ANALYSIS OF STOCHASTIC VECTOR-HOST EPIDEMIC MODEL WITH DIRECT TRANSMISSION

. In this paper, we consider the stochastic vector-host epidemic model with direct transmission. First, we study the existence of a positive global solution and stochastic boundedness of the system of stochastic diﬀerential equations which describes the model. Then we introduce the basic repro- ductive number R s 0 in the stochastic model, which reﬂects the deterministic counterpart, and investigate the dynamics of the stochastic epidemic model when R s 0 < 1 and R s 0 > 1. In particular, we show that random eﬀects may lead to extinction in the stochastic case while the deterministic model predicts persistence. Additionally, we provide conditions for the extinction of the in- fection and stochastic stability of the solution. Finally, numerical simulations are presented to illustrate some of the theoretical results.

1. Introduction. Infectious diseases are the leading cause of deaths of children and adolescents especially in the developing countries [14,15,16,31]. The burden of infectious diseases is manifested through death as well as socioeconomic impacts.
Vector-borne disease is transmitted by the bite of infected arthropods such as mosquitoes, ticks, mites, rats and blackflies. It is one of the most common infections accounting for more than 17% of infectious diseases. Many vector-borne diseases, such as malaria, dengue fever, West Nile virus, and recently Zika virus, are transmitted to the human population by insects (vectors) such as mosquitoes. Some vector-borne diseases may also be transmitted directly through blood transfusions, organ transplantation, exposure in a laboratory setting, or from mother to baby during pregnancy, delivery and breast feeding [16,17,29]. Direct transmission has an impact on the dynamics of many vector-borne diseases.
One way to improve control and ultimately eradicate infectious diseases is to choose an appropriate model that best describes the demography and epidemiology of the population being modeled. Indeed, significant improvement has been made in mathematical research of epidemic problems.
Epidemic models can be deterministic, in which the output of the model is fully determined by the parametric and initial values, or stochastic, where the model possesses some inherent randomness. Deterministic vector-host epidemic models have been studied by several authors [1,2,6,8,20,21,25,26,32,37]. In [8], Cai and Li analyzed a simple vector-host epidemic model with direct transmission. Yang and Wei discussed the global stability of an epidemic model for vector-borne disease and they showed that the global dynamics are completely determined by the basic reproductive number R 0 [39]. Belayneh, et al. provided a cost effective control effort for treatment of hosts and prevention of vector-host contacts for a non-autonomous model, while they establish global stability conditions for the autonomous case [6]. Particular vector-borne diseases such as malaria [21,24,25,26,32], dengue fever [1,2,34] and West Nile virus [5,38] have been modeled and studied by a number of authors.
The deterministic vector-host epidemic models with direct transmission have been studied by Cai and Li [8]. They showed that the stability of the equilibria can be controlled by the basic reproductive number. Similar studies can also be found in [6,32,37]. On the other hand, Jovanovic and Krstic [18] discussed the corresponding stochastic model and studied the stability conditions using Lyapunov functions.
Even though deterministic models are crucial in understanding more about the dynamics of the disease, they do not incorporate the effect of environmental fluctuations. In reality, the system is exposed to influences that are not completely understood and the spread of the disease is inherently random. Thus, for a better analysis of the model, we will include such stochastic influences [35].
There are several ways to include these fluctuations in the deterministic model. For example, [9,11,33] introduced parametric perturbations, since the parameters in the model are always altered due to continuous environmental fluctuations. Another approach, pioneered in the works of May and Beddington [4], assumes that the environmental noise is generated by an m-dimensional standard Brownian motion. Some other authors used this idea to study the properties of stochastic epidemic models in order to find a more efficient way to reduce infections [7,9,30,36,40,41].
The aim of this paper is to study the well-posedness of the stochastic vector born disease model and examine how the introduction of stochastic noise affects the dynamics of the vector-host epidemic model with direct transmission. We shall show that the solution of the stochastic model is ultimately bounded in probability. We also seek the sufficient conditions for the solution to be stochastically permanent. Furthermore, we will investigate how the basic reproductive number is affected by the introduction of the stochastic noises.
The organization of the paper is as follows: In section 2, we formulate the stochastic epidemic model from the corresponding deterministic one. In section 3, we analyze the stochastic system using different Lyapunov functions and Itô formula. In the last section we provide numerical simulation using the Milstein method to support the theoretical results in the previous sections.
2.1. Deterministic model. Let S H , I H respectively represent the number of susceptible and infected hosts and S V , I V respectively represent the number of susceptible and infected vectors. Assume that susceptible hosts can be infected both directly through contact with an infected host, such as blood transfusion, and indirectly by a bite from an infected vector, such as a mosquito. Similarly, we assume that if a susceptible vector bites an infected host, it will acquire the disease. The model does not assume disease-induced deaths in both species, that is, no one has died from the disease in the given time. However, it can be easily modified to include such assumptions [23,27,32]. The graph below depicts the transmission cycle of the vector-host epidemic model. Here µ and η are the mortality rates of the host and vector, φ is the recovery rate of infected hosts, β 1 is the direct transmission rate from an infected host to susceptible host, and β 2 is the indirect transmission rate from an infected vector to a susceptible host, and β represents the transition rate from infected host to susceptible vector. To begin with, let b 1 and b 2 be respectively the recruitment rates of the host and the vector. Based on the above discussion, we have the following system of non-linear differential equations [8,18]. (1) Then it is easy to see that Γ is positively invariant under system (1). Also, since the vector field associated with system (1) is bounded and continuous, and satisfies the Lipchitz condition on Γ, system (1) has a unique solution in Γ. Next we introduce the basic reproductive number R 0 which is defined as the expected number of secondary cases produced by a single infection in a completely susceptible population. Using the next-generation matrix introduced by Diekmann [12], R 0 is given by the expression: Now setting we obtain the disease-free equilibrium E 0 = (S H , I H , S V , I V ) = ( b1 µ , 0, b2 η , 0) and the endemic-equilibrium point and I * H is the positive solution of the equation, where The local and global stabilities of the equilibrium points can be summarized as follows [8].
Theorem 2.1. If R 0 < 1, then the disease-free equilibrium E 0 is both locally and globally asymptotically stable.
Theorem 2.2. If R 0 > 1, then the endemic equilibrium E 1 is both locally and globally asymptotically stable.
The above theorems indicate that R 0 is a sharp threshold to determine if the disease will die out or stay endemic (see Figure 2). 2.2. Stochastic model. In this subsection we derive the stochastic epidemic model from the corresponding deterministic one by including a random effect in the deterministic case. Throughout the rest of this paper, let (Ω, F, P ) be a complete probability space with a filtration {F t } t≥0 satisfying the usual conditions, that is, it is right continuous and increasing with F 0 containing all P -null sets. Also, we define the differential operator L associated with the stochastic differential equation be sequences of random variables defined on (Ω, F, P ) which are jointly independent to each other and each sequence is identically distributed such that, for any where E is the expectation and σ S H , σ I H , σ S V , σ I V are some non-negative constants which show the intensity of the fluctuations. Each sequence of random variables measures the effects of random influence on each compartment during [k∆t, (k + 1)∆t] for k ∈ {0, 1, 2, . . . }. Thus, during [k∆t, (k +1)∆t], each compartment changes according to the deterministic equation (1) and by a random amount, that is, for k ∈ {0, 1, 2, . . . }, Following a standard procedure [3,10,13,19] we can show that, as ∆t → ∞, X ∆t (t) converges to a diffusion process X(t) = (S H (t), I H (t), S V (t), I V (t)) which satisfies the following system of stochastic differential equations.
2.3. Existence of a global solution. Clearly, there exists a unique local solution X(t) on an interval [0, τ e ), where τ e is the explosion time [33,Theorem 3.1]. Next, we show that τ e = ∞, that is, this solution is in fact global.
. Now, for any k ∈ N such that k > k 0 define Next, we show that τ = ∞. This implies that the explosion time is infinity and thus we conclude the model in (5) has a unique solution and will remain in Γ with probability 1. Suppose on the contrary that τ < ∞. Then there exists T > 0 such that P (τ ≤ T ) > for all ∈ (0, 1). This implies that there exists Using Itô's formula we have Using (5), we have LV ≤ C 1 , where Integrating both sides of the above inequality on (0, τ k ∧ T ), taking the expectation, and noting that for any G ∈ L 2 (0, T ), then at least one of the following will hold true: Generally, since f (x) = x − 1 − ln x is increasing on (1, ∞) and decreasing on (0, 1) it follows Finally, letting k → ∞ we have which is a contradiction. Thus we conclude τ = ∞. We summarize the above result in the following theorem.
Theorem 2.3. For any initial value X(0) ∈ Γ, system (5) has a unique global solution on t ≥ 0 and the solution will remain in Γ with probability 1.

Stochastic boundedness and permanence.
Define Theorem 2.3 shows that for any initial condition X(0) ∈ Γ the solution of system (5) is always positive and remains in Γ. Next we exam how X(t) varies in Γ. First, we give the definition of a stochastically ultimately bounded solution.  Proof. Define V 1 (S H , I H ) = S θ H + I θ H . Then, we have .
Using the above lemma we show that the solution of system (5) is stochastically ultimately bounded. For any > 0 put χ( ) = ζ 2 2 . Then, by Chebyshev's inequality we get This concludes lim sup t→∞ P { X(t) > χ} ≤ ζ Lemma 3.4. Let k := min{µ, η}, σ 2 := max{σ 2 S H , σ 2 I H , σ 2 S V , σ 2 I V } and assume that b 1 + b 2 − k > 0. Then, for any initial value X(0) ∈ Γ the solution X = X(t) of system (5) satisfies Here ν > 0 and θ > 0 are any constants satisfying the following conditions: Then, by Itô's formula Let ν be as in the assumption, then The last inequality follows from the fact that Let θ satisfy the assumption of the lemma. Then, and also In conclusion, we have

YANZHAO CAO AND DAWIT DENU
Finally, by Itô's formula, Integrating both sides and taking the expectation will result Simplifying it further, we get Letting t → ∞, we conclude Thus, it follows that θ . In the study of epidemic models, an important property is the so called stochastic permanence, which indicates how the total population in the model changes in the long run. First, we give its definition and then we show that under some conditions, the system in (5) is stochastically permanent.
Theorem 3.6. Under the assumptions of Lemma (3.4), the solution of system (5) is stochastically permanent for any initial value X(0) ∈ Γ.
4. Extinction and stochastic stability. In section 2 we defined the basic reproductive number R 0 . According to [8], for the deterministic model the number of infected hosts I H and vectors I V will tend to zero in the long run provided that R 0 < 1. In this section we derive a similar condition for the stochastic model so that the number of infected hosts and vectors will decrease exponentially to zero almost surely in the long run. The following theorem provides a condition for the extinction of the infected hosts and vectors.
. If R s 0 < 1 and η < β, then for any initial value X(0) ∈ Γ, I H (t) will tend to zero exponentially almost surely. That is, lim sup t→∞ Proof. From system (5), and using S H = b1 µ − I H we get Now, by Itô's formula we have d(ln(I H )) Integrating both sides on [0, t], we get Then, M is a martingale [22,Theorem 5.14], with a quadratic variation given by Since lim sup t→∞ M,M t t = σ 2 I H < ∞, by the strong law of large numbers, we have lim sup t→∞ Now if R 0 s < 1 and η < β we have Using the above theorem and the following lemma [22,Theorem 3.3], we conclude that the number of infected vectors I V will also tend to zero exponentially almost surely.
Lemma 4.2. Given a stochastic differential equation Assume that there exists a function V ∈ C 2,1 (R d × [t 0 , ∞); R + ), and constants p > 0, c 1 > 0, c 2 ∈ R, c 3 ≥ 0, such that for all X = 0 and t ≥ t 0 , Proof. By Theorem 4.1, if R s 0 < 1, then I H (t) will tend to zero exponentially almost surely and since exponential stability implies asymptotic stability, we have that lim t→∞ I H (t) = 0 a.s.
Thus, for any > 0, there exists k 1 > 0 and a setΓ ⊂ Γ such that Remark. Note that in the deterministic case, if R 0 < 1 then the system has a disease-free equilibrium. In this case, R s 0 < 1 and by Theorem 4.1 and corollary 4.3, in the stochastic model, the number of infected hosts and vectors will go to zero exponentially almost surely. On the contrary, we might have cases where R s 0 < 1, but R 0 > 1. That is, a large environmental fluctuation can suppress the number of infected hosts (see Figures 3, 4 and 5 below).
Assume that f (0, t) = 0, g(0, t) = 0 for all t > 0 such that X(t) = 0 is a trivial solution. Then, the trivial solution is called 1. Stochastically stable if for all ∈ (0, 1) and r > 0 there exist δ = δ( , r, t 0 ) > 0 such that P { X(t; t 0 , x 0 ) < r for all t ≥ t 0 } ≥ 1 − , for X 0 ∈ R d such that X 0 < δ. 2. Stochastically asymptotically stable if it is stochastically stable and for every ∈ (0, 1), there exists a δ 0 = δ 0 ( , t 0 ) > 0 such that, The standard method of studying stability is through a Lyapunov function. However, it is difficult to construct such a function for a nonlinear system of stochastic differential equations. In this paper we analyze the stability of the corresponding linear system, and if the coefficients of the nonlinear system satisfy the condition given in the following theorem, then we conclude that it is asymptotically stable almost surely [28, Theorem 7.1].
Theorem 4.5. If the linear system dX(t) = M X(t) dt + σX(t)dB(t) with constant coefficients is asymptotically stable almost surely, and the coefficients of the nonlinear system dX(t) = b(X, t) dt + σ(X, t)dB(t) satisfy an inequality b(x, t) − M x + σ(x, t) − σ x < γ x , in a sufficiently small neighborhood of the point x = 0, and with a sufficiently small constant γ , then the solution X = 0 of the nonlinear system is asymptotically stable almost surely.
For S H = b1 µ − I H and S V = b2 η − I V , system (5) reduces to the following dI H = ( β1b1 µ I H − β 1 I 2 H + β2b1 µ I V − β 2 I H I V − (µ + φ)I H )dt + σ I H I H dB I H (t), The corresponding linearized system is given by Finally, we state the condition for the asymptotic stability of the system (7) as follows.
Theorem 4.6. Let X(t) = (I H (t), I V (t)) be the solution of (7) and suppose It is easy to check that all the hypotheses of Corollary 5.3 are satisfied, and we conclude that lim inf t→∞