Abstract and Applied Analysis

Volume 2013 (2013), Article ID 891765, 8 pages

http://dx.doi.org/10.1155/2013/891765

Research Article

Stochastic Extinction in an SIRS Epidemic Model Incorporating Media Coverage

Liyan Wang, Huilin Huang, Ancha Xu, and Weiming Wang

College of Mathematics and Information Science, Wenzhou University, Wenzhou 325035, China

Received 2 December 2013; Accepted 12 December 2013

Academic Editor: Kaifa Wang

Copyright © 2013 Liyan Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 -stability and global stability of the SDE model. One of the most interesting findings is that if the intensity of noise is large, then the disease is prone to extinction, which can provide us with some useful control strategies to regulate disease dynamics.

1. Introduction

Recent years, a number of mathematical models have been formulated to describe the impact of media coverage on the dynamics of infectious diseases [1–10]. Mass media (television, radio, newspapers, billboards, and booklets) has been used as a way of delivering preventive health messages as it has the potential to influence people’s behavior and deter them from risky behavior or from taking precautionary measures in relation to a disease outbreak [7, 11, 12]. Hence, media coverage has an enormous impact on the spread and control of infectious diseases [2, 3, 9].

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 [13], and population is subject to a continuous spectrum of disturbances [14, 15]. In epidemic dynamics, stochastic differential equation (SDE) models could be the more appropriate way of modeling epidemics in many circumstances and many realistic stochastic epidemic models can be derived based on their deterministic formulations [16–28].

In [10], Liu investigated an SIRS epidemic model incorporating media coverage with random perturbation. He assumed that stochastic perturbations were of white noise type, which were directly proportional to distance susceptible , infectious , and recover from values of endemic equilibrium point , influence on the , , , respectively. In fact, besides the possible equilibrium approach in [10], there are different possible approaches to introduce random effects in the epidemic models affected by environmental white noise from biological significance and mathematical perspective [28–30]. Some scholars [17, 28, 30, 31] demonstrated that one or more system parameter(s) can be perturbed stochastically with white noise term to derive environmentally perturbed system.

In [10], the author proved that the endemic equilibrium of the stochastic model is asymptotically stable in the large. Therefore, it is natural to ask how environmental fluctuations of the contact coefficient affect the extinction of the disease.

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 2, based on the results of Cui et al. [2] and [10], we derive the stochastic differential SIRS model incorporating media coverage. In Section 3, we give the conditions of existence of unique positive solution and the stochastic extinction of the SDE model. In Section 4, we provide some examples to support our research results. In the last section, we provide a brief discussion and the summary of main results.

2. Model Derivation and Related Definitions

2.1. Model Derivation

Let be the number of susceptible individuals, the number of infective individuals, and the number of removed individuals at time , respectively. Based on the work of Cui et al. [2] and [10], we consider the SIRS epidemic model incorporating media coverage as follows:

where is the recruitment rate, represents the natural death rate, is the loss of constant immunity rate, is the diseases induced constant death rate, and is constant recovery rate. is the usual contact rate without considering the infective individuals and is the maximum reduced contact rate due to the presence of the infected individuals. No one can avoid contacting with others in every case, so it is assumed that . The half-saturation constant reflects the impact of media coverage on the contact transmission. The function is a continuous bounded function which takes into account disease saturation or psychological effects.

For model (1), the basic reproduction number

is the threshold of the system for an epidemic to occur. Model (1) has a disease-free equilibrium and the endemic equilibrium if . The disease-free equilibrium is globally asymptotically stable if and unstable if . The endemic equilibrium is globally asymptotically stable if . These results of model (1) were studied in [10].

If we replace the contact rate in model (1) by , where is a white noise (i.e., is a Brownian motion), model (1) becomes as follows:

Obviously, the stochastic model (3) has the same disease-free equilibrium as model (1).

Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and increasing while contains all -null sets). Define a bounded set as follows:

2.2. Related Definitions

Consider the general -dimensional stochastic differential equation

on with initial value , the solution is denoted by . Assume that and for all , so (5) has the solution , which is called the trivial solution.

Let us first recall a few definitions.

Definition 1 (see [32]). The trivial solution of (5) is said to be

(i)stable in probability if, for all ,

(ii)asymptotically stable if it is stable in probability and moreover if

(iii)globally asymptotically stable if it is stable in probability and moreover if, for all

(iv)almost surely exponentially stable if for all ,

(v)exponentially -stable if there is a pair of positive constants and such that for all ,

3. Dynamics of the SDE Model (3)

In what follows, we first use the method of Lyapunov functions to find conditions of existence of unique positive solution of model (3).

3.1. Existence of Unique Positive Solution of Model (3)

In this subsection, we show the existence of the unique positive global solution of SDE model (3).

Theorem 2. Consider model (3), for any given initial value ; then there is a unique solution on and it will remain in with probability one.

Proof. The proof is almost identical to Theorem 2 of [33], but for completeness we repeat it here. Let . Summing up the three equations in (3) and denoting , we have

Then, if for all almost surely (briefly a.s.), we get

Hence, by integration, we check

Then, ., so,

Since the coefficients of model (3) satisfy the local Lipschitz condition, there is a unique local solution on , where is the explosion time. Therefore, the unique local solution to model (3) is positive by the Itô’s formula. Now, let us show that this solution is global; that is, a.s.

Let such that . For , define the stop-times

Then

Define a -function by

By the Itô’s formula, for all , , we obtain

where

By (14) we assert that for all . Hence

Substituting this inequality into (18), we see that

which implies that

Taking the expectations of the above inequality leads to

On the other hand, in view of (14), we have . It then follows that

where is the indicator function of . Note that there is some component of equal to ; therefore, . Thereby

Combining (23) with (25) gives, for all ,

Let ; we obtain, for all , . Hence, . As , then a.s. which completes the proof of the theorem.

From Theorem 2 and (14), we can conclude the following corollary.

Corollary 3. The set is almost surely positive invariant of model (3); that is, if , then for all .

3.2. Stochastic Extinction of Model (3)

In this subsection, we investigate stochastic stability of the disease-free equilibrium in almost sure exponential and exponential stability by using the suitable Lyapunov function and other techniques of stochastic analysis.

The following theorem gives a sufficient condition for the almost surely exponential stability of the disease-free equilibrium of model (3).

Theorem 4 (almost sure exponential stability). If , then disease-free of model (3) is almost surely exponentially stable in .

Proof. Define a function by

Using the Itô’s formula, we have

where . Since   , we obtain

Hence,

where is a martingale defined by . In virtue of Corollary 3, the solution of model (3) remains in . It then follows that

where is a positive constant which is dependent on , . By the strong law of large numbers for martingales [16], we have . It finally follows from (30) by dividing on the both sides and then letting that

which is the required assertion.

We now consider the concept of exponential -stability. The following lemma gives sufficient conditions for exponential -stability of stochastic systems in terms of the Lyapunov functions (see [32]).

Lemma 5 (see [32]). Suppose that there exists a function satisfying the following inequalities:

where and is positive constant. Then the equilibrium of mode (3) is exponentially -stable for . When , it is usually said to be exponentially stable in mean square and the the equilibrium is globally asymptotically stable.

From the above Lemma, we obtain the following theorem.

Theorem 6 (exponential -stability). Let . If the conditions and hold, the disease-free equilibrium of model (3) is th moment exponentially stable in .

Proof. Let and ; in view of Corollary 3, the solution of model (3) remains in . We define the Lyapunov function as follows:

where and are real positive constants that are to be chosen later. It is easy to check that inequalities (33) are true.

Furthermore, by the Itô’s formula, it follows from that

Using the fact that

we get

where

In view of , we have   . Hence, we chose sufficiently small and , are positive such that . According to Lemma 5 the proof is completed.

Under Lemma 5 and Theorems 6, we have in the case the following corollary.

Corollary 7 (globally asymptotically stable). If the conditions and hold, the disease-free equilibrium of model (3) is globally asymptotically stable in .

4. Numerical Simulations and Dynamics Comparison

In this section, as an example, we give some numerical simulations to show different dynamic outcomes of the deterministic model (1) versus its stochastic version (3) with the same set of parameter values by using the Milstein method mentioned in Higham [34]. In this way, model (3) can be rewritten as the following discretization equations:

where , , are the Gaussian random variables .

For the deterministic model (1) and its stochastic model (3), the parameters are taken as follows:

( 1) The Endemic Dynamics of the Deterministic Model (1). For the deterministic model (1), ; thus, it admits a unique endemic equilibrium which is globally stable for any initial values according to [10] (see, Figure 1).

Figure 1: The paths of , , and for the deterministic model (1) with initial values . The parameters are taken as (41) ().

( 2) The Stochastic Dynamics of Model (3). For the corresponding stochastic model (3), we choose ; then, we have . Thus, from Theorem 4, we can conclude that for any initial value , disease-free of model (3) is almost surely exponentially stable in (see Figure 2(a)).

(a)

(b)

Figure 2: The paths of , , and for the stochastic model (3) with initial values . The parameters are taken as (41) ().

To see the disease dynamics of model (3) more, we decrease the noise intensity to be and keep the other parameters unchanged. Then, we have . Therefore, the condition of Theorem 4 is not satisfied. In this case, our simulations suggest that model (3) is stochastically persistent (see Figure 2(b)).

5. Concluding Remarks

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 4 and [10] combined with numerical simulations (see Figures 1 and 2) provide us with a full picture on the dynamics of the deterministic model (1) and stochastic model (3). In [10], the authors showed that the deterministic model (1) admits a unique endemic equilibrium which is globally asymptotically stable if its basic reproduction number (see Figure 1). If the magnitude of the intensity of noise is large, that is, , the extinction of disease in the stochastic model (3) occurs whether is greater than 1 or less than 1 (see Figure 2(a)). While the magnitude of the intensity of noise is small, one of our most interesting findings is that disease may persist if , (see Figure 2(b)).

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 [10].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This research was supported by the National Science Foundation of China (61373005, 11201344, and 11201345) and Zhejiang Provincial Natural Science Foundation (LY12A01014).

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