Stochastic Extinction in an SIRS Epidemic Model Incorporating Media Coverage

and Applied Analysis 3 Then, if (S(s), I(s), R(s)) ∈ R3 + for all 0 ≤ s ≤ t almost surely (briefly a.s.), we get (Λ − (μ + α)N (s)) ds ≤ dN (s) ≤ (Λ − μN (s)) ds a.s. (12) Hence, by integration, we check Λ μ + α + (N (0) − Λ μ + α ) e −(μ+α)s ≤ N (s) ≤ Λ μ + (N (0) − Λ μ ) e −μs . (13) Then, 0 < Λ/(μ + α) < N(s) < Λ/μ a.s., so, (S (s) , I (s) , R (s)) ∈ (0, Λ μ ) 3 for all s ∈ [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 ε 0 > 0 such that S(0), I(0), R(0) > ε 0 . For ε ≤ ε 0 , define the stop-times τ ε = inf {t ∈ [0, τ e ] : S (t) ≤ ε or I (t) ≤ ε or R (t) ≤ ε} . (15) Then τ = lim


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][2][3][4][5][6][7][8][9][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][17][18][19][20][21][22][23][24][25][26][27][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][29][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.

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 Λ is the recruitment rate is the threshold of the system for an epidemic to occur.Model (1) has a disease-free equilibrium  0 = (Λ/, 0, 0) and the endemic equilibrium if  0 > 1.The disease-free equilibrium is globally asymptotically stable if  0 ≤ 1 and unstable if  0 > 1.The endemic equilibrium is globally asymptotically stable if  0 > 1.These results of model (1) were studied in [10].
Throughout this paper, let (Ω, F, P) be a complete probability space with a filtration {F  } ∈R + satisfying the usual conditions (i.e., it is right continuous and increasing while F 0 contains all P-null sets).Define a bounded set Γ as follows:
Let us first recall a few definitions.
Definition 1 (see [32]).The trivial solution () = 0 of ( 5) is said to be (i) stable in probability if, for all  > 0, (ii) asymptotically stable if it is stable in probability and moreover if lim (iii) globally asymptotically stable if it is stable in probability and moreover if, for all  0 ∈ R  P ( lim (iv) almost surely exponentially stable if for all  0 ∈ R  , lim sup (v) exponentially -stable if there is a pair of positive constants  1 and  2 such that for all  0 ∈ R  ,

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).

Stochastic Extinction of Model
The following theorem gives a sufficient condition for the almost surely exponential stability of the disease-free equilibrium  0 = (Λ/, 0, 0) of model (3).
By the strong law of large numbers for martingales [16], we have lim sup  → ∞ ()/ = 0 a.s.It finally follows from (30) by dividing  on the both sides and then letting  → ∞ that lim sup 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 (, ) ∈  2 (Ω) satisfying the following inequalities: where  > 0 and   ( = 1, 2, 3) is positive constant.Then the equilibrium of mode (3) is exponentially -stable for  ≥ 0. When  = 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.
Proof.Let  ≥ 2 and ((0), (0), (0)) ∈ Γ; in view of Corollary 3, the solution of model (3) remains in Γ.We define the Lyapunov function (, , ) as follows: where  1 > 0 and  2 > 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 , ,  ∈ (0, Λ/) that Using the fact that we get where In view of  0 + (( − 1)Λ Under Lemma 5 and Theorems 6, we have in the case  = 2 the following corollary.

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   ,  = 1, 2, . . ., , are the Gaussian random variables (0, 1).
To see the disease dynamics of model (3) more, we decrease the noise intensity  to be 0.02 and keep the other parameters unchanged.Then, we have 0.0004 =  2 <  2  1 /2 = 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)).

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  0 > 1 (see Figure 1).If the magnitude of the intensity of noise  is large, that is,  2 >  2  1 /2, the extinction of disease in the stochastic model (3) occurs whether  0 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  0 > 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].
,  represents the natural death rate,  is the loss of constant immunity rate,  is the diseases induced constant death rate, and  is constant recovery rate. 1 is the usual contact rate without considering the infective individuals and  2 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  1 >  2 .The half-saturation constant  > 0 reflects the impact of media coverage on the contact transmission.
is almost surely exponentially stable in Γ.