Dynamics Analysis of a Stochastic SIR Epidemic Model

Feng Rao

doi:10.1155/2014/356013

Abstract

We investigate an SIR epidemic model with stochastic perturbations. We assume that stochastic perturbations are of a white noise type which is directly proportional to the distances of three variables from the steady-state values, respectively. By constructing suitable Lyapunov functions and applying Itô’s formula, some qualitative properties are obtained, such as the existence of global positive solutions, stochastic boundedness, and permanence. A series of numerical simulations to illustrate these mathematical findings are presented.

Highlights

Almost all mathematical models for the transmission of infectious diseases descend from the classical susceptibleinfective-removed (SIR) model of Kermack and McKendrick [1]
We propose an SIR epidemic model with a nonlinear incidence rate of the form kSI/(1 + αI)
We extend to consider and analyze the epidemic model with stochastic perturbations

Summary

Introduction

Almost all mathematical models for the transmission of infectious diseases descend from the classical susceptibleinfective-removed (SIR) model of Kermack and McKendrick [1]. A classical model of an SIRS epidemic in an open population was considered by El Maroufy et al [12] They established the global stability of disease-free and endemic equilibrium points for both the deterministic and stochastic models. Based on the theory of stochastic differential equation, Cai et al [13] studied the dynamics of an SIRS epidemic model with a ratiodependent incidence rate. In [29], the authors extended the classical SIRS epidemic model incorporating media coverage from a deterministic framework to a stochastic differential equation and focused on how environmental fluctuations of the contact coefficient affect the extinction of the disease. The purpose of this paper is to study that the stochastic factor has a significant effect on the dynamics of SIR epidemic model with a saturated incidence rate.

Model and Dynamics Analysis

Conclusions