Dynamical Modeling And Analysis Of Epidemics

by Zhien Ma, Jia Li

Overview

This timely book covers the basic concepts of the dynamics of epidemic disease, presenting various kinds of models as well as typical research methods and results. It introduces the latest results in the current literature, especially those obtained by highly rated Chinese scholars. A lot of attention is paid to the qualitative analysis of models, the sheer variety of models, and the frontiers of mathematical epidemiology. The process and key steps in epidemiological modeling and prediction are highlighted, using transmission models of HIV/AIDS, SARS, and tuberculosis as application examples.

Product Details

ISBN-13:	9789812797490
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	05/25/2009
Pages:	512
Product dimensions:	5.90(w) x 9.00(h) x 1.30(d)

Table of Contents

Preface v

List of Contributors vii

1 Basic Knowledge and Modeling on Epidemic Dynamics 1

1.1 Introduction 1

1.2 The Fundamental Forms of Epidemic Models 6

1.2.1 Two fundamental dynamic models of epidemics 6

1.2.2 Fundamental forms of compartment models 10

1.3 Basic Concepts of Epidemiologic Dynamics 14

1.3.1 Adequate contact rate and incidence 14

1.3.2 Basic reproductive number and modified reproductive number 16

1.3.3 Average lifespan and average infection age 22

1.4 Epidemic Models with Various Factors 25

1.4.1 Epidemic models with latent period 25

1.4.2 Epidemic models with time delay 27

1.4.3 Epidemic models with prevention, control, or treatment 35

1.4.4 Epidemic models with multiple groups 46

1.4.5 Epidemic models with age structure 63

1.4.6 Epidemic models with impulses 71

1.4.7 Epidemic models with migration 74

1.4.8 Epidemic models with time-dependent coefficients 79

2 Ordinary Differential Equations Epidemic Models 83

2.1 Simple SIRS Epidemic Models with Vital Dynamics 84

2.1.1 SIRS models with constant immigration and exponential death 85

2.1.2 SIRS models with logistic growth 89

2.2 Epidemic Models with Latent Period 96

2.2.1 Preliminaries 97

2.2.2 Applications 102

2.3 Epidemic Models with Immigration or Dispersal 113

2.3.1 Epidemic models with immigration 113

2.3.2 Epidemic models with dispersal 120

2.4 Epidemic Models with Multiple Groups 126

2.4.1 The global stability of epidemic model only with differential susceptibility 131

2.4.2 The global stability of epidemic model only with differential infectivity 133

2.5 Epidemic Models with Different Populations 135

2.5.1 Disease spread in prey-predator system 136

2.5.2 Disease spread in competitive population systems 144

2.6 Epidemic Models with Control and Prevention 150

2.6.1 Epidemic models with quarantine 150

2.6.2 Epidemic models with vaccination 155

2.6.3 Epidemic models with treatment 164

2.7 Bifurcation 169

2.7.1 Backward bifurcation 170

2.7.2 Hopf and Bogdanov-Takens bifurcations 177

2.8 Persistence of Epidemic Models 186

2.8.1 Persistence of epidemic models of autonomous ordinary differential equations 187

2.8.2 Persistence of epidemic models of nonautonomous ordinary differential system 196

3 Modeling of Epidemics with Delays and Spatial Heterogeneity 201

3.1 Model Formulations 201

3.1.1 Models incorporating delays 201

3.1.2 Patchy models 205

3.2 Basic Techniques for Stability of Delayed Models 213

3.3 An SIS Epidemic Model with Vaccination 218

3.4 An SIS Epidemic Model for Vector-Borne Diseases 222

3.5 Stability Switches and Ultimate Stability 226

3.6 An SEIRS Epidemic Model with Two Delays 235

3.7 Quiescence of Epidemics in a Patch Model 240

3.8 Basic Reproductive Numbers in ODE Models 245

3.9 Basic Reproductive Numbers of Models with Delays 250

3.10 Fisher Waves in an Epidemic Model 256

3.11 Propagation of HBV with Spatial Dependence 266

4 The Epidemic Models with Impulsive Effects 273

4.1 Basic Theory on Impulsive Differential Equations 273

4.1.1 Differential equations with impulses 273

4.1.2 Existence and uniqueness of solutions 275

4.1.3 Comparison principle 277

4.1.4 Linear homogeneous impulsive periodic systems and Floquet theory 280

4.2 SIR Epidemic Model with Pulse Vaccination 282

4.2.1 SIR epidemic models with pulse vaccination and disease-induced death 282

4.2.2 SIR epidemic model without disease-induced death 292

4.3 SIRS Epidemic Model with Pulse Vaccination 295

4.3.1 SIRS model with pulse vaccination and standard incidence rate 296

4.3.2 SIRS model with pulse vaccination and nonmonotonic incidence rate 301

4.4 SIS Epidemic Model with Pulse Vaccination 306

4.5 SEIR Epidemic Model with Pulse Vaccination 309

4.6 SI Epidemic Model with Birth Pulse 312

4.6.1 The model with constant births 313

4.6.2 The model with birth pulse 314

4.7 SIR Epidemic Model with Constant Recruitment and Birth Pulse 317

4.7.1 The model with constant birth 317

4.7.2 The model with pulse birth 319

4.7.3 The comparison between constant and pulse births 329

4.8 SIR Epidemic Models with Pulse Birth and Standard Incidence 331

4.8.1 The existence and local stability of disease-free periodic solution 332

4.8.2 The global stability of disease-free periodic solution 333

4.8.3 The uniform persistence of the infection 337

4.9 SIR Epidemic Model with Nonlinear Birth Pulses 342

4.9.1 Existence and stability of the disease-free periodic solution 343

4.9.2 Existence of positive T-periodic solutions and bifurcation 351

4.10 SI Epidemic Model with Birth Pulses and Seasonality 361

4.10.1 Existence and local stability of disease-free periodic solution 362

4.10.2 Bifurcation analysis 366

4.10.3 Global stability of disease-free periodic solution 367

5 Structured Epidemic Models 371

5.1 Stage-Structured Models 372

5.1.1 A discrete epidemic model with stage structure 373

5.1.2 Epidemic models with differential infectivity structure 378

5.2 Age-Structured Models 385

5.2.1 Model Formulation 385

5.2.2 Existence of equilibrium 387

5.2.3 Stability of equilibria 389

5.3 Infection-Age-Structured Models 392

5.3.1 An infection-age-structured model with vaccination 392

5.3.2 An epidemic model with two age structures 396

5.4 Discrete Models 399

5.4.1 The model formulation 400

5.4.2 The existence of the endemic equilibrium 403

5.4.3 The stability of the disease-free equilibrium 404

5.4.4 The stability of the endemic equilibrium 406

5.4.5 Special cases 410

6 Applications of Epidemic Modeling 415

6.1 SARS Transmission Models 416

6.1.1 SARS epidemics and modeling 416

6.1.2 A simple model for SARS prediction 419

6.1.3 A discrete SARS transmission model 425

6.1.4 A continuous SARS model with more groups 431

6.2 HIV Transmission Models 437

6.2.1 The seventy of HIV transmission 437

6.2.2 An age-structured model for the AIDS epidemic 440

6.2.3 Discrete model with infection age structure 446

6.3 TB Transmission Models 453

6.3.1 Global and regional TB transmission 453

6.3.2 A TB model with exogenous reinfection 455

6.3.3 TB models with fast and slow progression, case detection, and two treatment stages 457

6.3.4 TB model with immigration 460

Bibliography 469

Index 493