A SIRS Epidemic Model Incorporating Media Coverage with Random Perturbation

Abstract

We investigate the complex dynamics of a SIRS epidemic model incorporating media coverage with random perturbation. We first deal with the boundedness and the stability of the disease—free and endemic equilibria of the deterministic model. And for the corresponding stochastic epidemic model, we prove that the endemic equilibrium of the stochastic model is asymptotically stable in the large. Furthermore, we perform some numerical examples to validate the analytical finding, and find that if the conditions of stochastic stability are not satisfied, the solution for the stochastic model will oscillate strongly around the endemic equilibrium.

1. Introduction

Epidemiology is the study of the spread of diseases with the objective of tracing factors that are responsible for or contribute to their occurrence. Mathematical modeling has become an important tool in analyzing the epidemiological characteristics of infectious diseases and can provide useful control measures. Various models have been used to study different aspects of diseases spreading [1–11].

Let 𝑆(𝑡) be the number of susceptible individuals, 𝐼(𝑡) the number of infective individuals, and 𝑅(𝑡) the number of removed individuals at time 𝑡, respectively. A general SIRS epidemic model can be formulated as 𝑑𝑆(𝐼)𝑑𝑡=𝑏−𝑑𝑆−𝑔𝑆+𝛾𝑅,𝑑𝐼(𝐼)𝑑𝑡=𝑔𝑆−𝑑+𝜇+𝛿𝐼,𝑑𝑅𝑑𝑡=𝜇𝐼−𝑑+𝛾𝑅, (1) where 𝑏 is the recruitment rate of the population, 𝑑 is the natural death rate of the population, 𝜇 is the natural recovery rate of the infective individuals, 𝛾 is the rate at which recovered individuals lose immunity and return to the susceptible class, and 𝛿 is the disease-induced death rate. The transmission of the infection is governed by the incidence rate 𝑔(𝐼)𝑆, and 𝑔(𝐼) is called the infection force.

In modelling of communicable diseases, the incidence rate 𝑔(𝐼)𝑆 has been considered to play a key role in ensuring that the models indeed give reasonable qualitative description of the transmission dynamics of the diseases. Some factors, such as media coverage, density of population, and life style, may affect the incidence rate directly or indirectly [12–18]. It is worthy to note that, during the spreading of severe acute respiratory syndrome (SARS) from 2002 to 2004 and the outbreak of influenza A (H1N1) in 2009, media coverage plays an important role in helping both the government authority make interventions to contain the disease and people response to the disease [12, 18]. And a number of mathematical models have been formulated to describe the impact of media coverage on the transmission dynamics of infectious diseases. Especially, Liu and Cui [15], Tchuenche et al. [17], and Sun et al. [16] incorporated a nonlinear function of the number of infective (2) in their transmission term to investigate the effects of media coverage on the transmission dynamics: 𝑔(𝐼)=𝛽₁−𝛽₂𝐼,𝑚+𝐼 (2) where 𝛽₁ is the contact rate before media alert; the terms 𝛽₂𝐼/(𝑚+𝐼) measure the effect of reduction of the contact rate when infectious individuals are reported in the media. Because the coverage report cannot prevent disease from spreading completely we have 𝛽₁≥𝛽₂. The half-saturation constant 𝑚>0 reflects the impact of media coverage on the contact transmission. The function 𝐼/(𝑚+𝐼) is a continuous bounded function which takes into account disease saturation or psychological effects. Then model (1) becomes 𝑑𝑆𝛽𝑑𝑡=𝑏−𝑑𝑆−₁−𝛽₂𝐼𝑚+𝐼𝑆𝐼+𝛾𝑅,𝑑𝐼=𝛽𝑑𝑡¹−𝛽₂𝐼𝑚+𝐼𝑆𝐼−𝑑+𝜇+𝛿𝐼,𝑑𝑅𝑑𝑡=𝜇𝐼−𝑑+𝛾𝑅, (3) where all the parameters are nonnegative and have the same definitions as before.

For model (3), the basic reproduction number 𝑅₀=𝑏𝛽₁𝑑𝑑+𝜇+𝛿 (4) is the threshold of the system for an epidemic to occur. Model (3) has a the disease-free equilibrium 𝑃₀=(𝑏/𝑑,0,0) which exists for all parameter values. And the endemic equilibrium 𝐸^∗=(𝑆^∗,𝐼^∗,𝑅^∗) of model (3) satisfies 𝑏−𝑑𝑆−𝛽₁−𝛽₂𝐼𝛽𝑚+𝐼𝑆𝐼+𝛾𝑅=0,₁−𝛽₂𝐼𝑚+𝐼𝑆𝐼−𝑑+𝜇+𝛿𝐼=0,𝜇𝐼−𝑑+𝛾𝑅=0 (5) which yields 𝑆^∗=𝑑+𝜇+𝛾𝑚+𝐼^∗𝛽₁(𝑚+𝐼^∗)−𝛽₂𝐼^∗,𝑅^∗=𝜇𝐼^∗,𝐻𝑑+𝛾₁𝐼^∗2+𝐻₂𝐼^∗+𝐻₃=0, (6) where 𝐻₁1=−𝛽𝑑+𝛾¹−𝛽₂𝛾(),𝐻𝑑+𝛿+𝑑𝑑+𝜇+𝛿₂=−𝑑𝛽₁𝑚𝜇𝑑+𝛾−𝛽₁𝑚(𝑑+𝛿)−𝑏𝛽₂+𝑏𝛽₁11−𝑅₀,𝐻₃=𝑑𝑚𝑑+𝜇+𝛿𝑅₀−1. (7) When 𝑅₀>1, we know that 𝐻₁<0, 𝐻₃>0; hence, model (3) has a unique endemic equilibrium 𝐸^∗=(𝑆^∗,𝐼^∗,𝑅^∗). These results of model (3) were studied in [15].

On the other hand, if the environment is randomly varying, the population is subject to a continuous spectrum of disturbances [19, 20]. That is to say, population systems are often subject to environmental noise; that is, due to environmental fluctuations, parameters involved in epidemic models are not absolute constants, and they may fluctuate around some average values. Therefore, many stochastic models for the populations have been developed and studied [21–40]. But, to our knowledge, the research on the dynamics of SIRS epidemic model incorporating media coverage with random perturbation seems rare.

In this paper, our basic approach is analogous to that of Beretta et al. [24]. They assumed that stochastic perturbations were of white noise type, which were directly proportional to distances 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡) from values of 𝑆^∗, 𝐼^∗, and 𝑅^∗, influenced the 𝑆(𝑡), 𝐼(𝑡), and 𝑅(𝑡), respectively. By this method, we formulate our stochastic differential equation corresponding to model (3) as follows: 𝑑𝑆=𝑏−𝑑𝑆−𝛽₁−𝛽₂𝐼𝑚+𝐼𝑆𝐼+𝛾𝑅+𝜎₁𝑆−𝑆^∗𝑑𝐵₁(𝑡),𝑑𝐼=𝛽₁−𝛽₂𝐼𝑚+𝐼𝑆𝐼−𝑑+𝜇+𝛿𝐼+𝜎₂𝐼−𝐼^∗𝑑𝐵₂(𝑡),𝑑𝑅=𝜇𝐼−𝑑+𝛾𝑅+𝜎₃𝑅−𝑅^∗𝑑𝐵₃(𝑡), (8) where 𝜎₁, 𝜎₂, and 𝜎₃ are real constants and known as the intensity of environmental fluctuations; 𝐵₁(𝑡), 𝐵₂(𝑡), and 𝐵₃(𝑡) are independent standard Brownian motions.

The aim of this paper is to consider the stochastic dynamics of model (8). The paper is organized as follows. In Section 2, we carry out the analysis of the dynamical properties of stochastic model (8). And in Section 3, we give some numerical examples and make a comparative analysis of the stability of the model with deterministic and stochastic environments and have some discussions.

2. Mathematical Properties of the Deterministic Model (3)

The following result shows that the solutions for model (3) are bounded and, hence, lie in a compact set and are continuable for all positive time.

Lemma 1. The plane 𝑆+𝐼+𝑅≤𝑏/𝑑 is an invariant manifold of model (3), which is attracting in the first octant.

Proof. Summing up the three equations in (3) and denoting 𝑁(𝑡)=𝑆(𝑡)+𝐼(𝑡)+𝑅(𝑡), we have 𝑑𝑁𝑑𝑡=𝑏−𝑑𝑁−𝛿𝐼≤𝑏−𝑑𝑁. (9) Hence, by integration, we check 𝑁(𝑡)≤𝑏𝑑+𝑁(0)−𝑏𝑑𝑒^−𝑑𝑡. (10) Hence, lim_𝑡→∞(𝑡)≤𝑏sup𝑁𝑑, (11) which implies the conclusion.

Therefore, from biological consideration, we study model (3) in the closed set Γ=(𝑆,𝐼,𝑅)∈ℝ³₊𝑏∶0<𝑆+𝐼+𝑅≤𝑑. (12)

Proposition 2 is proved in [15] and is here just recalled.

Proposition 2. (i) The disease-free equilibrium 𝐸₀=(𝑏/𝑑,0,0) is globally asymptotically stable if 𝑅₀<1 and unstable if 𝑅₀>1 in the set Γ.
(ii) The endemic equilibrium 𝐸^∗=(𝑆^∗,𝐼^∗,𝑅^∗) of model (3) is locally asymptotically stable if 𝑅₀>1 in the set Γ.

Next, we present the following theorem which gives condition for the global asymptotical stability of the endemic equilibrium 𝐸^∗ of model (3).

Theorem 3. If 𝑅₀>1, the endemic equilibrium 𝐸^∗=(𝑆^∗,𝐼^∗,𝑅^∗) of model (3) is globally asymptotically stable in the set Γ.

Proof. By summing all the equations of model (3), we find that the total population size verify the following equation: 𝑑𝑁𝑑𝑡=𝑏−𝑑𝑁−𝛿𝐼, (13) where 𝑁=𝑆+𝐼+𝑅.
It is convenient to choose the variable (𝑁,𝐼,𝑅) instead of (𝑆,𝐼,𝑅). That is, consider the following model: 𝑑𝑁𝑑𝑡=𝑏−𝑑𝑁−𝛿𝐼,𝑑𝐼=𝛽𝑑𝑡¹−𝛽₂𝐼()𝑚+𝐼𝑁−𝐼−𝑅𝐼−𝑑+𝜇+𝛿𝐼,𝑑𝑅𝑑𝑡=𝜇𝐼−𝑑+𝛾𝑅, (14) changing the variables such that 𝑥=𝑁−𝑁^∗, 𝑦=𝐼−𝐼^∗, and 𝑧=𝑅−𝑅^∗, where 𝑁^∗=𝑆^∗+𝐼^∗+𝑅^∗, so model (14) becomes as follows: 𝑑𝑥𝑑𝑡=−𝑑𝑥−𝛿𝑦,𝑑𝑦=𝛽𝑑𝑡¹−𝛽₂𝐼^∗𝑚+𝐼^∗𝐼^∗×𝑥−1+𝑚𝛽₂𝑑+𝜇+𝛿𝐼^∗𝛽₁−𝛽₂+𝛽₁𝑚𝑦−𝑧,𝑑𝑧𝑑𝑡=𝜇𝑦−𝑑+𝛾𝑧. (15) Consider the function 𝑉=1𝑥,𝑦,𝑧2𝑘₁𝑥²+𝑦²+𝑘₂𝑧², (16) where 𝑘₁ and 𝑘₂ are positive constants which will be chosen later. Then the derivative of 𝑉 along the solution for model (15) is given by 𝑑𝑉𝑑𝑡=𝑘₁𝑥𝑥_𝑡+𝑦𝑦_𝑡+𝑘₂𝑧𝑧_𝑡=𝑘₁𝑥−𝑑𝑥−𝛿𝑦+𝛽₁−𝛽₂𝐼^∗𝑚+𝐼^∗𝐼^∗𝑦×𝑥−1+𝑚𝛽₂𝑑+𝜇+𝛿𝐼^∗𝛽₁−𝛽₂+𝛽₁𝑚𝑦−𝑧+𝑘₂𝑧𝜇𝑦−𝑑+𝛾𝑧=−𝑑𝑘₁𝑥²−𝛽₁−𝛽₂𝐼^∗𝑚+𝐼^∗×1+𝑚𝛽₂𝑑+𝜇+𝛿𝐼^∗𝛽₁−𝛽₂+𝛽₁𝑚×𝐼^∗𝑦²−𝑘₂𝑑+𝛾𝑧²+𝐼^∗𝛽₁−𝛽₂+𝛽₁𝑚𝐼^∗𝑚+𝐼^∗−𝑘₁𝛿+𝑘𝑥𝑦₂𝐼𝜇−^∗𝛽₁−𝛽₂+𝛽₁𝑚𝐼^∗𝑚+𝐼^∗𝑦𝑧. (17) Let us choose 𝑘₁ and 𝑘₂ such that 𝐼^∗𝛽₁−𝛽₂+𝛽₁𝑚𝐼^∗𝑚+𝐼^∗−𝑘₁𝑘𝛿=0,₂𝐼𝜇−^∗𝛽₁−𝛽₂+𝛽₁𝑚𝐼^∗𝑚+𝐼^∗=0; (18) then  𝑘₁=(𝐼^∗(𝛽₁−𝛽₂)+𝛽₁𝑚)𝐼^∗/𝛿(𝑚+𝐼^∗)and𝑘₂=(𝐼^∗(𝛽₁−𝛽₂)+𝛽₁𝑚)𝐼^∗/𝜇(𝑚+𝐼^∗).Thus, we have 𝑑𝑉𝑑𝑡=−𝑑𝑘₁𝑥²−𝛽₁−𝛽₂𝐼^∗𝑚+𝐼^∗×1+𝑚𝛽₂𝑑+𝜇+𝛿𝐼^∗𝛽₁−𝛽₂+𝛽₁𝑚𝐼^∗𝑦²−𝑘₂𝑑+𝛾𝑧²≤0. (19) By applying the Lyapunov-LaSalle asymptotic stability theorem [41, 42], the endemic equilibrium 𝐸^∗ of model (3) is globally asymptotically stable. This completes the proof.

Example 4. We now use the parameter values 𝑏=5,𝑑=0.02,𝛽₁=0.002,𝛽₂=0.0018,𝑚=30,𝛿=0.1,𝜇=0.05,𝛾=0.01 (20) and show the stability of the endemic equilibrium 𝐸^∗ of model (3). Model (3) becomes 𝑑𝑆𝑑𝑡=5−0.02𝑆−0.002−0.0018𝐼30+𝐼𝑆𝐼+0.01𝑅,𝑑𝐼=𝑑𝑡0.002−0.0018𝐼()30+𝐼𝑆𝐼−0.02+0.05+0.1𝐼,𝑑𝑅()𝑑𝑡=0.05𝐼−0.02+0.01𝑅. (21) Note that 𝑅₀=𝑏𝛽₁𝑑𝑑+𝜇+𝛿=2.941>1. (22) From Theorem 3, one can therefore conclude that, for any initial values (𝑆(0),𝐼(0),𝑅(0)), the endemic equilibrium 𝐸^∗=(124.564,16.361,27.269) of model (21) is globally stable (see Figure 1).

Figure 1 

The global stability of the endemic equilibrium 𝐸∗=(𝑆∗,𝐼∗,𝑅∗) for model (21) with initial values 𝑆(0)=85, 𝐼(0)=15, and 𝑅(0)=0. The parameters are taken as (20).

3. Stochastic Stability of the Endemic Equilibrium of Model (8)

Throughout this paper, let (Ω,ℱ,𝒫) be a complete probability space with a filtration {ℱ_𝑡}_𝑡∈ℝ+ satisfying the usual conditions (i.e., it is right continuous and increasing while ℱ₀ contains all 𝒫-null sets).

Considering the general 𝑛-dimensional stochastic differential equation 𝑑𝑥(𝑡)=𝑓(𝑥(𝑡),𝑡)𝑑𝑡+𝜑(𝑥(𝑡),𝑡)𝑑𝐵(𝑡) (23) on 𝑡≥0 with initial value 𝑥(0)=𝑥₀, the solution is denoted by 𝑥(𝑡,𝑥₀). Assume that 𝑓(0,𝑡)=0 and 𝜑(0,𝑡)=0 for all 𝑡≥0, so (23) has the solution 𝑥(𝑡)=0. This solution is called the trivial solution.

Definition 5 ( see [43]). The trivial solution 𝑥(𝑡)=0 of (23) is said to be as follows:(i)stable in probability if for all 𝜀>0, lim_𝑥0→0𝒫sup_𝑡≥0𝑥𝑡,𝑥₀≥𝜀=0; (24)(ii)asymptotically stable if it is stable in probability and, moreover, lim_𝑥0→0𝒫lim_𝑡→∞𝑥𝑡,𝑥₀=0=1; (25)(iii)asymptotically stable in the large if it is stable in probability and, moreover, for all 𝑥₀∈ℝ^𝑛𝒫lim_𝑡→∞𝑥𝑡,𝑥₀=0=1. (26)Define the differential operator 𝐿 associated to (23) by 𝜕𝐿=+𝜕𝑡^𝑛_𝑖=1𝑓_𝑖()𝜕𝑥,𝑡𝜕𝑥_𝑖+12^𝑛_𝑖,𝑗=1𝜑^𝑇(𝑥,𝑡)𝜑(𝑥,𝑡)_𝑖𝑗𝜕²𝜕𝑥_𝑖𝜕𝑥_𝑗. (27) If 𝐿 acts on a function 𝑉(𝑥,𝑡)∈𝐶^2,1(ℝ^𝑑×(0,∞);ℝ₊), then 𝐿𝑉(𝑥,𝑡)=𝑉_𝑡(𝑥,𝑡)+𝑉_𝑥()𝑓()+1𝑥,𝑡𝑥,𝑡2trace𝜑^𝑇(𝑥,𝑡)𝑉_𝑥𝑥(𝑥,𝑡)𝜑(𝑥,𝑡), (28) where 𝑇 means transposition. For more definitions of stability we refer to [43].

In the following, we will give the result of the asymptotical stability in the large of the endemic equilibrium 𝐸^∗ of model (8).

If 𝑅₀>1, stochastic model (8) can center at its endemic equilibrium 𝐸^∗. By the change of variables 𝑢=𝑆−𝑆^∗,𝑣=𝐼−𝐼^∗,𝑤=𝑅−𝑅^∗, (29) we obtain the following system: 𝑑𝑧(𝑡)=𝑓₁(𝑧(𝑡))𝑑𝑡+𝑓₂(𝑧(𝑡))𝑑𝐵(𝑡), (30) where 𝑧(𝑡)=(𝑢(𝑡),𝑣(𝑡),𝑤(𝑡))^𝑇,𝑓₁(𝑧(𝑡=−𝛽))𝑑+₁−𝛽₂𝐼^∗𝐼^∗𝐼+𝑚^∗𝛽𝑢−𝑑+𝜇+𝛿−²𝑚𝑆^∗𝐼^∗(𝐼^∗+𝑚)²𝛽𝑣+𝛾𝑤₁−𝛽₂𝐼^∗𝐼^∗𝐼+𝑚^∗𝛽𝑢−²𝑚𝑆^∗𝐼^∗(𝐼^∗+𝑚)²𝑣𝑤𝑓𝜇𝑣−𝑑+𝛾₂(𝑧(𝑡=𝜎))¹𝑢(𝑡)000𝜎₂𝑣(𝑡)000𝜎₃𝑤(𝑡). (31)

It is easy to see that the stability of the endemic equilibrium 𝐸^∗ of model (8) is equivalent to the stability of the trivial solution for model (30).

Before proving the stochastic stability of the trivial solution for model (30), we put forward a Lemma in [44].

Lemma 6 (see [44]). Suppose that there exists a function 𝑉(𝑧,𝑡)∈𝐶²(Ω) satisfying the following inequalities: 𝐾₁|𝑧|^𝜔≤𝑉(𝑧,𝑡)≤𝐾₂|𝑧|^𝜔,𝐿𝑉(𝑧,𝑡)≤−𝐾₃|𝑧|^𝜔, (32) where 𝜔>0 and 𝐾_𝑖(𝑖=1,2,3) is positive constant. Then the trivial solution for model (30) is exponentially 𝜔-stable for all time 𝑡≥0. When 𝜔=2, it is usually said to be exponentially stable in mean square and the trivial solution 𝑥=0 is asymptotically stable in the large.

From the above Lemma, we obtain the following theorem.

Theorem 7. Assume that 𝑅₀=𝑏𝛽₁/𝑑(𝑑+𝜇+𝛿)>1. If the following conditions are satisfied: 𝜎²₁<2𝑑,𝜎²₃,2𝛾<2𝑑+𝛾₂+𝜇²2𝑑+𝛾−𝜎²₃𝛽<𝑑+𝜇+𝛿+²𝑚𝜃𝑆^∗𝐼^∗(𝐼^∗+𝑚)²,𝜎²₂<2𝛽1+𝜃𝑑+𝜇+𝛿+²𝑚𝜃𝑆^∗𝐼^∗(𝐼^∗+𝑚)²−2𝛾²+𝜇²2𝑑+𝛾−𝜎²₃, (33) where 𝐼𝜃=2𝑑+𝜇+𝛿^∗+𝑚𝛽₁−𝛽₂𝐼^∗+𝛽₁𝑚𝐼^∗, (34) then the trivial solution of model (30) is asymptotically stable in the large. And the endemic point 𝐸^∗ of model (8) is asymptotically stable in the large.

Proof. We define the Lyapunov function 𝑉(𝑢,𝑣,𝑤) as follows: 𝑉(𝑧(𝑡=1))2𝑐₁(𝑢+𝑣)²+12𝑐₂𝑣²+12𝑐₃𝑤²:=𝑉₁(𝑧(𝑡))+𝑉₂(𝑧(𝑡))+𝑉₃(𝑧(𝑡)), (35) where 𝑐₁>0, 𝑐₂>0 and 𝑐₃>0 are real positive constants to be chosen later. It is easy to check that inequalities (32) are true.
Furthermore, by the Itô formula, we have 𝐿𝑉₁=𝑐₁()+1𝑢+𝑣−𝑑𝑢−𝑑+𝜇+𝛿𝑣+𝛾𝑤2𝑐₁𝜎²₁𝑢²+12𝑐₁𝜎²₂𝑣²=−𝑐₁1𝑑−2𝜎²₁𝑢²−𝑐₁2𝑑+𝜇+𝛿𝑢𝑣−𝑐₁1𝑑+𝜇+𝛿−2𝜎²₂𝑣²+𝑐₁𝛾𝑢𝑤+𝑐₁𝛾𝑣𝑤,𝐿𝑉₂=𝑐₂𝑣𝛽₁−𝛽₂𝐼^∗𝐼^∗𝐼+𝑚^∗𝛽𝑢−²𝑚𝑆^∗𝐼^∗(𝐼^∗+𝑚)²𝑣+12𝑐₂𝜎²₂𝑣²=𝑐₂𝛽₁−𝛽₂𝐼^∗𝐼^∗𝐼+𝑚^∗𝑢𝑣−𝑐₂𝛽₂𝑚𝑆^∗𝐼^∗(𝐼^∗+𝑚)²−12𝜎²₂𝑣²,𝐿𝑉₃=𝑐₃𝑤𝑤+1𝜇𝑣−𝑑+𝛾2𝑐₃𝜎²₃𝑤²=𝑐₃𝜇𝑣𝑤−𝑐₃1𝑑+𝛾−2𝜎²₃𝑤². (36) Then we have 𝐿𝑉=𝐿𝑉₁+𝐿𝑉₂+𝐿𝑉₃=−𝑐₁1𝑑−2𝜎²₁𝑢²−𝑐₁+𝑐𝑑+𝜇+𝛿²𝛽₂𝑚𝑆^∗𝐼^∗(𝐼^∗+𝑚)²−12𝜎²₂𝑐₁+𝑐₂𝑣²−𝑐₂1𝑑+𝛾−2𝜎²₃𝑤²−𝑐₁2𝑑+𝜇+𝛿−𝑐₂𝛽₁−𝛽₂𝐼^∗𝐼^∗𝐼+𝑚^∗𝑢𝑣+𝑐₁𝛾𝑢𝑤+𝑐₃𝜇𝑣𝑤+𝑐₁𝛾𝑣𝑤. (37) Choose 𝑐₃=𝑐₁ and 𝑐₁2𝑑+𝜇+𝛿−𝑐₂𝛽₁−𝛽₂𝐼^∗𝐼^∗𝐼+𝑚^∗=0; (38) then 𝑐₂=𝑐₁2𝑑+𝜇+𝛿𝐼^∗+𝑚𝛽₁−𝛽₂𝐼^∗+𝛽₁𝑚𝐼^∗=𝑐₁𝜃. (39) Moreover, using Cauchy inequality to 𝛾𝑢𝑤, 𝛾𝑣𝑤, and 𝜇𝑣𝑤, we can obtain 𝛾𝛾𝑢𝑤≤²𝑢²𝑑+𝛾−(1/2)𝜎²₃+141𝑑+𝛾−2𝜎²₃𝑤²,𝜇𝜇𝑣𝑤≤²𝑣²𝑑+𝛾−(1/2)𝜎²₃+141𝑑+𝛾−2𝜎²₃𝑤²,𝛾𝛾𝑣𝑤≤²𝑣²𝑑+𝛾−(1/2)𝜎²₃+141𝑑+𝛾−2𝜎²₃𝑤². (40) Substituting (39) and (40) into (37), yields 1𝐿𝑉=−𝑑−2𝜎²₁−𝑐₁𝛾²𝑑+𝛾−(1/2)𝜎²₃𝑢²−143𝑐₃−2𝑐₁1𝑑+𝛾−2𝜎²₃𝑤²−𝑐₁+𝑐𝑑+𝜇+𝛿²𝛽₂𝑚𝑆^∗𝐼^∗(𝐼^∗+𝑚)²−12𝜎²₂𝑐₁+𝑐₂−𝑐₁𝛾²+𝑐₃𝜇²𝑑+𝛾−(1/2)𝜎²₃𝑣²=−𝐴𝑢²+𝐵𝑣²+𝐶𝑤², (41) where 1𝐴=𝑑−2𝜎²₁−𝑐₁𝛾²𝑑+𝛾−(1/2)𝜎²₃,𝐵=𝑐₁𝛽𝑑+𝜇+𝛿+²𝑚𝜃𝑆^∗𝐼^∗(𝐼^∗+𝑚)²−12𝜎²₂()−𝛾1+𝜃₂+𝜇²𝑑+𝛾−(1/2)𝜎²₃1𝐶=4𝑐₁1𝑑+𝛾−2𝜎²₃. (42) Let us choose 𝑐₁ such that 0<𝑐₁<1𝛾²1𝑑−2𝜎²₁1𝑑+𝛾−2𝜎²₃. (43)
On the other hand, the conditions in (33) are satisfied, so 𝐴, 𝐵, and 𝐶 are positive constants. Let 𝜆=min{𝐴,𝐵,𝐶}; then 𝜆>0. From (41), one sees that 𝐿𝑉(𝑧(𝑡))≤−𝜆|𝑧(𝑡)|². (44) According to Lemma 6, we therefore conclude that the trivial solution of model (30) is asymptotically stable in the large. We therefore have the assertion.

Next, for further studying the effects of noise on the dynamics of model (8), we give some numerical examples to illustrate the dynamical behavior of stochastic model (8) by using the Milstein method mentioned in Higham [45]. In this way, model (8) can be rewritten as the following discretization equations: 𝑆_𝑘+1=𝑆_𝑘+𝑏−𝑑𝑆_𝑘−𝛽₁−𝛽₂𝐼_𝑘𝑚+𝐼_𝑘𝑆_𝑘𝐼_𝑘+𝛾𝑅_𝑘×Δ𝑡+𝜎₁𝑆_𝑘−𝑆^∗√Δ𝑡𝜉_𝑘,𝐼_𝑘+1=𝐼_𝑘+𝛽₁−𝛽₂𝐼_𝑘𝑚+𝐼_𝑘𝑆_𝑘𝐼_𝑘−𝑑+𝜇+𝛿𝐼_𝑘×Δ𝑡+𝜎₂𝐼_𝑘−𝐼^∗√Δ𝑡𝜂_𝑘,𝑅_𝑘+1=𝑅_𝑘+𝜇𝐼_𝑘−𝑑+𝛾𝑅_𝑘×Δ𝑡+𝜎₃𝑅_𝑘−𝑅^∗√Δ𝑡𝜁_𝑘, (45) where 𝜉_𝑘, 𝜁_𝑘, and 𝜂_𝑘, 𝑘=1,2,…,𝑛, are the Gaussian random variables 𝑁(0,1).

The parameters of model (8) are fixed as (20). Then model (8) has the endemic point 𝐸^∗=(124.564,16.361,27.269). And model (8) becomes 𝑑𝑆=5−0.02𝑆−0.002−0.0018𝐼30+𝐼𝑆𝐼+0.01𝑅+𝜎₁(𝑆−124.564)𝑑𝐵₁(𝑡),𝑑𝐼=0.002−0.0018𝐼()𝐼30+𝐼𝑆𝐼−0.02+0.05+0.1+𝜎₂(𝐼−16.361)𝑑𝐵₂(𝑡),𝑑𝑅=0.05𝐼−(0.02+0.01)𝑅+𝜎₃(𝑅−27.269)𝑑𝐵₃(𝑡). (46) Choosing (𝜎₁,𝜎₂,𝜎₃)=(0.025,0.1,0.05) and noting that 𝑅₀=𝑏𝛽₁𝑑𝜎𝑑+𝜇+𝛿=2.941>1,₂₁=0.025²𝜎<2𝑑=0.04,₂₃=0.05²𝜎<2𝑑+𝛾=0.06,₂₂=0.1²<2𝛽1+𝜃𝑑+𝜇+𝛿+²𝑚𝜃𝑆^∗𝐼^∗(𝐼^∗+𝑚)²−2𝛾²+𝜇²2𝑑+𝛾−𝜎²₃=0.12739−0.00110.06−0.05²=0.108. (47)

It is easy to see that all the conditions of Theorem 7 are satisfied, and we can therefore conclude that the endemic point 𝐸^∗ of model (47) is asymptotically stable in the large. The numerical examples shown in Figure 2(b) clearly support these results. To further illustrate the effect of the noise intensity on model (47), we keep all the parameters in (20) unchanged but increase (𝜎₁,𝜎₂,𝜎₃) to (0.18,0.18,0.22). In this case, 𝜎²₁=0.0324<0.04,𝜎²₃𝜎=0.0484<0.06₂₂=0.0324<0.12739−0.00110.06−0.22²=0.033; (48) we can therefore conclude, by Theorem 7, that for any initial value (𝑆(0),𝐼(0),𝑅(0)), the endemic point 𝐸^∗ of model (47) is asymptotically stable in the large (see Figure 2(b)).

(a) 𝜎1=0.025,𝜎2=0.1,𝜎3=0.05

(b) 𝜎1=0.18,𝜎2=0.18,𝜎3=0.22

Figure 2 

The asymptotic behavior of the solutions to the stochastic model (47) around the endemic equilibrium 𝐸∗ with initial values 𝑆(0)=85, 𝐼(0)=15, and 𝑅(0)=0. The parameters are taken as (20).

In the above case, if we adopt 𝑑=0.01 and keep the other parameters unchanged, in this case, model (47) has the endemic point 𝐸^∗=(138.935,26.746,66.864). And it is easy to compute 𝑅₀=𝑏𝛽₁𝑑𝜎𝑑+𝜇+𝛿=6.25>1,₂₁𝜎=0.0324>2𝑑=0.02,₂₃=0.0484>2𝑑+𝛾=0.04. (49) Therefore, the conditions of Theorem 7 are not satisfied, and the solution of model (47) will oscillate strongly around the endemic point 𝐸^∗=(138.935,26.746,66.864), which is not asymptotically stable in the large (see Figure 3).

Figure 3 

The global stability of the endemic equilibrium 𝐸∗=(𝑆∗,𝐼∗,𝑅∗) for model (47) with initial values 𝑆(0)=85, 𝐼(0)=15, and 𝑅(0)=0. The parameters are taken as 𝑏=5, 𝑑=0.01, 𝛽1=0.002, 𝛽2=0.0018, 𝑚=30, 𝛿=0.1, 𝜇=0.05, 𝛾=0.01, 𝜎1=0.18, 𝜎2=0.18, and 𝜎3=0.22.

4. Conclusions and Discussions

In this paper, by using the theory of stochastic differential equation, we investigate the dynamics of a SIRS epidemic model incorporating media coverage with random perturbation. The value of this study lies in two aspects. First, it presents some relevant properties of the deterministic model (3), including boundedness and the stability of the disease-free and endemic points. Second, it verifies the stochastic stability in the large of the endemic equilibrium for the stochastic model (8).

From the theoretical and numerical results, we can know that, when the noise density is not large, the stochastic model (8) preserves the property of the stability of the deterministic model (3). To a great extent, we can ignore the noise and use the deterministic model (3) to describe the population dynamics. However, when the noise is sufficiently large, it can force the population to become largely fluctuating. In this case, we can not use deterministic model (3) but instead stochastic model (8) to describe the population dynamics. Needless to say, both deterministic and stochastic epidemic models have their important roles.

Acknowledgments

The author would like to thank the editors and referees for their helpful comments and suggestions. This research was supported by the National Science Foundation of China (61272018 & 61373005) and Zhejiang Provincial Natural Science Foundation (R1110261).