Stochastic Extinction in an SIRS Epidemic Model Incorporating Media Coverage

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 𝑝-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 𝑆(𝑡), infectious 𝐼(𝑡), and recover 𝑅(𝑡) from values of endemic equilibrium point (𝑆^∗,𝐼^∗,𝑅^∗), influence on the 𝑑𝑆(𝑡)/𝑑𝑡, 𝑑𝐼(𝑡)/𝑑𝑡, 𝑑𝑅(𝑡)/𝑑𝑡, 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 𝑆(𝑡) be the number of susceptible individuals, 𝐼(𝑡) the number of infective individuals, and 𝑅(𝑡) the number of removed individuals at time 𝑡, respectively. Based on the work of Cui et al. [2] and [10], we consider the SIRS epidemic model incorporating media coverage as follows: 𝑑𝑆𝛽𝑑𝑡=Λ−𝜇𝑆−₁−𝛽₂𝐼𝑏+𝐼𝑆𝐼+𝜂𝑅,𝑑𝐼=𝛽𝑑𝑡¹−𝛽₂𝐼𝑏+𝐼𝑆𝐼−𝜇+𝛼+𝜆𝐼,𝑑𝑅𝑑𝑡=𝜆𝐼−𝜇+𝜂𝑅, (1) where Λ is the recruitment rate, 𝜇 represents the natural death rate, 𝜂 is the loss of constant immunity rate, 𝛼 is the diseases induced constant death rate, and 𝜆 is constant recovery rate. 𝛽₁ is the usual contact rate without considering the infective individuals and 𝛽₂ 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 𝛽₁>𝛽₂. The half-saturation constant 𝑏>0 reflects the impact of media coverage on the contact transmission. The function 𝐼/(𝑏+𝐼) is a continuous bounded function which takes into account disease saturation or psychological effects.

For model (1), the basic reproduction number 𝑅₀=Λ𝛽₁𝜇𝜇+𝛼+𝜆 (2) is the threshold of the system for an epidemic to occur. Model (1) has a disease-free equilibrium 𝐸₀=(Λ/𝜇,0,0) and the endemic equilibrium if 𝑅₀>1. The disease-free equilibrium is globally asymptotically stable if 𝑅₀≤1 and unstable if 𝑅₀>1. The endemic equilibrium is globally asymptotically stable if 𝑅₀>1. These results of model (1) were studied in [10].

If we replace the contact rate 𝛽₁ in model (1) by 𝛽₁+𝜎(𝑑𝐵/𝑑𝑡), where 𝑑𝐵/𝑑𝑡 is a white noise (i.e., 𝐵(𝑡) is a Brownian motion), model (1) becomes as follows: 𝑑𝑆=Λ−𝜇𝑆−𝛽₁−𝛽₂𝐼(𝑡),𝛽𝑏+𝐼𝑆𝐼+𝜂𝑅𝑑𝑡+𝜎𝑆𝐼𝑑𝐵𝑑𝐼=₁−𝛽₂𝐼𝐼(𝑡),𝑅𝑏+𝐼𝑆𝐼−𝜇+𝛼+𝜆𝑑𝑡+𝜎𝑆𝐼𝑑𝐵𝑑𝑅=𝜆𝐼−𝜇+𝜂𝑑𝑡. (3)

Obviously, the stochastic model (3) has the same disease-free equilibrium 𝐸₀=(Λ/𝜇,0,0) as model (1).

Throughout this paper, let (Ω,ℱ,ℙ) be a complete probability space with a filtration {ℱ_𝑡}_𝑡∈ℝ+ satisfying the usual conditions (i.e., it is right continuous and increasing while ℱ₀ contains all ℙ-null sets). Define a bounded set Γ as follows: Γ=(𝑆,𝐼,𝑅)∈ℝ³₊Λ∶0<𝑆+𝐼+𝑅<𝜇⊂ℝ³₊. (4)

2.2. Related Definitions

Consider the general 𝑛-dimensional stochastic differential equation 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑡)𝑑𝑡+𝜑(𝑥(𝑡),𝑡)𝑑𝐵(𝑡) (5) on 𝑡≥0 with initial value 𝑥(0)=𝑥₀, the solution is denoted by 𝑥(𝑡,𝑥₀). Assume that 𝑓(0,𝑡)=0 and 𝜑(0,𝑡)=0 for all 𝑡≥0, so (5) has the solution 𝑥(𝑡)=0, which is called the trivial solution.

Let us first recall a few definitions.

Definition 1 (see [32]). The trivial solution 𝑥(𝑡)=0 of (5) is said to be(i)stable in probability if, for all 𝜀>0, lim_𝑥0→0ℙsup_𝑡≥0𝑥𝑡,𝑥₀≥𝜀=0; (6)(ii)asymptotically stable if it is stable in probability and moreover iflim_𝑥0→0ℙlim_𝑡→∞𝑥𝑡,𝑥₀=0=1; (7)(iii)globally asymptotically stable if it is stable in probability and moreover if, for all 𝑥₀∈ℝ^𝑛ℙlim_𝑡→∞𝑥𝑡,𝑥₀=0=1; (8)(iv)almost surely exponentially stable if for all 𝑥₀∈ℝ^𝑛, limsup_𝑡→∞1𝑡log𝑥𝑡,𝑥₀<0a.s.; (9)(v)exponentially 𝑝-stable if there is a pair of positive constants 𝐶₁ and 𝐶₂ such that for all 𝑥₀∈ℝ^𝑛, 𝔼𝑥𝑡,𝑥₀^𝑝≤𝐶₁𝑥₀^𝑝𝑒^−𝐶2𝑡on𝑡≥0. (10)

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 (𝑆(0),𝐼(0),𝑅(0))∈Γ; then there is a unique solution (𝑆(𝑡),𝐼(𝑡),𝑅(𝑡)) on 𝑡≥0 and it will remain in ℝ³₊ with probability one.

Proof. The proof is almost identical to Theorem 2 of [33], but for completeness we repeat it here. Let (𝑆(0),𝐼(0),𝑅(0))∈Γ. Summing up the three equations in (3) and denoting 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡), we have 𝑑𝑁(𝑡)=Λ−𝜇𝑁(𝑡)−𝛼𝐼(𝑡)𝑑𝑡. (11) Then, if (𝑆(𝑠),𝐼(𝑠),𝑅(𝑠))∈ℝ³₊ for all 0≤𝑠≤𝑡 almost surely (briefly a.s.), we get Λ−𝜇+𝛼𝑁(𝑠)𝑑𝑠≤𝑑𝑁(𝑠)≤Λ−𝜇𝑁(𝑠)𝑑𝑠a.s. (12) Hence, by integration, we check Λ+𝑁(0)−Λ𝜇+𝛼𝑒𝜇+𝛼^{−(𝜇+𝛼)𝑠}(𝑠)≤Λ≤𝑁𝜇+𝑁(0)−Λ𝜇𝑒^−𝜇𝑠. (13) Then, 0<Λ/(𝜇+𝛼)<𝑁(𝑠)<Λ/𝜇a.s., so, (𝑆(𝑠)(𝑠)(𝑠∈Λ,𝐼,𝑅))0,𝜇³forall𝑠∈[0,𝑡]a.s. (14)
Since the coefficients of model (3) satisfy the local Lipschitz condition, there is a unique local solution on [0,𝜏_𝑒), where 𝜏_𝑒 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, 𝜏_𝑒=∞ a.s.
Let 𝜖₀>0 such that 𝑆(0),𝐼(0),𝑅(0)>𝜖₀. For 𝜖≤𝜖₀, define the stop-times 𝜏_𝜖=inf𝑡∈0,𝜏_𝑒∶𝑆(𝑡)≤𝜖or𝐼(𝑡)≤𝜖or𝑅(𝑡)≤𝜖. (15) Then 𝜏=lim_𝜖→0𝜏_𝜖=inf𝑡∈0,𝜏_𝑒∶𝑆(𝑡)≤0or𝐼(𝑡)≤0or𝑅(𝑡)≤0. (16)
Define a 𝐶²-function 𝑉∶ℝ³₊→ℝ₊ by 𝑉(𝑆,𝐼,𝑅)=−log𝜇𝑆Λ−log𝜇𝐼Λ−log𝜇𝑅Λ. (17) By the Itô’s formula, for all 𝑡≥0, 𝑠∈[0,𝑡∧𝜏_𝜖], we obtain 1𝑑𝑉=−𝑆(𝑠)1𝑑𝑆+2𝑆(𝑠)²1𝑑𝑆𝑑𝑆−𝐼(𝑠)+1𝑑𝐼2𝐼(𝑠)²1𝑑𝐼𝑑𝐼−𝑅(𝑠)1𝑑𝑅+2𝑅(𝑠)²(𝐼(𝑠)(𝑠(𝑠),𝑑𝑅𝑑𝑅≜𝐿𝑉𝑑𝑠+𝜎−𝑆))𝑑𝐵 (18) where 𝐿𝑉=3𝜇+2𝜆+𝛼+𝛽₁𝛽𝐼+²𝑆𝐼+𝜎𝑏+𝐼2²𝑆²+𝐼²−𝛽₂𝐼²−𝑏+𝐼𝜂𝑅𝑆−𝛽₁Λ𝑆−𝑆−𝛼𝐼𝑅≤3𝜇+2𝜆+𝛼+𝛽₁𝐼+𝛽₂𝜎𝑆+²2𝑆²+𝐼². (19) By (14) we assert that (𝑆(𝑠),𝐼(𝑠),𝑅(𝑠))∈(0,Λ/𝜇) for all 𝑠∈[0,𝑡∧𝜏_𝜖]a.s. Hence Λ𝐿𝑉≤3𝜇+2𝜆+𝛼+𝜇𝛽₁+𝛽₂+𝜎²Λ𝜇∶=𝑀. (20) Substituting this inequality into (18), we see that 𝑑𝑉(𝑆,𝐼,𝑅)≤𝑀𝑑𝑠+𝜎(𝐼−𝑆)𝑑𝐵(𝑠), (21) which implies that ^{𝑡∧𝜏𝜖}₀(𝑆(𝑠)(𝑠)(𝑠≤𝑑𝑉,𝐼,𝑅))_{𝑡∧𝜏𝜖0}𝑀𝑑𝑠+𝜎^{𝑡∧𝜏𝜖}₀(𝐼(𝑠)−𝑆(𝑠))𝑑𝐵(𝑠). (22) Taking the expectations of the above inequality leads to 𝔼𝑉𝑆𝑡∧𝜏_𝜖,𝐼𝑡∧𝜏_𝜖,𝑅𝑡∧𝜏_𝜖≤𝑉(𝑆(0),𝐼(0),𝑅(0))+𝑀𝑡. (23)
On the other hand, in view of (14), we have 𝑉(𝑆(𝑡∧𝜏_𝜖),𝐼(𝑡∧𝜏_𝜖),𝑅(𝑡∧𝜏_𝜖))>0. It then follows that 𝔼𝑉𝑆𝑡∧𝜏_𝜖,𝐼𝑡∧𝜏_𝜖,𝑅𝑡∧𝜏_𝜖=𝔼𝕀_{(𝜏𝜖≤𝑡)}𝑉𝑆𝑡∧𝜏_𝜖,𝐼𝑡∧𝜏_𝜖,𝑅𝑡∧𝜏_𝜖+𝔼𝕀_{(𝜏𝜖>𝑡)}𝑉𝑆𝑡∧𝜏_𝜖,𝐼𝑡∧𝜏_𝜖,𝑅𝑡∧𝜏_𝜖≥𝔼𝕀_{(𝜏𝜖≤𝑡)}𝑉𝑆𝜏_𝜖,𝐼𝜏_𝜖,𝑅𝜏_𝜖, (24) where 𝕀_𝐴 is the indicator function of 𝐴. Note that there is some component of (𝑆(𝜏_𝜖),𝐼(𝜏_𝜖),𝑅(𝜏_𝜖)) equal to 𝜖; therefore, 𝑉(𝑆(𝜏_𝜖),𝐼(𝜏_𝜖),𝑅(𝜏_𝜖))≥−log(𝜇𝜖/Λ)>0. Thereby 𝔼𝑉𝑆𝑡∧𝜏_𝜖,𝐼𝑡∧𝜏_𝜖,𝑅𝑡∧𝜏_𝜖≥−log𝜇𝜖Λℙ(𝜏≤𝑡). (25) Combining (23) with (25) gives, for all 𝑡≥0, ℙ()𝑉(𝑆(0)(0)(0𝜏≤𝑡≤−,𝐼,𝑅))+𝑀𝑡.log𝜇𝜖/Λ (26) Let 𝜖→0; we obtain, for all 𝑡≥0, ℙ(𝜏≤𝑡)=0. Hence, ℙ(𝜏=∞)=1. As 𝜏_𝑒≥𝜏, then 𝜏_𝑒=𝜏=∞ a.s. which completes the proof of the theorem.

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

Corollary 3. The set Γ is almost surely positive invariant of model (3); that is, if (𝑆(0),𝐼(0),𝑅(0))∈Γ, then ℙ((𝑆(𝑡),𝐼(𝑡),𝑅(𝑡))∈Γ)=1 for all 𝑡≥0.

3.2. Stochastic Extinction of Model (3)

In this subsection, we investigate stochastic stability of the disease-free equilibrium 𝐸₀=(Λ/𝜇,0,0) in almost sure exponential and exponential 𝑝 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 𝐸₀=(Λ/𝜇,0,0) of model (3).

Theorem 4 (almost sure exponential stability). If 𝜎²>𝛽²₁/2𝜇, then disease-free 𝐸₀=(Λ/𝜇,0,0) of model (3) is almost surely exponentially stable in Γ.

Proof. Define a function 𝑉 by 𝑉()Λ𝑆,𝐼,𝑅=log𝜇−𝑆+𝐼+𝑅. (27) Using the Itô’s formula, we have 𝑑𝑉=𝜕𝑉𝜕𝑆𝑑𝑆+𝜕𝑉𝜕𝐼𝑑𝐼+𝜕𝑉+1𝜕𝑅𝑑𝑅2𝜕²𝑉𝜕𝑆²𝜕𝑑𝑆𝑑𝑆+²𝑉𝜕𝐼²𝜕𝑑𝐼𝑑𝐼+²𝑉𝜕𝑅²+𝜕𝑑𝑅𝑑𝑅₂𝑉𝜕𝜕𝑆𝜕𝐼𝑑𝑆𝑑𝐼+²𝑉𝜕𝜕𝑆𝜕𝑅𝑑𝑆𝑑𝑅+²𝑉=1𝜕𝐼𝜕𝑅𝑑𝐼𝑑𝑅()−1Λ/𝜇−𝑆+𝐼+𝑅−𝑑𝑆+𝑑𝐼+𝑑𝑅2Λ/𝜇−𝑆+𝐼+𝑅²()+1𝑑𝑆𝑑𝑆+𝑑𝐼𝑑𝐼Λ/𝜇−𝑆+𝐼+𝑅²=1𝑑𝑆𝑑𝐼×Λ/𝜇−𝑆+𝐼+𝑅−Λ+𝜇𝑆+2𝛽₁𝑆𝐼−2𝛽₂𝑆𝐼²−𝑅−𝑏+𝐼𝜇+𝛼𝐼−𝜇+2𝜂𝑑𝑡2𝜎²𝑆²𝐼²Λ/𝜇−𝑆+𝐼+𝑅²𝑑𝑡+2𝜎𝑆𝐼=Λ/𝜇−𝑆+𝐼+𝑅𝑑𝐵2𝛽₁𝑍−2𝜎²−1𝑍−𝜇𝑑𝑡+2𝜎𝑑𝐵𝑍Λ/𝜇−𝑆+𝐼+𝑅2𝛽₂𝑆𝐼²≤𝑏+𝐼+𝛼𝐼+2𝜂𝑅𝑑𝑡2𝛽₁𝑍−2𝜎²𝑍−𝜇𝑑𝑡+2𝜎𝑍𝑑𝐵, (28) where 𝑍(𝑆,𝐼,𝑅)=𝑆𝐼/((Λ/𝜇)−𝑆+𝐼+𝑅). Since 2𝛽₁𝑍−2𝜎²𝑍−𝜇=−2𝜎²(𝑍−(𝛽₁/2𝜎²))+  (𝛽²₁−2𝜎²𝜇)/2𝜎², we obtain 𝛽𝑑𝑉≤²¹−2𝜎²𝜇2𝜎²𝑑𝑡+2𝜎𝑍𝑑𝐵. (29) Hence, Λlog𝜇(𝑡)(𝑡)(𝑡)Λ−𝑆+𝐼+𝑅≤log𝜇(0)(0)(0)+𝛽−𝑆+𝐼+𝑅²¹−2𝜎²𝜇2𝜎²𝑡+𝐺(𝑡), (30) where 𝐺(𝑡) is a martingale defined by 𝐺(𝑡)=2𝜎∫^𝑡₀𝑍𝑑𝐵(𝑠). In virtue of Corollary 3, the solution of model (3) remains in Γ. It then follows that ⟨𝐺,𝐺⟩_𝑡=4𝜎^2𝑡₀𝑍²𝑑𝑠≤𝐶𝑡, (31) where 𝐶 is a positive constant which is dependent on Λ, 𝜇. By the strong law of large numbers for martingales [16], we have limsup_𝑡→∞𝐺(𝑡)/𝑡=0a.s. It finally follows from (30) by dividing 𝑡 on the both sides and then letting 𝑡→∞ that limsup_𝑡→∞1𝑡Λlog𝜇≤𝛽−𝑆+𝐼+𝑅²¹−2𝜎²𝜇2𝜎²<0a.s. (32) which is the required assertion.

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

Lemma 5 (see [32]). Suppose that there exists a function 𝑉(𝑧,𝑡)∈𝐶²(Ω) satisfying the following inequalities: 𝐾₁|𝑧|^𝑝≤𝑉(𝑧,𝑡)≤𝐾₂|𝑧|^𝑝,𝐿𝑉(𝑧,𝑡)≤−𝐾₃|𝑧|^𝑝, (33)(34) where 𝑝>0 and 𝐾_𝑖(𝑖=1,2,3) is positive constant. Then the equilibrium of mode (3) is exponentially 𝑝-stable for 𝑡≥0. When 𝑝=2, 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 𝑝-stability). Let 𝑝≥2. If the conditions 𝑅₀=𝛽₁Λ/𝜇(𝜇+𝛼+𝜆)<1 and 𝑅^𝑠₀∶=𝑅₀+((𝑝−1)Λ²𝜎²/2𝜇²(𝜇+𝛼+𝜆))<1 hold, the disease-free equilibrium 𝐸₀=(Λ/𝜇,0,0) of model (3) is 𝑝th moment exponentially stable in Γ.

Proof. Let 𝑝≥2 and (𝑆(0),𝐼(0),𝑅(0))∈Γ; in view of Corollary 3, the solution of model (3) remains in Γ. We define the Lyapunov function 𝑉(𝑆,𝐼,𝑅) as follows: 𝑉=𝑐₁Λ𝜇−𝑆^𝑝+1𝑝𝐼^𝑝+𝑐₂𝑅^𝑝, (35) where 𝑐₁>0 and 𝑐₂>0 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 𝑆,𝐼,𝑅∈(0,Λ/𝜇) that 𝐿𝑉=−𝑐₁𝑝Λ𝜇−𝑆^𝑝−1Λ−𝜇𝑆−𝛽₁𝛽𝑆𝐼+²𝑆𝐼²+1𝑏+𝐼+𝜂𝑅2𝑝𝑝−1𝑐₁𝜎²𝑆²𝐼²Λ𝜇−𝑆^𝑝−2+𝐼^𝑝−1𝛽₁𝛽𝑆𝐼−²𝑆𝐼²−𝐼+1𝑏+𝐼𝜇+𝛼+𝜆2𝑝−1𝜎²𝑆²𝐼^𝑝+𝑐₂𝑝𝑅^𝑝−1𝜆𝐼−𝜇+𝜂𝑅≤−𝑐₁Λ𝜇𝑝𝜇−𝑆^𝑝+𝑐₁𝛽₁𝑝Λ𝜇Λ𝜇−𝑆^𝑝−1𝐼+12𝜇²𝑝𝑝−1𝑐₁𝜎²Λ²Λ𝜇−𝑆^𝑝−2𝐼²−𝛽𝜇+𝛼+𝜆−¹Λ𝜇−12𝜇²𝑝−1𝜎²Λ²𝐼^𝑝−𝑐₂𝑝𝜇+𝜂𝑅^𝑝+𝑐₂𝑝𝜆𝐼𝑅^𝑝−1. (36) Using the fact that Λ𝜇−𝑆^𝑝−1𝐼≤𝑝−1𝑝𝜀Λ𝜇−𝑆^𝑝+1𝑝𝜀^1−𝑝𝐼^𝑝,Λ𝜇−𝑆^𝑝−2𝐼²≤𝑝−2𝑝𝜀Λ𝜇−𝑆^𝑝+2𝑝𝜀^(2−𝑝)/2𝐼^𝑝,𝑅^𝑝−1𝐼≤𝑝−1𝑝𝜀𝑅^𝑝+1𝑝𝜀^1−𝑝𝐼^𝑝, (37) we get 𝐿𝑉≤−𝐴₁Λ𝜇−𝑆^𝑝−𝐴₂𝐼^𝑝−𝐴₃𝑅^𝑝, (38) where 𝐴₁=𝛽𝜇𝑝−¹Λ𝑝−1𝜇+𝜎²Λ²𝑝−12𝜇²𝜀𝑐₁,𝐴₂𝛽=𝜇+𝛼+𝜆−¹Λ𝜇−12𝜇²𝑝−1𝜎²Λ²−𝛽₁Λ𝜇𝜀^1−𝑝+𝜎²Λ²𝑝−1𝜇²𝜀^(2−𝑝)/2𝑐₁−𝑐₂𝜆𝜀^1−𝑝𝐴₃=𝑐₂𝑝𝜇+𝜂−𝜆𝑝−1𝜀. (39) In view of 𝑅₀+((𝑝−1)Λ²𝜎²/2𝜇²(𝜇+𝛼+𝜆))<1, we have 𝜇+𝛼+𝜆−(𝛽₁Λ/𝜇)−  (1/2𝜇²)(𝑝−1)𝜎²Λ²>0. Hence, we chose 𝜀 sufficiently small and 𝑐₁, 𝑐₂ are positive such that 𝐴₁,𝐴₂,𝐴₃>0. According to Lemma 5 the proof is completed.

Under Lemma 5 and Theorems 6, we have in the case 𝑝=2 the following corollary.

Corollary 7 (globally asymptotically stable). If the conditions 𝑅₀=𝛽₁Λ/𝜇(𝜇+𝛼+𝜆)<1 and 𝑅^𝑠₀∶=𝑅₀+(Λ²𝜎²/2𝜇²(𝜇+𝛼+𝜆))<1 hold, the disease-free equilibrium 𝐸₀=(Λ/𝜇,0,0) of model (3) is globally asymptotically stable in Γ.

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: 𝑆_𝑘+1=𝑆_𝑘+Λ−𝜇𝑆_𝑘−𝛽₁𝑆_𝑘𝐼_𝑘+𝛽₂𝑆_𝑘𝐼²_𝑘𝑏+𝐼_𝑘+𝜂𝑅_𝑘Δ𝑡+𝜎𝑆_𝑘𝐼_𝑘√Δ𝑡𝜉_𝑘+𝜎²2𝑆_𝑘𝐼_𝑘𝜉²_𝑘𝐼−1Δ𝑡,_𝑘+1=𝐼_𝑘+𝛽₁𝑆_𝑘𝐼_𝑘−𝛽₂𝑆_𝑘𝐼²_𝑘𝑏+𝐼_𝑘−𝜇+𝛼+𝜆𝐼_𝑘Δ𝑡+𝜎𝑆_𝑘𝐼_𝑘√Δ𝑡𝜉_𝑘+𝜎²2𝑆_𝑘𝐼_𝑘𝜉²_𝑘𝑅−1Δ𝑡,_𝑘+1=𝑅_𝑘+𝜆𝐼_𝑘−𝜇+𝜂𝑅_𝑘Δ𝑡, (40) where 𝜉_𝑘, 𝑘=1,2,…,𝑛, are the Gaussian random variables 𝑁(0,1).

For the deterministic model (1) and its stochastic model (3), the parameters are taken as follows: Λ=1,𝜇=0.03,𝛽₁=0.02,𝛽₂=0.018,𝜂=0.01,𝛼=0.1,𝜆=0.05,𝑏=10. (41)

(1) The Endemic Dynamics of the Deterministic Model (1). For the deterministic model (1), 𝑅₀=𝛽₁Λ/𝜇(𝜇+𝛼+𝜆)=7.407>1; thus, it admits a unique endemic equilibrium 𝐸^∗=(8.1035,9.7664,12.2080) which is globally stable for any initial values (𝑆(0),𝐼(0),𝑅(0))∈Γ according to [10] (see, Figure 1).

Figure 1 

The paths of 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) for the deterministic model (1) with initial values (𝑆(0),𝐼(0),𝑅(0))=(9,1,0). The parameters are taken as (41) (𝑅0=7.407).

(2) The Stochastic Dynamics of Model (3). For the corresponding stochastic model (3), we choose 𝜎=0.1; then, we have 0.01=𝜎²>𝛽²₁/2𝜇=0.007. Thus, from Theorem 4, we can conclude that for any initial value (𝑆(0),𝐼(0),𝑅(0))∈Γ, disease-free 𝐸₀=(Λ/𝜇,0,0) of model (3) is almost surely exponentially stable in Γ (see Figure 2(a)).

(a) 𝜎=0.1

(b) 𝜎=0.02

Figure 2 

The paths of 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) for the stochastic model (3) with initial values (𝑆(0),𝐼(0),𝑅(0))=(9,1,0). The parameters are taken as (41) (𝑅0=7.407).

To see the disease dynamics of model (3) more, we decrease the noise intensity 𝜎 to be 0.02 and keep the other parameters unchanged. Then, we have 0.0004=𝜎²<𝛽²₁/2𝜇=0.007. 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 𝐸^∗ which is globally asymptotically stable if its basic reproduction number 𝑅₀>1 (see Figure 1). If the magnitude of the intensity of noise 𝜎 is large, that is, 𝜎²>𝛽²₁/2𝜇, the extinction of disease in the stochastic model (3) occurs whether 𝑅₀ is greater than 1 or less than 1 (see Figure 2(a)). While the magnitude of the intensity of noise 𝜎 is small, one of our most interesting findings is that disease may persist if 𝑅₀>1, (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).