Global dynamics in a class of discrete-time epidemic models with disease courses
Wang et al. Advances in Difference Equations 2013, 2013:57 http://www.advancesindifferenceequations.com/content/2013/1/57 RESEARCH
Open Access
Global dynamics in a class of discrete-time epidemic models with disease courses
Lei Wang, Qianqian Cui and Zhidong Teng*
*
Correspondence: zhidong@xju.edu.cn College of Mathematics and System
Sciences, Xinjiang University,
Urumqi, Xinjiang 830046, P.R. China
Abstract
In this paper, a class of discrete SIRS epidemic models with disease courses is studied.
The basic reproduction number R0 is computed. The main results on the permanence and extinction of the disease are established. That is, the disease-free equilibrium is globally attractive if R0 < 1, and there exists a unique endemic equilibrium and the disease is also permanent if R0 > 1.
MSC: 39A30; 92D30
Keywords: discrete epidemic model; disease course; basic reproduction number; permanence; global attractivity; extinction
1 Introduction
In recent years, more and more attention has been paid to the discrete-time epidemic models. There are several reasons for that. Firstly, since the statistic data about a disease is collected by day, week, month or year, it is more direct, more convenient and more accurate to describe the disease by using the discrete-time models than the continuoustime models; secondly, the discrete-time models have more wealthy dynamical behaviors; for example, the single-species discrete-time models have bifurcations, chaos and other more complex dynamical behaviors.
For a discrete-time epidemic model, we see that at the present time, the main research subjects are the computation of the basic reproduction number, the local and global stability of the disease-free equilibrium and endemic equilibrium, the extinction, persistence and permanence of the disease, and the bifurcations, chaos and more complex dynamical behaviors of the model, etc. Many important and interesting results can be found in
articles
Open Access
Global dynamics in a class of discrete-time epidemic models with disease courses
Lei Wang, Qianqian Cui and Zhidong Teng*
*
Correspondence: zhidong@xju.edu.cn College of Mathematics and System
Sciences, Xinjiang University,
Urumqi, Xinjiang 830046, P.R. China
Abstract
In this paper, a class of discrete SIRS epidemic models with disease courses is studied.
The basic reproduction number R0 is computed. The main results on the permanence and extinction of the disease are established. That is, the disease-free equilibrium is globally attractive if R0 < 1, and there exists a unique endemic equilibrium and the disease is also permanent if R0 > 1.
MSC: 39A30; 92D30
Keywords: discrete epidemic model; disease course; basic reproduction number; permanence; global attractivity; extinction
1 Introduction
In recent years, more and more attention has been paid to the discrete-time epidemic models. There are several reasons for that. Firstly, since the statistic data about a disease is collected by day, week, month or year, it is more direct, more convenient and more accurate to describe the disease by using the discrete-time models than the continuoustime models; secondly, the discrete-time models have more wealthy dynamical behaviors; for example, the single-species discrete-time models have bifurcations, chaos and other more complex dynamical behaviors.
For a discrete-time epidemic model, we see that at the present time, the main research subjects are the computation of the basic reproduction number, the local and global stability of the disease-free equilibrium and endemic equilibrium, the extinction, persistence and permanence of the disease, and the bifurcations, chaos and more complex dynamical behaviors of the model, etc. Many important and interesting results can be found in
articles
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