Global threshold dynamics of a stochastic epidemic model incorporating media coverage

Bin Yang,Weiming Wang,Kai Wang,Yongli Cai

doi:10.1186/s13662-018-1925-z

Abstract

In this paper, we investigate the global threshold dynamics of a stochastic SIS epidemic model incorporating media coverage. We give the basic reproduction number mathcal{R}_{0}^{s} and establish a global threshold theorem by Feller’s test: if mathcal{R}_{0}^{s}leq 1, the disease will die out a.s.; if mathcal{R}_{0}^{s}>1, the disease will persist a.s. In the case of mathcal{R}_{0}^{s}>1, we prove the existence, uniqueness, and global asymptotic stability of the invariant density of the Fokker–Planck equations associated with the stochastic model. Via numerical simulations, we find that the average extinction time decreases with the increase of noise intensity σ, and also find that the increasing σ will be beneficial to control the disease spread. Thus, in order to control the spread of the disease, we must increase the intensity of noise σ.

Highlights

It is widely believed that environmental variations have a critical influence on the spread of the disease [1,2,3,4], and stochastic noise plays an indispensable role in the transmission of diseases, especially in a small population
In the case of Rs0, i.e., the disease persists with probability one, by studying the Fokker–Planck equations (FPE) associated with stochastic differential equations (SDE) model (4), we prove the existence, uniqueness, and global asymptotic stability of the invariant density of the FPE, which can be useful for us to understand the profile of the distribution of the process I(t)
In the case of Rs0 ≤ 1, we find that the average extinction time decreases with the increase of noise intensity σ

Summary

Introduction

It is widely believed that environmental variations have a critical influence on the spread of the disease [1,2,3,4], and stochastic noise plays an indispensable role in the transmission of diseases, especially in a small population. In order to understand the role of media coverage towards the disease transmission dynamics in a random environment, based on the results in [21], Cai et al [22] studied the following stochastic differential equations (SDE) SIS model with a standard incidence rate:. It is well known that epidemic threshold theorem holds for most deterministic compartmental epidemic models by the basic reproduction number R0 [26]: if R0 < 1, there is a disease-free equilibrium which is globally asymptotically stable; if R0 > 1, there exists an endemic equilibrium which is globally asymptotically stable. There naturally comes a question: Is there any global threshold theorem for a stochastic epidemic model (e.g., SDE model (1) or (2)) incorporating media coverage?.

Proof Define

Simple computations reveal that

Findings

We can know that if