Global Analysis of SIRS Epidemic Model With General Incidence Function and Incomplete Recovery Rates Stochastical Model

In this paper, deterministic and stochastic models are developped for a class of SIRS epidemic models. Firstly, The conditions for the existence, local and global stability of the disease-free equilibrium and endemic equilibrium are obtained. Secondly, we built the stochastic model. The populations are computationally simulated under various conditions. Comparisons are made between the deterministic and stochastic model.


Introduction
Historically the mathematical modeling of epidemics has started since the time of Graunt (Graunt, 1662). In fact, Kermack and Mckendric (Kermack, Mckendric, 1927) describe some classical deterministic mathematical models of epidemiology by considering the total population into three classes namely of epidemiology susceptible (S) individuals, infected (I) individuals and recovered (R) individuals which is known to us as SIR epidemic models. This SIR epidemic model is very important in today's analysis of diseases. When the recovered lost immunity we say that we are an SIRS epidemic models.
Epidemic models have been studied by many authors. Most of them are interesting in the formulation of the incidence rate, i.e. the infection rate of susceptible individuals through their contacts with infective (see, for example, (Gao,Chen, Nieto,Torres, 2006), (Kyrychko, Blyuss,2005), (Li, Wang, Wang,Jin, 2007)). In order to model the disease transmission process several authors employ the following incidence functions. The first one is the bilinear incidence rate βS I, where S and I are respectively the number of susceptible and infective individuals in the population, and β is a positive constant ( (Jiang, Wei, 2008, (Zhang, Li, Zhang 2008, (Zhou, Liu, 2003)). The second one is the saturated incidence rate of the form βS I 1 + α 1 S , where α 1 is a positive constant. The effect of saturation factor (refer to α 1 ) stems from epidemic control (tacking appropriate preventive measures)( (Wei,Chen, 2008), (Zhang, Jin, Liu, Zhang, 2008)).
Stochastic differential equation (SDE) model is a natural generalization of ordinary differential equation (ODE) model. SDE became increasingly more popular in mathematical biology ((Allen, 2003), (Gard 1988) and the references therein).
In (Allen, Victory,2003), a SDE model for transmission of schistosomiasis was analyzed. That model assumes that births and deaths are neglected. So, the computational work is involved in a computation of √ Bη that one requires other schemes in which we solve an initial value problem.
In this paper, we consider SIR model of disease transmission that was presented and studied in (Connell, McCuskey, 2010). It is a refinement and generalization of earlier model that used incidence function βS I τ 1 + αI τ . The model given in (Connell, McCuskey, 2010) allows for saturation in the force of infection by using the general incidence function f (S , I).
In this paper, we consider the following SIRS epidemic model described by differential equations.
with initial conditions: where S (t), I(t) and R(t) denote the numbers of susceptible, infective and recovered individuals at time t, respectively, B is the recruitment rate of the population, µ i (i = 1, 2, 3) is the death rate of S (t), I(t) and R(t), respectively, γ is the recovery rate of the infective individuals, f (S , I) is the general incidence function, δ is the rate which the recovered individuals become susceptible again.
The organization of the paper is as follows. In Section 2, We give the positiveness and the boundedness of the different classes, the existence of equilibria is presented, We study the local and global stability of the free-equilibrium point and the global stability of the endemic equilibrium. We construct a Stochastic differential model in Section 3 and we derive an equivalent stochastic model. Section 4 is devoted to describe a numerical method to solve the equivalent stochastic model and numerical simulation. Finally, in Section 5, we end by a conclusion.

Positiveness, Eventual Boundedness
We consider the positiveness of system (1). We have the following basic lemma.
Lemma 1 For any solutions (S(t), I(t), R(t)) of system (1) with the initial conditions (2), and there solutions are bounded.

Existence of Equilibria
For model (1) we introduce the following assumptions. Let us denote by f 1 and f 2 the partial derivatives of f with respect to the first and to the second variable.
Remark 1 R 0 is the basic reproduction number evaluate the average number of new infections generated by a single infected individual in a completely susceptible population.
On the existence of the nonnegative equilibria of model (1), we have the following results: Theorem 1 (1) If R 0 ≤ 1, then model (1) has an unique disease-free equilibrium E 0 .
Proof. Let E = (S , I, R) be an equilibrium point of system (1).
By using the third equation of (1), we get By adding the first and the second equations of the model (1), we have . By using the second equation of (1), we get When R 0 > 1, we have lim I→0 + Φ(I) ≥ 0 so there exists I * ∈]0,Ī[. This implies that system (2) have a unique endemic equilibrium point E * .
2.3 Stability of the Disease-Free Equilibrium for R 0 ≤ 1 In this section, we study the local and global behaviour of the disease-free equilibrium.
Theorem 2 The disease-free equilibrium is locally asymptotically stable if R 0 ≤ 1 Proof. The characteristic equation of linearized system (1) at E 0 gives the following equation, It is exact to check that all solutions λ of equation (4) are a negative real parts.
Indeed, the equation (4) has negative root λ = −µ 3 − δ and other roots are given by By developping (5), we get Since f (S , 0) = 0, we have Since R 0 ≤ 1, we obtain Therefore, by the Routh-Hurwitz criterion all the roots of equation (7) have a negative real parts. This shows that equilibrium E 0 is locally asymptotically stable. This completes the proof.
Theorem 3 The disease-free equilibrium is globally asymptotically stable if R 0 ≤ 1 Proof. The proof is based on using a comparison theorem (Lakshmikantham, Leela, Martynyuk, 1989). Note that the equations of infected components in system (1) can be expressed aṡ So, we deduce that, f 2 (S 0 , 0) − (µ 2 + γ) is negative since R 0 ≤ 1.

Stability of the Endemic Equilibrium for R
In this section, we study the global dynamics for R 0 > 1.We make this additional assumption as in (Guiro, Ngom, Ouedraogo, 2017).
We recall that the endemic equilibrium E * exists if and only if R 0 > 1.
Theorem 4 If R 0 > 1 the endemic equilibrium E * is globally asymptotically stable.
Proof. We consider the system (1) when R 0 > 1, there exists a unique endemic equilibrium E * . We now establish the global asymptotic stability of this endemic equilibrium.
Evaluating both sides of (1) at E * gives Let Thus, V S ≥ 0, V I ≥ 0, V R ≥ 0 with equality if and only if S = S * , I = I * and R = R * . We will study the behaviour of the Lyapunov function We can see that V(t) ≥ 0 with equality if and only if S = S * , I = I * and R = R * .
The derivatives of V S , V I , and V R will be calculated separately and then combined to get the desired quantity dV dt .
Using the first equation of (10) to replace B gives Then, we may write Next, we calculate dV I dt .
Using the second equation of (10) to replace (µ 2 + γ)I * gives After that , we evaluate dV R dt .
Using the last equation of (10) to replace γI * gives Combining equations (13)- (15), by adding and substracting the quantity Since the function g is monotone on each side of point 1 and is minimized at this point 1,H4 implies for all (S , I) ∈ R 2 + with equality only for S = S * , I = I * and R = R * . Hence, the endemic equilibrium E * is the only positively invariant set of the system (1) contained in {(S , I) ∈ R 2 + ; S = S * , I = I * R = R * }. Then, it follows that E * is globally asymptotically stable (Lasalle, 1976).

Stochastic Differential Equation Model
To derive a stochastic model, we apply a similar procedure to that described in (Allen, 1999). Here, we neglect the possibility of multiple events of order (∆t) 2 . The possible changes in the populations over a short time ∆t, concern individual births, deaths and transformation. These changes are produced in Table 1, together with their corresponding probability. Let's denote this change by η = (∆S , ∆I, ∆R) T .
Neglecting terms of the order (∆t) 2 , the mean of system (1) is given by Further, the covariance matrix of system (1) is given by where B 11 = B + µ 1 S + f (S , I) + δR, B 22 = f (S , I) + (µ 2 + γ)I, It has been presented in (Allen, 1999) that the changes η are normally distributed. Then, where γ i ∈ N(0, 1) for i = 1, 2, 3. Furthermore, as ∆t → 0, Y(t) converges strongly to the solution of the stochastic system where Y(t) = (S , I, R) T and W(t) is the three-dimensional Wiener process in (Allen, 1999). The computational of (20) implies the calculation of √ B(Y(t)) at each time step that is difficult.
In the next section, we derive an equivalent stochatic model which seem to be easier to implement.

Equivalent Stochastic Differential Model
In this section, we develop a stochastic model to examine the changes occured on each vector individually. We use the vectors defined in the previous section but here the Poisson processes (P) are used to establish the different probabilities. Then, we have where u 1 ∼ P(B∆t), u 2 ∼ P(µ 1 S ∆t), u 3 ∼ P( f (S , I)∆t), u 4 ∼ P(δR∆t), u 5 ∼ P(µ 2 I∆t), u 6 ∼ P(γI∆t), u 7 ∼ µ 3 R∆t.
where Y(t) and µ are the same as in system (20), W is the seven-dimensional Wiener process and G is defined by where

Numerical Simulations
In this section, computational simulations are given for the stochastic system (24). We use the Euler-Maruyama method to solve the SDE model (24). Let h be a specified time step. The numerical method for system (24) is given by: , for k = 0, 1, 2, ... until the maximum time is reached.
Here, one case of computational simulation were studied. In this case R 0 < 1. In the computation, the functions f is chosen as follows f (S , I) = βS I(mass action).
The   Figure 1 illustrates the deterministic model (1) and the equivalent stochastic model (24) when R 0 < 1. We can see that, in the Figure 1 the trajectory of deterministic and stochastic graphs are approximately the same behaviour. Indeed, the infected extinction is effective if R 0 < 1. Figure 2 illustrates the deterministic model (1) and the equivalent stochastic model (24) when R 0 > 1. In Figure 2, we can see that the determinsitic graph are similar to those of the stochastic graph, the susceptible decrease is effective for this two models when R 0 < 1.

Conclusion
In this paper, an SIR epidemic model with the general incidence function is derived. In the first hand, the global behaviour of the model system was studied. We proved that, if R 0 ≤ 1 holds, then the disease-free equilibrium is globally asymptotically stable, Which implies that the disease fades out from the population. If R 0 > 1, then there exists a unique endemic equilibrium which is globally asymptotically stable, and this implies that the disease will persist in the population.
In a second part of this work, we construct a stochastic models derive from the deterministic models. The behavior of the stochastic models are studied. Computational simulations were presented to make comparison between deterministic and equivalent stochastic models. The behavior of the detrministic and equivalent stochastic models are approximately the same.