We extend the classical SIRS epidemic model incorporating media coverage
from a deterministic framework to a stochastic differential equation (SDE) and focus
on how environmental fluctuations of the contact coefficient affect the extinction of the
disease. We give the conditions of existence of unique positive solution and the stochastic
extinction of the SDE model and discuss the exponential
Recent years, a number of mathematical models have been formulated to describe the impact of media coverage on the dynamics of infectious diseases [
On the other hand, for human disease, the nature of epidemic growth and spread is inherently random due to the unpredictability of person-to-person contacts [
In [
In [
In this paper, we will focus on the effects of environmental fluctuations on the disease’s extinction through studying the stochastic dynamics of an SIRS model incorporating media coverage. The rest of this paper is organized as follows. In Section
Let
For model (
If we replace the contact rate
Obviously, the stochastic model (
Throughout this paper, let
Consider the general
Let us first recall a few definitions.
The trivial solution stable in probability if, for all asymptotically stable if it is stable in probability and moreover if globally asymptotically stable if it is stable in probability and moreover if, for all almost surely exponentially stable if for all exponentially
In what follows, we first use the method of Lyapunov functions to find conditions of existence of unique positive solution of model (
In this subsection, we show the existence of the unique positive global solution of SDE model (
Consider model (
The proof is almost identical to Theorem 2 of [
Since the coefficients of model (
Let
Define a
On the other hand, in view of (
From Theorem
The set
In this subsection, we investigate stochastic stability of the disease-free equilibrium
The following theorem gives a sufficient condition for the almost surely exponential stability of the disease-free equilibrium
If
Define a function
We now consider the concept of exponential
Suppose that there exists a function
From the above Lemma, we obtain the following theorem.
Let
Let
Furthermore, by the Itô’s formula, it follows from
Under Lemma
If the conditions
In this section, as an example, we give some numerical simulations to show different dynamic outcomes of the deterministic model (
For the deterministic model (
The paths of
The paths of
To see the disease dynamics of model (
In this paper, we propose an SIRS epidemic model with media coverage and environment fluctuations to describe disease transmission. It is shown that the magnitude of environmental fluctuations will have an effective impact on the control and spread of infectious diseases. In a nutshell, we summarize our main findings as well as their related biological implications as follows.
Theorem
Needless to say, both equilibrium possible approach and parameter possible approach in the present paper have their important roles to play. Obviously, our results in the present paper may be a useful supplement for [
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research was supported by the National Science Foundation of China (61373005, 11201344, and 11201345) and Zhejiang Provincial Natural Science Foundation (LY12A01014).