A stochastic threshold for an epidemic model with isolation and a non linera incidence

COMMUNICATIONS ON doi:10.3934/cpaa.2020110 PURE AND APPLIED ANALYSIS Volume 19, Number 5, May 2020 pp. 2513–2531 A STOCHASTIC THRESHOLD FOR AN EPIDEMIC MODEL WITH ISOLATION AND A NON LINEAR INCIDENCE Tom´ as Caraballo∗ Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Facultad de Matem´aticas, Universidad de Sevilla c/ Tarfia s/n, 41012-Sevilla, Spain Mohamed El Fatini and Idriss Sekkak D´epartement de Math´ematiques, Facult´e des Sciences, Universit´e Ibn Tofail BP 133, K´enitra, Morocco Regragui Taki Universit´e Chouaib Doukkali, EST Sidi Benour, El Jadida, Morocco Aziz Laaribi Universit´e Sultan Moulay Slimane, Facult´e Polydisciplinaire, Beni Mellal, Morocco (Communicated by Hermen Jan Hupkes) Abstract. In this paper, we study a stochastic epidemic model with isolation and nonlinear incidence. In particular, we propose a stochastic threshold for the model without any sharp sufficient assumptions on model parameters as compared to existing works. Firstly, we establish the uniqueness of the global positive solution according to Lyapunov function method. Secondly, we prove stochastic permanence of the solutions. Then, we establish sufficient condition for the extinction. Thirdly, we investigate necessary and sufficient conditions for persistence in mean of the disease. Finally, we provide some numerical simulations to illustrate our theoretical results. 1. Introduction. Diseases are caused by pathogens such as viruses, bacteria, epiphytes or parasites like protozoans and worms. Epidemiology is the study of the distribution and determinants of disease prevalence in populations like natural environment (migration, vegetation, water ways, ...), human environment (infrastructure, intensive agriculture,...), socio-demographic drivers social inequality, prevention,...) and public health systems (healthcare system, food and water quality. One objective of epidemiology is to describe the distribution of the disease, i.e., to find out who has how much of what, where and when. Another objective is to identify the causes or risk factors for diseases in order to find out why everyone does not have the same thing uniformly [8]. A third objective of mathematical modeling in 2010 Mathematics Subject Classification. 92B05, 60J60, 60J65. Key words and phrases. Epidemic model, threshold, stochastic process, stochastic permanence, extinction, persistence in mean. The first author is supported by Ministerio de Ciencia. Innovaci´on y Universidades (Spain), FEDER (European Community) under grant PGC2018-096540-B-I00, and Consejer´ıa de Innovaci´on Ciencia y Empresa de la Junta de Andaluc´ıa (Spain) under grant P12-FQM-1492. The second and the third authors are supported by Ibn Tofail University of K´enitra (Morocco). ∗Corresponding author. 2513 2514 T. CARABALLO, M. FATINI, I. SEKKAK, R. TAKI AND A. LAARIBI epidemiology is to develop models that will assist the decision making process [2]. In order to limit the spread of certain severe diseases, infected people are isolated as a preventive measure, this operation is called quarantine. As a helping hand to build strategies for governments to control the public health, several compartmental epidemic models have been established, to understand the mechanism and the behavior of the isolation tool. Hethcote et al. [9] introduced a deterministic SIQS model where they studied the asymptotic stability of the equilibrium points. The local and global stability of the model were also discussed by Chen [4]. The stability analysis with saturated incidence rate was investigated by Adebimpe et al. [1]. In [29], the authors studied the deterministic and stochastic dynamics of an SIQS epidemic model with non linear incidence. Yang et al. [26,27] studied a deterministic SIQS model with saturated incidence ψ(S, I) = βSI 1+rI , where βis the disease transmission coefficient, and ris the saturation constant. The term βI represents the infection force of the disease and 1 1+rI represents the inhibition effect from the behavioral change of the susceptible individuals when their number increases or from the crowding effect of the infective individuals. The resulting system is ˙ S=A−ψ(S, I)−µS +γI +εQ, ˙ I=ψ(S, I)−(µ+α2+δ+γ)I, (1) ˙ Q=δI −(µ+α3+ε)Q, where S(t), I(t) and Q(t) are the population fractions of susceptible, infective and quarantined. The parameters A, β, γ, ε, α2, α3, δ, are positive constants. The constant Ais the recruitment rate of susceptible corresponding to births and immigration, µis the natural death rate of population, δis the rate constant for individuals leaving the compartment Ifor the quarantine compartment Q,α2and α3are the disease-related death rate constant in compartments Iand Qrespectively, γis the rate at which individuals recover and return to Susceptible Sfrom compartment Iand εis the rate at which individuals recover and return to Susceptible Sfrom compartment Q. The corresponding basic reproduction number [9] related to (1) is given by R0=Aβ µ(µ+α2+δ+γ). In deterministic models, the output of the model is fully determined by the parameter values and the initial conditions. In the real world, epidemic dynamics is inevitably perturbed by the environmental noise, see references for white noise [3,7,17,24,20] and [6,31] for L´evy noise. Stochastic models possess some inherent randomness. The same set of parameter values and initial conditions will lead to an ensemble of different outputs. In this work, we are interested in a stochastic model perturbed by white noise. In the following, B(R3 +) denotes the Borel σ-algebra on R3 +,and (R3 +,B(R3 +),{Ft}t≥0,P) is a complete probability space with a filtration {Ft}t≥0satisfying the usual conditions (i.e., it is rightly continuous and increasing while F0contains all P-null sets). Then, to characterize the effects of stochastic noises on the deterministic system (1) and to make it reasonable and realistic, a stochastic model driven by Brownian motion is proposed as follows dS = [A−ψ(S, I)−µS +γI +εQ]dt +σ1SdB1(t), dI = [ψ(S, I)−(µ+α2+δ+γ)I]dt +σ2IdB2(t),(2) A STOCHASTIC THRESHOLD FOR AN EPIDEMIC MODEL WITH ISOLATION 2515 dQ = [δI −(µ+α3+ε)Q]dt +σ3QdB3(t), where B1(t), B2(t), B3(t) are independent Brownian motions, and σ1, σ2, σ3are the intensities of the white noises. See also a recent work [22], which is concerned with the dynamical behavior of a stochastic SIQR epidemic model with both white and telegraph noises. Define RS=R0−σ2 2 2(µ+α2+δ+γ). In [25], Wei and Chen considered an SIQS epidemic model with a saturated incidence rate, and the authors discussed the permanence and ultimately boundedness with random perturbations related to death rates. The stochastic model with bilinear incidence, when r= 0, has been discussed in Zhang et al. [30], where the following behaviors of model solutions, according to the value of the threshold RS, were shown: •If RS<1, then the disease will die out. •If RS>1, then the disease will prevail provided that µ > σ2 2,(H1) where σ2= max(σ2 1, σ2 2, σ2 3). In [31], the authors presented a stochastic model driven by L´evy noise with a mass action incidence and they discussed the existence of a stochastic threshold under the assumption of existence of ρ > 2 such that µ−1 2(ρ−1)σ2−ζ ρ>0.(H2) where ζis a dependent parameter of the intensities of Lvy jumps see [31] Our motivation, is to extend the results introduced in [30,31]. We improve these works by deriving a stochastic threshold which does not impose any condition on the stochastic volatility as (H1) in [30] for white noise, and our analysis can be extended to the L´evy noise case without (H2) hypothesis as in [31] . The rest of this paper is organized as follows. The next section is devoted to the existence and uniqueness of the global positive solution. In the third section, we explore the stochastic boundedness and permanence of the solutions of the stochastic system (2). The fourth section is dedicated to the investigation of the stochastic threshold between the extinction and the persistence in mean of the epidemic. In the last section the presented results are demonstrated by numerical simulations. Finally, some conclusions are stated too. We define the differential operator L,associated with the following general ddimensional stochastic system dX(t) = F(t, X(t))dt+G(t, X(t))dB(t),(3) where F(t, X(t)) is a function in Rddefined in [t0,+∞[×Rd, G(t, X(t)) is a d×m matrix, Fand Gare locally Lipschitz functions in xand B(t) is a d-dimensional Wiener process. Let Sh={x∈Rd:|x|< h}.The differential operator L,acts on 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).(4) 2516 T. CARABALLO, M. FATINI, I. SEKKAK, R. TAKI AND A. LAARIBI 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),(5) where Vt=∂V ∂t , Vx=∂V ∂x1 ,∂V ∂x2 ,..., ∂V ∂xdVxx =∂V 2 ∂xi∂xjd×d .(6) For any X∈R3,the norm |X|,as usual, is given by |X|=pX2 1+X2 2+X2 3. 2. Existence and uniqueness of the global positive solution. In this section, using the Lyapunov analysis method (mentioned in [23]), we shall show that the model (2) has a local positive solution, then we show that this solution is global positive. Denote R3 +=(S, I, Q)∈R3:S > 0, I > 0, Q > 0. Theorem 2.1. For any given initial value (S(0), I(0), Q(0)) ∈R3 +, there is a unique positive solution (S(t), I(t), Q(t)) of model (2)on t≥0and the solution will remain in R3 +with probability 1. Proof. Since the coefficients of the system (2) are locally Lipschitz continuous, for any given initial value (S(0), I(0), Q(0)) ∈R3 +there is a unique local solution positive (S(t), I(t), Q(t)) on t∈[0, τe),where τeis the explosion time. Now, we show that the solution is global, we have only to prove that τe=∞a.s. Consider 0>0 such that S(0) > 0, I(0) > 0, Q(0) > 0,then we define the stopping time, for all  > 0 such that ≤0,as follows τ= inf t∈[0, τe) : S(t)6∈ , 1 or I(t)6∈ , 1 ,or Q(t)6∈ , 1 .(7) Throughout this paper we set inf ∅=∞(∅denotes the empty set). It is clear that, τis increasing as →0. Set τ0= lim →0τ, therefore τ0< τea.s. If τ0=∞a.s. is true, then τe=∞a.s. and (S(t), I(t), R(t)) ∈R3 +a.s. for t≥0.In other words, to complete the proof it is required to show that τ0=∞a.s. If this statement is false, then there exist a pair of constants T > 0 and δ∈(0,1) such that P{τ0≤T}> δ. Thus there is 1>0 such that P{τ≤T} ≥ δ∀≤1.(8) Consider a C2-function V1:R3 +→R+as follows V1(X(t)) = S(t) + 1 S(t)+I(t) + 1 I(t)+Q(t) + 1 Q(t),(9) where X(t) denote (S(t), I(t), Q(t)) Applying Itˆo’s formula and using the expectation (see [14] and [23]) we obtain EV1(X(t∧τ)) = V1(X(0)) + EZt∧τ 0 LV1(X(u))du, (10) where LV1is given by LV1(X(u)) =A−µS −1 S2A−µS −βSI 1 + rI +γI +εQ −(µ+α2+δ+γ)I−1 I2βS(u)I(u) 1 + rI(u)−(µ+α2+δ+γ)I(u) A STOCHASTIC THRESHOLD FOR AN EPIDEMIC MODEL WITH ISOLATION 2517 −(µ+α3+ε)Q−1 Q2(δI −(µ+α3+ε)Q) + σ2 1 S+σ2 2 I+σ2 3 Q ≤A+µ S+β rS +µ+α2+δ+γ I+µ+α3+ε Q+σ2 1 S+σ2 2 I+σ2 3 Q ≤c1V(X(u)),(11) where c1= max(µ+β/r +σ2 1, µ +α2+δ+γ+σ2 2, µ +α3+ε+σ2 3).Substituting (11) in (10) we obtain EV1(X(t∧τε)) ≤V1(X(0)) + EZt∧τε 0 c1A c1 +V1(X(u))du EA c1 +V1(X(t∧τε))≤A c1 +V1(X(0)) + c1Zt 0 EA c1 +V1(X(u))du. Using the Gronwall inequality we obtain for t=T A c1 +E[V1(X(T∧τ))] ≤A c1 +V1(X(0))ec1T.(12) Let Ω={τ≤T}. Then, from (8) we have, P(Ω)≥δ. On the other hand, from (9), we have V1(X(T∧τe)) >0,thus E[V1(X(T∧τ))] =E[1ΩV1(X(T∧τ)] + E1ΩC V1(T∧τ) ≥E[1ΩV1(X(τ))] ,(13) where 1Ωis the characteristic function of Ω. For every ω∈Ωsome component of (X(τ, ω)) is equal to , or 1/, then V1(X(τ)) ≥1 ,(14) Combining (12)-(14) yields A c1 +V1(X(0))ec1T≥E[1ΩV1(X(τ))] ≥1 P(Ω)≥δ  Letting goes to 0 we obtain ∞>A c1 +V1(X(0))ec1T=∞, which is a contradiction, and therefore we must have τe=∞a.s. Consequently S(t), I(t) and Q(t) are positive and global. 3. Stochastic permanence. In this section, we will investigate the stochastic permanence of (2). First of all, we present the following definitions introduced in [13]. Definition 3.1. The solutions of SDE (2) are called stochastically ultimately bounded, if for any ˜ε∈(0,1),there is a positive constant χ(= χ(˜ε)),such that the solution of SDE (2) with any positive initial value has the property that lim sup t→∞ P(|X(t)|> χ)≤˜ε. Definition 3.2. SDE (2) is said to be stochastically permanent if for any ˜ε∈(0,1), there exist positive constants χ(= χ(˜ε)), δ(= δ(˜ε)),such that lim inf t→∞ P(|X(t)| ≤ χ)≥1−˜ε, lim inf t→∞ P(|X(t)| ≥ δ)≥1−˜ε. 2518 T. CARABALLO, M. FATINI, I. SEKKAK, R. TAKI AND A. LAARIBI Theorem 3.3. The solutions of model SDE (2)are stochastically ultimately bounded for any initial value X(0) ∈R3 +. Proof. The total population is denoted by N(t) = S(t)+I(t)+Q(t).Consider a C2 function V2=ϑp 2,where ϑ2(X) = 1 + Nand pa positive real to be chosen below. By Itˆo’s formula we have dV2(t) =pϑp−1 2(t)[A−µN(t)−α2I(t)−α3Q(t)]dt +p(p−1) 2ϑp−2 2(t)[σ2 1S2(t) + σ2 2I2(t) + σ2 3Q2(t)]dt +pϑp−1 2(t)[σ1S(t)dB1(t) + σ2I(t)dB2(t) + σ3Q(t)dB3(t)].(15) Integrating (15) from 0 to tand taking expectations of both sides and using Fubini’s Theorem, we obtain that E(V2(t)) −V2(0) =pZt 0 E(ϑp−1 2(u)[A+µ−µϑ2(u)−α2I(u)−α3Q(u)])du +pZt 0 Ep−1 2ϑp−2 2(u))[σ2 1S2(u) + σ2 2I2(u) + σ2 3Q2(u)]du. Then we have dE(V2(t)) dt =pE(ϑp−1 2(t)[A+µ−µϑ2(t)−α2I(t)−α3Q(t)]) +p(p−1) 2E[ϑp−2 2(t)(σ2 1S2(t) + σ2 2I2(t) + σ2 3Q2(t))]. Let σ2= max(σ2 2, σ2 2, σ2 3) and p > 1.Since I, Q > 0,we obtain dE(V2(t)) dt ≤pp−1 2σ2−µE(ϑp 2(t)) + (A+µ)E(ϑp−1 2(t)). We choose psuch that 1 < p < 1 + 2µ σ2,and using the fact that x7→ E(|.|x)1 xis increasing for all x > 0 (see [23, page 23]). Consequently we have dE(V2(t)) dt ≤p−µ−p−1 2σ2[E(ϑp−1 2(t))] p p−1+ (A+µ)E(ϑp−1 2(t)) ≤pE(ϑp−1 2((t)) (A+µ)−µ−p−1 2σ2[E(ϑp−1 2(t))] 1 p−1 Let, ϑ(t) = E(ϑp−1 2(t)),then we have d(V2(t)) dt ≤pϑ(t)(A+µ)−µ−p−1 2σ2ϑ1 p−1(t). We recall that the solution of ODE defined as follow (dϑ(t) = pϑ(t)h(A+µ)−µ−p−1 2σ2ϑ1 p−1(t)i, ϑ(0) = V2(0) is ϑ(t) = 2µ−(p−1)σ2 2(A+µ)+1 V2(0) −2µ−(p−1)σ2 2(A+µ)e−(A+µ)t−(p−1) , which leads to ϑ(t)→2(A+µ) 2µ−(p−1)σ2p−1 ,as t→ ∞. A STOCHASTIC THRESHOLD FOR AN EPIDEMIC MODEL WITH ISOLATION 2519 Then by a comparison theorem we have lim sup t→∞ E(V2(t)) ≤2(A+µ) 2µ−(p−1)σ2p−1 := L(p), Which implies that there exists a T > 0,such that E(V2(t)) ≤2L(p), t > T. Since the function E(V2(t)) is continuous, we deduce that there exists a constant ` > 0 such that for all t > 0, E(1 + N(t))p=E(V2(t)) ≤`. (16) We choose a constant χsufficiently large such that ` χp<1.Using Chebyshev’s inequality, we have P(|X(t)|> χ)≤ P((1 + N(t))p)> χp)≤1 χpE((1 + N(t))p), and this implies that lim sup t→∞ P(|X(t)|> χ)≤` χp:= ˜ε. (17) Theorem 3.4. The SDE (2)is stochastically permanent for any initial value X(0) ∈R3 +. Proof. Its easy to see that inequality (17) leads to lim inf t→∞ P(|X(t)| ≤ χ)≤1−˜ε. Then to complete the proof we need to show lim inf t→∞ P(|X(t)| ≥ δ)≤1−˜ε. By the Itˆo formula, we have d1 N(u)=−N−2(u)[A−µN(u)−α2I−α3Q(u)]du +N−3(u)[σ2 1S(u)2+σ2 2I(u)2+σ2 3Q(u)2]du −N−2(u)[σ1S(u)dB1(u) + σ2I(u)dB2(u) + σ3Q(u)dB3(u)].(18) Integrating (18) from 0 to t, taking expectations of both sides and using Fubini’s Theorem, we obtain E1 N(u)−1 N(0)=Zt 0 EN−2(u)[−A+µN(u) + α2I(u) + α3Q(u)]du +Zt 0 EN−3(u)[σ2 1S2(u) + σ2 2I2(u) + σ2 3Q2(u)]du. Then we have dE(N−1(t)) dt =EN−2(t)[−A+µN(t) + α2I(t) + α3Q(t)] +EN−3(t)[σ2 1S2(t) + σ2 2I2(t) + σ2 3Q2(t)] ≤−AE(N−2(t)) + (µ+α2+ε+σ2)E(N−1(t)) ≤−A[E(N−1(t))]2+ (µ+α2+ε+σ2)E(N−1(t)) =E(N−1(t))((µ+α2+ε+σ2)−A[E(N−1(t))]). 2520 T. CARABALLO, M. FATINI, I. SEKKAK, R. TAKI AND A. LAARIBI Therefore, letting z(t) = E(N−1(t)), dz(t) dt ≤z(t)[(µ+α2+ε+σ2)−Az(t)]. Recall that the solution of the following ODE (dz(t) dt =z(t)[(µ+α2+ε+σ2)−Az(t)], z(0) = 1 N(0) , is given by z(t) = 1 A µ+α2+ε+σ2+N(0) −A µ+α2+ε+σ2e−(A µ+α2+ε+σ2)t. Therefore lim t→∞ z(t) = µ+α2+ε+σ2 A:= l. Thus by the comparison theorem we obtain lim sup t→∞ E1 N(t)≤l. By the same approach as before we deduce that there is a positive real number `0 such that for all t > 0 E1 N(t)≤`0. Besides, we choose δsufficiently small such that √3δ`0<1.By Chebyshev’s inequality and using the inequality N2≤3|X|2we deduce that P(|X| ≤ δ) = P1 |X|≥1 δ≤δE1 |X(t)|≤δE √3 N(t)!≤√3δ`0:= ˜ε. Hence lim sup t→∞ P(|X|< δ)≤˜ε, which implies that lim inf t→∞ P(|X| ≥ δ)≥1−˜ε. 4. Investigation of a stochastic threshold. In this section, we investigate a stochastic threshold between the extinction and the persistence in mean of the stochastic system (2). Lemma 4.1. Let (S(t), I(t), Q(t)) be a solution of system (2)with any initial value (S(0), I(0), Q(0)) ∈R3 +,then lim t→∞ S(t) t= lim t→∞ I(t) t= lim t→∞ Q(t) t= 0 a.s. (19) Proof. From (15) and for all psuch that 1 <p<1 + 2µ σ2,we can derive that d(V2(s)) ≤pAϑp−1 2(s) + (p−1)σ2 2ϑp 2(s)ds pϑp−1 2(s)[σ1S(s)dB1(s) + σ2I(s)dB2(s) + σ3Q(s)dB3(s)], A STOCHASTIC THRESHOLD FOR AN EPIDEMIC MODEL WITH ISOLATION 2521 which implies that Esup k≤u≤k+1 V2(u)≤EV2(k) + pZk+1 k EAϑp−1 2(s) + (p−1)σ2 2ϑp 2(s)ds +pEsup k≤u≤k+1 Zu k ϑp−1 2(s)[σ1S(s)dB1(s) + σ2I(s)dB2(s) + σ3Q(s)dB3(s). (20) Let abe a positive real number such that 1 −4apσ2>1 2.By Burkholder-Davis- Gundy’s inequality (see [23]) and H¨older’s inequality, we derive that Esup k≤u≤k+1 Zu k ϑp−1 2(s)(σ1S(s)dB1(s) + σ3I(s)dB2(s) + σ3Q(s)dB3(s)) ≤4√2"Zk+1 kϑ2(p−1) 2(s)[σ2 1S2(s) + σ2 2I2(s) + σ2 3Q2(s)]ds#1 2 ≤4√2σ2E"Zk+1 k ϑ2p 2(s)ds#1 2 ≤4√2σ2E"sup k≤s≤k+1 ϑp 2(s)Zk+1 k ϑp 2(s)ds#1 2 ≤4σ2E asup k≤s≤k+1 ϑp 2(s)2 + 1 aZk+1 k ϑp 2(s)ds!2  1 2 ≤4aσ2Esup k≤s≤k+1 ϑp 2(s)+4 aσ2Zk+1 k Eϑp 2(s)ds = 4aσ2Esup k≤s≤k+1 V2(s)+4 aσ2Zk+1 k EV2(s)ds. This together with (20) and (16) means that there is a constant H > 0 such that Esup k≤u≤k+1 V2(u)≤2`1 + pA+p−1 2σ2+4 aσ2:= H, k = 1 ,2, . . . Let  > 0 be arbitrary. Then, by Chebyshev’s inequality, we have Psup k≤u≤k+1 V2(u)> k1+≤H k1+, k = 1 ,2, . . . Now, the Borel-Cantelli Lemma implies that, with probability 1, we have sup k≤u≤k+1 V2(u)≤k1+, for all but finitely many k. Thus for almost all ω∈Ω, there exists a positive integer k0(ω) such that sup k≤u≤k+1 V2(u)≤k1+,whenever k≥k0. Hence, for almost all ω∈Ω,if k≥k0and k≤t≤k+ 1,we obtain log(N(t))p log(t)≤log(1 + N(t))p log(t)≤(1 + ) log(k) log(k)= 1 + . 2528 T. CARABALLO, M. FATINI, I. SEKKAK, R. TAKI AND A. LAARIBI 0 500 1000 1500 2000 2500 3000 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 x 105 time Susceptible deterministic stochastic 0 500 1000 1500 2000 2500 3000 0 1000 2000 3000 4000 5000 6000 time Infected deterministic stochastic 0 500 1000 1500 2000 2500 3000 0 50 100 150 200 250 300 350 400 450 500 time Quarantned deterministic stochastic Figure 3. Paths of stochastic and deterministic systems as given in Example 3 Theorem 4.3, all positive solutions of the system (2) if Rs<1 the disease extinct a.s. Fig 5a clearly support the theoretical result. The presented illustration for the extinction of the Pneumococcus disease shows the impact of the quarantine measurement upon the time of the stochastic epidemic model. Thus the adopted strategy of increasing the isolation measure will lead to a quick extinction of the disease. Example 6. In this example, we choose step ∆ = 0.01 with initial values S(0) = 156015, I(0) = 1990, Q(0) = 995 and the parameters r= 0, α2= 0.001, α3= 0, δ = [0 0.005 0.01], σ1= 0.01, σ2= 0.02, σ3= 0.01.By computation, we get that, R0= [2,004 1,6227 1.3650] and RS= [1.9911 1.6152 1.3586]. Hence, according to Theorem 4.6, all positive solutions of the system (2) are persistent in mean in both cases of no quarantine measure taken and the use of isolation with different rate of individuals leaving the infected compartment I to the quarantine Q. Fig 5b clearly support the theoretical result. The performed simulation shows the role of the isolation measurement to maintain the persistence of the disease under a reasonable number of infected for a population of children under two years in Scotland. In other words, as the adoption of the quarantine method has been proven efficient for the limitation of the disease spread. A STOCHASTIC THRESHOLD FOR AN EPIDEMIC MODEL WITH ISOLATION 2529 0 500 1000 1500 2000 2500 3000 1.46 1.48 1.5 1.52 1.54 1.56 1.58 1.6 1.62 1.64 x 105 time Susceptible deterministic stochastic 0 500 1000 1500 2000 2500 3000 0 1000 2000 3000 4000 5000 6000 time Infected deterministic stochastic 0 500 1000 1500 2000 2500 3000 0 50 100 150 200 250 300 350 400 450 500 time Quarantned deterministic stochastic Figure 4. Paths of stochastic and deterministic systems as given in Example 4 0 50 100 150 200 250 300 0 1 2 3 4 5 6 7 8x 104 time Infected δ= 0.00090 δ= 0.00500 δ= 0.01500 (a) Paths of stochastic and deterministic systems as given in Example 5 0 500 1000 1500 2000 2500 3000 0 2 4 6 8 10 12 x 104 time Infected δ= 0.000 δ= 0.005 δ= 0.010 (b) Paths of stochastic and deterministic systems as given in Example 6 Figure 5. Effect of quarantine Concluding remarks and discussions. In this paper we have established the existence and uniqueness of the global positive solution for a stochastic solution SIQS epidemic model with saturated incidence rate. We showed the solutions are 2530 T. CARABALLO, M. FATINI, I. SEKKAK, R. TAKI AND A. LAARIBI ultimately bounded and stochastic permanence for the stochastic SIQS model.We investigated the extinction and persistence of the disease according to the stochastic threshold RS. Our main motivation was to alleviate the assumptions on the threshold in the extinction and the persistence in mean cases. Throughout this paper, this work provides theoretical and numerical guidance for prevention and control of infectious diseases such as pneumococcus. More precisely the results show that the stochastic threshold Rscompletely determines the global behavior of the stochastic SIQS model; if Rs<1 the diseases goes to extinction; if Rs>1 the disease will persist in mean. 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