A Stochastic SIR Epidemic System with a Nonlinear Relapse

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 systemwith the fractionP(t) of recovered individuals remaining in the recovered class t units after the recovery. In [5] the displacement of recovered individuals to the infective class due to relapse is given by ηRt. 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 σβRtIt. We consider the following SIR compartmental model in a population of varying size with a bilinear relapse rate.


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.
= (Λ −  1   −     ) ,   = (− ( 2 + )   +     +     ) ,   = (− 3   +   −     ) . (1) 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.Λ is the rate at which new individuals enter the population. is the rate at which the infective individuals become recovered. ∈ [0, 1] is the parameter that measure the intensity of the relapse.The positive constants  1 ,  2 , and  3 satisfying represent 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 [6][7][8][9][10].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   = Λ/( +  + (1/2) 2 ), they showed that if   < 1, the disease will die out from the population with the probability one and the disease will persist if   > 1.In the case of persistence, they proved the existence of a stationary distribution.
Let (Ω, F, {F  } ≥0 , P) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is right continuous and F 0 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,  1  is replaced with  1  +  1  1 ; that is, the rate is perturbed by Gaussian white noise.The rates  2 and  3 are similarly perturbed by an independent Gaussian white noises.Therefore, the corresponding stochastic system to (1) can be described by the Itô equation: where  1  ,  2  , and  3  are independent Brownian motions;  1 ,  2 ,  3 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 ( 0 ,  0 ,  0 ).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).

Preliminaries
Throughout the rest of this paper, we denote In general, consider the −dimensional stochastic differential equation: where : R  → R  ,  : R  → R × , 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 L associated with (5) as follows.For any twice continuously differentiable where ∇V, ∇ 2 V denote the gradient, Hessian of V, respectively.
The following theorem gives a criterion for positive recurrence in terms of Lyapunov function (see [13] (ii) There exists a nonnegative function V : Δ  → R such that V is twice continuously differentiable and that for some  > 0 Moreover, the positive Markov process () has a unique ergodic stationary distribution .That is, if ℎ is a function integrable with respect to the measure , then

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.
() From ( 13) and ( 14) we have, for all  ∈ [,  + 1], where In view of (11) and the continuity of   , there exists  > 0 such that sup So, there exists a positive constant  1 such that Applying Itô's formula leads to From Letting   > 1 we have where  3 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.
Assume that (1/2) 2 <  1 .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   < 1 − 1/.
() Let We know that lim →∞ ( 3 /) = 0. Then lim From the fact that lim sup we deduce that for any  > 0 and for almost all  ∈ Ω there exists () such that for all  ≥ where   ,  = 1, 2, 3 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 ℎ  > 0,   < 1 and (59) we obtain lim →∞   = 0.

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.
First of all we have Second, we have By using the elementary inequality  ≤  2 + 1/4 2 , where  is a positive constant to be choosen later, we have Now we compute By using the elementary inequalities −1/ ≤  log  ≤  2 and  ≤ (1/2)( 2 +  2 ) we have Next, we compute where  is a positive constant.It is easy to check that W 1 (, , ) has a minimum point ( Ŝ, Î, R) in R 3 + , and by choosing  sufficiently large we can assure the positivity of W 2 (, , ).

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 R  which determines the dynamical behavior of the system (3) is found.Precisely, if R  < 1 the disease will die out from the population with the probability one, while R  > 1 leads to the persistence of the disease with a unique positive stationary distribution.We remark that R  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.