Dynamical Modeling and Analysis of Epidemics

The following sections are included:

• Preface

• List of Contributors

• Contents

The following sections are included:

• Introduction

• The Fundamental Forms of Epidemic Models
  • Two fundamental dynamic models of epidemics
    • Kermack–Mckendrick SIR compartment model
    • Kermack–Mckendrick SIS compartment model
  • Fundamental forms of compartment models
    • Models without vital dynamics
    • Models with vital dynamics

• Basic Concepts of Epidemiologic Dynamics
  • Adequate contact rate and incidence
  • Basic reproductive number and modified reproductive number
    • Basic reproductive number
    • Modified reproductive number
  • Average lifespan and average infection age

• Epidemic Models with Various Factors
  • Epidemic models with latent period
  • Epidemic models with time delay
    • Ideas for the modeling
    • Examples of models with time delay
  • Epidemic models with prevention, control, or treatment
    • Models with quarantine
    • Models with vaccination
    • Models with treatment
  • Epidemic models with multiple groups
    • Models with different subgroups
    • Models with multiple populations
    • Models with vector-host
  • Epidemic models with age structure
    • Population models with age structure
    • Epidemic models with age structure
  • Epidemic models with impulses
    • Concepts of impulsive differential equations
    • Epidemic models consist of impulsive differential equations
  • Epidemic models with migration
    • Epidemic models with migration among different patches
    • Epidemic models with continuous diffusion in space
  • Epidemic models with time-dependent coefficients
    • SIR model with time-dependent coefficients

The following sections are included:

• Simple SIRS Epidemic Models with Vital Dynamics
  • SIRS models with constant immigration and exponential death
    • SIRS models with bilinear incidence
    • SIRS models with standard incidence
  • SIRS models with logistic growth
    • Equilibrium and threshold
    • Stability analysis
    • Global stability of the equilibria for specific case: α = 0 or a = 0

• Epidemic Models with Latent Period
  • Preliminaries
    • Method I: Proving global stability using the Poincaré–Bendixson property
    • Method II: Proving global stability using autonomous convergence theorems
  • Applications
    • Application of method I
    • Application of method II

• Epidemic Models with Immigration or Dispersal
  • Epidemic models with immigration
    • SIR model with no immigration of infectives
    • SIR model with immigration of infectives
  • Epidemic models with dispersal

• Epidemic Models with Multiple Groups
  • The global stability of epidemic model only with differential susceptibility
  • The global stability of epidemic model only with differential infectivity

• Epidemic Models with Different Populations
  • Disease spread in prey–predator system
    • Disease spread only in the prey population
    • Disease spread in prey–predator populations
  • Disease spread in competitive population systems

• Epidemic Models with Control and Prevention
  • Epidemic models with quarantine
    • SIQS model with bilinear incidence
    • SIQR model with quarantine-adjusted incidence
  • Epidemic models with vaccination
    • The existence and local stability of equilibria
    • Global analysis of (2.97)
  • Epidemic models with treatment

• Bifurcation
  • Backward bifurcation
  • Hopf and Bogdanov–Takens bifurcations
    • Hopf bifurcation
    • Bogdanov–Takens bifurcations

• Persistence of Epidemic Models
  • Persistence of epidemic models of autonomous ordinary differential equations
    • Preliminaries
    • Applications
  • Persistence of epidemic models of nonautonomous ordinary differential system

The following sections are included:

• Model Formulations
  • Models incorporating delays
  • Patchy models

• Basic Techniques for Stability of Delayed Models

• An SIS Epidemic Model with Vaccination

• An SIS Epidemic Model for Vector-Borne Diseases

• Stability Switches and Ultimate Stability

• An SEIRS Epidemic Model with Two Delays

• Quiescence of Epidemics in a Patch Model

• Basic Reproductive Numbers in ODE Models

• Basic Reproductive Numbers of Models with Delays

• Fisher Waves in an Epidemic Model

• Propagation of HBV with Spatial Dependence

The following sections are included:

• Basic Theory on Impulsive Differential Equations
  • Differential equations with impulses
  • Existence and uniqueness of solutions
  • Comparison principle
  • Linear homogeneous impulsive periodic systems and Floquet theory

• SIR Epidemic Model with Pulse Vaccination
  • SIR epidemic models with pulse vaccination and disease-induced death
    • Existence and local stability of the disease-free periodic solution
    • Global stability of the disease-free periodic solution
    • Comparison between constant and pulse vaccinations
  • SIR epidemic model without disease-induced death

• SIRS Epidemic Model with Pulse Vaccination
  • SIRS model with pulse vaccination and standard incidence rate
    • Existence and local stability of the disease-free periodic solution
    • Global stability of the disease-free periodic solution
  • SIRS model with pulse vaccination and nonmonotonic incidence rate
    • Existence and stability of the disease-free solution
    • Bifurcation and existence of epidemic periodic solutions

• SIS Epidemic Model with Pulse Vaccination

• SEIR Epidemic Model with Pulse Vaccination

• SI Epidemic Model with Birth Pulse
  • The model with constant births
  • The model with birth pulse

• SIR Epidemic Model with Constant Recruitment and Birth Pulse
  • The model with constant birth
  • The model with pulse birth
    • Existence and local stability of the disease-free periodic solution
    • Global asymptotic stability of the disease-free solution
  • The comparison between constant and pulse births

• SIR Epidemic Models with Pulse Birth and Standard Incidence
  • The existence and local stability of disease-free periodic solution
  • The global stability of disease-free periodic solution
  • The uniform persistence of the infection

• SIR Epidemic Model with Nonlinear Birth Pulses
  • Existence and stability of the disease-free periodic solution
  • Existence of positive T-periodic solutions and bifurcation

• SI Epidemic Model with Birth Pulses and Seasonality
  • Existence and local stability of disease-free periodic solution
  • Bifurcation analysis
  • Global stability of disease-free periodic solution

The following sections are included:

• Stage-Structured Models
  • A discrete epidemic model with stage structure
  • Epidemic models with differential infectivity structure

• Age-Structured Models
  • Model formulation
  • Existence of equilibrium
  • Stability of equilibria

• Infection-Age-Structured Models
  • An infection-age-structured model with vaccination
  • An epidemic model with two age structures

• Discrete Models
  • The model formulation
  • The existence of the endemic equilibrium
  • The stability of the disease-free equilibrium
  • The stability of the endemic equilibrium
  • Special cases
    • The case of m = 2
    • The case of m = 3

The following sections are included:

• SARS Transmission Models
  • SARS epidemics and modeling
  • A simple model for SARS prediction
  • A discrete SARS transmission model
  • A continuous SARS model with more groups

• HIV Transmission Models
  • The severity of HIV transmission
  • An age-structured model for the AIDS epidemic
  • Discrete model with infection age structure

• TB Transmission Models
  • Global and regional TB transmission
  • A TB model with exogenous reinfection
  • TB models with fast and slow progression, case detection, and two treatment stages
  • TB model with immigration

The following sections are included:

• Bibliography

• Index