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

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: 𝑑𝑆𝑑𝑡=𝐴−𝜇𝑆−𝛽𝑆𝐼1+𝛼₁𝑆+𝛼₂𝐼+𝛼₃,𝑆𝐼𝑑𝐼=𝑑𝑡𝛽𝑆𝐼1+𝛼₁𝑆+𝛼₂𝐼+𝛼₃−𝑆𝐼𝜇+𝑑+𝑟𝐼, (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+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: 𝑑𝑆=Λ−𝜇𝑆−𝛽𝑆𝐼1+𝛼₁𝑆+𝛼₂𝐼+𝛼₃𝑆𝐼𝑑𝑡+𝜎₁𝑆−𝑆^∗𝑑𝐵₁,𝑑𝐼=𝛽𝑆𝐼1+𝛼₁𝑆+𝛼₂𝐼+𝛼₃−𝐼𝑆𝐼𝜇+𝑑+𝑟𝑑𝑡+𝜎₂𝐼−𝐼^∗𝑑𝐵₂, (2) where 𝑆^∗, 𝐼^∗ are the positive points of equilibrium for the corresponding deterministic system (1), 𝐵_𝑖(𝑖=1,2) are independent standard Brownian motions, and 𝜎²_𝑖(𝑖=1,2) represent the intensities of 𝐵_𝑖, 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 𝑅₀=𝛽𝐴𝜇+𝛼₁𝐴.𝜇+𝑑+𝑟 (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 𝐸_𝑓(𝐴/𝜇,0) which is globally asymptotically stable; otherwise, if 𝑅₀>1, the disease-free equilibrium 𝐸_𝑓 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: 𝑢₁=𝑆−𝑆^∗,𝑢₂=𝐼−𝐼^∗. (4) Hence, the linearized version corresponding to the stochastic model (2) around 𝐸^∗ is given by the following form: 𝑑𝑢(𝑡)=𝑓(𝑢(𝑡))𝑑𝑡+𝑔(𝑢(𝑡))𝑑𝐵(𝑡), (5) where𝑢(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^∗2𝑔(𝑢)=𝜎−𝑎𝑢,₁𝑢₁00𝜎₂𝑢₂, (6) and the superscript “𝑇” represents transposition.

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

Let 𝐶^1,2([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)∈𝐶^1,2([0,+∞)×ℝ²;ℝ⁺) by ()=()𝐿𝑊𝑡,𝑢𝜕𝑊𝑡,𝑢𝜕𝑡+𝑓^𝑇(𝑢)𝜕𝑊(𝑡,𝑢)+1𝜕𝑢2Tr𝑔^𝑇(𝑢)𝜕²𝑊(𝑡,𝑢)𝜕𝑢²𝑔(𝑢), (7) where 𝜕𝑊/𝜕𝑢=(𝜕𝑊(𝑡,𝑢)/𝜕𝑢₁,𝜕𝑊(𝑡,𝑢)/𝜕𝑢₂)^𝑇 and 𝜕²𝑊(𝑡,𝑢)/𝜕𝑢²=(𝜕²𝑊/𝜕𝑢_𝑖𝜕𝑢_𝑗)_𝑖,𝑗.

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

Theorem 1. Suppose that there exists a function 𝑊(𝑡,𝑢)∈𝐶^1,2([0,+∞)×ℝ²;ℝ⁺) satisfying the following inequalities: 𝐾₁|𝑢|^𝑝≤𝑊(𝑡,𝑢)≤𝐾₂|𝑢|^𝑝,𝐿𝑊(𝑡,𝑢)≤−𝐾₃|𝑢|^𝑝, (8) where 𝐾_𝑖,  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, 𝜎²₁≤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: 𝑊()=1𝑡,𝑢2𝑤₁𝑢²₁+𝑤₂𝑢²₂, (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 𝐿𝑊(𝑡,𝑢)=−𝜇+𝛽𝐼^∗1+𝛼₂𝐼^∗1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗2𝑤₁𝑢²₁−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^∗2𝑤₂𝑢²₂+12Tr𝑔^𝑇(𝑢)𝜕²𝑊(𝑡,𝑢)𝜕𝑢²𝑔(𝑢)=−𝜇+𝛽𝐼^∗1+𝛼₂𝐼^∗1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗2−12𝜎²₁×𝑤₁𝑢²₁−𝛽𝑆^∗1+𝛼₁𝑆^∗1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗2𝑤₁−𝛽𝐼^∗1+𝛼₂𝐼^∗1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗2𝑤₂𝑢₁𝑢₂−𝑎−𝛽𝑆^∗1+𝛼₁𝑆^∗1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗2−12𝜎²₂×𝑤₂𝑢²₂. (10) If we choose 𝑤₁𝑆^∗(1+𝛼₁𝑆^∗)=𝑤₂𝐼^∗(1+𝛼₂𝐼^∗), then 𝐿𝑊(𝑡,𝑢)=−𝜇+𝛽𝐼^∗1+𝛼₂𝐼^∗1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗2−12𝜎²₁×𝑤₁𝑢²₁−𝑎−𝛽𝑆^∗1+𝛼₁𝑆^∗1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗2−12𝜎²₂×𝑤₂𝑢²₂=−𝑢^𝑇𝐵𝑢, (11) where𝐵=𝜇+𝛽𝐼^∗1+𝛼₂𝐼^∗1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗2−12𝜎²₁𝑤₁00𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗𝑎1+𝛼₁𝑆^∗+𝛼₂𝐼^∗+𝛼₃𝑆^∗𝐼^∗−12𝜎²₂𝑤₂. (12) From the assumptions of the theorem, we deduce that 𝑏_𝑖𝑖>0,𝑖=1,2 and |𝐵|>0. Hence, 𝐵 is a symmetric positive definite matrix. Let 𝜆_𝑚 denote the minimum of its two positive eigenvalues 𝜆₁ and 𝜆₂; then, we can easily get 𝐿𝑊(𝑡,𝑢)≤−𝜆_𝑚|𝑢|². (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, 𝜎²₁=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.

(a)

(b)

Figure 1 

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].