A stochastic SIRI epidemic model with relapse and media coverage

Manuscript submitted to Website: http://AIMsciences.org AIMS’ Journals Volume X, Number 0X, XX 200X pp. X–XX A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE Tom´ as Caraballo Departamento de Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla c/ Tarfia s/n, 41012 Sevilla, Spain Mohamed El Fatini Ibn Tofail University, FS, Department of Mathematics, BP 133, K´enitra, Morocco. Roger Pettersson Linnaeus University, Department of Mathematics, 351 95 V¨axj¨o, Sweden Regragui Taki Ibn Tofail University, FS, Department of Mathematics, BP 133, K´enitra, Morocco. (Communicated by the associate editor name) Abstract. This work is devoted to investigate the existence and uniqueness of a global positive solution for a stochastic epidemic model with relapse and media coverage. We also study the dynamical properties of the solution around both disease-free and endemic equilibria points of the deterministic model. Furthermore, we show the existence of a stationary distribution. Numerical simulations are presented to confirm the theoretical results. 1. Introduction. Infectious disease is a major public health problem in the World. Mathematical models have been widely used in providing quantitative predictions and evaluating interventions. Models in systems biology help to explain and understand many aspects of biological processes occurring at viral or population level. In mathematical epidemiology, the compartmental approach of Kermack and McKendrick (1927) uses the deterministic dynamics to make predictions of infectious diseases in a number of different biological contexts. Deterministic models have the advantage of simplicity, as they rely on the law of large numbers to longitudinally describe the average trends in a collection of molecules or viruses or other populations of interest. However, the dynamics of such 2000 Mathematics Subject Classification. 92D25, 34C60, 92B05, 60J60, 60J65. Key words and phrases. Epidemic model, Media coverage, Stochastic process, Stationary distribution, Ergodicity. This work has been partially supported by FEDER and the Spanish Ministerio de Econom´ıa y Competitividad project MTM2015-63723-P, the Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492 and the Faculty of Sciences, Ibn Tofail University. 1 2 A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE collections are inherently stochastic phenomena and there are many reasons why one should use stochastic rather than deterministic modeling to capture their dynamics. Hence, the development of stochastic methods in the study of infectious diseases is of great significance to make predictions that can guide public policy. In this work, we investigate stochastic SIRI models, which consist of systems with three compartments: susceptibles, infectives, and removed, labeled by S, I and R. In such models, susceptibles become infectious, then are removed with temporary immunity and then become infectious again. This recurrence of disease is an important feature of some animal and human diseases, for example tuberculosis including human and bovine and herpes [3,37]. In [32] an epidemic model with relapse, which incorporates bilinear incidence rate and constant total population, was formulated. This system was extended to include nonlinear incidence functions by Moreira and Wang [28]. Blower [3] developed a compartmental model for genital herpes, assuming standard incidence for the disease transmission and constant recruitment rate. A more general SIRI model, formulated as an integrodifferential system with the fraction P(t) of recovered individuals remaining in the recovered class, ttime units after the recovery, expressed in an abstract form, has been proposed and analyzed by van den Driessche and Zou [33], with certain threshold stability results being obtained by particularizing P(t). See also van den Driessche et al. [34] for an analysis of a related SEIRI model. For other works see [10,23,38] and the references therein. A deterministic SIRI disease can be modeled as follows [35]: ˙ S=µ−βSI −µS, ˙ I=βSI −(γ+µ)I+δR, ˙ R=γI −(µ+δ)R. Recently, various mathematical models have been used to investigate the impact of the media coverage. Li and Cui [20], Tchuenche et al. [31], and Sun et al. [30] used deterministic models to investigate the effects of media coverage on the transmission dynamics, by means of the following incidence function g(S, I) = β1−β2I m+ISI, where β1is the contact rate before media alert; the term β2I m+Imeasures 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>0. The half-saturation constant m > 0 reflects the impact of media coverage on the contact transmission. The function I m+Iis a continuous bounded function that takes into account disease saturation or psychological effects. The deterministic SIRS epidemic model can be modelled as follows: dS =Λ−β1−β2I m+ISI −µSdt, dI =β1−β2I m+ISI −(µ+γ)I+δRdt, (1.1) dR = [γI −(µ+δ)R]dt, where, Λ, µ, β1, β2, m, γ, δ are all positive constants, S(t) is the numbers of the individuals susceptible to the disease, I(t) are the infected members and R(t) represents the members who have recovered from the infection. In this model, the parameters have the following features: Λ is the total numbers of the susceptible, µrepresents the natural death rate, γrepresents the rate of A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE 3 recovery from infection and δis the rate of temporary immunity. The basic reproduction number [11] for this model is given by R0=β1Λ µµ+γ−γδ µ+δ. It is a threshold quantity which determines whether an epidemic occurs or the disease simply dies out. In the few past years, epidemic models with stochastic perturbation have attracted much attention (see e.g. [12,13,18,22,40]). For stochastic models with white noise incorporating media coverage (see Cai et al. [6]), where the effects of stochastic dynamics of an SIS model incorporating media coverage was investigated. Lui and Zheng [25] investigated the stochastic disease dynamics of an SIS epidemic model on two patches incorporating media coverage. Zhao et al. [40] studied the basic dynamical features of a stochastic SIR epidemic model incorporating media coverage. In the real world, epidemic dynamics is unavoidably perturbed by some kind of environmental noise. To reveal the effects of stochasticity, many authors have introduced randomness into epidemic models [2,12,13,18,21,22,26,36,41]. Amongst authors who studied SIRI epidemic model, we can cite [19] for an epidemic model perturbed by white noise, [39] for a nonautonomous periodic SIRI model, [8] where authors discussed stability and instability for an epidemic model with relapse, and [?] for a stochastic SIRI model driven by l´evy noise. Then to make model (1.1) more reasonable and realistic, we introduce a corresponding stochastic perturbation as follows:        dS =hΛ−β1−β2I m+ISI −µSidt +σ1SdB1(t), dI(t) = hβ1−β2I m+ISI −(µ+γ)I+δRidt +σ2IdB2(t), dR(t) = [γI −(µ+δ)R]dt +σ3RdB3(t), (1.2) where B1(t), B2(t), B3(t) are independent Brownian motions, and σ1, σ2, σ3are the intensities of the white noises. Notice that this model appears when we assume that the parameter µis affected by some stochastic perturbation in each equation, in other words, we replace µby µ−σi˙ Bi(t) in each equation. The rest of this paper is organized as follows. In the next section we prove existence and uniqueness of a global positive solution. In Section 3 we study asymptotic behavior around the disease-free equilibrium of the deterministic model and we show that the solution of the stochastic model (1.2) oscillates around the disease-free equilibrium of the deterministic model (1.1) if R0⩽1. In Section 4 we show that when R0>1,the solution of stochastic model (1.2) oscillates around the endemic equilibrium of deterministic model (1.1). In Section 5 we prove that the solution of stochastic model (1.2) has a stationary distribution and is ergodic. In the last section we study the persistence in mean. The presented results are illustrated and confirmed by numerical simulations. Throughout this paper, we let (Ω,F,P) be a complete probability space with a filtration {Ft}t⩾0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0contains all P-null sets). In addition, we let Bi(t),(i= 1,2,3),be defined on the above probability space. A general d-dimensional stochastic system is given by dX(t) = F(t, X(t))dt+G(t, X(t))dB(t),(1.3) where F(t, X(t)) is a function in Rddefined in [t0,+∞[×Rd,and G(t, X(t)) is a d×mmatrix, Fand Gare locally Lipschitz functions in xand B(t) is a d-dimensional Wiener process. Let Sh={x∈Rd:|x|< h},where |.|denote the euclidean norm. The differential operator Lacts on 4 A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE a function V∈C1,2(R+×Sh;R+) as follows LV(t, x) = Vt(t, x) + Vx(t, x)F(t, x) + 1 2trace GT(t, x)Vxx(x, t)G(x, t).(1.4) By Itˆo’s formula, if x(t)∈Sh, dV (t, x(t)) = LV(t, x(t))dt+Vx(t, x(t))G(t, x(t))dB(t),(1.5) where Vt=∂V ∂t , Vx=∂V ∂x1 ,∂V ∂x2 ,..., ∂V ∂xdVxx =∂V 2 ∂xi∂xjd×d .(1.6) 2. Existence and uniqueness of the global positive solution. Assume that S(t), I(t) and R(t) in the model (1.2) are population densities of the susceptible, infected, and removed at time trespectively. In this section, using the Lyapunov analysis method ([27]) we shall show that the model (1.2) has a unique local positive solution. Then we show that this solution is global, and obtain the following results Theorem 2.1. For any given initial value (S(0), I(0), R(0)) ∈R3 +, the model (1.2)has a unique global solution (S(t), I(t), R(t)) ∈R3 +for all t≥0almost surely. Proof. Since the drift and the diffusion are locally Lipschitz, then for any given initial value (S(0), I(0), R(0)) ∈R3 +, there is a unique local solution (S(t), I(t), R(t)) for t∈[0, τe),where τeis the explosion time (see [26,27]). To show that this solution is global, we need to show that τe=∞a.s. At first, we prove that S(t), I(t), R(t) do not explode to infinity in a finite time. Let k0>0 be sufficiently large so that S(0), I(0) and R(0) lie within the interval [ 1 k0, k0]. For each integer k≥k0, we define the stopping time: τk= inf t∈[0, τe) : S(t)/∈1 k, kor I(t)/∈1 k, kor R(t)/∈1 k, k, where τkis increasing as k↑ ∞. Set τ∞= lim k→∞ τk, whence, τ∞≤τea.s. Showing that τ∞=∞, means that τe=∞and (S(t), I(t), R(t)) ∈R3 +a.s. If this statement is false, then there exists a pair of constants T > 0 and ε∈(0,1) such that P{τ∞≤T}> ε. Thus there is an integer k1≥k0 such that P{τk≤T}⩾ε, ∀k⩾k1.(2.1) Consider the C2-function V1:R3 +→R+as follows V1(S, I, R) = S−a−alog S a+ (I−1−log(I)) + (R−1−log R).(2.2) Let X(t) = (S(t), I(t), R(t)) then by (1.5) and using the expectation (see [14] and [27]), we obtain for all k⩾k0 EV1(X(T∧τk)) = V1(X(0)) + EZT∧τk 0 LV1(X(u∧τk))du, (2.3) A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE 5 where LV1is given by LV1=1−a SΛ−µS −β1−β2I m+ISI+aσ2 1 2 +1−1 Iβ1−β2I m+ISI −(µ+γ)I+δR+σ2 2 2 +1−1 R[γI −(µ+δ)R] + σ2 3 2 = Λ −µS −aΛ S+aµ +aβ1−β2I m+II+aσ2 1 2−β1−β2I m+IS +2µ+γ+δ−µI −δR I+σ2 2 2−µR −γI R+σ2 3 2 ≤Λ + aµ +aβ1I+aσ2 1 2+ 2µ+γ+δ−µI +σ2 2 2+σ2 3 2, choosing a=µ β1, we obtain LV1≤Λ + aµ +aσ2 1 2+ 2µ+γ+δ+σ2 2 2+σ2 3 2:= K0>0.(2.4) Substituting (2.4) in (2.3) yields 0≤E[V1(X(τk∧T))] ≤V1(X(0)) + K0T. (2.5) Let Ωk={τk⩽T}. By (2.1) we have P(Ωk)⩾ε, for k≥k1.On the other hand, from (2.2), we have V1(X(T∧τk)) ⩾0, thus E[V1(X(T∧τk))] = E[1ΩkV1(X(T∧τk)] + Eh1ΩC kV1(X(T∧τk))i ⩾E[1ΩkV1(X(τk))] ,(2.6) where 1Ais the characteristic function of a Borel-set A. For every w∈Ωksome component of X(τk, w)) = ((S(τk, w), I(τk, w), R(τk, w)) is equal either 1/k or k, so V1(X(τk, w)) ⩾A(k),(2.7) where A(k) = min(f(1, k), f(1,1/k), f(µ/β1, k), f(µ/β1,1/k), f(µ/β2, k), f(µ/β2,1/k)) and f(a, x) = x−a−alog x a.Combining (2.5)-(2.7) yields V1(X(0)) + K0T⩾E[1ΩkV1(X(T∧τk))] ⩾A(k)P(Ωk) ⩾A(k)ε. By taking limit when ktends to ∞we obtain ∞> V1(X(0)) + K0T=∞ which is a contradiction, so we must have τe=∞a.s. Consequently S(t), I(t) and R(t) are positive and the solution to (1.2) is global. The proof is complete. 6 A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE 3. The Dynamic Property Around the Disease-free Equilibrium. In this section we are interested in the behavior of the global positive solution (S(t), I(t), R(t)) around the disease-free equilibrium E0. Theorem 3.1. If R0⩽1and the following condition is satisfied σ2 1< µ, σ2 2<2µ, σ2 3<2µ, then for any given initial condition (S(0), I(0), R(0)) ∈R3 +,the solution of model (1.2)has the property lim sup τ→+∞ 1 tEZt 0"S(u)−Λ µ2 +I2(u) + R2(u)#du ≤K1σ2 1 m,(3.1) where m= min 2(µ−σ2 1)1 + 2µ δ,(2µ−σ2 2)1 + 2µ δ,2µ−σ2 3 and K1= 2 1 + 2µ δΛ µ2 Proof. Define the C2function V2(S, I, R) = ϑ1+xϑ2+yϑ3, with ϑ1=S−Λ µ+I+R2 , ϑ2=S−Λ µ+I2 , ϑ3=I+zR, where x,yand zare positive real constants to be chosen in the following. Using Itˆo formula, we obtain dϑ1=Lϑ1dt + 2 S−Λ µ+I+R[σ1SdB1(t) + σ2IdB2(t) + σ3RdB3(t)] , dϑ2=Lϑ2dt + 2 S−Λ µ+I[σ1SdB1(t) + σ2IdB2(t)],(3.2) dϑ3=Lϑ3dt +σ2IdB2+zσ3RdB3, dV2=dϑ1+xdϑ2+ydϑ3. We have, Lϑ1= 2 S−Λ µ+I+R−µS−Λ µ−µI −µR+ (σ1S)2+ (σ2I)2+ (σ3R)2 =−2µS−Λ µ2 −2µ−σ2 2I2−2µ−σ2 3R2−4µI S−Λ µ −4µR S−Λ µ−4µIR +σ2 1S−Λ µ+Λ µ2 , A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE 7 beside Lϑ2= 2 S−Λ µ+I−µS−Λ µ−(γ+µ)I+δR+ (σ1S)2+ (σ2I)2 =−2µS−Λ µ2 −2(µ+γ)−σ2 2I2−2(γ+ 2µ)IS−Λ µ +2δR S−Λ µ+ 2δIR +σ2 1S2 ≤ −2µS−Λ µ2 −2µ−σ2 2I2−2(γ+ 2µ)IS−Λ µ +2δR S−Λ µ+ 2δIR +σ2 1S−Λ µ+Λ µ2 . Using the inequality (a+b)2≤2a2+ 2b2for all a, b ∈Rwe obtain Lϑ1≤ −2(µ−σ2 1)S−Λ µ2 −2µ−σ2 2I2−2µ−σ2 3R2 −4µI S−Λ µ−4µR S−Λ µ−4µIR + 2 Λσ1 µ2 ,(3.3) and Lϑ2≤ −2(µ−σ2 1)S−Λ µ2 −2µ−σ2 2I2 −2(γ+ 2µ)IS−Λ µ+ 2δR S−Λ µ+ 2δIR + 2 Λσ1 µ2 .(3.4) However Lϑ3=β1−β2I m+ISI −(µ+γ)I+δR +z[γI −(µ+δ)R]. Since S, I > 0, Lϑ3≤β1SI + [γz −(µ+γ)]I+ [δ−z(µ+δ)]R =β1IS−Λ µ+Λβ1 µ−(γ+µ) + γzI+ [δ−z(µ+γ]R. Since R0≤1 we can choose zsuch that δγ µ+δ≤γz ≤(γ+µ)−Λβ1 µ=δγ µ+δ+µ(µ+δ+γ) µ+δ(1 − R0) Using the fact that I, R > 0, Lϑ3≤β1IS−Λ µ.(3.5) 8 A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE By choosing x=2µ δand β1y= 4µ+x(2µ+γ),then combining Eq (3.3)-(3.5), we obtain LV2≤ −2(µ−σ2 1)1 + 2µ δS−Λ µ2 −(2µ−σ2 2)1 + 2µ δI2 −2µ−σ2 3R2+ 2 1 + 2µ δΛσ1 µ2 .(3.6) Substituting m= min 2(µ−σ2 1)1 + 2µ δ,(2µ−σ2 2)1 + 2µ δ,2µ−σ2 3into dV2, dV2⩽−m"S−Λ µ2 +I2+R2#+ 2 1 + 2µ δΛσ1 µ2 + 2σ12Λ µ−S+ 2I+RSdB1(t) +2σ22S−Λ µ+ 2I+RIdB2(t) + σ3S−Λ µ+I+RRdB3(t). By integration we deduce that 0⩽V2(t)⩽V2(0) −mZt 0"S(u)−Λ µ2 +I(u)2+R(u)2#du +M(t) +2 1 + 2µ δΛσ1 µ2 t, (3.7) where M(t) is a martingale defined by M(t) = 2σ1Zt 0S−Λ µ+I+R+xS−Λ µ+ISdB1(t) + 2σ2Zt 0S−Λ µ+I+R+xS−Λ µ+I+yIdB2(t) +σ3Zt 02S−Λ µ+I+R+yzRdB3(t). Using Expectation, dividing both sides of (3.7) by tand letting t→ ∞,it follows that lim sup t→∞ 1 tEZt 0"S−Λ µ2 +I2+R2#ds ⩽2 m1 + 2µ δΛσ1 µ2 a.s. (3.8) The proof is therefore complete. Remark 1. 1. Note that, in general, the disease-free state is not an equilibrium of the stochastic model. Consequently, we mainly estimate the average oscillation around this equilibrium to exhibit whether the disease will die out. In a biological interpretation, one expects any solution to be close to the deterministic equilibrium most of the time if the stochastic effects are small. 2. If R0≤1 and under the conditions of Theorem 3.1 we conclude that the solution fluctuates around the disease-free equilibrium. Furthermore if σ1= 0 then the disease-free equilibrium E0= (Λ/µ, 0,0) is an equilibrium of stochastic model (1.2).In this case, (3.6) becomes LV2≤ −2(µ−σ2 1)1 + 2µ δS−Λ µ2 −(2µ−σ2 2)1 + 2µ δI2 −2µ−σ2 3R2. A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE 9 Then under assumption σ1= 0 and using the Lyapunov method, we deduce the following result If R0⩽1, σ2 2<2µand σ2 3<2µthen the disease-free equilibrium E0= (Λ/µ, 0,0) of system (1.2) is globally asymptotically stable, namely the disease will extinct with probability 1. 3. If σ1=σ2=σ3= 0 then Equation (3.6) becomes LV2≤ −2µ1 + 2µ δS−Λ µ2 −2µ1 + 2µ δI2−2µR2. By Lyapunov’s method, the disease-free equilibrium E0of the deterministic model (1.1) is globally asymptotically stable, if R0⩽1. 4. The Dynamic Property Around the Endemic Equilibrium. In this section we assume that R0>1. Let us first prove that there is only one endemic equilibrium. Indeed, if we let the left-hand side of each differential equation of system (1.1) be zero, the endemic (S⋆, I⋆, R⋆) satisfies S⋆>0, I⋆>0, R⋆>0 and Λ−µS⋆−β1−β2I⋆ m+I⋆S⋆I⋆= 0, β1−β2I⋆ m+I⋆S⋆I⋆−(µ+γ)I⋆+δR⋆= 0,(4.1) γI⋆−(µ+δ)R⋆= 0. This implies that S⋆=γ+µ−δγ µ+δ β1−β2I⋆ m+I⋆ , R⋆=γI⋆ µ+δ. Substituting the above expressions into the first equation of system (4.1), we obtain AI⋆2+BI⋆+C= 0, where A=−µ(β1−β2)1 + γ µ+δ, B= Λ(β1−β2)−µ(µ+mβ1)1 + γ µ+δ, C=mµ2(R0−1) 1 + γ µ+δ. Since R0>1 and β1⩾β2we have A⩽0 and C > 0,hence system (1.1) has a unique endemic equilibrium E⋆= (S⋆, I⋆, R⋆).Then, before studying the behavior of the global positive solution (S(t), I(t), R(t)) of the system (1.2) around the endemic equilibrium E⋆, we define K2and m2as follows    m2= min 1⩽i⩽3(µ−σ2 i) K2= (2µ δ+ 1)(σ2 1(S⋆)2+σ2 2(I⋆)2) + (σ3R⋆)2+ 2µ δ(2µ+γ)+2µ β1−β2I⋆ m+I⋆σ2 2 2I⋆+σ2 3δ(R⋆)2 2γI⋆.(4.2) 16 A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 S Density 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 I Density 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 R Density Figure 3. The kernel density function estimations of S(t), I(t) and R(t) of stochastic system (1.2) at time t= 9000, based on 10000 stochastic simulation with the parameters values given in Example 3 0.8 1 1.2 1.4 1.6 1.8 2 2.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2x 104 S Density 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 2000 4000 6000 8000 10000 12000 14000 16000 I Density 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 2000 4000 6000 8000 10000 12000 14000 16000 R Density Figure 4. The kernel density function estimations of S(t), I(t) and R(t) of stochastic system (1.2) at time t= 9500, based on 10000 stochastic simulation with the parameters values given in example 3 Example 6.4. The impact of media coverage To illustrate the impact of media on the spread of an epidemic, we consider a variation of the parameter of the media coverage β2= 0.01,0.1, and 0.15 with Λ = 0.8, µ = 0.25, β1= 0.18, δ = 0.2, γ = 0.2, m = 1, σ1= 0.01, σ2= 0.04, σ3= 0.3 and initial values S(0) = 0.5, I(0) = 0.4 and R(0) = 0.1.We remark that increasing the media coverage parameter, decreases magnitude of infected individuals (Figure 4). This states that the media coverage can reduce the propagation of the disease among the population. Conclusion. In this work we have established the existence and the uniqueness of a global positive solution for a stochastic SIRI epidemic model and studied its the dynamical behavior. The analysis of the stochastic system shows the dynamical behavior around disease-free and endemic equilibria. This implies, in particular, that when the intensity of the stochastic perturbation is small enough, the epidemic will tend to die out provided that the reproduction number does not exceed a critical level. However, small stochastic disturbance can lead to persistence of the disease, when the reproduction number is larger than a critical level. Numerical results support the conclusions of our theoretical studies. Furthermore, simulations show that the role of media is crucial for reducing incidence. Increasing the media coverage rate reduces the basic reproduction number, and hence A STOCHASTIC SIRI EPIDEMIC MODEL WITH RELAPSE AND MEDIA COVERAGE 17 0 20 40 60 80 100 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 time Infected β2= 0.010 β2= 0.100 β2= 0.150 Figure 5. Paths simulations of I(t) for stochastic model with the parameters values as in Example 4 and β2= 0.01,0.1,0.15 respectively. the magnitude of the infectious individuals in the population. 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