Stationary Distribution and Dynamic Behaviour of a Stochastic SIVR Epidemic Model with Imperfect Vaccine

We consider a stochastic SIVR (susceptible-infected-vaccinated-recovered) epidemicmodel with imperfect vaccine. First, we obtain critical condition under which the disease is persistent in the mean. Second, we establish sufficient conditions for the existence of an ergodic stationary distribution to the model. Third, we study the extinction of the disease. Finally, numerical simulations are given to support the analytical results.


Introduction
Mathematical models have been an unavoidable tool in analyzing the mechanisms of infectious diseases.Most modelers were inspired by the works in [1][2][3].For controlling the spread of diseases, vaccination is considered the most effective way to reduce both the morbidity and mortality of individuals (see [4][5][6]).In this paper, we consider the deterministic SIVR (susceptible-infected-vaccinated-recovered) model: where S, I, and R denote the densities of susceptible, infected, and recovered individuals, respectively.V denotes the density of individuals who are immune to an infection as result of vaccination.
The parameters involved in the system are described below: : the average number of contacts per infected per unit time : the recruitment rate and the death rate : recovery rate of infected individuals : the rate at which susceptible individuals are moved into the vaccination process : a positive factor satisfying 0 ≤  ≤ 1, and  = 0 means that the vaccine is perfectly effective and  = 1 means that the vaccine has no effect In model (1), the fundamental parameter that governs the spread of the disease into a population is the basic reproduction number denoted by  0 .It can be thought of as the number of cases one case generates on average over the course of its infectious period in an otherwise uninfected population (see [7]).
Let us denote by  0 = /( + ) the basic reproduction number of model (1) and by   =  0 (( + )/( + )) the basic reproduction number in a population in which a proportion  had been vaccinated.It is known that, in the absence of the disease, there is a unique disease-free equilibrium ( 0 ,  0 ,  0 ,  0 ) = (/(+), 0, /(+), 0) which is globally asymptotically stable if   < 1.If   > 1 for some parameters values, the model exhibits a backward bifurcation leading to the existence of multiple endemic equilibria and news subthreshold, which may be important when it comes to designing vaccination strategies (see [8,9]).
In this paper, we assume that the multiplicative noise sources are linear in (), () and (), according to [20].Note that recovered population has no effect on the dynamics of , , and .Then, following this approach, we obtain the following reduced stochastic SIVR model: where  1 (),  2 (), and  3 () are standard independent Brownian motions and  2  is a positive constant, for all  ∈ {1, 2, 3}.This paper is organized as follows.In Section 2, we present some lemmas concerning the existence of a global positive solution and ergodic stationary distribution.In Section 3, we prove that the disease is persistent under one condition.In Section 4, we establish sufficient conditions for the existence of a unique ergodic stationary distribution.In Section 5, we determine a condition under which the disease goes to extinction.In the last section, we introduce some examples and numerical simulations to confirm our results.

Preliminaries
Throughout this paper, let (Ω, F, (F  ) ≥0 , P) be a complete probability space with a filtration (F  ) ≥0 satisfying the usual conditions (i.e., it is increasing and right continuous, while F 0 contains all P-null sets).Moreover, let R  + = {( 1 , . . .,   ) ∈ R  :   > 0,  = 1, . where By Itô's formula, we get Next, we shall present a lemma that gives a criterion for the existence of an ergodic stationary distribution to system (3).
Let () be a homogeneous Markov process in   (  denotes -dimensional Euclidian space) and it is described by the following stochastic differential equation: The diffusion matrix is defined as follows: Lemma 2 (see [23], Chapter 4).
for all  ∈   , where Φ(.) is a function integrable with respect to the measure .Now, we consider the following one-dimensional homogeneous Markov process: where () and () are measurable functions from R to R and () is the Brownian motion.It is assumed that the functions (), (), and 1/() are locally bounded.(H.3): the functions () and () are such that Lemma 3 (see [24], Theorem 1.13).If (H.3) is satisfied, then the stochastic process ( 14) has ergodic properties with the density given by

Persistence
Theorem 5.If   0 > 1, then the disease will be persistent in mean; that is, where Journal of Applied Mathematics

Proof.
Set where positive constants , , and  will be determined later.
We apply Itô's formula on  1 ; then we get 2 ) 2 ) Let Then By integrating the last inequality, we obtain By applying Theorem 1 and the strong law of large numbers for martingales [22], we get the desired result.

Stationary Distribution
In this section, by using the theory of Has'minski [18], we prove the existence of a unique ergodic stationary distribution, which indicates that the disease is persistent.(i) To verify (.2), we show that there exists a neighborhood  ⊂ R 3 + and a nonnegative C 2 -function  such that () is negative for any  ∈ R 3 + \ .To this end, we define a C 2 -function in the form where  1 is the same function defined in Section 3,  2 š (1/(1 + ))( +  + ) 1+ , and ,  are positive constants that satisfy where It is easy to see that lim (, , ) = +∞.
From the proof of Theorem 5, we have The operator  defined in Section 2 acts on  2 , − log , and − log  as follows: where (34) Next, we construct the following compact subset: where  is a sufficiently small positive number, satisfying the following inequalities: where the constant  will be determined later.Then with Now, we will show that  is negative for any (, , ) ∈ R From the previous discussion, we have  (, , ) < 0, (, , ) ∈ R 3 + \ .
The threshold ∽   0 is defined as follows: Proof.By integrating system (3), we obtain It follows that where From the first equation of system (3), we obtain where Applying Itô's formula to system (3), one obtains log  () − log  (0 2 ) 2 ) From the strong law of large numbers for martingales [22], we get lim By Theorem 1, we get lim By taking the superior limit of both sides of (61), we obtain lim sup This finishes the proof.
One can see that where Γ is the Gamma function defined by Thus, condition (.3) in Lemma 3 is verified.So, system (70) has the ergodic property and the invariant density given by Let compute .
According to one has In view of the exponential martingales inequality [22], for any positive constants , , and ], we have Choosing  = ,  = 1 and ] = 2 log , one has Applying the Borel-Cantelli Lemma [22] leads to the fact that, for almost all  ∈ Ω, there exists a random integer  0 =  0 () such that, for any  ≥  0 , we obtain it follows that the distribution of the process () + () converges weakly to the measure that has the density .This completes the proof.

Numerical Simulations
In this section, we present the numerical simulations to support the above analytical results, illustrating persistence in mean and extinction of the disease.
Example 10.We choose the parameters as follows: Theorem 7 implies that system (3) has disease extinction (see Figure 3).It follows that the distribution of the process () + () converges weakly to the measure that has the density  (see Figure 4).

Conclusion
This paper is concerned with the dynamics of a stochastic SIVR epidemic model with multiplicative noise sources.By constructing a convenient positive function, we establish sufficient conditions for the existence of a unique ergodic stationary distribution to model (3) and we prove that the disease will be permanent.In addition, we also establish sufficient conditions for extinction of the disease.Many works have been done to study the continuous stochastic models by using the white noise.Sometimes, population systems may suffer sudden environmental perturbations, such as SARS, floods, and toxic pollutants.These phenomena cannot be modeled by stochastic continuous   models.During the last years, there was an increasing interest in Lévy noise that has non-Gaussian statistics.A lot of works have been realized and have shown the effectiveness of the Lévy noise in studying such phenomena (see [25][26][27]).We investigate in our future works the impact of the Lévy noise on the dynamic of complicated population systems.

Theorem 6 .
If   0 > 1, then system (3) has a unique stationary distribution and it has the ergodic property.Proof.To prove Theorem 6, it is sufficient to verify assumptions (.1) and (.2).

Example 9 .
In model(3), we choose the parameters as follows: