Abstract
We consider a stochastic SIVR (susceptible-infected-vaccinated-recovered) epidemic model 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.
1. Introduction
Mathematical models have been an unavoidable tool in analyzing the mechanisms of infectious diseases. Most modelers were inspired by the works in [1–3]. For controlling the spread of diseases, vaccination is considered the most effective way to reduce both the morbidity and mortality of individuals (see [4–6]). 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 , and means that the vaccine is perfectly effective and 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 . 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 the basic reproduction number of model (1) and by 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 which is globally asymptotically stable if . If 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]).
It is well known that epidemic models are inevitably affected by the environmental noise that influences the dynamic behaviour of the epidemic models (see [10–14]). For this, inspired by the works in [15–17], Tornatore et al. [18] have formulated and studied a stochastic version of model (1) by replacing the contact rate in system (1) by , where denotes the standard Brownian motion and denotes the intensity of the white noise. Then, system (1) becomes as follows:
Dianli Zhao and Sanling Yuan [19] have obtained conditions ensuring the persistence and extinction of model (2) and have found a threshold whose value below 1 or above 1 can determine the extinction and persistence of the epidemic under mild extra conditions.
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 , and are standard independent Brownian motions and is a positive constant, for all .
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.
2. Preliminaries
Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is increasing and right continuous, while contains all -null sets). Moreover, let . If is a bounded function on , define and . We define also .
The following theorem concerns the existence and uniqueness of the global positive solution. Since the proof is almost the same as that in [21, Lemma 2.1 and Lemma 2.2], we omit it here.
Theorem 1. For any initial value , there is a unique solution of system (3) on and the solution will remain in with probability one. The solution has the following properties: Furthermore, if , then
Now, we consider the -dimensional stochastic differential equation:with initial value . denotes an n-dimensional standard Brownian motion defined on the complete probability space . Let us denote by the family of all nonnegative functions defined on such that they are continuously twice differentiable in and once in . The differential operator of (6) is defined by [22] If acts on a function , then 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). The Markov process has a unique ergodic stationary distribution if there exists a bounded domain with regular boundary and
(H.1): there is a positive number such that ,
(H.2): there exists a nonnegative -function such that is negative for any . Then for all , where 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 to and is the Brownian motion. It is assumed that the functions , , and 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
3. Persistence
Definition 4. System (3) is said to be persistent in the mean, if
Theorem 5. If , then the disease will be persistent in mean; that is, where
Proof. Set where positive constants , and will be determined later.
We apply Itô’s formula on ; then we get 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.
4. 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.
Theorem 6. If , 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 and .
(i) To verify , we show that there exists a neighborhood and a nonnegative -function such that is negative for any . To this end, we define a -function in the form where is the same function defined in Section 3, , and are positive constants that satisfy where It is easy to see that Moreover, is a continuous function. Hence, must have a minimum point in the interior of . Then we define a nonnegative -function as From the proof of Theorem 5, we have The operator defined in Section 2 acts on , , and as follows: where Therefore where 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 .
Case 1. If , we obtain from (36) thatCase 2. If , (37) implies that Case 3. If , from (38) it follows that Case 4. If , (39) implies that Case 5. If , from (40) we get where Case 6. If , from (41) we obtain From the previous discussion, we have (ii) Now, we verify assumption . The diffusion matrix of system (3) is There is such that for and . That is to say, assumption holds.
Consequently, system (3) has an ergodic stationary distribution.
5. Extinction
Theorem 7. We assume that . Let be the solution of system (3). If , then the disease dies out with probability one.
The threshold is defined as follows:
Proof. By integrating system (3), we obtain It follows that where From the first equation of system (3), we obtainwhere Applying Itô’s formula to system (3), one obtainsBy injecting (56) and (58) into (60), we haveFrom the strong law of large numbers for martingales [22], we get By Theorem 1, we get By taking the superior limit of both sides of (61), we obtain This finishes the proof.
Theorem 8. We assume that and . Let be the solution of system (3). If , then If , then and the distribution of converges weakly to the measure that has the density given by where
Proof. We assume that and .
From system (3), we havewherewith the initial value .
We compute that where B is a constant.
One can see that where is the Gamma function defined by If , then Thus, condition in Lemma 3 is verified. So, system (70) has the ergodic property and the invariant density given by Let us compute .
According to one has From the ergodic theorem, it follows that On the other hand, by applying Itô’s formula to and then integrating, one haswhere whose quadratic variation is In view of the exponential martingales inequality [22], for any positive constants , and , we have Choosing and , one has Applying the Borel-Cantelli Lemma [22] leads to the fact that, for almost all , there exists a random integer such that, for any , we obtain That is, Substituting this inequality into (79) yields Applying the comparison theorem of stochastic differential equations on inequality (69) gives It follows that Taking the superior limit on both sides of the previous inequality leads to If , we conclude that As a result, for any small , there exist and a set such that and for all and . From it follows that the distribution of the process converges weakly to the measure that has the density .
This completes the proof.
6. 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.
In the two following examples, we choose the initial value as .
Example 9. In model (3), we choose the parameters as follows: We compute that Then, according to Theorem 5, the disease is persistent (see Figure 1). Furthermore, by Theorem 6, system (3) has a unique stationary distribution (see Figure 2).
Example 10. We choose the parameters as follows: Then Theorem 7 implies that system (3) has disease extinction (see Figure 3).
Example 11. We choose the parameters as follows: Then According to Theorem 8, we getIt follows that the distribution of the process converges weakly to the measure that has the density (see Figure 4).
7. 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 [25–27]). We investigate in our future works the impact of the Lévy noise on the dynamic of complicated population systems.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.