Dynamics of a Stochastic SIRS Epidemic Model with Regime Switching and Specific Functional Response

,e purpose of this work is to investigate the dynamic behaviors of the SIRS epidemic model with nonlinear incident rate under regime switching. We establish the existence of a unique positive solution of our system. Furthermore, we obtain the conditions for the extinction of diseases, and we show the existence of the stationary distribution for our stochastic SIRS model under regime switching. Numerical simulations are employed to illustrate our theoretical analysis.


Introduction
Several of mathematicians have developed various epidemic models to prevent and control the spread of transmissible diseases in the community. e classical SIR model presented by Kermack and McKendrick [1] has played an important role in mathematical epidemiology. e SIR model are used to study the disease spread between three groups of population to know the susceptible S, the infective I, and the recovered R.
In this work, we introduce a switched stochastic SIRS epidemic model with specific functional response. en, we consider the following deterministic SIRS epidemic model with specific functional response: dS dt � Λ − μS − βSI 1 + α 1 S + α 2 I + α 3 SI + cR, dI dt � βSI 1 + α 1 S + α 2 I + α 3 SI − (μ + λ + δ)I, where S(t) denotes the number of susceptible individuals, I(t) denotes the number of infective individuals, and R(t) represents the number of removed individuals. Λ is the recruitment rate of the population, μ is the natural death rate of the population, c is the rate at which recovered individuals loss immunity and return to the susceptible class, λ denotes the natural recovery rate of the infectious individuals, and δ denotes the disease inducing death rate. e infection transmission process in (1) is modeled by the specific functional response (βSI/1 + α 1 S + α 2 I + α 3 SI), where β is the transmission coefficient between compartments S and I, and α 1 , α 2 , α 3 ≥ 0 are the saturation factors measuring the psychological or inhibitory effect. In addition, this functional response generalizes many common types existing in the literature such as the Crowley-Martin functional response introduced in [2] and used in [3] when α 3 � α 1 α 2 and the Beddington-DeAngelis functional response proposed in [4] and used in [5] when α 3 � 0.
Environmental fluctuations have been indicated to play an important role in the propagation of disease [6,7]. In effect, disease infestation is highly stochastic, and stochastic noise can raise the probability of disease extinction in the early phase of epidemics. By running an ODE system, we can get only a certain sample solution, whereas by running an SDE system, we can obtain the stochastic distribution of disease dynamics [8]. Lately, dynamic modeling of infectious diseases based on stochastic differential equations (SDE) has received considerable attention from experts and academics [9][10][11]. e SIRS epidemic model with white noise is expressed by where B 1 (t), B 2 (t), and B 3 (t) are independent Brownian motions and σ 1 , σ 2 , and σ 3 are their intensities. Besides environment white noise, in this paper, we will also consider another noise, namely, telegraph noise ( [12][13][14][15]). e latter can be described as a switching between two or more regimes of environment, which differ in terms of factors such as nutrition, climatic characteristics, or sociocultural factors. e switching among different environments is memoryless and the waiting time for the next switch is exponentially distributed. e regime switching can hence be modeled by a finite-state Markov chain ς(t), t ≥ 0 { } taking values in a finite-state space M � 1, 2, . . . , N { }. e stochastic system (1) with regime switching can be described by the following model: (3) roughout this paper, we let (Ω, F, F t t≥0 , P) be a complete probability space with a filtration F t t≥0 satisfying the usual conditions (i.e. it is increasing and right continuous while F 0 contains all P-null sets). Let ς(t), t ≥ 0 { } be a right-continuous Markov chain on the probability space (Ω, F, F t t≥0 , P) taking values in a finite-state space M � 1, 2, . . . , N { } with the generator Φ � (ϕ uv ) 1≤u,v≤N given, for δ > 0, by Here, ϕ uv is the transition rate from u to v and Suppose that the Markov chain ς(t) is independent of the Brownian motion B(·) and it is irreducible. Under this condition, the Markov chain has a unique stationary (probability) distribution π � (π 1 , . . . , π N ), which can be determined by solving the linear equation πΦ � 0, subject to N i�1 π i � 1, and π i > 0, ∀i ∈ M.
ereafter, for any vector h � (h(1), . . . , }. e rest of the paper is organized as follows. In Section 2, we show that there exists a unique global positive solution of system (3). In Section 3, we give sufficient conditions for the extinction of the disease. In Section 4, sufficient conditions for the existence of the ergodic stationary distribution are established for model (3). Finally, numerical simulations are carried out to support the theoretical results.

Existence and Uniqueness of the Global Positive Solution
In this section, we will prove that model (3) has a unique global positive solution. We also denote us, we established the following theorem.
Proof. Since the coefficients of system (3) are locally Lipshitz continuous, for any initial value X(0) ∈ R 3 + , there exists a unique local solution X(t) on t ∈ [0, τ e ), where τ e is the explosion time. We need to show that this solution is global almost surely that is, τ e � ∞ a.s. Let m 0 be sufficiently large such that every component of X(0) lies within the interval 2 Discrete Dynamics in Nature and Society For each integer m ≥ m 0 , define a sequence of stopping times by Hence, there is an integer m 1 ≥ m 0 such that By Itô's formula, we have where which implies that Hence, Integrating both sides of the above inequality from 0 to τ m ∧T, and taking the expectations, we get Set Ω m � τ m ≤ T for m ≥ m 1 and by (4), we have P(τ ∞ ≤ T) ≥ ε for each m ≥ m 1 . For every ω ∈ Ω m , we have Discrete Dynamics in Nature and Society en, we obtain

Extinction
Our goal in this section is to study the extinction and give the extinction threshold of system (3). en, the following theorem gives a sufficient condition for extinction of the disease.
Proof. Applying Itô's formula, we can get Integrating (19) from 0 to t and then dividing by t into both sides leads to where Making use of the strong law of large numbers for martingales ( [16]) yields Taking the superior limit on both sides of (20) and applying the ergodicity of Markov chain ς(t), we get

Existence of Ergodic Stationary Distribution
In this section, we shall discuss sufficient conditions for the existence of an ergodic stationary distribution to model (3). e following lemma gives a criterion for positive recurrence in terms of lyapunov function [17].
Let (X(t), ς(t)) is the diffusion Markov process and satisfy the following equation where f: (x, k)). For each k ∈ M, and for any twice continuously differentiable function V(., k), the operator L can be defined by Lemma 1. If the following conditions are satisfied: (1) ϕ ij > 0 for any i ≠ j.
Combining with (10), it can be achieved that which follows from (11).
which together with (44) implies that Discrete Dynamics in Nature and Society 7 Case 5. If (S, I, R, k) ∈ D 5 × M, we obtain which follows from (44). Hence, In view of (44), we arrive at LH ≤ − 1 for all(S, I, R, k) ∈ D 6 × M.
(57) erefore, we have proof that us, condition 3 in Lemma 1 has been satisfied, and system (3) has a unique stationary distribution and ergodicity holds. is completes the proof.

Remark 1.
Assume the condition R s 0 < 1 holds. Disease I goes to extinction exponentially with probability one, eorem 2, and if R s 0 > 1 there is a unique ergodic stationary distribution μ(·,·) of system (3), which implies that disease I persists eorem 3. en, the number R s 0 can be considered as a threshold to identifying the stochastic extinction and persistence of system (3).

Simulations
Numerical solutions of stochastic differential equations are very important in the study of real examples of epidemic. In this section, we present some numerical results to illustrate the theoretical one. For numerical simulations of the SDEMS model (3), we use the Euler-Maruyama (EM) method ( [19] Obviously, the Markov chain ς(t) has a unique stationary distribution π � (π 1 , π 2 , π 3 ) � (0.1666, 0.6666, 0.1666). Given a step size Δ � 0.0001, the Markov chain can be simulated by computing the one-step transition probability matrix P � e ΔΓ ( [20]), and the transition probability matrix is given by (60) Figure 1 shows a result of one simulation run of the Markov chain ς(t). Example 1. To illustrate eorem 2, we choose the parameter values in system (3) as follows:

(61)
Simple computations result, as a consequence result of eorem 2. Disease I dies out exponentially with probability one. Figure 2 confirms this.

Conclusion
is article discusses the dynamic behavior of a SIRS epidemic model with a regime switching and nonlinear incidence rate. We obtain sufficient conditions for the extinction of system (3) if R s 0 < 1. We prove the stochastic system (3) under regime switching has a unique stationary distribution which is ergodic and positive recurrent by using the Lyapunov function method. In future works, it is interesting to study the effect of Lévy noise and a color noise (telegraph noise) in the stochastic SIRS epidemic model (2). We will investigate this case in our future works.