we obtain that the solution (s(a,t), i(a,t), r(a,t),V0 (t),V (t))T ,t ∈ [0, m) is continuously differentiable and satisfies the system (6.1)–(6.2), where either m = +∞ or m < ∞ and lim ( s(a,t) + i(a,t) + r(a,t) + |V0 (t)| + |V (t)|) = +∞.

t→m

216

6 Vector-Borne Age-Structured Models

From N(a) = N(a,t) = N and V¯ (t) = V0 (t) + V (t) = 1, we easily obtain m = +∞. Thus we have the following result: Proposition 6.1. The initial-boundary value problem (6.1)–(6.2) has a unique mild solution on X with respect to (s0 (a), i0 (a), r0 (a),V0 ,V1 )T ∈ X. Now, we are going to prove the positivity of the solution of (6.1)–(6.2). In order to do this, we just show that the solution of (6.9) is positive. Then we take an equivalent transformation and let Aκ (x) = (A − κ I)(x), Fκ (x) = (κ I + F)(x), where I denotes an identity operator. Define a positively invariant set

Ω = {(i, r,V ) ∈ X+ |i + r ≤ 1, V ≤ 1}.

Proposition 6.2. Let Assumption 6 hold. The operator Fκ is Lipchitz continuous and Fκ (Ω ) ⊂ Ω . Proof. The Lipschitz continuality of the operator is a direct result based on Assumption 6 following the proof process of Sect. 2.2 in Chap. 2. For any x = (x1 , x2 , x3 ) ∈ Ω , we denote xˆ = (xˆ1 , xˆ2 , xˆ3 )T = eAκ t x +

 t 0

eAκ (t−s) Fκ (x)(s)ds.

Then the positivity of xˆ readily follows the components of Fκ and Assumption 6. By the definition of Fκ , we have xˆ1 + xˆ2 =eAκ t (x1 + x2 ) +

 t 0

eAκ (t−s) [κ (x1 (s) + x2 (s)) + (P(x2 (s)) + η x3 (s))

(1 − x1 (s) − x2 (s))]ds ≤e−κ t +

 ∞ 0

e−(κ (t−s) (κ + P¯ + η )ds

κ + P¯ + η P¯ + η −κ t − e κ κ κ + P¯ + η ≤ . κ =

So that if we take κ large enough, then process, we obtain xˆ3 = eAκ t x3 + This completes the proof.

 t 0

¯ η κ +P+ κ

approaches 1. Repeating the similar

eAκ (t−s) H(x2 )(1 − x3 )(s)ds ≤ 1.

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host

217

6.2.3 Existence of Steady States Let u∗ = (s∗ (a), i∗ (a),V ∗ )T be the steady state solution of the system (6.6). That is, u∗ satisfies ⎧ ∗ ds (a) ⎪ ⎪ = −(γ1 (a)P(i∗ ) + η V ∗ )s∗ (a), ⎪ ⎪ ⎨ da di∗ (a) (6.15) = (γ1 (a)P(i∗ ) + η V ∗ )s∗ (a) − α i∗ (a), ⎪ ⎪ ⎪ da ⎪ ⎩ 0 = δ1 H(i∗ )(1 −V ∗ ) − μ¯ V ∗ where ∗

P(i ) =

 ∞ 0

∗

 ∞

∗

γ2 (σ )i (σ )d σ ,

H(i ) =

0

δ2 (a)i∗ (a)da.

(6.16)

From (6.15), it follows that s∗ (a) = e−

a

0 (γ1 (σ )P(i

i∗ (a) = e−α a V∗ =

 a 0

∗ )+η V ∗ )d σ

, σ

[γ1 (σ )P(i∗ ) + η V ∗ ]e

0

[α −γ1 (τ )P(i∗ )−η V ∗ ]d τ

dσ ,

(6.17)

δ1 H(i∗ ) . μ¯ + δ1 H(i∗ )

Substituting the second equation of (6.17) into P(i∗ ) and V ∗ , we have P(i∗ ) = V∗ =

 ∞ 0

δ1

γ2 (a)N(a)e−α a  ∞ 0

μ¯ + δ1

δ2 (a)e−α a

0

∞

 a 0 a



Let

0

a

(α −γ1 (τ )P∗ (i∗ )−η V ∗ )d τ

(α −γ1 (τ )P∗ (i∗ )−η V ∗ )d τ

0

σ

[γ1 (σ )P∗ (i∗ ) + η V ∗ ]e

0

d σ da,

d σ da

(α −γ1 (τ )P∗ (i∗ )−η V ∗ )d τ

d σ da (6.18)

x = P(i∗ ),

Φ1 (τ ) = eατ Φ2

0

σ

[γ1 (σ )P∗ (i∗ ) + η V ∗ ]e

0

δ2 (a)e−α a

σ

[γ1 (σ )P∗ (i∗ ) + η V ∗ ]e

y = V ∗

(τ ) = eατ

∞

τ ∞ τ

γ2 (a)e−α a da, −α a

δ2 (a)e

(6.19)

da.

Exchange the orders of the integration in the expression of P(i∗ ) and V ∗ , and using the notations in (6.19), we get two equations for x, y. x= y=

 ∞ 0

[γ1 (σ )x + η y]e−

δ1

 ∞ 0

μ¯ + δ1

σ 0

(γ1 (τ )x+η y)d τ

[γ1 (σ )x + η y]e− ∞

0

σ 0

[γ1 (σ )x + η y]e−

Φ1 (σ )d σ ,

(γ1 (τ )x+η y)d τ σ 0

Φ2 (σ )d σ

(γ1 (τ )x+η y)d τ

Φ2 (σ )d σ

.

(6.20)

.

218

6 Vector-Borne Age-Structured Models

It is clear that one solution of (6.20) is (x, y) = (0, 0), which corresponds to the equilibrium point with no disease. In order to investigate a nontrivial solution for (6.20), we define a nonlinear operator F(x, y) in R2 with the positive cone R2+ : ! F(x, y) = F1 (x, y) = F2 (x, y) =

F1 (x, y)

" ,

 F∞2 (x, y) 0

(x, y)T ∈ R2+ ,

[γ1 (σ )x + η y]e−

δ1

 ∞ 0

μ¯ + δ1

σ 0

(γ1 (τ )x+η y)d τ

[γ1 (σ )x + η y]e−

0

∞

σ 0

Φ1 (σ )d σ ,

(γ1 (τ )x+η y)d τ

(x, y)T ∈ R2+ ,

Φ2 (σ )d σ

σ

[γ1 (σ )x + η y]e− 0 (γ1 (τ )x+η y)d τ Φ2 (σ )d σ

,

(x, y)T ∈ R2+ .

(6.21) The range of F(x, y) is included in and the solutions of (6.20) correspond to fixed points of F(x, y). Observe that the operator F(x, y) has a positive linear majorant T (x, y), that is, F(x, y) ≤ T (x, y), ∀(x, y) ∈ R2+ , defined by: R2+ ,

! T (x, y) = T1 (x, y) = T2 (x, y) =

T1 (x, y)

T2 (x, y)  +∞ 0

δ1 μ¯

" ,

(x, y)T ∈ R2+ ,

(γ1 (σ )x + η y)Φ1 (σ )d σ ,

 +∞ 0

(x, y)T ∈ R2+ ,

(γ1 (σ )x + η y)Φ2 (σ )d σ ,

(6.22)

(x, y)T ∈ R2+ .

Definition 6.1. A positive operator T ∈ B(X) is called semi-nonsupporting if and only if for every pair ψ ∈ X+ \ {0}, F ∈ X+∗ \ {0}, there exists a positive integer p = p(ψ , F) such that F, T p ψ  > 0. A positive operator T ∈ B(X) is called nonsupporting if and only if for every pair ψ ∈ X+ \ {0}, F ∈ X+∗ \ {0}, there exists an integer p = p(ψ , F) such that F, T n ψ  > 0 for all n ≥ p. The reader may refer to [156] for the proof of the following theorem: Lemma 6.1. Let the cone X+ be total, T ∈ B(E) be semi-nonsupporting with respect to X+ and let r(T ) be a pole of the resolvent R(λ , T ). Then the following holds (1) r(T ) ∈ Pσ (T ) \ {0}, Pσ (T ) denotes the point spectrum, r(T ) is a simple pole of the resolvent. (2) The eigenspace corresponding to r(T ) is one-dimensional and the corresponding eigenvector ψ ∈ X+ is a nonsupporting point. The relation T φ = μφ with φ ∈ X+ implies that φ = cψ for some constant c > 0. (3) The eigenspace of T ∗ corresponding to r(T ) is also a one-dimensional subspace of X ∗ spanned by a strictly positive functional F ∈ X ∗ . (4) Assume that X is a Banach lattice. If T ∈ B(X) is nonsupporting, then the peripheral spectrum of T consists only of r(T ), i.e., |λ | < r(T ) for λ ∈ σ (T ) \ {r(T )}. The following comparison theorem is due to [129].

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host

219

Lemma 6.2. Suppose that X is a Banach lattice. Let S and T be positive operator in B(X). (1) If S ≤ T , then r(S) ≤ r(T ). (2) If S and T are semi-nonsupporting operators, then S ≤ T, S = T implies that r(S) < r(T ). After the above preparations, we first consider the nature of the majorant operator T (x, y) defined by (6.22). In the following, we shall make an assumption: Assumption 7 Supp{γ1 (a)}, Supp{γ2 (a)}, Supp{δ2 (a)} = 0, ¯ ∈ X| f (x) = 0}. where Supp of f, denoted the support of f, is the set supp f = {x Lemma 6.3. Under the Assumption 7, the operator T : X → X is nonsupporting and compact. Proof. Let a= c=

 +∞ 0

δ1 μ¯

γ1 (τ )Φ1 (τ )d τ < ∞,

 +∞ 0

b=η

γ1 (τ )Φ2 (τ )d τ < ∞,

 +∞

d=

0

δ1 η μ¯

Φ1 (τ )d τ < ∞,  +∞ 0

Φ2 (τ )d τ < ∞,

m = min{a, b, c, d}. !

Then T (x, y) =

T1 (x, y) T2 (x, y)

"

! =

ax + by cx + dy

"

! ≥m

x+y x+y

" .

Under the Assumption 7, it is clear that m > 0 holds. Let  x+y Ψ (x, y) = m , (x, y)T ∈ R2+ . x+y Then by iteration we get T n (x, y) ≥ 2n−1 mnΨ (x, y). n Taking any pair (x, y)T ∈ R2+ − {0}, F ∈ R2∗ + − {0}, we obtain F, T (x, y) > 0, that is, T is nonsupporting. Since x, y ∈ R, it is clear that T (x, y) is compact.

From Lemma 6.1 we obtain that the spectral radius r(T ) of operator T is the only positive eigenvalue with a positive eigenvector and also an eigenvalue of the dual operator T ∗ with a strictly positive eigenfunctional. Proposition 6.3. Let r(T ) be the spectral radius of the operator T . Then the following results hold: (1) If r(T ) ≤ 1, then (0, 0)T is unique nonnegative solution of (x, y)T = F(x, y). (2) If r(T ) > 1, then (x, y)T = F(x, y) has at least one nonzero positive solution.

220

6 Vector-Borne Age-Structured Models

Proof. Suppose r(T ) ≤ 1, we have T (x, y) − F(x, y) ∈ R2+ − {0} for (x, y)T ∈ R2+ − {0}. If there exists a (x0 , y0 )T ∈ R2+ − {0} which satisfies (x0 , y0 )T = F(x0 , y0 ), then F(x0 , y0 ) ≤ T (x0 , y0 ). Let F ∗ ∈ R2∗ + − {0} be the eigenvector of the adjoint operator T ∗ corresponding to eigenvalue r(T ). Taking duality pairing, we get F ∗ , T (x0 , y0 )−(x0 , y0 )T  = (T ∗ −I ∗ )F ∗ , (x0 , y0 )T  = (r(T )−1)F ∗ , (x0 , y0 )T  > 0. Since T (x0 , y0 ) − (x0 , y0 ) = T (x0 , y0 ) − F(x0 , y0 ) ∈ R2+ − {0} and F ∗ is strictly positive, thus we have r(T ) > 1. That is a contradiction. This shows that the statement (1) holds in Proposition 6.3. It is easy to see F(x, y) is a compact (completely continuous) operator in R2+ . Moreover, if we define the number M by M = max{M1 ,

δ1 M2 }, μ¯

where M1 = sup Φ1 (a),

M2 = sup Φ2 (a).

0≤a 0, β = β (x) > 0 ), and 0 < t < 1, there exists η = η (x,t) > 0 such that A(tx) ≥ tAx + η x0 ,

(6.23)

then A has at most one positive fixed point. Proposition 6.4.

If the following condition is satisfied:

α eασ α eασ

 ∞ σ  ∞ σ

γ2 (τ )e−ατ d τ − γ2 (σ ) ≥ 0,

for all σ ≥ 0

δ2 (τ )e−ατ d τ − δ2 (σ ) ≥ 0,

for all σ ≥ 0.

(6.24)

Which is equivalent to Φ1 (σ ), Φ2 (σ ) is increasing functions, and r(T ) > 1, then F(x, y) has only one positive fixed point. Proof. We first prove that F(x, y) is a monotone operator in R2+ . From (6.21) and (6.19) we have  ∞

σ

[γ1 (σ )x + η y]e− 0 (γ1 (τ )x+η y)d τ Φ1 (σ )d σ 0 σ ∞ d =− Φ1 (σ ) e− 0 (γ1 (τ )x+η y)d τ d σ dσ 0 σ 0  ∞ −  σ (γ (τ )x+η y)d τ d Φ1 (σ ) = Φ1 (σ )e− 0 (γ1 (τ )x+η y)d τ ∞ + e 0 1 dσ dσ  ∞  ∞  0  ∞  σ = γ2 (τ )e−ατ d τ + e− 0 (γ1 (τ )x+η y)d τ α eασ γ2 (a)e−α a da σ 0 0  − γ2 ( σ ) d σ , (6.25)

F1 (x, y) =

222

6 Vector-Borne Age-Structured Models

F2 (x, y) =

δ1

 ∞ 0

μ¯ + δ1

= 1−

[γ1 (σ )x + η y]e− ∞

0

σ 0

[γ1 (σ )x + η y]e−

(γ1 (τ )x+η y)d τ σ 0

(γ1 (τ )x+η y)d τ

μ¯

 ∞

Φ2 (σ )d σ Φ2 (σ )d σ

σ

(γ1 (σ )x + η y)e− 0 (γ1 (τ )x+η y)d τ Φ2 (σ )d σ 0 μ¯ = 1−  ∞ d − 0σ (γ1 (τ )x+η y) μ¯ − Φ2 ( σ ) e dτ dσ 0 μ¯ = 1− σ 0  ∞ −  σ (γ (τ )x+η y)d τ d Φ2 (σ ) d σ μ¯ + Φ2 (σ )e− 0 (γ1 (τ )x+η y)d τ ∞ + e 0 1 dσ 0 μ¯  ∞  ∞   ∞ = 1− . σ μ¯ + δ2 e−ατ d τ + e− 0 (γ1 (τ )x+η y)d τ [α eασ δ2 (τ )e−ατ d τ − δ2 (σ )]d σ

μ¯ + δ1

0

σ

0

(6.26)

If the inequality (6.24) holds, it is clear that F1 (x, y), F2 (x, y) are decreasing with respect to (x, y)T ∈ R2+ . We conclude that F(x, y) is decreasing for (x, y)T ∈ R2+ . Let u0 = (1, 1)T ,

α (x, y) = min{α1 (x, y),

α2 (x, y),

β1 (x, y),

β2 (x, y)},

β (x, y) = max{α1 (x, y),

α2 (x, y),

β1 (x, y),

β2 (x, y)},

α1 (x, y) = x α2 (x, y) = y

β1 (x, y) =

β2 (x, y) =

 +∞ 0

 +∞ 0

γ1 (τ )e− η e−

δ1 x μ¯ + δ1

(xγ1 (s)+η y)ds

(xγ1 (s)+η y)ds

γ1 (τ )e−

 +∞

 +∞ 0

0

τ 0

0

η e−

τ 0

Φ1 (τ )d τ ,

Φ1 (τ )d τ ,

(xγ1 (s)+η y)ds

[(xγ1 (τ ) + η y)e−

0

δ1 y μ¯ + δ1

0

 +∞

0 +∞



τ

τ

τ 0

(xγ1 (s)+η y)ds

(xγ1 (s)+η y)ds

[(xγ1 (τ ) + η y)e−

τ 0

Φ2 (τ )d τ ]Φ2 (τ )d τ

Φ2 (τ )d τ

(xγ1 (s)+η y)ds

]Φ2 (τ )d τ

,

.

We have that α (x, y)u0 ≤ F(x, y) ≤ β (x, y)u0 , and we can get F(tx,ty) ≥ tF(x, y) for 0 < t < 1. From Definition 6.2, we obtain that F(x, y) is a concave operator. Next we prove that F(x, y) satisfies the condition (6.23). From (6.21), for 0 < t < 1, we get

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host

223

F1 (tx,ty) − tF1 (x, y) =t

 +∞ 0

−t =t =t

(γ1 (σ )x + η y)e−t

 +∞ 0

 ∞ 0

 ∞ 0

σ 0

(γ1 (σ )x + η y)e−

(γ1 (τ )x+η y)d τ

σ 0

Φ1 (σ )d σ

(γ1 (τ )x+η y)d τ

Φ1 (σ )d σ

σ   σ (γ1 (σ )x + η y) e−t 0 (γ1 (τ )x+η y)d τ − e− 0 (γ1 (τ )x+etay)d τ Φ1 (σ )d σ

(γ1 (σ )x + η y)e−

σ 0

(γ1 (τ )x+η y)d τ

 (1−t)  σ (γ (τ )x+η y)d τ  0 1 e − 1 Φ1 (σ )d σ

= η1 (x, y,t) > 0. Let A= B=

 +∞ 0

 +∞ 0

(xγ1 (σ ) + η y)e−t (xγ1 (σ ) + η y)e−

σ 0

σ 0

(xγ1 (τ )+η y)d τ

(xγ1 (τ )+η y)d τ

Φ2 (σ )d σ

Φ2 (σ )d σ .

Then F2 (tx,ty) − tF2 (x, y) =

t δ1 A A−B t δ1 B − = t δ1 μ¯ μ¯ + δ1 A μ¯ + δ1 B (μ¯ + δ1 A)(μ¯ + δ1 B) ∞

= t δ1 μ¯

0

(γ1 (σ )x + η y)e−

= η2 (x, y,t) > 0. Let

σ 0

(γ1 (τ )x+η y)d τ

[e(1−t)

σ 0

(γ1 (τ )x+η y)d τ

− 1]Φ2 (σ )d σ

(μ¯ + δ1 A)(μ¯ + δ1 B) !

η (x, y,t) =

η1 (x, y,t) η2 (x, y,t)

" .

We obtain that F(x, y) satisfies the condition (6.23). From Lemma 6.4, we get F(x, y) has only one positive fixed point. This theorem is proved. It is easy to see that if γ2 (a) and δ2 (a) is constants, then (6.24) holds.

6.2.4 Local Stability of Equilibria Since r(a,t) = 1 − s(a,t) − i(a,t),V0 (t) = 1 − V (t), it is sufficient to consider system (6.6)–(6.7) in terms of only s(a,t), i(a,t),V (t).

224

6 Vector-Borne Age-Structured Models

In order to investigate the local stability of the equilibrium (s∗ (a), i∗ (a), of the (6.6)–(6.7), we first rewrite (6.6)–(6.7) into equations for small perturbations. Let

V ∗)

s(a,t) = s∗ (a) + x(a,t), i(a,t) = i∗ (a) + y(a,t),V (t) = V ∗ + z(t), we get a linearized system around the equilibrium (s∗ (a), i∗ (a),V ∗ ): ⎧ ∂x ∂x ⎪ ⎪ + = −(γ1 (a)P(i∗ ) + η V ∗ )x(a,t) − (γ1 (a)P(y) + η z)s∗ (a), ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎨ ∂y ∂y + = (γ1 (a)P(i∗ ) + η V ∗ )x(a,t) + (γ1 (a)P(y) + η z)s∗ (a) − α y(a,t), (6.27) ⎪ ∂ t ∂ a ⎪ ⎪ ⎪ ⎪ dz(t) ⎪ ⎩ = −(δ1 H(i∗ ) + μ¯ )z(t) + δ1 H(y)(1 −V ∗ ), dt with boundary and initial conditions: x(0,t) = 0,

y(0,t) = 0;

x(a, 0) = 0,

y(a, 0) = 0,

z(0) = 0, (6.28)

where P(i∗ ) = γ1 (a)

 ∞ 0

γ2 (σ )i∗ (σ )d σ .

Therefore, we can formulate (6.27)–(6.28) as an abstract Cauchy problem: ⎧ ⎨ dΨ (t) = BΨ (t) +CΨ (t), dt ⎩ Ψ (0) = Ψ ,

(6.29)

0

where

Ψ = (x, y, z)T ∈ X, ⎛

−(

B : D(B) → X,

d + γ1 (a)P(i∗ ) + η V ∗ )u1 da

⎞

⎜ ⎟ ⎜ ⎟ ⎜ ⎟, d BU = ⎜ ∗ ∗ ⎟ + −( α )u + ( γ (a)P(i ) + η V ))u 2 1 1 ⎝ da ⎠ ∗ −(δ H(i ) + μ¯ )u3 with domain D(B) = {(u1 , u2 , u3 )T , u1 , u2 ∈ W 1,1 [0, +∞), u3 ∈ R, u1 (0) = 0, u2 (0) = 0}. The inhomogeneous term C is defined as

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host

⎛

−(λ (a,t) + η u3 )s∗ (a)

⎞

⎜ ⎟ ∗ (a) ⎟ λ (a,t) + η u )s ( C=⎜ 3 ⎝ ⎠,

C : X → X,

225

D(C) = X.

δ1 H(u2 )(1 −V ∗ ) In the following, we make an assumption: Assumption 8 (1) γ1 (a) ∈ C[0, ∞) is uniformly continuous. (2) There exists a M3 > 0 such that M3 = max{γ1 (a), γ2 (a)} for a ∈ [0, +∞). Now let us consider the characteristic equation of B +C:

λ U = (B +C)U,U = (u1 , u2 , u3 ) ∈ D(B), λ ∈ C. We obtain ⎧ du1 (a) ⎪ ⎪ = −(λ + γ1 (a)P(i∗ )(a) + η v∗ )u1 − (γ1 (a)P(u2 ) + η u3 )s∗ (a), ⎪ ⎪ da ⎨ du2 (a) (6.30) = (γ1 (a)P(i∗ ) + η v∗ )u1 − (λ + α )u2 + (γ1 (a)P(u2 ) + η u3 )s∗ (a), ⎪ ⎪ ⎪ da ⎪ ⎩ λ u3 = −(δ1 H(i∗ ) + μˆ )u3 + (1 −V ∗ )δ1 H(u2 ), with initial conditions u1 (0) = 0, From (6.30)–(6.31), we have u1 (a) = −e−

a

0 (λ +γ1 (s)P(i

u2 (a) = E(a, λ )

 a

 a 0

(6.31)

¯ 2 ) + η u3 )eλ τ d τ , (γ1 (τ )P(u

[(γ1 (τ )P(i∗ ) + η V ∗ )u1 (τ )e(λ +α )τ

0 a

+E(a, λ )

∗ )+η V ∗ )ds

u2 (0) = 0.

0

(γ1 (τ )P(u2 ) + η u3 )e−

τ 0

(γ1 (s)P(i∗ )+η V ∗ )ds (λ +α )τ

e

(1 −V ∗ )δ1 u3 = H(u2 ) λ + δ1 H(i∗ ) + μ¯ where E(a, λ ) = e−(λ +α )a . Let Y = {x1 γ1 (a) + η x2 : x1 , x2 ∈ R} with norm

x ¯ = max x , ¯ x¯ = |x1 |γ1 (a) + η |x2 | ∈ Y. a∈[0,+∞)

Then Y is complete Banach lattice. Denote

Θ (a) = γ1 (a)P(u2 ) + η u3 .

,

(6.32)

226

6 Vector-Borne Age-Structured Models

Substituting u2 (a), u3 into Θ (a), using (6.17), we get

Θ (a) = γ1 (a)

 ∞ 0

−γ1 (a) × + − ×

 τ 0

γ2 (a)E(a)

 ∞ 0

 a 0

γ2 (a)E(a)

 ∞ 0

0

 a 0

0

(λ +α −γ1 (s)P(i∗ )−η V ∗ )ds τ

(γ1 (τ )P(i∗ ) + η V ∗ )e

0

d τ da

(α −γ1 (s)P(i∗ )−η V ∗ )ds

Θ (σ )eλ σ d σ d τ da

η (1 −V ∗ )δ1  λ + δ1 H(i∗ ) + μ¯

 τ

τ

Θ (τ )e

δ2 (a)E(a) λσ

Θ (σ )e

 a 0

 ∞ 0

δ2 (a)E(a)

 a 0

τ

Θ (τ )e

(γ1 (τ )P(i∗ ) + η V ∗ )e−

τ 0

0

(λ +α −γ1 (s)P(i∗ )−η V ∗ )ds

d τ da

(λ +γ1 (s)P(i∗ )+η V ∗ )ds

d σ d τ da . (6.33)

If λ is an eigenvalue of B +C, then there exists Θ = 0 ∈ Y such that

Θ = Tλ (Θ ),

(6.34)

where Tλ (Θ ) denote the right-hand side of (6.33). So we investigate λ such that the eigenvalue of Tλ is 1. Let δ ∗ = δ1 H(i∗ ) + μ¯ . If α ≥ δ ∗ , as λ ∈ (−δ ∗ , +∞), exchanging the order of the integration, the right-hand side of (6.33) can be expressed as Tλ (Θ ) = γ1 (a)

 ∞ 0

−γ1 (a) 2

× + − ×

 τ 0

γ2 (a)

 ∞ 0

 a 0

Θ (σ )eλ σ d σ

 ∞ 0

0

τ

(γ1 (τ )P(i∗ ) + η V ∗ )e

η (1 −V ∗ )δ1  λ +δ∗

 τ

Θ (τ )e(λ +α )(τ −a) e−  ∞ τ

 ∞ 0

δ2 (a)

 ∞ τ

0

(γ (s)P(i∗ )+η V ∗ )ds

d τ da

(α −γ1 (s)P(i∗ )−η V ∗ )ds

γ2 (a)e−(λ +α )a dad τ  a 0

(γ1 (τ )P(i∗ ) + η V ∗ )e−

Θ (σ )eλ σ d σ

0

τ

Θ (τ )e(λ +α )(τ −a) e−

τ 0

τ 0

(γ1 (s)P(i∗ )+η V ∗ )ds

d τ da

(λ +γ1 (s)P(i∗ )+η V ∗ )ds

δ2 (a)e−(λ +α )a dad τ .

(6.35) Using (6.24), it is clear to see that Tλ (Θ ) is decreasing as a function of λ ∈ (−δ ∗ , +∞), θ ∈ Y+ . From Assumption 8, we get that Tλ (θ ), λ ∈ (−δ ∗ , +∞), is compact and nonsupporting.

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host

227

From (6.35), we have Tλ (Θ ) ≥

η (1 −V ∗ )δ1  λ +δ∗ − ×

= Gλ , Θ  =

 ∞ 0

 τ 0

 ∞ 0

δ2 (a)

 a

(γ1 (τ )P(i∗ ) + η V ∗ )e−

Θ (σ )eλ σ d σ

 ∞ τ

0

− ×

δ2 (a)

 ∞ 0

 τ 0

 a 0

τ 0

τ

(γ1 (s)P(i∗ )+η V ∗ )ds

d τ da

λ ∈ (−δ ∗ , +∞),

(γ1 (τ )P(i∗ ) + η V ∗ )e−  ∞

0

(λ +γ1 (s)P(i∗ )+η V ∗ )ds

Θ (τ )e(λ +α )(τ −a) e−

Θ (σ )eλ σ d σ

τ

δ2 (a)e−(λ +α )a dad τ

ηδ1 (1 −V ∗ ) Gλ , θ  · 1, λ +δ∗

 ∞

0

Θ (τ )e(λ +α )(τ −a) e−

τ 0

τ 0

(γ1 (s)P(i∗ )+η V ∗ )ds

d τ da

(λ +γ1 (s)P(i∗ )+η V ∗ )ds

δ2 (a)e−(λ +α )a dad τ ,

Θ ∈ Y+ .

Taking duality pairing with the eigenfunctional Fλ of Tλ that corresponds to r(Tλ ), we obtain ηδ1 (1 −V ∗ ) r(Tλ )Fλ , θ  ≥ Gλ , θ Fλ , 1. λ +δ∗ If taking θ = 1, we have r(Tλ ) ≥

ηδ1 (1 −V ∗ ) Gλ , 1. λ +δ∗

If λ ∈ (−δ ∗ , +∞), then 0 < Gλ , 1 < +∞, lim r(Tλ ) = +∞.

λ →−δ ∗

We can also prove that if α < δ ∗ , then there exists −∞ < α ∗ < 0 such that lim r(Tλ ) = +∞.

λ →α ∗

228

6 Vector-Borne Age-Structured Models

From (6.35), we can obtain Tλ (θ ) ≤ γ1 (a) +

 +∞ 0

−(λ +α )a

γ2 (a)e

ηδ1 (1 −V ∗ ) λ +δ∗

< (M32 +

 +∞

 a 0

e−(λ +α )a

 a

0

0

ηδ1 (1 − v∗ ) ) λ +δ∗

Let G∗λ , Θ  = (M32 +

τ

θ (τ )e

 +∞

e−λ a

(λ +α −γ1 (s)P(i∗ )−η V ∗ )ds

τ

θ (τ )e

 a

0

0

ηδ (1 −V ∗ ) ) λ +δ∗

0

(λ +α −γ1 (s)P(i∗ )−η V ∗ )ds

d τ da

θ (τ )eλ τ d τ da, θ ∈ Y+ .

 +∞ 0

0

d τ da

e−λ a

 a 0

Θ (τ )eλ τ d τ da.

We have Tλ (Θ ) ≤ G∗λ , Θ  · 1, Θ ∈ Y+ , r(Tλ ) ≤ G∗λ , 1, lim r(Tλ ) ≤ lim G∗λ , 1 = 0.

λ →+∞

λ →+∞

From (6.35), we get that Tλ (Θ ) is decreasing as a function of λ ∈ (δ ∗ , +∞) or (α ∗ , +∞), Θ ∈ Y+ . Using Lemma 6.2, we know that the function λ → r(Tλ ) is strictly decreasing. If there exists λ ∈ (δ ∗ , +∞) or (α ∗ , +∞) such that r(Tλ ) = 1, then λ ∈ Pσ (B +C). So, we get the following lemma: Lemma 6.5. Let T0 = Tλ |λ =0 . If the inequality (6.24) holds, under Assumption 8, there exists unique λ0 ∈ (δ ∗ , +∞) ∩ Pσ (B + C) or (α ∗ , +∞) ∩ Pσ (B + C) such that the following results hold: (1) If r(T0 ) > 1, then λ0 > 0. (2) If r(T0 ) = 1, then λ0 = 0. (3) If r(T0 ) < 1, then λ0 < 0. Using the similar argument as Theorem 6.13 in [81], we can prove that λ0 is a dominant singular point. We have the following lemma: Lemma 6.6. Under the condition of Lemma 6.5, if there exists a λ , r(Tλ ) = 1, λ = λ0 , then Reλ < λ0 . We define: T¯ (x1 γ1 (a) + x2 η ) = γ1 (a)T1 (x1 , x2 ) + η T2 (x1 , x2 ), T1 , T2 are defined by (6.22). It is clear to see that r(T¯ ) = r(T ) holds. From (6.32), if P(i∗ )(a) = 0, v∗ = 0, λ = 0, we have r(T0 ) = r(T¯ ) = r(T ). So, we obtain the following results.

6.2 A Vector-Borne Disease Model with Chronological Age Structure of Host

229

If r(T ) < 1, then sup{Reλ , r(Tλ ) = 1} = λ0 < 0. If r(T ) > 1, then exists λ such that r(Tλ ) = 1 and Reλ > 0. From (6.21), for (x1 , x2 )T ∈ R2+ , we define ¯ 1 γ1 (a) + η x2 ) = γ1 (a) F(x

 +∞ 0

(x1 γ1 (τ ) + η x2 )e−

 +∞

τ 0

(x∗ γ1 (s)+η y∗ )ds

Φ1 (τ )d τ



τ ∗ ∗ (x1 γ1 (τ ) + η x2 )e− 0 (x γ1 (s)+η y )ds Φ2 (τ )d τ τ +  +∞0 , ∗ ∗ δ1 0 (x∗ γ1 (τ ) + η y∗ )e− 0 (x γ1 (s)+η y )ds φ2 (τ )d τ + μ¯

ηδ1

where (x∗ , y∗ )T is a nontrivial positive solution of (x, y)T = F(x, y). We easily get ¯ holds. On the other hand, that F¯ is a nonsupporting operator, and that r(T0 ) < r(F) ∗ ∗ T since (x , y ) is a nontrivial positive solution of (x, y)T = F(x, y), it implies that F¯ has a positive eigenfunction x∗ γ1 (a) + η y∗ corresponding to eigenvalue 1. Since a nonsupporting operator has only one positive eigenfunction corresponding to its ¯ = 1 and r(T0 ) < 1. spectral radius, we get that r(F) From above arguments, we conclude that Lemma 6.7. Let operator T is defined in (6.22). If the inequality (6.24) holds, under Assumption 8, the following results hold: (1) Suppose that (x∗ , y∗ )T is a trivial solution of (x, y)T = F(x, y). If r(T ) < 1, then sup{Reλ , r(Tλ ) = 1} < 0. If r(T ) > 1, there exists λ , r(Tλ ) = 1, Reλ > 0. (2) Suppose (x∗ , y∗ )T is a nontrivial positive solution of (x, y)T = F(x, y), if r(T ) > 1, then sup{Reλ , r(Tλ ) = 1} < 0. Quasi-compact is defined as following: Definition 6.3. J(t) is called quasi-compact if J(t) = J1 (t) + J2 (t) with operator families J1 (t),J2 (t), where J1 (t) → 0, as t → 0, J2 (t) is eventually compact, that is, there exists t0 > 0 such that J2 (t) is a compact operator for all t > t0 . Next, we prove that the operator B +C generates a quasi-compact C0 -semigroup. We first make an assumption: Assumption 9 Suppose γ1 (a), i∗ (a), p satisfy the condition: ¯ ∗ ) < μ0 + p, sup γ1 (a)P(i a∈[0,+∞)

where μ0 =

inf

a∈[0,+∞)

μ (a).

There exist γ1 (θ ), p such that Assumption 9 holds. In fact, from (6.20) we obtain ¯ ∗ ) satisfies that x = P(i x=

 +∞ 0

(xγ1 (τ ) + η y)e−

Φ1 (τ ) = eατ

 +∞ τ

τ 0

(xγ1 (s)+η y)ds

γ2 (a)eα a da.

φ1 (τ )d τ .

230

6 Vector-Borne Age-Structured Models

So we get z = γ1 (a)μ ∗ N

 +∞ 0

(z(τ ) + η y)e−

τ 0

(z(s)+η y)ds

φ1 (τ )d τ ,

¯ ∗ ). From above equation, if γ1 (a) is small enough, we easily where z = γ1 (a)P(i ¯ ∗ ) may be small enough such that Assumption 9 holds. obtain that γ1 (a)P(i In order to investigate the stability of equilibria we prove a lemma. Lemma 6.8. The operator B is a closed linear operator and exists ε > 0 such that λ − B has bounded inverse for λ > −ε and

(λ − B)−n ≤

1 . (λ + ε )−n

Proof. For f = ( f1 , f2 , f3 )T ∈ X+ , consider the equation (λ − B)u = f with λ > −ε , we have du1 (a) = −(λ + γ1 (a)P(i∗ ) + η V ∗ )u1 + f1 , da du2 (a) = (γ1 (a)P(i∗ ) + η V ∗ )u1 − (λ + α )u2 + f2 , da 0 = −(λ + δ H(i∗ ) + μ¯ )u3 + f3 ,

(6.36)

u1 (0) = 0, u2 (0) = 0. From (6.36), we obtain u1 (a) = e− u2 (a) = e− u3 =

a

0 (λ +γ1 (s)P(i

a

0 (λ +α )ds

∗ )+η V ∗ )ds

 a 0

f3 λ +δ ∗ ,

 a 0

τ

f1 (τ )e

0

(λ +γ1 (s)P(i∗ )+η V ∗ )ds

dτ ,

[(γ1 (τ )P(i∗ ) + η V ∗ )u1 (τ ) + f2 (τ )]e(λ +α )τ d τ , (6.37)

and

u ≤

1 (λ

+ λ ∗ + ηV ∗ )

f1

1 P∗ + η V ∗

f1 +

f2 (λ + α )(λ + λ ∗ + η V ∗ ) λ +α 1 +

f3 , λ +δ∗

+

where λ ∗ = inf P(i∗ )(a), P∗ = sup P(i∗ )(a). 0≤a 0 if and only if all eigenvalues of B¯ have strictly negative real part. From Lemmas 6.7 and 6.9, we have the following theorem: Proposition 6.5. If the inequality (6.24) holds, under Assumptions 8 and 9, the following results hold: (1) If r(T ) < 1, then the trivial equilibrium point of the system (6.1)–(6.2) is locally asymptotically stable. (2) If r(T ) > 1, then the trivial equilibrium point of the system (6.1)–(6.2) is unstable. (3) If r(T ) > 1, then the endemic equilibrium point of the system (6.1)–(6.2) is locally asymptotically stable.

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection 6.3.1 Introduction In many infectious diseases, such as HIV, schistosomiasis, tuberculosis, the infectiousness of an infected individual can be very different at various stages of infection. Hence, the age of infection may be an important factor to consider in modeling transmission dynamics of infectious diseases. In the epidemic model of Kermack and Mckendrick [101], infectivity is allowed to depend on the age of infection. Because the age-since-infection epidemic model is described by first order PDEs, it is more difficult to theoretically analyze the dynamical behavior of the PDE models, particularly the global stability. Several recent studies [169] have focused on agestructured models, and the results show that age of infection may play an important role in the transmission dynamics of infectious diseases.

232

6 Vector-Borne Age-Structured Models

The vector-borne epidemic model formulated in [40] only incorporates a single strain. In reality, many diseases are caused by more than one antigenically different strains of the causative agent [124]. For instance, the dengue virus has four different serotypes [53]. Therefore, it is necessary to study infection-age-structured epidemic models with multiple strains. In this section, we consider the following infection-age-structured vector-borne epidemic model with two strains: ⎧  ∞ 2 dSv (t) ⎪ ⎪ = Λ − S (t) ⎪ v ∑ v 0 βvj (a)Ihj (a,t)da − μv Sv (t), ⎪ ⎪ dt ⎪ j=1 ⎪ ⎪ ⎪  ∞ ⎪ j ⎪ dIv (t) ⎪ ⎪ ⎪ = Sv (t) βvj (a)Ihj (a,t)da − μv Ivj (t), ⎪ ⎪ dt 0 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ Sh (t) dSh (t) ⎪ ⎪ = Λ − h ⎪ dt ∑ βhj N (t) Ivj (t) − μh Sh (t), ⎨ j=1

h

⎪ ⎪ ∂ Ihj (a,t) ∂ Ihj (a,t) ⎪ ⎪ = −(μh + rhj (a))Ihj (a,t), ⎪ ⎪ ∂a + ∂t ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ Sh (t) j ⎪ ⎪ ⎪ Iv (t), Ihj (0,t) = βhj ⎪ ⎪ N h (t) ⎪ ⎪ ⎪ ⎪ ⎪ 2  ∞ ⎪ ⎪ ⎪ dRh (t) = ⎪ rhj (a)Ihj (a,t)da − μh Rh (t). ⎩ ∑ dt j=1 0

(6.38)

In the model (6.38), Sh (t), Ihj (a,t), Rh (t) represent the number/density of the susceptible hosts, infected hosts with strain j( j = 1, 2), and recovered hosts at time t, respectively. Nh (t) represent the total number/density of the hosts at time t. Sv (t) and Ivj (t) denote the number of the susceptible vectors and infected vectors with strain j at time t, respectively. Λv , Λh are the birth /recruitment rates of the vectors and hosts, respectively; μv , μh are the natural death rates of the vectors and hosts, respectively. The parameter rhj (a) denotes the recovery rate of the infected hosts of infection age a with strain j; βvj (a) is the transmission coefficient of the infected host individuals with strain j at age of infection a, and βhj is the adequate bitting rate from an infected vector with strain j to the susceptible host individuals. The dynamics of the epidemic model involving multiple strains has fascinated researchers for a long time (see [23, 49, 53, 64, 151] and the references therein), and one of the important results is the competitive exclusion principle. In epidemiology, the competitive exclusion principle states that if multiple strains circulate in the population, only the strain with the largest reproduction number persists and the strains with suboptimal reproduction numbers are eliminated [120]. Using a multiple-strain ODE model Bremermann and Thieme [19] first proved that the principle of competitive exclusion is valid under the assumption that infection with one strain precludes

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

233

additional infections with other strains. In 2013, Martcheva and Li [120] extended the competitive exclusion principle to a multi-stain age-since-infection structured model of SIR/SI-type. Here we extend this principle to model (6.38). As we all know, the proof of competitive exclusion principle is based on the global stability of the single-strain equilibrium. The stability analysis of nonlinear dynamical systems has always been an important topic theoretically and practically since global stability is one of the most important issues related to their dynamic behaviors. Due to the lack of generically applicable tools proving the global stability is very challenging, especially for the continuous age-structured models which are described by first order PDEs. Although there are various approaches for some general nonlinear systems, the method of Lyapunov functions is the most common tool used to prove the global stability. In this section, we will apply a class of Lyaponuv functions to study the global dynamics of system (6.38) and draw on the results to derive the competitive exclusion principle for infinite-dimensional systems.

6.3.2 The Reproduction Numbers and Threshold Dynamics In this section, we mainly derive the reproduction numbers for each strain, and show that the stain will die out if its basic reproduction number is less than one. Adding the first and all equations for Ivj in (6.38) yields   2 2 d Sv (t) + ∑ Ivj (t) ≤ Λv − μv Sv (t) + ∑ Ivj (t) . dt j=1 j=1 Hence,

 2 Λv lim sup Sv (t) + ∑ Ivj (t) ≤ . μv t→+∞ j=1

Similarly, adding the equation for Sh and all equations for Ihj , Rh , we have  2  ∞ d d j Nh (t) = Ih (a,t)da + Rh (t) Sh (t) + ∑ dt dt j=1 0  2  ∞ j ≤ Λh − μh Sh (t) + ∑ Ih (a,t)da + Rh (t) , j=1 0

and it then follows that  2  lim sup Nh (t) = lim sup Sh (t) + ∑ t→+∞

t→+∞

j=1 0

∞

Λh Ihj (a,t)da + Rh (t) ≤  Nh . μh

By the comparison principle, we study the dynamics of the limiting system of (6.38) as follows:

234

6 Vector-Borne Age-Structured Models

⎧  ∞ 2 dSv ⎪ ⎪ = Λv − ∑ Sv βvj (a)Ihj (a,t)da − μv Sv , ⎪ ⎪ ⎪ dt 0 ⎪ j=1 ⎪ ⎪ ⎪ ⎪  j ∞ ⎪ dIv ⎪ ⎪ ⎪ βvj (a)Ihj (a,t)da − μv Ivj , ⎪ dt = Sv ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎪ ⎪ Sh dSh ⎪ ⎪ = Λ − βhj Ivj − μh Sh , ⎪ h ∑ ⎨ dt N h j=1 ⎪ ⎪ ∂ Ihj (a,t) ∂ Ihj (a,t) ⎪ ⎪ + = −(μh + rhj (a))Ihj (a,t), ⎪ ⎪ ⎪ ∂a ∂t ⎪ ⎪ ⎪ ⎪ ⎪ Sh ⎪ ⎪ ⎪ Ihj (0,t) = βhj Ivj , ⎪ ⎪ N h ⎪ ⎪ ⎪ ⎪ ⎪ 2  ∞ ⎪ ⎪ dRh ⎪ ⎪ rhj (a)Ihj (a,t)da − μh Rh . ⎩ dt = ∑ j=1 0

(6.39)

Since the equations for the recovered individuals are decoupled from the system, it follows that the dynamical behavior of system (6.39) is equivalent to the dynamical behavior of the following system: ⎧  ∞ 2 dSv ⎪ ⎪ = Λ − S βvj (a)Ihj (a,t)da − μv Sv , ⎪ v v ∑ ⎪ ⎪ dt 0 ⎪ j=1 ⎪ ⎪ ⎪  ∞ ⎪ j ⎪ ⎪ dIv ⎪ ⎪ = S βvj (a)Ihj (a,t)da − μv Ivj , v ⎪ ⎪ dt 0 ⎪ ⎪ ⎪ ⎪ ⎪ 2 ⎨ dSh Sh = Λh − ∑ βhj Ivj − μh Sh , dt N ⎪ h j=1 ⎪ ⎪ ⎪ ⎪ j j ⎪ ⎪ ∂ Ih (a,t) ∂ Ih (a,t) ⎪ ⎪ + = −(μh + rhj (a))Ihj (a,t), ⎪ ⎪ ⎪ ∂ a ∂ t ⎪ ⎪ ⎪ ⎪ ⎪ Sh ⎪ ⎪ I j (0,t) = βhj Ivj . ⎪ ⎪ ⎩ h Nh

(6.40)

Model (6.40) is equipped with the following initial conditions: Sv (0) = Sv0 ,

Ivj (0) = Ivj0 ,

Sh (0) = Sh0 ,

Ihj (a, 0) = ψ j (a).

All parameters are nonnegative, Λv > 0, Λh > 0, and μv > 0, μh > 0. We make the following assumptions on the parameter-functions. Assumption 10 1. The function βvj (a) is bounded and uniformly continuous for every j. When βvj (a) is of compact support, the support has nonzero Lebesgue measure;

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

235

2. The functions rhj (a) belong to L∞ (0, ∞); 3. The functions ψ j (a) is integrable. Let us define

2

2

j=1

j=1

X = R × ∏ R × R × ∏ L1 (0, ∞). It is easily verified that solutions of (6.40) with nonnegative initial conditions belong to the positive cone for t ≥ 0. Therefore, the following set is positively invariant for system (6.40): 

2  Λv Ω = (Sv , Iv1 , Iv2 , Sh , Ih1 , Ih2 ) ∈ X+  Sv (t) + ∑ Ivj (t) ≤ , μv j=1   2  ∞ Λh j Sh (t) + ∑ Ih (a,t)da ≤ . μh j=1 0

(6.41)

In what follows, we only consider the solutions of the system (6.40) with initial conditions which lie in the region Ω . The reproduction number is one of most important concepts in epidemiological modeling. Next, we will express the basic reproduction numbers for each strain. To simplify expression, let us introduce one notation. Definition 6.4. The exit rate of infected individuals with strain j from the infective compartment is given by μh + rhj (a), the probability of still being infectious after a time units, denoted by π j (a), is given by

π j (a) = e−μh a e−

a j

0 rh (σ )d σ

.

(6.42)

Then we can give the expression for the basic reproduction number of strain j which can be expressed as R0j

βhj Λv = μv2

 ∞ 0

βvj (a)π j (a)da.

(6.43)

The reproduction number of strain j gives the number of secondary infections produced in an entirely susceptible population by a typical infected individual with strain j during its entire infectious period. R0j gives the strength of strain j to invade into the system when rare and alone. The reproduction number of strain j consists of two terms:  βj Λv ∞ j βv (a)π j (a)da, Rvj = h . Rhj = μv 0 μv The first term Rhj represents the reproduction number of human-to-vector transmission of strain j, and the second term Rvj is the reproduction number of vector-tohuman transmission of strain j.

236

6 Vector-Borne Age-Structured Models

Now we are able to state the results on a threshold dynamics of strain j: Theorem 6.1. If R0j < 1, strain j will die out. Proof. Let

B j (t) = Ihj (0,t).

Integrating along the characteristic lines of system (6.40) yields ⎧ B (t − a)π j (a), t > a, ⎪ ⎨ j j Ih (a,t) = π (a) ⎪ ⎩ ψ j (a − t) j , t < a. π j (a − t)

(6.44)

From the first and the third equations of system (6.40), we obtain

Λv , μv

lim sup Sv (t) ≤ t→+∞

lim sup Sh (t) ≤ t→+∞

Λh . μh

(6.45)

Thus, from system (6.40) and inequalities (6.45), we have ⎧  ∞ j ⎪ ⎪ ⎨ dIv (t) ≤ Λv β j (a)Ihj (a,t)da − μv Ivj , dt μv 0 v ⎪ ⎪ ⎩ I j (a,t) = I j (0,t − a)π (a), t > a. j h h

(6.46)

From the first inequality of (6.46), we obtain that Ivj (t) ≤ Ivj (0)e−μv t + ≤

Ivj (0)e−μv t +

 ∞ s

Λv μv

Λv + μv

 t

e−μv (t−s)

0

 ∞ 0

 t

−μv (t−s)

βvj (a)Ihj (a, s)dads



s

e 0

0

βvj (a)Ihj (0, s − a)π j (a)da

(6.47)



βvj (a)ψ j (a − s)

π j (a) da. π j (a − s)

Notice that  t

lim sup t→+∞

 ≤

e−μv (t−s)

0

0

 t

lim sup t→+∞

 s

−μv (t−s)

e 0

βvj (a)Ihj (0, s − a)π j (a)dads 

ds 0

∞



βvj (a)π j (a)da

  1 ∞ j j = β (a)π j (a)da lim sup Ih (0,t) , μv 0 v t→+∞

lim sup Ihj (0,t) t→+∞

(6.48)

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

 t

lim sup t→+∞

e−μv (t−s)

0

≤ β¯ lim sup t→+∞

≤ β¯ lim sup t→+∞

s

 t

e−μv (t−s)

 t

s

e−μv (t−s)

0

 t

0



 t

ψ j (a − s)e−

π j (a) dads π j (a − s)

a

j a−s ( μh +rh (σ ))d σ

dads

∞s ψ j (a − s)e−μh s dads

e(μv −μh )s

0

t→+∞

 ∞

βvj (a)ψ j (a − s)

 ∞

0

= β¯ lim sup e−μv t = β¯

 ∞

237

 ∞ 0



(6.49)

ψ j (a)dads

 e(μv −μh )t − 1 ψ j (a)da lim sup e−μv t μv − μh t→+∞

= 0, It then follows from (6.48) and (6.49) that lim sup Ivj (t) ≤ t→+∞

≤

Since R0j < 1 and Ivj (t), that

Λv μv2

 ∞ 0

t→+∞

 Λv ∞

μv2

0

 βvj (a)π j (a)da lim sup Ihj (0,t)  Sh βvj (a)π j (a)da lim sup βhj Ivj Nh t→+∞

 ∞

≤

Λv βhj 2 μv

≤

R0j lim sup Ivj (t). t→+∞

0

(6.50)

βvj (a)π j (a)da lim sup Ivj (t) t→+∞

j = 1, 2, are all bounded, the above expression implies lim sup Ivj (t) = 0, t→+∞

j = 1, 2.

(6.51)

Hence, we have lim sup Ihj (0,t) = 0, t→+∞

lim sup Ihj (a,t) = lim sup Ihj (0,t − a)π j (a) = 0. t→+∞

(6.52)

t→+∞

Therefore, (Ivj (t), Ihj (a,t)) → 0 as t → ∞. This means that strain j will die out. The proof of Theorem 6.1 is completed.

6.3.3 Global Stability of the Disease-Free Equilibrium In this section, we mainly define the disease reproduction number and show that the disease-free equilibrium is globally asymptotically stable if the disease reproduction number R0 is less than one, where R0 = max{R01 , R02 }.

238

6 Vector-Borne Age-Structured Models

System (6.40) always has a unique disease-free equilibrium E0 , which is given by

 E0 = Sv∗0 , 0, Sh∗0 , 0 ,

where Sv∗0 =

Λv , μv

Sh∗0 =

Λh , μh

and 0 = (0, 0) is a 2-dimensional zero vector. Now let us establish the local stability of the disease-free equilibrium. Let Sv (t) = Sv∗0 + xv (t), Ivj (t) = yvj (t), Sh (t) = Sh∗0 + xh (t), Ihj (a,t) = yhj (a,t). Then the linearized system of system (6.40) at the disease-free equilibrium E0 can be expressed as ⎧  ∞ 2 dxv (t) ⎪ ∗ ⎪ = − S ⎪ ∑ v0 0 βvj (a)yhj (a,t)da − μv xv (t), ⎪ ⎪ dt ⎪ j=1 ⎪ ⎪ ⎪ ⎪  ∞ j ⎪ dyv (t) ⎪ ∗ ⎪ ⎪ = S βvj (a)yhj (a,t)da − μv yvj (t), ⎪ v0 ⎪ dt 0 ⎪ ⎪ ⎪ ⎪ ⎨ ∗ 2 dxh (t) j Sh0 j (6.53) = − β ∑ h N yv (t) − μh xh (t), ⎪ dt ⎪ h j=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ yhj (a,t) ∂ yhj (a,t) ⎪ ⎪ + = −(μh + rhj (a))yhj (a,t), ⎪ ⎪ ⎪ ∂ a ∂ t ⎪ ⎪ ⎪ ⎪ ⎪ S∗ ⎪ ⎪ ⎩ y j (0,t) = β j h0 yvj (t). h h N h Let

yvj (t) = y¯vj eλ t , yhj (a,t) = y¯hj (a)eλ t ,

(6.54)

where y¯vj and y¯hj (a) are to be determined. Substituting (6.54) into (6.53), we obtain ⎧  ∞ ⎪ ⎪ λ y¯vj = Sv∗ βvj (a)y¯hj (a)da − μv y¯vj , ⎪ 0 ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎨ d y¯ j (a) h = −(λ + μh + rhj (a))y¯hj (a), (6.55) da ⎪ ⎪ ⎪ ∗ ⎪ ⎪ j j Sh0 j ⎪ ⎪ ⎪ ⎩ y¯h (0) = βh Nh y¯v . Solving the differential equation, we obtain y¯hj (a) = y¯hj (0) e−λ a π j (a) = βhj

Sh∗0 Nh

y¯vj e−λ a π j (a).

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

239

Substituting the above expression for y¯hj (a) into the first equation of (6.55), we can obtain  ∞ λ + μv = βhj Sv∗0 βvj (a)e−λ a π j (a)da. (6.56) 0

Now we are able to state the following result. Theorem 6.2. If R0 = max{R01 , R02 } < 1, then the disease-free equilibrium is locally asymptotically stable. If R0 > 1, it is unstable. Proof. We first prove the first result. Let us assume R0 < 1. For ease of notation, set def LHS = λ + μv , (6.57)  def RHS = G1 (λ ) = βhj Sv∗0 0∞ βvj (a)e−λ a π j (a)da. We can easily verify that |LHS| ≥ μv , |RHS| ≤ G1 (ℜλ ) ≤ G1 (0) = βhj Sv∗0

 ∞



βhj Λv ∞ j = β (a)π j (a)da μv 0 v j = R0 μv < |LHS|,

0

βvj (a)π j (a)da

for any λ , ℜλ ≥ 0. This is a contradiction. The contradiction implies that Eq. (6.56) cannot have any roots with nonnegative real parts. Hence, the disease-free equilibrium is locally asymptotically stable. j Next, let us assume max{R01 , R02 } = R00 > 1. We rewrite the characteristic equation (6.56) in the form G2 (λ ) := (λ + μv ) − βh 0 Sv∗0 j

 ∞ 0

βvj0 (a)e−λ a π j0 (a)da = 0.

(6.58)

It is easily verified that G2 (0) = μv − βh 0 Sv∗0 j

 ∞ 0

βvj0 (a)π j0 (a)da

j

= μv (1 − R00 ) < 0, and lim G2 (λ ) = +∞.

λ →+∞

Hence, the characteristic equation (6.58) has a real positive root. Therefore, the disease-free equilibrium E0 is unstable. This concludes the proof.

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6 Vector-Borne Age-Structured Models

We have proved that the disease-free equilibrium is locally stable if R0 < 1. It also follows from Theorem 6.1 that strain j will die out if R0j < 1. Therefore, we have the following result. Theorem 6.3. If R0 = max{R01 , R02 } < 1, then the disease-free equilibrium E0 is globally asymptotically stable.

6.3.4 Existence and Stability of Boundary Equilibria In this section, we mainly investigate the existence and stability of the boundary equilibria. For ease of notation, let

Δj = bj =

βhj Λv , μv2

 ∞ 0

b j (λ ) =

βvj (a)π j (a)da,  ∞ 0

(6.59)

βvj (a)e−λ a π j (a)da.

From Theorem 6.1, it follows that strain j will die out if R0j < 1. Thus in later sections we always assume that R0j > 1 for all j, j = 1, 2. If R0j > 1, straightforward computation yields that system (6.40) has a corresponding single-strain equilibrium E j which is given by E1 = (Sv1∗ , Iv1∗ , 0, Sh1∗ , Ih1∗ (a), 0),

E2 = (Sv2∗ , 0, Iv2∗ , Sh1∗ , 0, Ih2∗ (a)).

The nonzero components Ivj∗ and Ihj∗ are given as follows: Ivj∗ =

μv μh Nh (R0j − 1) βhj (μh b j + μv )

,

Svj∗ =

Λv − μv Ivj∗ , μv

Ihj∗ (a) = Ihj∗ (0)π j (a),

Ihj∗ (0) =

Shj∗ =

Λh j∗ βhj INvh

S j∗ βhj h Ivj∗ . Nh

+ μh

, (6.60)

The results on the local stability of single-strain equilibrium E j0 are summarized below: j

Theorem 6.4. Assume R00 > 1 for a fixed j0 and j

R0j < R00

for

j = j0 .

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

241

Then the single-strain equilibrium E j0 is locally asymptotically stable. If the reproduction number of the other strain j satisfies R0j > R0j0 , then the single-strain equilibrium E j0 is unstable. Proof. Without loss of generality, we assume that j = 2, j0 = 1. Let ∗

∗

Sv (t) = Sv1 + xv (t), Iv1 (t) = Iv1 + y1v (t), Iv2 (t) = y2v (t), ∗

∗

Sh (t) = Sh1 + xh (t), Ih1 (a,t) = Ih1 (a) + y1h (a,t), Ih2 (a,t) = y2h (a,t). Then the linearization system of system (6.40) at the equilibrium E1 can be expressed as ⎧  ∞  ∞ ∗ dxv (t) ⎪ 1∗ 1 1 ⎪ = −S β (a)y (a,t)da − x (t) βv1 (a)Ih1 (a)da v ⎪ v v h ⎪ dt ⎪ 0 0 ⎪ ⎪ ⎪  ∞ ⎪ ⎪ ∗ ⎪ ⎪ −Sv1 βv2 (a)y2h (a,t)da − μv xv (t), ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ ⎪  ∞  ∞ ⎪ ∗ ⎪ dy1v (t) 1∗ 1 1 ⎪ ⎪ = S β (a)y (a,t)da + x (t) βv1 (a)Ih1 (a)da − μv y1v (t), v v v ⎪ h ⎪ dt 0 0 ⎪ ⎪ ⎪ ⎪  ⎪ ∞ ⎪ ∗ dy2v (t) ⎪ ⎪ = Sv1 βv2 (a)y2h (a,t)da − μv y2v (t), ⎪ ⎪ ⎪ dt 0 ⎪ ⎨ 1∗ 1∗ (6.61) ⎪ dxh (t) = −β 1 Sh y1 (t) − β 1 xh (t) I 1∗ − β 2 Sh y2 (t) − μh xh (t), ⎪ v v v h h h ⎪ ⎪ dt Nh Nh Nh ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ∂ y j (a,t) ∂ yhj (a,t) ⎪ ⎪ h + = −(μh + rhj (a))yhj (a,t), ⎪ ⎪ ∂ a ∂ t ⎪ ⎪ ⎪ ⎪ ⎪ 1∗ ⎪ ⎪ xh (t) 1∗ ⎪ 1 1 Sh 1 ⎪ y (0,t) = β yv (t) + βh1 I , ⎪ h h ⎪ N Nh v ⎪ h ⎪ ⎪ ⎪ ⎪ ⎪ 1∗ ⎪ ⎪ 2 2 Sh 2 ⎪ y (0,t) = βh y (t). ⎪ ⎪ Nh v ⎩ h Here we assume that the stability of the equilibria is determined by the eigenvalues. In the chapter of class age structured models we develop the abstract techniques necessary to prove this fact rigorously and we do that there. An approach similar to [123] (see Appendix B in [123]) also show that the linear stability of the system is determined by the eigenvalues of the linearized system (6.61). In order to investigate the linear stability of the linearized system (6.61), we consider exponential solutions (see the case of the disease-free equilibrium) and obtain a linear eigenvalue problem. For the whole system, we only consider the equations for strains 2, and obtain

242

6 Vector-Borne Age-Structured Models

the following eigenvalue problem: ⎧ 2  ∞ dyv (t) ⎪ 1∗ ⎪ = S βv2 (a)y2h (a,t)da − μv y2v (t), ⎪ v ⎪ ⎪ dt 0 ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ∂ y2h (a,t) ∂ y2h (a,t) + = −(μh + rh2 (a))y2h (a,t), ∂ a ∂ t ⎪ ⎪ ⎪ ⎪ ⎪ 1∗ ⎪ ⎪ 2 2 Sh 2 ⎪ y (0,t) = β y (t). ⎪ h h ⎪ Nh v ⎩

(6.62)

By using the same argument to Eq. (6.56), we obtain the following characteristic equation: ∗  ∞ β 2 S1∗ S1 (6.63) λ + μv = h v h βv2 (a)e−λ a π2 (a)da. Nh 0 Notice that Svj∗ and Shj∗ satisfy

βhj Svj∗ Shj∗ Nh

 ∞ 0

βvj (a)π j (a)da = μv ,

(6.64)

for j = 1, 2. It then follows from (6.43) and (6.59) that we have ∗

∗

Sv1 Sh1 =

μv Nh Λv Nh = . βh1 b1 μv R01

(6.65)

Substituting (6.65) into Eq. (6.63), we get the following characteristic equation:

λ + μv = βh2

Λv b2 (λ ), μv R01

(6.66)

where b2 (λ ) is defined in (6.59). First, assume that R02 > R01 , and set def

G 2 (λ ) = (λ + μv ) − βh2

Λv b2 (λ ). μv R01

Straightforward computation yields that G 2 (0) = μv − βh2

R2 Λv b2 = μv (1 − 01 ) < 0. 1 μ v R0 R0

Furthermore, for λ real, G 2 (λ ) is an increasing function of λ such that lim G 2 (λ ) → +∞ as λ → +∞. Hence intermediate value theorem implies that Eq. (6.66) has a unique real positive solution. We conclude that in that case E1 is unstable.

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

243

Next, assume R02 < R01 , and set def

G3 (λ ) = λ + μv ,

def

G4 (λ ) = βh2

Λv b2 (λ ). μv R01

(6.67)

Consider λ with ℜλ ≥ 0. For such λ , following from (6.67), we have |G3 (λ )| ≥ μv , |G4 (λ )| ≤ G4 (ℜλ ) ≤ G4 (0) = =

R02 μv < |G3 (λ )|. R01

1 2 Λv β R01 h μv

 ∞ 0

βv2 (a)π2 (a)da

This gives a contradiction. Hence, Eq. (6.66) has no solutions with positive real part and all eigenvalues of these equations have negative real parts. Therefore, the stability of E1 depends on the eigenvalues of the following system:  ∞  ∞ ⎧ ∗ 1∗ 1 1 ⎪ λ x = −S β (a)y (a)da − x βv1 (a)Ih1 (a)da − μv xv , ⎪ v v v v h ⎪ ⎪ 0 0 ⎪  ∞  ∞ ⎪ ⎪ ∗ ⎪ 1 1∗ 1 1 ⎪ ⎪ λ y = Sv βv (a)yh (a)da + xv βv1 (a)Ih1 (a)da − μv y1v , ⎪ ⎪ v 0 0 ⎪ ⎪ ⎪ ⎨ λ x = −y1 (0) − μ x , h h h h (6.68) ⎪ ⎪ 1 ⎪ dyh (a) ⎪ ⎪ = −(λ + μh + rh1 (a))y1h (a), ⎪ ⎪ ⎪ da ⎪ ⎪ ∗ ⎪ 1∗ ⎪ Sh1 1 ⎪ 1 1 1 Iv ⎪ ⎪ ⎩ yh (0) = βh N yv + βh N xh . h h Solving the differential equation, we have y1h (a) = y1h (0) e−λ a π1 (a). Substituting the above expression for y1h (a) into the first and the second equations of (6.68) yields that  ∞ ⎧ ∗ ∗ ⎪ ( λ + μ + βv1 (a)Ih1 (a)da)xv + Sv1 b1 (λ )y1h (0) = 0, v ⎪ ⎪ ⎪ 0 ⎪  ∞ ⎪ ⎪ ⎪ 1 1∗ 1 1∗ 1 ⎪ ⎪ ⎨ −xv 0 βv (a)Ih (a)da + (λ + μv )yv − Sv b1 (λ )yh (0) = 0,

⎪ (λ + μh )xh + y1h (0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ∗ ⎪ 1∗ ⎪ S1 ⎪ ⎪ −βh1 Iv xh − βh1 h y1v + y1h (0) = 0. ⎩ Nh Nh

(6.69)

244

6 Vector-Borne Age-Structured Models

Direct calculation yields the following characteristic equation: ( λ + μv + =

1∗ ∞ 1 1∗ 1 Iv 0 βv (a)Ih (a)da)(λ + μv )(λ + μh + βh N ) h

∗

∗

βh1 Sh1 Sv1 b1 (λ ) (λ Nh

(6.70)

+ μv )(λ + μh ).

Dividing both sides by (λ + μv )(λ + μh ) gives G5 (λ ) = G6 (λ ),

(6.71)

where G5 (λ ) = G6 (λ ) =

( λ + μv +

1∗ ∞ 1 1∗ 1 Iv 0 βv (a)Ih (a)da)(λ + μv )(λ + μh + βh N )

(λ + μv )(λ + μh ) ∗ ∗ βh1 Sh1 Sv1

Nh

b1 (λ ) =

∗ ∗ βh1 Sh1 Sv1

 ∞

Nh

0

h

, (6.72)

βv1 (a)e−λ a π1 (a)da.

If λ is a root with ℜλ ≥ 0, it follows from Eq. (6.72) that |G5 (λ )| > |λ + μv | ≥ μv .

(6.73)

From (6.64), we have ∗

β 1 S1 S1 |G6 (λ )| ≤ |G6 (ℜλ )| ≤ G6 (0) = h h v Nh

∗

 ∞ 0

βv1 (a)π1 (a)da = μv < |G5 (λ )|.

(6.74) This leads to a contradiction. The contradiction implies that (6.71) has no roots such that ℜλ ≥ 0. Thus, the characteristic equation for strain one has only roots with negative real parts. Thus, the single-strain equilibrium E1 is locally asymptotically stable if R01 > 1 and R02 < R01 . This concludes the proof.

6.3.5 Preliminary Results and Uniform Persistence In the previous section, we proved that if the reproduction number is less than one, all strains are eliminated and the disease dies out. Our next step is to show that the competitive exclusion principle holds for system (6.40). In the later sections, we always assume that R0 > 1. Without loss of generality, we assume that R01 = max{R01 , R02 } > 1. In the following we will show that strain 1 persists, while strain 2 dies out if R02 /R01 < b2 /b1 < 1, where b j is the force of infection of humans at equilibrium. Hence, under suitable condition strain 1 eliminates the other strain and the competitive exclusion principle will be established for system (6.40).

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

245

Mathematically speaking, establishing the competitive exclusion principle means establishing the global stability of the single-strain equilibrium E1 . From Theorem 6.4 we know that if R02 /R01 < 1, the equilibrium E1 is locally asymptotically stable. In the following we only need to show that E1 is a global attractor. The method used here to show this result is similar to the one used in [18, 118, 120, 189]. Set f (x) = x − 1 − ln x. It is easy to check that f (x) ≥ 0 for all x > 0 and f (x) reaches its global minimum value f (1) = 0 when x = 1. Next, let us define the following Lyapunov function: U(t) = U1 (t) +U21 (t) +U22 (t) +U3 (t) +U41 (t) +U42 (t),

(6.75)

where  1 ∗ Sv Iv Iv1 Iv2 1 2 (t) = f (t) = , , U , U ∗ ∗ ∗ ∗ 2 2 Sv1 Sv1 b1 Iv1 Sv1 b1   1  ∞ Ih (a,t) ∗ Sh 1 1 1∗ U3 (t) = Sh1 f (t) = q (a)I (a) f ,U da, 1 ∗ ∗ 4 h Sh1 R01 0 Ih1 (a) U1 (t) =

U42 (t) =

1 f b1

1 R02



 ∞ 0

q2 (a)Ih2 (a,t)da

and q j (a) = Δ j

 ∞

βvj (s)e−

s

j a ( μh +rh (σ ))d σ

ds.

(6.77)

qj (a) = −βvj (a)Δ j + (μh + rhj (a))q j (a).

(6.78)

a

Direct computation gives and

(6.76)

q j (0) = R0j ,

The main difficulty with the Lyapunov function U above is to show that the Lyapunov function U is well defined. Thus in the following we first show that strain one persists both in the hosts and in the vectors as strain two dies out. Let 

  ∞  1 Xˆ = ψ1 ∈ L+ (0, ∞)∃s ≥ 0 : βv1 (a + s)ψ1 (a)da > 0 0

and define

2

X0 = R+ × ∏ R+ × R+ × Xˆ × L1 (0, ∞), j=1

Ω0 = Ω ∩ X0 . Note that Ω0 is forward invariant because (6.41) show that Ω is forward invariant. To see X0 is forward invariant, we firstly demonstrate that Xˆ is forward invariant. Let us assume that the inequality holds for the initial condition. The inequality says

246

6 Vector-Borne Age-Structured Models

that the support of βv1 (a) will intersect the support of the initial condition if it is translated s units to the right. Since the support of the initial condition only moves to the right, the intersection will take place for any other time if that happens for the initial time. Therefore, Ω0 is forward invariant. Now let us recall two important definitions. Definition 6.5. Strain one is called uniformly weakly persistence if there exists some γ > 0 independent of the initial conditions such that  ∞

whenever

lim sup Iv1 (t) > γ

whenever

lim sup 0

t→∞

 ∞

Ih1 (a,t)da > γ

0

ψ1 (a)da > 0,

and t→∞

Iv10 > 0,

for all solutions of system (6.40). One of the important implications of uniform weak persistence of the disease is that the disease-free equilibrium is unstable. Definition 6.6. Strain one is uniformly strongly persistence if there exists some γ > 0 independent of the initial conditions such that  ∞

lim inf t→∞

0

 ∞

Ih1 (a,t)da > γ

whenever

lim inf Iv1 (t) > γ

whenever

0

ψ1 (a)da > 0,

and t→∞

Iv10 > 0,

for all solutions of model (6.40). It is evident from the definitions that, if the disease is uniformly strongly persistent, it is also uniformly weakly persistent. Now we are able to state the main results in this section. Theorem 6.5. Assume R01 > 1 and R02 < R01 . Furthermore, assume that stain 2 will die out, i.e., lim sup Iv2 (t) = 0, and t→+∞

 ∞

lim sup t→+∞

0

Ih2 (a,t)da = 0.

Then strain 1 is uniformly weakly persistent for the initial conditions that belong to Ω0 , i.e., there exists γ > 0 such that lim sup t→+∞

βh1 Iv1 (t) ≥ γ, Nh

 ∞

lim sup t→+∞

0

βv1 (a)Ih1 (a,t)da ≥ γ .

Proof. We argue by contradiction. Assume that strain 1 also dies out. For any ε > 0 and every initial condition in Ω0 such that

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

lim sup t→+∞

 ∞

βh1 Iv1 (t) < ε, Nh

lim sup 0

t→+∞

247

βv1 (a)Ih1 (a,t)da < ε .

Following that there exist T > 0 such that for all t > T we have

βhj Ivj (t) < ε, Nh

 ∞ 0

βvj (a)Ihj (a,t)da < ε ,

j = 1, 2.

We may assume that the above inequality holds for all t ≥ 0 by shifting the dynamical system. From the first equation in (6.40) we have Sv (t) ≥ Λv − 2ε Sv − μv Sv ,

Sh (t) ≥ Λh − 2ε Sh − μh Sh .

Exploiting the comparison principle, we have lim sup Sv (t) ≥ lim inf Sv (t) ≥ t→+∞

t→+∞

Λv , 2ε + μv

lim sup Sh (t) ≥ lim inf Sh (t) ≥ t→+∞

t→+∞

Λh . 2ε + μh

Since B1 (t) = Ih1 (0,t), it then follows from system (6.40) that

Λv dIv1 (t) ≥ dt 2ε + μv

 ∞ 0

βv1 (a)Ih1 (a,t)da − μv Iv1 (t).

(6.79)

By using the equations in (6.44), we can easily obtain the following inequalities on B1 (t) and Iv1 (t): ⎧ Λh 1 1 ⎪ ⎪ ⎨ B1 (t) ≥ βh (2ε + μ )N Iv (t), h h (6.80)  t ⎪ dI 1 (t) Λv ⎪ ⎩ v ≥ βv1 (a)B1 (t − a)π1 (a)da − μv Iv1 (t). dt 2ε + μv 0 Let us take the Laplace transform of both sides of inequalities (6.80). Since all functions above are bounded, the Laplace transforms of the functions exist for λ > 0. Denote the Laplace transforms of the functions B1 (t) and Iv1 (t) by Bˆ 1 (λ ) and Iˆv1 (λ ), respectively. Furthermore, set ˆ λ) = K(

 ∞ 0

βv1 (a)π1 (a)e−λ a da.

(6.81)

Using the convolution property of the Laplace transform, we obtain the following inequalities for Bˆ 1 (λ ) and Iˆv1 (λ ): ⎧ Λh ⎪ 1 ⎪ Iˆ1 (λ ), ⎨ Bˆ 1 (λ ) ≥ βh (2ε + μh )Nh v (6.82) Λv ˆ ⎪ ⎪ ⎩ λ Iˆv1 (λ ) − Iv1 (0) ≥ K(λ )Bˆ 1 (λ ) − μv Iˆv1 (λ ). 2ε + μv

248

6 Vector-Borne Age-Structured Models

Eliminating Iˆv1 (λ ) yields ˆ λ) βh1ΛvΛh K( βh1Λh Bˆ 1 (λ ) + I 1 (0). (2ε + μv )(2ε + μh )(λ + μv )Nh (2ε + μh )(λ + μv )Nh v (6.83) This is impossible since ˆ βh1Λv K(0) := R01 > 1, 2 μv Bˆ 1 (λ ) ≥

we can choose ε and λ small enough such that ˆ λ) βh1ΛvΛh K( > 1. (2ε + μv )(2ε + μh )(λ + μv )Nh The contradiction implies that there exists γ > 0 such that for any initial condition in Ω0 , we have lim sup t→+∞

βh1 Iv1 (t) ≥ γ, Nh

 ∞

lim sup t→+∞

0

βv1 (a)Ih1 (a,t)da ≥ γ .

In addition, the equation for Iv1 can be rewritten in the form

Λv γ dIv1 ≥ − μv Iv1 , dt 2γ + μv which implies a lower bound for Iv1 . This concludes the proof. Next, we claim that system (6.40) has a global compact attractor T. Firstly, define the semiflow Ψ : [0, ∞) × Ω0 → Ω0 generated by the solutions of system (6.40)   1 2 1 2 1 2 Ψ t; Sv0 , Iv0 , Iv0 , Sh0 , ψ1 (·), ψ2 (·) = Sv (t), Iv (t), Iv (t), Sh (t), Ih (a,t), Ih (a,t) . Definition 6.7. A set T in Ω0 is called a global compact attractor for Ψ if T is a maximal compact invariant set and for all open sets U containing T and all bounded sets B of Ω0 there exists some T > 0 such that Ψ (t, B) ⊆ U holds for t > T . Theorem 6.6. Under the hypothesis of Theorem 6.5, there exists T, a compact subset of Ω0 , which is a global attractor for the semiflow Ψ on Ω0 . Moreover, we have

Ψ (t, x0 ) ⊆ T for every

x0 ∈ T, ∀t ≥ 0.

Proof. We split the solution semiflow into two components. For an initial condition x0 ∈ Ω0 , let Ψ (t, x0 ) = Ψˆ (t, x0 ) + Ψ˜ (t, x0 ), where   1 2 1 2 ˆ ˆ ˆ Ψ t; Sv0 , Iv0 , Iv0 , Sh0 , ψ1 (·), ψ2 (·) = 0, 0, 0, 0, Ih (a,t), Ih (a,t) , (6.84)

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

249

  Ψ˜ t; Sv0 , Iv10 , Iv20 , Sh0 , ψ1 (·), ψ2 (·) = Sv (t), Iv1 (t), Iv2 (t), Sh (t), I˜h1 (a,t), I˜h2 (a,t) , (6.85) and Ihj (a,t) = Iˆhj (a,t) + I˜hj (a,t) for j = 1, 2. Iˆhj (a,t) and I˜hj (a,t) are the solutions of the following equations: ⎧ j j ⎪ ⎪ ∂ Iˆh ∂ Iˆh ⎪ + = −(μh + rhj (a))Iˆhj (a,t), ⎪ ⎪ ⎨ ∂t ∂a (6.86) Iˆhj (0,t) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ˆj Ih (a, 0) = ψ j (a), and

⎧ j ∂ I˜h ∂ I˜hj ⎪ ⎪ ⎪ + = −(μh + rhj (a))I˜hj (a,t), ⎪ ⎪ ∂a ⎨ ∂t Sh I˜hj (0,t) = βhj Ivj , ⎪ ⎪ ⎪ N h ⎪ ⎪ ⎩ ˜j I (a, 0) = 0.

(6.87)

h

We can easily see that system (6.86) is decoupled from the remaining equations. Using the formula (6.44) to integrate along the characteristic lines, we obtain ⎧ 0, t > a, ⎪ ⎨ ˆI j (a,t) = (6.88) h π (a) ⎪ ⎩ ψ j (a − t) j , t < a. π j (a − t) Integrating Iˆhj with respect to a yields  ∞ t

ψ j (a − t)

π j (a) da = π j (a − t)

 ∞ 0

ψ j (a)

π j (t + a) da ≤ e−μh t π j (a)

 ∞ 0

ψ j (a)da → 0

as t → ∞. This implies that Ψˆ (t, x0 ) → 0 as t → ∞ uniformly for every x0 ∈ B ⊆ Ω0 , where B is a ball of a given radius. In the following we need to show Ψ˜ (t, x) is completely continuous. We fix t and let x0 ∈ Ω0 . Note that Ω0 is bounded. We have to show that the family of functions defined by  0 1 2 1 2 ˜ ˜ ˜ Ψ (t, x ) = Sv (t), Iv (t), Iv (t), Sh (t), Ih (a,t), Ih (a,t) is a compact family of functions for that fixed t, which are obtained by taking different initial conditions in Ω0 . The family {Ψ˜ (t, x0 )|x0 ∈ Ω0 ,t − fixed} ⊆ Ω0 ,

250

6 Vector-Borne Age-Structured Models

and, therefore, it is bounded. Thus, we have established the boundedness of the set. To show that Ψ˜ (t, x) is precompact, we first see the third condition of limt→∞ t∞ I˜hj (a,t)da = 0 in the Frechet-Kolmogorov Theorem of [193]. The third condition in [193] is trivially satisfied since I˜hj (a,t) = 0 for a > t. To use the second condition of the Frechet-Kolmogorov Theorem in [193], we must bound by two constants the L1 -norms of ∂ Ihj /∂ a. Notice that ⎧ ˜ t > a, ⎪ ⎨ B j (t − a)π j (a), j ˜I (a,t) = (6.89) h t < a, ⎪ ⎩ 0, where

Sh (t) j B˜ j (t) = βhj I (t). Nh v

(6.90)

B˜ j (t) is bounded because of the boundedness of Sh and Ivj . Hence, the B˜ j (t) satisfies B˜ j (t) ≤ k1 . Next, we differentiate (6.89) with respect to a:   |(B˜ (t − a)) |π (a) + B˜ (t − a)|(π (a)) |,  j j j j j  ∂ I˜h (a,t)     ∂a  ≤ 0,

t > a, t < a.

We see that |(B˜ j (t − a)) | is bounded. Differentiating (6.90), we obtain (B˜ j (t)) =

 βhj  j j  S (t)I (t) + Sh (t)(Iv (t)) . Nh h v

(6.91)

Taking an absolute value and bounding all terms, we can rewrite the above equality as the following inequality: |(B˜ j (t)) | ≤ k2 . Putting all these bounds together, we have

∂a I˜hj ≤ k2

 ∞ 0

π j (a)da + k1 (μh + r¯h )

 ∞ 0

π j (a)da < b,

where r¯h = supa, j {rhj (a)}. To complete the proof, we notice that  ∞ 0

|I˜hj (a + h,t) − I˜hj (a,t)|da ≤ ∂a I˜hj |h| ≤ b|h|.

Thus, the integral can be made arbitrary small uniformly in the family of functions. That establishes the second condition of the Frechet-Kolmogorov Theorem. We conclude that the family is asymptotically smooth.

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

251

Equation (6.41) means that the semigroup Ψ (t) is point dissipative and the forward orbit of boundedness sets is bounded in Ω0 . Thus, we prove Theorem 6.6 in accordance with Lemma 3.1.3 and Theorem 3.4.6 in [80]. Now we have all components to establish the uniform strong persistence. The next proposition states the uniform strong persistence of Iv1 and Ih1 . Theorem 6.7. Under the hypothesis of Theorem 6.5 strain one is uniformly strongly persistent for all initial conditions that belong to Ω0 , that is, there exists γ > 0 such that  ∞ β 1 I 1 (t) ≥ γ , lim inf βv1 (a)Ih1 (a,t)da ≥ γ . lim inf h v t→+∞ t→+∞ 0 Nh Proof. We apply Theorem 2.6 in [167]. We consider the solution semiflow Ψ on Ω0 . Let us define three functionals ρl : Ω0 → R+ , l = 1, 2 as follows:

ρ1 (Ψ (t, x0 )) = ρ2 (Ψ (t, x0 )) =

βh1 Iv1 (t) , Nh

 ∞ 0

βv1 (a)I˜h1 (a,t)da.

Theorem 6.5 implies that the semiflow is uniformly weakly ρ -persistent. Theorem 6.6 shows that the solution semiflow has a global compact attractor T. Total orbits are solutions to the system (6.40) defined for all times t ∈ R. Since the solution semiflow is nonnegative, we have

βh1 Iv1 (t) βh1 1 = I (s)e−μv (t−s) , Nh Nh v  ∞ 0

βv1 (a)I˜h1 (a,t)da =

 t 0

= k1 ≥ k2 =

βv1 (a)B˜ 1 (t − a)π1 (a)da ≥ k1

 t 0

 t 0

B˜ 1 (a)da = k1 Iv1 (a)da = k2

 t 0

 t 0

βh1

 t 0

B˜ 1 (t − a)da

Sh (a) 1 I (a)da Nh v

Iv1 (s)e−μv (a−s) da

k2 Iv1 (s) μv s e (1 − e−μv t ), μv 

β 1 I 1 (t)

for any s and any t > s. Therefore, hNv > 0, 0∞ βv1 (a)I˜h1 (a,t)da > 0 for all t > s h provided I˜v1 (s) > 0. Theorem 2.6 in [167] now implies that the semiflow is uniformly strongly ρ -persistent. Hence, there exists γ such that lim inf t→+∞

βh1 Iv1 (t) ≥ γ, Nh

 ∞

lim inf t→+∞

0

βv1 (a)Ih1 (a,t)da ≥ γ .

252

6 Vector-Borne Age-Structured Models

According to Theorem 6.7, we obtain that for all initial conditions that belong to Ω0 , strain 1 persists. Furthermore we had verified that the solutions of (6.40) with nonnegative initial conditions belong to the positive cone for all t ≥ 0. All the solutions are in a positively invariant set. Therefore, we can obtain the following Theorem 6.8 from Theorem 6.7. Theorem 6.8. Under the hypothesis of Theorem 6.5, there exists constants ϑ > 0 and M > 0 such that

ϑ ≤ Sv (t) ≤ M, and

ϑ≤

βh1 Iv1 (t) ≤ M, ϑ ≤ Nh

ϑ ≤ Sh (t) ≤ M, ∀t ∈ R,  ∞ 0

βv1 (a)Ih1 (a,t)da ≤ M, ∀t ∈ R.

for each orbit (Sv (t), Iv1 (t), Iv2 (t), Sh (t), Ih1 (a,t), Ih2 (a,t)) of Ψ in T.

6.3.6 Principle of Competitive Exclusion In this section we mainly state the main result of the paper. Theorem 6.9. Assume R01 > 1, R02 /R01 < b2 /b1 < 1. Then the equilibrium E1 is globally asymptotically stable. Proof. From Theorem 6.4 we know that the endemic equilibrium E1 is locally asymptotically stable. In the following we only need to show that the endemic equilibrium E1 is global attractor. From Theorem 6.6 there exists an invariant compact set T which is global attractor of system (6.40). Furthermore, it follows from Theorem 6.8 that there exist ε1 > 0 and M1 > 0 such that

ε1 ≤

Iv1 ∗ ≤ M1 , Iv1

ε1 ≤

Ih1 (a,t) ≤ M1 ∗ Ih1 (a)

for any solution in Ψ . This makes the Lyapunov function well-defined in (6.75). Differentiating U(t) along the solution of system (6.40) yields that  ∗   ∞ ∗ ∗ ∗ Sv1 1 dU1 (t) 1− Sv1 βv1 (a)Ih1 (a)da + μv Sv1 = 1∗ dt Sv b1 Sv 0 −Sv

 ∞ 0

βv1 (a)Ih1 (a,t)da − μv Sv − Sv

 ∞ 0



βv2 (a)Ih2 (a,t)da

 ∗ ∗  Sv Ih1 (a,t) Ih1 (a,t) 1 ∞ 1 Sv1 μv (Sv − Sv1 )2 1∗ + β (a)I (a) 1 − − + da =− ∗ ∗ ∗ ∗ h Sv1 Sv b1 b1 0 v Sv Sv1 Ih1 (a) Ih1 (a)  ∞  ∞ ∗ 1 Sv βv2 (a)Ih2 (a,t)da − Sv1 βv2 (a)Ih2 (a,t)da ; − 1∗ Sv b1 0 0 (6.92)

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

253

∗

I1 ∗ ∗ 1 − Iv1   ∞ Sv1 0∞ βv1 (a)Ih1 (a)da 1 dU21 (t) 1 1 v = 1∗ βv (a)Ih (a,t)da − Iv Sv ∗ dt Sv b1 Iv1 0   ∗  ∞ Sv I 1 (a,t) Iv1 ∗ ∗ 1 I1 (6.93) = 1∗ βv1 (a)Ih1 (a) 1∗h 1∗ 1 − v1 Sv1 − ∗ da Sv b1 Iv Sv Ih (a) Iv1 0  1 ∗  Sv I (a,t) Iv1 Sv I 1 (a,t)I 1 ∗ 1 ∞ 1 = βv (a)Ih1 (a) 1∗h 1∗ − 1∗ − 1∗h 1∗ v1 + 1 da; b1 0 Sv Ih (a) Iv Sv Ih (a)Iv   ∞ dU22 (t) 1 (6.94) = 1∗ βv2 (a)Ih2 (a,t)da − μv Iv2 ; Sv dt Sv b1 0  ∗  S1 ∗ ∗ dU3 (t) = 1− h Ih1 (0) + μh Sh1 − Ih1 (0,t) − μh Sh − βh2 Sh Iv2 dt Sh  ∗ ∗ ∗ μh (Sh − Sh1 )2 Sh1 1∗ Sh1 1 1∗ 1 =− + Ih (0) − Ih (0,t) − I (0) + I (0,t) Sh Sh h Sh h  ∗ S1 − Ih2 (0,t) − βh2 h Iv2 , Nh (6.95) and

Ih1 (a,t) 1 ∂ Ih1 (a,t) da ∗ ∗ 1 1 ∂t Ih (a) Ih (a) 0  1  1  ∞ I (a,t) ∂ Ih (a,t) ∗ 1 1 1 1 =− 1 + ( q1 (a)Ih1 (a) f  h1∗ μ + r (a))I (a,t) da ∗ h h h ∂a R0 0 Ih (a) Ih1 (a)  1  ∞ I (a,t) ∗ 1 =− 1 q1 (τ )Ih1 (a)d f h1∗ R0 0 Ih (a)  1      ∞  1 I (a,t) ∞ Ih (a,t) ∗ 1 1∗ = − 1 q1 (a)Ih1 (a) f h1∗ − f (a)I (a) d q 1 ∗ h R0 Ih (a) 0 Ih1 (a) 0  1  1    ∞ I (0,t) I (a,t) ∗ ∗ 1 = 1 q1 (0)Ih1 (0) f h1∗ βv1 (a)Ih1 (a) f h1∗ − Δ1 da R0 Ih (0) Ih (a) 0   1  I (a,t) I 1 (0,t) ∗ ∗ ∗ 1 ∞ 1 = Ih1 (0,t) − Ih1 (0) − Ih1 (0) ln h1∗ βv (a)Ih1 (a) f h1∗ − da. b1 0 Ih (0) Ih (a) (6.96)

dU41 (t) 1 = 1 dt R0

 ∞

∗

q1 (a)Ih1 (a) f 



The above equality follows from (6.59) and the fact ∗

∗

q1 (a)Ih1 (a) + q1 (a)(Ih1 (a))   ∗ ∗ 1 1 = − βv (a)Δ1 + (μh + rh (a))q1 (a) Ih1 (a) − q1 (a)(μh + rh1 (a))Ih1 (a) ∗

= −βv1 (a)Δ1 Ih1 (a).

254

6 Vector-Borne Age-Structured Models

Noting (6.78), we differentiate the last term (6.75) in with respect to t, and have  2   ∞ ∂ Ih (a,t) dU42 (t) 1 =− 2 + (μh + rh2 (a))Ih2 (a,t) da q2 (a) dt ∂a R0 0  q2 (0)Ih2 (0,t) Δ2 0∞ βv2 (a)Ih2 (a,t)da (6.97) = − R02 R02  1 ∞ 2 = Ih2 (0,t) − β (a)Ih2 (a,t)da. b2 0 v Adding all five components of the Lyapunov function, we have U  (t) = U 1 (t) +U 2 (t), where ∗

μv (Sv − Sv1 )2 ∗ Sv1 Sv b1  ∗  Sv I 1 (a,t) Ih1 (a,t) ∗ 1 ∞ 1 S1 + 1∗ da + βv (a)Ih1 (a) 1 − v − 1∗ h 1∗ b1 0 Sv Sv Ih (a) Ih (a)  1 ∗  Sv I (a,t) Sv I 1 (a,t)I 1 ∗ I1 1 ∞ 1 − 1v∗ − 1∗h 1∗ v1 + 1 da + βv (a)Ih1 (a) 1∗ h 1∗ b1 0 Sv Ih (a) Iv Sv Ih (a)Iv  ∗ ∗ 1∗ μh (Sh − Sh1 )2 S S1 ∗ ∗ + Ih1 (0) − Ih1 (0,t) − h Ih1 (0) + h Ih1 (0,t) − Sh Sh Sh   ∞ 1 (0,t) I I 1 (a,t) ∗ ∗ ∗ 1 − )da, βv1 (a)Ih1 (a) f ( h1∗ + Ih1 (0,t) − Ih1 (0) − Ih1 (0) ln h1∗ b1 0 Ih (0) Ih (a) (6.98)

U 1 (t) = −

and

 ∞  ∞ 1 2 2 1∗ 2 2 β (a)I (a,t)da − S β (a)I (a,t)da S v ∗ v v v h h Sv1 b1 0 0  ∞ 1 + 1∗ βv2 (a)Ih2 (a,t)da − μv Iv2 Sv Sv b1 0    ∗ 1 ∞ 2 Sh1 2 2 2 2 2 − Ih (0,t) − βh N Iv + Ih (0,t) − β (a)Ih (a,t)da . h b2 0 v

U 2 (t) = −

(6.99)

Canceling terms, (6.98) can be simplified as ∗

∗

μv (Sv − Sv1 )2 μh (Sh − Sh1 )2 − ∗ Sv1 Sv b1 Sh  ∗ ∗  Sv I 1 (a,t)I 1 I 1 (a,t) ∗ 1 ∞ 1 S1 I1 + βv (a)Ih1 (a) 3 − v − 1v∗ − 1∗h 1∗ v1 + ln h1∗ da b1 0 Sv Iv Sv Ih (a)Iv Ih (a)  ∗ ∗ S1 S1 I 1 (0,t) I 1 (0,t) ∗ +Ih1 (0) − h + h h1∗ − ln h1∗ Sh Sh Ih (0) Ih (0)

U 1 (t) = −

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

255

∗

∗

μv (Sv − Sv1 )2 μh (Sh − Sh1 )2 − ∗ Sv1 Sv b1 Sh   1∗  1  1 ∗   ∞ Sv Ih (a,t)Iv1 Sv Iv 1 1 1∗ − β (a)Ih (a) f + f 1∗ + f da ∗ ∗ b1 0 v Sv Iv Sv1 Ih1 (a)Iv1   1∗  1∗ 1  Sh Sh Ih (0,t) ∗ +Ih1 (0) − f +f . ∗ Sh Sh Ih1 (0)

=−

∗

∗

(6.100)

∗

Noting that Ih1 (0,t) = βh1 Sh Iv1 , Ih1 (0) = βh1 Sh1 Iv1 , we get ∗

∗

Sh1 Ih1 (0,t) S1 β 1 Sh I 1 I1 = h 1 h 1∗ 1v∗ = 1v∗ . ∗ 1 Iv Sh Ih (0) Sh βh Sh Iv

(6.101)

Furthermore, from (6.60) and (6.77) we have  1  1   ∗ Iv I 1 ∞ 1 I1 1 ∞ 1 1∗ 1∗ βv (a)Ih (a) f 1∗ da = βv (a)Ih (a)da f 1v∗ = Ih1 (0) f ( 1v∗ ). b1 0 Iv b1 0 Iv Iv (6.102) Finally, simplifying (6.100) with (6.101) and (6.102), we obtain  1∗ ∗ ∗ Sh μv (Sv − Sv1 )2 μh (Sh − Sh1 )2 1∗ − − I (0) f ∗ h 1 Sv Sv b1 Sh Sh   1∗  1 ∗   ∞ Sv Ih (a,t)Iv1 ∗ S 1 − βv1 (a)Ih1 (a) f v + f da. ∗ ∗ b1 0 Sv Sv1 Ih1 (a)Iv1

U 1 (t) = −

(6.103)

Canceling terms, (6.99) can be simplified as  U 2 (t) =

1 1 − b1 b2



∞ 0



βv2 (a)Ih2 (a,t)da +

∗ βh2 Sh1 μv − 1∗ I2. Nh Sv b1 v

(6.104)

Simplifying (6.104) with (6.43) and (6.65), we get  U 2 (t) =

1 1 − b1 b2

 0

∞

βv2 (a)Ih2 (a,t)da +

βh2 Sh1 Nh

∗

 1−

R01 b2 2 I . R02 b1 v

(6.105)

Hence, by using (6.103) and (6.105) we obtain  1∗ ∗ ∗ Sh μv (Sv − Sv1 )2 μh (Sh − Sh1 )2 1∗ − − Ih (0) f U (t) = − ∗ 1 Sv Sv b1 Sh Sh   1∗  1 ∗   ∞ Sv Ih (a,t)Iv1 ∗ S 1 − βv1 (a)Ih1 (a) f v + f da ∗ ∗ b1 0 Sv Sv1 Ih1 (a)Iv1   ∞ ∗ β 2 S1 R 1 b2 2 1 1 + − βv2 (a)Ih2 (a,t)da + h h 1 − 02 I . b1 b2 Nh R0 b1 v 0 

(6.106)

256

6 Vector-Borne Age-Structured Models

Since f (x) ≥ 0 for x > 0, R02 /R01 < b2 /b1 < 1 we have U  ≤ 0. Define, 

  1 2 1 2 Θ2 = (Sv , Iv , Iv , Sh , Ih , Ih ) ∈ Ω0 U  (t) = 0 . We want to show that the largest invariant set in Θ2 is the singleton E1 . First, we ∗ ∗ notice that equality in (6.106) occurs if and only if Sv (t) = Sv1 , Sh (t) = Sh1 , Iv2 = 0, and, therefore Ih2 (a,t) = 0. ∗ Ih1 (a,t)Iv1 = 1. (6.107) ∗ Ih1 (a)Iv1 Thus, we obtain

Ih1 (a,t) Iv1 (t) = 1∗ . ∗ Iv Ih1 (a)

It is obvious that the left term term have

Iv1 (t) ∗ Iv1

Ih1 (a,t) ∗ Ih1 (a)

(6.108)

of (6.108) is a function with a,t, while the right ∗

is a function with t. So we can assume that Ih1 (a,t) = Ih1 (a)g(t). Thus we ∗

Iv1 = Iv1 g(t).

(6.109)

It follows from (6.40) we can also obtain  Iv1 (t)

= Sv =

 ∞ 0

βv1 (a)Ih1 (a,t)da − μv Iv1

∗ g(t)Sv1

 ∞ 0

=

∗ Sv1

∗ βv1 (a)Ih1 (a)da − μv Iv1

 ∞ 0

∗

βv1 (a)Ih1 (a)g(t)da − μv Iv1 ∗

= g(t)μv Iv1 − μv Iv1

∗

= μv (Iv1 g(t) − Iv1 ) = 0. (6.110) Therefore, we can get

∗

Iv1 = Iv1 . Subsequently, it follows from (6.108) we have ∗

Ih1 (a,t) = Ih1 (a). At last we conclude that the largest invariant set in Θ2 is the singleton E1 . Since Ψ (t)Ω0+ ⊂ Ω0+ , the global attractor, T, is actually contained in Ω0+ . Furthermore, the interior global attractor T is invariant. By using the above result, we show that the compact global attractor T = {E1 }. This completes the proof of Theorem 6.9. In this chapter, we formulate a two-strain partial differential equation (PDE) model describing the transmission dynamics of a vector-borne disease that incorporates infection age of the infectious hosts. The formulas for the reproduction number R0j of strain j, j = 1, 2 are obtained from the biological meanings of models. We define the basic reproduction number of the disease as the maximum of the reproduction numbers of each strain. We show that if R0 < 1, the disease-free equilibrium is locally and globally asymptotically stable. That means the disease dies out and

6.3 A Two-Strain Vector-Host Epidemic Model with Age of Infection

257

the number of infected with each strain goes to zero. If R0 > 1, without loss of generality, assuming R01 = max{R01 , R02 } > 1, we show that the single-strain equilibrium E1 corresponding to strain one exists. The single-strain equilibrium E1 is locally asymptotically stable when R01 > 1 and R02 < R01 . We then extend the competitive exclusion result established by Bremermann and Thieme in [19], who using a multiple-strain ODE model derived that if multiple strains circulate in the population only the strain with the largest reproduction number persists, to the case of age-structured vector-borne diseases. The proof of the competitive exclusion principle is based on the proof of the global stability of the single-strain equilibrium E1 . We approach the result by using a Lyapunov function under a stronger condition that R02 b2 < < 1. 1 b1 R0

(6.111)

Our results do not include the case of max{R01 , R02 } = R01 = R02 > 1. Simulation have suggested (see [149]) that if there is no mutation between two strains and if the basic reproduction numbers corresponding to the two strains are the same, then for the two-strain epidemic model there exist many coexistence equilibria. We surmise that coexistence of the two strains may occur in case of model (6.39) and competitive exclusion in this case is impossible. From the expression (6.43) of the basic reproduction number R0j corresponding to strain j and the inequality R02 /R01 < b2 /b1 , it follows that r2 < r1 , where rj =

βhj , μv

for j = 1, 2.

r j represents the transmission rate of an infectious vector with strain j during its entire infectious period. The condition (6.111) implies that the following three inequalities hold at the same time: R02 < R01 ,

r2 < r1 ,

b2 < b1 .

Recall that b j denotes force of infection of humans to susceptible vectors. Then the condition (6.111) for the occurrence of competition exclusion of strain 1 means that the basic reproduction number corresponding to strain 1, the basic human and vector reproduction numbers are bigger than the corresponding reproduction numbers for strain 2.

Chapter 7

Metapopulation and Multigroup Age-Structured Models

7.1 Metapopulation Age-Structure Model Richard Levins [111] used the term “metapopulation” to describe a population of populations. Metapopulations naturally occur in fragmented habitats where each component of the habitat is occupied by one population and the populations are connected by migration. Levins proposed a very simple equation model to investigate the dynamics of the metapopulation in a temporally varying environment. Later, Hanski and Gilpin [67] gave more details from ecological view of point. As shown in Arino [8], the simplest style of metapopulation models originated from the discrete cellular automata. Actually, metapopulation models are related to patch models, every patch generates a node, and the evolution of every patch node is affected by their neighbor nodes. Therefore, a metapopulation is a graph with patches (nodes) containing a population possibly structured into subpopulations and linked by migration as arcs. For convenience, we give some definitions: Assume that a set of patches is denoted with M . Each patch X ∈ M contains a certain number of species belonging to a common set E of species. A multi-graph is defined as G(M , A ), where A is the set of arcs. Define a binary relation Re (X,Y ) for e ∈ E and X,Y ∈ M which takes values between 0 and 1. Definition 7.1. Direct Access We say that spices e ∈ E has a direct access, if Re (X,Y ) = 1, that is, if for e ∈ E and X,Y ∈ M , there exists an arc A ∈ A from X to Y . If A does not exist, then Re (X,Y ) = 0. Indirect Access We say that species e ∈ E has indirect access in E , patches X,Y ∈ M if there exists a patch X1 ∈ M such that Re (X, X1 ) = 1, Re (X1 ,Y ) = 1 but Re (X,Y ) = 0. Access We say species e ∈ E has access from X to Y if it has a direct or indirect access. If we number the patches, then the graph G(M , A ) becomes a simple labeled graph. Let m be the number of patches in M . Examples of simple labeled graphs are given in Fig.7.1. Each such graph corresponds a matrix B, called adjacency matrix, © Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1_7

259

260

7 Metapopulation and Multigroup Age-Structured Models 4 2 3.5 3 2.5 1

2 1.5

3 1

5

0.5 4 0

0

0.5

1

1.5

2

2.5

3

3.5

4

Fig. 7.1 A simple labeled graph

whose element is assume to be 1 in position bi j according to whether node i and node j are adjacent or not. For instance, ⎛ ⎞ 00010 ⎜0 0 0 0 1⎟ ⎜ ⎟ ⎟ B=⎜ ⎜0 0 0 1 1⎟ ⎝1 0 1 0 0⎠ 01100 For a simple graph with no self-loops, the adjacency matrix must have 0 on the diagonal. Irreducibility of a Matrix For an undirected graph the adjacency B is called irreducible if for all i, j = 1, · · · , m, there exists a k such that bkij , the (i, j)-entry in Bk = B × B × · · · × B, is strictly positive. Let that entry be denoted by bkij (here k is *+ , ) k times

not a power). Considering our example in Fig.7.1, G is strongly connected because there is a path between each two vertices of the graph and B is irreducible. Reducibility of a Matrix A matrix is reducible if there exists a permutation matrix P such that PT BP is block triangular, ⎞ ⎛ B11 B12 · · · B1n ⎜ 0 B22 · · · B2n ⎟ ⎟ ⎜ ⎜ .. .. . . .. ⎟ ⎝ . . . . ⎠ 0 0 · · · Bnn with every block Bii square and either irreducible or a 1 × 1 null matrix. Z-matrix B is a Z-matrix if bi j ≤ 0 for i = j,i, j = 1, 2, · · · , m. Non-singular M-matrix B is a non-singular M-matrix if and only if one of the following equivalent statements holds: (1) B has positive principle minors. (2) B + D is non-singular for each D-diagonal.

7.1 Metapopulation Age-Structure Model

261

(3) Every real eigenvalue of B is positive and the real parts of the complex eigenvalues are positive. (4) B−1 is nonnegative. (5) B is monotone, that is, BX ≥ 0 ⇒ x ≥ 0. (6) B has a representation −B = F −V where V −1 ≥ 0, F ≥ 0 and ρ (FV −1 ) < 1. Lemma 7.1 ([7]). If (−Ms ) is a singular M-matrix, all eigenvalues of Ms have nonpositive real parts. 0 is an eigenvalue of Ms , and one of the eigenvectors associated with the eigenvalue 0 is the vector 1T = (1, 1, · · · , 1). In the case that Ms is irreducible, then 0 is an eigenvalue with multiplicity 1, 1T is the only strongly positive left eigenvector associated with Ms , and all other eigenvalues have negative real part. Bartlett [12] used a patch approach to modeling the dynamics of metapopulation with two patches. Rvachev and Longini [153] proposed a multi-city transmission model for the spread of influenza between patches. They also used their model to fit data by the least square method and pointed out that the average duration is a key parameter in controlling the disease spread. Faddy [50] gave a KMK-type SI model and obtain the expression for the final size of the epidemic. Rodriguez and Torres-Sorando [155] investigated the effect of different migration patterns and of the environment heterogeneity on the dynamics of a malaria epidemic model. Wang and Zhao [173] proposed an epidemic model with general birth rate and pointed out that the population movement can either intensify or reduce the spread of disease. Fromont et al. [62] studied a model for Feline leukemia virus among domestic cats and found that spatial heterogeneity promotes disease persistence. There is no doubt that mathematical problems for metapopulation models are still challenging. Most of these models are described by ordinary differential equations. Age is distinct features in modeling the diseases transmission such as measles, handfoot-mouth disease, brucellosis, and others. Studying age-structured metapopulation models may be helpful for us to design optimal control strategies which will enable us to target control measures to a certain age-group.

7.1.1 Monotone Dynamical System Approach The main idea is inspired by Kuniya et al. [108]. First, we here summarize a monotone dynamical system approach to solve the global stability problem for structured population dynamics [21]. Let E be a Banach lattice and E+ be its positive cone. Let z(t) be a population vector that takes a value in a closed convex subset C ⊂ E+ . We say that x, y ∈ E+ satisfy x ≤ y if y − x ∈ E+ . Suppose that the dynamics of the population vector z(t) is written in a semilinear Cauchy problem: d z(t) = Az(t) + F(z(t)), dt

t > 0,

z(0) = z0 ,

(7.1)

262

7 Metapopulation and Multigroup Age-Structured Models

where A is a linear operator describing the survival process and F is a nonlinear perturbation describing the production of new individuals. We assume that differential operator A is the infinitesimal generator of a strongly continuous positive semigroup {etA }t≥0 on E which satisfies (7.2) etA (C) ⊂ C. In addition, we assume that there exists a positive constant α > 0 such that (I − α A)−1 (C) ⊂ C,

(I + α F) (C) ⊂ C.

(7.3)

Furthermore, we assume that the following monotonicity and concavity hold: (I − α A)−1 ϕ ≤ (I − α A)−1 ψ for all ϕ , ψ ∈ C such that ϕ ≤ ψ ,

(7.4)

(I + α F) ϕ ≤ (I + α F) ψ for all ϕ , ψ ∈ C such that ϕ ≤ ψ ,

(7.5)

ξ (I + α F) ϕ ≤ (I + α F) ξ ϕ for all ϕ ∈ C and ξ ∈ (0, 1),

(7.6)

where I denotes the identity operator. Then the basic equation (7.1) can be rewritten as a positively perturbed equation:  d 1 1 z(t) = A − z(0) = z0 (7.7) z(t) + (I + α F) z(t), t > 0, dt α α and hence, its mild solution can be obtained as a solution of the integral equation 1 e z0 + α

− α1 t tA

z(t) = e

 t 0

1

e− α (t−s) e(t−s)A (I + α F) z(s)ds.

(7.8)

Under conditions (7.2)–(7.6), we can adopt a positive iterative procedure to show that the mild solution is given as z(t) = U(t)z0 , where {U(t)}t≥0 is the nonnegative semiflow satisfying the following monotonicity and concavity: U(t)(C) ⊂ C

and U(t)ϕ ≤ U(t)ψ for all ϕ , ψ ∈ C such that ϕ ≤ ψ ,

ξ U(t)ϕ ≤ U(t)ξ ϕ for all ϕ ∈ C and ξ ∈ (0, 1) .

(7.9) (7.10)

In particular, if z0 belongs to the domain of A, then z(t) = U(t)z0 becomes the global classical solution for the original equation (7.1). The positively perturbed equation (7.7) is also useful for showing the existence of a positive equilibrium. Let z∗ denote an equilibrium. Then, we have  1 ∗ 1 z + (I + α F)z∗ = 0. A− α α Because − (A − 1/α ) is positively invertible by (7.3), we have the fixed point equation for z∗ :  1 1 −1 z∗ = − (I + α F) z∗ = (I − α A)−1 (I + α F)z∗ =: Φ (z∗ ), A− α α

7.1 Metapopulation Age-Structure Model

263

where Φ is a positive nonlinear operator preserving the subset C invariant. If Φ has a positive fixed point, it gives a positive equilibrium of the basic system (7.1). Definition 7.2. Let V and W be Banach spaces, and U ⊂ V be an open subset of V. A function f : U → W is called Fréchet differentiable at x ∈ U if there exists a bounded linear operator A : V → W such that

f (x + h) − f (x) − Ah W = 0.

h V

h →0 lim

Definition 7.3. Suppose V and W are locally convex topological vector spaces (for example, Banach spaces), U ⊂ V is open, and f : V → W . The Gâteaux differential d f (u; ψ ) of f at x ∈ U in the direction ψ ∈ V is defined as d f (x; ψ ) = lim

τ →0

 f (x + τψ ) − f (x) d  = f (x + τψ ) . τ dτ τ =0

Compared with the two definition, if f is Fréchet differentiable, then it is also Gâteaux differentiable, and its Fréchet and Gâteaux derivatives agree. Define the linearized operator at the origin:

Φ  [0] := Kα = (I − α A)−1 (I + α F  [0]), where F  [0] is the Fréchet derivative of the operator F at the origin. Using the wellknown Krasnoselskii’s fixed point theorem, we can expect that the fixed point equation has at least one positive solution if Φ and Φ  [0] are compact and nonsupporting operators and r(Kα ) > 1, whereas there is no positive fixed point if r(Kα ) ≤ 1. Uniqueness of the positive fixed point will be proved by using some strong concave property of Φ [92]. Even when the existence and uniqueness of such a positive equilibrium z∗ ∈ C \ {0} of Eq. (7.1) are guaranteed, it is not necessarily easy to show the global stability of the unique steady state. However if there exists a maximal point z† ∈ C such that z ≤ z† for all z ∈ C and U(t) is eventually z∗ -positive, that is, there exist 0 positive constants ξ ∈ (0, 1) and t0 > 0 such that

ξ z∗ ≤ U(t0 )z0 ,

(7.11)

then it is easy to show the global attractivity of z∗ . In fact, if (7.11) holds, then it follows from the monotonic and concave properties (7.9) and (7.10) of the semiflow that

ξ z∗ = ξ U(t)z∗ ≤ U(t)ξ z∗ ≤ U(t)U(t0 )z0 ≤ U(t)z† ≤ z† ,

t ≥ 0.

Hence, we can construct a nondecreasing sequence {U(t)n ξ z∗ }+∞ n=0 and a nonin, both of which are bounded and converge to the creasing sequence {U(t)n z† }+∞ n=0 unique z∗ . Consequently, U(t)U(t0 )z0 = U(t +t0 )z0 also converges to z∗ as t → +∞. As shown above, we can use the spectral radius r(Kα ) of operator Kα as the threshold value for the global attractivity of the positive equilibrium z∗ . Since

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7 Metapopulation and Multigroup Age-Structured Models

Kα = (I − α A)−1 (I + α F  [0]), the spectral radius r(Kα ) depends on the choice of α . However, this dependence does not affect the threshold property of Kα . To see this fact, let us introduce the next generation operator (NGO) for our basic system as follows ([95]): K := F  [0] (−A)−1 . The basic reproduction number is given by its spectral radius: R0 := r(K). For epidemic models, R0 can be interpreted as the expected number of newly infected individuals produced by a typical infected individual during its entire period of infectiousness in a fully susceptible population (see, e.g., [41, 95]). Indeed, in the linearized system at the trivial steady state, R0 = r(K) denotes the per generation asymptotic growth factor and if the spectral mapping theorem holds, the spectral bound ω (A + F  [0]) of the linearized generator gives the Malthusian parameter (asymptotic exponential growth rate) of infective population. From the renewal theorem, the following sign relation holds [95, 168]:   sign (R0 − 1) = sign ω (A + F  [0]) . (7.12) On the other hand, by applying similar ideas to the perturbed operator A − I/α + (I + α F  [0]) /α , we have the following relation:    1 1  I + α F [0] sign (r(Kα ) − 1) = sign ω A − I + . (7.13) α α Since the right-hand sides of (7.12) and (7.13) are the same, we have the sign relation as (7.14) sign (R0 − 1) = sign (r(Kα ) − 1) , which implies that the threshold property of Kα is independent of the choice of α . In summary, we can expect that if R0 > 1 and under conditions (7.9), (7.10), then the basic system has a unique positive equilibrium z∗ ∈ C \ {0} and it attracts all solutions z(t) = U(t)z0 with nontrivial initial datum z0 ∈ C \ {0}, that is, U(t)z0 → z∗ in E as t → +∞, whereas U(t)z0 → 0 as t → +∞ if R0 < 1. In the following, we apply the above general recipe to an age-structured SIS epidemic model with spatial heterogeneity.

7.1.2 SIS Epidemic Model with Age and Patch Structures In this section, we are concerned with the SIS epidemic model with age and patch structures. Let n ∈ N be the number of patches and let N := {1, 2, · · · , n}. Let Pj (t, a) be the population of age a ∈ [0, a† ] at time t ≥ 0 in patch j ∈ N , where a† ∈ (0, +∞) denotes the maximum attainable age of individuals. For j ∈ N , let μ j (a) and β j (a) be the age-specific mortality rate and birth rate in patch j, respectively. For j, k ∈ N such that j = k, let m jk (a) be the age-specific migration rate from patch k to patch j. We assume that the population in each patch j obeys the following Lotka-McKendrick system:

7.1 Metapopulation Age-Structure Model

265

⎧ ! " ⎪ ∂ ∂ ⎪ ⎪ + Pj (t, a) = − μ j (a) + ∑ mk j (a) Pj (t, a) + ∑ m jk (a)Pk (t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪ k= j k= j ⎪  ⎨ a† Pj (t, 0) = β j (a)Pj (t, a)da, ⎪ 0 ⎪ ⎪ Pj (0, a) = Pj,0 (a), ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ t > 0, a ∈ (0, a† ), j ∈ N , (7.15) For simplicity, we use the vector and matrix representations P(t, a) := (P1 (t, a), P2 (t, a), · · · , Pn (t, a))T , P0 (a) := (P1,0 (a), P2,0 (a), · · · , Pn,0 (a))T , M(a) := (m jk (a)) j,k∈N ,

B(a) := diag(β j (a)) j∈N ,

where T denotes the transpose operation and m j j (a) := −μ j (a) − ∑k= j mk j (a), j ∈ N . Then, (7.15) can be rewritten as follows: ⎧ ∂ ∂ ⎪ ⎪ ⎪ ⎨ ∂ t + ∂ a P(t, a) = M(a)P(t, a), t > 0, a ∈ (0, a† ), a† (7.16) P(t, 0) = B(a)P(t, a)da, t > 0, ⎪ ⎪ ⎪ 0 ⎩ P(0, a) = P0 (a), a ∈ (0, a† ). Here, we assume that the population is in the demographic steady state. For this purpose, we make the following assumptions: a

1 (0, a† ) and 0 † μ j (a)da = +∞ for all j ∈ N ; (A1) μ j (·) ∈ Lloc,+ ∞ (A2) m jk (·) ∈ L+ (0, a† ) for all j, k ∈ N such that j = k; ∞ (0, a ) for all j ∈ N ; (A3) β j (·) ∈ L+ † a (A4) The net reproduction matrix 0 † B(a)L(a)da is irreducible and its spectral radius is 1, where L(a) denotes the survival matrix such that dL(a)/da = M(a)L(a) and L(0) is equal to the identity matrix.

Under (A1)–(A4), we see from Propositions 3.2, 3.3, and 4.2 in Inaba [91] that (7.16) has a positive demographic steady state P∗ (a) := (P1∗ (a), P2∗ (a), · · · , Pn∗ (a))T , which satisfies ⎧ ∗ ⎪ ⎨ dP (a) = M(a)P∗ (a), a ∈ (0, a† ), da  a† ⎪ ⎩ P∗ (0) = B(a)P∗ (a)da. 0

It has been shown in [91, 96] that P∗ is globally asymptotically stable with intrinsic growth rate 0. Therefore, in what follows, we assume that the demographic steady state has been already reached at t = 0, so P(t, a) ≡ P∗ (a) = L(a)b∗ for all t > 0, where b∗ denotes the positive eigenvector (the birth rate vector at the steady state) a corresponding to the eigenvalue 1 of the net reproduction matrix 0 † B(a)L(a)da. Here, we focus on an infectious disease such that recovered individuals do not have immunity and can be susceptible again. For instance, some sexually transmit-

266

7 Metapopulation and Multigroup Age-Structured Models

ted diseases such as gonorrhea have this behavior. Such a disease can be modeled by the SIS epidemic model (see, for instance, [109]). Let S j (t, a) and I j (t, a) be the susceptible and infective populations of age a ∈ [0, a† ] at time t ≥ 0 in patch j ∈ N , respectively. We assume that the population is divided into these two classes: Pj∗ (a) ≡ S j (t, a) + I j (t, a), j ∈ N . For j ∈ N , let λ j (t, a) be the force of infection to susceptible individuals of age a at time t, and let γ j (a) be the age-specific recovery rate. We assume that λ j (t, a) is given by the following form:

λ j (t, a) = κ0 j (a)I j (t, a) +

 a† 0

κ j (a, σ )I j (t, σ )dσ ,

j∈N ,

where κ0 j denotes the infection rate for pure intra-cohort interaction, κ j (a, σ ) denotes the infection rate at which an infected individual with age σ transmits the disease with an susceptible individual with age a. The main model in this section is formulated as the following SIS epidemic model with age and patch structure. ⎧ n ∂ ∂ ⎪ ⎪ + (t, a) = − λ (t, a)S (t, a) + γ (a)I (t, a) + S j j j j j ⎪ ∑ m jk (a)Sk (t, a), ⎪ ∂t ∂a ⎪ k=1 ⎪  ⎪ n ⎪ ∂ ∂ ⎪ ⎪ ⎪ + I j (t, a) = λ j (t, a)S j (t, a) − γ j (a)I j (t, a) + ∑ m jk (a)Ik (t, a), ⎨ ∂ t ∂ a k=1 a† ∗ ⎪ ⎪ S j (t, 0) = β j (a)Pj (a)da, I j (t, 0) = 0, ⎪ ⎪ 0 ⎪ ⎪ ⎪ ⎪ S j (0, a) = S j,0 (a), I j (0, a) = I j,0 (a), ⎪ ⎪ ⎪ ⎩ t > 0, a ∈ (0, a† ), j ∈ N . (7.17) As a metapopulation model (7.17) is one of the Eulerian metapopulation models, that is, it models long-term movement between patches. Note that the diseaseinduced death and reduction of the migration rate are not considered in (7.17). As stated in [8, Section 3.3], these assumptions would be suitable for mild diseases. In particular, some sexually transmitted diseases such as gonorrhea (7.17) can be suitable for. Furthermore, note that model (7.17) is different from the multi-group SIS epidemic model studied in [55], which has the interaction of individuals in different groups in the force of infection term but does not have the movement of individuals between groups. We make the following assumption: ∞ (0, a ), and there exist positive constants ε and nonnega(A5) γ j (·), κ0 j (·) ∈ L+ j † tive functions κ1 j (a) and κ2 j (σ ) such that

ε j κ1 j (a)κ2 j (σ ) ≤ κ j (a, σ )Pj∗ (a) ≤ κ1 j (a)κ2 j (σ ), for all j ∈ N , κ1 j ∈ L1 (0, a† ), κ2 j ∈ L∞ (0, a† ). There are 0 ≤ a1 j , a2 j , b1 j , b2 j ≤ a† such that a1 j < b1 j ,

a2 j < b2 j ,

a1 j < b2 j ,

7.1 Metapopulation Age-Structure Model

267

κ1 j (a) > 0, if a1 j < a < b1 j , κ2 j (a) > 0, if a2 j < a < b2 j . In what follows, we consider the normalization of the solution of system (7.17): s j (t, a) :=

S j (t, a) , Pj∗ (a)

i j (t, a) :=

I j (t, a) , Pj∗ (a)

j∈N .

Since s j (t, a) = (Pj∗ (a) − I j (t, a))/Pj∗ (a) = 1 − i j (t, a), we can rewrite (7.17) as follows: ⎧ n ∂ i j (t, a) ∂ i j (t, a) ⎪ ⎪ = − + λ (t, a)(1 − i (t, a)) − γ (a)i (t, a) + m¯ jk (a)ik (t, a), ⎪ j j j j ∑ ⎪ ∂a ⎨ ∂t k=1 t > 0, a ∈ (0, a† ), j ∈ N , ⎪ ⎪ ⎪ i (t, 0) = 0, ⎪ ⎩ j i j (0, a) = i j,0 (a). (7.18) Here, λ j (t, a) = κ¯ 0 j i j (t, a) + m¯ jk (a) := m jk (a)

 a† 0

Pk∗ (a) , Pj∗ (a)

κ¯ j (a, σ )i j (t, σ )dσ ,

j∈N , m¯ j j (a) := − ∑ m¯ jk (a),

j, k ∈ N such that j = k,

κ¯ 1 j (a) = κ0 j (a)Pj∗ (a), κ¯ j (a, σ ) := κ j (a, σ )Pj∗ (σ ),

k= j

j∈N ,

j∈N .

In fact,   ∂ Pj∗ (a) ∂ i j (t, a) ∂ i j (t, a) ∂ I j (t, a) 1 ∂ I j (t, a) 1 ∗ P + = ∗ + ∗ − I (a) (t, a) j ∂t ∂a Pj (a) ∂ t (Pj (a))2 j ∂a ∂a %  i j (t, a) 1 ∂ ∂ = ∗ −(μ j (a)+ ∑ mk j (a))Pj∗ (a) + I j (t, a)− ∗ Pj (a) ∂ t ∂ a Pj (a) k= j & + ∑ m jk (a)Pk∗ (a) k= j

P∗ (a) Ik (t, a) S j (t, a) I j (t, a) − γ j (a) ∗ + ∑ m jk (a) k∗ = λ j (t, a) ∗ Pj (a) Pj (a) k= j Pj (a) Pk∗ (a) ! " I j (t, a) − μ j (a) + ∑ mk j (a) Pj∗ (a) k= j ! " P∗ (a) + μ j (a) + ∑ m jk (a) i j (t, a) − ∑ m jk (a) k∗ i j (t, a) Pj (a) k= j k= j = λ j (t, a)(1 − i j (t, a)) − γ j (a)i j (t, a) + ∑ m¯ jk (a)ik (t, a) k= j

− ∑ m¯ jk (a)i j (t, a), k= j

j∈N

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7 Metapopulation and Multigroup Age-Structured Models

and we obtain (7.18). Moreover, we make the following technical assumption: (A6)

∞ (0, a ) for all j, k ∈ N such that j = k. m¯ jk (·) ∈ L+ †

For example, (A6) is satisfied if the migration rate m jk (a) is zero at the left neighborhood of the maximum attainable age a† , or m jk (a) is proportional to Pj∗ (a).

7.1.3 Definition of the Semiflow To apply the monotone dynamical system approach of Busenberg et al. [21] (see Sect. 7.1.1), we define the semiflow generated by (7.18). Let E := L1 (0, a† ; Rn ), 1 (0, a ; Rn ) and C be the state space for system (7.18) defined as follows: E+ := L+ † . C := ϕ = (ϕ1 , ϕ2 , . . . ϕn )T ∈ E+ : 0 ≤ ϕ j (a) ≤ 1 a.e. for all j ∈ N . (7.19) To rewrite (7.18) as an abstract Cauchy problem in E, we define the following two operators: Aϕ (a) := −

dϕ (a) , da

/ 0 ϕ ∈ D(A) := ϕ ∈ E : ϕ ∈ W 1,1 (0, a† ; Rn ), ϕ (0) = 0 ,

! F(ϕ )(a) :=

(7.20)

"T

n

λ j [a|ϕ ](1 − ϕ j (a)) − γ j (a)ϕ j (a) + ∑ m¯ jk (a)ϕk (a) k=1

,

ϕ ∈ E,

j∈N

where

λ j [a|ϕ ] := κ¯ 0 j (a)ϕ j (a) +

 a† 0

κ¯ j (a, σ )ϕ j (σ )dσ ,

ϕ ∈ E, j ∈ N .

Then, system (7.18) can be rewritten to the following abstract form in E: di(t) = Ai(t) + F(i(t)), t > 0, dt

i(0) = i0 ∈ E,

(7.21)

where i(t) = (i1 (t), i2 (t), · · · , in (t))T ∈ E and i0 = (i1,0 , i2,0 , · · · , in,0 )T ∈ E. Now, we are in a position to show that (7.21) defines the semiflow {U(t)}t≥0 such that i(t) = U(t)i0 , satisfying the following monotonicity and concavity conditions, listed in Sect. 7.1.1. U(t)(C) ⊂ C

and U(t)ϕ ≤ U(t)ψ for all ϕ , ψ ∈ C if ϕ ≤ ψ , (7.22)

ξ U(t)ϕ ≤ U(t)ξ ϕ for all ϕ ∈ C and ξ ∈ (0, 1) .

(7.23)

7.1 Metapopulation Age-Structure Model

269

Proposition 7.1. Let C be defined by (7.19). The abstract Cauchy problem (7.21) has the global classical solution i(t) ∈ C for any i0 ∈ C ∩ D(A). Furthermore, it defines the positive semiflow {U(t)}t≥0 satisfying (7.22) and (7.23). Proof. The operator A defined by (7.20) generates the following C0 -semigroup {etA }t≥0 : E → E.

ϕ (a − t), a − t > 0, ϕ ∈ E. etA ϕ (a) := 0, t − a ≥ 0. Then, it is easy to see that etA (C) ⊂ C and etA ϕ ≤ etA ψ for all ϕ , ψ ∈ C if ϕ ≤ ψ . For any positive constant α > 0, the resolvent (I − α A)−1 : E → E is given as follows. (I − α A)−1 ϕ (a) = ψ . That is

d ψ (a) 1 1 = − ψ (a) + ϕ (a). da α α

We solve it and obtain

ψ (a) =

1 α

 a 0

e− α (a−σ ) ϕ (σ )dσ . 1

Hence, we can define the resolvent operator as follows: (I − α A)−1 ϕ (a) :=

1 α

 a 0

e− α (a−σ ) ϕ (σ )dσ , 1

ϕ ∈ E.

(7.24)

Moreover, for all ϕ , ψ ∈ C such that ϕ ≤ ψ , it holds that (I − α A)−1 ϕ ≤ (I − α A)−1 ψ , Let α > 0 satisfy the following inequality: !

α< max j∈N

1

λ j+ + γ + j +

",

∑

k= j

(7.25)

m¯ +jk

where λ j+ , γ + ¯ +jk ( j, k ∈ N , j = k) denote the essential positive upper j ( j ∈ N ) and m bounds for the respective functions, which existence follows from (A5) and (A6). For any ϕ , ψ ∈ C such that ϕ ≤ ψ , we have

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7 Metapopulation and Multigroup Age-Structured Models

((I + α F)ϕ ) (a) !

/ 0 ϕ j (a) + αλ j [a|ψ ] 1 − ϕ j (a) − αγ j (a)ϕ j (a) + α

≤ ! ≤ ! ≤

"T

n

∑ m¯ jk (a)ϕk (a)

k=1

/ 0 ϕ j (a) 1 − α (λ j [a|ψ ] + γ j (a) − m¯ j j (a)) + αλ j [a|ψ ] + α 

#

j∈N

∑ m¯ jk (a)ψk (a)

k= j

ψ j (a) 1 − α (λ j [a|ψ ]+γ j (a) + ∑ m¯ jk (a)) +αλ j [a|ψ ]+α k= j

"T j∈N "T

∑ m¯ jk (a)ψk (a)

k= j

j∈N

= ((I + α F)ψ ) (a).

That is, I + α F is a monotone operator. For any ϕ ∈ C and ξ ∈ (0, 1), we have ! / 0 ξ ((I + α F) ϕ ) (a) =ξ ϕ j (a) + αλ j [a|ϕ ] 1 − ϕ j (a) − αγ j (a)ϕ j (a) +α !

n

"T

∑ m¯ jk (a)ϕk (a)

k=1

j∈N

/ 0 ≤ ξ ϕ j (a) + αλ j [a|ξ ϕ ] 1 − ξ ϕ j (a) − αγ j (a)ξ ϕ j (a) +α

n

"T

∑ m¯ jk (a)ξ ϕk (a)

k=1

j∈N

= ((I + α F) ξ ϕ ) (a). That is, (7.6) in Sect. 7.1.1 holds. For any ϕ ∈ C, we have

T 1 0 = (I − α A)−1 0 ≤ (I − α A)−1 ϕ ≤ (I − α A)−1 1 = 1 − e− α a ≤ 1, j∈N  T n 0 = (I + α F) 0 ≤ (I + α F) ϕ ≤ (I + α F) 1 = 1 − αγ j (a) + α ∑k=1 m¯ jk (a) j∈N = (1 − αγ j (a))Tj∈N ≤ 1,

where the second to the last equality follows from (7.24). Hence, (I − α A)−1 (C) ⊂ C and (I + α F)(C) ⊂ C hold. Consequently, from the arguments in Sect. 7.1.1, we see that the abstract Cauchy problem (7.21) has the global classical solution i(t) ∈ C for any i0 ∈ C ∩D(A), and it defines the semiflow {U(t)}t≥0 satisfying (7.22) and (7.23).  

7.1 Metapopulation Age-Structure Model

271

7.1.4 Existence and Uniqueness of the Endemic Equilibrium Let i∗ = (i∗1 , i∗2 , · · · , i∗n )T ∈ C denote the equilibrium of the basic system (7.18). From (7.21), it satisfies the following equality: 0 = Ai∗ + F(i∗ ).

(7.26)

By using the positive constant α > 0 as in the proof of Proposition 7.1, we can rewrite the above equality as follows (see also Sect. 7.1.1). i∗ = (I − α A)−1 (I + α F)i∗ =: Φ (i∗ ). Hence, to show the existence of the endemic equilibrium, we can restrict our attention to the fixed point problem of the operator Φ = (I − α A)−1 (I + α F). From the proof of Proposition 7.1, we see that Φ (C) ⊂ C and Φ (ϕ ) ≤ Φ (ψ ) for all ϕ , ψ ∈ C such that ϕ ≤ ψ . Let Φ  [0] := Kα = (I − α A)−1 (I + α F  [0]) be the Fréchet derivative of Φ at the origin, where F  [0] : E → E is the Fréchet derivative of F at the origin. For convenience, we define an operator as ! H ϕ (a) :=

"T

n

λ j [a|ϕ ] − γ j (a)ϕ j (a) + ∑ m¯ jk (a)ϕk (a)

.

k=1

j∈N

It follows from the definition of the Fréchet derivative of the operator F that

λ j [a|h]h

F(h)(a) − F(0)(a) − Hh = lim = 0.

h

h

h →0

h →0 lim

(7.27)

This indicates that F  [0] = H. Hence, we can define the linear operators as follows: (Kα ϕ )(a) :=

1 α

 a 0

e− α (a−σ ) [ϕ (σ ) + α F  [0]ϕ (σ )]d σ , 1

ϕ ∈ E,

(7.28)

where ! F  [0]ϕ (a) :=

n

"T

λ j [a|ϕ ] − γ j (a)ϕ j (a) + ∑ m¯ jk (a)ϕk (a) k=1

,

ϕ ∈ E. (7.29)

j∈N

We can expect that the spectral radius r(Kα ) of the operator Kα plays the role of the threshold value for the existence of the nontrivial fixed point of Φ (see also Sect. 7.1.1). Lemma 7.2. The operator I + α F  [0] is a positive operator. Proof. Let ϕ (a) ∈ C+ \{0}.

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7 Metapopulation and Multigroup Age-Structured Models



 I + α F  [0] (ϕ )(a) =ϕ j (a) + α [λ j [a|ϕ ] − γ j (a)ϕ j (a) + ∑ m¯ jk (a)ϕk (a)] j=k

=ϕ j (a)[I − α (γ j (a) + ∑ m¯ jk (a))] j=k

(7.30)

+ α [λ j [a|ϕ ] + ∑ m¯ jk (a)ϕk (a)] j=k

>0. Choosing α ≤

1 , γ+ ¯+ j + ∑ m jk

we have I + α F  [0](C) ⊂ C.

j=k

By the definition of I − α A and Lemma 7.2, we have Kα is also a positive operator. On the other hand,

Φ =(I − α A)−1 (I + α F) =(I − α A)−1 (I + α (F  [0] + F − F  [0])) =(I − α A)−1 (I + α F  [0]) + α (I − α A)−1 (F − F  [0]) =Kα + α (I − α A)−1 (F − F  [0]). and [F − F  [0]](ϕ )(a) =λ j [a|ϕ ](1 − ϕ ) − γ j (a)ϕ j (a) + ∑ m¯ jk (a)ϕk (a) j=k

− λ j [a|ϕ ] + γ j (a)ϕ j (a) − ∑ m¯ jk (a)ϕk (a) j=k

= − λ j [a|ϕ ]ϕ . Hence 1 Φ (ϕ )(a) = α

 a

#

 − α1 (a−σ )

e 0

[ϕ (σ )[I − α (γ j (σ ) + ∑ m¯ jk (σ ))] + ∑ m¯ jk (σ )ϕk (σ )] d σ . k= j

If we pick up α ≤

1 , γ+ ¯+ j + ∑ m jk

j=k

it is easy to see that Φ is a positive operator. For

j=k

convenience, we define two functions as follows: D1 j (a) = and D2 j (a) =

 a 0

 a 0

−(γ + ¯ jk )s j + ∑ m

κ1 j (s)e

j=k

ds +

[c2 j κ1 j (s) + ∑ m¯ jk (s)]ek0 j s ds. j=k

7.1 Metapopulation Age-Structure Model

273

Proposition 7.2. There exists some positive constant εl j , l = 1, 2, j = 1, 2, · · · , n, such that ε1 j D1 j (a) ≤ i∗j (a) ≤ D2 j (a), j ∈ N. (7.31) Proof. Assume that i∗j (a) is a solution of (7.31), the incidence rate λ j satisfies the following equality: ∗

λ [a|i ] =

κ0 (a)Pj∗ (a)i∗j +

 a† 0

κ (a, σ )Pj∗ (σ )i∗j (a)da.

By the definition of κ j , we know that

ε1 j κ1 j (a)

 a† 0

κ2 j (σ )i∗j (σ )d σ ≤ λ [a|i∗ ] ≤ k0+j i∗j (a) + κ1 j (a)

or  a†

 a† 0

κ2 (σ )i∗j (σ )d σ ,

c1 j κ1 j (a) ≤ λ [a|i∗ ] ≤ k0+j i∗j (a) + c2 j κ1 j (a),

where c1 j = ε1 j 0 κ2 j (σ )i∗j (σ )d σ and c2 j = satisfies the following inequality:

 a† 0

(7.32)

κ2 j (σ )i∗j (σ )d σ . Therefore, i∗j (a)



¯ +jk )i∗j (a) ≤ i∗j (a) ≤ k0+j i∗j (a) + c2 j κ1 j (a) + ∑ m¯ jk (a). c1 j κ1 j (a) − (γ + j +∑m j=k

j=k

(7.33) By integration, we have c1 j D1 j (a) ≤ i∗j (a) ≤ D2 j (a).

(7.34)

  Lemma 7.3. The operator F  [0] is a bounded operator. Proof. Assume that φ j ∈ C,

F  [0]ϕ (a) =

 a† 0

(λ j (a) − γ j (a)ϕ j (a) + ∑ m¯ jk (a)ϕk (a))da n

k=1

¯ jk ) ϕ (a) . ≤ (κ0+j + c2 j κ1+j + γ + j + ∑ m

(7.35)

j=1

.  Lemma 7.4. Kα is a compact operator.  Proof.  ∞Lemma 7.3 implies that I + α F [0] is a bounded operator. Hence lim A Kα (a)da = 0. On the other hand,

A→∞

274

7 Metapopulation and Multigroup Age-Structured Models

1

Kα ϕ (a + h) − Kα ϕ (a) = [ α +

0  a 0

1 = [ α +

 a†  a+h

0

1

1

0

(e−1/α (a+h−σ ) − e−1/α (a−σ ) )[ϕ (σ ) + α F  [0]ϕ (σ )]d σ da

 a†  a+h 0

 a† 0

e− α (a+h−σ ) [ϕ (σ ) + α F  [0]ϕ (σ )d σ da

e− α (a−σ ) [ϕ (σ ) + α F  [0]ϕ (σ )]d σ da]

 a†  a

1 α

1 ≤ [ α

0

a

e−1/α (a+h−σ ) [ϕ (σ ) + α F  [0]ϕ (σ )d σ da

eσ /α |e−h/α − 1|[ϕ (σ ) + α F  [0]ϕ (σ )]





 a† σ

e−a/α dad σ

h 1 a† σ /α e [ϕ (σ ) + α F  [0]ϕ (σ )]d σ e−1/α (a+h) da α 0 0   σ +h 1 a† σ /α + e [ϕ (σ ) + α F  [0]ϕ (σ )]d σ e−1/α (a+h) da α 0 σ  a† h a† ≤ 2 [ϕ (σ ) + α F  [0]ϕ (σ )]d σ α 0  ea† −h h a† + [ϕ (σ ) + α F  [0]ϕ (σ )]d σ α 0  a† h + [ϕ (σ ) + α F  [0]ϕ (σ )]d σ . α 0

+

(7.36) By the boundedness of I + α F  [0], we obtain that lim Kα ϕ (a + h) − Kα ϕ (a) = h→0

0. It follows from Kolmogorov Theorem that Kα is compact. Using Krein-Rutman Theorem and the positivity of the operator Φ , we prove the following proposition. Proposition 7.3. Let C and Kα be defined by (7.19) and (7.28), respectively. If r(Kα ) > 1, then system (7.18) has at least one endemic equilibrium i∗ ∈ C \ {0}. Proof. By Lemma 7.4, we obtain that the operator Kα is linear, positive, and compact. It follows from the Krein-Rutman Theorem A.8 that there exists a positive eigenfunction ϕ ∗ ∈ E+ \ {0} which corresponds to r(Kα ) > 1. Without loss of generality since if ϕ ∗ is not in C we can normalize it, and then assume that ϕ ∗ ∈ C. For any ξ ∈ (0, 1), we have

Φ (ξ ϕ ∗ )(a) =(I − α A)−1 (I + α (F  [0] + F − F  [0]))(ξ ϕ ∗ )(a) =Kα (ξ ϕ ∗ )(a) + α (I − α A)−1 (F − F  [0])(ξ ϕ ∗ )(a)  a T 1 ∗ − α1 (a−σ ) ∗ ∗ =Kα ξ ϕ (a) − e αλ j [σ |ξ ϕ ]ξ ϕ j (σ )dσ α 0 j∈N  a T 1 =ξ r(Kα )ϕ ∗ (a) − ξ 2 e− α (a−σ ) λ j [σ |ϕ ∗ ]ϕ ∗j (σ )dσ 0

≥ξ r(Kα )ϕ ∗ (a) − ξ 2 λ¯

 0

a

ϕ ∗j (σ )dσ

j∈N

T , j∈N

(7.37)

7.1 Metapopulation Age-Structure Model

275

 where λ¯ := max j∈N λ j+ . On the other hand, to bound 0a ϕ ∗j (σ )d σ from above, we have

r(Kα )ϕ ∗ (a) =Kα ϕ ∗ (a)  ! "# "T !  1 − 1 a† a + + ∗ e α ϕ j (σ )dσ 1 − α γ j + ∑ m¯ jk ≥ α 0 k= j j∈N  T  a 1 − 1 a† ≥ e α (1 − α γ¯) ϕ ∗j (σ )dσ , α 0 j∈N

¯ +jk > 0. From the choice of α as in (7.25), we where γ¯ := max j∈N γ + j + ∑k= j m see that 1 − α γ¯ > 0 and hence,

α r(Kα ) 1

e− α a† (1 − α γ¯)

ϕ ∗j (a) ≥

 a 0

ϕ ∗j (σ )dσ ,

j∈N .

(7.38)

Since r(Kα ) > 1, we take η small enough so that r(Kα ) − 1 − η λ¯

α r(Kα ) − α1 a†

e

(1 − α γ¯)

> 0.

(7.39)

From (7.37), (7.38), and (7.39), we have

Φ (ηϕ ∗ )(a) ≥η r(Kα )ϕ ∗ (a) − η 2 λ¯  ∗

α r(Kα ) − α1 a†

e

(1 − α γ¯)

=ηϕ (a) + r(Kα ) − 1 − η λ¯

ϕ ∗ (a)

α r(Kα ) 1

e− α a† (1 − α γ¯)

#

ηϕ ∗ (a) ≥ ηϕ ∗ (a).

Hence, from the monotonicity of Φ , we can construct a monotone nondecreasing sequence {Φ n (ηϕ ∗ )}+∞ n=0 . Since Φ (C) ⊂ C, this sequence is increasing bounded above and hence, there exists a ϕ ∞ ∈ C \ {0} such that Φ (ϕ ∞ ) = ϕ ∞ . This is the desired fixed point of Φ , that is, the endemic equilibrium i∗ ∈ C \ {0} of system (7.18). This completes the proof.   Next we prove the uniqueness of the endemic equilibrium i∗ = (i∗1 , i∗2 , · · · , i∗n )T ∈ C \ {0}. We see from (7.18) that it satisfies the following system of ordinary differential equations: ⎧ ∗ n ⎨ di j (a) = λ [a|i∗ ] 1 − i∗ (a) − γ (a)i∗ (a) + m¯ jk (a)i∗k (a), a ∈ (0, a† ), j j ∑ j j da k=1 ⎩ ∗ i j (0) = 0, j∈N . (7.40)

276

7 Metapopulation and Multigroup Age-Structured Models

Lemma 7.5. Let C be defined by (7.19) and i∗ ∈ C \ {0} be an endemic equilibrium of system (7.18). For any ξ ∈ (0, 1) and v ∈ C \ {0}, there exists a positive functional c(v) > 0 such that Φ (ξ i∗ + v) (a) ≥ ξ i∗ (a) + c(v)(a). Proof. For any ξ ∈ (0, 1) and v ∈ C \ {0}, we have (I + α F) (ξ i∗ + v) !

"T n     = ξ i∗ + v + α λ j [·|ξ i∗ + v] 1 − ξ i∗j − v j − γ j (·) ξ i∗j + v j + ∑ m¯ jk (·) (ξ i∗k + vk ) !





 ∗

!

k=1

j∈N

"#

= ξ i∗ + α F(ξ i∗ ) + αλ j [·|v] 1 − ξ i j + v j 1 − α λ j [·|ξ i∗ + v] + γ j (·) + ∑ m¯ jk (·) k= j

"T +α

∑ m¯ jk (·)vk

k= j

j∈N

≥ ξ i∗ + ξ α F(i∗ ) + α (1 − ξ ) (λ j (·|v))Tj∈N .

Hence, it follows from the monotonicity of (I − α A)−1 that

Φ (ξ i∗ + v) (a) =(I − α A)−1 (I + α F)(ξ i∗ + v)(a)  a T ∗ − α1 (a−σ ) ≥ξ Φ (i ) (a) + (1 − ξ ) e λ j (·|v) dσ ≥ξ i∗ (a) + (1 − ξ ) ∗

 0

0

a

e−(a−σ )/α c1 j (v)κ1 j (σ )d σ

− α1 a†

≥ξ i (a) + (1 − ξ ) e =:ξ i∗ (a) + c(v)(a),

j∈N

T

j∈N

c1 (v)D1 (a),

where c1 (v) := min j∈N c1 j (v), D1 (a) = inf j∈N D1 j (a) and c(v)(a) = (1 − ξ ) 1  c1 (v)D1 (a)e− α a† .  Using Lemma 7.5, we prove the following proposition on the uniqueness of the endemic equilibrium i∗ ∈ C \ {0} of (7.18). Proposition 7.4. Let C be defined by (7.19). System (7.18) has at most one endemic equilibrium i∗ ∈ C \ {0}. Proof. Suppose that there exists another endemic equilibrium j∗ = ( j1∗ , j2∗ , · · · , jn∗ )T ∈ C \ {0} of system (7.18). From the inequality (7.31), for j ∈ N we have i∗j (a) ≥

c1 j D1 j (a) c1 j D1 j (a) ∗ D2 j (a) ≥ j (a). D2 j (a) D2 j (a) j

This implies that there exists a ξ ∈ (0, 1) such that i∗ > ξ j∗ . Let η := sup{ξ ∈ R+ : i∗ > ξ j∗ } and suppose that η ∈ (0, 1). From the monotonicity of Φ and the inequality (7.31), we have

7.1 Metapopulation Age-Structure Model

277

i∗ (a) =Φ (i∗ )(a) ≥ Φ (η j∗ )(a)  a T ∗ − α1 (a−σ ) ∗ ∗ =ηΦ ( j )(a) + η (1 − η ) e λ j [σ | j ] j j (σ )dσ 0

1



≥ηΦ ( j∗ )(a) + η (1 − η ) e− α a† c21 j ! 1

=η j∗ (a) + η (1 − η ) e− α a†

c21 j !

1

≥η j∗ (a) + η (1 − η ) e− α a†

c21 j

 a 0

j∈N

T D1 j (σ )κ1 j (σ )d σ

D21 j (σ ) a  2 0 "T D21 j (a) 2

j∈N

"T j∈N

.

(7.41)

j∈N

Regarding the last term in the right-hand side of this inequality as v in Lemma 7.5, we see that there exists a c(v) > 0 such that Φ (η j∗ + v) (a) ≥ η j∗ (a) + c(v)(a). Hence, mapping Φ on both sides of (7.41), we have

Φ (i∗ )(a) ≥ Φ (η j∗ + v)(a) ≥ η j∗ (a) + ε1 c(v)D1 (a) c(v)(a) c(v) ∗ D2 (a) ≥ η j∗ (a) + j (a), a ∈ (0, a† ). ≥ η j∗ (a) + D2 (a) D2 (a) Recalling that i∗ = Φ (i∗ ), we have  c(v) ∗ i (a) ≥ η + j∗ (a), D2 (a)

a ∈ (0, a† ),

which contradicts the definition of η . Therefore, η ≥ 1 and we have i∗ ≥ η j∗ ≥ j∗ . Exchanging the roles of i∗ and j∗ , ensures j∗ ≥ i∗ in a similar way. Consequently,  we have i∗ = j∗ and the proof is complete. 

7.1.5 Global Attractivity of Equilibria From (7.21) and (7.29), we have the following differential inequality:  di(t)  ≤ A + F  [0] i(t), t > 0, dt

i(0) = i0 ∈ E.

(7.42)

When r(Kα ) < 1, as stated in Sect. 7.1.1, we have that ω (A + F  [0]) < 0, where ω (·) denotes the spectral bound of an operator and it equals to the asymptotic exponential growth rate. Therefore, from (7.42), we see that U(t)i0 → 0 in E as t → +∞, that is, the disease-free equilibrium 0 is globally attractive. Consequently, we have the following proposition.

278

7 Metapopulation and Multigroup Age-Structured Models

Proposition 7.5. Let C and Kα be defined by (7.19) and (7.28), respectively. If r(Kα ) < 1, then the disease-free equilibrium 0 of system (7.18) is globally attractive in C. Next we investigate the global attractivity of the endemic equilibrium i∗ ∈ C \{0} when r(Kα ) > 1. Lemma 7.6. Let C be defined by (7.19) and suppose that system (7.18) has the unique endemic equilibrium i∗ ∈ C \ {0}. Then, there exist positive constants ξ ∈ (0, 1) and t0 > 0 such that ξ i∗ ≤ U(t0 )i0 , (7.43) provided that i0 ∈ C \ {0}. Proof. By integrating the first equation in (7.18) along the characteristic line t − a = const., one arrives at  ⎧ − 0t {λ j (ρ ,a−t+ρ )+γ j (a−t+ρ )+∑k= j m¯ jk (a−t+ρ )}dρ ⎪ i j,0 (a − t)e dσ # ⎪  ⎪ ⎪  t ⎪ ⎪ ⎪ ⎪ + λ j (σ , a−t+σ ) + ∑ m¯ jk (a−t+σ )ik (σ , a − t + σ ) a−t > 0; ⎪ ⎪ 0 ⎪ k= j ⎪ t ⎪ ⎨ ×e− σ {λ j (ρ ,a−t+ρ )+γ j (a−t+ρ )+ ∑k= j m¯ jk (a−t+ρ )}dρ dσ , i j (t, a) = ⎪ ⎪  # ⎪ ⎪  a ⎪ ⎪ ⎪ ⎪ λ j (t − a + σ , σ ) + ∑ m¯ jk (σ )ik (t − a + σ , σ ) ⎪ ⎪ 0 t−a ≥ 0. ⎪ k= j ⎪ ⎪  ⎩ − σa {λ j (t−a+ρ ,ρ )+γ j (ρ )+ ∑k= j m¯ jk (ρ )}dρ dσ , ×e It suffices to consider the case t > a† . From the second case of the above equation, it is easy to see that ij

 a n − λ j+ +γ + ¯+ j +∑k=1 m jk a† (t, a) ≥e λ

n − λ j+ +γ + ¯+ j +∑k=1 m jk a†

≥e

0

j (t − a + σ , σ ) dσ

c1 j D1 j (a),

j∈N .

(7.44)

Let t0 > a† . Since i0 ∈ C \ {0}, it follows from Lemma 7.2 and (7.44), we have i j (t0 , a)

n − λ j+ +γ + ¯+ j +∑k=1 m jk a† ≥e c1 j D1 j (a)

+ + + n − λ j +γ j +∑k=1 m¯ jk a†

=

e

≥

e

D2 j (a)

n − λ j+ +γ + ¯+ j +∑k=1 m jk a† D2 j (a)

c1 j D1 j (a)D2 j (a) c1 j D1 j (a)i∗j (a),

j∈N ,

which implies the existence of a sufficiently small ξ ∈ (0, 1) such that (7.43) holds.  

7.2 Multigroup Epidemic Model with Age Structure

279

Lemma 7.6 implies that the nontrivial solution will become comparable with i∗ after finite time (see Sect. 7.1.1). Hence, since the semiflow {U(t)}t≥0 satisfies (7.22) and (7.23), we can prove that U(t)i0 → i∗ in E as t → +∞ for any i0 ∈ C \ {0}, that is, the endemic equilibrium i∗ is globally attractive in C \ {0} as stated in Sect. 7.1.1. Consequently, from Propositions 7.3 and 7.4 and Lemma 7.6, we have the following proposition. Proposition 7.6. Let C and Kα be defined by (7.19) and (7.28), respectively. If r(Kα ) > 1, then system (7.18) has the unique endemic equilibrium i∗ ∈ C \ {0} and it is globally attractive in C \ {0}. Finally in this section, we show the relationship between r(Kα ) and the basic reproduction number R0 . As stated in Sect. 7.1.1, R0 can be defined by R0 = r(K),

K = F  [0](−A)−1 ,

(7.45)

and sign(R0 − 1) = sign(r(Kα ) − 1) (see (7.14)). Consequently, from Propositions 7.5 and 7.6, we have the following main theorem in this section. Theorem 7.1. Let C and R0 be defined by (7.19) and (7.45), respectively. 1. If R0 < 1, then the disease-free equilibrium 0 of system (7.18) is globally attractive in C. 2. If R0 > 1, then system (7.18) has the unique endemic equilibrium i∗ ∈ C \ {0} and it is globally attractive in C \ {0}.

7.2 Multigroup Epidemic Model with Age Structure Multi-group epidemic models focus on a heterogeneous host population divided into homogeneous subpopulations based on a heterogeneity such as age, sex, position, and so on. Contact heterogeneity is one of the main characters in modeling intra-group and inter-group infections. Lajmanovich and Yorke first proposed a multi-group SIS epidemic model to investigate the transmission of gonorrhea in a community. Since then, a number of publications using multi-group mathematical epidemic models have been investigated. The global stability of the endemic equilibrium has become a popular subject and challenging issue. Guo et al. [73] were the first to use the graph-theoretical method to derive the global stability of the nontrivial equilibrium under the theory of nonnegative irreducible matrices. Employing this idea, the uniqueness and global stability of nontrivial equilibrium have been solved and some open problems have been resolved.

280

7 Metapopulation and Multigroup Age-Structured Models

7.2.1 Multigroup SIR Vector-Borne Epidemic Model Some diseases such as influenza, measles, mumps, and so on can lead to permanent immunity even if people have been infected by these diseases. From the compartment mathematical modeling point of view, SIR structure is an appropriate choice. The multigroup SIR epidemic models have been investigated in literature [103, 106, 163]. Almost of them used ordinary differential equation to describe the behavior contact heterogeneities and ignore the individual age heterogeneities. Recently, Kuniya et al. [107] used an SIR structured epidemic model to combine age structure and multi-group structure, which is described as follows: ⎧ ∂ ∂ ⎪ ⎪ + S j (t, a) = −λ j (t, a)S j (t, a) − μ j (a)S j (t, a), ⎪ ⎪ ∂ t ∂a ⎪ ⎪ ⎪  ⎪ ⎪ ∂ ∂ ⎪ ⎪ ⎪ + I j (t, a) = λ j (t, a)S j (t, a) − μ j (a) + γ j (a)I j (t, a)(t, a), ⎪ ⎪ ∂ t ∂ a ⎪ ⎨ (7.46) ∂ ∂ ⎪ + R j (t, a) = γ j (a)I j (t, a) − μ j (a)R j (t, a),t > 0, a > 0 ⎪ ⎪ ∂ t ∂ a ⎪ ⎪ ⎪ ⎪ n  ∞ ⎪ ⎪ ⎪ ⎪ S (t, 0) = ∑ f jk (a)Pk (t, a)da, I j (t, 0) = R j (t, 0) = 0,t > 0, ⎪ ⎪ j ⎪ k=1 0 ⎪ ⎩ S j (0, a) = S j,0 (a), I j (0, a) = I j,0 (a), R j (0, a) = R j,0 (a), j ∈ N, where the total population is divided as three classes: susceptible, infected, recovered, denoted by S j (t, a), I j (t, a), and R j (t, a), respectively. μ j (a) and γ j (a) denote the natural death rate and the recovery rate with respect to age. f jk (a) denotes the per capita fertility rate at which an individual of age a in group k gives birth to an offspring in group j. Infection incidence rate is defined as follows:

λ j (t, a) =

n

∑

 ∞

k=1 0

β jk (a, σ )Ik (t, σ )d σ , j ∈ N.

Total population is denoted as Pj (t, a) = S j (t, a) + I j (t, a) + R j (t, a), which satisfies the following equations: ⎧ ∂ ∂ ⎪ ⎪ + Pj (t, a) = −μ j (a)Pj (t, a), ⎪ ⎪ ⎨ ∂t ∂a  n ∞ (7.47) P (t, 0) = f jk (a)Pk (t, a)da,t > 0, ⎪ j ∑ ⎪ ⎪ ⎪ k=1 0 ⎩ Pj (0, a) = Pj,0 (a), j ∈ N. Denote Q1 (a) := diag(μ1 (a), μ2 (a), · · · , μn (a)) and

B(a) = ( f jk (a))1≤ j,k≤n .

7.2 Multigroup Epidemic Model with Age Structure



281 a

If the net reproduction matrix 0∞ B(a)e−λ a e− 0 Q1 (σ )d σ da is irreducible, and it has a unique real root λ = 0. It follows from Propositions 3.2 and 3.3 [107] that system (7.47) has a nonnegative steady state Pj∗ (a), which is globally asymptotically stable. Then system (7.46) can be changed as ⎧ ∂ ∂ ⎪ ⎪ + S j (t, a) = −λ j (t, a)S j (t, a) − μ j (a)S j (t, a), ⎪ ⎪ ∂ t ∂a ⎪ ⎪  ⎪ ⎪ ⎪ ∂ ∂ ⎪ ⎪ + I j (t, a) = λ j (t, a)S j (t, a) − (μ j (a) + γ j (a))I j (t, a)(t, a), ⎪ ⎪ ⎪ ⎨ ∂t ∂a ∂ ∂ (7.48) + R j (t, a) = γ j (a)I j (t, a) − μ j (a)R j (t, a), ⎪ ⎪ ⎪ ∂ t ∂ a ⎪ ⎪ ⎪ n  ∞ ⎪ ⎪ ⎪ ⎪ S j (t, 0) = b j = ∑ f jk (a)Pk∗ (a)da, I j (t, 0) = R j (t, 0) = 0,t > 0, ⎪ ⎪ 0 ⎪ k=1 ⎪ ⎩ S j (0, a) = S j,0 (a), I j (0, a) = I j,0 (a), R j (0, a) = R j,0 (a), j ∈ N. Based on model (7.48), we introduce a vector-borne infection into this model. The vector population is divided into two classes: susceptible mosquitoes (Sv ) and infected mosquitoes (Iv ). Then the model is described as follows: ⎧ ∂ ∂ ⎪ ⎪ + S j (t, a) = −λ j (t, a)S j (t, a) − μ j (a)S j (t, a), ⎪ ⎪ ∂t ∂a ⎪ ⎪  ⎪ ⎪ ⎪ ∂ ∂ ⎪ ⎪ + I j (t, a) = λ j (t, a)S j (t, a) − (μ j (a) + γ j (a)) I j (t, a), ⎪ ⎪ ∂ t ∂ a ⎪ ⎪ ⎪  ⎪ ⎪ ∂ ∂ ⎪ ⎪ + R j (t, a) = γ j (a)I j (t, a) − μ j (a)R j (t, a), ⎪ ⎪ ⎪ ∂ t ∂ a ⎪ ⎪ ⎪ ⎨ dSv (t) = Λ − Sv (t)λ (t) − μv Sv (t), (7.49) dt ⎪ ⎪ ⎪ ⎪ ⎪ dIv (t) ⎪ ⎪ = Sv (t)λ (t) − μv Iv (t), ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ S j (t, 0) = b j , I j (t, 0) = R j (t, 0) = 0,t > 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S j (0, a) = S j,0 (a), I j (0, a) = I j,0 (a), R j (0, a) = R j,0 (a), ⎪ ⎪ ⎪ ⎪ ⎩ Sv (0) = Sv0 , Iv (t, 0) = Iv0 , j ∈ N, where Λ is input rate at which all the mosquitoes give the susceptible newborns, μv is death rate of mosquitoes.

λ j (t, a) = λ j (t, a) + βvhj (a)Iv (t), and

λ (t) =

n

∑

 ∞

j=1 0

βhvj (a)I j (t, a)da,

282

7 Metapopulation and Multigroup Age-Structured Models

where βvhj (a) denotes the transmission rate from an infected vector to a susceptible individual, βhvj (a) denotes the transmission rate from an infected individual to a susceptible vector. We denote the total vector population as Nv (t) = Sv (t) + Iv (t). It follows from the last two equations of (7.49) that Nv (t) = Λ − μv Nv (t). Solving this equation, we have lim Nv (t) = μΛv . Without loss of generality, we assume that t→∞

Nv (t) = Nv∗ = μΛv at any time. Therefore, we can consider the following reduced 3n + 1 dimensional system: ⎧ ∂ ∂ ⎪ ⎪ + S j (t, a) = −λ j (t, a)S j (t, a) − μ j (a)S j (t, a), ⎪ ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ ⎪ ∂ ∂ ⎪ ⎪ ⎪ + I j (t, a) = λ j (t, a)S j (t, a) − μ j (a) + γ j (a)I j (t, a)(t, a), ⎪ ⎪ ∂ t ∂ a ⎪ ⎪  ⎪ ⎪ ⎪ ∂ ∂ ⎪ ⎪ ⎪ ⎨ ∂ t + ∂ a R j (t, a) = γ j (a)I j (t, a) − μ j (a)R j (t, a),t > 0, a > 0 (7.50) dIv (t) ⎪ ⎪ = (Nv∗ − Iv (t))λ (t) − μv Iv (t), ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ S j (t, 0) = b j , I j (t, 0) = R j (t, 0) = 0, ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ S j (0, a) = S j,0 (a), I j (0, a) = I j,0 (a), R j (0, a) = R j,0 (a), ⎪ ⎪ ⎪ ⎩ Iv (t, 0) = Iv0 , j ∈ N.

In order to investigate the dynamical behavior of system (7.50), we define the functional space for system (7.50) as X := L1 (0, +∞; Rn ), with norm

ϕ X =

Y := X × X × R,

Z = X ×X ×X ×R

 ∞ n 0

∑ |ϕ j (a)|da, ϕ = (ϕ1 , ϕ2 , · · · , ϕn )T ∈ X.

j=1

ψ Y = ψ1 X + ψ2 X + |ψv |, ψ Z = ψ1 X + ψ2 X + ψ3 X + |ψv |,

ψi = (ψi,1 , ψi,2 , · · · , ψi,n ) ∈ X, i = 1, 2, 3. Define a set for system (7.50) is

 Λ Ω = (S, I, R, Iv )T ∈ Z+ : S + I + R = P∗ , Iv ≤ , μv where S = (S1 , S2 , · · · , Sn )T ∈ X, I = (I1 , I2 , · · · , In )T ∈ X, R = (R1 , R2 , · · · , Rn )T ∈ X. So system (7.50) can be reduced as a 2n + 1 dimensional system as follows:

7.2 Multigroup Epidemic Model with Age Structure

283

⎧ ∂ ∂ ⎪ ⎪ + I j (t, a)=λ j (t, a)(Pj∗ (a)−I j (t, a)−R j (t, a))− (μ j (a)+γ j (a)) I j (t, a)(t, a), ⎪ ⎪ ∂ t ∂ a ⎪ ⎪  ⎪ ⎪ ∂ ∂ ⎪ ⎪ ⎪ + R j (t, a) = γ j (a)I j (t, a) − μ j (a)R j (t, a),t > 0, a > 0 ⎪ ⎨ ∂t ∂a dIv (t) ⎪ = (Nv∗ − Iv (t))λ (t) − μv Iv (t),t > 0, a > 0 ⎪ ⎪ ⎪ dt ⎪ ⎪ ⎪ ⎪ I j (t, 0) = R j (t, 0) = 0,t > 0, ⎪ ⎪ ⎪ ⎪ ⎩ I j (0, a) = I j,0 (a), R j (0, a) = R j,0 (a), Iv (t, 0) = Iv0 , j ∈ N. (7.51) For the parameters in system (7.51), we make the following assumptions:

β jk (a, σ ) ∈ L∞ (R+ × R+ ),

βhvj (a) ∈ L∞ (R+ ),

βvhj (a) ∈ L∞ (R+ ).

For death rate and recovery rate functions with respect to a, we assume that

γk (a) ∈ L∞ (R+ ), and there exists a positive value μ such that μk (a) > μ > 0. We can also define a newly positively invariant set

 Λ T ∗ Ω1 = (I, R, Iv ) ∈ Y+ : I + R ≤ P , Iv ≤ . μv In what follows, the initial condition of system (7.47) is chosen from Ω1 . Define a liner operator A1 : D(A1 ) ⊂ Y → Y by A1 ψ (a) = −

d ψ (a) − Q1 (a)ψ (a), da

D(A1 ) = {ψ ∈ Y : ψ  is absolutely continuous, ψ2 (0) = ψ3 (0) = 0}. On the other hand, define anther linear operator as A2 ψv = −μv ψv , D(A2 ) = R. Furthermore, define a nonlinear operator F : Ω ⊂ Y → Y by ⎞ ⎛ (Λ1 ψ2 )(P∗ (a)−ψ2 −ψ3 )−Γ (a)ψ2 (a)+βvh (a)(P∗ (a)−ψ2 −ψ3 )ψv ⎠ Γ (a)ψ2 (a) F(ψ )(a)= ⎝ (Nv∗ − ψv )Λv where Λ1 is a linear operator on Y defined by n

(Λ1 φ )(a) = diag1≤ j≤n ( ∑

 ∞

k=1 0

β jk (a, σ )φk (σ )d σ ) + βvhj (a)φv ), φ ∈ Y.

(7.52)

284

7 Metapopulation and Multigroup Age-Structured Models

Γ is a diagonal matrix on X defined by Γ (a) = diag(γ1 (a), γ2 (a), · · · , γn (a)). Λv is also a linear operator on R defined by (Λv φ )(a) =

n

∑

 ∞

j=1 0

βhvj (a)φ j (a)da.

(7.53)

Define A = diag(A1 , A2 ), and Q = diag(Q1 , −μv ). Set u = (I, R, Iv ), system (7.51) can be rewritten as a Cauchy problem d u(t) = Au(t) + F(t), u(0) = u0 ∈ Ω , dt

(7.54)

where u0 = (I10 , I20 , · · · , In0 , R10 , R20 , · · · , Rn0 , Iv0 )T ∈ Y. The operator A is an infinitesimal generator of a C0 -semigroup {eAt }t≥0 : Y → Y defined by

0,  t > a, a (etA )(ψ (a)) = (7.55) − a−t Q(σ )d σ e ψ (a − t), t ≤ a. By (7.55), the set Ω1 is positively invariant under the semiflow etA for all t ≥ 0. Lemma 7.7. The operator F : Ω1 → Y is Lipschitz continuous and there exists a constant α such that (7.56) (I + α F)(Ω1 ) ⊂ Ω1 , where I denotes the identity operator. Proof. First, we prove the Lipschitz continuity of F. Let u and u¯ be in Ω1 and note that

F(u) − F(u) ¯ = Λ1 (ψ2 )(P∗ − ψ2 − ψ3 ) − Λ1 (ψ¯ 2 )(P∗ − ψ¯ 2 − ψ¯ 3 ) + ψv

 ∞ 0

βvh (a)ψ2 (a)da − ψ¯ v

 ∞ 0

βvh (a)ψ¯ 2 (a)da

+ (Nv∗ − ψv )Λv (ψ2 ) − (Nv∗ − ψ¯ v )Λv (ψ¯ 2 )

≤

 ∞ 0

P∗ (a)daΛ1 (|ψ2 − ψ¯ 2 |) + Λ1 (ψ¯ 2 )( ψ2 − ψ¯ 2 + ψ3 − ψ¯ 3 )

+ ψv

 ∞ 0

βvh (a)|ψ2 (a) − ψ¯ 2 (a)|da +

+ Nv∗Λv |ψ2 − ψ¯ 2 | + Λv (ψ¯ 2 )|ψv − ψ¯ v | ≤2bβ ψ2 − ψ¯ 2 + bβ ψ3 − ψ¯ 3

 ∞ 0

βvh (a)ψ¯ 2 (a)da|ψv − ψ¯ v |

+ +

ψ2 − ψ¯ 2 + bβvh |ψv − ψ¯ v | + Nv∗ βvh + +

ψ2 − ψ¯ 2 + bβhv |ψv − ψ¯ v | + Nv∗ βhv

≤L u − u , ¯ (7.57)

7.2 Multigroup Epidemic Model with Age Structure

285

+ + + where b = max{b j }, β = max β jk , βhv = max{βhvj } and βvh = max{βvhj }. L = j∈N

k, j∈N

j∈N

j∈N

+ + + + max{2bβ + Nv∗ (βvh + βhv ), b(βhv + βvh )}. Equation (7.57) indicates that F is Lipschitz continuous. Next, we will show that I + α F preserves the set Ω1 invariant. Note that

λ j (t, a) ≤

n

∑

+ β jk

 ∞

+

Pk∗ (s)ds  λ  j < +∞, j = 1, 2, · · · , n,

k=1

0

λv =

∑ βhvk+

and

n

 ∞

k=1

0

P∗ (s)ds  λv+ < +∞.

(7.58)

(7.59)

For any ψ = (ψ2 , ψ3 , ψv )T ∈ Ω1 , let ⎛ ⎞ ⎛ ⎞ ψˆ 2 ψ2 ⎝ ψˆ 3 ⎠ := (I + α F) ⎝ ψ3 ⎠ . ψˆ v ψv It follows from the definition of F that

ψˆ 2 + ψˆ 3 =ψ2 + ψ3 + αΛ1 (ψ2 )(P∗ − ψ2 − ψ3 ) + αβvh (a)ψv (P∗ − ψ2 − ψ3 ) ≤ψ2 + ψ3 + α (λ where λ

+

+

+ ∗ + βvh Nv )(P∗ − ψ2 − ψ3 ).

= max{λ j+ }. Choosing α ≤ jN

other hand,

λ

+

1 + ∗, +βvh Nv

we have ψˆ 2 + ψˆ 3 ≤ P∗ . On the

ψˆ v = ψv + αΛv (ψ2 )(Nv∗ − ψv ) ≤ ψv + αλv+ (Nv∗ − ψv ).

Similarly, if we pick up α
0, ⎪ ⎪ ⎪ ⎪ ⎪ S I j (0, a) = I j,0 (a), j (0, a) = S j,0 (a), ⎪ ⎪ ⎩ Iv (t, 0) = Iv0 , j ∈ N.

(7.66)

Obviously, equilbria of (7.66) satisfy the following equations ⎧ ∗ dS j (a) ⎪ ⎪ = −λ j∗ (a)S∗j (a) − μ j (a)S∗j (a), ⎪ ⎪ da ⎪ ⎪ ⎪ ⎪ dI ∗j (a) ⎪ ⎪ ⎪ = λ j∗ (a)S∗j (a) − {μ j (a) + γ j (a)}I ∗j (a), ⎪ ⎪ da ⎪ ⎪ ⎪ ⎪ ⎨ 0 = (Nv∗ − Iv∗ )λv∗ − μv Iv∗ , n

 ∞

⎪ ⎪ λ j∗ (a) = ∑ β jk (a, σ )Ik∗ (σ )d σ + βvhj (a)Iv∗ , ⎪ ⎪ 0 ⎪ k=1 ⎪ ⎪ ⎪ n ⎪ ⎪ ⎪ k ⎪ λv∗ = ∑ βhv (a)Ik∗ (a)da, ⎪ ⎪ ⎪ k=1 ⎪ ⎪ ⎩ ∗ S j (0) = b j , I ∗j (0) = 0, j = 1, 2, · · · , n.

(7.67)

Equation (7.67) has a disease-free equilibrium E0 = (P∗ (a), 0, 0) and an endemic equilibrium E ∗ = (S∗ (a), I ∗ (a), Iv∗ ), whose components are given by Iv∗ =

λv∗ Nv∗ , λv∗ + μv

S∗j (a) =Pj∗ (a)e−  a

a ∗ 0 λ (s)ds a

,

λ1j∗ (s)e− s {μ j (ξ )+γ j (ξ )}d ξ ds, 0 λ1j∗ (a) =S∗j (a)λ j∗ (a)  σ n  ∞ σ λ j∗ (a) = β jk (a, σ ) λ1k∗ (ρ )e− ρ {μk (s)+γk (s)}ds d ρ d σ 0 k=1 0 I ∗j (a) =

∑

+

βvhj (a)λv∗ , λv∗ + μv

7.2 Multigroup Epidemic Model with Age Structure n

λv∗ = ∑

 ∞

k=1 0 n  ∞

=∑

k=1 0

289

βvhj (a)I ∗j (a)da βhvj (a)

 a 0

λ1j∗ (s)e−

a s

(μ j (ξ )+γ j (ξ ))d ξ

dsda.

(7.68)

From the expression of S∗j (a), it follows that

λ1j∗ (a) = Pj∗ (a)H j (2 λ j∗ )(a)e−

a 0

H j (2 λ j∗ (σ ))d σ

,

where H j (ϕ )(a) =

n

∑

 ∞

β jk (a, σ )

 σ

ϕk (ρ )e−

σ ρ

{μk (ξ )+γk (ξ )}d ξ

dρ dσ

0 k=1 0 a n   j ∞ j ∗ βvh (a)Nv ∑ 0 βhv (a) 0a ϕ j (s)e− s (μ j (ξ )+γ j (ξ ))d ξ dsda j=1 + n  . a a ∞ j ∑ 0 βhv (a) 0 ϕ j (s)e− s (μ j (ξ )+γ j (ξ ))d ξ dsda + μv j=1

Note that λ j∗ = H j (λ˜ j∗ ), j = 1, 2, · · · , n. Then

λ˜ j∗ (a) = S∗j (a)λ j∗ (a) = λ j∗ (a)Pj∗ (a)e−

a ∗ 0 λ j (s)ds

= Pj∗ (a)H j (λ˜ ∗ )(a)e−

a 0

H j (λ˜ ∗ )(σ )d σ

.

Therefore, we can define a nonlinear operator

Φ (ϕ )(a) = (Φ1 (ϕ )(a), · · · , Φ (ϕ )(a))T ∈ X, ϕ ∈ X where

Φ j (ϕ )(a) = Pj∗ (a)H j (ϕ )(a)e−

a 0

H j (ϕ j (σ ))d σ

(7.69)

.

If (7.66) has an endemic equilibrium E ∗

then the operator Φ has a positive nontrivial fixed point. Noting that H j (0) = 0, the Fréchet derivative of Φ at ϕ = 0 is defined by

Φ  [0](ϕ )(a) = (P1∗ (a)H1 (ϕ )(a), P2∗ (a)H2 (ϕ )(a), · · · , Pn∗ (a)Hn (ϕ )(a))T , ϕ ∈ X (7.70) where (ϕ )(a) Pj∗ (a)H j = Pj∗ (a) +

n

∑

 ∞

β jk (a, σ )

k=1 0  βvhj (a)Nv∗ n ∞

μv

∑

j=1 0

βhvj (a)

 σ 0

 a 0

ϕk (ρ )e−

ϕ j (s)e−

σ

a s

ρ

{μk (ξ )+γk (ξ )}d ξ

(μ j (ξ )+γ j (ξ ))d ξ

dρ dσ #

dsda .

Actually, Φ  [0] is the next generation operator K defined by (7.70). In what follows, we determine the existence of the positive fixed point of Φ .

290

7 Metapopulation and Multigroup Age-Structured Models

Assumption 11 (i) For every j, k ∈ {1, 2, · · · , n}, the following equality holds uniformly for all σ ≥ 0  ∞

lim

h→0 0

|β jk (a + h, σ ) − β jk (a, σ )|da = 0;

 ∞

lim

h→0 0

|βvhj (a + h) − βvhj (a)|da = 0;

(ii) For every j, k ∈ {1, 2, · · · , n}, there exists ε0 and a˜ ∈ (0, +∞) such that

β jk (a, σ ) ≥ ε0 , for almost everywhere (a, σ ) ∈ [0, +∞) × [a, ˜ +∞), βvhj (a) ≥ ε0 , for almost everywhere a ∈ [a, ˜ +∞). Using Assumption 11, we give the following lemma. Theorem 7.2 (Kolmogorov Theorem, [193]). Let B be a bounded set in L1 (Rn+ ). Then the subset B is relative compact if and only if the following properties hold: (i) lim

∞

A→∞

A

f (a)da = 0 uniformly on B.

(ii) lim f (a + h) − f (a) = 0 uniformly on B. h→0

Lemma 7.8. K is compact and nonsupporting. Proof. Assume that B0 is a closed subset of X. Then there exists a positive constant c0 > 0 such that ϕ ≤ c0 for all ϕ ∈ B0 . By Kolmogorov Theorem 7.2 we need to show that for all ϕ ∈ B0 lim (K ϕ )(a + h) − (K ϕ )(a) X = 0.

h→0

Note that

(K ϕ )(a + h) − (K ϕ )(a) X = ≤ +

n  ∞ ∞ ∞

∑

j,k=1 0 Nv∗ n

μv

∑

ρ

 ∞

j=1 0

|(K j ϕ )(a + h) − (K j ϕ )(a)|da

|Pj∗ (a + h)β jk (a + h, σ ) − Pj∗ (a)β jk (a, σ )|dae−

ρ 0  ∞ ∞ ∞

j,k=1 0

n

∑

0

k |βvhj (a + h) − βvhj (a)|daβhv (σ )e−

σ ρ

σ ρ

(μk (ξ )+γk (ξ ))d ξ

(μk (ξ )+γk (ξ ))d ξ

d σ |ϕk (ρ )|d ρ

d σ |ϕk (ρ )|d ρ .

Since Pj∗ (a) is a continuous, monotone nonincreasing and lim Pj∗ (a) = 0. Based on a→∞

Assumption 11, for any ε > 0 there exists a δ > 0 such that  ∞

ε |β jk (a + h) − β jk (a)|da < , 2 0 ∞ ε |βvhj (a + h) − βvhj (a)|da < +, 2βhv 0

(7.71)

7.2 Multigroup Epidemic Model with Age Structure

291

Equation (7.71) implies that 



∞ ∞ ε e−μ (σ −ρ ) d σ |ϕk (ρ )|d ρ ∑ 2 k, j=1 0 ρ   εβ + ∞ ∞ −μ (σ −ρ ) + hv e d σ |ϕk (ρ )|d ρ 2 0 ρ ε ε = ϕ X ≤ c0 , μ μ

(K ϕ )(a + h) − (K ϕ )(a) X ≤

(7.72)

which implies that the first condition of Kolmogorov Theorem holds.  Next, we show that lim ∑ A∞ |(K j ϕ )(a)|da = 0 uniformly for ϕ ∈ B0 . In fact, A→∞ j=1

 ∞

|(K j ϕ )(a)|da A→∞  ∞ A n  ∞ ∞ ρ |Pj∗ (a) β jk (a, σ )e− σ (μk (ξ )+γk (ξ ))d ξ d σ ϕk (ρ )d ρ |da ρ A k=1 0  n  ∞ ∞ ρ Nv∗ ∞ ∗ k |P (a) βvhj (a)βhv (σ )e− σ (μk (ξ )+γk (ξ ))d ξ d σ ϕk (ρ )d ρ |da + μv A ρ k=1 0  ∞ n  ∞ ∞  − 0a μ j (s)ds + |e e−μ (σ −ρ ) d σ ϕk (ρ )d ρ |da b jβ j ρ A 0 k=1 + + ∗ ∞ n  ∞ ∞  b j βvh βhv Nv − 0a μ j (s)ds + |e e−μ (σ −ρ ) d σ ϕk (ρ )d ρ |da μv A 0 ρ k=1 + + ∗ βhv Nv /μv  ∞ −μ a b j β j+ + b j βvh e da ϕ X μ A + + ∗ βhv Nv /μv −μ A b j β j+ + b j βvh e c0 . 2 lim

=

∑

∑

≤

∑

∑

≤ ≤

μ

By the above inequality, it is easy to show that lim

A→∞

∞ A

(7.73) |(K j ϕ )(a)|da converges to

zero when A goes to infinity for any ϕ ∈ B0 . This implies that the condition (ii) of Kolmogorov Theorem holds. Therefore, the linear operator K is compact. In what follows, we will show that the operator K is nonsupporting. Let X ∗ be the dual space of X.  f , ϕ  denotes the inner product for f ∈ X ∗ and ϕ ∈ X. Then X+∗ denotes all the positive functionals. This implies that  f , ϕ  is strictly positive for all ϕ ∈ X+ and f ∈ X+∗ . For every j, k ∈ {1, 2, · · · , n}, define

ε˜0 , a ∈ [a, ˜ +∞), ˜ β jk (a) = 0, otherwise,

and

β˜hvj (a) =

εˆ0 , a ∈ [a, ˆ +∞), 0, otherwise.

292

7 Metapopulation and Multigroup Age-Structured Models

Then an associated positive linear functional is defined by  f˜, ϕ  :=

n

∑

 ∞ ∞

ρ k=1 0 β˜vhj Nv∗ n

+

μv

β˜ jk (σ )e−

∑

 ∞ ∞ ρ

k=1 0

σ

(μk (ξ )+γk (ξ ))d ξ

ρ

k β˜hv (σ )e−

σ ρ

d σ ϕk (ρ )d ρ

(μk (ξ )+γk (ξ ))d ξ

d σ ϕk (ρ )d ρ .

By definitions of β˜ jk and β˜hvj , we know that  f˜, ϕ  > 0 for all ϕ ∈ X+ \{0}. It follows from the definition of K that K ϕ ≥  f˜, ϕ P∗ . Then, for any positive integer n ∈ N, we have K n ϕ ≥  f˜, ϕ  f˜, ϕ n−1 P∗ which implies for every pair ϕ ∈ X+ \{0} and f ∈ X+∗ \{0}  f , K n ϕ  > 0. This proves that K is nonsupporting.   Lemma 7.9. Let Φ be defined by (7.69). Then Φ is compact, Φ (X+ ) is bounded. This means that there exists a positive constant M˜ 0 > 0 such that for all ϕ ∈ X+ ,

Φ ≤ M˜ 0 holds. Proof. In order to prove this lemma, we write the nonlinear operator as composition Φ = Φ˜ ◦ H where

Φ˜ (ϕ )(a) = (Φ˜ (ϕ )(a), Φ˜ (ϕ )(a), · · · , Φ˜ (ϕ )(a))T ∈ X, ϕ ∈ X, and

Φ˜ j (ϕ )(a) = Pj∗ (a)ϕ j (a)e−

a 0

ϕ j (s)ds

, j = 1, 2, · · · , n, ϕ ∈ X.

First, it follows from Lemma 5.2 [107] that Φ˜ (X+ ) is bounded. In what follows, we just need to show that H is a compact operator. Note that

H (ϕ )(a + h) − H (ϕ )(a) = ≤

n

∑

 ∞

j=1 0 n  ∞

∑

j=1 0

|H j (ϕ )(a + h) − H j (ϕ )(a)|da |K j (ϕ )(a + h) − K j (ϕ )(a)|da.

By (7.72) that H (ϕ )(a + h) − H (ϕ )(a) converges to zero when h goes to zero. On the other hand,  ∞

lim

A→∞ A

|H j (ϕ )(a)|da ≤ lim

 ∞

A→∞ A

Employing (7.73), we have lim

A→∞

∞ A

|K j (ϕ )(a)|da.

|H j (ϕ )(a)|da = 0. The above discussion in-

dicates that the operator H is compact. Since Φ is a composition of a bounded operator Φ˜ and a compact operator H , Φ is also a compact operator. This finishes the proof.

7.2 Multigroup Epidemic Model with Age Structure

293

Lemmas A.6 and A.7 together with Theorem A.9 yield the following proposition on the existence of the endemic steady state. Proposition 7.8. If R0 > 1, then Φ has at least one nontrivial fixed-point in X+ \{0}. Proof. It follows from Lemma 7.9 that the nonlinear operator Φ is compact and Φ (X+ ) is bounded. Since Φ (0) = 0, (i) of Lemma A.6 holds. (ii) of Lemma A.6 immediately follows by the definition of K = Φ  [0]. By the compactness and nonsupporting property of K , we employ Lemma A.7 to obtain that r(K ) > 1 is the only positive eigenvalue of K corresponding to a positive eigenvector and there is no eigenvector of K corresponding to eigenvalue 1. This implies (iii) of Lemma A.6 is satisfied. Therefore, Φ has at least one nontrivial fixed-point in X+ \{0}. Proposition 7.8 implies that if R0 > 1 system (7.46) has at least one nontrivial endemic equilibrium (S∗ , I ∗ , Iv∗ ) ∈ X+ \{0}. In order to obtain the uniqueness of the equilibrium of (7.46), we make the following assumptions. Assumption 12 For each j, k ∈ {1, 2, · · · , n}, there exist positive bounded functions ∞ (0, ∞) such that β (a, σ ) = β 1 (a)β 2 (σ ). β j1 , β j2 ∈ L+ j,k j j It follows from the definition of λ j∗ (a) that

λ j∗ (a) =

n

∑

 ∞

k=1 0

β jk (a, σ ) n

+

βvhj (a)Nv∗ ∑ n

∑

n

=∑

k=1 ∞



0

λk∗ (ρ )Pj∗ (ρ )e−

ρ 0



λk∗ (ξ )d ξ − ρσ (μk (ξ )+γk (ξ ))d ξ e dρ dσ

σ ρ ∞ k σ ∗ − 0 λk∗ (ξ )d ξ − ρ (μk (ξ )+γk (ξ ))d ξ e dρ dσ 0 βhv (σ ) 0 Pk (ρ )e

k=1 σ ρ ∗  k βhv (σ ) 0σ Pk∗ (ρ )e− 0 λk (ξ )d ξ e− ρ (μk (ξ )+γk (ξ ))d ξ d ρ d σ

β jk (a, σ )Pj∗ (σ )

k=1 0

n

+

 σ

βvhj (a)Nv∗ ∑

k=1

 σ 0

λk∗ (ρ )e−

ρ 0

 λk∗ (ξ )d ξ − ρσ γk (ξ )d ξ

e

+ μv

dρ dσ

σ ρ ∞ k σ ∗ − 0 λk∗ (ξ )d ξ − ρ γk (ξ )d ξ ∗ e dρ dσ 0 βhv (σ )Pk (σ ) 0 λk (ρ )e

σ ρ ∗ ∞ k  (σ )Pk∗ (σ ) 0σ λk∗ (ρ )e− 0 λk (ξ )d ξ e− ρ γk (ξ )d ξ d ρ d σ + μv ∑ 0 βhv

n

k=1

n

=∑

 ∞ ∞ ρ

k=1 0

β jk (a, σ )Pj∗ (σ )e− n

+

βvhj (a)Nv∗ ∑

k=1

σ ρ

γk (ξ )d ξ

d σ λk∗ (ρ )e−

ρ 0

λk∗ (ξ )d ξ

dρ

 ρ ∗ ∞∞ k − ρσ γk (ξ )d ξ ∗ d σ λk∗ (ρ )e− 0 λk (ξ )d ξ d ρ 0 ρ βhv (σ )Pk (σ )e

σ ρ ∗ ∞∞ k (σ )Pk∗ (σ )e− ρ γk (ξ )d ξ d σ λk∗ (ρ )e− 0 λk (ξ )d ξ d ρ + μv ∑ 0 ρ βhv

n

.

k=1

φ jk (a, ρ ) =  − ρσ γk (ξ )d ξ ∗ Pk (σ )e dσ , Define

 − ρσ γk (ξ )d ξ ∗ dσ , ρ β jk (a, σ )Pk (σ )e ∗ j, k = 1, 2, · · · , n. Then λ j (a) can be

∞

(7.74)

 k (σ ) ψk (ρ ) = ρ∞ βhv

rewritten as

294

7 Metapopulation and Multigroup Age-Structured Models

λ j∗ (a) =

n

∑

 ∞

φ jk (a, ρ )λk∗ (σ )e−

ρ 0

λk∗ (ξ )d ξ

dρ

k=1 0 ρ ∗ n  βvhj (a)Nv∗ ∑ 0∞ ψk (ρ )λk∗ (ρ )e− 0 λk (ξ )d ξ d ρ + n  k=1 , ρ ∗ ∞ ∑ 0 ψk (ρ )λk∗ (ρ )e− 0 λk (ξ )d ξ d ρ + μv k=1

(7.75) j = 1, 2, · · · , n.

Therefore, we can define a nonlinear operator Φ˜ = (Φ˜ 1 , Φ˜ 2 , · · · , Φ˜ n )T where n

Φ˜ j (ϕ )(a) = ∑

k=1

+

∞

φ jk (a, ρ )ϕk (ρ )e−

0

n

βvhj (a)Nv∗ ∑ ∞

n

∑

k=1

0

k=1

∞ 0

ρ 0

ϕk (ξ )d ξ

ψk (ρ )ϕk (ρ )e−

ψk (ρ )ϕk (ρ )e−

ρ 0

dρ ρ 0

ϕk (ξ )d ξ

ϕk (ξ )d ξ

dρ , j = 1, 2, · · · .

d ρ + μv

Proposition 7.9. If the Assumption 12 holds, Φ has at most one nontrivial fixedpoint in X+ \{0}. Proof. If Assumption 12 holds, it is obvious to see that n

Φ˜ j (ϕ )(a) = β j1 (a) ∑

 ρ ∞∞ 2 − ρσ γk (ξ )d ξ ∗ d σ ϕk (ρ )e− 0 ϕk (ξ )d ξ d ρ 0 ρ βk (σ )Pk (σ )e

k=1 ρ n  j βvh (a)Nv∗ ∑ 0∞ ψk (ρ )ϕk (ρ )e− 0 ϕk (ξ )d ξ d ρ + n  k=1 . ρ ∞ ∑ 0 ψk (ρ )ϕk (ρ )e− 0 ϕk (ξ )d ξ d ρ + μv k=1

(7.76)

The solution of the fixed-point problem λ ∗ = Φ (λ ∗ ) is given by

λ ∗ = (cβ11 , cβ21 , · · · , cβn1 ),

(7.77)

Substituting (7.77) into (7.76) yields 1=

n

∑

 ∞ ∞

βk2 (σ )Pk∗ (σ )e−

σ ρ

γk (ξ )d ξ

d σ βk1 (ρ )e−c

ρ k=1 0 ρ n  1 j βvh (a)Nv∗ ∑ 0∞ ψk (ρ )βk1 (ρ )e− 0 cβk (ξ )d ξ d ρ + n  k=1 . ρ 1 ∞ ∑ 0 ψk (ρ )βk1 (ρ )e− 0 cβk (ξ )d ξ d ρ + μv k=1

ρ 0

βk1 (ξ )d ξ

dρ

It is easy to see that the right of the above equation is a decreasing function with respect to c. This implies that there exists a unique solution of λ ∗ . Proposition 7.9 indicates that if R0 > 1 and Assumption 12 hold, system (7.46) has a unique nontrivial endemic equilibrium (S∗ , I ∗ , R∗ , Iv∗ ) ∈ X+ \{0}.

7.2 Multigroup Epidemic Model with Age Structure

295

7.2.4 Global Stability of the Disease-Free Equilibria In this section, we show that if R0 < 1, the solutions of (7.50) converge to the disease-free equilibrium. By the definition of the operator B1 , the second equation and third equation of system (7.50) can be rewritten as ⎧ dI ⎨ dt = B1 I(t) + (Λ I(t) + βvh(a) Iv (t))S(t), dIv (7.78) = −μv Iv + Λv Sv (t), ⎩ dt I(0) = I0 ∈ X, Iv (0) = Iv0 ∈ R+ where S, Sv , I, I0 , Iv0 denote the vector function and Λ defined as (7.52). The life span of mosquitos is much shorter than the human life span. So, we assume the infected Λ (I)N ∗ v (t) mosquitos approach a quasi-steady state, i.e. Iv = Λv (I)S ≤ v μv v . Substituting μv the formula Iv into the first equation of (7.78), we obtain dI(t) N ∗ βvh (a) ≤ B1 I(t) + (Λ I(t) + v Λv (I))P∗ . dt μv

(7.79)

Define a linear operator P(ϕ )(a) = (P1 (ϕ )(a), P2 (ϕ )(a), · · · , Pn (ϕ )(a))T , j ∈ {1, 2, · · · , n}.

(7.80)

where n

Pj (ϕ )(a) = Pj∗ ( ∑

 ∞

k=1 0

β jk (a, σ )ϕk (σ )d σ +

Nv∗ βvhj (a) n ∑ μv k=1

 ∞ 0

k βhv (a)ϕk (a)da).

Solving the first equation in (7.78), together with S ≤ P∗ , we have I(t) = etB I0 +

 t 0

e(t−s)B (Λ I(s))S(s)ds ≤ etB I0 +

 t 0

e(t−s)B (PI(s))ds.

(7.81)

Thus, 0 ≤ I(t) ≤ W (t)I0 , ∀t ≥ 0, W (t) is a C0 -semigroup generated by the perturbed operator B+P. By the definition of growth bound and the α -growth bound of W (t), we have ln W (t) , ω0 (W ) = lim t→∞ t and ln α [W (t)] , ω1 (W ) = lim t→∞ t where α [A] denotes the measure of noncompactness of a bounded linear operator A. Lemma 7.10 (Proposition 4.13 [179]). Let T (t) t ≥ 0 be a strongly continuous semigroup of bounded operators in the Banach space in X with infinitesimal generator B. The following hold:

296

7 Metapopulation and Multigroup Age-Structured Models

(i) rσ (T (t)) = exp[ω0 (B)t], for t ≥ 0; (ii) rEσ (T (t)) = exp[ω1 (B)t], for t > 0; (iii) sup ≤ ω0 (B); λ ∈σ (B)

(vi)

sup λ ∈Eσ (B)

≤ ω1 (B);

(v) ω0 = max{ω1 (B),

sup λ ∈σ (B) Eσ (B)

Reλ }.

In this lemma, σ (A) and Eσ (A) denote the spectral and the essential spectral of a closed linear operator A, respectively. By Lemma 7.10, we have that  #

ω0 (B1 + P) = max ω1 (B1 + P),

sup z∈σ (B1 +P)\Eσ (B1 +P)

Re z .

(7.82)

Next, it is necessary to investigate the growth bound ω0 (B1 + P) < 0 if R0 < 1. Lemma 7.11 (Theorem 4.6 [179]). Let β : L1 → Rn be the birth function and μ : L1 → Rn be death function, S (t) is the strongly continuous semigroup of bounded linear operators with infinitesimal generator B. If β ∈ L∞ and μ ∈ L∞ , the following hold: (i) ω1 (B) ≤ −μ ; (ii) If Reλ > −μ , and det Δ (λ ) = 0, then λ ∈ Pσ (B); (iii) If Reλ > −μ , and λ ∈ ρ (B), then for ψ ∈ L1 , a ≥ 0 ((λ I − B)−1 ψ )(a) =

 a

eλ (a−b) π (a, b)ψ (b)db + e−λ a π (a, 0)Δ (λ )−1

0

×[ where Δ (λ )x = x −

b

0

 ∞ 0

β (b)e−λ b

eλ τ π (b, τ )ψ (τ )d τ ]db

b  ∞ −λ a e β (a)π (a, 0)xda and π (a, b) = e− a μ (s)ds .

0

Proposition 7.10 (Proposition 4.14 [179]). Let T (t) be a strongly continuous semigroup of bounded linear operators with infinitesimal generator B and C is a bounded operator in the Banach space X. Then B + C is the infinitesimal generator of a strongly continuous semigroup of bounded linear operator S (t),t ≥ 0 in X. S (t)x = T (t)x +

 t 0

T (t − s)CS (s)xds,t ≥ 0, x ∈ X.

Moreover, if C is compact then ω1 (B) = ω1 (B +C). We will combine Lemma 7.11 and Proposition 7.10 to prove the following Proposition. Proposition 7.11. Let B1 and P be defined by (7.61) and (7.80).

ω1 (B1 + P) = ω1 (B1 ) ≤ −μ , where μ is defined in the original assumptions.

7.2 Multigroup Epidemic Model with Age Structure

297

Proof. Note that the strongly continuous semigroup T (t) generated by the linear bounded operator B1 is defined as follows:

0, t ≥ a, (T (t)ϕ )(a) = eB1 (t−a) ϕ (a − t), t < a. It follows from the Lemma 7.11 that α ([T (t)]) ≤ T (t) ≤ e−μ t . Moreover, employing bound P∗ ≤ b, together with Assumption 11 and Proposition 7.10, we have ω1 (B1 + P) = ω1 (B1 ). For m ∈ C with Re m > −μ , the following operator is defined on X: ⎛

n

∑

∞

n

φ1k (m, a, ρ )ϕk (ρ )d ρ + ∑

∞

φˆ1k (m, a, ρ )ϕk (ρ )d ρ

⎞

0 ⎟ ⎜ k=1 0 k=1 ⎟ ⎜ . ⎟ . (Lm ϕ )(a) := ⎜ . ⎟ ⎜ ⎠ ⎝ n  n  ∞ ∞ ˆ ∑ 0 φnk (m, a, ρ )ϕk (ρ )d ρ + ∑ 0 φnk (m, a, ρ )ϕk (ρ )d ρ k=1

k=1

(7.83) with

φ jk (m, a, ρ ) := Pj∗ (a)

 ∞ ρ

β jk (a, σ )e−

σ ρ

{m+μk (ξ )+γk (ξ )}d ξ

d σ , j, k = 1, 2, · · · , n,

and

φˆ jk (m, a, ρ ) :=

Pj∗ (a)βvhj Nv∗ 

μv

∞

ρ

k βhv (σ )e−

σ ρ

{m+μk (ξ )+γk (ξ )}d ξ

d σ , j, k = 1, 2, · · · , n.

When m = 0, L0 = K , where K is the next generation operator defined by (7.65). Proposition 7.12. Let

ϒ = {m ∈ C : 1 ∈ Pσ (Lm ), Re m > −μ },

(7.84)

ϒ = Pσ (B1 + P) ∩ {m ∈ C : Re m > −μ }.

(7.85)

then Proof. If m ∈ ϒ , define an eigenvector

ψm := (ψm,1 , · · · , ψm,n )T ∈ X with respect to 1, i.e. ψm = ϒ ϕm . Again, we define a function

ψ˜ = (ψ˜ 1 , · · · , ψ˜ 2 )T , with

ψ˜ j (a) =

 a 0

e−

a

σ {m+ μ j (ξ )+γ j (ξ )}d ξ

ψm, j (σ )d σ , j = 1, 2, · · · , n.

298

7 Metapopulation and Multigroup Age-Structured Models

By the definition of B1 defined in (7.61), we obtain B1 ψ˜ (a) = −ψm, j (a) + mψ˜ j (a), j = 1, 2, · · · , n and (Pψ˜ ) j (a) = (ϒm ψm ) j (a) = ψm, j , j = 1, 2, · · · , n. This means that (B1 + P)ψ˜ = mψ˜ , ψ˜ is an eigenvector of operator B1 + P with respect to m ∈ C and Re m > −μ . Next, we want to show that given m ∈ Pσ (B1 + P) and Re m > −μ , there exists a function ϕ˜ ∈ D(B1 ) such that mϕ˜ = (B1 + P)ϕ . By the definition of B1 , we have

ϕ˜  (a) = −mϕ˜ − {Q1 (a) + Γ (a)}ϕ˜ (a) + P ϕ˜ (a). Integrating the above equality yields

ϕ˜ j (a) =

 a 0

e−

a

σ (m+ μ j (ξ )+γ j (ξ ))d ξ

(P ϕ˜ ) j (σ )d σ , j = 1, 2, · · · , n

with ⎛

⎞ σ n   P1∗ ∑ 0∞ β1k (a, σ ) 0σ e− ρ {m+μk (ξ )+γk (ξ )}d ξ (P ϕ˜ )k d ρ d σ ⎜ ⎟ k=1 ⎜ ⎟  σ −  σ {m+μk (ξ )+γk (ξ )}d ξ ⎜ Nv∗ βvh1 (a) n  ∞ k ⎟ ρ (Pϕ˜ )k d ρ da ⎟ ∑ 0 βhv (a) 0 e ⎜ + μv ⎜ ⎟ k=1 ⎜ ⎟ .. ⎟ P ϕ˜ (a) = ⎜ . ⎜ ⎟ ⎜ ⎟  ⎜ P∗ n  ∞ β (a, σ )  σ e− ρσ {m+μk (ξ )+γk (ξ )}d ξ (P ϕ˜ ) d ρ d σ ⎟ k ⎜ n ∑ 0 nk ⎟ 0 ⎜ ⎟ k=1  n ⎝ Nv∗ β n (a) ⎠ ∞ k  σ − σ {m+μk (ξ )+γk (ξ )}d ξ vh ρ + μv (Pϕ˜ )k d ρ da ∑ 0 βhv (a) 0 e k=1 ⎛ n  ⎞ n  ∞ φ (m, a, ρ )(P(ρ˜ ))k d ρ + ∑ 0∞ φˆ1k (m, a, ρ )(P(ρ˜ ))k d ρ ∑ 0 1k ⎜ k=1 ⎟ k=1 ⎜ ⎟ . ⎜ ⎟ . =⎜ . ⎟ ⎝ n  ⎠ n  ∞ ∞ ˆ ˜ ˜ φnk (m, a, ρ )(P(ρ ))k d ρ + ∑ φnk (m, a, ρ )(P(ρ ))k d ρ ∑ =

k=1 Lm Pϕ˜ ,

0

k=1

0

which indicates that P ϕ˜ is an eigenfunction associated with an eigenvalue unity of Lm , i.e. m ∈ ϒ . Observing Assumption 11, we obtain that Lm is nonsupporting which proof follows from Lemma 7.8.

7.2 Multigroup Epidemic Model with Age Structure

299

Lemma 7.12 (Lemma 7.3 [107]). sign(R0 − 1) = sign(ω0 (B1 + P)). Lemma 7.12 implies that ω0 (B1 + P) < 0 if R0 < 1. Hence lim W (t)I0 = 0. Theret→∞ fore, we can give the following theorem. Theorem 7.3. If R0 < 1, then the disease-free equilibrium (P∗ , 0, 0, Nv∗ , 0) ∈ X+ of system (7.46) is globally asymptotically stable. From what we have been discussed, we just focus on the global attractivity of the disease-free equilibrium E0 . For the local stability and attractivity of the endemic equilibrium, we leave them to readers. If we choose β jk (a, σ ) = β jk (σ ), μ j (a) = μ j and γ j (a) = γ j similar to [107], the global attractivity of the endemic equilibrium can be obtained by constructing suitable Lyapunov functional.

Chapter 8

Class Age-Structured Epidemic Models

8.1 Introduction In most epidemiological models for the transmission of infectious diseases, the infectious individuals are assumed to have the same infectivity. This assumption is reasonable in modeling communicable diseases such as influenza [24] and sexually transmitted diseases such as gonorrhea [82]. However, in the study of diseases such as HIV/AIDS, tuberculosis, and hepatitis C that have a long-term latent or chronic stage, it is necessary to incorporate the infection age (that is, the time that has passed since infection) into the model. Beside infection age, other class ages, such as latent age, vaccination age, immunity age, etc., are also important factors affecting the transmission of infectious diseases. In this chapter, we formulate and analyze several class age-structured epidemic models. It is our goal to illustrate more advanced mathematical techniques for the analysis of such models such as integrated semigroup techniques. We illustrate the same technique on several different examples and demonstrate how different class-age models can be analyzed within the same framework. Because the classage models are often associated with nonlinear boundary conditions, the natural setup to handle this difficulty is the contexts of integrated semigroups. The abstract approach is absolutely necessary to justify inferring local stability of the equilibria from the roots of the characteristic equation—a result that is not automatically true for PDE models. Most of the techniques in this chapter come from [118].

8.2 SIRS Model with Age of Infection In the classic SIRS models, the total population is divided into three epidemiological classes: susceptible, infected, and recovered (or removed). The numbers of individuals in these classes at time t are denoted, respectively, by S(t), I(t), and R(t). Our class-age structured models are based on the following SIRS ODE model, © Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1_8

301

302

8 Class Age-Structured Epidemic Models

⎧ dS(t) ⎪ ⎪ = Λ − β S(t)I(t) − μ S(t) + δ R(t), ⎪ ⎪ ⎨ dt dI(t) = β S(t)I(t) − (μ + k)I(t), ⎪ dt ⎪ ⎪ ⎪ ⎩ dR(t) = kI(t) − (μ + δ )R(t). dt Here Λ is the recruitment rate of susceptibles, β is the transmission coefficient, μ is the natural death rate, 1/k is the infectious period, and δ is the loss of immunity rate. To incorporate the age of infection (for simplicity, referred to as age in the remaining of this chapter), let a denote the age-since-infection and let i(t, a) denote the age density of infected individuals at time t. Then we obtain the following model described by a system of ordinary and partial differential equations, ⎧  ∞ dS(t) ⎪ ⎪ = Λ − S(t) β (a)i(t, a)da − μ S(t) + δ R(t), ⎪ ⎪ ⎪ 0 ⎨ dt ∂ i(t, a) ∂ i(t, a) (8.1) + = −(μ + k(a))i(t, a), ⎪ ∂t ⎪  ∞∂ a ⎪ ⎪ dR(t) ⎪ ⎩ = k(a)i(t, a)da − (μ + δ )R(t), dt 0 with the boundary and initial conditions i(t, 0) = S(t)

 ∞ 0

β (a)i(t, a)da,

t > 0,

R(0) = R0 ≥ 0, S(0) = S0 ≥ 0, 1 i(0, a) = i0 (a) ∈ L+ (R+ , R), where β (a) is the transmission coefficient, which is dependent on age a, and the infectious period is given by  ∞

e−

a 0

k(σ )d σ

da.

0

Model (8.1) uses mass action incidence. Assumption 13 All the parameters satisfy all the following conditions: (i) Λ , δ , and μ are nonnegative. (ii) β (a), k(a) ∈ CBU (R+ , R+ ), and |β (a + h) − β (a)| ≤ Lβ h, a ∈ R+ . (iii) β (a) ∈ L∞ (R+ ), that is, there exists a constant β¯ such that

β¯ = ess sup β (a). a∈R+ 1 (R , R ) for all a ∈ R . (iv) k(a) ∈ L+ + + +

8.2 SIRS Model with Age of Infection

303

Notice that the total population at time t is N(t) = S(t) +

 ∞ 0

i(t, a)da + R(t).

We can easily see that N(t) satisfies the following ordinary differential equation, dN(t) = Λ − μ N(t). dt It follows that lim N(t) =

t→∞

Λ . μ

(8.2)

We will rewrite (8.1) in a form convenient to put in abstract formulation. Because the boundary condition in (8.1) is nonlinear, and the boundary condition is part of the domain of the C0 -semigroup generator, we cannot put problem (8.1) into the context of C0 -semigroups with densely defined generator. In general there are two ways to avoid this problem that emerges with nonlinear boundary conditions: (1) to change variables and put the boundary condition(s) as part of the main system; or (2) to use integrated semigroups that have non-densely defined generators. We choose to follow the second approach. For the first approach, see Iannelli and Martcheva well-posedness of two-sex model [86]. We rewrite (8.1) in a more convenient form: ⎧ ∂ i(t, a) ∂ i(t, a) ⎪ ⎪ + = −(μ + k(a))i(t, a), ⎪ ⎪ ∂t ∂a ⎪ ⎪ ⎪ dV (t) ⎪ ⎨ = G(i(t, a),V (t)) − EV (t), dt  ∞ (8.3) ⎪ i(t, 0) = S(t) β (a)i(t, a)da, t > 0, ⎪ ⎪ ⎪ 0 ⎪ ⎪ 1 (R , R), ⎪ i(0, a) = i0 (a) ∈ L+ + ⎪ ⎩ V (0) = V0 ∈ R2+ , where

 V (t) =

S(t) , R(t)

 E=

μ 0 0 μ +δ

,

⎞  ∞ Λ − S(t) β (a)i(t, a)da + δ R(t) ⎟ ⎜ 0 ∞ G(i(t, a),V (t)) = ⎝ ⎠. k(a)i(t, a)da ⎛

0 1 We extend the main $  space $ of the PDE (L ) with the real axes and define Y = R × $ $ α $ 2 L1 (R+ , R) with $ $ φ $ = |α | + φ L1 (R+ ,R) . We set X = Y × R . Furthermore, Y we define

X+ = Y+ × R2+ ,

X0 = Y0 × R2 ,

X0+ = X0 ∩ X+ ,

304

8 Class Age-Structured Epidemic Models

where 1 (R+ , R), Y+ = R+ × L+

Y0 = {0} × L1 (R+ , R).

We define the generator of (8.3) A1 : D(A1 ) ⊂ Y → Y by   0 −φ (0) A1 = φ −φ  − (μ + k(a))φ with D(A1 ) = {0} ×W 1,1 (R+ , R). Hence, A1 is not densely defined on Y as its first component is only defined at zero. If λ ∈ C with Reλ > −μ , then λ ∈ ρ (A1 ), where ρ (A1 ) is the resolvent set of A1 . Moreover, if λ ∈ ρ (A1 ) and   θ 0 −1 = , (λ I − A1 ) φ ψ then

φ (a) = e−(λ +μ )a e−

a 0

k(σ )d σ

θ+

 a

e−

a s

k(l)dl −(λ +μ )(a−s)

0

e

ψ (s)ds.

Now we can rewrite (8.3) as ⎧    d 0 0 0 ⎪ ⎪ + F1 ,V (t) , = A1 ⎪ ⎪ i(t, .) ⎪ dt i(t, .) ⎪  i(t, .) ⎨ dV (t) 0 = −EV (t) + F2 ,V (t) , i(t, .) ⎪ dt ⎪ ⎪ 1 ⎪ i(0, a) = i0 (a) ∈ L+ (R+ , R), ⎪ ⎪ ⎩ V (0) = V0 ∈ R2+ , where

 F1

(8.4)

 0 B(i(t, .),V (t)) ,V (t) = , i(t, .) 0  ∞

β (a)i(t, a)da, B(i(t, .),V (t)) = S(t)   0  Λ − S(t) 0∞ β (a)i(t, a)da + δ R(t) 0 F2 . ,V (t) = ∞ i(t, .) 0 k(a)i(t, a)da To express (8.4) as an ordinary differential equation on a Banach space, we write  0 u(t) = ,V (t) . i(t, .)

8.2 SIRS Model with Age of Infection

305

Let L : D(L) ⊂ X → X be the linear operator defined by   0 L(u(t)) = A1 , −EV (t) , i(t, .) where D(L) = Z × R2 with Z = {0} ×W 1,1 (R+ , R), where the Sobolev space W 1,1 is defined in Appendix A. It follows that X0 = D(L) and X0+ = D(L) ∩ X+ . So D(L) = X0 is not  dense in X. We consider the nonlinear map F : D(L) → X defined F1 (u(t)) by F(u(t)) = . Then (8.4) can be rewritten as F2 (u(t)) ⎧ du(t) ⎨ dt =  Lu(t) + F(u(t)), 0 ⎩ u(0) = ,V0 ∈ D(L). i0 (·)

(8.5)

Define Ω := {λ ∈ C : Re(λ ) ≥ −μ }.To prove the existence and uniqueness of system (8.5), we have to show that L is a Hill-Yosida operator defined in Appendix A. We prove that L is a Hille-Yosida operator in Lemma 8.1 below (this result can be also found in [165]).

λ ∈ Ω , then λ ∈ ρ (L). Lemma 8.1.  If  More precisely, for any λ > −μ , any  α ψ1 ϕ1 0 , ∈ X, and , ∈ D(L), we have ψ2 ϕ ϕ2 ψ     α ψ1 ϕ1 0 −1 , = , (λ I − L) ψ2 ϕ ϕ2 ψ if and only if

ϕ (a) = e−

a

0 (λ + μ +k(l))dl

α+

 a

ψ1 , λ +μ ψ2 ϕ2 = . λ +μ +δ

0

e−

a s

(λ +μ +k(l))dl

ψ (s)ds,

ϕ1 =

Moreover, L is a Hille-Yosida operator and M for Re(λ ) > −μ and n ≥ 1. (Re(λ ) + μ )n   α 0 Proof. For any ∈ Y and ∈ D(A1 ), we have ϕ ψ   0 −1 α (λ I − A1 ) = , ϕ ψ

(λ I − L)−n ≤

(8.6)

306

8 Class Age-Structured Epidemic Models

that is,

ϕ (0) = α , ϕ  (a) = −(λ + μ + k(a))ϕ + ψ . Then (8.6) follows easily. From the definition of L, one directly gets (λ + μ )ϕ1 = ψ1 , (λ + μ + δ )ϕ2 = ψ2 . Thus

ψ1 , λ +μ ψ2 ϕ2 = . λ +μ +δ ϕ1 =

For any λ > −μ , we have $   $ $ $ α ψ1 $(λ − L)−1 $ , $ $ ψ2 ψ X $! ! ψ1 ""$ $ $ 0 $ $ λ +μ a  a −  a (λ +μ +k(l))dl =$ , $ ψ2 − ( λ + μ +k(l))dl e 0 α+ 0 e s ψ (s)ds $ $ λ + μ +δ X    ∞     a ∞ a   − 0a (λ +μ +k(l))dl  − ( λ + μ +k(l))dl  e s ≤ ψ (s)ds da e  da|α | +  0 0 0 |ψ1 | |ψ2 | + |λ + μ | |λ + μ + δ |  ∞  ∞ |α | |ψ2 | |ψ1 | ≤ + + |ψ (τ )| |e−(λ +μ )(a−τ ) |dad τ + |λ + μ | |λ + μ | |λ + μ + δ | 0 τ 1 (|α | + ψ L1 + |ψ1 | + |ψ2 |). ≤ |Re(λ ) + μ | +

The results immediately follow. The lemma below shows that F is a locally Lipschitz operator. Lemma 8.2. For all C > 0, there exists K(C) > 0 such that

F(ξ ) − F(ξˆ ) X ≤ K(C) ξ − ξˆ (8.7) ⎛ ⎞ ⎛ ⎞ 0 0 ⎜ ψ ⎟ ⎜ ψˆ ⎟ ˆ ⎟ ⎟ ⎜ ¯ ¯ for all ξ := ⎜ ⎝ ψ1 ⎠ ∈ X0 ∩ BC (0), and ξ := ⎝ ψˆ 1 ⎠ ∈ X0 ∩ BC (0), where ψ2 ψˆ 2 B¯C (0) := {ξ ∈ X0 : ξ ≤ C}.

8.2 SIRS Model with Age of Infection

307

Proof. For any ξ , ξˆ ∈ X0 ∩ B¯C (0), we have   ⎛ ⎞ ψ1 0∞ β (a)ψ (a)da − ψˆ 1 0∞ β (a)hat ψ (a)da ⎜ ⎟ 0 ⎟  

F(ξ )−F(ξˆ ) X = ⎜ ⎝−ψ1 ∞ β (a)ψ (a)da + δ ψ2 + ψˆ 1 ∞ β (a)ψˆ (a)da − δˆ ψ2⎠ X 0 0 ∞ ∞ ˆ 0 k(a)φ (a)da − 0 k(a)φ (a)da ≤ 2|ψ1 +

 ∞ 0

 ∞ 0

β (a)ψ (a)da − ψˆ 1

0

β (a)hat ψ (a)da| + δ |ψ2 − ψˆ 2 |

k(a)|ψ (a) − ψˆ (a)|da

≤ 2|ψ1 − ψˆ 1 | ¯ + k|

 ∞

 ∞ 0

 ∞ 0

β (a)ψ (a)da + |ψˆ 1 |

 ∞ 0

β (a)|ψ (a) − ψˆ (a)|da + δ |ψ2 − ψˆ 2 |

|ψ (a) − ψˆ (a)|da

¯ ≤ 2β¯ C|ψ1 − ψˆ 1 | + (2β¯ C + k)

 ∞ 0

|ψ (a) − ψˆ (a)|da + δ |ψ2 − ψˆ 2 |.

¯ δ }, then Eq. (8.7) holds. So that, if we choose K(C) := max{2β¯ C + k, Lemma 8.3 ([115, Lemma 3.1]). Suppose Lemmas 8.1 and 8.2 hold. For each x ∈ X0 , system (8.5) has at most one solution u ∈ C([0, T ], X0 ) for some T > 0. From Lemmas 8.1 to 8.3, it follows that the following proposition holds on the existence and uniqueness of the integral solution of system (8.5). Proposition 8.1. There  exists a uniquely determined semiflow {U(t)}t≥0 on X0,+ 0 such that, for each u = ,V (t) ∈ X0+ , there exists a unique continuous i(t, .) map U ∈ C(R+ , X0+ ) which is an integral solution of the Cauchy problem (8.5), that is, for all t ≥ 0,  t 0

U(s)uds ∈ D(L)

and

U(t)u = u + L

 t 0

U(s)uds +

 t

F(U(s)u)ds. 0

To investigate the differentiability of the integral solution of U(t)x0 , we define the following set D(L0 ) = {ψ ∈ D(L) : Lψ + F(ψ ) ∈ D(L)}. We readily observe that D(A0 ) is dense in X0,+ . Besides, for any ξˆ = (0, ψ , ψ1 , ψ2 )T ∈ X0 , the differentiation of F at ξˆ is given by ⎛ ⎞ ⎛ ⎞ ⎛ ⎞   ψˆ 1 0∞ β (a)ψ (a)da + ψ1 0∞ β (a)ψˆ (a)da 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ψ ⎟ 0  ˆ ⎜ψ ⎟ ⎜ ⎟ ⎜ ⎟  ∞ F [ξ ] ⎝ ⎠ := ⎝ ⎠ , ⎝ψ1⎠ ∈X0 . ˆ − ψ β (a) ψ (a)da + δ ψ ψ1 −ψˆ 1 0∞ β (a)ψ (a)da 1 2 0 ∞ ψ2 ψ2 0 k(a)ψ (a)da Therefore, the nonlinear operator F is differentiable based on the Assumption 13. Using the presentation in [115, Section 4], we obtain the following lemma.

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8 Class Age-Structured Epidemic Models

Lemma 8.4. If u0 ∈ D(L0 ) ∩ X0,+ , then the integral solution U(t)x0 defined in Proposition 8.1 is continuously differentiable with respect to t, that is, u(t) = U(t)u0 is the global classical solution of the abstract Cauchy problem (8.5).

8.2.1 Volterra Formulation a

Define π (a) = e−μ a− 0 k(s)ds . By Volterra formulation, we understand the following integral form of the problems. Solving the second equation in (8.4) along the characteristic line t − a = const yields ⎧ t ≥ a, ⎨ B(t − a)π (a), (8.8) i(t, a) = π (a) ⎩ i0 (a − t) , t < a, π (a − t) where B(t) = S(t)

 ∞ 0

β (a)i(t, a)da. By the definition of bounded dissipative in Ap-

pendix (Definition A.6), we readily see that {U(t)}t≥0 is bounded dissipative by using (8.2). Hence, we can define an invariant set for system (8.1)

Γ = {(0, ψ , ψ1 , ψ2 )T ∈ X0 | (0, ψ , ψ1 , ψ2 ) =

Λ } μ

(8.9)

which attracts each point in X0 . Theorem 8.1. Suppose that Assumption 13 holds. The following estimates also hold: Λ ¯ N(t) ≤ max{N0 , } := M, B(t) := i(t, 0) ≤ β¯ M¯ 2 , (8.10) μ for all t ≥ 0. The upper bounds satisfy, lim sup N(t) = t→∞

Finally, A ≤ lim inf S(t) ≤ t→∞

Λ μ

Λ Λ 2 β¯ , lim sup B(t) ≤ 2 . μ μ t→∞

with A =

Λ ¯ μ + βΛ μ

(8.11)

.

Proof. By Eq. (8.2), inequalities (8.10) and (8.11) hold after a simple computation. Denote S∞ = lim inf S(t), and B∞ = lim sup B(t). For the estimation for S(t), it folt→∞

t→∞

lows from Lemma A.12 that there exists a sequence tk such that S (tk ) → 0 and lim S(tk ) = S∞ . From the first equation of system (8.1),

k→∞

0 ≥ Λ − μ S∞ − β¯ S∞

Λ , μ

8.2 SIRS Model with Age of Infection

309

with a notice (8.11), we have

Λ 0 ≥ Λ − (μ + β¯ )S∞ , μ Hence, S∞ ≥

¯ Λ βΛ + . μ μ

8.2.2 Asymptotical Smoothness In this subsection, we apply the approach developed by Hale in [80] to prove the semiflow U(t)x0 is asymptotically smooth defined in Appendix A. From Sect. 8.2.1, the semiflow U(t) is point-dissipative and eventually bounded on bounded sets in X0 . To apply Theorem A.3, we define U1 (t) and U2 (t) as follows: U1 (t)x0 = (0, iˆ(t, ·), 0, 0),U2 (t)x0 = (0, i˜(t, ·), S(t), R(t)). where

t ≥ a, π (a) ⎩ i0 (a − t) , t < a, π (a − t)  B(t − a)π (a), t ≥ a, i˜(t, a) = 0, t < a,

iˆ(t, a) = and

⎧ ⎨ 0,

where B(t) = i(t, 0). Lemma 8.5. U1 (t) satisfies the condition (i) of Theorem A.3 in Appendix A. Proof. Let K ∈ D(L) ∩ X0,+ be any bounded closed set that is forward invariant under U(t). Let C > 0 be a positive constant such that ξ X < C holds for any ξ ∈ K.

U1 (t, X0 ) = iˆ(t, ·) 1 ≤ Ce−μ t . Obviously, U1 (t)(K ) approaches to zero as t goes to infinity. Lemma 8.6. For any ε > 0, there exists δ > 0 such that |B(t + h) − B(t)| ≤ LB h for all t ∈ R+ , 0 < h < δ , and (0, i0 , S0 , R0 ) ∈ Γ ,

 Λ ¯ Λ ¯ . LB = β (LS + + k) + Lβ ) μ μ (8.12) Proof. The proof is a direct result from Lemma A.10 in Appendix A. Lemma 8.7. U2 (t) satisfies the condition (ii) of Theorem A.3 in Appendix A.

310

8 Class Age-Structured Epidemic Models

Proof. We employ Theorem B.2 in [161] or Frechét-Kolmogorov Theorem in [80] to prove this lemma. By the two theorems, for any (0, S0 , i0 (·), R0 ) ∈ K , we need to verify that the following four conditions hold.  ∞

(i) sup i(t,a)∈K

(ii) lim

 ∞

r→∞ r

0

|i(t, a)|da = 0, uniformly in i(t, ·) ∈ K ,

 ∞

(iii) lim

h→0+ 0  h

(iv) lim

h→0+ 0

|i(t, a)|da < ∞,

|i(t, a + h) − i(t, a)|da = 0 uniformly in i(t, ·) ∈ K ,

|i(t, a)|da = 0 uniformly in i(t, ·) ∈ K .

Note that i˜(t, a) = B(t − a)π (a) = S(t − a) a direct calculation,  ∞

sup 0

i(t,a)∈K

|i(t, a)|da ≤ β¯ (

∞

Λ 2 ) μ

0

β (a)i(t, a)daπ (a) ≤ β¯ ( Λμ )2 π (a). With

 ∞ 0

π (a)da ≤

β¯ Λ 2 ( ) < +∞. μ μ

Condition (i) immediately holds. For condition (ii),  ∞

lim

r→∞ r

Λ |i(t, a)|da = lim β¯ ( )2 r→∞ μ

 ∞ r

Λ e−μ a da = β¯ ( )2 lim e−μ r = 0. μ r→∞

Next, we concern condition (iv).  h

lim

h→0+

0

Λ |i(t, a)|da = lim β¯ ( )2 + μ h→0

 h 0

Λ e−μ a da = β¯ ( )2 lim (1 − e−μ h ) = 0. μ h→0+

In what follows, we need to verify that condition (iii) is still satisfied.

i˜(t, ·) − i˜(t, · + h) 1 = = + ≤ +

 ∞ 0

|i˜(t, a) − i˜(t, a + h)|da

 t−h 0

 t

B(t − a)π (a)da

t−h  t−h 0  t−h 0

≤ 2β¯ (

|B(t − a − h)π (a + h) − B(t − a)π (a)|da

Λ B(t − a − h)|π (a + h) − π (a)|da + β¯ ( )2 μ

|B(t − a − h) − B(t − a)|π (a)da

Λ 2 ) h + LB (t − h)h. μ

 t t−h

e−μ a da

8.2 SIRS Model with Age of Infection

311

1 (R ), which implies Therefore, i˜(t, ·) remains in a pre-compact subset K i of L+ + Λ Λ i that Φ (t, K ) ⊆ K × [0, μ ] × [0, μ ] has a compact closure in Γ .

8.2.3 Existence of Equilibria In this section, we consider the existence of equilibria of (8.5). Denote K=

 ∞ 0

−μ a

β (a)e

π (a)da,

K1 =

 ∞ 0

k(a)e−μ a π (a)da.



Integration by parts shows that K1 ≤ 0∞ k(a)π (a)da = 1. We define the basic reproduction number (8.13) R0 = Λ K/μ . The reason for this definition will become clear below. Theorem 8.2. The following statements are true.

  Λ 0 , μ . 0 0 (ii) If R0 ≤ 1, then P0 is the only equilibrium of (8.5) while if R0 > 1 then it also has an endemic equilibrium P∗ given by ! ! 1 "" 0 K P∗ = , (8.14) , i(0)K 1 i(0)e−μ a π (a) μ +δ (i) Equation (8.5) always has a disease-free equilibrium P0 =

where i(0) =

Λ (1 − R10 )

. (8.15) δ 1 − μ+ δ K1   0 S Proof. Suppose that u = ∈ X0 is an equilibrium of (8.5). Then , i(a) R we have (8.16)

−i (a) − (μ + k(a))i(a) = 0,

(8.17)

β (a)i(a)da − μ S + δ R = 0,

(8.18)

k(a)i(a)da − (μ + δ )R = 0.

(8.19)



Λ −S

 ∞  0∞ 0

 ∞

β (a)i(a)da = 0,

− i(0) + S

0

It follows from (8.17) that i(a) = i(0)e−μ a π (a). If i(0) = 0, then we easily get the disease-free equilibrium P0 . Now, suppose that i(0) = 0. It follows from (8.16) and (8.19), respectively that S = 1/K and R = i(0)K1 /(μ + δ ). These, combined with (8.18), tell us that i(0) is given by (8.15), that is,

312

8 Class Age-Structured Epidemic Models

i(0) =

Λ − Kμ 1−

δ K1 μ +δ

=

μ (R0 − 1) K(1 − μδ+K1δ )

.

Noting K1 < 1, we see that i(0) > 0 if and only if R0 > 1. Thus if R0 ≤ 1, then (8.5) only has the disease-free equilibrium P0 ; if R0 > 1, then, besides P0 , (8.5) also has an endemic equilibrium P∗ given by (8.14). This completes the proof. Notice that e−μ a π (a) is the probability of an infected person with age a to remain in the infected compartment independently of death and treatment. So the biological meaning of the basic reproduction number R0 is clear and same as in the above Chapters.

8.2.4 Stability of Equilibria In the previous section, we obtained the existence of equilibria. In this section, we study their stability. The local stability is obtained by analyzing the spectrum of the linearized operator around each equilibrium.

8.2.4.1 Linearized Systems and Their Characteristic Equations Let u be an equilibrium of (8.5). Linearizing (8.5) about u to obtain the following system,

dy dt = (L + DF(u))y(t), y(0) = y0 ∈ D(L). Denote the part of L in D(L) by L0 , that is, D(L0 ) ={x ∈ D(L) : Lx ∈ D(L)} and 0 ϕ1 L0 u = Lu for u ∈ D(L0 ). It follows that, for , ∈ D(L0 ), we have ϕ ϕ2     0 ϕ1 0 −μϕ1 L0 , = , , ϕ ϕ2 −(μ + δ )ϕ2 A' 10 ϕ  1,1 (R , R) : ϕ (0) = 0}. ' where A' + 10 ϕ = −ϕ − ( μ + k(a))ϕ with D(A10 ) = {ϕ ∈ W Then we can easily get the following result.

Lemma 8.8. The linear operator L0 is the infinitesimal generator of the strongly continuous semigroup {TL0 (t)}t≥0 of bounded linear operators on D(L) and for each t ≥ 0 the linear operator TL0 (t) is defined by     −μ t 0 e ϕ1 0 ϕ1 , = , −(μ +δ )t , TL0 (t) ϕ ϕ2 e ϕ2 TA10 (t)ϕ

8.2 SIRS Model with Age of Infection

where TA10 (t)ϕ (a) =

e− 0,

313

a

a−t ( μ +k(l))dl

ϕ (a − t),

a ≥ t, otherwise.

Now we estimate the essential growth bound of the C0 -semigroup generated by + DF(u)) 0 , which is the part of L + DF(u) in D(L). Note that for any  (L ϕ1 0 , ∈ D(L), we have ϕ ϕ2   0 ϕ1 , DF(u) ϕ ϕ2   DB(i(·),V )(ϕ , ϕ1 )  = , DF2 (u)(ϕ , ϕ1 , ϕ2 ) 0   ∞  ϕ1 0 β (a)i(a)da + S 0∞ β (a)ϕ (a)da = , 0   ∞ − S β (a) ϕ (a)da + δ ϕ −ϕ1 0∞ β (a)i(a)da 2 0 ∞ . 0 k(a)ϕ (a)da Definition 8.1. A Fredholm operator T : X → Y is bounded with a finite-dimensional ker T and coker T = Y /ran T, and with closed ran T . By Definition 8.1, we readily see that DF(u) is a Fredholm operator. It follows from Lemma 8.1 and Definition A.1 that w0,ess (L0 ) ≤ w0 (L0 ) ≤ −μ . From Proposition 4.13 in [179], we obtain  w0 (L + DF(u)) = max w1 (L + DF(u)),

# sup z∈σ (L+DF(u))\Eσ (L+DF(u))

Re z , (8.20)

where σ (A) and Eσ (A) denote the spectrum and the essential spectrum of a closed linear operator A. Based on Assumption 13, together with the property of DF(u), we obtain that w1 (L + DF(u)) = w1 (L). As in the proof of Theorem 2.6 in [179], α ([TL (t)]) ≤ TL (t) ≤ e−μ t . Hence, w1 (L + DF(u)) < −μ . It follows from Eq. (8.20) w0,ess (L + DF(u))0 ≤ −μ < 0. In summary, we have proved the following result. Theorem 8.3. The essential growth rate of the strongly continuous semigroup generated by (L + DF(u))0 is strictly negative, that is, ω0,ess ((L + DF(u))0 ) < 0. From what has been discussed, the disease-free equilibrium P0 is locally asymptotically stable.

314

8 Class Age-Structured Epidemic Models

To study the stability of the endemic equilibria, we denote Q := DF(u) for simplicity of notation. Let λ ∈ Ω . Since λ I − L is invertible, it follows that λ I − (L + DF(u)) = λ I − (L + Q) is invertible if and only if I − Q(λ I − L)−1 is invertible. In addition, (λ I − (L + Q))−1 = (λ I − L)−1 [I − Q(λ I − L)−1 ]−1 . Consider −1

(I − Q(λ I − L) )



α ϕ

   ϕ1 γ ψ1 , = , . ϕ2 ψ2 ψ

By Lemma 8.1, we obtain ⎛ ⎞ ⎞ ⎛ ⎞ 0  α γ a  a ⎜ e− 0 (λ +μ +k(l))dl α + 0a e− s (λ +μ +k(l))dl ψ (s)ds ⎟ ⎜ ψ ⎟ ⎜ϕ ⎟ ⎟=⎜ ⎟ ⎜ ⎟−Q⎜ ϕ1 ⎝ ⎠ ⎝ ψ1 ⎠ ⎝ ϕ1 ⎠ λ +μ ϕ2 ϕ2 ψ2 λ + μ +δ ⎛

Noting definitions of DF and DB, we obtain ⎧ ⎛ ⎞ ⎛ ⎞ α γ ⎪ ⎪ ⎨ Δ (λ ) ⎝ ϕ1 ⎠ = Ψ (λ , ψ ) + ⎝ ψ1 ⎠ , ϕ2 ψ2 ⎪ ⎪ ⎩ ϕ = ψ, where

⎛

∞

β (a)i(a)da

0  λ) SK( ⎜  ∞ λ +μ Δ (λ ) = I − ⎜  λ ) − 0 β (a)i(a)da ⎝ −SK( λ +μ '1 (λ ) K 0

 λ) = K( '1 (λ ) = K

 ∞ 0

 ∞ 0

⎞ 0 δ λ + μ +δ

⎟ ⎟, ⎠

(8.21)

0

β (a)π (a)e−(λ +μ )a da, k(a)π (a)e−(λ +μ )a da,

a   ⎞ S 0∞ β (a) 0a e− s (λ +μ +k(l))dl ψ (s)dsda   a Ψ (λ , ψ ) = ⎝ −S 0∞ β (a) 0a e− s (λ +μ +k(l))dl ψ (s)dsda ⎠ . ∞  a − a (λ +μ +k(l))dl s ψ (s)dsda 0 k(a) 0 e

⎛

Lemma 8.9. The following statements are true. (i) σ (L + Q) ∩ Ω = σ p (L + Q) ∩ Ω = {λ ∈ Ω : det Δ (λ ) = 0}. (ii) Suppose λ ∈ ρ (L + Q) ∩ Ω . Then     0 α ϕ1 ψ1 −1 , = , (λ I − (L + Q)) ϕ2 ψ ψ2 ϕ

(8.22)

8.2 SIRS Model with Age of Infection

315

if and only if

ψ (a) = e−(λ +μ )a π (a)(Δ (λ )−1 ((α , ϕ1 , ϕ2 )T + Ψ (λ , ϕ ))1 +

 a 0

e−

a s

(λ +μ +k(l))dl

ϕ (s)ds,

(Δ (λ )−1 ((α , ϕ1 , ϕ2 )T + Ψ (λ , ϕ ))2 , λ +μ (Δ (λ )−1 ((α , ϕ1 , ϕ2 )T + Ψ (λ , ϕ ))3 ψ2 = , λ +μ +δ

ψ1 =

where Δ (λ ) and Ψ (λ , ϕ ) are defined, respectively, by (8.21) and (8.22), and ()i denotes the ith element of a vector (). Proof. Assume that λ ∈ Ω and det(Δ (λ )) = 0. Then we have   α ϕ1 , (λ I − (L + Q))−1 ϕ2 ϕ   α ϕ1 = (λ I − L)−1 (I − Q(λ I − L)−1 )−1 , ϕ2 ϕ   0 ψ1 = , . ψ ψ2 Thus (ii) follows from Lemma 8.1. We also see that {λ ∈ Ω : det Δ (λ ) = 0} ⊂ ρ (L + Q) and hence σ (L + Q) ∩ Ω ⊂ {λ ∈ Ω : det Δ (λ ) = 0}. Now, assume that λ ∈ Ω and det Δ (λ ) = 0. To show that λ ∈ σ p (L + Q),  this needs to find a nonzero 0 ϕ1 solution to (L + Q)u = λ u. In fact, for u = , ∈ D(L), we have ϕ ϕ2   −μϕ1 −ϕ (0) , (L + Q)u = −(μ + δ )ϕ2 −ϕ  − (μ + k(a))ϕ  B(u)u + , DF2 (u)u . 0 Then (L + Q)u = λ u gives ⎧ −ϕ (0) + B(u)u = 0, ⎪ ⎪ ⎨ −ϕ  − (μ + k(a))ϕ = λ ϕ ,   0 ϕ1 −(λ + μ ) ⎪ ⎪ + DF2 (u)u = 0. ⎩ ϕ2 0 −(λ + μ + δ )

(8.23)

From the second equation of (8.23) we get ϕ (a) = ϕ (0)e−(λ +μ )a π (a). Substituting it into the other equations of (8.23) yields Δ (λ )(ϕ (0), ϕ1 , ϕ2 )T = 0. This implies that (L + Q)u = λ u has a nonzero solution if and only if we can find (ϕ (0), ϕ1 , ϕ2 )T = 0 such that Δ (λ )(ϕ (0), ϕ1 , ϕ2 )T = 0, which is true since det Δ (λ ) = 0. This proves that λ ∈ σ p (L + Q) and hence the proof is complete.

316

8 Class Age-Structured Epidemic Models

With the above preparation, we are ready to study the stability of equilibria.

8.2.4.2 Stability of the Disease-Free Equilibrium P0 Theorem 8.4. The disease-free equilibrium P0 is locally asymptotically stable if R0 < 1 and it is unstable if R0 > 1. Proof. At P0 , we have ⎛

Λ ˆ μ K(λ ) ⎜ Λ ˆ Δ (λ ) = I − ⎝ − μ K(λ )

0 0 ' K1 (λ ) 0

⎞

0 δ λ + μ +δ

⎟ ⎠.

0

ˆ λ )| ≤ R0 Notice that | Λμ K( ˆ λ )| < 1. for all λ ∈ C with nonnegative real parts. Therefore, if R0 < 1, then | Λμ K( This implies that all solutions of det(Δ (λ )) = 0 have negative real parts if R0 < 1 and hence P0 is locally asymptotically stable if R0 < 1. Now suppose that R0 > 1. ˆ λ ) is a decreasing function with Λ K(0) ˆ = R0 and Note that if λ ∈ R, then Λμ K( μ Λ ˆ K(λ ) → 0 as λ → ∞. Therefore, if R0 > 1, then there always exists a positive Obviously, det(Δ (λ )) = 0 if and only if 1 =

Λ ˆ μ K(λ ).

μ

solution to det(Δ (λ )) = 0. This means that P0 is unstable if R0 > 1 and the proof is complete. To deal with the global stability of P0 , we need some notations. For a function ϕ : R+ → R, we denote

ϕ∞ = lim inf ϕ (t) t→∞

and

ϕ ∞ = lim sup ϕ (t). t→∞

Theorem 8.5. Suppose that R0 < 1. Then the disease-free equilibrium P0 is a global attractor, i.e., lim (i, S, R) = (0, Λμ , 0). t→∞

Proof. Note Eq. (8.2), S∞ ≤ N ∞ = Λμ . Obviously, we have S∞ ≤ Λμ . Integrating the equation for i in (8.1) along the characteristic lines, t − a = constant, yields  a < t, B(t − a)e−μ a π (a), (8.24) i(t, a) = (a) , a ≥ t, i0 (a − t)e−μ t ππ(a−t) where B(t) = S(t)

∞ 0

β (a)i(t, a)da. Substitute i(t, a) into B(t) to obtain

B(t) = S(t) 

 t 0

β (a)B(t − a)e−μ a π (a)da + F(t),

(8.25)

(a) where F(t) = S(t) t∞ i0 (a − t)e−μ t ππ(a−t) da. By Lemma A.12, there exists a sequence {tn } such that tn → ∞ and B(tn ) → B∞ as n → ∞. Then in (8.25) letting t = tn

8.2 SIRS Model with Age of Infection

317

and n → ∞ gives B∞ ≤ S∞ KB∞ since lim F(t) = 0. This implies that B∞ = 0 since t→∞

B∞ ≥ 0 and S∞ K ≤ Λ K/μ = R0 < 1. So lim B(t) = 0. This, combined with (8.24), gives lim i(t, a) = 0 for all a ∈ R+ .

t→∞

t→∞

Now, we get from the equation on R(t) in (8.1) that dR(t) = dt

 t 0

+

k(a)B(t − a)e−μ a π (a)da − (μ + δ )R(t)

 ∞ t

k(a)i0 (a − t)e−μ t

π (a) da. π (a − t)

With the help of Lemma A.12 again, there is a sequence {vn } such that vn → ∞, ∞ R(vn ) → R∞ , and dtd R(vn ) → 0. Then we can get R∞ ≤ Kμ1+Bδ = 0. Finally, we apply Lemma A.12 to estimate S∞ . It follows from the first equation of (8.1) that S∞ ≥ Λ /μ . This, combined with S∞ ≤ Λ /μ , gives lim S(t) = Λμ and t→∞ hence the proof is complete.

8.2.4.3 Persistence When R0 > 1, P0 is unstable. We show that in this case the system is persistent and hence the disease will establish. By Volterra formulation in Sect. 8.2.1, B(t) = S(t)

 ∞ 0

β (a)B(t − a)π (a)da + f (t),



(a) where f (t) := 0∞ β (a +t)i0 (a) ππ(a−t) da. Observe that π (a) is always positive for all a ∈ R. From Growall inequality, it follows that

B(t) ≤ f (t) eS(t)

t

0 β (a)π (a)da

.

Hence, if β (a + t)i0 (a) = 0 for any t, then B(t) approaches to zero when t goes to infinity. Then we have the following proposition. Proposition 8.2. Either i0 (·) = 0 ∈ L1 , then i(t, ·) = 0 for all t ≥ 0, or B(t) takes on positive values for t large enough. In order to establish the persistence of system (8.1), we need to assume that β (a + t)i0 (a) > 0 for some t. This implies that the support of the transmission function β lies to the right of the initial density i0 . To achieve this goal, we let Π : X0 → L1 (R+ , R) be the Poincare projector defined   by  ∞ 0 v Π (v)(t) = β (a)v1 (t, a)da, for v = , 2 ∈ X0 . v3 v1 0 Set M0 = {v ∈ M : Π (v) = 0}, ∂ M0 = M \ M0 . M = X0+ , Following [115], we have the following lemma.

318

8 Class Age-Structured Epidemic Models

Lemma 8.10. The subsets M0 and ∂ M0 are both positively invariant under the semiflow {U(t)}t≥0 , namely, U(t)M0 ⊂ M0 and U(t)∂ M0 ⊂ ∂ M0 for t ≥ 0. The persistence of (8.5) is established by using the results in Magal and Zhao [117]. For this purpose, since P0 is globally asymptotically stable in ∂ M0 , it is sufficient to prove that there exists ε > 0 with the property that for each v ∈ {y ∈ M0 : P0 − y ≤ ε } there exists t0 ≥ 0 such that

P0 −U(t)v > ε

for t > t0 .

This shows that P0 is the largest invariant set for U in the neighborhood of P0 and also leads to / W s (P0 ) ∩ M0 = 0, where W s (P0 ) = {v ∈ X0+ : lim U(t)v = P0 }. t→∞

Theorem 8.6. Assume that R0 > 1. The semiflow {U(t)}t≥0 is uniformly persistent with respect to the pair (∂ M0 , M0 ), that is, there exists ε > 0 such that lim Π v(t) ≥ ε for x ∈ M0 . Moreover, there exists a compact subset A0 of M0

t→∞

which is a global attractor for {U(t)}t≥0 in M0 . Proof. By way of contradiction, assume that for every n ∈ N there exists vn ∈ {y ∈ 1 } such that M0 : P0 − y ≤ n+1

P0 −U(t)vn ≤ 

Write U(t)vn as

0 vn1

1 n+1

for t ≥ 0.

(8.26)

 n v , 2n . Then we have v3

|vn2 (t) − S0 | = |Sn (t) − S0 | ≤  Moreover, the map t →

0 vn1



1 n+1

for t ≥ 0.

is an integral solution of the Cauchy problem

⎧    n ∞ 0 0 v2 (t) 0 β (a)vn1 (t, a)da ⎪ d ⎪ = A + ⎪ 1 ⎪ 0 vn1 ⎨ dt vn1 (t, ·) for t ≥ 0,   ⎪ ⎪ 0 0 ⎪ ⎪ = . ⎩ vn10 vn1 (0, ·) 1 , by the comparison principle, we deduce that Since vn2 (t) ≥ S0 − n+1

vn1 (t, ·) ≥ vn1 (t, ·), where vn1 (t, ·) is a solution of the linear abstract problem

(8.27)

8.2 SIRS Model with Age of Infection

319

⎧    0  ∞ 1 n 0 0 S − n+1 ⎪ d 0 β (a)v1 (t, a)da ⎪ = A + ⎪ 1 ⎪ vn1 ⎨ dt vn1 (t, ·) 0 for t ≥ 0,   ⎪ ⎪ ⎪ 0 0 ⎪ ⎩ = . vn1 (0, ·) vn10

(8.28)

We can easily see that, for all n large enough, the dominated eigenvalue of the linear equation (8.28) satisfies the characteristic equation   λ0n ) S0 − 1 K( = 1. n+1 Since R0 > 1, using similar arguments as in the proof of Theorem 8.4, we can easily show that λ0n > 0 for all n large enough. Since vn0 ∈ M0 , we have Π (vn0 ) = ∞ n λ0n t da = 0. Thus 0 β (a)v10 (a)e lim vn1 (t, ·) = lim vn10 (a)eλ0n t =

t→∞

t→∞

 ∞ 0

vn10 (a)daeλ0n t = ∞

and hence it follows from (8.27) that lim vn1 (t, ·) ≥ lim vn1 (t, ·) = ∞.

t→∞

t→∞

This contradicts with (8.26) and the proof is complete. 8.2.4.4 Stability of the Endemic Equilibrium P∗ As noted earlier, P∗ exists if and only if R0 > 1. Theorem 8.7. If k(a) = k, the endemic equilibrium P∗ is locally asymptotically stable if R0 > 1. Proof. The characteristic equation of (8.1) at the endemic equilibrium is ! " ∞ '1 (λ ) β (a)i(a)da K δ 0  λ) = 1+ SK( 1− λ +μ λ +μ +δ or

 λ) K( = 1+ K

∞ 0

β (a)i(a)da λ +μ

!

'1 (λ ) δK 1− λ +μ +δ

" (8.29)

as S = 1/K. It suffices to show that, if k(a) = k, then all roots of (8.29) have negative real parts. This is done by the following proof. '1 (λ ) = k . It is obvious that, if Reλ ≥ 0, then Note that if k(a) = k, then K λ +μ +k    K(λ )   K  < 1. In order to prove that all the roots of (8.29) have negative real parts, it is sufficient to prove that the real part of

320

8 Class Age-Structured Epidemic Models

!

'1 (λ ) δK 1− λ +μ +δ

1 λ +μ

"

is greater than 0 for any λ = x + iy, x ≥ 0. In fact, ! " '1 (λ ) 1 δK 1− λ +μ λ +μ +δ 1 (λ + μ + δ )(λ + μ + k) − δ k ) = (1 − ( λ +μ (λ + μ + δ )(λ + μ + k) λ +μ +δ +k = (λ + μ + δ )(λ + μ + k) (x + μ + k + δ )(x + μ + k) + y2 − iδ y 1 = λ +μ +δ (x + μ + k)2 + y2 [(x + μ + k + δ )(x + μ + k) + y2 − iδ y][(x + μ + δ ) − iy] = [(x + μ + δ )2 + y2 ][(x + μ + k)2 + y2 ] Therefore, ! " '1 (λ ) 1 δK [(x + μ + k)2 + δ (x + μ + k)](x + μ + δ ) + (x+μ )y2 Re 1− = λ +μ λ +μ +δ [(x+μ +δ )2 +y2 ][(x+μ + k)2 +y2 ] ≥ 0. Hence the proof is complete. From Lemma 8.10 and Section 9.3 in [161], there exists a total trajectory of U(t), which is a function H : R → Y such that U(H(t + s)) = H(t + s) for all t ∈ R and all s ≥ 0. For a total trajectory, i(t, a) = i(t − a, 0)π (a), for all a ∈ R and a ∈ R+ . Lemma 8.11 (Corollary B.6, [161]). Let w, r, and g be negative and locally integral,  w(t) ≥

t

0

k(s)w(t − s)ds + g(t),

Assume that k is almost nonzero. Then there exists  some b > 0, which only depends on b nor g, such that w(t) > 0 for all t > b with 0t−b g(s)ds > 0. Following the approach found in [161, Lemma 9.12] and Lemma 8.11, we obtain the following lemma. Lemma 8.12. For a total trajectory Y (·) in Y , J(t) := positive and either is identically zero.

∞ 0

β (a)i(t, a)da is strictly

Proof. Let Y (·) be a total trajectory in Y , with Y (t) = (S(t), i(t, ·), R(t))T . First, we will show that if S(T ) = 0 for some T defined as follows: T = inf {t ∈ R+ |S(t) = 0 and S(t) > 0 for t ∈ [0, T )} . t∈R

8.2 SIRS Model with Age of Infection

321



) ∞ Λ It follows from the first equation (8.1) that dS(T dt = Λ + γ ( μ − 0 i(T, a)da) > 0. So that, there exists a sufficiently small ε > 0 such that S(T − ε ) < 0, contradicting the definition of T . Secondly, suppose there exists a T ∈ R such that i(T, ·) = 0. Then by Lemma 8.2, J(t) is identically zero for all t ≥ T. On the other hand, for any t < T, we have 0 = i(T, T − t) = i(t, 0)π (T − t) = S(t)J(t)π (T − t). For the positivity of π and S(t) for all t ∈ R, J(t) = 0. Hence, for any t ∈ R, J(t) is identically zero. Finally, we assume that i(t, ·) is nonzero for any T ∈ R. Then there exists a sequence {Tn } tending to −∞ such that i(Tn , ·) is nonzero for each n. So that, there exists an > 0 such that 0 = i(Tn , an ) = i(Tn − an , 0)π (an ). Thus, we have i(Tn∗ , 0) = 0 where Tn∗ = Tn − an tends to −∞. For each n ∈ N, let Jn (t) = J(Tn∗ + t), then

Jn (t) = =

 ∞ 0

 t 0

≥A

β (a)i(Tn∗ + t, a)da =

 t 0

β (a)i(Tn∗ + t, a)da +

β (a)S(tn∗ +t−a)J(Tn∗ +t−a)π (a)da+

 t 0

 ∞ t

 ∞ t

β (a)i(Tn∗ + t, a)da

β (a)i0 (a − Tn∗ −t)

π (a) da π (a−t)

β (a)Jn (t − a)π (a)da + J˜n (t) ∞



(a) β (a)i(Tn∗ , a − t) ππ(a−t) da. Note that J˜n (0) = 0∞ β (a)i(Tn∗ , a)da = ∗ i(Tn ,0) ˜ Sn (Tn∗ ) > 0. So that, we observe Lemma 8.11 that Jn (t) and Jn (t) are positive for small t. We employ Lemma 8.11 that Jn (t) is positive for all t > b, where b is a positive value. Therefore, Jn (t) > 0 if we let Tn∗ tend to −∞.

where J˜n (t) =

t

Finally, we study the global stability of P∗ . The following result can be readily proved by applying Lemma 8.12, together with Theorem 4.9 and Lemma 3.1 of [118]. Lemma 8.13.  There exist constants M > ε > 0 such that for each complete orbit

 v2 0 of U in A0 and the following estimations hold , v3 v1 t∈R

ε ≤ S(t), R(t),

 ∞ 0

β (a)i(t, a)da ≤ M

for t ∈ R.

Observe the global trajectory of system (8.1), we have i(t, a) B(t − a) S(t − a) = = i(a) i(0) and hence

∞ 0

β (τ )i(t − τ , τ )d τ i(a)

ε2 i(t, a) M2 ≤ ≤ . i(0) i(a) i(0)

322

8 Class Age-Structured Epidemic Models

The following result tells us that P∗ is globally asymptotically stable under some additional conditions. Similar results were obtained the delayed version of the model in [47, 139]. Theorem 8.8. Let R0 > 1. If μ S > δ R and k(a) = k, then the endemic equilibrium P∗ is globally asymptotically stable. Proof. We use the Lyapunov functional approach to prove the result. For the simplicity of notation, we denote x = S/S and z = R/R. Then we construct the Lyapunov functional as follows: U(t) = US (t) +Ui (t) +UR (t), where US (t) = g(x),

Θ (a) =

 ∞ a

Ui (t) =

β (l)i(l)dl,

 ∞ 0

Θ (a)g(

i(t, a) )da, i(a) 2

UR (t) = δ (R(t) − R)2 /(2kS ) with g(x) = x − 1 − ln x.

First, we have 

1 1 dS x S dt    ∞ 1 1 = 1− Λ − μ S − S(t) β (a)i(t, a)da + δ R x S 0    ∞ 1 1 = 1− μS + S β (a)i(t, a)da − δ R x S 0   ∞ −μ S − S(t) β (a)i(t, a)da + δ R 0   δR 1 1 = −μ 1 − 1− (x − 1) + (z − 1) x x S   ∞ i(t, a) i(t, a) 1 + β (a)i(a) 1 − − x da. + x i(a) i(a) 0

dUS (t) = dt

1−

Then due to the integral transformation and character of Θ (a) we have    ∞ i(t, 0) dUi i(t, 0) i(t, a) i(t, a) − ln − + ln = β (a)i(a) da. dt i(0) i(0) i(a) i(a) 0 Note that  ∞ 0

β (a)i(a)



  ∞ i(t, 0) i(t, a) i(t, 0) ∞ −x β (a)i(a)da − x β (a)i(t, a)da da = i(0) i(a) i(0) 0 0 i(t, 0) i(t, 0) − = S S =0.

8.2 SIRS Model with Age of Infection

323

Thus   d(US (t) +Ui (t)) δR 1 1 = −μ 1 − 1− (x − 1) + (z − 1) dt x x S    ∞ i(t, 0) i(t, a) 1 + ln β (a)i(a) 1 − − ln da + x i(0) i(a) 0   δR 1 1 1− = −μ 1 − (x − 1) + (z − 1) x x S      ∞ 1 xi(t, a)i(0) β (a)i(a) g − da. +g x x i(a)i(t, 0) 0 Moreover, it follows from k(a) = k, Λ μ

− S(t) − R(t) that

Λ μ

= S+

μ +δ +k R, k

and I(t) =

∞ 0

i(t, a)da =

   dUR (t) δ Λ = 2 (R(t) − R) k − S(t) − R(t) − (μ + δ )R(t) dt μ kS δ = 2 (R(t) − R)[−k(S(t) − S) − (μ + δ + k)(R(t) − R)] kS 2 δR δ R (μ + δ + k) = − (z − 1)(x − 1) − (z − 1)2 . S kS Therefore,   dU(t) δR 1 1 = −μ 1 − (z − 1) 2 − x − (x − 1) + dt x x S 2

δR (μ + k + δ )(z − 1)2 kS      ∞ 1 i(t, a) i(0) − β (a)i(a) g +g x da x i(a) i(t, 0) 0   2 δR δR 1 (z − 1) 2 − x − (μ + k + δ )(z − 1)2 − = μ+ x S kS      ∞ 1 i(t, a) i(0) − β (a)i(a) g +g x da x i(a) i(t, 0) 0   2 δR δR 1 2−x− (μ + k + δ )(z − 1)2 − ≤ μ− x S kS      ∞ 1 i(t, a) i(0) − β (a)i(a) g +g x da x i(a) i(t, 0) 0 ≤ 0. −

324

8 Class Age-Structured Epidemic Models

Let M be the largest invariant set of {(S(t), i(t, a), R(t)) : dU(t) dt = 0}. We show that M consists of only the endemic equilibrium P∗ . In fact, since g(u) = 0 if and only if u = 1, we have that dU(t) dt = 0 if and only if S(t) = S, Then

dS(t) dt

R(t) = R,

i(t, a) i(0) = 1 for a ∈ (0, ∞). i(a) i(t, 0)

and

(8.30)

= 0. This, combined with (8.30), yields 0=

dS = Λ − μS − S dt

=S =S =S

 ∞ 0 ∞ 0

 ∞ 0

 ∞ 0

β (a)i(t, a)da + δ R

β (a)(i(a) − i(t, a))da β (a)(i(a) − i(t, 0)e−μ a π (a))da β (a)i(a)da − i(t, 0)S

 ∞ 0

β (a)e−μ a π (a)da

= i(0) − i(t, 0), or i(t, 0) = i(0). It follows from (8.30) that i(t, a) = i(a) for a ∈ (0, ∞). This completes the proof.

8.3 SIRS Model with Age of Recovery Many diseases, such as influenza and tuberculosis (TB), the immunity of recovered individuals acquired from infection is temporary and varies with different stages. In this section we formulate and analyze an SIRS epidemic model with age of recovery: ⎧  ∞ dS(t) ⎪ ⎪ = Λ − β S(t)I(t) − μ S(t) + δ (a)R(a,t)da, ⎪ ⎪ ⎪ dt 0 ⎪ ⎪ dI(t) ⎪ ⎪ ⎨ = β S(t)I(t) − (μ + ν + γ )I(t), dt (8.31) ∂ R(a,t) ∂ R(a,t) ⎪ ⎪ + = −( δ (a) + μ )R(a,t), ⎪ ⎪ ∂a ∂t ⎪ ⎪ ⎪ R(0,t) = γ I(t), ⎪ ⎪ ⎩ S(0) = S , I(0) = I , R(a, 0) = R (a) ∈ L1 (0, +∞), 0 0 0 + where S(t) denotes the number of susceptible individuals at time t, I(t) the number of infective individuals, R(a,t) is the density of recovered individuals with respect to the age of recovery a at time t and it is assumed that the newly recovered individuals enter the recovered class R(a,t) with recovery-age equal to zero. Λ is the recruitment rate of the susceptible population, β is the contact rate, γ is the recovery rate from the infected compartment, ν is the death rate induced by the disease, μ is the

8.3 SIRS Model with Age of Recovery

325

natural death rate, δ (a) is the age-dependent loss of immunity rate of the recovered. ∞ ((0, +∞), R) \ {0}. We further assume that the function δ (a) belongs to L+

8.3.1 Preliminary Results By setting N(t) = S(t) + I(t) +

 +∞

R(a,t)da, 0

we deduce from (8.31) that N(t) satisfies the following ordinary differential equation: N  (t) = Λ − μ N(t) − ν I, and therefore in the absence of disease, N(t) converges to Λ /μ as t tends to infinity. Denote

  +∞ Λ 1 Ω = (S, I, R) ∈ R+ × R+ × L+ ((0, +∞), R) : S + I + R(a, ·)da ≤ . μ 0 (8.32) Then Ω is the positively invariant set of system (8.31) and we can study the dynamics of model (8.31) in the bounded set Ω . In order to prove the main results of the current chapter, we need to reformulate system (8.31) as a Cauchy problem. We can rewrite system (8.31) as the following ⎧ ∂ R(a,t) ∂ R(a,t) ⎪ ⎪ + = −(δ (a) + μ )R(a,t), ⎪ ⎪ ∂a ∂t  ⎪  ∞ ⎪ ⎪ dV (t) ⎨ = −CV (t) + G V (t), δ (a)R(a,t)da , dt 0 ⎪ ⎪ R(0,t) = γ I(t), ⎪ ⎪ ⎪ 1 ((0, +∞), R), ⎪ R(a, 0) = R0 (a) ∈ L+ ⎪ ⎩ 2 V (0) = V0 ∈ R ,

(8.33)

where  ∞ "  !  ∞ Λ − β S(t)I(t) + δ (a)R(a,t)da G V (t), δ (a)R(a,t)da = , 0 0 β S(t)I(t)    S0 μ 0 S(t) , V (t) = ∈ R2 . , V (0) = V0 = C= I0 0 μ +ν +γ I(t)

In system (8.33), by setting V (t) :=

 ∞ 0

V (a,t)da,

326

8 Class Age-Structured Epidemic Models



V1 (a,t) . We can rewrite the ordinary differential equation V2 (a,t) in (8.33) as an age-structured model ⎧ ∂ V (a,t) ∂ V (a,t) ⎪ ⎪ + = −CV (a,t), ⎪ ⎨ ∂a  ∂t

where V (a,t) =

 ∞

δ (a)R(a,t)da , V (0,t) = G V (t), ⎪ ⎪ 0 ⎪ ⎩ 1 V (a, 0) = V0 (a) ∈ L+ ((0, +∞), R2 ).  R(a,t) Thus by setting w(a,t) = , we obtain the following system V (a,t) ⎧ ∂ w(a,t) ∂ w(a,t) ⎪ ⎪ = −D(a)w(a,t), ⎪ ∂a + ∂t ⎨ w(0,t) = B (w(·,t)), ⎪ ⎪ ⎪ 1 ((0, +∞), R3 ), ⎩ w(a, 0) = w0 (a) = R0 (a) ∈ L+ V0 (a) where

(8.34)

⎞ μ + δ (a) 0 0 ⎠, μ 0 D(a) = ⎝ 0 0 0 μ +ν +γ ⎞ ⎛ γ I(t)  ∞ ⎟ ⎜ δ (a)R(a,t)da ⎠ . B (w(·,t)) = ⎝ Λ − β S(t)I(t) + ⎛

0

β S(t)I(t) In order to take into account the boundary condition, followed the results developed in Magal [118], we consider the following Banach space X = R3 × L1 ((0, +∞), R3 ) $ $ $ α $ $ with $ $ ψ $ = α R3 + ψ L1 ((0,+∞),R3 ) Furthermore, we let 1 ((0, +∞), R3+ ). X+ = R3 × L+

Define the linear operator L : Dom(L) → X by   0 −ϕ (0) L = ϕ −ϕ  − D(a)ϕ with Dom(L) = {0} ×W 1,1 ((0, +∞), R3 ) ⊂ X, and the operator f : Dom(L) → X by   B(ϕ (t)) 0 = . f ϕ (t) 0

8.3 SIRS Model with Age of Recovery

327

The linear operator L is non-densely defined due to Dom(L) = {0} × L1 ((0, +∞), R3 )(:= X0 ). By denoting that

 x(t) =

(8.35)

0 , w(·,t)

we can rewrite system (8.34) as the following non-densely abstract Cauchy problem ⎧ dx(t) ⎪ ⎨ = Lx(t) + f (x(t)), dt  (8.36) ⎪ ⎩ x(0) = 0 ∈ Dom(L). w0 Let 1 X0+ := X0 ∩ X+ = {0}3 × L+ ((0, ∞), R3+ ) and Ω0 := X0+ ∩ Ω .

The well-posedness of problem (8.36) follows from Theorem 2.3 in [165] if Assumptions 2.1 and 2.2 in [165] are satisfied. Those assumptions require the following. (1) L is a closed linear operator on X. (λ − L) has a bounded linear inverse on X and M (8.37)

(λ − L)−n ≤ (λ − ω )n for all n ∈ N, λ > ω and appropriate constants M and ω . (2) f is a bounded operator with f : Ω0 → X. Furthermore, f is locally Lipschitz, that is, there exist constants C > 0 and ε > 0 such that

f (y) − f (z) ≤ C y − z for all y, z ∈ Ω0 and y − z ≤ ε . (3) λ (λ − L)−1 maps Ω into itself for sufficiently large λ > ω . (4) For every u ∈ Ω0 1 dist(u + h f (u), Ω ) → 0 as h → 0. h To complete the proof we check the following assumptions. (1) L is a closed linear operator. To check the estimate, let ζ ∈ X with coordinates 

ζ=

ξ0 ζ0

  T ξ ξ , 1 , 2 ζ1 ζ2

and we consider the equation (λ − L)u = ζ . Let π (a) := e−

a

0 ( μ +δ (ς ))d ς

. Then

328

8 Class Age-Structured Epidemic Models

⎛! ⎜ ⎜ ⎜ ⎜ −1 u = (λ − L) ζ = ⎜ ⎜ ⎜ ⎝

"⎞

0

 ξ0 e−λ a π (a) + 0a e−λ (a−s) ππ(a) (s) ζ0 (s)ds



0



1 λ +μ





0

⎟ ⎟ ⎟ ⎟ ⎟. ⎟ ⎟ ⎠

(8.38)

1 λ + μ +ν +γ

Assuming λ > −μ , we integrate the second and the last terms with respect to a and add. We obtain the following estimate

(λ − A)−1 ≤

1 . (λ + μ )

Iterating the above estimate n times we obtain inequality (8.37). (2) Since δ (a) is bounded function, it is not hard to see that f is a bounded operator. To see that it is Lipschitz, we need only consider that f (u) − f (u). For instance, from the first part of f , we have that −β SI +

 +∞ 0

δ (a)R(a, ·)da + β SI −

= β S(I − I) + β I(S − S) −

 +∞ 0

 +∞ 0

δ (a)R(a, ·)da

δ (a)(R(a, ·) − R(a, ·))da.

(8.39)

It is easy to see that f (y) − f (z) ≤ C y − z . This establishes that f is Lipschitz. (3) For ζ ∈ Ω , using (8.38) and multiplying by λ , we have that λ (λ − L)−1 ζ ≤

ζ . Since ζ ∈ Ω , so does λ (λ − L)−1 ζ . Hence, λ (λ − L)−1 maps Ω into itself for λ > μ . (4) To see the last condition, let u ∈ Ω0 . We define a new operator f(u) as follows: f(u) = f (u) + ε u, where ε > β Λμ . Then, the operator f : X0+ → X+ . Therefore, for any h > 0 and sufficiently small and any u ∈ Ω0 we have 1 1 dist(u + h f (u), Ω ) = dist(u − ε hu + h f(u), Ω ) = 0. h h  ∈ The last equality follows since for h sufficiently small u − ε hu ∈ Ω and hF(u) Ω. By applying the results given in [115], we derive the existence and uniqueness of the semiflow {U(t)}t≥0 on X0+ generated by system (8.36). Lemma 8.14. System (8.36) generates a unique continuous semiflow {U(t)}t≥0 on X0+ . Define R0 =

βΛ μ (μ + ν + γ )

(8.40)

8.3 SIRS Model with Age of Recovery

329

as the basic reproduction  number of the disease described by system (8.31). 0 If x(a) = ∈ X0 is a steady state of system (8.36), we then have that w(a) a ⎛ ⎞ ⎞ γ Ie− 0 (μ +δ (ς ))d ς   R(a) ∞ ⎜ ⎟ −μ a ⎟ w(a) = ⎝ V1 (a) ⎠ = ⎜ ⎝ Λ − β SI + 0 δ (a)R(a)da e ⎠ V2 (a) −( μ + ν + γ )a β SIe

⎛

with S⎛=

 +∞

V⎞ 1 (a)da and I =

 +∞

0 V2 (a)da. There is a disease-free steady state 0 E 0 = ⎝ Λ e−μ a ⎠ with I = 0. If R0 > 1, there is an endemic steady state E ∗ = 0 ⎞ ⎛ ∗ R (a) ⎝ V1∗ (a) ⎠ with I = 0 where V2∗ (a)   +∞ a R∗ (a) = γ I ∗ e− 0 (μ +δ (ς ))d ς , V1∗ (a) = Λ − β S∗ I ∗ + δ (a)R∗ (a)da e−μ a , 0

V2∗ (a) = (μ + ν + γ )I ∗ e−(μ +ν +γ )a ,

μ +ν +γ S = , β

0

∗

Λ 1 a I = 1−  +∞ − ( μ + δ ( ς ))d ς R 0 (μ + ν + γ ) − γ 0 δ (a)e da 0 ∗

. (8.41)

Theorem 8.9. If R0 ≤ 1, system (8.31) only has a disease-free steady state E 0 ; if R0 > 1, besides E 0 , system (8.31) has a unique endemic steady state E ∗ .

8.3.2 Stability of the Disease-Free Equilibrium In this section, we need to get the linearized equation around a feasible steady state and then consider the stability of the disease-free steady state E 0 . By using the following change of variable y(t) := x(t) − x(a), we obtain

⎧ dy(t) ⎪ ⎨ = Ly(t) + f (y(t) + x(a)) − f (x(a)), dt  0 ⎪ ⎩ y(0) =  y0 ∈ Dom(L). w0 − w(a)

(8.42)

Therefore the linearized equation (8.42) around the disease-free steady state 0 is given by

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8 Class Age-Structured Epidemic Models

dy(t) = Ay(t), dt

(8.43)

where A = L + D f (x) is a linear operator. Then we can rewrite system (8.42) as dy(t) = Ay(t) + H(y(t)), dt where H(y(t)) = f (y(t) + x) − f (x) − D f (x)y(t) satisfying H(0) = 0 and DH(0) = 0. Denote that Ω1 := {λ ∈ C : Re(λ ) > −μ }. By applying the results of Liu et al. [113], we obtain the following result. Lemma 8.15. For λ ∈ Ω1 , λ ∈ ρ (L) and    a  a a ξ 0 e− ς (λ I+D(s))ds η (ς )d ς = ⇔ ϕ (a) = e− 0 (λ I+D(s))ds ξ + (λ I−L)−1 ϕ η 0   ξ 0 with ∈ X and ∈ Dom(L). Moreover, L is a Hille-Yosida operator and η ϕ $ $ $(λ I − L)−n $ ≤

1 , ∀ λ ∈ Ω , ∀ n ≥ 1. (Re(λ ) + μ )n

Let L0 be the part of L, that is, L0 : Dom(L0 ) ⊂ X → X. Then we get for Dom(L0 ) L0

(8.44)  0 ∈ ϕ

  0 0 =  ϕ L0 (ϕ )

0 (ϕ ) = −ϕ  − D(·)ϕ with Dom(L 0 ) = {ϕ ∈ W 1,1 ((0, +∞), R3 ) : ϕ (0) = 0}. where L  0 Noticing that for any ∈ Dom(L), ϕ   0 DB(w)(ϕ ) , (8.45) = D f (x) ϕ 0 where ⎞ ⎛  +∞ 0 0 γ 0 DB(w)(ϕ ) = ⎝ 0 − β I − β S ⎠ ϕ (a)da + ⎝ 1 0 0 0 βI βS ⎛

0 0 0

⎞ 0  +∞ 0⎠ δ (a)ϕ (a)da. 0 0

Then D f (x) : Dom(L) → X is a compact operator. Define the part of A in Dom(A) by A0 ,

8.3 SIRS Model with Age of Recovery

331

A0 = L0 + D f (x) : Dom(A0 ) → X, with Dom(A0 ) = {x ∈ Dom(A) : Ax ∈ Dom(A)} and where L0 x = Lx for x ∈ Dom(L0 ) = {x ∈ Dom(L) : Lx ∈ Dom(L)}. Then the linear operator A0 can generate a strongly continuous semigroup {TA0 (t)}t≥0 . By Lemma 3.3 in [172], the linear operator L0 generates a strongly continuous semigroup {TL0 (t)}t≥0 . From (8.44) we have that $ $ $TL (t)$ ≤ e−μ t , ∀ t ≥ 0. 0 Thus we have

ω0,ess (L0 ) ≤ ω0 (L0 ) ≤ −μ . Now we estimate the essential growth bounded of the strongly continuous semigroup {TA0 (t)}t≥0 generated by the linear operator A0 . Applying the perturbation results in Thieme [166], we can deduce that

ω0,ess (L + D f (x))0 ≤ −μ < 0.

(8.46)

Hence we obtain the following proposition. Proposition 8.3. The linear operator A is a Hille-Yosida operator, and its part A0 in Dom(A) satisfies ω0,ess (A0 ) < 0. Let λ ∈ Ω1 . To obtain the characteristic equation of problem (8.43), we need to concern the invertibility of (λ I − A). Since (λ I − L) is invertible for λ ∈ Ω1 , we set G := I − D f (x)(λ I − L)−1 . It then follows that (λ I − A) is invertible if and only if G is invertible. (λ I − A)−1 = (λ I − (L + D f (x)))−1 = (λ I − L)−1 (I − D f (x)(λ I − L)−1 )−1 = (λ I − L)−1 G−1 . (8.47) Let   ξ α G = . η ψ It follows that

   ξ ξ α − D f (x)(λ I − L)−1 = . η η ψ

From Lemma 8.15, we have that    0 ξ α = − D f (x) . ϕ (a) η ψ Then we obtain that

ξ − DB(w)ϕ (a) = α η =ψ

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8 Class Age-Structured Epidemic Models

i.e., 

ξ − DB(w)e−

a

0 (λ I+D(s))ds

ξ = α + DB(w)

η = ψ.

 a

e−

a ς

(λ I+D(s))ds

0

η (ς )d ς ,

From the formula of DB(w) we know that

Δ (λ )ξ = α + K(λ , ψ ), η = ψ, ⎛

where

0 Δ (λ ) = I − ⎝ 0 ⎛ 0 0 −⎝ 1 0

0 −βI βI 0 0 0

⎞  +∞  γ a −βS⎠ e− 0 (λ I+D(ς ))d ς da 0 β ⎞S 0  +∞ a 0⎠ δ (a)e− 0 (λ I+D(ς ))d ς da 0 0

and K(λ , ψ ) = DB(w)

 a

e−

a ς

(λ I+D(s))ds

0

ψ (ς )d ς .

(8.48)

(8.49)

Whenever Δ (λ ) is invertible, we have

ξ = (Δ (λ ))−1 (α + K(λ , ψ )),

(8.50)

and G is invertible and therefore λ I − A is invertible. From the above discussion, we obtain the following lemma. Lemma 8.16. Let λ ∈ Ω1 . Then we have the following results: (i) If det Δ (λ ) = 0, we have the following formula for resolvent   α 0 = , (λ I − A)−1 ϕ ψ where

ϕ (a) = e−

a

0 (λ I+D(s))ds

(Δ (λ ))−1 (α + K(λ , ψ )) +

 a 0

e−

a ς

(λ I+D(s))ds

ψ (ς )d ς

with Δ (λ ) and K(λ , ψ ) defined in (8.48) and (8.49). Therefore, λ ∈ ρ (A) ∩ Ω1 . (ii) If det Δ (λ ) = 0, then λ ∈ σ (A) ∩ Ω1 = σ p (A) ∩ Ω1 and det Δ (λ ) = 0 is the characteristic equation of problem (8.42). Proof. (i) Since λ ∈ Ω1 . Assume that det Δ (λ ) = 0. It then follows from (8.47) that have     0 −1 α −1 −1 α −1 ξ = (λ I − A) = (λ I − L) G = (λ I − L) . ϕ ψ ψ η

8.3 SIRS Model with Age of Recovery

333

It follows from Lemma 8.15 and (8.50) that

ϕ (a) = e− =

a

0 (λ I+D(s))ds

a

ξ+

 a

e−

a ς

(λ I+D(s))ds

0

η (ς )d ς

e− 0 (λ I+D(s))ds (Δ (λ ))−1 (α + K(λ , ψ )) +

 a

e−

a ς

(λ I+D(s))ds

0

ψ (ς )d ς .

Therefore, λ ∈ ρ (A) ∩ Ω (i.e., λ I − A is invertible). (ii) From the above discussion, we have that {λ ∈ Ω1 : det(Δ (λ )) = 0} ⊂ ρ (A)∩ Ω1 and thus σ (A) ∩ Ω1 ⊂ {λ ∈ Ω1 : det(Δ (λ )) = 0}. Conversely, if det(Δ (λ )) = 0, we need to prove that λ ∈ σ p (A) ∩ Ω1 . Since (λ I − A)(λ I − L)−1 = I − D f (x)(λ I − L)−1 = G   ξ 0 (λ I − L) = . ϕ η  0 ∈ Dom(L) \ {0} such that Then we find that there exists ϕ  0 =0 (λ I − A) ϕ  ξ ∈ X \ {0} such that if and only if there exists η

 ξ Δ (λ )ξ = 0, =0⇔ G η = 0, η

and

−1

(8.51)

(8.52)

= 0 (i.e., ξ is an In fact, if det(Δ (λ )) = 0, there exists ξ = 0 such that Δ (λ )ξ  ξ eigenvector corresponding to the eigenvalue λ ). So the pair = 0 satis0  0 fies (8.52) and we find a ∈ Dom(L) \ {0} satisfying (8.51). Thus we have ϕ that λ ∈ σ p (A) ∩ Ω1 and therefore {λ ∈ Ω1 : det(Δ (λ )) = 0} ⊂ σ p (A) ∩ Ω1 and (ii) follows. Now we consider the stability of the disease-free steady state E 0 = (R0 , S0 , I 0 ) = (0, Λμ , 0). Theorem 8.10. If R0 < 1, the disease-free steady state E 0 of system (8.31) is locally and globally asymptotically stable. Proof. It follows from (8.48) that the characteristic equation at the steady state E 0 is β Λμ = 0. (8.53) det(Δ (λ )) = 1 − λ +μ +ν +γ

334

8 Class Age-Structured Epidemic Models

Therefore, λ = (μ + ν + γ )(R0 − 1) and it is negative if R0 < 1. Then the steady state E 0 of system (8.31) is locally asymptotically stable. We employ Lemma A.12 to prove the global stability of the disease-free equilibrium. By the second equation of (8.31), it follows from S(t) ≤ N(t) ≤ Λμ that 

 Λ I (t) ≤ β − (μ + ν + γ ) I(t), μ =(μ + ν + γ ) (R0 − 1) I(t). 

We readily obtain that I(t) → 0 as t → ∞ when R0 < 1. Solving the third equation of (8.31) along the characteristic line t − a =const, we obtain  γ I(t − a)π (a), t > a, R(a,t) = (8.54) (a) , t ≤ a. R0 (a − t) ππ(a−t) Substituting R(a,t) into Δ (a) :=

Δ (t) = γ

∞ 0

 t 0

δ (a)R(a,t)da, we arrive at

δ (a)I(t − a)π (a)da + F(t),

(8.55)

∞

δ (a + t)R0 (a) ππ(a+t) (a) da. From the assumption for the relapse rate δ , we obtain F(t) ≤ δ¯ R0 (a) 1 e−μ t . So that F(t) approaches to zero as t goes to infinity. Taking the super limitation on both sides of (8.55), we admit where F(t) =

0

Δ∞ ≤ γ

 ∞ 0

δ (a)π (a)daI ∞ → 0.

By Lemma A.12, there exists a sequence {tn } such that S(tn ) → S∞ and S (tn ) → ∞ when n → ∞. Observing the first equation of (8.31), we obtain 0 ≥ Λ − β S∞ I ∞ − μ S∞ + Δ∞ , So that Λμ ≥ S∞ ≥ Λμ . This implies that S(t) → Λμ as t goes to infinity. Therefore, system (8.31) converges to the disease-free equilibrium E0 if R0 < 1.

8.3.3 Local Hopf Bifurcation In this section, we would consider the local Hopf bifurcation of system (8.31). From here on, we make the following assumption. Assumption 14 Assume that

δ (a) :=

δ ∗ , if a ≥ τ , 0, if a ∈ (0, τ ).

8.3 SIRS Model with Age of Recovery

335

Since δ (a) denotes the loss of immunity rate of the removed individuals, it is reasonable to assume that the protection period τ ≥ 0. If R0 > 1, the endemic steady state E ∗ exists. Under Assumption 14, it follows from (8.41), we have that I ∗ is replaced by  (μ + δ ∗ )Λ 1 ∗ I = 1− . (μ + δ ∗ )(μ + ν + γ ) − γδ ∗ e−μτ R0 It then follows that  +∞

e−

a

0 (λ I+D(ς ))d ς

0

⎛ ⎜ da = ⎝

and

⎛  +∞ 0

δ (a)e−

a

0 (λ I+D(ς ))d ς

∗

eδ τ λ + μ +δ ∗

⎜ da = ⎜ ⎝

0 0 δ ∗ e−(λ +μ )τ λ + μ +δ ∗

0 0

0

⎞

0 0

1 λ +μ

⎟ ⎠

(8.56)

1 λ + μ +ν +γ

0

0 δ ∗ e−(λ +μ )τ λ +μ

0

0 0 δ ∗ e−(λ +μ +ν +γ )τ λ + μ +ν +γ

⎞ ⎟ ⎟. ⎠

(8.57) It follows from (8.48), (8.56), and (8.57), we have that the characteristic equation at the endemic steady state (S∗ , I ∗ , R∗ (a)) is  γ  1 0 − λ +μ + ν +γ   − δ ∗ e−(λ +μ )τ β S∗ β I∗ 1 + λ +μ det(Δ (λ )) =  λ +μ +δ ∗ λ + μ +ν +γ  ∗ ∗  0 − λβ+I μ 1 − λ +μβ+S ν +γ

     = 0.  

(8.58)

After a simple calculation, we have that det(Δ (λ )) =

f (λ , τ ) λ 3 + a(τ )λ 2 + b(τ )λ + c(τ ) + d(τ )e−(λ +μ )τ  = 0, (8.59) (λ + d1 )(λ + d2 )(λ + d3 ) g(λ )

where

and

a(τ ) = d1 + d3 + β I ∗ > 0, b(τ ) = d1 d3 + (d2 + d3 )β I ∗ > 0, c(τ ) = d2 d3 β I ∗ > 0, d(τ ) = −β I ∗ γδ ∗ < 0, d1 = μ , d2 = μ + ν + γ , d3 = μ + δ ∗

(8.60)

f (λ , τ ) = λ 3 + a(τ )λ 2 + b(τ )λ + c(τ ) + d(τ )e−(λ +μ )τ , g(λ ) = (λ + d1 )(λ + d2 )(λ + d3 ).

(8.61)

It is easy to see that {λ ∈ Ω1 : det(Δ (λ )) = 0} = {λ ∈ Ω1 : f (λ , τ ) = 0} and f (0, τ ) > 0 for any τ ≥ 0.

336

8 Class Age-Structured Epidemic Models

If τ = 0, then f (λ , 0) = λ 3 + aλ 2 + bλ + c + d. By the Routh-Hurwitz criterion, when τ = 0 all the roots of f (λ , 0) = 0 have negative real parts only if a(τ )b(τ )|τ =0 > (c(τ ) + d(τ ))|τ =0 . In fact, by direct calculations, we have that (a(τ )b(τ ) − (c(τ ) + d(τ )))|τ =0 =((β I ∗ )2 (d2 + d3 ) + β I ∗ [d1 (d2 + d3 + 1) + d3 (d3 + 1) + γδ ∗ ])|τ =0 + (d1 + d3 )d1 d3 > 0. Therefore, we have the following theorem. Lemma 8.17. With Assumption 14 and τ = 0, the endemic steady state E ∗ of system (8.31) is locally asymptotically stable if it exists. Now we are concerned with the existence of Hopf bifurcation. We will use the idea in Beretta and Kuang [16]. The equation f (λ , τ ) = 0 takes the general form P(λ , τ ) + Q(λ , τ )e−λ τ = 0,

(8.62)

where P(λ , τ ) = λ 3 + a(τ )λ 2 + b(τ )λ + c(τ ), Q(λ , τ ) = d(τ )e−μτ . In the following, we will investigate the existence of purely imaginary roots λ = iω (ω > 0) to Eq. (8.62). Equation (8.62) takes the form of a transcendental function of λ , with all the coefficients of P and Q depending on τ . The authors established in [16] a geometrical criterion which gives the existence of purely imaginary roots of a characteristic equation with delay dependent coefficients. We will use the same notation as in [16] to make things easier for the reader. In order to apply the criterion due to [16], one can easily verify the following conclusions: (i) P(0, τ ) + Q(0, τ ) = 0; + Q(iω ,τ ) = 0; (ii) P(iω , τ )- .  λ ,τ )  : | (iii) lim sup  Q( λ | → ∞, Re λ ≥ 0 < 1;  P(λ ,τ ) (iv) F(ω , τ ) = |P(iω , τ )|2 − |Q(iω , τ )|2 has a finite number of zeros; (v) Each positive root ω (τ ) of F(ω , τ ) = 0 is continuous and differentiable in τ whenever it exists. Let λ = iω (ω > 0) be a purely imaginary root of f (λ , τ ) = 0. Substituting it into the equation f (λ , τ ) = 0 and separating the real and imaginary parts yield that sin(ωτ ) =

−ω 3 + ω b(τ ) a(τ )ω 2 − c(τ ) , cos( ωτ ) = . d(τ )e−μτ d(τ )e−μτ

(8.63)

Furthermore, we have F(ω , τ ) := |P(iω , τ )|2 − |Q(iω , τ )|2 = ω 6 + p(τ )ω 4 + q(τ )ω 2 + r(τ ).

(8.64)

8.3 SIRS Model with Age of Recovery

337

where p(τ ) = a2 (τ ) − 2b(τ ), q(τ )=b2 (τ )−2a(τ )c(τ ) and r(τ )=c2 (τ )−d 2 (τ )e−2μτ > 0. To obtain a positive root of F(ω , τ ) = 0, we set h(x, τ ) = x3 + p(τ )x2 + q(τ )x + r(τ ) = 0. ∂ h(x) 2 τ )x + q(τ ). ∂ x = 3x + 2p(√ −p(τ )+ p2 (τ )−3q(τ ) roots, x∗ (τ ) = 3

The derivative of h(x, τ ) with respect to x is

The

equation 3x2 + 2p(τ )x + q(τ ) = 0 has two and √ 2 −p( τ )− p ( τ )−3q( τ ) x∗∗ (τ ) = . Based on [162], the following lemma gives the results 3 on the positive root of the equation h(x, τ ) = 0. Lemma 8.18 ([162]). (i) If r < 0, then h(x) = 0 has at least one positive root. (ii) If r ≥ 0 and p2 ≤ 3q, then h(x) = 0 has no positive root. (iii) If r ≥ 0 and p2 > 3q, then h(x) = 0 has two positive roots if and only if x∗ > 0 and h(x∗ ) ≤ 0. From Lemma 8.18 (iii), for some τ , equation h(x, τ ) = 0 has two positive real τ )). Thus, Eq. (8.64) has two posiroots denoted by x+ (τ ) and  x− (τ ) (x+ (τ ) > x− ( tive real roots: ω+ (τ ) = x+ (τ ) and ω− (τ ) = x− (τ ). From Lemma 8.18, we define I = {τ ≥ 0 : p2 (τ ) > 3q(τ ), x∗ > 0, h(x∗ , τ ) < 0}. For τ ∈ I , let θ± (τ ) ∈ [0, 2π ] be defined by sin θ± (τ ) =

−ω±3 (τ ) + ω± (τ )b(τ ) aω±2 (τ ) − c(τ ) , cos θ ( τ ) = . ± d(τ )e−μτ d(τ )e−μτ

Define two sequences of functions on I by Sn+ (τ ) = τ −

θ+ + 2nπ θ− + 2nπ , Sn− (τ ) = τ − , ω+ (τ ) ω− (τ )

(8.65)

where n ∈ N = {0, 1, 2, 3, · · ·}. One can verify that iω ∗ (ω ∗ > 0) is a purely imaginary root of Eq. (8.59) if and only if τ ∗ is a root of the function Sn+ or Sn− for some n ∈ N. Based on the above analysis, it has the following result on the existence of Hopf bifurcations. Theorem 8.11. For some τ ∈ R+ , (i) if either p2 (τ ) ≤ 3q(τ ), p2 (τ ) ≥ 3q(τ ) and x∗ < 0 or p2 (τ ) ≥ 3q(τ ), x∗ > 0 and h(x∗ , τ ) > 0 then Eq. (8.59) has no purely imaginary root; (ii) if p2 (τ ) ≥ 3q(τ ), x∗ > 0 and h(x∗ , τ ) ≤ 0 holds, then iω ∗ (ω ∗ = ω (τ ∗ ) > 0) is a purely imaginary root of Eq. (8.59) if and only if τ ∗ is a zero of the function Sn+ (τ ) or Sn+ (τ ) for some n ∈ N. Lemma 8.19. Assume τ ∈ I and iωk (k ∈ {+, −}) are roots of Eq. (8.59). Then d f (λ , τ )  = 0,  dλ λ =iωk and iω+ , iω− are, respectively, simple roots of (8.59).

338

8 Class Age-Structured Epidemic Models

Proof. By the expression of f (λ , τ ) = 0, we know that d f (λ , τ )  = −3ωk2 + b(τk ) + i2a(τk )ωk − d(τk )τk e−(iωk +μ )τk .  dλ λ =iωk Then we find that

d f (λ , τ )  −3ωk2 + b(τk ) − d(τk )τk e−μτk cos(ωk τk ) = 0, =0 ⇔  2aωk + d(τk )τk e−μτk sin(ωk τk ) = 0. dλ λ =iωk That is, tan(ωk τk ) =

−2a(τk )ωk . −3ωk2 + b(τk )

It then follows from (8.63) that −ωk3 + ωk b(τk ) −2a(τk )ωk ω τ ) = = tan( , k k −3ωk2 + b(τk ) aωk2 − c(τk ) that is, 3ωk4 + 2p(τk )ωk2 + q(τk ) = 0. This is a contradiction since τ ∈ I and iωk (k ∈ {+, −}) are roots of Eq. (8.59). Then d f (λ , τ )  = 0, k ∈ {+, −}.  dλ λ =iωk Lemma 8.20. Assume τ ∈ I and iωk (k ∈ {+, −}) are roots of Eq. (8.59). Then the transversality condition holds, i.e., d(Reλ )  = 0.  dτ τ =τk Proof. For the sake of convenience, we study ddλτ instead of ddλτ . From the expression of f (λ , τ ) = 0, we have that   τ 3λ 2 + 2a(τ )λ + b(τ ) d τ   = − − .   d λ λ =iωk λ + μ (λ + μ )(λ 3 + a(τ )λ 2 + b(τ )λ + c(τ )) λ =iωk Rationalizing denominator, we have   dτ  G(ωk , τk ) −μτk Re − , = 2  d λ λ =iωk μ + ωk2 B21 (τk ) + B22 (τk ) where

(8.66)

8.3 SIRS Model with Age of Recovery

339

G(ωk , τk ) = −3ωk6 + A2 (τk )ωk4 + A1 (τk )ωk2 + A0 (τk ), B1 (τk ) = ωk4 − (a(τk )μ + b(τk ))ωk2 + μ c(τk ), B2 (τk ) = (b(τk )μ + c(τk ))ωk − (a(τk ) + μ )ωk3 , A2 (τk ) = a(τk )μ + 4b(τk ) − 2a2 (τk ), A1 (τk ) = μ a(τk )b(τk ) + 2a(τk )c(τk ) − 3μ c(τk ) − b2 (τk ),

A0 (τk ) = μ b(τk )c(τk ) > 0.

Then, substituting ωk values, obtained in the proof of Lemma 8.18, into (8.66), we can check whether the transversality condition holds. We conclude that  dτ  Re d λ  = 0 under some certain conditions. λ =iωk

From (8.63), for τ ∈ I , we define θ± (τ ) ∈ [0, 2π ] by sin θ± (τ ) =

−ω±3 (τ ) + ω± b(τ ) a(τ )ω±2 (τ ) − c(τ ) , cos θ± (τ ) = . − μτ d(τ )e d(τ )e−μτ

Then we set for τ ∈ I that Sn+ (τ ) = τ −

θ+ + 2nπ θ− + 2nπ , Sn− (τ ) = τ − , ω+ (τ ) ω− (τ )

(8.67)

 (ω  > 0) is a purely imaginary where n ∈ N = {0, 1, 2, 3, · · ·}. One can verify that iω root of Eq. (8.59) if and only if τ is a root of the function Sn+ or Sn− for some n ∈ N. From Lemmas 8.19 and 8.20, we know that there exists at least one τn± satisfying dSn± (τn± )/d τ = 0 if Sn± (τ ) equals to zero at some τn± ∈ I for some n ∈ N. Define (8.68) J = {τ j | τ j < τ j+1 , j = 0, 1, 2, · · ·, jC }, where jC = Card{τn± }. Let ω0 be the imaginary part of the purely imaginary root corresponding to τ0 . It is easy to see that S0+ (τ0 ) = 0 and dS0+ (τ0 )/d τ > 0. The following lemma is from [16]. Lemma 8.21 ([16]). Assume that the function Sn+ (τ ) or Sn− (τ ) has a positive root τ ∗ ∈ I , for some n ∈ N, then a pair of simple purely imaginary roots ±iω ∗ of Eq. (8.59) exist at τ = τ ∗ . In addition, (i) if Sn+ (τ ∗ ) = 0 holds, then this pair of simple conjugate pure imaginary roots cross the imaginary axis from left to right if ζ+ (τ ∗ ) > 0 and from right to left if ζ+ (τ ∗ ) < 0, where



+  dRe(λ ) dSn (τ ) ζ+ (τ ∗ ) := Sign |λ =iω ∗ = Sign |τ = τ ∗ ; dτ dτ (ii) if Sn− (τ ∗ ) = 0 holds, then this pair of simple conjugate pure imaginary roots cross the imaginary axis from left to right if ζ− (τ ∗ ) > 0 and from right to left if ζ− (τ ∗ ) < 0, where

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8 Class Age-Structured Epidemic Models

ζ− (τ ∗ ) := Sign

dRe(λ ) |λ =iω ∗ dτ



= −Sign

 dSn− (τ ) |τ = τ ∗ ; dτ

− + (τ ) and Sn− (τ ) > Sn+1 (τ ) for all n ∈ N; (iii) Sn+ (τ ) > Sn+1 − + + (iv) if S0 (τ ) > S0 (τ ) holds, then Sn (τ ) > Sn− (τ ) for all n ∈ N.

It follows from Lemmas 8.15, 8.16 and 8.21, we have that all conditions of Hopf bifurcation theorem in [72] can be satisfied. Then Hopf bifurcation results for the non-densely defined abstract Cauchy problem (8.36) can be expressed as follows: Theorem 8.12. Assume R0 > 1 and Assumption 14 are satisfied. (i) if S0+ (τ ) and S0− (τ ) have no positive zero on I , then the endemic steady state E ∗ is locally asymptotically stable for all τ ≥ 0; (ii) if either Sn+ (τ ) or Sn− (τ ) has positive zeros on I for some n ∈ N, then the endemic equilibrium E ∗ is locally stable for τ ∈ [0, τ0 ), and becomes unstable for τ staying in the right neighborhood of τ0 . In addition, system (8.31) undergoes Hopf bifurcations when τ = τ j for τ j ∈ J .

8.3.4 Global Hopf Bifurcation In the above section, we have proved that the periodic solution of model (8.31) generated by the Hopf bifurcation exists locally when τ near τ j , τ j ∈ J . To get the global continuation of the local bifurcation for system (8.31), we will use global Hopf bifurcation theorem developed by Guo and Wu in [72] listed in the following. Consider the following functional differential equation, ˙ = F(xt , α , p) x(t)

(8.69)

with (α , p) ∈ R × R+ , and F : X × R × R+ → Rn is completely continuous. Define a map Fˆ = F Rn ×R×R+ →Rn : Rn × R × R+ → Rn . We need the following conditions hold. (1) Fˆ is twice continuously differentiable. ˆ α , p) with respect to the first variable x, evaluated at (2) The derivative of F(x, (xˆ0 , α0 , p0 ), is an isomorphism of Rn . (3) F(ϕ , α , p) is differentiable with respect to ϕ , and the n × n complex matrix function Δ(y( ˆ α ,p),α ,p) (λ ) is continuous in (ϕ , α , p) ∈ Bε0 (α0 , p0 ) × C, Here, Δ(xˆ0 ,α0 ,p0 ) (λ ) = λ Id − DF(xˆ0 , α0 , p0 )(eλ · Id), where DF(xˆ0 , α0 , p0 ) is the complexification of the derivative of F(ϕ , α , p) with respect to ϕ , evaluated at (xˆ0 , α , p0 ). (4) There exist ε ∈ (0, ε0 ) and δ ∈ (0, ε0 ) so that on [α0 − δ , α0 + δ ] × ∂ Ωε ,p0 , 2π det Δ(y( ˆ α ,p),α ,p) (u + im p ) = 0 if and only if α = α0 , u = 0, p = p0 , where Ωε0 ,p0 = {(u, p); 0 < u < ε , p0 − ε < p < p0 + ε } .

8.3 SIRS Model with Age of Recovery

341

(5) There exist an isolated center (xˆ0 , α0 , p) and an integer m ∈ J(xˆ0 , α , p) and γm (xˆ0 , α0 , p0 ) = 0, where J(xˆ0 , α0 , p0 ) denote the set of all positive integers m such that im 2pπ is a characteristic value of (xˆ0 , α , p0 ) and γm (xˆ0 , α0 , p0 ) denotes the mth crossing number of (xˆ0 , α , p0 ), defined by

γm (xˆ0 , α , p0 ) = degB (Hm− (xˆ0 , α , p0 ), Ωε ,p0 ) − degB (Hm+ (xˆ0 , α , p0 ), Ωε ,p0 ), here H ± (xˆ0 , α , p0 ) = det Δy( ˆ α0 ±δ ,p),α0 ±δ ,p (u + im

2π ). p

Theorem 8.13 (Theorem 3.3, [72]). Let

∑(F) = Cl {(x, α , p); x is a

p - periodic solution of (8.69)} ⊂ X × R × R,

and N(F) = {(x, ˆ α , p); F(x, ˆ α , p) = 0} . Assume that (x, ˆ α , p) is an isolated center satisfying the above conditions (1)–(5). ˆ α , p) in ∑(F). Then either C(x, ˆ α , p) denotes the connected component of (x, (i) C(xˆ0 , α0 , p0 ) is unbounded or (ii) C(xˆ0 , α0 , p0 ) is bounded, C(xˆ0 , α0 , p0 ) ∩ N(F) is finite and

∑

γm (x, ˆ α , p) = 0

(x, ˆ α ,p)∈C(x, ˆ α ,p)∩N(F)

for all m = 1, 2, · · · , where γm (x, ˆ α , p) is the mth crossing number of (x, ˆ α , p) if m ∈ J(x, ˆ α , p), or it is zero if otherwise. 

From Assumption 14, we let RL (t) = τ∞ R(a,t)da and also denote it as R(t) for simplicity. By some calculations, we can rewrite model (8.31) as the following equivalent model ⎧ dS(t) ⎪ ⎪ = Λ − β S(t)I(t) − μ S(t) + δ ∗ R(t), ⎪ ⎪ ⎨ dt dI(t) = β S(t)I(t) − (μ + ν + γ )I(t), ⎪ dt ⎪ ⎪ ⎪ ⎩ dR(t) = γ e−μτ I(t − τ ) − (μ + δ ∗ )R(t), dt

(8.70)



with the initial conditions that S(0) = S0 , I(0) = I0 , R(0) = τ∞ R0 (a)da. For more details about the transformation between models, one can refer to the reference [45]. Once the global Hopf bifurcation results of model (8.70) are proved, it then gives the global Hopf bifurcation results of model (8.31). Let Y := C([−τ , 0], R3+ ) be equipped with the supremum norm, and ut = (St , It , Rt ) with ut = u(t + θ ), t ≥ 0, θ ∈ [−τ , 0]. Model (8.70) can be modified as the following equivalent functional differential equations u˙t = F(ut , τ , T ),

(8.71)

342

8 Class Age-Structured Epidemic Models

⎞ Λ − β φ1 (0)φ2 (0) − μφ1 (0) + δ ∗ φ3 (0) F(Φ , τ , T ) = ⎝ β φ1 (0)φ2 (0) − (μ + ν + γ )φ2 (0) ⎠ . γ e−μτ φ2 (−1) − (μ + δ ∗ )φ3 (0) ⎛

where

T is the period of the periodic solution, and Φ = (φ1 , φ2 , φ3 ) ∈ Y . The mapping F : Y × R+ × R+ → R3+ is completely continuous. By restricting F to the subspace of Y consisting of all constant mappings with R3+ , we define a mapping F = F|R3 ×R+ ×R+ → R3+ where +

⎞ Λ − β SI − μ S + δ ∗ R  τ , T ) = ⎝ β SI − (μ + ν + γ )I ⎠ . F(u, γ e−μτ I − (μ + δ ∗ )R ⎛

 ∗ , τ ∗ , T ∗ ) = 0. A point (u∗ , τ ∗ , T ∗ ) is called a stationary solution of (8.71) if F(u ∗ ∗  τ, T ) Clearly, there is a unique positive steady state u = E . The derivative of F(u, ∗ with respect to u at the positive steady state u is ⎞ ⎛ −β I− μ − β S δ∗  τ , T )|u=u∗ = ⎝ β I ⎠. Du F(u, 0 0 − μτ ∗ 0 γe − (μ + δ ) The mappings F and F satisfy the following conditions: (A1), (A2), and (A3). (A1) (A2) (A3)

F ∈ C2 (R3+ × R+ × R+ , R3+ ), i.e., it is twice continuously differentiable;  τ , T ) is an isomorphism at the steady state (u∗ , τ , T ); Du F(u, F(Φ , τ , T ) is differentiable with respect to Φ .

The characteristic matrix of model (8.70) at any stationary solution (u∗ , τ ∗ , T ∗ ) is obtained as following: ⎛

λ + β I+ μ β S Δ (u∗ , τ ∗ , T ∗ )(λ ) = ⎝ −β I λ ∗ 0 − γ e−(μ +λ )τ

⎞ −δ∗ ⎠. 0 ∗ λ +μ +δ

From the above characteristic matrix, we have that det(Δ (u∗ , τ , T )(λ )) = λ 3 + a(τ )λ 2 + b(τ )λ + c(τ ) + d(τ )e−(λ +μ )τ ,

(8.72)

where a(τ ), b(τ ), c(τ ), and d(τ ) are given in (8.60). Equation (8.72) is equivalent to that f (λ , τ ) = 0. If Δ (u∗ , τ ∗ , T ∗ )(i T2π∗ m) = 0 for some integer m, the stationary solution (u∗ , τ ∗ , T ∗ ) is called a center. A center is said to be an isolated center if it is the only center in some neighborhood of (u∗ , τ ∗ , T ∗ ). It follows from the analysis in Sect. 8.3.3 that (u∗ , τ j , ω2π0 ) ( j = 0, 1, 2, · · ·, jC ) are isolate centers. Meanwhile, there exist ζ > 0, ε > 0, and a smooth curve λ : (τ j − ζ , τ j + ζ ) → C, such that det(u∗ , τ ∗ , T ∗ )(λ (τ )) = 0, |λ (τ ) − iω0 | < ε for all

8.3 SIRS Model with Age of Recovery

343

τ ∈ [τ j − ζ , τ j + ζ ] and λ (τ j ) = iω0 ,

dRe(λ ) d τ |τ = τ j

= 0. Let

Ωε = {(η , T ) | 0 < η < ε , |T −

2π | < ε }. ω0

Then, it is easily to prove that (A4) det((u∗ , τ , T )(η + i 2Tπ )) = 0 if and only if τ = τ j , η = 0 and T = τ ∈ [τ j − ζ , τ j + ζ ] × ∂ Ωε , j = 0, 1, 2, · · ·, jC .

2π ω0

for

Therefore, the hypotheses (A1)–(A4) in [72] are satisfied. For a periodic solution of Eq. (8.71), we define

Σ (F) := Cl{(u, τ , T ) ∈ Y × R+ × R+ : ut+T = ut }, N(F) := {( u, τ , T ) ∈ R3+ × R+ × R+ : F( u, τ , T ) = 0R3 }, where Σ (F) and N(F) are, respectively, the set of all T -periodic solutions and the set of all trivial periodic

solutions(equilibria) of Eq. (8.71).

2π Let L u, τ j , ω0 be the connected component of the centers u, τ j , ω2π0 (in Σ (F)) of (8.71). Then, we have the following lemma.



Lemma 8.22. L u, τ j , ω2π0 is unbounded for each center u, τ j , ω2π0 .

Proof. Noticing that L u, τ j , ω2π0 is the connected component of the centers

u, τ j , ω2π0 and it is nonempty. From the idea of Ref. [184], we define     2π 2π 2π η +i (η , T ) = det Δ u∗ , τ j ± ζ , . H ± u∗ , τ j , ω0 ω0 T

It follows from (A4) that H ± u∗ , τ j , ω2π0 (η , T ) = 0 for (η , T ) ∈ Ωε . Then the first crossing number ρ (u∗ , τ j , ω2π0 ) of the isolated center (u∗ , τ j , ω2π0 ) can be obtained as    2π 2π ∗ − ∗ ρ u , τ j, = degB H u , τ j, , Ωε ω0 ω0   2π + ∗ − degB H u , τ j, , Ωε ω0 = −1. From the above discussion and the notation of Σ (F), we have that

Σ (u,τ ,T )∈L u,τ

2π j, ω 0



2 ∩N(F)

ρ ( u, τ , T )

< 0.

344

8 Class Age-Structured Epidemic Models

By use of Theorem 3.3 in [72], we obtain that L u, τ j , ω2π0 is unbounded. This completes the proof. From the maximum positively invariant set Ω in (8.32), we obtain the following lemma. Lemma 8.23. All the periodic solutions of system (8.70) are uniformly bounded in R3+ . Lemma 8.24. There is no nonconstant τ -periodic solution for model (8.70). Proof. If (S(t), I(t), R(t)) is a τ -periodic solution of model (8.70), then it is a periodic solution of the following model ⎧ dS(t) ⎪ ⎪ = Λ − β S(t)I(t) − μ S(t) + δ ∗ R(t), ⎪ ⎪ dt ⎪ ⎪ ⎨ dI(t) = β S(t)I(t) − (μ + ν + γ )I(t), ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dR(t) = γ e−μτ I(t) − (μ + δ ∗ )R(t). dt

(8.73)

2 = S(t) + I(t) + R(t) and N 2 ∗ = S∗ + I ∗ + R∗ . By summing the equations Let N(t) of (8.73), we have that 2 d N(t) 2 − (ν + γ − γ e−μτ )I(t), = Λ − μ N(t) dt and model (8.73) is equivalent to the following model ⎧ 2 d N(t) ⎪ ⎪ 2 − (ν + γ − γ e−μτ )I(t), = Λ − μ N(t) ⎪ ⎪ ⎪ dt ⎪ ⎨ dI(t) 2 − I(t) − R(t)) − (μ + ν + γ )I(t), = β I(t)(N(t) ⎪ dt ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ dR(t) = γ e−μτ I(t) − (μ + δ ∗ )R(t). dt

(8.74)

Define a Lyapunov function 2 I, R)(t) = 1 (N(t) 2 −N 2 ∗ ) 2 + κ1 I ∗ V (N, 2 with

κ1 =



I I − 1 − ln ∗ ∗ I I



1 + κ2 (R(t) − R∗ )2 , 2

κ1 β 1 (ν + γ − γ e−μτ ), κ2 = −μτ . β γe

8.3 SIRS Model with Age of Recovery

345

Note that (N − N ∗ )

  dN(t) =(N(t) − N ∗ ) Λ − μ N(t) − (ν + γ − γ e−μτ ) I(t) dt   =(N(t) − N ∗ ) μ (N ∗ − N(t)) − (ν + γ + γ e−μτ )(I ∗ − I(t)) = − μ (N(t) − N ∗ )2 + (ν + γ + γ e−μτ )(I ∗ − I(t))(N(t) − N ∗ ), (8.75)

κ1 (1 −

I ∗ dI(t) I∗ ) =κ1 (1 − ) [β I(N(t) − I(t) − R(t) − (μ + ν + γ )I(t)] I(t) I(t) dt I(t) =κ1 (1 −

I∗ ) [β I(N(t) − I(t) − R(t)) − β I(N ∗ − I ∗ − R∗ )] I(t)

=κ1 (I − I ∗ )(N(t) − N ∗ ) − κ1 (I(t) − I ∗ )2 + κ1 (I(t) − I ∗ )(R∗ − R(t)), (8.76)

and   dR(t) =κ2 (R(t) − R∗ ) γ e−μτ (I(t) − I ∗ ) − (μ + δ ∗ )(R(t) − R∗ ) dt =κ2 (R(t) − R∗ )γ e−μτ (I(t) − I ∗ ) − κ2 (μ + δ ∗ )(R(t) − R∗ )2 . (8.77) With a substitution, we have that the derivative of V along the solutions of model (8.74)

κ2 (R(t) − R∗ )

2 I, R)(t) dV (N, 2 −N 2 ∗ )2 − κ1 β (I − I ∗ )2 − κ2 (μ + δ ∗ )(R − R∗ )2 ≤ 0 = − μ (N dt 2 I, R) = (N 2 ∗ , I ∗ , R∗ ). Thus, the equilibrium with equality holding if and only if (N, ∗ ∗ ∗ 2 (N , I , R ) is globally asymptotically attractive. The contradiction between the global attractivity and existence of a periodic solution implies that model (8.70) has no nontrivial τ -periodic solution, which completes the proof of Lemma 8.24. Recalling that the endemic steady state of model (8.70) is (u∗ , τ ∗ , ω2π∗ ), where = τ j , ω ∗ = ω (τ ∗ ) for τ j ∈ J , and ±iω ∗ is a pair of simple purely imaginary roots of (8.64) when τ = τ j . It follows from (8.67) and the expressions of p(τ ), q(τ ) and r(τ ), that we have

τ∗

ω ∗ τ ∗ > 2π , j ≥ 1, i.e., τ ∗ >

2π for j ≥ 1. ω∗

From Lemma 8.24, similarly as the proof of Lemma 5.4 in [98], we have the following lemma.

346

8 Class Age-Structured Epidemic Models

Lemma 8.25. Periods of periodic solutions of model (8.70) are uniformly bounded. From Lemmas 8.23

and 8.25, we know that the projections of the connected component L u, τ j , ω2π0 onto the u-space and the T -space are, respectively, bounded. Fur

thermore, it follows from Lemma 8.22 we have that the projection of L u, τ j , ω2π0 onto the τ -space is unbounded. Based on the above discussion, we have the global bifurcation of model (8.70) in the following theorem. Theorem 8.14. Assume that R0 > 1 and Theorem 8.12 (ii) hold, then for each τ j ∈ J model (8.70) has at least one nonconstant positive periodic solution when τ varying between τ j and τ j+1 , j ≥ 1. Remark 8.1. Theorem 8.14 shows that, if R0 > 1, under certain additional conditions, the disease will occur in periodic outbreaks with the immunity protection period τ changing.

8.3.5 Simulations In this section, some numerical simulation results of system (8.31) at different parameter values are given. Firstly, pick up Λ = 2.0, β = 0.25, μ = 0.10, ν = 0.20, γ = 8, δ ∗ = 0.5 and τ = 3.5. By a computation, one obtains R0 = 0.6024 < 1 and the endemic steady state is nonexistent. By the analysis of Sect. 8.3.2, one knows that the disease-free steady state E 0 (20, 0, 0) is globally asymptotically stable (see Fig. 8.1). For system (8.74), one chooses Λ = 3.6, β = 0.5, μ = 0.10, ν = 0.20, γ = 8, and δ ∗ = 0.85. The functions Sn± (τ ) are plotted in Fig. 8.2. In Fig. 8.2, there are four Hopf bifurcation values for τ , say τ0 < τ1 < τ2 < τ3 . The first occurs when S0− (τ ) crosses 0 at τ = τ0 = 1.6206, the second occurs when S0+ (τ ) crosses at τ =

I(t)

20 10 0 30

20

E0

10

0 0

S(t)

20

10

30

R(t)=∫∞R(θ,t)dθ 0

Fig. 8.1 The disease-free steady state of system (8.74) is asymptotically stable when R0 ≤ 1. The initial conditions are S0 = 20, I0 = 13 and R0 (a) = 0 E0

8.4 SVIR Model with Age of Vaccination

347

Fig. 8.2 The plots of Sn± (τ )

τ1 = 3.5028, the third occurs when S1− (τ ) crosses at τ = τ2 = 7.7263, and the fourth occurs when S1− (τ ) crosses at τ = τ3 = 8.4717. The stability of the endemic steady state E ∗ switches at τ0 , τ1 , τ2 and τ3 . In Fig. 8.3, one shows the trajectory plots of the (S(t), I(t)) with the same initial functions for five values of τ = 1(< τ0 ), τ = 2.5(∈ (τ0 , τ1 )), τ = 6(∈ (τ1 , τ2 )), τ = 8.1(∈ (τ2 , τ3 )) and τ = 10(> τ3 ). One shows that the endemic steady state E ∗ is asymptotically stable for both τ = 1, τ = 6 and τ = 10 and a stable periodic solution appears for τ = 2.5 and τ = 8.1. One finds that even for delays not chosen in the neighborhood of the Hopf critical points, multiple families of stable periodic orbits occur in a large region delay τ . Namely, stable bifurcation periodic orbits globally exist.

8.4 SVIR Model with Age of Vaccination 8.4.1 Introduction and Model Formulation Vaccination is the administration of antigenic material (a vaccine) to stimulate an individual’s immune system to develop adaptive immunity to a pathogen. Vaccines can prevent or ameliorate infectious disease. When a sufficiently large percentage of a population has been vaccinated, a herd immunity may result. The effectiveness of vaccination has been widely studied and verified. Vaccination is the most effective method of preventing infectious diseases; widespread immunity due to vaccination is largely responsible for the worldwide eradication of smallpox and the elimination of diseases such as polio, measles, and tetanus from much of the world. For infectious diseases caused by single strain, it is possible to prevent them by inventing suitable vaccines. But, for multi-strains infectious diseases such as HIV,

348

8 Class Age-Structured Epidemic Models

*

E

10

∞

R(t)=∫0 R(θ,t)dθ

15

5

0 1.5 1 0.5

I(t)

0 12

(a)

20

18

16

14

S(t)

15 10

0

R(t)=∫∞R(θ,t)dθ

20

5 0 3 2 1

I(t)

(b)

0 12

18

16

14

22

20

24

S(t)

∞ 0

R(t)=∫ R(θ,t)dθ

20 *

E 10

0 2

25

1.5 1

20

0.5 I(t)

∞

R(t)=∫0 R(θ,t)dθ

(c)

0 10

S(t)

20 10 0 1 0.5

(d)

15

I(t)

0 12

16

14

18

20

S(t)

(e)

Fig. 8.3 The endemic steady state E ∗ of system (8.74) is locally asymptotically stable for τ = 1 < τ1 (for a), unstable and there exists a stable periodic solution for τ = 2.5 ∈ (τ0 , τ1 ) (for b) and stable for τ = 6 ∈ (τ1 , τ2 ) (for c), there exists a stable periodic solution for τ = 8.1 ∈ (τ2 , τ3 ) (for d) and stable for τ = 10 > τ3 (for e). The initial conditions are S0 = 13, I0 = 20 and R0 (a) = 0

8.4 SVIR Model with Age of Vaccination

349

HCV, etc., it is very difficult to prevent them by inventing vaccines because the pathogens causing such infectious diseases are mutating constantly. Many are not immune to the disease permanently, and as the time of vaccination goes on, the effect of the vaccine will be reduced. In this section, we formulate and analyze an SVIR epidemic model with age of vaccination. Assume that the total population N is divided into four classes: susceptible, vaccinated, infectious and removed, denoted, respectively, by S,V, I, and R. Consider the following SIR model with age of vaccination: ⎧  ∞ d ⎪ ⎪ S = Λ − β SI − (μ + ψ )S + α (θ )v(θ ,t)d θ , ⎪ ⎪ dt ⎪ 0 ⎪ ⎪ ∂ v( θ ,t) ∂ v( θ ,t) ⎪ ⎪ = −(α (θ ) + μ )v(θ ,t), + ⎪ ⎪ ⎪ ∂θ ∂t ⎨ d I = β SI − (μ + γ )I, (8.78) ⎪ dt ⎪ ⎪ d ⎪ ⎪ R = γ I − μ R, ⎪ ⎪ dt ⎪ ⎪ ⎪ v(0,t) = ψ S(t), ⎪ ⎪ ⎩ 1 (0, +∞), S(0) = S0 , I(0) = I0 , R(0) = R0 , v(θ , 0) = v0 (θ ) ∈ L+ where the vaccinated class (V ) is structured by vaccination age θ . The parameters Λ , β , μ , ψ , γ , and α (θ ) (0 ≤ α (θ ) ≤ αmax ) have the similar biological meanings as in model (8.31). We have that for all t, lim v(θ ,t) = 0.

θ →∞

Obviously, the total number of individuals in the vaccinated class at time t is V (t) =

 +∞ 0

v(θ ,t)d θ .

Summing the equations in (8.78), we have that the total population N(t) satisfies the differential equation (8.79) N  (t) = Λ − μ N(t). It is clear that limt→∞ N(t) = Λ /μ is a solution of (8.79) and for any N(0)(= N0 ) ≥ 0 the general solution of (8.79) is N(t) = N0 e−μ t +

Λ (1 − e−μ t ). μ

So we can assume that the initial values satisfy N0 = S0 + I0 +

 ∞ 0

v0 (θ )d θ + R0 =

Λ μ

such that the population remains a constant size N = Λμ . We only focus on system (8.78) in the plane S + I +V + R = Λ /μ . Thus, we focus on the reduced system

350

8 Class Age-Structured Epidemic Models

 ∞ ⎧ d ⎪ S = Λ − β SI − ( μ + ψ )S + α (θ )v(θ ,t)d θ , ⎪ ⎪ ⎪ dt 0 ⎪ ⎪ ⎪ ∂ v(θ ,t) ∂ v(θ ,t) ⎪ ⎨ = −(α (θ ) + μ )v(θ ,t), + ∂θ ∂t d ⎪ ⎪ I = β SI − (μ + γ )I, ⎪ ⎪ dt ⎪ ⎪ ⎪ v(0,t) = ψ S(t), ⎪ ⎩ 1 (0, +∞). S(0) = S0 , I(0) = I0 , v(θ , 0) = v0 (θ ) ∈ L+

(8.80)

In the following, we will investigate only the dynamics of system (8.80). For ease of notation, let θ

(μ + α (s))ds , K0 (θ ) := exp − 0

K(θ ) := α (θ )ψ K0 (θ ),

K0 

 ∞

K 

0

K0 (θ )d θ ;

 ∞ 0

K(θ )d θ .

The rate of a susceptible person becoming a vaccinated person and leaving the vaccinated class with lost immunity but alive is called the return rate, K . Then the relationship between K and K0 satisfies K =

 ∞ 0

K(θ )d θ = ψ − ψ μ K0 .

Furthermore, we notice that μ K0 is the probability of dying and staying in the vaccinated class. System (8.80) always has a disease-free equilibrium E 0 = (S0 , v0 (θ ), 0), where S0 =

Λ Λ , = μ +ψ −φ μ (1 + ψ K0 )

where

φ=

 ∞ 0

v0 (θ ) =

ψΛ K0 (θ ), μ (1 + ψ K0 )

α (θ )K0 (θ )d θ .

Define the basic reproduction number as follows: R0 (ψ ) = Letting

β A−μ (μ +γ ) μ (μ +γ )K0

βΛ S0 β . = μ +γ μ (μ + γ )(1 + ψ K0 )

(8.81)

= ψ0 , then we have R0 (ψ ) ≤ 1 corresponds to ψ ≥ ψ0

and R0 (ψ ) > 1 corresponds to ψ < ψ0 . Now let us investigate the positive equilibrium of system (8.80). A positive equilibrium (S∗ , v∗ (θ ), I ∗ ) of system (8.80) should satisfy the following equations:

8.4 SVIR Model with Age of Vaccination

351

⎧  ∞ ⎪ ⎪ Λ = (μ + ψ )S∗ + β S∗ I ∗ − α (a)v∗ (a)da, ⎪ ⎨ 0 d ∗ v (θ ) = −(α (θ ) + μ )v∗ (θ ), ⎪ ⎪ d θ ⎪ ⎩ 0 = β S∗ I ∗ − (μ + γ )I ∗ , v∗ (0) = ψ S∗ .

(8.82)

Solving the second equation of (8.82) yields that v∗ (θ ) = ψ K0 (θ )S∗ .

(8.83)

Substituting (8.83) into the first equation of (8.82), we have

Λ = (μ + ψ )S∗ + β S∗ I ∗ − K S∗ .

(8.84)

From (8.83) and (8.84), we can easily see that S∗ =

μ +γ , β

v∗ (θ ) =

ψ (μ + γ ) K0 (θ ), β

I∗ =

1

Λ 1− . μ +γ R0 ( ψ )

Then there is a unique positive equilibrium E ∗ = (S∗ , v∗ (θ ), I ∗ ) if R0 (ψ ) > 1. Summarizing the above discussion, we can reach the following theorem. Theorem 8.15. System (8.80) has a unique disease-free equilibrium E 0 = (S0 , v0 (θ ), 0), if R0 (ψ ) ≤ 1; it has two equilibria E 0 = (S0 , v0 (θ ), 0) and E ∗ = (S∗ , v∗ (θ ), I ∗ ), if R0 (ψ ) > 1. In order to prove the main results of the current chapter, we need to make the following preparations. We will use the framework of integral semigroups. In order to take into account the boundary condition, we enlarge the functional space X = R × R × L1 ((0, +∞), R) × R, 1 X+ = R+ × R × L+ ((0, +∞), R) × R+

and consider the linear operator A : Dom(A ) ⊂ X → X defined by ⎛ ⎞ ⎛ ⎞ −(μ + ψ )S S  ⎜ 0 ⎟ ⎜ ⎟ −v(0) ⎟ ⎜ ⎟ A⎜ ⎝ v ⎠ = ⎝ −v − (α (a) + μ )v ⎠ I −(μ + γ )I with Dom(A) = R × {0} ×W 1,1 ((0, +∞), R) × R, where W 1,1 is a Sobolev space defined in the Appendix. Then Dom(A) = R × {0} × L1 ((0, +∞), R) × R is not dense in X . We consider a nonlinear map F : Dom(A) → X which is defined by

352

8 Class Age-Structured Epidemic Models

 ∞

⎞ Λ − β S(t)I(t) + α (a)v(a,t)da S  ⎟ 0  ⎜ 0 ⎟ ⎜ ⎟ ⎟=⎜ ψ S(t) F⎜ ⎟, ⎜ ⎝ v ⎠ ⎝ ⎠ 0L1 I β SI ⎛

and let

⎞

⎛

  u(t) = S(t),

T 0 , I(t) . v(·,t)

Set X0 := Dom(A) = R × {0} × L1 ((0, +∞), R) × R and 1 X0+ := Dom(A) ∩ X+ = R+ × {0} × L+ ((0, +∞), R) × R+ .

Based on above, we can reformulate system (8.80) as the following abstract Cauchy problem: du(t) = Au(t) + F(u(t)) for t ≥ 0, dt

with u(0) = x ∈ X0+ .

(8.85)

Therefore, problem (8.85) has a solution with the following integral form: u(t) = u(0) + A

 t 0

u(s)ds +

 t

F(u(s))ds.

(8.86)

0

A continuous solution of (8.86) is called a mild solution of problem (8.85). Define the closed convex set 

 ∞  Λ  v0 (θ )d θ ≤ . C = u(0) ∈ X0+  S0 + I0 + μ 0 It is not hard to see that C is forward invariant. Furthermore, define C0 = C ∩ X0+ . The well-posedness of problem (8.85) follows from Theorem 2.3 in [115] if Assumptions 2.1 and 2.2 in [115] are satisfied. Those required assumptions are familiar as and the validating process is also same as that presented in Sect. 8. So, we ignore the details and leave these for the readers as an exercise. By applying the results given in [115, 165], we derive the existence and uniqueness of the semiflow {U(t)}t≥0 on X0+ generated by system (8.85). By using the theory for dynamical system (see [115]), we can further obtain the following lemma. Lemma 8.26. System (8.85) generates a unique continuous semiflow {U(t)}t≥0 on X0+ that is asymptotically smooth and bounded dissipative. Furthermore, the semiflow {U(t)}t≥0 has a compact global attractor A ⊂ X0+ .

8.4 SVIR Model with Age of Vaccination

353

8.4.2 Local Stability Analysis Now, we will emphasize that through the change of variables given above, the general form of system (8.80) is equivalent to the special case (8.90) below. By using Volterra Formulation (see Iannelli [84]) or in Appendix 2, for all t > 0, the term v(θ ,t) can be solved as ⎧ if t ≥ θ , ⎨ ψ S(t − θ )K0 (θ ), K0 (θ ) v(θ ,t) = (8.87) v(θ − t, 0), if θ ≥ t. ⎩ K0 (θ − t) Let x ∈ A . Since U(t)A = A for all t ≥ 0, we can find a complete orbit {u(t)}t∈R through x in A . Set PS u(t) = S(t) and Pv u(t) = v(t)(θ ), for all t ∈ R. Then S(t) > 0, for all t ∈ R and for t ≥ 0, r ∈ R, we have (see [161])  +∞ 0

 +∞

α (θ )v(t + r)(θ )d θ =

t

+

 t 0

α (θ )

K0 (θ ) v(r)(θ − t)d θ K0 (θ − t)

α (θ )ψ K0 (θ )S(t − θ + r)d θ .

Setting tˆ = t + r, it follows that for t ≥ r,  +∞ 0

α (θ )v(t)(θ )d θ =

 +∞ t−r

+ and for t ≥ r,  +∞   α (θ )  t−r

α (θ )

 t−r 0

K0 (θ ) v(r)(θ − (t − r))d θ K0 (θ − (t − r))

α (θ )ψ K0 (θ )S(t − θ )d θ

  K0 (θ ) v(r)(θ − (t − r))d θ  ≤ e−μ (t−r) v(r) L1 (0,+∞) . K0 (θ − (t − r))

It follows that (as r → −∞) for t ∈ R,  +∞ 0

α (θ )v(t)(θ )d θ =

 +∞ 0

α (θ )ψ K0 (θ )S(t − θ )d θ .

(8.88)

For the sake of convenience, we adopt a symbol K(θ ) = ψα (θ )K0 (θ ). Then system (8.80) becomes ⎧  ∞ d ⎪ ⎪ S = Λ − β SI − ( μ + ψ )S + K(θ )S(t − θ )d θ , ⎪ ⎨ dt 0 d I = β SI − (μ + γ )I, ⎪ ⎪ ⎪ dt ⎩ S(0) = S0 , I(0) = I0 .

(8.89)

(8.90)

354

8 Class Age-Structured Epidemic Models

Once system (8.90) is solved, we can use (8.87) to obtain v(θ ,t). In the sequel, we consider system (8.90). Once the stability of system (8.90) is obtained, it then gives the stability of system (8.80). Theorem 8.16 (Local Stability). (i) The disease-free equilibrium E 0 is locally stable if R0 (ψ ) < 1 and is unstable if R0 (ψ ) > 1; (ii) The endemic equilibrium E ∗ is locally stable if R0 (ψ ) > 1. Proof. Linearizing system (8.90) at the equilibrium (S0 , 0) and looking for exponential solutions, we get the following characteristic equation:    λ +μ +ψ −K '1 (λ ) β S0  = 0,  (8.91)  0 λ + ( μ + γ ) − β S0  Λ '1 (λ ) denotes the Laplace transform of K(θ ). Noting that S0 = where K μ (1+ψ K0 ) , and with the help of the expression of R0 (ψ ), Eq. (8.91) can be written as follows:

'1 (λ )] = 0. [λ − (μ + γ )(R0 (ψ ) − 1)][λ + μ + ψ − K If R0 (ψ ) > 1, then the above characteristic equation has at least one positive root, and therefore the disease-free equilibrium E 0 is unstable. If R0 (ψ ) < 1, all the solutions of the characteristic equation have negative real parts provided that all the roots of the equation '1 (λ ) = 0 λ +μ +ψ −K (8.92) have negative real parts. In fact, if λ is a root of Eq. (8.92) with ℜλ ≥ 0, then we have   ∞ θ   ' α (θ )e−λ θ e−μθ e− 0 α (s)ds d θ  |K1 (λ )| = ψ 0  ∞  θ   < ψ α (θ )e− 0 α (s)ds d θ  = ψ 0

and also |λ + μ + ψ | ≥ μ + ψ > ψ , which leads to a contradiction. So we conclude that all the roots of Eq. (8.92), and therefore Eq. (8.91) have negative real parts. Thus the equilibrium (S0 , 0) of system (8.90) (or system (8.80)) is locally stable. Similarly, linearizing system (8.90) at the equilibrium (S∗ , I ∗ ) and looking for exponential solutions, we get the following characteristic equation:    λ +μ +ψ −K '1 (λ ) + β I ∗ β S∗  = 0.  (8.93) ∗ ∗  −β I λ + μ +γ −βS 

8.4 SVIR Model with Age of Vaccination

355

Notice that β S∗ = μ + γ , and '1 (λ ) = ψ − (λ + μ )ψ K '0 (λ ), K

(8.94)

'0 (λ ) is the Laplace transform of K0 (θ ). Equation (8.93) can be written as where K '0 (λ )) + (λ + μ + γ )β I ∗ = 0. (λ + μ )(λ + λ ψ K

(8.95)

Clearly, λ = −μ is not a root of Eq. (8.95). Therefore, Eq. (8.95) can also be written as '0 (λ ) = 0, λ + lβ I∗ + λ ψ K (8.96) where l =

λ + μ +γ λ +μ .

By appealing to (8.94), it follows from Eq. (8.96) that '0 (λ ). '1 (λ ) + μψ K λ + lβ I∗ + ψ = K

(8.97)

Now, let ℜλ ≥ 0, if we can prove the real parts of l is nonnegative, it follows from (8.97) that we have

ψ < |l β I ∗ + ψ | ≤ |λ + l β I ∗ + ψ | '1 (λ ) + μψ K '0 (λ )| ≤ K + μψ K0 = ψ . = |K There is a contradiction. This means that all the roots of Eq. (8.93) have negative real parts, and consequently the equilibrium (S∗ , I ∗ ) of system (8.90) (or system (8.80)) is locally stable. In fact, if we let λ = x + iy with x ≥ 0, then the real parts of l can be worked out as 

 x + μ + γ + iy λ +μ +γ =ℜ ℜl =ℜ λ +μ x + μ + iy

 (x + μ )(x + μ + γ ) + y2 − iγ y =ℜ (x + μ )2 + y2 =

(x + μ )(x + μ + γ ) + y2 ≥ 0. (x + μ )2 + y2

This completes the proof of Theorem 8.16.

8.4.3 Global Stability Analysis In this section, we mainly prove the global stability results for system (8.90). We use the Lemma A.12 to establish the global stability of the disease-free steady state E 0 . To this end, we first introduce the notation g∞ = lim inf g(t) and g∞ = lim sup g(t). t→∞

t→∞

356

8 Class Age-Structured Epidemic Models

Theorem 8.17. If R0 (ψ ) < 1, then the disease-free equilibrium E 0 = (S0 , v0 (θ ), 0) is the unique equilibrium of system (8.84), and it is globally stable. Proof. Theorem 8.16 shows that the disease-free equilibrium E 0 of system (8.90) is locally stable if R0 (ψ ) < 1. Choose the sequences tn1 → ∞ such that S(tn1 ) → S∞  and S (tn1 ) → 0. Then F1 (t) = 0 α (θ + t)ν0 (θ ) KK0 (θ(θ+t) d θ → 0 as t → ∞. With the 0 ) assistance of Lemma A.12, it follows from the first equation of (8.90) that 0 ≤ Λ − (μ + ψ )S∞ + S∞ K , and S∞ ≤

Λ . μ +ψ −K

It follows from the second equation of (8.90) that dI(t) = β S(t)I(t) − (μ + γ )I(t) dt βΛ I(t) − (μ + γ )I(t) = (μ + γ )[R0 (ψ ) − 1]I(t). ≤ μ +ψ −K This leads to I ∞ → 0 as R0 (ψ ) < 1. Choose the sequences tn2 → ∞ such that S(tn2 ) → S∞ and S (tn2 ) → 0. Note that ∞ I = 0 and lim F1 (tn2 ) = 0. Thus, it follows from and the first equation of (8.90) that n→∞

0 ≥ Λ − β S∞ I ∞ − (μ + ψ )S∞ + K S∞ = Λ − (μ + ψ )S∞ + K S∞ . Then we have that

Λ Λ ≤ S∞ ≤ S∞ ≤ . μ +ψ −K μ +ψ −K That is, lim S(t) = t→∞

Λ μ +ψ −K

. It follows from (8.87) that

lim v(θ ,t) =

t→∞

ψΛ K0 (θ ) = v0 (θ ). μ +ψ −K

1 × R as t → ∞, This completes the proof of Therefore, (S, v, I) → E 0 in R+ × L+ + Theorem 8.17.

Lemma 8.27. If R0 (ψ ) > 1, then there exists a global attractor A for the solutions semiflow {U(t)}t≥0 of (8.85) on C . Following a familiar steps as the above section, we give the following lemma to present the persistence of system (8.84). Lemma 8.28. If R0 (ψ ) > 1, then system (8.84) is weakly uniformly persistent, i.e., there exists a constant ε > 0 such that lim supt→∞ I(t) > ε .

8.4 SVIR Model with Age of Vaccination

357

Proof. If R0 (ψ ) > 1, we can choose an ε0 > 0 such that

βΛ − β ε0 ≥ (μ + γ ). μ + ψ − K + β ε0

(8.98)

Assume by contradiction that for any small enough ε0 > 0, there exists a t1 , such that I(t) ≤ ε0 when t > t1 . Furthermore, if {tn } is a sequence such that S(tn ) → S∞ , Lemma A.12 together with (8.87) shows that dS(tn ) ≥ Λ − β ε0 S(tn ) − (μ + ψ )S(tn ) + dt

 tn

Then it follows from Lemma A.12 that we have S∞ ≥

K(θ )S(tn − θ )d θ .

Λ μ +ψ −K +β ε0 . Thus there exists

Λ μ +ψ −K +β ε0 − ε0 . Again, by replacing the initial condition, S(t) ≥ μ +ψ −ΛK +β ε − ε0 for t ∈ R+ . It then follows that 0

t2 such that S(t) ≥ can assume that

0

we



Λ − ε0 β I − (μ + γ )I μ + ψ − K + β ε0  βΛ − β ε0 − ( μ + γ ) I = μ + ψ − K + β ε0

dI(t) = β SI − (μ + γ )I ≥ dt

for t ∈ R+ . It follows from the positivity of I0 and (8.98) that lim inf I(t) = +∞. This t→∞ contradicts with the boundedness of I. The proof is complete. Theorem 8.18. The unique endemic equilibrium E ∗ = (S∗ , v∗ (θ ), I ∗ ) of system (8.84) is globally asymptotically stable provided it exists. Proof. When R0 (ψ ) > 1, we define the following Lyapunov function U22 (t) = S(t) − S∗ − S∗ ln

S(t) I(t) + I(t) − I ∗ − I ∗ ln ∗ . S∗ I

(8.99)

Obviously, U22 (t) is positive-definite with (S∗ , I ∗ ) as its global minimum point. Calculating the time derivative of U22 (t) along the solution of system (8.90), we obtain that  ∞

S∗ dU22 (t)  = 1− Λ − β SI − (μ + ψ )S + K(θ )S(t − θ )d θ  dt S(t) (8.90) 0

I ∗ + 1− β SI − (μ + γ )I . I(t) (8.100) ∞ ∗ ∗ ∗ ∗ ∗ ∗ Noting also that Λ = β S I + (μ + ψ )S − 0 K(θ )S d θ and β S I = (μ + γ )I ∗ , it follows that

358

8 Class Age-Structured Epidemic Models

dU22 (t)  S∗

(μ + ψ )(S∗ − S(t)) = 1−  dt S(t) (8.90) S∗

(μ + γ )I ∗ − β S(t)I(t) + β S∗ I + β S(t)I(t) − β S(t)I ∗ + 1− S(t) 

S∗ ∞ + 1− K(θ ) S(t − θ ) − S∗ d θ − (μ + γ )(I(t) − I ∗ ) S(t) 0

2 μ +ψ ∗ =− S − S(t) S(t)  ∞ S(t)

S∗ − ∗ (μ + γ )I ∗ + + 2− K(θ )S(t − θ )d θ S 0  ∞S(t)

∗ S(t − θ ) KS dθ + −S∗ K(θ ) S∗ − S(t) . S(t) S(t) 0 (8.101) Let  ∞  θ S(t − r)

U2 (t) = U22 (t) + K(θ ) (8.102) S(t − r) − S∗ − S∗ ln drd θ , S∗ 0 0 which is also positive-definite with (S∗ , I ∗ ) as its global minimum point. Then we have  ∞  ∞

dU22 (t)  S(t − θ ) dU2 (t)  dθ = + K(θ ) S(t) − S(t − θ ) d θ + S∗ K(θ ) ln   dt (8.90) dt S(t) (8.90) 0 0  ∞  ∞  dU22 (t)  S(t − θ ) dθ . = + K S(t) − K(θ )S(t − θ )d θ + S∗ K(θ ) ln  dt S(t) (8.90) 0 0

Thus, it follows from (8.101) that

2 S(t)

μ +ψ ∗ dU2 (t)  S∗ − ∗ (μ + γ )I ∗ S − S(t) + 2 − =−  dt (8.90) S(t) S(t) S

K S∗ ∗ S − S(t) + K S(t) − K S∗ + S(t)  ∞ S(t − θ )

S(t − θ ) + ln +S∗ K(θ ) 1 − dθ S(t) S(t) 0

2 S(t)

μ (1 + ψ K0 ) ∗ S∗ − ∗ (μ + γ )I ∗ S − S(t) + 2 − =− S(t) S(t) S  ∞

S(t − θ ) S(t − θ ) + ln +S∗ K(θ ) 1 − dθ . S(t) S(t) 0 Notice that

S∗ S(t)

+ S(t) S∗ ≥ 2 and the equality holds if and only if S(t) = S∗

θ) S(t−θ ) and also that for S(t − θ ) and S(t) > 0, 1 − S(t− S(t) + ln S(t) ≤ 0 with equality holding if and only if S(t − θ ) = S(t).

8.4 SVIR Model with Age of Vaccination

Therefore, we have that

dU dt

359

≤ 0. Define

 dU .  = 0 = {(S, I)| S = S∗ , I ≥ 0}. Ω = (S, I) dt We can show that its largest invariant subset M only contains the endemic equilibrium (S∗ , I ∗ ) when R0 (ψ ) > 1. In fact, let (S(t), I(t)) be the solution with initial data in M . From the invariance of M , we know that S(t) = S∗ for any t. It then follows from the first equation of system (8.90) that I(t) = I ∗ for any t. Therefore, we have that M = {(S∗ , I ∗ )}. According to the LaSalle’s invariance principle, the equilibrium (S∗ , I ∗ ) of system (8.90) is globally asymptotically stable when it exists. Combining with (8.87), we can see that v(θ ,t) trends to v∗ (θ ) as time goes to infinity. Then the endemic equilibrium E ∗ of system (8.84) is globally asymptotically stable, and it completes the proof of Theorem 8.18.

Appendix A

In this appendix, we will show some existing results to simplify written styles of the book and make it compact. All the notations are enclosed in main chapters. We begin from some definitions for the operator. In the one-dimensional case, the Sobolev space W k,p is defined to be the subset of functions f in L p (R) such that the function f and its weak derivatives up to some order k have a finite L p norm, for given p(1 ≤ p < +∞). For example, f (a) ∈ W 1,1 , a ∈ R implies that f (a) ∈ L1 (R) and f  (a) ∈ L1 (R). Definition A.1 ([113]). Let A : D(A) ⊂ X → X be the infinitesimal generator of a linear C0 -semigroup {TA (t)}t≥0 on a Banach space X. We define the growth bound w0 (L) ∈ [−∞, ∞) of A by w0 (A) = lim

t→∞

ln( TA (t) X ) . t

The essential growth bound w0,ess (A) ∈ [−∞, ∞) of A is defined by w0,ess (A) = lim

t→∞

ln( TA (t) ess ) , t

where TL (t) ess is the essential norm of TA (t) defined by

TA (t) ess = κ (TA (t)BX (0, 1)). Here BX (0, 1) = {x ∈ X : x X ≤ 1} and, for each bounded set B ⊂ X,

α (B) = inf{ε > 0 : B can be covered by a finite number of balls of radius ≤ ε } is the Kuratowski measure of non-compactness. The α -growth bound of TA (t) w1 (A) denotes the measure of noncompactness of a bounded linear operator A, and it is defined as follows: ln(α [TA (t)]) . w1 (A) = lim t→∞ t © Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1

361

362

Appendix A

A.1 Hille-Yosida Theorem Let X be a Banach space, and L (X) be the space of bounded operators on X with the usual operator norm. Definition A.2 (Hille-Yosida Operator, [148]). Let A be a bounded operator on a Banach space X. If there exist M ≥ 1 and w0 ∈ R such that (w0 , ∞) ∈ ρ (A), ρ (A) is the resolvent A

ρ (A) = {λ ∈ C|ker(λ I − A) = {0}, (λ I − A)−1 ∈ L (X)}, and

(λ I − A)−n ≤

M . (λ − w0 )n

Then A is called a Hille-Yosida operator of (M, w0 ). Lemma A.1. Supposed a linear operator A is the infinitesimal generator of a C0 semigroup {T (t)}t≥0 on a Banach space (X, · ). Then A is a Hille-Yosida operator and there exists a norm | · | on X, such that |(λ I − A)−1 |L (x) ≤

1 , ∀ t > ω0 . λ − ω0

Theorem A.1. A linear operator A : D(A) ⊂ X → X is the infinitesimal generator of ¯ = X. a C0 -semigroup {T (t)}t≥0 if and only if A is a Hille-Yosida operator with D(A) Moreover,

T (t) ≤ Meω0 t , t ≥ 0 where M and ω0 are defined in Definition A.2. Proof. Let us define a norm | · | on X |x| = sup e−ω0 t T (t)x , ∀ x ∈ X. t≥0

Based on the norm definition, we have that x ≤ |x| ≤ M x . Therefore,

(λ I − A)−n L (X) ≤ M|(λ I − A)−1 nL (x) ≤

M , (λ − ω0 )n

, ∀ λ ∈ (ω0 , +∞). (A.1)

¯ = X. Lemma A.2. Let A : D(A) ⊂ X → X be a Hille-Yosida operator with D(A) Then for any x ∈ X, lim λ (λ I − A)−1 x = x. λ →+∞

The following lemma directly follows from the Hille-Yosida Theorem A.1. Lemma A.3. Let A be a linear operator on a Banach space X. The part AD(A) ¯ of A is ¯ the infinitesimal of a C0 -semigroup {TD(A) ¯ }t≥0 on D(A) if and only if the following two conditions hold:

Appendix A

363

(1) For any x ∈ X, lim (λ I − A)−1 x = 0; λ →+∞

(2) There exist two positive constants ω0 and M such that for λ > ω0 and n ≥ 1

(λ I − A)−n L (D(A)) ≤ ¯

M . (λ − ω0 )n

A.2 Volterra Formulation Consider the following age-structured model

∂ p(t, a) ∂ p(t, a) + = −μ (a)p(t, a),t ≥ 0, a > 0, ∂t ∂a  p(t, 0) = h(

∞

0

(A.2)

b(a)p(t, a)da),t ≥ 0

1 (R , R), b(·) ∈ with initial value p(0, ·) = φ (·) ∈ L1 (R, R+ ). Assume that μ ∈ L+ + ∞ (R , R) denotes the natural death rate and the birth rate, respectively. Thus, L + + ∞ 0 b(a)p(t, a)da represents the total newborns. The function h(x) represents the mechanism of the newborns, which can be defined as the Ricker’s type h(x) = xe−δ x with δ > 0. Let us define a map P : (t, a) → p(t, a) as follows:

lim

t→+∞

p(t + h, a + h) − p(t, a) ∂ p(t, a) ∂ p(t, a) = + . h ∂t ∂a

Therefore, (A.2) can be written as P (h) = p (t + h, a + h) = −μ (a)P(h). Solving it along the characteristic line t − a = c, we have P(0) = P(−h)e− If we set B(t) =

∞ 0

a

a−h μ (s)ds

b(a)p(t, a)da and π (a) = e−

a 0

.

μ (s)ds ,

then

⎧ ⎨ h(B(t − a))π (a),t ≥ a, p(t, a) = π (a) ⎩ φ (a − t) ,t < a. π (a − t)

(A.3)

Replacing p in the formula B by (A.3), we have B(t) =

 t 0

where F(t) =

b(a)h(B(t − a))π (a)da + F(t),  ∞ t

b(a)φ (a − t)

(A.4)

π (a) da. π (a − t)

Therefore, Eq. (A.4) can be rewritten as the following Volterra integral equation:

364

Appendix A

B(t) = (K ∗ h(B))(t) + F(t),

(A.5) 

where K(a) = b(a)π (a) and the convolution operator K ∗ h(B) = 0t K(a)h(B(t − a))da. Moreover, F(t) denotes the initial data and K(t) denotes a locally integrable function. For the sake of convenience, let us take h(x) = x and the initial data F(t) = I where I is an identity map. Taking coevolution on both sides of (A.5), we obtain that B ∗ K = F ∗ K + B ∗ K ∗ K = K + B ∗ K ∗ K. (A.6) Now we define R(t) = B ∗ K. (A.6) can be rewritten by R(t) = K(t) + R ∗ K = K(t) + K ∗ R. Hence, we can call R(t) as the resolvent associated with the kernel function K(t). ∞

(A.6) implies that R(t) = ∑ K ( j) (t) where K (1) = K(t), K ( j+1) (t) = K ( j) ∗ K(t) = j=1

t

( j) 0 K (t − s)K(s)ds. If h(x) = x, the solution of (A.4) can be given by

B(t) = F(t) + (R ∗ F)(t).

(A.7)

 Let us define  f (λ ) = 0∞ eλ t f (t)dt, a Laplace transform of f . From Theorem 5.2, [136], we have the following lemma on the integrable feature of the resolvent R.

Lemma A.4 ([136, Paley-Wiener Theorem, Theorem 5.2]). The resolvent R(·) is integrable if and only if the following condition holds:  det(I − K( λ )) = 0,

∀ Reλ ≥ 0.

(A.8)

 where r(·) denotes the spectral radius of K(0).  Now, let us define R0 = r(K(0)), From [96, Proposition 10.38], we have the following theorem. Theorem A.2. Let K and F be both nonnegative and integrable on R+ . Then (1) if R0 < 1,

∞ 0

  −1 F(0); B(t)dt = (I − K(0))

1  t − λ0 t e B(t)dt t→∞ t 0

(2) if R0 ≥ 1, lim

=

 v0 ,F( λ0 ) v0 ,K1 u0  u0 ,

 where λ0 is a unique real solution of r(K( λ )) = 1, v0 and u0 are the left and right   eigenvectors of K(λ0 ) with respect to its eigenvalue of unity, and K1 := 0∞ tK(t)dt. The following lemma shows the distributions of the characteristic roots of (A.4), which has been proved by Inaba in [96].  λ ) be a compact operator for any λ ∈ C. Moreover, there exLemma A.5. Let K( ists a quasi-interior point e and a strictly positive functional Fλ associated with a 1 such that positive cone L+

Appendix A

365

 λ )ψ ≥ Fλ , ψ e, K(

lim Fλ , e = +∞.

λ →−∞

Then, the following statements hold.  λ ))}, (1) Λ = {λ ∈ C|1 ∈ Pσ (K(  λ ) is strictly nonsupporting, (2) For any λ ∈ R, the operator K(  λ )) is monotone decreasing with respect to (3) For λ ∈ R, the spectral radius r(K( λ,  (4) The characteristic equation r(K( λ )) = 1 has a unique real root λ0 such that  − 1), sign(λ0 ) = sign(r(K(0)) (5) λ0 > sup{Reλ |λ ∈ Λ \{λ0 }}.

A.3 Positive Operator Let E and E ∗ be a real or a complex Banach space and its dual space, respectively. Definition A.3. A closed set B ⊂ E is the cone if the following conditions hold. (1) B + B ⊂ B; (2) for each λ ∈ R+ , λ B ⊂ B; (3) B ∩ (−B) = {0}; (4) B = 0. The cone is total if the set {ψ − φ |ψ , φ ∈ B} is dense in E. A nonzero operator T ∈ L (E) is positive if T (B) ⊂ B. A positive operator T is semi-nonsupporting if for any ψ ∈ B+ and f ∈ B∗ \{0}, there exists an integer p = p(ψ , f ) such that  f , T p ψ  > 0. A positive operator T is nonsupporting if T is semi-nonsupporting and for each n > p such that  f , T n ψ  > 0. Lemma A.6 (Corollary 5.2 [107], Proposition 4.6 [92]). Let E be a real Banach space and E+ be its positive cone. Let Φ be a positive nonlinear operator on E. Suppose that (i) Φ is compact, Φ (E+ ) is bounded, and Φ (0) = 0, where 0 ∈ E+ is the origin of the Banach space E. (ii) Φ has the strong Fréchet derivative T := Φ  [0] : E → E at the origin. (iii) T has a positive eigenvector v0 ∈ E+ \{0} corresponding to an eigenvalue λ0 > 1, and it has no eigenvector corresponding to eigenvalue 1. Then, Φ has at least one nontrivial fixed-point in E+ \{0}. Lemma A.7 (Corollary 7.6 [95]). Let Ω1 be a cone of a Banach space X and K be a bounded linear operator from Ω1 to itself. Assume that Ω1 is total, the spectral radius r(K ) > 0 of operator K is positive and K is compact and nonsupporting with respect to Ω1 . Then the following holds:

366

Appendix A

(i) r(K ) ∈ Pσ (K )\{0}, where Pσ denotes the point spectrum of an operator. Moreover, r(K ) is a simple pole of the resolvent R(λ , K ) := (λ − K )−1 . (ii) The eigenvalue corresponding to r(K ) is one dimensional and its eigenvector ψ ∈ C is a quasi-interior point, that is,  f , ψ  > 0 for all f ∈ Ω1∗ \{0}. Moreover, any eigenvector of K is proportional to ψ . (iii) r(K ) is a dominant point of the spectrum σ (K ), that is, |μ | < r(K ) holds for all μ ∈ σ (K )\{r(K )}. (iv) B1 = lim r(K )−n T n converges in the topology of the operator norm and it is n→∞ a strictly nonsupporting operator given by B1 =

1 2π i

 Γ0

R(λ , K )d λ ,

where Γ0 denotes the positively oriented circle centered at r(K ) such that no points of the spectrum σ (K ) except r(K ) lie on and inside Γ0 . Lemma A.8 ([105, Krein-Rutman Theorem]). Suppose that B is total and the positive linear operator T : B → B is compact and r(T ) > 0. Then, r(T ) is an eigenvalue of T associated with a positive eigenvector ψ ∈ B+ . The following lemma gives the existence of the fixed point of a positive operator Ψ . Lemma A.9 ([104, Theorem 4.11]). Let Ψ be a positive operator from a cone B in a real Banach lattice into itself. Suppose that Ψ (0) = 0, Ψ has a strong Fréchet derivative T : Ψ  (0) and T has a positive eigenvector v0 ∈ B with respect to the eigenvalue λ0 > 1, but has no eigenvector in C associated with unity. Moreover, Ψ has a strong asymptotic derivative Ψ  (∞) associated with a cone B and the spectrum of the operator Ψ  (∞) lies in the circle |λ | ≤ ρ < 1. Then, Ψ has at least one nonzero fixed point in B if Ψ is completely continuous.

A.4 Some Features and Definitions In this subsection, we give some basic features of age-since-infection model for constructing Lyapunov functional to prove the global behaviors of the models. For the sake of convenience, we assume that f (t, a) ∈ L1 (R+ ) for t ∈ R+ is a solution of the following system: ⎧ ∂ f (t, a) ∂ f (t, a) ⎪ ⎪ + = −l(a) f (t, a), ⎪ ⎪ ⎨ ∂t ∂ a∞ (A.9) β (a) f (t, a)da, f (t, 0) = S(t) ⎪ ⎪ 0 ⎪ ⎪ ⎩ f (0, a) = 0. We want to show that f is a equicontinuous function for t, a ∈ R+ .

Appendix A

367



Lemma A.10. Suppose that S (t) ≤ M, 0∞ f (t, a)da ≤ M, and β ∈ L∞ (R+ ) for some constant M. There exists a constant M1 and sufficient small value h > 0 such that for all t > a  t−h 0

Proof. Denote B(t) = S(t) we readily obtain that

| f (a,t + h) − f (t, a)|da ≤ M1 h.

∞ 0

β (a) f (t, a). Then by the assumption of this lemma,

|B(t)| = |S(t)

 ∞ 0

β (a) f (t, a)da| ≤ β¯ M 2 := MB .

For all t > a, we integrate Eq. (A.9) along the characteristic line and obtain that a f (t, a) = B(t)π (a), π (a) = e− 0 l(s)ds . Deviating f (t, a) with respect to a yields

∂ f (t, a) = −B (t − a)π (a) − l(a)π (a)B(t − a). ∂a

(A.10)

On the other hand, for any t ∈ R+ ,  ∞

 ∞

∂ f (t, a)da da| ∂t    ∞  ∞ ∂ f (t, a)  ≤|S (t)|| β (a) f (t, a)da| + |S(t) β (a) −l(a) f (t, a) − da| ∂a 0 0  ∞ ∂ f (t, a) ¯ + β¯ M |da. | ≤β¯ M 2 + β¯ lM ∂a 0 (A.11) Then we plug (A.11) into (A.10) and obtain |B (t)| =|S (t)

|

0

β (a) f (t, a)da + S(t)

0

β (a)

∂ f (t, a) | =| − B (t − a)π (a) − l(a)π (a)B(t − a)| ∂a ¯ B ≤|B (t − a)| + lM  ∞ ∂ f (t, a) ¯ + lM ¯ B ) + β¯ M |da ≤(β¯ M 2 + β¯ lM | ∂a 0  t ∂ f (t, a) =M f + β¯ M | |da, ∂a 0

(A.12)

¯ + lM ¯ B . From Gronwall inequality, we have where M f = β¯ M 2 + β¯ lM |

∂ f (t, a) ¯ | ≤ M f eβ Mt . ∂a

Now, we are in a position to prove t ∈ R+ ,  t−h 0

 t−h 0

| f (a,t + h) − f (t, a)|da ≤

where ξ ∈ (0,t − h).

| f (a,t + h) − f (t, a)|da ≤ M1 h. For any fixed

 t−h ∂ i(t, ξ ) 0

|

∂a

¯

|hda ≤ M f eβ Mt (t − h)h,

368

Appendix A

Lemma A.11 (Proposition 3.1, [85]). Suppose h : R+ → R+ is a bounded function and k ∈ L1 (R+ ). Then   t lim sup k(θ )h(t − θ )d θ ≤ k 1 lim sup h(t) . t→∞

0

t→∞

Lemma A.12 (Fluctuation Lemma [83]). Let ϕ : R+ → R be a bounded and continuously differentiable function. Then there exist sequences {sn } and {tn } such that sn → ∞, tn → ∞, ϕ (sn ) → ϕ∞ , ϕ  (sn ) → 0, ϕ (tn ) → ϕ ∞ , and ϕ  (tn ) → 0 as n → ∞. Let B be a bounded set of X. The asymptotically compact is defined as follows. Definition A.4 (Asymptotically Compact, [157]). The semiflow Φ (t) is called asymptotically compact on B ⊂ X. If for any sequence xn ∈ B and tn → ∞, there exist subsequences, which can be relabeled as xn and tn , with property that y = lim Φ (tn )xn and y ∈ M.

n→∞

Moreover, asymptotical smoothness is given as follows. Definition A.5. The semiflow {Φ (t)}t≥0 is called asymptotically smooth if {Φ (t)}t≥0 is asymptotically compact on every forward invariant bounded closed set. Definition A.6. The semiflow {U(t)}t≥0 is called bounded dissipative if there exists a bounded subset M ∈ X such that M attracts all points in X0 . Lemma A.13 ([41, Theorem 3.4.6]). Let T (t) be a semigroup acting on X = 1 (0, ∞) × R . If T (t) : X → X , t ∈ R is asymptotically smooth, point disR+ × L+ + + sipative, and orbits of bounded sets are bounded, then there exists a global attractor. A semiflow is asymptotically smooth if each forward invariant bounded closed set is attracted by a nonempty compact set. Lemma A.14 ([41, Theorem 3.2.3]). For each t ∈ R+ , suppose T (t) = Ψ (t) + Φ (t), where Ψ (t), Φ (t) : X → X . Suppose Φ (t) is completely continuous and there is a continuous function η : R+ × R+ → R+ such that η (t, h) → 0 as t → ∞ and Ψ (t)x ≤ η (t, h) if x < h. Then T (t), t ∈ R+ , is asymptotically smooth. Theorem A.3 (Theorem 2.33, [80]). The semiflow Φ : R+ × X0,+ → X0,+ is asymptotically smooth if there are maps Φˆ 1 (t), Φ˜ (t) : R+ × X0,+ → X0,+ such that

Φ (t)x0 = Φˆ (t)x0 + Φ˜ x0 , and the following hold for any bounded closed set K that is forward invariant under Φ (t) : (i) lim inf diamΦˆ (t)(K ) = 0; t→∞

(ii) there exists a t ≥ 0 such that Φ˜ (t)(K ) has a compact closure for each t ≥ tK .

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Index

A Access, 259 Adjacent matrix, 189 Adjoint system 44–48, 50 α -growth bound, 295 Asymptotically compact, 370 Asymptotically smooth, 130, 160, 183, 184, 250, 309, 355, 370 B Backward bifurcation 21, 30–34, 36–38, 60, 93, 94, 103 Banach lattice, 219, 226, 261, 368 Banach space, 23, 162, 214, 218, 221, 263, 295, 296, 304, 313, 327, 363, 364, 367 Barabasi-Albert (BA) networks, 174 Benefit-cost, 134, 135 Between-host models 69, 72, 73, 74, 76, 77, 78, 80, 84, 85, 87, 90, 91, 95, 96, 97, 99 Bidirectionally linked models 84, 85–94 Bounded dissipative, 308, 355, 370 C Cauchy sequence, 113 Characteristic equation 7, 8, 14, 29, 39, 57, 59, 78, 79, 82, 91, 92, 98, 99, 102, 116, 117, 142, 144, 167, 181, 182, 194, 199, 201, 225, 239, 242, 244, 301, 312, 319, 320, 332, 333, 334, 336, 337, 356, 357, 367 Characteristic lines 3, 40, 48, 164, 236, 249, 278, 308, 317, 335, 365, 369 Cholera model 10, 89–94 Coexistence, 21, 53, 59, 60, 71, 72, 107, 191–193, 197, 201, 203, 207, 257 Compact closure, 162, 311, 370

Competitive exclusion principle, 232, 233, 244, 245, 257 Complex networks, vi, 153–209 Concave operator, 221, 223 Cost, 44, 53, 108–110, 132–135, 148, 149 C0-semigroup, vi, vii, 211, 215, 229, 231, 269, 284, 295, 303, 313, 363, 364 D Death rate 3, 19, 70, 90, 95, 109, 176, 212, 232, 280, 281, 283, 302, 325, 365 Degree, 9, 12, 13, 15, 153–156, 163, 174–176, 190–192, 196, 197, 203 Degree correlations, 154 Densely defined operator, 231 Direct access, 259 Directed networks, 153 Disease-free equilibrium 20, 28–30, 35, 38, 39, 54–58, 77, 78, 80, 81, 86, 88, 91, 92, 97, 98, 101, 102, 114, 137, 138, 140, 162–165, 175, 178–180, 181, 190, 193, 194, 198–200, 205, 237–241, 246, 256, 277, 278, 279, 288, 295, 299, 311, 312, 314, 316, 317, 330, 335, 351, 352, 356, 358 Disease-induced mortality 16–18, 73, 83, 84 Dynamical networks, 196 E Edge-based compartmental model, 154 The essential growth bound, 313, 332, 363 Eventually compact, 229 F Frechet-Kolmogorov Theorem, 250 Fredholm operator, 313

© Springer Nature Switzerland AG 2020 X.-Z. Li et al., Age Structured Epidemic Modeling, Interdisciplinary Applied Mathematics 52, https://doi.org/10.1007/978-3-030-42496-1

381

382 G Game theory, vi, 105–151 Gateaux differentiable, 44 Global attractor, 129, 130, 148, 245, 248, 252, 256, 318, 355, 359, 370 Global Hopf bifurcation, 341, 342 Graph theory, 153, 279 Growth bound, 295, 296, 313, 332, 363 H Herd immunity, 348 Hille-Yosida operator, 305, 331, 332, 364 Hille-Yosida theory, 364 HIV model 19, 68, 70, 72, 73, 80–83, 85–89, 196, 231 Homogenous age-structured epidemic model 155 Hopf bifurcation, 119, 124, 144, 151, 196, 335–346 I Imitation dynamics, 108–150 Immuno-epidemiological models vi, 69–103 Incidence mass action 11, 15, 16, 95, 302 standard 12, 79, 80, 155 Indirect access, 259 Infinitesimal generator, 231, 262, 284, 295, 296, 313, 363, 364 Inter-cohort 12, 15, 16, 19, 20, 21, 27, 32, 38, 42, 54 Intermediate value theorem, 164, 242 Intra-cohort 12, 15, 19, 20, 21, 266 Invasion numbers 58–67, 71, 72, 206 Irreducibility, 260 K Kermack-McKendrick model v, 10, 11, 73 Krein-Rutman Theorem, 274, 368 Kuratowski measure, 363 L Laplace transform, 129, 147, 148, 169, 185, 247, 356, 357, 366 Linking parameters 73, 76, 100 Lotka model 1, 3, 4 M Malthusian parameter/Intrinsic growth rate 7, 8, 13, 15 McKendrick age-structured model v, 1–4, 13 Mean-field model, 154 Model with re-infection 32–39 Monotone operator, 222, 270

Index Mortality rate 2, 10, 11, 264 Mutation, 96, 99, 191, 196–203, 257, 260 N Nested models 69, 76, 103 Net reproduction rate 7 Non-densely defined operator, vii, 303, 327, 341 Non-singular M-matrix, 260 Nonsupporting, 218–220, 227, 229, 263, 290–292, 299, 367, 368 Non-vaccinator, 110, 116, 133, 134, 141 Numerical methods 40–42, 48, 51, 60 O Optimal control vi, 21, 42–53, 60, 261 P Pairwise model, 154 Paley-Wiener theorem, 366 Payoff, 105–110, 133–135, 149, 151 Perron-Frobenius theory, 218 Persistent solutions 8, 9, 13, 16 Poincare projector, 318 Population density 6, 13, 16 Prevalence, 43, 72, 76, 84, 85, 108, 109, 118, 132, 134, 149 Probability of survival 2, 17, 77, 80, 95, 97 Pure vaccinator, 116, 118, 133, 141, 144 R Random graph, 154 Reducibility, 260 Relative compact, 290 Renewal equation 4, 5 Replicator dynamics, 106, 107 Resolvent set, 304 Ro/basic reproduction number 77, 119, 142, 151, 162, 190, 197, 257, 264, 312, 351 S Scale of networks, 154 Separable force of infection 20, 32, 42, 54 solutions 8, 9, 13, 15 Sharpe-Lotka Theorem 9, 10 Side-effects, 105, 107, 108, 132–134, 148–151 SI model 17, 73, 83, 96 Spectral radius, 218, 220, 229, 263, 264, 271, 366, 367 Static networks, 155, 156, 196 Stationary population 15, 16 Superinfection, 191, 203–209

Index Susceptible-infected-removed (SIR) model, v, 10, 11, 20, 21, 27, 32–39, 60, 174–190, 348 T Total, 1, 2, 6, 10, 11–13, 16–18, 20, 27, 31, 33, 42, 43, 54, 87, 90, 99, 105, 109, 113, 115, 133–136, 150, 153, 157, 163, 167, 168, 170, 186, 212, 218, 219, 232, 251, 280, 281, 301, 303, 312, 321, 348, 350, 365, 367, 368 Total birth rate 2, 6 Total trajectory, 167, 168, 186, 321 Transversal condition, 120, 122 Two-strain 53–67, 96–99, 191–209, 231–257 U Undirected networks, 153

383 Uniformly strongly ρ -persistent, 128, 131, 170, 186, 246, 251 Uniformly weakly ρ -persistent, 128, 251 V Vector borne disease, 211–231, 256, 257 Volterra formulation, 308–312, 317, 355, 365–367 Voluntary vaccination, 107, 134, 151 W Weighted digraph, 188 Well-posedness vi, 3, 21, 23–26, 33, 34, 76, 111, 177, 303, 328, 353 Within-host models 69–77, 84, 85, 89, 90, 95, 96, 99, 100, 103 Z Z-matrix, 260