# A SIRS Epidemic Model Incorporating Media Coverage with Random Perturbation

Authors: Wenbin Liu
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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].LetS(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
(1)dSdt=b-dS-g(I)S+γR,dIdt=g(I)S-(d+μ+δ)I,dRdt=μI-(d+γ)R,
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 rateg(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:
(2)g(I)=β1-β2Im+I,
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
(3)dSdt=b-dS-(β1-β2Im+I)SI+γR,dIdt=(β1-β2Im+I)SI-(d+μ+δ)I,dRdt=μI-(d+γ)R,
where all the parameters are nonnegative and have the same definitions as before.For model (3), the basic reproduction number
(4)R0=bβ1d(d+μ+δ)
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
(5)b-dS-(β1-β2Im+I)SI+γR=0,(β1-β2Im+I)SI-(d+μ+δ)I=0,μI-(d+γ)R=0
which yields
(6)S*=(d+μ+γ)(m+I*)β1(m+I*)-β2I*,R*=μI*d+γ,H1I*2+H2I*+H3=0,
where
(7)H1=-1d+γ(β1-β2)(γ(d+δ)+d(d+μ+δ)),H2=-dβ1mμd+γ-β1m(d+δ)-bβ2+bβ1(1-1R0),H3=dm(d+μ+δ)(R0-1).
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:
(8)dS=b-dS-(β1-β2Im+I)SI+γR+σ1(S-S*)dB1(t),dI=(β1-β2Im+I)SI-(d+μ+δ)I+σ2(I-I*)dB2(t),dR=μI-(d+γ)R+σ3(R-R*)dB3(t),
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 planeS+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
(9)dNdt=b-dN-δI≤b-dN.
Hence, by integration, we check
(10)N(t)≤bd+(N(0)-bd)e-dt.
Hence,
(11)limt→∞supN(t)≤bd,
which implies the conclusion.Therefore, from biological consideration, we study model (3) in the closed set
(12)Γ={(S,I,R)∈ℝ+3:0<S+I+R≤bd}.Proposition2 is proved in [15] and is here just recalled.Proposition 2.
(i) The disease-free equilibriumE0=(b/d,0,0) is globally asymptotically stable if R0<1 and unstable if R0>1 in the set Γ.
(ii) The endemic equilibriumE*=(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 equilibriumE* of model (3).Theorem 3.
IfR0>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:
(13)dNdt=b-dN-δI,
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:
(14)dNdt=b-dN-δI,dIdt=(β1-β2Im+I)(N-I-R)I-(d+μ+δ)I,dRdt=μI-(d+γ)R,
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:
(15)dxdt=-dx-δy,dydt=(β1-β2I*m+I*)I*×(x-(1+mβ2(d+μ+δ)I*(β1-β2)+β1m)y-z),dzdt=μy-(d+γ)z.
Consider the function
(16)V(x,y,z)=12(k1x2+y2+k2z2),
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
(17)dVdt=k1xxt+yyt+k2zzt=k1x(-dx-δy)+(β1-β2I*m+I*)I*y×(x-(1+mβ2(d+μ+δ)I*(β1-β2)+β1m)y-z)+k2z(μy-(d+γ)z)=-dk1x2-(β1-β2I*m+I*)×(1+mβ2(d+μ+δ)I*(β1-β2)+β1m)×I*y2-k2(d+γ)z2+((I*(β1-β2)+β1m)I*m+I*-k1δ)xy+(k2μ-(I*(β1-β2)+β1m)I*m+I*)yz.
Let us choose k1 and k2 such that
(18)(I*(β1-β2)+β1m)I*m+I*-k1δ=0,k2μ-(I*(β1-β2)+β1m)I*m+I*=0;
then  k1=(I*(β1-β2)+β1m)I*/δ(m+I*)andk2=(I*(β1-β2)+β1m)I*/μ(m+I*).Thus, we have
(19)dVdt=-dk1x2-(β1-β2I*m+I*)×(1+mβ2(d+μ+δ)I*(β1-β2)+β1m)I*y2-k2(d+γ)z2≤0.
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(20)b=5,d=0.02,β1=0.002,β2=0.0018,m=30,δ=0.1,μ=0.05,γ=0.01
and show the stability of the endemic equilibrium E* of model (3). Model (3) becomes
(21)dSdt=5-0.02S-(0.002-0.0018I30+I)SI+0.01R,dIdt=(0.002-0.0018I30+I)SI-(0.02+0.05+0.1)I,dRdt=0.05I-(0.02+0.01)R.
Note that
(22)R0=bβ1d(d+μ+δ)=2.941>1.
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 equilibriumE*=(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 {ℱt}t∈ℝ+ satisfying the usual conditions (i.e., it is right continuous and increasing while ℱ0 contains all 𝒫-null sets).Considering the generaln-dimensional stochastic differential equation
(23)dx(t)=f(x(t),t)dt+φ(x(t),t)dB(t)
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 solutionx(t)=0 of (23) is said to be as follows:(i)
stable in probability if for allɛ>0,
(24)limx0→0𝒫(supt≥0|x(t,x0)|≥ɛ)=0;(ii)
asymptotically stable if it is stable in probability and, moreover,(25)limx0→0𝒫(limt→∞x(t,x0)=0)=1;(iii)
asymptotically stable in the large if it is stable in probability and, moreover, for allx0∈ℝn(26)𝒫(limt→∞x(t,x0)=0)=1.Define the differential operator L associated to (23) by
(27)L=∂∂t+∑i=1n‍fi(x,t)∂∂xi+12∑i,j=1n‍[φT(x,t)φ(x,t)]ij∂2∂xi∂xj.
If L acts on a function V(x,t)∈C2,1(ℝd×(0,∞);ℝ+), then
(28)LV(x,t)=Vt(x,t)+Vx(x,t)f(x,t)+12trace[φT(x,t)Vxx(x,t)φ(x,t)],
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 equilibriumE* of model (8).IfR0>1, stochastic model (8) can center at its endemic equilibrium E*. By the change of variables
(29)u=S-S*,v=I-I*,w=R-R*,
we obtain the following system:
(30)dz(t)=f1(z(t))dt+f2(z(t))dB(t),
where (31)z(t)=(u(t),v(t),w(t))T,f1(z(t))=(-(d+(β1-β2I*I*+m)I*)u-(d+μ+δ-β2mS*I*(I*+m)2)v+γw(β1-β2I*I*+m)I*u-β2mS*I*(I*+m)2vμv-(d+γ)w)f2(z(t))=(σ1u(t)000σ2v(t)000σ3w(t)).It is easy to see that the stability of the endemic equilibriumE* 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 functionV(z,t)∈C2(Ω) satisfying the following inequalities:
(32)K1|z|ω≤V(z,t)≤K2|z|ω,LV(z,t)≤-K3|z|ω,
where ω>0 and Ki(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 thatR0=bβ1/d(d+μ+δ)>1. If the following conditions are satisfied:
(33)σ12<2d,σ32<2(d+γ),2(γ2+μ2)2(d+γ)-σ32<d+μ+δ+β2mθS*I*(I*+m)2,σ22<21+θ(d+μ+δ+β2mθS*I*(I*+m)2-2(γ2+μ2)2(d+γ)-σ32),
where
(34)θ=(2d+μ+δ)(I*+m)((β1-β2)I*+β1m)I*,
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 functionV(u,v,w) as follows:
(35)V(z(t))=12c1(u+v)2+12c2v2+12c3w2∶=V1(z(t))+V2(z(t))+V3(z(t)),
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(36)LV1=c1(u+v)(-du-(d+μ+δ)v+γw)+12c1σ12u2+12c1σ22v2=-c1(d-12σ12)u2-c1(2d+μ+δ)uv-c1(d+μ+δ-12σ22)v2+c1γuw+c1γvw,LV2=c2v((β1-β2I*I*+m)I*u-β2mS*I*(I*+m)2v)+12c2σ22v2=c2(β1-β2I*I*+m)I*uv-c2(β2mS*I*(I*+m)2-12σ22)v2,LV3=c3w(μv-(d+γ)w)+12c3σ32w2=c3μvw-c3(d+γ-12σ32)w2.
Then we have
(37)LV=LV1+LV2+LV3=-c1(d-12σ12)u2-(c1(d+μ+δ)+c2β2mS*I*(I*+m)2-12σ22(c1+c2)c2β2mS*I*(I*+m)2)v2-c2(d+γ-12σ32)w2-(c1(2d+μ+δ)-c2(β1-β2I*I*+m)I*)uv+c1γuw+c3μvw+c1γvw.
Choose c3=c1 and
(38)c1(2d+μ+δ)-c2(β1-β2I*I*+m)I*=0;
then
(39)c2=c1(2d+μ+δ)(I*+m)((β1-β2)I*+β1m)I*=c1θ.
Moreover, using Cauchy inequality to γuw, γvw, and μvw, we can obtain
(40)γuw≤γ2u2d+γ-(1/2)σ32+14(d+γ-12σ32)w2,μvw≤μ2v2d+γ-(1/2)σ32+14(d+γ-12σ32)w2,γvw≤γ2v2d+γ-(1/2)σ32+14(d+γ-12σ32)w2.
Substituting (39) and (40) into (37), yields
(41)LV=-(d-12σ12-c1γ2d+γ-(1/2)σ32)u2-14(3c3-2c1)(d+γ-12σ32)w2-(c1γ2+c3μ2d+γ-1/2σ32c1(d+μ+δ)+c2β2mS*I*(I*+m)2-12σ22(c1+c2)-c1γ2+c3μ2d+γ-(1/2)σ32)v2=-(Au2+Bv2+Cw2),
where
(42)A=d-12σ12-c1γ2d+γ-(1/2)σ32,B=c1(γ2+μ2d+γ-1/2σ32d+μ+δ+β2mθS*I*(I*+m)2-12σ22(1+θ)-γ2+μ2d+γ-(1/2)σ32)C=14c1(d+γ-12σ32).
Let us choose c1 such that
(43)0<c1<1γ2(d-12σ12)(d+γ-12σ32).
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
(44)LV(z(t))≤-λ|z(t)|2.
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:
(45)Sk+1=Sk+(b-dSk-(β1-β2Ikm+Ik)SkIk+γRk)×Δt+σ1(Sk-S*)Δtξk,Ik+1=Ik+((β1-β2Ikm+Ik)SkIk-(d+μ+δ)Ik)×Δt+σ2(Ik-I*)Δtηk,Rk+1=Rk+(μIk-(d+γ)Rk)×Δt+σ3(Rk-R*)Δtζk,
where ξk, ζk, and ηk, 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
(46)dS=5-0.02S-(0.002-0.0018I30+I)SI+0.01R+σ1(S-124.564)dB1(t),dI=(0.002-0.0018I30+I)SI-(0.02+0.05+0.1)I+σ2(I-16.361)dB2(t),dR=0.05I-(0.02+0.01)R+σ3(R-27.269)dB3(t).
Choosing (σ1,σ2,σ3)=(0.025,0.1,0.05) and noting that
(47)R0=bβ1d(d+μ+δ)=2.941>1,σ12=0.0252<2d=0.04,σ32=0.052<2(d+γ)=0.06,σ22=0.12<21+θ(d+μ+δ+β2mθS*I*(I*+m)2-2(γ2+μ2)2(d+γ)-σ32)=0.12739-0.00110.06-0.052=0.108.It is easy to see that all the conditions of Theorem7 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,
(48)σ12=0.0324<0.04,σ32=0.0484<0.06σ22=0.0324<0.12739-0.00110.06-0.222=0.033;
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)).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).
(a)
σ
1
=
0.025
,
σ
2
=
0.1
,
σ
3
=
0.05
(b)
σ
1
=
0.18
,
σ
2
=
0.18
,
σ
3
=
0.22In the above case, if we adoptd=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
(49)R0=bβ1d(d+μ+δ)=6.25>1,σ12=0.0324>2d=0.02,σ32=0.0484>2(d+γ)=0.04.
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 equilibriumE*=(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.



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