International Journal of Stochastic Analysis
Volume 2013, Issue 1 431257
Research Article
Open Access

Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate

Jihad Adnani, 

Corresponding Author

Jihad Adnani

Department of Mathematics and Computer Science, Faculty of Sciences Ben M′sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco univh2m.ac.ma

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Khalid Hattaf, 

Corresponding Author

Khalid Hattaf

Department of Mathematics and Computer Science, Faculty of Sciences Ben M′sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco univh2m.ac.ma

Centre Régional des Métiers de l′Education et de la Formation (CRMEF), Derb Ghalef, Casablanca, Morocco crmefcasablanca.org

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Noura Yousfi, 

Noura Yousfi

Department of Mathematics and Computer Science, Faculty of Sciences Ben M′sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco univh2m.ac.ma

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First published: 22 September 2013
https://doi.org/10.1155/2013/431257
Citations: 11
Academic Editor: Josefa Linares-Pérez

Abstract

We investigate a stochastic SIR epidemic model with specific nonlinear incidence rate. The stochastic model is derived from the deterministic epidemic model by introducing random perturbations around the endemic equilibrium state. The effect of random perturbations on the stability behavior of endemic equilibrium is discussed. Finally, numerical simulations are presented to illustrate our theoretical results.

1. Introduction

Many mathematical models have been developed in order to understand disease transmissions and behavior of epidemics. One of the earliest of these models was used by Kermack and Mckendrick [1], by considering the total population into three classes, namely, susceptible (S) individuals, infected (I) individuals, and recovered (R) individuals which is known to us as SIR epidemic model. This SIR epidemic model is very important in today′s analysis of diseases.

The disease transmission process is unknown in detail. However, several authors proposed different forms of incidences rate in order to model this disease transmission process. In this paper, we consider the following model with specific nonlinear incidence rate:
(1)
where A is the recruitment rate of the population, μ is the natural death rate of the population, d is the death rate due to disease, r is the recovery rate of the infective individuals, β is the infection coefficient, and βSI/(1 + α1 S + α2 I + α3 SI) is the incidence rate, where α1, α2, α3 ≥ 0 are constants. It is very important to note that this incidence rate becomes the bilinear incidence rate if α1 = α2 = α3 = 0, the saturated incidence rate if α1 = α3 = 0 or α2 = α3 = 0, the modified saturated incidence rate proposed in [2, 3] when α3 = 0, and Crowley-Martin functional response presented in [4–6] if α3 = α1α2.
On the other hand, environmental fluctuations have great influence on all aspects of real life. The aim of this work is to study the effect of these environmental fluctuations on the model (1). We assume that the stochastic perturbations are of white noise type and that they are proportional to the distances of S and I, respectively. Then, the system (1) will be extended to the following system of stochastic differential equation:
(2)
where S*, I* are the positive points of equilibrium for the corresponding deterministic system (1), Bi  (i  = 1,2) are independent standard Brownian motions, and represent the intensities of Bi, respectively.

The rest of paper is organized as follows. In the next section, we present the stability analysis of our stochastic model (2). In Section 3, we present the numerical simulation to illustrate our result. The conclusion of our paper is in Section 4.

2. Stability Analysis of Stochastic Model

Clearly, the system (1) has a basic reproduction number given by
(3)
Using the results presented by Hattaf et al. in [7], it is easy to show that if R0 ≤ 1, the system (1) has just one disease-free equilibrium Ef(A/μ, 0) which is globally asymptotically stable; otherwise, if R0 > 1, the disease-free equilibrium Ef is still present and is unstable, but there is also a unique positive endemic equilibrium E*(S*, I*), where , I* = (A − μS*)/a with a = μ + d + r, and δ = (β + α2μ − α1a − α3A) 2 + 4α3μ(a + α1A). This endemic equilibrium is globally asymptotically stable.
The system (2) has the same equilibria as the system (1). We assume that R0 ≤ 1, and we discuss the stability of the endemic equilibrium E* of (2). The stochastic system (2) can be centered at its interior endemic equilibrium E* by the changes of the variables as follows:
(4)
Hence, the linearized version corresponding to the stochastic model (2) around E* is given by the following form:
(5)
where
(6)
and the superscript “T” represents transposition.

Clearly the endemic equilibrium E* corresponds to the trivial solution u(t) = 0 in (5).

Let C1,2([0, +∞) × ℝ2; ℝ+) be the family of nonnegative functions W(t, u) defined on [0, +∞) × ℝ2 such that they are continuously differentiable with respect to t and twice with respect to u. From [8], we define the differential operator L for a function W(t, u) ∈ C1,2([0, +∞) × ℝ2; ℝ+) by
(7)
where and .

According to [8], we have the following theorem.

Theorem 1. Suppose that there exists a function W(t, u) ∈ C1,2([0, +∞) ×   ℝ2; ℝ+) satisfying the following inequalities:

(8)
where Ki,   i = 1,2, and p are positive constants. Then, the trivial solution of (5) is exponentially p-stable for t ≥ 0. Moreover, if p = 2, then the trivial solution is also called asymptotically mean square stable and it is globally asymptotically stable in probability.

From Theorem 1, we get the conditions for stochastic asymptotic stability of trivial solution of (5) which are given by the following theorem.

Theorem 2. Assume that R0 > 1, , and hold. Then, the trivial solution of (5) is asymptotically mean square stable.

Proof. We consider the following Lyapunov function:

(9)
where w1 and w2 are nonnegative constants that will be chosen later. It is easy to verify that inequality (8) holds true with p = 2.

By applying the operator L on W(t, u), we get

(10)
If we choose w1S*(1 + α1S*) = w2I*(1 + α2I*), then
(11)
where
(12)
From the assumptions of the theorem, we deduce that bii > 0,   i = 1,2 and |B | > 0. Hence, B is a symmetric positive definite matrix. Let λm denote the minimum of its two positive eigenvalues λ1 and λ2; then, we can easily get
(13)
According to Theorem 1, we conclude that the trivial solution of system (5) is globally asymptotically stable.

3. Numerical Simulations

In this section, we present the numerical simulations to illustrate our theoretical results.

We use the following parameter values: A = 0.9, μ = 0.1, β = 0.5, d = 0.1, r = 0.1, α1 = 0.1, α2 = 0.1, α3 = 0.01, σ1 = 0.1, and σ2 = 0.01. In this case, we have R0 = 7.8947 > 1, , and . By applying Theorem 2, we deduce that the endemic equilibrium E* is globally asymptotically stable. Figure 1 demonstrates the above analysis.

Details are in the caption following the image
Figure 1 (a)
Open in figure viewerPowerPoint
Deterministic and stochastic trajectories of models (1) and (2) with parameter values A = 0.9, μ = 0.1, β = 0.5, d = 0.1, r = 0.1, α1 = 0.1, α2 = 0.1, α3 = 0.01, σ1 = 0.1, and σ2 = 0.01.
Details are in the caption following the image
Figure 1 (b)
Open in figure viewerPowerPoint
Deterministic and stochastic trajectories of models (1) and (2) with parameter values A = 0.9, μ = 0.1, β = 0.5, d = 0.1, r = 0.1, α1 = 0.1, α2 = 0.1, α3 = 0.01, σ1 = 0.1, and σ2 = 0.01.

4. Conclusion

The purpose of this work is to study the effects of the environmental fluctuations on dynamical behavior of a deterministic SIR epidemic model with specific nonlinear incidence rate by considering the white noise perturbation around the endemic equilibrium state. We have shown that our stochastic model is globally asymptotically stable in probability when the intensities of white noise are less than certain threshold of parameters. However, if these intensities of white noise are zero, which meant that there is no environmental stochastic perturbation, then the conditions of Theorem 2 are reduced to the condition R0 > 1, which gives a nonlinear stability condition for the deterministic model (1).

From our analytical and numerical results, we conclude that the main factor that affects the stability of the stochastic model is the intensities of white noise. In addition, our main results extend the corresponding results in paper [3] and those in [9] when the value of the parameter  ′h′ is equal to one into the stochastic model [9].

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