We investigate the conditions that control the extinction and the existence of a unique stationary distribution of a nonlinear mathematical spread model with stochastic perturbations in a population of varying size with relapse. Numerical simulations are carried out to illustrate the theoretical results.

1. Introduction

In medicine, relapse is the return of a disease or the signs and symptoms of a disease after a period of improvement. Relapse also refers to returning to the use of an addictive substance or behavior, such as cigarette smoking [1]. For example, for human tuberculosis, incomplete treatment can lead to relapse, but relapse can also occur in patients who take a full course of treatment and are declared cured. Recently, considerable attention has been paid to model the relapse phenomenon. In [2] Tuder developed one of the first epidemic models with relapse in a constant population with bilinear incidence rate. Moreira and Wang [3] included a nonlinear incidence rate in the model. Van Den Driessche and Zou in [4] formulated a SIRI epidemic model as an integro-differential system with the fraction of recovered individuals remaining in the recovered class units after the recovery. In [5] the displacement of recovered individuals to the infective class due to relapse is given by . Inspired by the works cited above and the fact that relapse is due to contact with infected, it is more reasonable to consider a bilinear relapse rate . We consider the following SIR compartmental model in a population of varying size with a bilinear relapse rate.In this model each letter refers to a compartment in which an individual can reside. Let denote the number of members of a population susceptible to the disease at time , the number of infective members, and the number of members who have been removed from the possibility of infection with permanent or temporary immunity. The parameters that occur in the model have the following meaning. is the rate at which new individuals enter the population. is the rate at which the infective individuals become recovered. is the parameter that measure the intensity of the relapse. The positive constants , , and satisfyingrepresent the natural death rate of susceptible, infected, and recovered individuals, respectively. Another addition in the modeling of population dynamics of diseases is the introduction of stochasticity into epidemic models. Many scholars have studied the effect of stochasticity on epidemic models [610]. For instance, to include stochastic demographic variability, Allen [6] studied SDEs for simple SIS and SIR epidemic models with constant population size that was derived from a continuous time Markov chain model. In [7, 10], the situation of a white noise stochastic perturbations around the endemic equilibrium state was considered. Lahrouz et al. in [11] formulated a stochastic version of the classical SIS epidemic model with varying population size. The authors studied the long time behavior of the stochastic system. They also gave conditions for extinction and persistence of the disease in the population. According to the value of the threshold , they showed that if , the disease will die out from the population with the probability one and the disease will persist if . In the case of persistence, they proved the existence of a stationary distribution.

Let be a complete probability space with a filtration satisfying the usual conditions (i.e., it is right continuous and contains all Prob-null sets). In this paper, we assume that fluctuations in the environment will manifest themselves mainly as fluctuations in the death rates of , , and . Specifically, is replaced with ; that is, the rate is perturbed by Gaussian white noise. The rates and are similarly perturbed by an independent Gaussian white noises. Therefore, the corresponding stochastic system to (1) can be described by the Itô equation:where , , and are independent Brownian motions; , , represent the intensities of the white noises. By using the same method as in [10, 12], the existence and uniqueness of positive solution for system (3) hold with probability 1, if we start from any positive initial value The main concern of the present paper is to establish a sufficient condition for the extinction and the persistence of solutions of the system (3).

2. Preliminaries

Throughout the rest of this paper, we denoteIn general, consider the dimensional stochastic differential equation: where, , and denotes a -dimensional standard Brownian motion defined on the underlying probability space. If is a vector or matrix, its transpose is denoted by The matrix is called the diffusion matrix. For the convenience of a later presentation, we introduce the generator associated with (5) as follows. For any twice continuously differentiable where , denote the gradient, Hessian of , respectively.

The following theorem gives a criterion for positive recurrence in terms of Lyapunov function (see [13] Theorem 3.13, p. 1164, Theorem 4.3, p. 1168, and Theorem 4.4, p. 1169).

Theorem 1. The system (5) is positive recurrent if there is a bounded open subset of with a regular boundary, and the following holds.
(i) There exist some such that, for all , (ii) There exists a nonnegative function such that is twice continuously differentiable and that for some Moreover, the positive Markov process has a unique ergodic stationary distribution That is, if is a function integrable with respect to the measure , then

3. Extinction of the Disease

In this section, we followed the methods of Lahrouz et al. [11] to establish sufficient condition for the extinction of the disease. Before this, let us prepare two useful lemmas. In the following lemma, we show that the positive solutions to (3) have finite moments.

Lemma 2. Let be the total population size in system (3) for an initial positive value Then for any such that , where , we have

Proof. Let such that and let where
Applying Itô’s formula leads towhereNote that the function is bounded in with integrating (13); then taking expectation on both sides and using (14) and (15), we obtain Therefore, for all , we have and for the fact that we have ().
() From (13) and (14) we have, for all , so where In view of (11) and the continuity of , there exists such that So, there exists a positive constant such that Applying Itô’s formula leads to Fromwe haveSo, there exists a positive constant such thatLetting we have where is a positive constant independent of , and the first inequality is derived from the maximal inequality for martingales and the second by (27) and Jensen’s inequality.
Choosing where we have So, by (22) we get , and by (23) we have Under the conditions of the preceding lemma we have the following lemma.

Lemma 3. Let be the solution of (3) with the positive initial condition
Assume that . Then

Proof. In view of (12) and Markov’s inequality there exists a positive constant such that where verifies the conditions of Lemma 3 and is a positive constant such that
An application of Borel-Cantelli lemma yields to for almost and there is a random integer such that for hence for all and Let , then so We shall prove that , and the other limits can be obtained in the same way. Put . In view of the assumption and (11), one can derive that there exists such that Therefore Beside, let such that and . Thanks to Doob’s martingale inequality, and using (37), we getSince , the Borel-cantelli lemma implies that for almost , there is such that for Hence, for all and Since , we have and leads to That isNow, we are in the position to establish the threshold criterion for disease extinction in terms of the positive number:

Theorem 4. Let be the solution of system (3) with any initial value . if and , then we have the following property:

Proof. Applying Itô’s formula leads to and so Integrating (47) and dividing by we obtain and using Lemma 3 yields Hence by condition we have a.s.
Let We know that Then From the fact that we deduce that for any and for almost all there exists such that for all where are almost surely finite positive random variables. By (53) we have Let be the solution of stochastic equation: and we can write So from (53) we have By comparison theorem for stochastic differential equations we have Since is arbitrary, we get From , and (59) we obtain .

4. Stationary Distribution and Positive Recurrence

In many papers, the existence of stationary distribution needs the construction of suitable Lyapunov functions that are based on the positive equilibrium state of the deterministic system, which gives strong sufficient conditions [14]. In the following theorem, to investigate the existence of an asymptotically invariant distribution for the solution of model (3), we did not use equilibrium state to construct Lyapunov functions.

Theorem 5. The solutions are positive recurrent and admit a unique ergodic stationary distribution, provided that the following conditions hold: and

Proof. We construct a positive function such that for some , where
First of all we have Second, we have By using the elementary inequality , where is a positive constant to be choosen later, we have Now we compute By using the elementary inequalities and we have Next, we compute and and Now let and where is a positive constant. It is easy to check that has a minimum point in , and by choosing sufficiently large we can assure the positivity of
Finally consider the positive function: Combining (63), (65), (67), (68), (69), and (70) we get Simplification and reorganization of the above inequality give where Since (61), then we can choose such that so, for sufficiently large , we have andFrom (75) we have and by (79) we get and from condition (60) which is equivalent to we have In the following, we will prove (62).
Firstly, since and continuity of , has a maximum point in So, let Similarly let Case 1 (). From (75), we have , hence Case 2 (). From (75), we have , hence Case 3 (). From (75), we have and since , then we can choose sufficiently large such that Hence for a sufficiently large we have Thereby and from (80) and (81) we deduce that, for a sufficiently large , So the condition of Theorem 1 holds.
The diffusion matrix associated with the system (3) is given byIt is easy to see that satisfy the uniform ellipticity condition of Theorem 1.

5. Conclusion and Numerical Simulation

In the current paper, a stochastic SIR epidemic model with nonlinear relapse rate is considered to model the influence of infected individuals on the recovered ones. The threshold which determines the dynamical behavior of the system (3) is found. Precisely, if the disease will die out from the population with the probability one, while leads to the persistence of the disease with a unique positive stationary distribution. We remark that is independent of the relapse coefficient . Hence, relapse under the pressure of the infected individuals has no influence on the dynamics of the stochastic model (3). However, one can remark from numerical simulation that the nonlinear relapse phenomenon plays a crucial role in the speed of the extinction and the spread of the disease in the population. Indeed, in Figure 1, we show that disease dies out quickly as long as is small enough. Furthermore, in Figure 2, we observe that, for higher coefficient , the modes of the invariant stationary distributions of infected individuals become larger.

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.