STABILITY ANALYSIS OF A STOCHASTIC SIS MODEL WITH DOUBLE EPIDEMIC HYPOTHESIS AND SPECIFIC NONLINEAR INCIDENCE

The purpose of this work is to investigate the almost surely exponentially stable of a stochastic SIS model with double epidemic hypothesis and specific nonlinear incidence rate. We establish the global existence and positivity of solution. Furthermore, the stability of the disease-free equilibrium of the model are showed. The analytical results are illustrated by computer simulations.


Introduction
Epidemiology is the study of the spread of diseases with the objective to trace factors that are responsible for or contribute to their occurrence.Consequently, it has been investigated by several mathematicians through establishing mathematical models for a long time (see, for example, [1,2,3,4]).Particularly, the susceptible-infected-susceptible SIS epidemic model is often used to model the dynamics of the diseases such as the bacterial diseases and some sexually transmitted diseases where individuals start off susceptible, at some stage catch the disease, and after a short infectious period become susceptible again [5].
In the classic SIS epidemic model, the disease is caused by one virus.In many cases, the disease is frequently not caused by one certain kind of viruses, but two or more kinds of viruses.Recently, the authors of [6,7,8] investigated the epidemic model with double epidemic hypothesis which has two epidemic diseases caused by two different viruses.In this paper, we consider a deterministic SI model with double epidemic hypothesis and cure rate described by the following differential system (1) where S(t) represents the number of susceptibles at time t, I 1 and I 2 are the total population of the infectives with virus V 1 and V 2 at time t, respectively.A represents the recruitment rate of the population, d is the natural death rate of the population, a i is the disease-related death rate, r i is the treatment cure rate, and β i is the infection coefficient, i = 1, 2. The incidence rate of disease I i is modeled by the specifc functional response β i SI i /1 + α i S + γ i I i + µ i SI i , where α i , γ i , µ i ≥ 0. This specific functional response was introduced by Hattaf et al [9], and he is becomes the bilinear incidence rate if α i = γ i = µ i = 0, the saturated incidence rate if [10,11] if µ i = 0, and the Crowley-Martin functional response [12] if α i γ i = µ i .
In the reality, epidemic systems are inevitably effected by environmental white noise.Therefore it is necessary to study that how the noise influences on the epidemic models.Consequently, many authors have studied stochastic epidemic models, see [13,14,15].For this, we consider the case in which the rates β i is subject to random fluctuations.namely, β i dt is replaced by , where B i independent standard Brownian motions and σ i represent the intensities of the white noises of B i .Therefore, the corresponding stochastic system to (1) can be described by the Itô's equation dB 2 (t), dB 1 (t), with f i (S, The rest of the paper is organized as follows.In the next section , we present the global stability analysis of the disease-free equilibrium for deterministic model (1).In Section 3, we prove that the stochastic system (2) has a unique global positive solution and we give a sufficient condition for the almost sure exponential stability of the disease-free equilibrium.Numerical simulation to illustrate our theoretical result will be presented in Section 4. Finally, we close the paper with discussions and future directions.

Deterministic SIS model
For biological reasons, we assume that the initial conditions of system (1) satisfy Thus the system (1) is positive [16], that is, S (t) ≥ 0, I 1 (t) ≥ 0 and I 2 (t) ≥ 0 for all t > 0. In fact by Proposition 2.1 in [17], we have By summing all the equations of the system (1) we find that the total population size N (t) = S (t) + I 1 (t) + I 2 (t) satisfies the inequality The standard comparison theorem [18] can be used to deduce that Thus, the feasible solution set of the system equation of the model enters and remains in the region Therefore, the model ( 1) is well posed epidemiologically and mathematically [19].Hence, it is sufficient to study the dynamics of the model (1) in Γ.
It is easy to see that system (1) has a disease-free equilibrium state E 0 = ( A d , 0, 0).Therefore, the basic reproduction numbers are .
We mention that the expression of R 01 and R 02 can also be obtained by applying the next generation matrix method provided by van den Driessche and Watmough [20].
Biologically, R 0i (i = 1, 2) represents the average number of secondary infections that occur when one infectious individual is introduced into a completely susceptible population.Now, we investigate the local stability of the disease-free equilibrium E 0 .The Jacobian matrix of system (1) at the equilibrium E 0 is as follows and . Hence, the equilibrium E 0 will be locally asymptotically stable if R 01 < 1 and R 02 < 1, and unstable when R 01 > 1 or R 02 > 1.
The following theorem discusses the global stability of the disease-free equilibrium E 0 .
Proof.Let V be the Lyapunov function defined as Differentiating V along the solutions of positive system (1), we have Using the inequalitys S 1 Therefore, R 01 ≤ 1 and R 02 ≤ 1 ensures that V (S, I 1 , I 2 ) ≤ 0. Furthermore, it is easy to verify that the singleton {E 0 } is the largest compact invariant set in (S, and hence by the LaSalle's invariance principle [21], every solution to equations of system (1), with initial conditions in Γ, approaches E 0 as t → ∞.Thus E 0 is globally asymptotically stable.

Stochastic SIS model
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 : (3) where f (x,t) is a function defined in R d × [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 (3).
Definition 3.1.[22] The trivial solution x = 0 of system ( 3) 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 where

Existence and uniqueness of the global positive solution
The following theorem shows that the solution of our system (2) is global and positive.
Proof.Since the coefficients of system (2) are locally Lipschitz continuous, then for any initial value (S 0 , I 10 , I 20 ) ∈ Γ there is a unique local solution (S(t), I 1 (t), I 2 (t)) on t ∈ [0, τ e ), where τ e is the explosion time.To show that this solution is global, we only need to prove τ e = ∞ a.s.

Exponentially stability
The goal of this subsection is to establish a sufficient conditions for the exponentially stability of the disease-free equilibrium.For this, we consider Theorem 3.1.Ψ(S(t), I 1 (t), I 2 (t)) converges exponentially to zero if the following condition holds lim sup t→∞ L V (S(t), I 1 (t), I 2 (t)) < 0 a.s.
Proof.By Itô's formula, we have Integrating both sides from 0 to t yields that Ψ f i (S(s),I i (s)) dB i (s) is a continuous local martingale and M(0) = 0.Moreover, whose quadratic variation is So, the strong law of large number for local martingales [23] implies that lim t→∞ 1 t t 0 2σ i S(s)I i (s) It follows that lim sup the proposition is proved.
In order to establish the conditions for the exponentially stability of the disease-free equilibrium of system (2), we need the following lemma [24].
Lemma 3.1.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−1,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 ). Let then the disease-free equilibrium of system ( 2) is almost surely exponentially stable.
Proof.By It ô's formula, we have For every sample path w of the process B (t), there exists an unbounded increasing sequence t n of positive time values for which Then by Lemma 3.1, there exists a subsequence t n for which the following limits exists and as well the conditions x + y 1 + y Then from (5), we obtain Therefore, by substituting the following inequality −4σ 2

Numerical examples and simulations
In this section, we give some numerical simulations in order to illustrate our theoretical results.
Example 4.1.As a numerical example, for By In this case, the conditions of Theorem 3.2 is not satisfied (see Figure (c)).

Conclusion
In this work, we have proposed and analyzed a new stochastic SIS with double epidemic hypothesis and specific functional response by introducing random perturbations of white noise type directly to β i .Firstly, in absence of noise we have proved that the disease dies out if the basic reproduction numbers R 0i ≤ 1 (Theorem 2.1).Next, we have proved the global existence and positivity of solution for our stochastic model (Theorem 3.1).In addition, we give a sufficient conditions for exponentially stable of our stochastic model (Theorem 3.2).What is more, we can also investigate the stochastic SIS epidemic model with two infectious diseases with Markov chain.We will investigate these cases in our future work.
Figure (a)