DYNAMICAL BEHAVIORS OF A STOCHASTIC SIRS EPIDEMIC MODEL

In this paper, we study the dynamical behavior of a stochastic SIRS epidemic model with specific nonlinear incidence rate and vaccination. We show the existence and positivity of the solution of the SIRS stochastic differential equation. We defined a number R and we prove the disease free equilibrium is almost sure exponentially stable if R < 1. We studying the behavior around the endemic equilibrium E*. Numerical simulations presented our theoretical results.


Introduction
Mathematical epidemiology play an important role in the study and control of the infectious diseases, the objective of this study is to implement measures to combat and terminate the spread of these diseases.
In recent years many authors have been developed the mathematics model for the transmission dynamics of infectious diseases, amid these models there are the classical deterministic SIR epidemic model [1,2], the population are divide into three classes or S represents the number of the individuals susceptible, I represents the number of infective individuals, R represents the number of recovered individuals with temporary immunity acquired from a disease.
In the other hand the modeling of population dynamics of diseases have recognized the introduction of stochastic term into deterministic models witch do incorporate the effect of fluctuating environment.To formulate stochastic differential equation (SDE) there is many approach, D. Greenhalgh et al. [3] have used The technique of parameter perturbation by the white noise.
They proved Almost sure exponential stability of the disease free equilibrium and stability in probability.Adnani et al. [4] and Lahrouz et al. [13] have utilized The technique of perturbations stochastic by withe noise around the endemic equilibrium state.They have proved the asymptotically mean square stable of stochastic linearized system.The case of a color noise was introduced by Lahrouz et al., and Gray et al. [5], [6].They have made a full analysis on asymptotic behavior of an SIS epidemic model under a finite regimes-switching.
Men et al. [7], have studied the SIR Models with horizontal and vertical transmission described by the system of differential equations : (1) where β is the contact rate, b is the mortality rate in the susceptible and the recovered individuals, d is the mortality rate in the infective individuals, r is the recover rate in the infective individuals into recovered individuals, p is the proportion of the offspring of infective parents that are susceptible individuals, and q is the proportion of the offspring of infective parents that are infective individuals, p, q verify p + q = 1, m is the successful vaccination proportion to the newborn from S and R, Men et al. have defined the reproduction number of system (1) by R 0 = (1−m)β pd+r , and proved that If R 0 < 1 then the infection-free equilibrium E 0 (1 − m, 0) is globally stable, else if R 0 > 1 then the epidemic equilibrium E * (S * , I * ) is globally asymptotically stable.For biological reasons, we assume that b − pd > 0 .We consider the following SIRS model with non-linear incidence rate : (2) where f (S, have been introduced by Hattaf et al. [8], where α 1 , α 2 , α 3 ≥ 0 are constants, this incidence rate generalise the incidence rate existing in the literature, if α 1 = α 2 = α 3 = 0 then we get the bilinear incidence rate β SI, if we put α 2 = α 3 = 0 then we have the saturated incidence rate β SI/(1 + α 1 I), we get functional response of Crowley Martin [9] if α 3 = α 1 α 2 , and if α 3 = 0 we obtained Beddington-DeAnglis functional response [10], [11].
In this paper, we consider the stochastic version of SIRS model (2)with a general incidence rate find it by perturbing the parameter β by the white noise : (3) The system is constant, so we normalized to unity S(t) + I(t) + R(t) = 1.therefor, we only need to consider the model defined as follows: (4) The stochastic version defined by : (5) It is important to note that system (3) includes many special case existing in the literature.
the remnant of the paper is organized as flows.In section 3 we prove that system (4) allows a unique global and positive solution starting from a initial value in Γ, in section 4 we proved Almost sure exponential stability of the disease free equilibrium, in section 5 we investigate its asymptotic behavior around the endemic equilibrium of system ( 5), in section 6 we show the numerical simulation to illustrate our theoretical result.Finally, the conclusion of our paper is in Section 5.

Preliminaries
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).
Next, we consider the d-dimensional stochastic system : where f (x,t) is a function in R d defined in [t 0 , +∞) and g(x,t) is an d × m matrix, f and g are locally Lipschitz functions in x. {B(t)} t≥0 is an d-dimensional standard Wiener process defined on the above probability space.
Let us suppose that f (t, 0) = g(t, 0) = 0 for all t ≥ 0. We assume that x = 0 is a solution of the system (2.1).
Definition 2.1.[13] The trivial solution x = 0 of system (2.1) is said to be almost surely exponentially stable if for all x(0) = x 0 ∈ R d : If the differential operator L acts on a function And In order to establish the conditions for the exponentially stability of the disease-free equilibrium of system (4), we need the following lemma lemma 2.2.For k ∈ N, let X(t) = (X 1 (t), X 2 (t), .., X k (t)) be a bounded R k -valued function.Let (t 0,n ) be any increasing unbounded sequence of positive real numbers.Then there is a family of sequences (t s,n ) such that for each s ∈ 1, 2, .., k, (t s,n ) is a subsequence of (t s,n ) and the sequence X s (t s,n ) converges to the largest limit point of the sequence X s (t s−1,n ).

Global existence and positivity
In this section, we will prove that model (4) has a unique global positive solution for any initial value in Γ.Where Theorem 3.1.For any given initial value (S 0 , I 0 , R 0 ) ∈ Γ, there is a unique positive solution (S(t), I(t), R(t)) of (3) on t ≥ 0 and the solution will remain in Γ with probability 1, namely (S(t), I(t), R(t)) ∈ Γ for all t ≥ 0 almost surely.
Proof.Since the coefficients of system (3) are locally Lipschitz continuous, then for any initial value (S 0 , I 0 , R 0 ) ∈ Γ there is a unique local solution (S(t), where τ e is the explosion time.To show that this solution is global, we only need to prove τ e = ∞ a.s.
Define the stopping time : We set inf / 0 = ∞, as usual / 0 denotes the empty set.We have τ ≤ τ e , If τ = ∞ a.s., then τ e = ∞ a.s. and (S(t), I(t), R(t)) ∈ Γ for all t ≥ 0. In addition, to complete the proof we only need to prove τ = ∞.Assume that τ < ∞, then there exists a T > 0 such that P(τ < T ) > 0. Consider the C 2 -function Q, defined by the expression : Using Itô's Formula, we have for all We have f (S, I) ≥ 0, implies that dQ(S, I, R) ≤ G(S, I)dt + σ (I − S) f (S, I) dB(t), where G(S, I) Integrating the above inequality, we obtain There is some element of X(τ) equal 0. Then Which contradicts our assumption.So we must therefore have τ = ∞ a.s.This completes the proof of Theorem 3.1.

Exponential stability
In this section, we give a sufficient conditions for the exponentially stability of the diseasefree equilibrium.We set X(t) = (S(t), I(t)).We define the following stochastic process Ψ(t) and U(X(t)) as following where c = b(1 − m) + γ b + γ .We note that Ψ(t) > 0 a.s.for all t, also we defined the following invariant R with a constant 0 < k ≤ 1 and l = 1 + α 1 + α 2 + α 3 , we will employed in the main theorem of the stability : In the rest of this section we will shown that if R < 1 and kσ 2 < min{β l 2 , 4(b + γ)l 2 } then the disease free equilibrium will almost sure exponentially stable.
Theorem 4.1.Suppose that the following inequality holds: If R < 1 then the disease-free equilibrium of the system ( 5) is almost surely exponentially stable. Proof.
Step 1 : We prove that Ψ(t) converges exponentially to zero a.s.By using Itô's formula we obtain integrating both sides from 0 to t yields that Where According to Lemma 2.1 we obtain Consequently, Step 2 : We prove that lim sup t→∞ L U(X(t)) < 0 a.s.Applying the operator L to U we obtain : Using the inequality we get In view of Lemma 2.2 we can define the following limits for a suitable increasing, unbounded sequence t n as And with In particular then we have z + ax = 1, and y ≤ 1.We notice Therefore, we can write (8) as follows using the inequality Hence, there exist 0 < k ≤ 1 such that the following inequality hold From ( 9), yields that By the inequality kσ 2 < β l 2 , we find Using the inequality (10), we obtain Since kσ 2 < 4(b + γ)l 2 implide that the coefficients of z are negative.as though R < 1 we deduce that the coefficients of x are negative, gold : we choose a number a such that the following inequality hold, We have z + ax = 1, since the limits z, x cannot all be zero.Consequently, we obtain that Π < 0, the proof of theorem is completed.

Asymptotic Behavior Around the Endemic Equilibrium
By study epidemic dynamical systems, we are interested by extinction, and persistent in a population.In the deterministic models, the second problem is solved by showing that the endemic equilibrium of the model is globally asymptotic stable.But, there is none of endemic equilibrium in system (5).So in this section we study the behavior around the endemic equilibrium E * to indicate that whether the disease will prevail.We find the following result.
we can choose the number a such that From( 10) and ( 15), we obtain integrating both sides from 0 to t yields that This completes the proof of Theorem 4.1.

Numerical simulations
In this section, we illustrate our theoretical results presented above, the system (5) is simulated for various sets of parameters and by using the Euler-Maruyama method.Fig. 1 illustrates that the almost sure exponential stabile of stochastic SIRS model (5), whenever the conditions of theorem is realised R = 0.9375 < 1 and kσ 2 < min{β l 2 , 4(b + γ)l 2 }.In Fig. 2 little values of intensity, will still render the free equilibrium unstable.

Conclusion
The introduction of stochastic effects into deterministic models gives us a more realistic way of modeling epidemic models.In this paper we have considerate a stochastic SIRS epidemic model with vaccination and non-linear incidence rate, we have studied the effects of the environmental fluctuation in SIRS epidemic model.we first proved the existence and positivity of solutions of our stochastic model which implies that the model is well posed.Then, we showed the stability exponentially almost surely of the disease free equilibrium E 0 as R < 1.To indicate that whether the disease will prevail we study the behavior around the endemic equilibrium E * .Finally we have simulated our theoretical result.