Global stability of the endemic equilibrium of a discrete SIR epidemic model.

Yicang Zhou,Hui Cao,Xia Ma

doi:10.1186/1687-1847-2013-42

Abstract

The basic reproductive number of a discrete SIR epidemic model is defined and the dynamical behavior of the model is studied. It is proved that the disease free equilibrium is globally asymptotically stable if , and the persistence of the model is obtained when . The main attention is paid to the global stability of the endemic equilibrium. Sufficient conditions for the global stability of the endemic equilibrium are established by using the comparison principle. Numerical simulations are done to show our theoretical results and to demonstrate the complicated dynamics of the model.Electronic supplementary materialThe online version of this article (doi:10.1186/1687-1847-2013-42) contains supplementary material, which is available to authorized users.

Highlights

Mathematical models have played a significant role in describing the dynamical evolution of infectious diseases
We will study the global stability of the endemic equilibrium of a discrete SIR model and get sufficient conditions
A lot of results on the existence and global stability of the endemic equilibrium have been obtained for continuous SIR models with various transmission rate

Summary

Introduction

Mathematical models have played a significant role in describing the dynamical evolution of infectious diseases. The transmission dynamics of an infectious disease is described by modeling the population movements among those epidemiological compartments. We will study the global stability of the endemic equilibrium of a discrete SIR model and get sufficient conditions. The discrete SIR model, the biological requirement of the model, the basic reproductive number, and the invariant domain of the model are given . Let S(t), I(t), and R(t) be the number of individuals in the susceptible, infective, and removed/recovered compartments at time t, respectively. There is abundant amount of research into SIR models since they capture the basic evolution mechanism of the infectious diseases when the recovered individuals will acquire life-long immunity. A lot of results on the existence and global stability of the endemic equilibrium have been obtained for continuous SIR models with various transmission rate. The magnitude of plays a crucial role in determining the dynamical behavior of model ( )

Extinction and persistence for the disease

Conclusion