# Dynamical Behaviors of a StochasticSIQR Epidemic Model with Quarantine-Adjusted Incidence

Authors: Zhongwei Cao; Shengjuan Zhou
Journal: Discrete Dynamics in Nature and Society
 (2018)

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## Abstract

We study the dynamics of a stochasticSIQR epidemic disease with quarantine-adjusted incidence in this article. In order to find the sufficient conditions for the ergodicity and extermination of the model, we construct suitable stochastic Lyapunov functions and find the results of the stochasticSIQR epidemic model. From the results, we find that when the white noise is relatively large, the infectious diseases will become extinct; this also shows that the intervention of white noise will play an important part in controlling the spread of the disease.


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## Body

## 1. Introduction

Recently, owing to the negative impact of infectious diseases on population growth, understanding the dynamic behavior of these diseases and predicting what will happened have become an important research topic (see e.g., [1–8]). Therefore, the establishment of mathematical models has become an important method to study the properties of infectious diseases. For more contagious diseases such as smallpox, measles, plague, mumps, and Ebola, the most direct and effective methods of interference are to isolate those who have already been infected, in order to decrease transmissions to susceptible individuals. From then on, one of the famous disease models,SIQR (see [9]), has been established, which can be described as follows:(1)S˙t=Λ-βSIS+I+R-dS,I˙t=βSS+I+R-γ+δ+d+α1I,Q˙t=δI-μ+d+α2Q,R˙t=γI+μQ-dR. In this model, one assumed that the infection is given a permanent immunization after recovery. S is the susceptible individual. When these people are infected with the disease, some enter I compartment, which will be infected; other people can quickly and completely recover and access R compartment. In addition, when the susceptible individual enters I compartment, it may be quarantined directly to enter Q compartment. Before they recover, they all will go into R compartment. Here, the total population of the model varies, because vulnerable parts of the population can be received through birth or immigrants and people will die of natural and disease deaths. Besides, in this model, the incidence given by βSI/S+I+R is the quarantine-adjusted incidence. The total contacts of a susceptible person using this form of incidence are βSI/N-Q=β(N-Q)/N-Q=β [9], so that, during quarantine process, the total number of contacts per day remains at β. From the model, the parameters can be summarized in the following list:Λ is the influx of people into the susceptible person’s compartment.d is the natural death rate of compartmentsS,I,Q, and R.β is transmission coefficient from compartment S to compartment I.γ is the recovery rate of infectious individuals.δ is the isolation rate from I to Q.μ is the recovery rate of isolated individuals.α1 is the disease-caused death rate of infectious individuals.α2 is the disease-caused death rate of isolated individuals.Assume that all parameters are nonnegative parameters. In particular,Λ and d are positive constants.In model (1), the quarantine reproduction number is R0=β/γ+δ+d+α1 [9], which determines whether the disease occurs. If R0≤1, system (1) has a unique disease-free equilibrium P0=(S0,0,0,0)=(Λ/d,0,0,0) and P0 is globally asymptotically stable in invariant set D, where D={S,I,Q,R∈R+4∣S+I+Q+R≤Λ/d}. This reveals that the disease will die out and all people are susceptible to it. If R0>1 and α1=α2=0, then P0 is locally asymptotically stable in the region D and system (1) has only a positive endemic equilibrium P∗=(S∗,I∗,Q∗,R∗) which we can find in [9], where (2)S∗=Λ/dγ+dμ+d+α2+δμμ+d+α2R0γ+δ+d+α1-δ-α1+μδ,I∗=ΛR0-1μ+d+α2μ+d+α2R0γ+δ+d+α1-δ-α1+μδ,Q∗=δΛR0-1μ+d+α2R0γ+δ+d+α1-δ-α1+μδ,R∗=Λ/dR0-1γμ+d+α2+δμμ+d+α2R0γ+δ+d+α1-δ-α1+μδ. In addition, for some parameter values the Hopf bifurcation may occur.In real life, disease systems are often affected by white noise (see [10–17]). So it is important to include the effect of stochastic perturbation in estimation of parameters. In many cases, stochastic systems can better describe the spread of infectious diseases (see [16–18]). For instance, the stochastic model can account for the stochastic infectious contact during latent and infectious period [19]. Compared with the deterministic system, this is more practical (see [10, 20–25]). Reference [10] cleared that the stochastic systems are more adapted to the problem of extinction of the disease. Paper [21] showed that the unique equilibrium in a deterministic model may disappear in corresponding stochastic system due to stochastic fluctuations [3].There are so many methods to introduce stochastic perturbations in this system. From biological perspective, random effects can be expressed in Itô or Stratonovich stochastic integrals [26]. From mathematical perspective, random effects are manifested directly in input parameters which are assumed to have specific probability distributions such as uniform, beta, exponential distribution, and gamma [27]. In this paper, we used the stochasticSIQR epidemic model introduced by the approach of Hethcote et al. [9]; following this approach, we can establish a SDESIQR epidemic model with quarantine-adjusted incidence.Think the effects of random fluctuations out, we assume that the fluctuations in the environment are represented as a parameterβ change to random variable β+σ1B˙1(t) and we assume that random fluctuations are linear perturbations corresponding to the rate of change for each population. Then corresponding to model (1), we can establish the stochastic model:(3)dS=Λ-βSIS+I+R-dSdt-σ1SIS+I+RdB1t+σ2SdB2t,dI=βSIS+I+R-γ+δ+d+α1Idt+σ1SIS+I+RdB1t+σ3IdB3t,dQ=δI-μ+d+α2Qdt+σ4QdB4t,dR=γI+μQ-dRdt+σ5RdB5t, where Bi(t) are independent standard Brownian motions and σi2>0(i=1,2,3,4,5) represent the intensities of the white noise. The other coefficients are the same as system (1).In this part, we will give some theoretical knowledge about stochastic differential equations. First of all, letB(t) be l-dimensional standard Brownian motion which is defined on the complete probability space (Ω,F,{F}t≥0,P) adapted to the filtration {F}t≥0 and let X(t) be l-dimensional Itô’s process on t≥0 with the stochastic differential equation [13](4)dXt=fXt,tdt+gXt,tdBt,Xt0=X0∈Rl. Let V∈C2,1(Rl×[t0,∞];R+); then V(X,t) is again Itô’s process with the stochastic differential equation [13] which can be defined as follows: (5)dVX,t=VtX,t+VXX,tfX,t+12tracegTX,tVXXX,tgX,tdt+VXX,tgX,tdBt, where Vt=∂V/∂t,VX=(∂V/∂x1,…,∂V/∂xl),VXX=(∂2V/∂Xi∂Xj)l×l.Lemma 1.
The Markov processX(t) has a unique ergodic stationary distribution π(·) if there exists a bounded domain D⊂El with regular boundary Γ and if(A1) there is a positive number M such that ∑i,j=1laij(x)ξiξj≥Mξ2,x∈D,ξ∈Rl,(A2) there exists a nonnegative C2-function V that LV is nonpositive for any [25] El∖D.


## 2. Existence and Uniqueness of Positive Solution

In this section, we will study the existence and uniqueness of positive solutions in system (3), which is also the premise of studying the long-term behavior of the model.Theorem 2.
For any initial valueX(0)=(S(0),I(0),Q(0),R(0))∈R+4, there is a unique positive solution X(t)=(S(t),I(t),Q(t),R(t))∈R+4 of system (3) for t≥0 and the solution will maintain in R+4 with probability 1.Proof.
Since the coefficients of system (3) satisfy the local Lipschitz condition [3], then for any initial value (S0,I0,Q0,R0)∈R+4, there is a unique local solution (S(t),I(t),Q(t),R(t)) on [0,τe), where τe is the explosion time [13]. To find that this solution is global, we only need to prove that τe=∞ a.s. Since the following argument is similar to that of [17], here we simply have to prove the difference with it. Let n0>0 which make every component of X(0) sufficiently large in interval [1/n0,n0]. For every integer n>n0, define the stopping time(6)τn=inf⁡t∈0,τe:min⁡St,It,Qt,Rt≤1normax⁡St,It,Qt,Rt≥n,where we set inf⁡Ø=∞ throughout this article. Through the definition, we can get that τn increases with n→∞. Let τ∞=limsupn→∞τn; then τ∞≤τe a.s. Next, we have to prove that τ∞=∞ a.s. If the assertion is against this, then there will be a pair of parameters T>0 and ϵ∈(0,1) so that P{τ∞≤T}>ϵ. Therefore, here exists an integer n1>n0 so that (7)Pτn≤T≥ϵ,n≥n1. Define a C2-function V:R+4→R+ as follows:(8)VS,I,Q,R=S-1-ln⁡S+I-1-ln⁡I+Q-1-ln⁡Q+R-1-ln⁡R.Using Itô’s formula, one yields that (9)
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## 3. Ergodic Properties

In this section, we define(10)R0s=dβd+σ12/2+σ22/2γ+δ+d+α1+σ12/2+σ32/2.Theorem 3.
Assume thatR0s>1; there exists a stationary distribution π(·) and the ergodicity holds for any initial value (S(0),I(0),Q(0),R(0))∈R+4 in system (3).Proof.
Let(11)D=ε1≤S,ε12≤I,ε13≤Q,ε13≤R,S+I+Q+R≤1ε1, where ε1>0 is a sufficiently small constant. In the set R+4∖D, one can find that the following conditions hold:(12)c1Mε1β<1(13)-Λε1+K≤-1,(14)-δε1+K≤-1,(15)-γε1+K≤-1,(16)-dε1+K≤-1, where M>0 satisfying(17)-Mλ+Λ+3d+μ+β+α2+σ12+σ22+σ42+σ522≤-2; here λ=3Λ[(R0s)1/3-1]>0. In order to be more intuitive, we divide R+4∖D into the following five regions: (18)D1=S,I,Q,R∈R+4,0<S<ε1,D2=S,I,Q,R∈R+4,0<I<ε12,S≥ε1,D3=S,I,Q,R∈R+4,0<Q<ε13,I≥ε12,D4=S,I,Q,R∈R+4,0<R<ε13,I≥ε12,D5=S,I,Q,R∈R+4,S+I+Q+R>1ε1. The diffusion matrix of (3) is given by (19)A=σ1SIS+I+R2+σ22S2-σ1SIS+I+R200-σ1SIS+I+R2σ1SIS+I+R2+σ32I20000σ42Q20000σ52R2. Choosing M=min(S,I,Q,R)∈Dσ¯⊂R+4{σ22S2,σ32I2,σ42Q2,σ52R2}, we can get (20)∑i,j=15aijS,I,Q,Rξiξj=σ1SIS+I+R2ξ1-ξ22+σ22S2ξ12+σ32I2ξ22+σ42Q2ξ32+σ52R2ξ42≥σ22S2ξ12+σ32I2ξ22+σ42Q2ξ32+σ52R2ξ42≥Mξ2,S,I,Q,R∈Dσ¯,ξ=ξ1,ξ2,ξ3,ξ4∈R+4. Thus condition (A1) in Lemma 1 holds.
Now, we can define(21)V1=S+I+R+Q-c1ln⁡S-c2ln⁡I, where c1,c2 are the positive constants to be determined. Using Itô’s formula, one gets (22)LV1=Λ-dS+I+R-δI-α1I+μQ+δI-μ+d+α2Q+c1-ΛS+βIS+I+R+d+σ12I22S+I+Q2+σ222+c2γ+δ+d+α1-βSS+I+R+σ12S22S+I+Q2+σ322=-dS+I+R+c1ΛS+c2βSS+I+R+Λ-α1I-dQ-α2Q+c1βIS+I+R+c1d+σ12I22S+I+Q2+σ222+c2γ+δ+d+α1+σ12S22S+I+Q2+σ322<-3dΛβc1c21/3+Λ+c1βIS+I+R+c1d+σ122+σ222+c2γ+δ+d+α1+σ122+σ322. Let (23)c1d+σ122+σ222=c2γ+δ+d+α1+σ122+σ322=Λ.Then (24)c1=Λd+σ12/2+σ22/2,c2=Λγ+δ+d+α1+σ12/2+σ32/2. Thus we can get (25)LV1≤-3dβΛ3d+σ12/2+σ22/2γ+δ+d+α1+σ12/2+σ32/21/3-Λ+c1βIS+I+R≤-3ΛR0s1/3-1+c1βIS+I+R. In addition, we can define (26)V^=MV1-ln⁡S-ln⁡R-ln⁡Q+S+I+Q+R; combined with (17), we can easily find that (27)lims→∞,S,I,Q,R∈R+4∖UsV^S,I,Q,R=+∞; here Us=(1/s,s)×(1/s,s)×(1/s,s)×(1/s,s). Moreover, V^(S,I,Q,R) is also a continuous function [25]. Therefore V^(S,I,Q,R) must have a minimum point [25] (S¯0,I¯0,Q¯0,R¯0) which is inside R+4. Then we define a nonnegative C2-function V: R+4→R+ as follows: (28)VS,I,Q,R=V^S,I,Q,R-V^S¯0,I¯0,Q¯0,R¯0. Combined with the calculation method of stochastic differential, we get (29)LV≤M-3ΛR0s1/3-1+c1βIS+I+R-ΛS+βIS+I+R+d+σ122-δIQ+ε+d+α2+σ322-rIR-εQR+d+σ422+Λ-dS+I+Q+R+α1I+α2Q≤-Mλ+Mc1βIS+I+R+βIS+I+R-ΛS-γIR-εQR-δIQ-dS+I+Q+R-α1I-α2Q+Λ+3d+ε+β+α2+σ12+σ32+σ422≔-Mλ+Mc1βIS+I+R-ΛS-γIR-δIQ-dS+I+Q+R+Λ+3d+ε+β+α2+σ12+σ32+σ422. Next we consider the following five cases. Let K=Mc1β+Λ+3d+μ+β+α2+σ12+σ22+σ42+σ52/2, where K is positive constant.
Case  1 (if (S,I,Q,R)∈D1). We get(30)LV≤K-ΛS≤K-Λε1; combined with (13), we have (31)LV≤-1foranyS,I,Q,R∈D1.
Case  2 (if (S,I,Q,R)∈D2). We can obtain (32)LV≤-Mλ+Mc1βIS+Λ+3d+μ+β+α2+σ12+σ22+σ42+σ522≤-Mλ+Mc1βε12ε1+Λ+3d+μ+β+α2+σ12+σ22+σ42+σ522. Thus one can see that(33)LV≤-Mλ+Mc1βε1+Λ+3d+μ+β+α2+σ12+σ22+σ42+σ522; combined with (12) and (17), for any sufficiently small ε1, we get (34)LV≤-1foranyS,I,Q,R∈D2.
Case  3 (if (S,I,Q,R)∈D3). We find that(35)LV≤-Mλ+Mc1βIS+I+R-ΛS-γIR-δIQ-dS+I+Q+R+Λ+3d+μ+β+α2+σ12+σ22+σ42+σ522≤K-δIQ≤K-δε12ε13≤K-δε1; combined with (14), we have (36)LV≤-1foranyS,I,Q,R∈D3.
Case  4 (if (S,I,Q,R)∈D4). One can see that(37)LV≤-Mλ+Mc1βIS+I+R-ΛS-γIR-δIQ-dS+I+Q+R+Λ+3d+μ+β+α2+σ12+σ22+σ42+σ522≤K-γIR≤K-γε12ε13≤K-γε1; combined with (15), we get (38)LV≤-1foranyS,I,Q,R∈D4.
Case  5 (if (S,I,Q,R)∈D5). We can obtain(39)LV≤K-dS+I+Q+R≤K-dε1; according to (16), one can see that (40)LV≤-1foranyS,I,Q,R∈D5. Obviously, from (30), (33), (35), (37), and (39), we can easily find that for a sufficiently small ε1, (41)LVS,I,Q,R≤-1∀S,I,Q,R∈R+4∖D.
Thus we can easily find that (A2) in Lemma 1 is satisfied. From Lemma 1, we get that model (3) has a uniquely stationary distribution and satisfies ergodicity. Hence, the proof of Theorem 3 has been carried out.


## 4. Disease Extinction

In this section, we will investigate the extinction of disease under the following conditions:(C1)
σ12≤β and R0s^<1, where R0s^=β/γ+δ+d+α1+σ12/2+σ32/2,(C2)
σ12>max⁡{β,β2/2(γ+δ+d+α1+σ32/2)}.Theorem 4.
If(C1) or (C2) holds, the disease I(t) will die out exponentially with probability 1; that is,(42)limt→∞sup⁡ln⁡Itt<0a.s.Proof.
By Itô’s formula, one can easily find that(43)dln⁡It=βSS+I+R-γ+δ+d+α1-σ122SS+I+R2-σ322dt+σ1SS+I+RdB1t+σ3dB3t.
Integrating (43) from 0 to t, one can obtain that (44)ln⁡It-ln⁡I0=∫0tβSuSu+Iu+Ru-γ+δ+d+α1+σ322-σ122SuSu+Iu+Ru2du+∫0tσ1SuSu+Iu+RudB1u+σ3B3u. Let (45)x=SuSu+Iu+Ru,x∈0,1,C=γ+δ+d+α1+σ322. We can obtain (46)βSuSu+Iu+Ru-γ+δ+d+α1+σ322-σ122SuSu+Iu+Ru2=βx-C+σ122x2=fx,(47)f′x=β-σ2x. Next, let f′(x)=0, then one can get x=β/σ12.
Case  1. Suppose that (C1) holds; we can get (48)fx≤f1=β-C+σ122; thus we have(49)ln⁡It-ln⁡I0≤∫0tβ-γ+δ+d+α1+σ322+σ122du+σ1B1t+σ3B3t=γ+δ+d+α1+σ322+σ122R0s^-1t+σ1B1t+σ3B3t.
By the strong law of large numbers (see [8]), we get that(50)limt→∞Bitt=0i=1,3a.s. Both sides simultaneously take maximum of (49) and combining with (50), we have (51)limsupt→∞ln⁡Itt≤γ+δ+d+α1+σ322+σ122R0s^-1<0a.s.; this means that limt→∞I(t)=0a.s.. In other words, the disease I(t) will tend to zero exponentially with probability 1.
Case  2. If (C2) holds, this conclusion is also correct. In fact, by the condition, we can get (52)fx≤fβσ12=β2σ12-C+σ122βσ122=β22σ12-C; thus we have(53)ln⁡It-ln⁡I0≤∫0tβ22σ12-γ+δ+d+α1+σ322du+σ1B1t+σ3B3t=β22σ12-γ+δ+d+α1+σ322t+σ1B1t+σ3B3t. Both sides simultaneously take maximum of (53) and combining with (50), we get (54)limsupt→∞ln⁡Itt≤β22σ12-γ+δ+d+α1+σ322<0a.s.
By system (3), one can easily obtain that when limt→∞I(t)=0a.s., then limt→∞Q(t)=0 and limt→∞R(t)=0a.s.. This completes the proof.Remark 5.
From condition(C1), we can easily find that if R0s^<1 and white noise is small, the disease will be extinct. From condition (C2) we get that if white noise is large enough, the disease will also be extinct. Otherwise, the manifestation of the disease is uncertain, which does not happen in deterministic system (1). Furthermore, we notice that R0s^ in C1 is smaller than the basic reproduction number of system (1).


## 5. Examples and Numerical Simulations

In this section, we will give some numerical examples to illustrate our main results by using Milstein’s Higher Order Method [28].Example 1.
We choose the parameter values in system (3) as follows: Λ=1;d=0.2;β=0.66;γ=0.02;δ=0.1;α1=0.01;μ=0.1;α2=0.2;σ1=0.01;σ2=0.002;σ3=0.2;σ4=0.3;σ5=0.2. By calculation, we can get R0s=1.8850>1. That is to say, the conditions of Theorem 3 are satisfied. In Figure 1, we choose the different initial values to illustrate that wherever S(t),I(t),R(t) and Q(t) start from, the density functions of S(t),I(t),R(t) and Q(t) converge to the same functions, respectively. In Figure 2, selecting different iterations, one can easily find that the density functions of S(t),I(t),R(t), and Q(t) also converge to the same functions, respectively. Hence, Figures 1 and 2 verify Theorem 3 very well so that there exists a unique ergodic stationary distribution μ(·) of system (3). In Figure 3, the blue line and red line are almost the same. This strongly illustrates ergodicity.Figure 1
Based on the 10000 sample paths, after iterating 10,000 times, we get the density functions ofS(t),I(t),R(t) and Q(t) with different initial values. Here Δt=0.01.Figure 2
Based on the 10000 sample paths, after iterating 10,000 times, 20,000 times and 30,000 times, respectively, we get the density functions ofS(t),I(t),R(t) and Q(t) with S(0)=0.3,I(0)=0.3,Q(0)=0.2, and R(0)=0.2. Here Δt=0.01.Figure 3
The blue line corresponds to the density function of the solution withS(0)=0.3,I(0)=0.3,Q(0)=0.2, and R(0)=0.2. Based on the 10000 sample paths, after iterating 10,000 times. The red line corresponds to the density function of all states that one trajectory of model (3) reaches. Here Δt=1.Example 2.
In order to obtain the extinction of the model, we give numerical simulations under which one of the conditions ofC1 and C2 hold. In addition, neither C1 nor C2 holds; we also give numerical simulations to verify that in this case, the disease may be persistent in Figure 6 or extinct in Figure 7.
In Figure4, the parameters in (3) are chosen by Λ=0.2;d=0.1;β=0.7;γ=0.2;δ=0.2;α1=0.1;μ=0.1;α2=0.2;σ1=0.4;σ2=0.1;σ3=0.4;σ4=0.5;σ5=0.5. By calculation, we know that σ12=0.16<β and R0s^=0.9211<1. This means that C1 holds. In Figure 5, we keep all the parameters unchanged but increase σ1 to 0.85. Note that β2/2(γ+δ+d+α1+σ32/2)=0.3603 and σ12=0.7225>β>0.3603, which means that C2 holds in Theorem 4. Figures 4 and 5 verify Theorem 4 very well so that if (C1) or (C2) holds, I(t) will tend to zero exponentially with probability one.
In Figure6, by changing γ=0.1, δ=0.2, and σ1=0.1 and not changing other parameters in Figure 4, we calculate β2/2(γ+δ+d+α1+σ32/2)=0.5833 and 0.01=σ12<0.5833<β=0.7 and R0s^=1.6471>1, which means that both (C1) and (C2) in Theorem 4 do not hold. From the line in Figure 6, we can easily find that I(t) persist. In Figure 7, we also change some parameters γ=0.1,α1=0.2,σ1=0.2,σ3=0.2 and other parameters are unchanged in Figure 4. By calculation, we have σ12<max⁡{β,β2/2(γ+δ+d+α1+σ32/2)} and R0s^=1.0937>1, which also means that both (C1) and (C2) in Theorem 4 do not hold. From Figure 7, it is shown that I(t) become extinct. Thus, combining Figure 6 with Figure 7, one can easily find that both (C1) and (C2) do not hold; the direction of I(t) is uncertain and may persist or become extinct.Figure 4
Computer simulation of the pathsS(t),I(t),Q(t), and R(t) for models (1) and (3) for parameter values Λ=0.2;d=0.1;β=0.7;γ=0.2;δ=0.2;α1=0.1;σ1=0.4;σ2=0.1;σ3=0.4;μ=0.1; this way, σ12≤β and R0s^<1 hold.
(a)
Beta = 0.4 (b)
Beta = 0.4 (c)
Beta = 0.4 (d)
Beta = 0.4Figure 5
Computer simulation of the pathsS(t),I(t),Q(t), and R(t) for models (1) and (3) for parameter values Λ=0.2;d=0.1;β=0.7;γ=0.2;δ=0.2;α1=0.1;σ1=0.85;σ2=0.1;σ3=0.4;μ=0.1, such that σ12>max⁡{β,β2/2(γ+δ+d+α1+σ32/2)} hold.
(a)
Beta = 0.4 (b)
Beta = 0.4 (c)
Beta = 0.4 (d)
Beta = 0.4Figure 6
Computer simulation of the pathsS(t),I(t),Q(t), and R(t) for models (1) and (3) for parameter values Λ=0.2;d=0.1;β=0.7;γ=0.1;δ=0.1;α1=0.1;σ1=0.1;σ2=0.1;σ3=0.2;μ=0.1, such that σ12<max⁡{β,β2/2(γ+δ+d+α1+σ32/2)} and R0s^>1 hold.
(a)
Beta = 0.4 (b)
Beta = 0.4 (c)
Beta = 0.4 (d)
Beta = 0.4Figure 7
Computer simulation of the pathsS(t),I(t),Q(t), and R(t) for models (1) and (3) for parameter values Λ=0.2;d=0.1;β=0.7;γ=0.1;δ=0.2;α1=0.2;σ1=0.2;σ2=0.1;σ3=0.2;μ=0.1, such that σ12<max⁡{β,β2/2(γ+δ+d+α1+σ32/2)} and R0s^>1 hold.
(a)
Beta = 0.4 (b)
Beta = 0.4 (c)
Beta = 0.4 (d)
Beta = 0.4


## 6. Conclusion Remarks and Future Directions

In this work, we study the dynamics of a stochasticSIQR epidemic disease with quarantine-adjusted incidence. After formulating the Lyapunov functions, we have confirmed existence and uniqueness of solutions of stochasticSIQR systems and basic reproduction number R0s. If R0s>1, there exists a stationary distribution π(·) and the ergodicity holds for any initial value in system (3). In addition, in the fourth part, when condition (C1) or (C2) holds, the disease I(t) will be extinct.In the future, we can propose a class of practical and complex models, such as considering the effects of regime switching onSIQR epidemic model (see [29]) or considering the dynamical properties of stochasticSIQR epidemic model with different incidence and time delays (see [30, 31]) or the persistence and the existence of periodic solutions of stochasticSIQR epidemic model. We leave these investigations for future work.



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