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

Authors: Jihad Adnani; Khalid Hattaf; Noura Yousfi
Journal: International Journal of Stochastic Analysis
 (2013)

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## 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.


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## Body

## 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)dSdt=A-μS-βSI1+α1S+α2I+α3SI,dIdt=βSI1+α1S+α2I+α3SI-(μ+d+r)I,
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+α1S+α2I+α3SI) 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)dS=[Λ-μS-βSI1+α1S+α2I+α3SI]dt+σ1(S-S*)dB1,dI=[βSI1+α1S+α2I+α3SI-(μ+d+r)I]dt+σ2(I-I*)dB2,
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 σi2(i=1,2) 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)R0=βA(μ+α1A)(μ+d+r).
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 S*=2μ(a+α1A)/(β+α2μ-α1a-α3A+δ), 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)u1=S-S*,u2=I-I*.
Hence, the linearized version corresponding to the stochastic model (2) around E* is given by the following form:
(5)du(t)=f(u(t))dt+g(u(t))dB(t),
where(6)u(t)=(u1(t),u2(t))T,f(u)=(μ-βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2-βS*(1+α1S*)(1+α1S*+α2I*+α3S*I*)2βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2βS*(1+α1S*)(1+α1S*+α2I*+α3S*I*)2-a)u,g(u)=(σ1u100σ2u2), and the superscript “T” represents transposition.Clearly the endemic equilibriumE* corresponds to the trivial solution u(t)=0 in (5).LetC1,2([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)LW(t,u)=∂W(t,u)∂t+fT(u)∂W(t,u)∂u+12Tr[gT(u)∂2W(t,u)∂u2g(u)],
where ∂W/∂u=(∂W(t,u)/∂u1,∂W(t,u)/∂u2)T and ∂2W(t,u)/∂u2=(∂2W/∂ui∂uj)i,j.According to [8], we have the following theorem.Theorem 1.
Suppose that there exists a functionW(t,u)∈C1,2([0,+∞)×ℝ2;ℝ+) satisfying the following inequalities:
(8)K1|u|p≤W(t,u)≤K2|u|p,LW(t,u)≤-K3|u|p,
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 Theorem1, we get the conditions for stochastic asymptotic stability of trivial solution of (5) which are given by the following theorem.Theorem 2.
Assume thatR0>1, σ12≤2[μ+(βI*(1+α2I*)/(1+α1S*+α2I*+α3S*I*)2)], and σ22≤2(α2I*+α3S*I*)a/(1+α1S*+α2I*+α3S*I*) hold. Then, the trivial solution of (5) is asymptotically mean square stable.Proof.
We consider the following Lyapunov function:(9)W(t,u)=12(w1u12+w2u22),
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 operatorL on W(t,u), we get
(10)LW(t,u)=-[μ+βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2]w1u12-βS*(1+α1S*)(1+α1S*+α2I*+α3S*I*)2w1u1u2+βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2w2u2u1-[a-βS*(1+α1S*)(1+α1S*+α2I*+α3S*I*)2]w2u22+12Tr[gT(u)∂2W(t,u)∂u2g(u)]=-[μ+βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2-12σ12]×w1u12-[βS*(1+α1S*)(1+α1S*+α2I*+α3S*I*)2w1-βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2w2]u1u2-[a-βS*(1+α1S*)(1+α1S*+α2I*+α3S*I*)2-12σ22]×w2u22.
If we choose w1S*(1+α1S*)=w2I*(1+α2I*), then
(11)LW(t,u)=-[μ+βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2-12σ12]×w1u12-[a-βS*(1+α1S*)(1+α1S*+α2I*+α3S*I*)2-12σ22]×w2u22=-uTBu,
where(12)B=([μ+βI*(1+α2I*)(1+α1S*+α2I*+α3S*I*)2-12σ12]w100[(α2I*+α3S*I*)a1+α1S*+α2I*+α3S*I*-12σ22]w2). 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)LW(t,u)≤-λm|u|2.
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, σ12=0.01<2.4539, and σ22=0.0001<0.00033502. By applying Theorem 2, we deduce that the endemic equilibrium E* is globally asymptotically stable. Figure 1 demonstrates the above analysis.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.
(a)
(b)


## 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 Theorem2 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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