Dynamical Modeling and Anaylsis of Epidemics 1st Edition by Zhien Ma, Jia Li 9812797491 9789812797490

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ISBN 10: 9812797491 

ISBN 13: 9789812797490

Author: Zhien Ma, Jia Li

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.

Table of contents:

1. Basic Knowledge and Modeling on Epidemic Dynamics

1.1 Introduction

1.2 The Fundamental Forms of Epidemic Models

1.2.1 Two fundamental dynamic models of epidemics

1.2.1.1. Kermack–Mckendrick SIR compartment model

1.2.1.2. Kermack–Mckendrick SIS compartment model

1.2.2 Fundamental forms of compartment models

1.2.2.1. Models without vital dynamics

1.2.2.2. Models with vital dynamics

1.3 Basic Concepts of Epidemiologic Dynamics

1.3.1 Adequate contact rate and incidence

1.3.2 Basic reproductive number and modified reproductive number

1.3.2.1. Basic reproductive number

1.3.2.2. Modi.ed reproductive number

1.3.3 Average lifespan and average infection age

1.4 Epidemic Models with Various Factors

1.4.1 Epidemic models with latent period

1.4.2 Epidemic models with time delay

1.4.2.1. Ideas for the modeling

1.4.2.2. Examples of models with time delay

1.4.3 Epidemic models with prevention, control, or treatment

1.4.3.1. Models with quarantine

1.4.3.2. Models with vaccination

1.4.3.3. Models with treatment

1.4.4 Epidemic models with multiple groups

1.4.4.1. Models with di.erent subgroups

1.4.4.2. Models with multiple populations

1.4.4.3. Models with vector-host

1.4.5 Epidemic models with age structure

1.4.5.1. Population models with age structure

1.4.5.2. Epidemic models with age structure

1.4.6 Epidemic models with impulses

1.4.6.1. Concepts of impulsive differential equations

1.4.6.2. Epidemic models consist of impulsive differential equations

1.4.7 Epidemic models with migration

1.4.7.1. Epidemic models with migration among di.erent patches

1.4.7.2. Epidemic models with continuous di.usion in space

1.4.8 Epidemic models with time-dependent coe.cients

1.4.8.1. SIR model with time-dependent coefficients

2. Ordinary Differential Equations Epidemic Models

2.1 Simple SIRS Epidemic Models with Vital Dynamics

2.1.1 SIRS models with constant immigration and exponential death

2.1.1.1. SIRS models with bilinear incidence

2.1.1.2. SIRS models with standard incidence

2.1.2 SIRS models with logistic growth

2.1.2.1. Equilibrium and threshold

2.1.2.2. Stability analysis

2.1.2.3. Global stability of the equilibria for speci.c case: a = 0 or a = 0

2.2 EpidemicModels with Latent Period

2.2.1 Preliminaries

2.2.1.1. Method I: Proving global stability using the Poincar´e–Bendixson property.

2.2.1.2. Method II: Proving global stability using autonomous convergence theorems

2.2.2 Applications

2.2.2.1. Application of method I

2.2.2.2. Application of method II

2.3 Epidemic Models with Immigration or Dispersal

2.3.1 Epidemic models with immigration

2.3.1.1. SIR model with no immigration of infectives

2.3.1.2. SIR model with immigration of infectives

2.3.2 Epidemic models with dispersal

2.4 Epidemic Models with Multiple Groups

2.4.1 The global stability of epidemic model only with differential susceptibility

2.4.2 The global stability of epidemic model only with differential infectivity

2.5 Epidemic Models with Different Populations

2.5.1 Disease spread in prey–predator system

2.5.1.1. Disease spread only in the prey population

2.5.1.2. Disease spread in prey–predator populations

2.5.2 Disease spread in competitive population systems

2.6 Epidemic Models with Control and Prevention

2.6.1 Epidemic models with quarantine

2.6.1.1. SIQS model with bilinear incidence

2.6.1.2. SIQR model with quarantine-adjusted incidence

2.6.2 Epidemic models with vaccination

2.6.2.1. The existence and local stability of equilibria

2.6.2.2. Global analysis of (2.97)

2.6.3 Epidemic models with treatment

2.7 Bifurcation

2.7.1 Backward bifurcation

2.7.2 Hopf and Bogdanov–Takens bifurcations

2.7.2.1. Hopf bifurcation

2.7.2.2. Bogdanov–Takens bifurcations

2.8 Persistence of Epidemic Models

2.8.1 Persistence of epidemic models of autonomous ordinary differential equations

2.8.1.1. Preliminaries

2.8.1.2. Applications

2.8.2 Persistence of epidemic models of nonautonomous ordinary differential system

3. Modeling of Epidemics with Delays and Spatial Heterogeneity

3.1 Model Formulations

3.1.1 Models incorporating delays

3.1.2 Patchy models

3.2 Basic Techniques for Stability of Delayed Models

3.3 An SIS Epidemic Model with Vaccination

3.4 An SIS Epidemic Model for Vector-Borne Diseases

3.5 Stability Switches and Ultimate Stability

3.6 An SEIRS Epidemic Model with Two Delays

3.7 Quiescence of Epidemics in a Patch Model

3.8 Basic Reproductive Numbers in ODE Models

3.9 Basic Reproductive Numbers of Models with Delays

3.10 FisherWaves in an EpidemicModel

3.11 Propagation of HBV with Spatial Dependence

4. The Epidemic Models with Impulsive Effects

4.1 Basic Theory on Impulsive Differential Equations

4.1.1 Differential equations with impulses

4.1.2 Existence and uniqueness of solutions

4.1.3 Comparison principle

4.1.4 Linear homogeneous impulsive periodic systems and Floquet theory

4.2 SIR Epidemic Model with Pulse Vaccination

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

4.2.1.1. Existence and local stability of the disease-free periodic solution

4.2.1.2. Global stability of the disease-free periodic solution

4.2.1.3. Comparison between constant and pulse vaccinations

4.2.2 SIR epidemic model without disease-induced death

4.3 SIRS Epidemic Model with Pulse Vaccination

4.3.1 SIRS model with pulse vaccination and standard incidence rate

4.3.1.1. Existence and local stability of the disease-free periodic solution

4.3.1.2. Global stability of the disease-free periodic solution

4.3.2. SIRS model with pulse vaccination and nonmonotonic incidence rate

4.3.2.1. Existence and stability of the disease-free solution

4.3.2.2. Bifurcation and existence of epidemic periodic solutions

4.4 SIS Epidemic Model with Pulse Vaccination

4.5 SEIR Epidemic Model with Pulse Vaccination

4.6 SI Epidemic Model with Birth Pulse

4.6.1 The model with constant births

4.6.2 The model with birth pulse

4.7 SIR Epidemic Model with Constant Recruitment and Birth Pulse

4.7.1 The model with constant birth

4.7.2 The model with pulse birth

4.7.2.1. Existence and local stability of the disease-free periodic solution

4.7.2.2. Global asymptotic stability of the disease-free solution

4.7.3 The comparison between constant and pulse births

4.8 SIR Epidemic Models with Pulse Birth and Standard Incidence

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

4.8.2 The global stability of disease-free periodic solution

4.8.3 The uniform persistence of the infection

4.9 SIR Epidemic Model with Nonlinear Birth Pulses

4.9.1 Existence and stability of the disease-free periodic solution

4.9.2 Existence of positive T-periodic solutions and bifurcation

4.10 SI Epidemic Model with Birth Pulses and Seasonality

4.10.1 Existence and local stability of disease-free periodic solution

4.10.2 Bifurcation analysis

4.10.3 Global stability of disease-free periodic solution

5. Structured Epidemic Models

5.1 Stage-StructuredModels

5.1.1 A discrete epidemic model with stage structure

5.1.2 Epidemic models with di.erential infectivity structure

5.2 Age-StructuredModels

5.2.1 Model formulation

5.2.2 Existence of equilibrium

5.2.3 Stability of equilibria

5.3 Infection-Age-Structured Models

5.3.1 An infection-age-structured model with vaccination

5.3.2 An epidemic model with two age structures

5.4 Discrete Models

5.4.1 The model formulation

5.4.2 The existence of the endemic equilibrium

5.4.3 The stability of the disease-free equilibrium

5.4.4 The stability of the endemic equilibrium

5.4.5 Special cases

5.4.5.1. The case of m = 2

5.4.5.2. The case of m = 3

6. Applications of Epidemic Modeling

6.1 SARS Transmission Models

6.1.1 SARS epidemics and modeling

6.1.2 A simple model for SARS prediction

6.1.3 A discrete SARS transmission model

6.1.4 A continuous SARS model with more groups

6.2 HIV Transmission Models

6.2.1 The severity of HIV transmission

6.2.2 An age-structured model for the AIDS epidemic

6.2.3 Discrete model with infection age structure

6.3 TB Transmission Models

6.3.1 Global and regional TB transmission

6.3.2 A TB model with exogenous reinfection

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

6.3.4 TB model with immigration

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