# A Stochastic SEIRS Epidemic Model with Infection Forces and Intervention Strategies

Authors: Zhang Lijuan; Wang Fuchang; Liang Hongri
Journal: Journal of Healthcare Engineering
 (2022)

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## Abstract

The spread of epidemics has been extensively investigated using susceptible-exposed infectious-recovered-susceptible (SEIRS) models. In this work, we propose a SEIRS pandemic model with infection forces and intervention strategies. The proposed model is characterized by a stochastic differential equation (SDE) framework with arbitrary parameter settings. Based on a Markov semigroup hypothesis, we demonstrate the effect of the proliferation numberR0S on the SDE solution. On the one hand, when R0S<1, the SDE has an illness-free solution set under gentle additional conditions. This implies that the epidemic can be eliminated with a likelihood of 1. On the other hand, when R0S>1, the SDE has an endemic stationary circulation under mild additional conditions. This prompts the stochastic regeneration of the epidemic. Also, we show that arbitrary fluctuations can reduce the infection outbreak. Hence, valuable procedures can be created to manage and control epidemics.


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## Body

## 1. Introduction

Many biological and human populations have been facing the threat of viral epidemics. The spread of such epidemics typically leads to large death tolls and significant economic and healthcare costs. The Ebola outbreak in early 2014 led to the loss of thousands of lives in Africa [1–3]. Thousands of people died as victims of SARS in early 2002 [4–7]. The H7N9 [8–11] and H5N6 [12, 13] epidemics emerge every year in southern areas of China, causing excessive poultry losses.Recently, perturbations have been incorporated into deterministic models of pandemics under reasonable conditions. Subsequent models have been proposed under stochastic assumptions. Gray et al. [14] proposed a stochastic susceptible-infectious-susceptible (SIS) model and investigated the influence of perturbations on the contact rate. Tornatore et al. [15] devised a stochastic susceptible-infectious-recovered (SIR) framework and demonstrated the presence of a limit on the reproduction incentive. A stochastic susceptible-infected-vaccinated-susceptible (SIVS) model was created by Tornatore et al. in [16]. Ji and Jiang [17] examined the characteristics of a stochastic susceptible-infected-recovered-susceptible (SIRS) model under low perturbations. Lahrouz and Omari [18] addressed the extinction conditions within a nonlinear stochastic SIRS framework. Zhao et al. [19] examined a stochastic SIS model with inoculation. For this stochastic SIS model, Lin et al. [20] demonstrated the presence of stationary dispersion. Cai et al. [21] extended the SIRS model to account for the force of infection and the stochastic nature of the problem. Stochastic differential equations (SDEs) were used for the model construction. Mummert and Otunuga [22] investigate the scalability of an approach for solving a nonlinear system of ODEs by Euler’s method. The system describes susceptible-exposed-infectious-recovered-susceptible (SEIRS) epidemic disease in the prey where the predator-prey interaction is given by the Lotka–Volterra type. All parameters grouping in the above 4 groups are discretized with a fixed step in a given interval. The parallel algorithm allows to receive a large number of solutions of the system of ODEs. Using these solutions, we can select those cases of system’s parameters in which the dynamics of the population is stable and the disease is controlled. Talkibing [22] has proposed a stochastic version of a SEIRS epidemiological model for infectious diseases evolving in a random environment for the propagation of infectious diseases. This random model takes into account the rates of immigration and mortality in each compartment, and the spread of these diseases follows a four-state Markov process. Mummert and Otunuga [22] adapted generalized method of moments to identify the time-dependent disease transmission rate and time-dependent noise for the stochastic susceptible- exposed- infectious- temporarily immune- susceptible disease model (SEIRS) with vital rates. The stochasticity appears in the model due to fluctuations in the time-dependent transmission rate of the disease. The method is demonstrated with the US influenza data from 2004-2005 through 2016-2017 influenza seasons. The transmission rate and noise intensity stochastically work together to generate the yearly peaks in infections. There has been much work already done on the stochastic aspects of the epidemic model. For example, Norden [23, 24] described the stochastic SIS model as a logistic population model and investigated the distribution of the extinction times both numerically and theoretically. Ref. [25] introduced environmental stochasticity into the disease transmission term in a model for AIDS and condom use with two distinct states. In a second paper, Dalal et al. [26] introduced stochasticity into a deterministic model of internal HIV viral dynamics via the same technique of parameter perturbation into the death rate of healthy cells, infected cells, and viral particles. Another way to introduce stochasticity into deterministic models is telegraph noise where the parameters switch from one set to another according to a Markov switching process. As a special period of the development of infectious diseases, the incubation period has a far-reaching impact on the spread trend of different infectious diseases, some of which are very short and some of which are very long. However, in this study the SEIR model with stochasticity is missing or rare.In this study, the main contributions are introducing a susceptible-exposed-infectious-recovered-susceptible (SEIRS) epidemic model with infection forces and investigating how changes in conditions, hatching time, and other parameter settings affect the epidemic dynamics. In particular, we extend the SDE formulation of Cai et al. [21] and fine-tune critical structural parameters. The remainder of this study is as follows. We infer a general deterministic SEIRS model (without perturbation) and its stochastic counterpart (with an infection force) in Section 2. In Section 3, we express the primary outcomes of our model. We briefly review the Markov semigroups in Section 4, while itemized evidences of the model primary outcomes are given in Section 5. In Section 6, we show our model outcomes on two SEIR models with contamination. In Section 7, we give a short discussion and a summary of the primary outcomes.


## 2. SEIR Epidemic Representation

We consider a pandemic of the SEIR type, where we indicate the numbers of susceptible, exposed, infectious, and recovered people byS,E,I, and R, respectively. The total population N is given by N=S+E+I+R. The SEIR model accepts that the recovered people might lose their immunity and reemerge in the susceptible state. The SEIR model is applicable to numerous infectious epidemics such as H7N9, bacterial loose bowels, typhoid fever, measles, dengue fever, and AIDS [21, 22, 27].An epidemic is expected to cause increased mortality. According to theSEIR model, the epidemic dynamics are governed by the following equation:(1)dSdt=Λ−μS−βI+αEf1S+γR,dEdt=βI+αEfIS−μE−ηE,dIdt=ηE−μ+ν+δI,dRdt=νI−μ+γR,where Λ,μ,γ,δ,ν, and α are all positive real constants. Λ is the population enrollment rate, μ is the normal population death rate, ν is the rate of recovery for infected people, γ is the rate of recovered people who lose immunity and become susceptible again, δ is the epidemic transmission rate, and α is a coefficient for the exposed people. See [28, 29] for more details. The infection force HI affects the infected people and has been proposed in earlier models as a key factor in deciding the epidemic transmission. The infection force HI in the model incorporates the adaptation of people to epidemics. For instance, HI might diminish as the number of infected people rises because of the way that the population may in general lessen the contacts rate. This has been translated as the “mental” impact [3]. This effect could be enforced by necessitating that the epidemic force HI expands for small I, while this force diminishes for large I. The infection force HI can be expressed as βI/fI, where 1/fI represents the reduction in the valid contact coefficient β due to the intervention strategy [2]. With no such strategy, fI=1, the incidence rate reduces to the bilinear transmission rate βSI. To guarantee a non-monotonic epidemic force, two assumptions are made:(H1)
f0>0 and f′1>0, for I>0.(H2)
There is a strictly positiveξ>0 for which I/fI′>0, for 0<I<ξ and I/fI′<0 for I>ξ.In the study of epidemic transmission, these assumptions portray the impact of intervention systems: if0<I<ξ, the frequency rate increases, while for I>ξ, the rate decreases.To fuse the impacts of ecological changes, we define the stochastic model by bringing multiplicative force terms into the development conditions of both the susceptible and exposed populations. In this work, we assume that the epidemic transmission coefficientβ varies about some normal incentive because of the persistent ecological variations [30]. Hence, we incorporate uncertainty into the deterministic model (1) through the perturbation of the dimensionless substantial contact coefficient β to become β+σζt. This perturbation leads to a system of stochastic differential equations:(2)dSdt=Λ−μS−β+σζtI+αEfIS+γR,dEdt=β+ζtI+αEfIS−μE−ηE,dIdt=ηE−μ+ν+δI,dRdt=νI−μ+γR,where ζt is a zero-mean unit-variance Gaussian white noise: ζt=0,ζt⋅ζ′t=δt−t′, where · denotes the ensemble mean, δ⋅ is the Dirac δ function, and σ is the ecological perturbation power. The system of stochastic differential equations can be rewritten as follows:(3)dSt=Λ−μSt−βIt+αEtfItSt+γRtdt−σIt+αEtStfItdBt,dEt=βIt+αEtfItSt−μEt−ηEtdt+σIt+αEtfItdBt,dItdt=ηEt−μ+ν+δIt,dRdt=νIt−μ+γRt,where Bt is the typical 1-dimensional autonomous Wiener process demarcated on the whole probability space Ω,F,Ftt≥0,Prob. The white noise is related to the Wiener process by dBt=ζtdt.


## 3. Main Results

First, we address the epidemic dynamics for a deterministic model with no perturbation [31]. We can obtain the reproduction number as follows:(4)R0=βη+αμ+ν+δΛμ2ηf0.The dynamics of SEIRS model is bounded by the following equation:(5)Γ=S,E,I,R∈X:0<S+E+I+R≤Λμ⊂XTheorem 1.
(I)
IfR0≤1, the epidemic-free equilibrium E0=Λ/μ,0,0,0 of the deterministic model (1) is globally asymptotically stable.(II)
IfR0>1, model (1) admits a unique equilibrium E∗=S∗,E∗,I∗,R∗, which is globally asymptotically stable.(6)R∗=νμ+γI∗,E∗=μ+ν+δηI∗,S∗=μ+ημ+ν+δfI∗βη+αμ+ν+δ.Remark 1.
Theorem1 shows that the reproduction number R0 highly influences the endemic behavior of the deterministic model. Moreover, Theorem 1 (II) implies, for R0>1, the persistence (or endemicity) of model (1) with simple dynamics. This, however, does not hold for the stochastic model as shown by the subsequent theorem.
Secondly, we investigate the epidemic dynamics associated with stochastic models. We define the stochastic reproduction numberR0S as follows:(7)R0S:=R0−βΛη+αv+δμ2ηf0−σ2α2Λ22μ2μ+ηf20,R0S=2μf0αΛβ−σ2α2Λ22μ2μ+ηf20.
The next theorem describes the epidemic-free extinction states and the endemic persistent states for the stochastic model (2).Theorem 2.
LetSt,Et,It,Rt be a solution of model (3) with arbitrary initial values S0,E0,I0,R0∈Γ. If R0S<0, and σ2<βμf0/Λ, then the model solution St,Et,It,Rt satisfies the following properties:(8)limt⟶∞suplogEtt≤−c<0,a.s.limt⟶∞suplogItt≤min−μ+ν+δ,−c<0,a.s.limt⟶∞suplogRtt≤min−μ+γ,−λ˜<0,a.s.limt⟶∞1t∫0tSsds=Λμ,a.s.,where c=μ+η1−R0S. Eventually, the epidemic disappears with a likelihood of 1.Remark 2.
Adequate conditions are given by Theorem2 when the solutions for model (1) are epidemic-free states a.s.; that is, practically all solutions of (1) are of the form Λ/μ,0,0,0.Remark 3.
The number of infected peopleIt of the deterministic model vanishes at any point where R0≤1 (cf. Theorem 1 (I)), while the contamination is constant at any point where R0≥1 (cf. Theorem 1 (II)).
Theorem2, The aforementioned outcomes do not affect the stochastic model. We can easily discover precedents in which R0≥1 yet R0≤1 to the extent of the epidemic episode.


## 4. Proofs of Theorems1 and 2

### 4.1. Preliminaries

Basic definitions and remarks on the Markov semigroups and their asymptotic characteristics [32–38] are given here to facilitate the demonstration of our results.


#### 4.1.1. Markov Semigroups

Let∑=BX be the σ− algebra of the Borel subsets of X, and let m be the Lebesgue measure on X,Σ. For the space L1=L1X,Σ,m, let D=DX,Σ,m denote the subset of all density functions, i.e.,(9)D=g∈L1:g≥0,g=1,where the norm · is defined in L1. A linear operator P: L′⟶L′ is of the Markov type if PD⊂D.Letk: X×X⟶0,∞ be a measurable function that satisfies ∫Xkx,ymdx=1 for essentially all y∈X. The operator PgX=∫Xkx,ygymdy is thus an integral Markov operator, with a kernel k. Let Ptt≥0 be a family of the Markov-type operators that fulfills the conditions:(1)
P0=Id;(2)
Pt+s=PtPs for all s,t>0; and(3)
The functiont⟶Ptg is continuous for each g∈L′. Then, the operator family Ptt≥0 is called a Markov semigroup. This semigroup is called essential if the operator Pt is a vital Markov operator for every t>0. That is, a measurable function k:0,∞×X×X⟶0,∞ exists so that PtgX=∫Xkt,x,ygymdy for each g∈D.Key terms follow for the asymptotic analysis of Markov semigroups. Firstly, a densityg∗ is said to be invariant under the Markov semigroup Ptt≥0 if Ptg∗=g∗ for every t>0. Secondly, the Markov semigroup Ptt≥0 is asymptotically stable if an invariant density g∗ exists such that limt⟶∞Ptg−g∗=0 for any g∈D. If a differential equation system (e.g., a SDE model) generates the semigroup, then the asymptotic stability implies the convergence of all solutions starting from any density in D to the invariant density. Thirdly, a Markov semigroup Ptt≥0 is sweeping (or zero type) with respect to a set A∈Σ if, for each g∈D, limt⟶∞∫APtgXmdX=0.Remark 4.
A Markov semigroup that is sweeping with respect to limited measure sets possesses no invariant density [32, 34]. Thus, a positive kernel vital Markov semigroup with no invariant density can be non-sweeping with respect to smaller sets. Sweeping with respect to minimal sets is not identical to sweeping with respect to limited measure sets. While a Markov semigroup could be both repetitive and sweeping, it should be noted that dissipativity does not necessarily imply sweeping.
The next lemma characterizes Markov semigroups as asymptotically stable or sweeping [38].Lemma 1.
AssumePtt≥0 is an integral Markov semigroup having a continuous kernel kt,x,y for t>0 and that ∫Xkx,ymdX=1 for all y∈X. Assume for every density g∈D that ∫0∞PtgXdt>0. Then, this semigroup is either asymptotically stable or sweeping with respect to minimal sets.
The fact that a Markov semigroupPtt≥0 is asymptotically stable or sweeping from an adequately large family of sets (e.g., from every minimal set) is known as the Foguel alternative [33].


#### 4.1.2. Fokker–Planck Equation

For anyA∈Σ, let Pt,x,y,z,A denote the progress likelihood work for the dissemination procedure St,It,Et, where(10)Rt=N−St−Et−It,pt,x,y,z,A=probSt,It,Et∈A.Assume thatSt,It,Et is a solution of (3) such that the distribution S0,I0,E0 is uniformly continuous and with density νx,y,z. Thus, St,It,Et has a density Ut,x,y,z that satisfies the Fokker–Planck equation [35, 37]:(11)∂U∂t=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z,where φx,y,z=x2y+αz2/f2y and f1x,y,z=Λ−μx−βy+αz/fyx+γN−x−y−z,(12)f2x,y,z=ηz−μ+ν+δy,f3x,y,z=βy+αzfyx−μ+ηz.Define the operatorPt by setting Ptνx,y,z=Ut,x,y,z for ν∈D. Because the operator pt is a contraction on D, it may be protracted to a contraction on L1. Thus, the operator family Ptt≥0 creates a Markov semigroup, whose infinitesimal generator A satisfies (12), i.e.,(13)ΑV=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z.The adjoint ofA is given by the following equation:(14)Α∗V=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z.


### 4.2. Proofs of Theorems1 and 2

We give here rigorous proofs for the theoretical results of Section3 using the preliminaries.The deterministic SEIRS model (1) has two equilibrium states: one is the epidemic-free equilibrium E0=Λ/μ,0,0,0, which can be obtained for any parameter settings, while the other state is the endemic equilibrium E−∗=S∗,E∗,I∗,R∗, which is a positive solution of the following scheme:(15)Λ−μs−βI+αEfIS+γR=0,βI+αEfIS−μ+ηE=0,ηE−μI−νI−δI=0,νI−μ+γR=0.The endemic equilibrium terms, namely,S∗,E∗,I∗, and R∗, can be expressed as follows:(16)S∗=μI∗fI∗βI∗+αE∗,R∗=νμ+γ,E∗=μ+ν+δηI∗,and Λ−ημ2fI∗/βη+αμ+ν+δ+μγ/μ+γ−βμI∗=0.DefineFI=Λ−ημ2fI∗/βη+αμ+ν+δ+μγ/μ+γ−βμI∗. Based on the assumption (H1), the function FI is decreasing. Since (17)F0=Λ−μ2ηβη+αμ+ν+δf0=μ2ηβη+αμ+ν+δβη+αμ+ν+δΛμ2ηf0−1f0=μ2ηβη+αμ+ν+δR0−1f0.The equationFI=0 possesses a unique positive solution I∗ if R0>1. Therefore, a unique endemic equilibrium E−∗=S∗,E∗,I∗,R∗ exists for model (1).The next lemma demonstrates that the solutions for model (1) are limited, contained in a reduced set, and continuous for all t>0.Lemma 2.
Model (1) is decidedly invariant where pulls of each solution with initial conditions begin in its state space X. Also, every direction of model (1) will in the long run remain in a reduced subset of Γ.Proof.
Joining all conditions in (1) and considering Nt=St+Et+It+Rt, we have the following:(18)Λ−μ+δN≤dNdt=Λ−μN−δI≤Λ−μN.
Hereafter, by integrating (18), we obtain the following equation:(19)Λμ+δ+N0−Λμ+δe−μ+δt≤Nt≤Λμ+N0−Λμe−μt.
This concludes the proof of the lemma.Remark 5.
Lemma2 shows in particular that the dynamics of model (1) can be studied in the restricted set Γ obtained in (7).


#### 4.2.1. Epidemic-Free Dynamics of Model (1)

Here, the global asymptotic stability of the epidemic-free equilibrium E0 is investigated. In particular, we prove Theorem1 (I).Proof.
Construct the following Lyapunov function:VS,E,I,R=1/2S−Λ/μ2+θ1E+θ2I,
whereθ1=Λ/μ and θ2=θ1/ημ+η1−R0−εifR0<10ifR0=1 for each adequately small ε>0.
Hence, the time derivative ofV for a solution of model (1) is as follows:(20)dVdt=S−ΛμdSdt+θ1βI+αEfIS−μ+ηE+θ2ηE−μ+v+δI=−µS−Λμ2−S−ΛμβSI+αEfI+γRS−Λμ+θ1βSI+αEfI−θ1μ+η−θ2ηE−θ2μ+v+δI≤−μ+βI+αEfIS−Λμ2+θ1Λβ−μfIθ1μ+η−θ2ημfII−θ2μ+v+δI,where the following is applied:(21)βSI−αEfI=βI−αEfIS−Λμ+ΛβI−αEμfI,γs−ΛμR≤0.
IfR0<1, and since fI=f0+f′0I+oI, we get the following:(22)θ1Λβ−µfIθ1μ+η−θ2η=θ1Λβ−µf0μ+η+μf0θ2η−µθ1µ+η−θ2ηf′0I+oI≤−θ1µf0μ+η1−R0+μf0θ2η−μf′0θ1μ+η−θ2ηI.
Moreover, sinceθ2=θ1/ημ+η1−R0−εifR0<10ifR0=1,
we have the following:(23)−θ1μf0μ+η1−R0+μf0θ2η=−μηf0ε,θ1μ+η−θ2η=R0θ1μ+η+εη.
Hence,dV/dt≤−μ+βI+αE/fIS−Λ/μ2−ηf0ε/fII−θ2μ+v+δI−R0θ1μ+η+εη/fIf′0I2. Note the nonnegativity of the functions S,E,I,andR. Also, note that the relationships in the right side of the last inequality are nonpositive; i.e., dV/dt≤0, if and only if dV/dt=0 Consequently, the best invariant S=Λ/μ,E=0,I=0,andR=0 set in S,E,I,R:dV/dt=0 is a singleton E0.
IfR0=1, then(24)dVdt≤−μ+βI+αEfIS−Λμ2−θ1μ+ηfII2f′0.
By LaSalle’s invariance principle [39, 40], the solutions of model (1) tend to M⊂S,E,I,R||S=Λ/μ,E=0,I=0,R=0, the biggest invariant subset of model (1). From the description of model (1), M=E0 is a singleton set. Thus, E0 is universally asymptotically constant on the set Γ if R0≤0.
WhenR0>1, the Jacobian of model (1) at E0 is given by the following equation:(25)JE0=−μ−Λαμf0−Λβμf0γ0−μ+η+βαΛμf0βΛμf000η−μ+ν+δ000ν−μ+γ,with eigenvalues −µ<0,−µ+v+δ<0,−μ+γ<0, and(26)−μ+η+βαΛμf0βαΛμf0η−μ+ν+δ=μ+ημ+v+δ−R0μη=μημ+ημ+v+δμη−R0.
Therefore, the epidemic-free equilibrium is perturbed ifR0>μ+ημ+v+δ/μη>1. This concludes the proof.


#### 4.2.2. Endemic Dynamics of Model (1)

Here, the global asymptotic stability of the endemic equilibriumE∗ is addressed. In particular, Theorem 1 (II) is proved.Proof.
The Jacobian of (2) at E∗ is as follows:(27)JE∗=−μ−βI∗+αE∗fI∗βαS∗fI∗βS∗I∗1−I∗f2I∗γβI∗+αE∗fI∗−μ+η000η−μ−ν−δ000ν−μ+γ.
The characteristic polynomial of the JacobianJE∗ is as follows:(28)λ4+c1λ3+c2λ2+c3λ+c4=0,c1=4μ+η+v+δ+γ+βI∗+αE∗fI∗>0,c2=μ+v+δμ+γ+2μ+v+δ+γβI∗+αE∗fI∗+2µ+η+µ+ηµ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0,c3=μ+v+δμ+γβI∗+αE∗fI∗+2μ+η+2μ+v+δ+γμ+ημ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0,c4=μ+v+δμ+γβI∗+αE∗fI∗+2μ+η+μ+v+δ+μ+γμ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0.
It can be verified thatc1c2−c3>0,c3c1c2−c3−c12c4>0. Hence, the asymptotic stability of E∗ can be determined by exploiting the Routh–Hurwitz criterion.
Now, by proving thatS∗,E∗,andI∗ of model (1) are globally asymptotically stable, we will immediately prove the same type of stability for the endemic equilibrium of model (1).


### 4.3. Proof of Theorem2

For proving Theorem2, we first prove the existence and uniqueness of a positive global solution for model (2).Theorem 3.
For some random initial solutionsS0,E0,I0,R0∈Γ, there is an nontrivial positive solution St,Et,It,Rt of model (2) for t≥0, which stays in X with a likelihood of 1.Proof.
LetS0,E0,I0,R0∈Γ. Adding up the three equations in model (2) and using Nt=St+Et+Rt+It, we have dNt=Λ−μN−δIdt.
Then, ifS0,E0,I0,R0∈X for all 0≤t1≤t almost surely (briefly a.s.), then we get Λ−μ+δNt1dt≤dNt1≤Λ−μNt1dta.s.
By integration, we obtainΛ/μ+δ≤Ns≤Λ/μ. Thus, St1,Et1,It1,Rt1∈0,Λ/μ for all t1∈0,t a.s. Because the model coefficients for (2) fulfill the neighborhood Lipschitz condition, an extraordinary nearby solution exists on t1∈0,τe, where τe is the blast time. In this manner, the unique nearby solution of model (2) is certain by Itô’s equation. Now, the global nature of this solution is shown, i.e., τe=∞ a.s. Let n0>0 be appropriately big so that S0,I0, and R0 lie inside the interval 1/n0,n0. For every integer n>n0, the stop times are obtained:(29)τn=inft∈0,τe:minSt,Et,It,Rt≤1normaxSt,Et,It,Rt≥n.
Set infϕ=∞ (∞ represents the empty set). τn grows as n⟶∞. Let τ∞=limn⟶∞τn. Then, τ∞≤τe a.s.
In the following, we demonstrate thatτ∞=∞. Assume on the contrary that this is not true. Thus, there exists a steady T>0 such that Prob τ∞≤t>ε for any ε∈0,1. As a result, a whole number n1≥n0 exists for which(30)Probτn≤T≥ε,n≥n1.
Describe the positiveC2 function V:D⟶ℝ++by the following equation:(31)VS,E,I,R=S−1−lnS+I−1−lnI+R−1−lnR+I−1−lnI.
IfSt,Et,It,Rt∈X, then by the Itô formulation, we obtain the following equation:(32)dV=1−1SΛ−μS−βIt+αEtfItSt+γRt+σ2It+αEt2St22f2Itdt+1−1EβIt+αEtfItSt−μ+ηE+σ2It+αEt2St2f2Itdt+1−1IηE−μ+ν+δIdt+1−1RνI−μ+γRdt−1−1SσIt+αEtfItSt+1−1IσIt+αEtfItStdBt=LVdt+σIt−St+αEtfItdBt,where(33)LV=Λ+4μ+γ+η+v+δ−μSt+Et+It+Rt+3σ2It+αEt2f2ItSt2−βIt+αEtEfItSt+βIt+αEtfIt+γR+γRs−ηIE−1Rν−ΛS−δI−γR−ηE<Λ+4μ+γ+η+ν+δ+3σ2ξ2+α2+2αf2ξ+2βξ+αfξ≔K.
Replacing this inequality in equation (32), we get the following equation:(34)dVSt,Et,It,Rt≤Kdt+σIt−StfItdBt,which implies that(35)∫0τn∧TdVSr,Er,Ir,Rr≤∫0τn∧TKdt+∫0τn∧TσIr−SrfIrdBr,where τnΛT=minτn,T. Evaluating the integrals of the last inequality gives the following equation:(36)EVSτn∧T,Iτn∧T,Eτn∧T,Rτn∧T≤VS0,E0,I0,R0+KT.
SetΩn=τn≤T. From (35), we have Prob Ωn≥ε. For each w∈Ωn, at least one exists among Sτnw,Eτnw,Iτnw, and Rτnw with a value of either n or 1/n. Hence,(37)VSτnw,Eτnw,Iτnw,Rτnw≥n−1−lnn∧1n−1−ln1n.
Next, from (34), we have the following:(38)VS0,E0,I0,R0≥Eχτnw·VSτnw,Eτnw,Iτnw,Rτnw+KT≥εn−1−lnn∧1n−1−ln1n,where χΩn is the characteristic function of Ωn.Asn⟶∞, the following contradiction is obtained:(39)∞>VS0,E0,I0,R0+KT=∞a.s.
Therefore,τ∞=∞, and the solution of model (2) shall not blast within a limited time with a probability of one. The proof is complete.Remark 6.
From Theorem1, the set is an almost surely positive invariant of the SDE (2). That is, for S0,E0,I0,R0∈Γ,(40)ProbSt,Et,It,Rt∈Γ=1∀t≥0.


#### 4.3.1. Disease Extinction in the SDE Model

Here, Theorem2 (I) on the disease extinction in the stochastic model (3) will be proved.Proof.
Based on the Itô formulation,(41)dlnEt=φSt,Et,It,Rtdt+σαStfItdBt,where φ:R3+⟶R is given by the following equation:(42)φu,v=βαufν−μ+η+σ2α2u22f2v.
Hence,(43)lnEt=lnI0+∫0tφSs,Isds+∫0tσαSsfIsdBs.
SettingGt≔∫0tσαSs/fIsdBs, we have the following equation:(44)G,Gtt≔1t∫0tσ2α2Ss2f2IsdBs≤σ2α2Λ2μ2f20<+∞.
From the strong law of large numbers for martingales [38], we obtain limt⟶∞supGt/t=0a.s.
Based on (9), we have the following equation:(45)φSs,Is=βαStfIt−μ+η+σ2α2Ss22f2Is=−σ2α22StfIt−βσ2α2+β22σ2−µ+η≤β22σ2−µ+η.
It then follows from (29) that(46)lnEt≤lnI0+∫0tβ22σ2−μ+ηds+Gt.
If we divide both sides of (46) by t and let t⟶∞, we get the following equation:(47)limt⟶∞lnItt≤β22σ2−µ+ηa.s.
Now, consider the case whenσ2<βμf0/Λ. Thus,(48)φSs,Is=−σ2α22StfIt−βσ2α2+β22σ2−μ+η≤−σ2α22Λμf0−βσ2α2+β22σ2−µ+η=Λαβμf0−σ2α2Λ22μ2f20−µ+η=µ+η2μf0αΛβ−σ2α2Λ22μ2µ+ηf20−1=µ+ηR0s−1,where(49)R0s=2μf0αΛβ−σ2α2Λ22μ2µ+ηf20.
It then follows from (49) that(50)lnIt≤lnI0+μ+ηR0s−1t+Gt.
Therefore, from the last inequality and (10),(51)limt⟶∞suplnEtt≤µ+ηRs0−1<0a.s.
The reason is thatR0<1,R0s=2μf0αΛβ−σ2α2Λ2/2μ2µ+ηf20<μf0αΛβ/μ2ηf0=μαΛβ/μ2ηf0<1, and there exists a null set N1 for which Prob N1=0 and for any ω∉N1,(52)limt⟶∞supEtωt<−c.
Thus, for each adequately smallε>0, there is T1=T1ω for which(53)Etω≤e−c+εt,∀t≥T1.
From the 3rd equation of the stochastic model (3), for each ω∈Ω, if t≥T1ω,(54)Itω=e−μ+v+δt∫0teµ+v+δsIsds+I0≤I0e−μ+v+δt+ηe−μ+v+δt∫0teµ+v+δsIsds+νe−μ+v+δt∫T1teμ+v+δ−c+εsds.
Thus, for any(55)ω∉N1,limsup1tlnItω≤min−μ+v+δ,−c+εa.s.
Lettingε⟶0, we get limsup1/tlnItω≤min−μ+v+δ,−c,a.s. Correspondingly, there is a null set N2 such that Prob N2=0 and for each ω∉N2,(56)limsupItωt<−λ∼a.s.for a constant λ∼>0. Therefore, for each adequately small ε>0, there is T2=T2ω such that Itω≤e−λ∼+εt,∀t≥T2. Similarly, we have the following equation:(57)Rtω=e−μ+γtv∫0teµ+γsIsds+R0≤R0e−µ+γt+ve−μ+γt∫0T2eµ+γtIsds+ve−µ+γt∫T2teµ+γ−λ+ε∼sIsds.
It follows that for anyω∉N2,(58)limsup1tlnRtω≤min−μ+γ,−λ˜+εa.s.
Lettingε⟶0, we get linsup1/tlnRtω≤min−μ+γ,−λ˜,a.s. Likewise, a null set N3 exists so that Prob N3=0 and for all ω∉N3,(59)linsupRtωt<−λ˜,a.s.,for some constant −λ˜>0. Thus, for any adequately small ε>0, there exists T3=T3ω for which Rtω≤e−λ˜+εt,∀t≥T3. Finally, we consider St. In view of the above analysis, there exists the null set N=N1∪N2∪N3 and T=Tω=maxT1,T2,T3 for which
ProbN=0 and for all ω∉N,(60)dSt+Et+It+Rt=Λ−δIt−μSt+Et+It+Rtdt≥Λ−δe−λ˜+εt−μSt+Et+It+Rtdt,∀t≥T.
This implies(61)1t∫0tSt+Et+It+Rtds≥Λμ−δt∫0te−λ˜+εtds−φt,where (62)φt=1μSt+Et+It+Rtt−S0+E0+I0+R0tandlimt⟶∞φt=0a.s.
For a randomε, we have the following equation:(63)limt⟶∞inf1t∫0tSs+Is+Es+Rsds≥Λμa.s.
From Remark6, we deduce that(64)limt⟶∞1t∫0tSs+Is+Es+Rsds=Λμa.s.
Together with the aforementioned results, we getlimt⟶∞1/tSsds=Λ/μa.s. Hence, the proof is finished.


#### 4.3.2. Stochastic Asymptotic Stability

In this subsection, we show that under mild additional conditions the solutions of model (2) converge to the endemic state a.s., and in particular, we prove Theorem 2 (II). We will initially demonstrate the asymptotic stability of the Markov semigroup by showing the existence of an invariant density for the semigroup.Lemma 3.
For each pointx0,y0,z0∈X and t>0, the progress likelihood work Pt,x0,y0,z0,A possesses a continuous density kt,x,y,z;x0,y0,z0.Proof.
For proving this lemma, we utilize the Hörmander hypothesis [41] on the presence of smooth densities of the change likelihood work for dispersion processes.
Let(65)a0S,E,I=Λ−µS−βI+αEfIS+γRβI+αEfIS−μE−ηEµE−μv+δI,and(66)a1S,E,I=−σI+αEfISσI+αEfIS0.
By straight computations, the Lie bracketa0,a1 is a vector field expressible as follows:(67)a1=a0,a1=−σA11f2IσA11f2I−SI+αEησfI,where(68)A11=−I2γ+EEαγ−Nαγ+Sαγ−Sη+Sαη−αΛ+Sαμ+IE−N+Eαγ−Λ+Sγ+δ+μfI+SI+αE−Eη+Iv+δ+ωf′I.
Seta3=a1,a2. The vector fields a1,a2,a3 are linearly independent on the space X. Hence, for all S,E,I∈X, a1,a2,anda3 span the space X. Based on the Hörmander theorem [42, 43], the transition probability function Pt,x0,y0,z0,A has a continuous density kt,x,y,z;x0,y0,z0 and k∈C∞0,∞×X×X. Next, the positivity of k is examined using sustenance theorems [41, 44].
Pick a fixed pointx0,y0,z0∈X and a function φ∈L20,T,ℝ. Note this system of integral equations:(69)xϕt=x0+∫0tf1xϕs,yϕs,zϕs−σϕxϕs+αyϕsfyϕsxϕsds,yϕt=y0+∫0tf2xϕs,yϕs,zϕsσϕxϕs+αyϕsfyϕsxϕsds,zϕt=z0+∫0tf3xϕs,yϕs,zϕsds,where f1x,y,z,f2x,y,z,andf3x,y,z are given in (11).
LetX=x,y,zTandX0=x0,y0,z0T. Let DX0;ϕ be the Fréchet derivative of the h⟶Xϕ+hT function from L20,T;ℝ to X. If for some ϕ∈L20,T;ℝ the derivative DX0;ϕ has a rank of 3, then kT,x,y,z;x0,y0,z0>0 for X=XϕT.Letψt=f′Xϕt+φg′Xϕt, where f′ and g′ are, respectively, the Jacobians of f=f1x,y,zf2x,y,zf3x,y,z and −σxy/fyσxy/fy0.
LetQt,t0, for 0≤t0≤t≤T, be a matrix function for which Qt0,t0=Id and ∂Qt,t0/∂t=ψtQt,t0. Then, DX0;ϕh=∫0TQT,sgshsds.Lemma 4.
For everyx0,y0,z0∈Πandx,y,z∈Π, there exists kT,x,y,z;x0,y0,z0>0, where Π is characterized as in (68).Proof.
Since a continuous control workϕ is considered, the inequality (35) could be supplemented by these differential equations:(70)x′ϕt=f1xϕs,yϕs,zϕs−σϕxϕs+αyϕsfyϕsxϕs,y′ϕt=f2xϕs,yϕs,zϕs+σϕxϕs+αyϕsfyϕsxϕs,z′ϕt=f3xϕs,yϕs,zϕs.
Firstly, the rank ofDX0;ϕ is shown to be 3. Let
ht=χT−ε,Ttfyϕt/χϕt+αyϕt,t∈0,T, where χ is a characteristic function. Since(71)QT,s=Id+ψTs−T+12ψ2Ts−T2+os−T2,we obtain DX0;φh=εV−1/2ε2ψTv+1/6ε3ψ2Tv+oε3, where ν=−σσ0, and(72)ψT.v=σ−x+y+αzβfy+μf2y+xy+αzβf′yf2y−ν+δ+μσβσx−y−αzfy−xy+αzf′yf2y.
Thus,v,ψTv,andψ2Tv are straightly autonomous and the subsidiary DX0;ϕ has a rank of 3.
Next, for anyX0∈Ωand X∈Ω, we demonstrate the existence of a control work ϕ and T > 0 for which Xφ0=X0andXφT=X. Set ωϕ=xϕ+yϕ+zϕ. Model (37) becomes(73)g1x,w,z=Λ−μx−βz+αw−x−zfzx+γN−ω−σφz+αω−z−zStf,g2x,w,z=Λ+μz−μz−vz−δz−γ+μw,g3x,w,z=ηw−x−z−μ+v+δz.
LetΠ0=x,w,z∈:0<x,z<Λ/μ,Λ/Λ+μ<w<Λ/μandx,z,<w. For any x0,w0,z0∈Π0and.x1,w1,z1∈Π0, it can be claimed that there exists a control function ϕandT>0 for which
xϕ0,wϕ0,zϕ0=x0,w0,z0andxϕT,wϕT,zϕT=x1,w1,z11. We create the function ϕ in the next steps. First of all, we determine a positive constant T and a differentiable function wϕ:0,T⟶Λ/μ+γ,Λ/μ, for which(74)wϕ0=w0,wϕ′T=w1,wϕ′0=g2x0,w0,z0=w0d,wϕ′T=g2x1,w1,z1=wTdandΛ−μ+γwϕt<wϕ′t<Λ−μwϕtfort∈0,T.
For achieving this, the domain of the functionwϕ is divided into 3 segments 0,ε,ε,Tε,andT−ε,T, where 0<ε<T/2. Let(75)η=12minω0−Λμ+γ,ω1−Λμ+γ,Λμ−ω0,Λμ−ω1.
Ifωϕ∈Λ/μ+γ+m,Λ/μ−m, then we have the following equation:(76)Λ−μ+γωϕt<−μ+γm<0andΛ−μωϕt>μm>0fort∈0,t.
Based on (41), a C2 function ωϕ:0,ε⟶Λ/μ+γ+m,Λ/μ−m can be obtained for which(77)ωϕ0=ω0,ωϕ′0=ω0d,ωϕ′ε=0,where ωϕ satisfies (40) for t∈0,t. Similarly, a C2 function ωϕ:T−ε,T⟶Λ/μ+γ+m,Λ/μ−m is constructed so that(78)ωϕT=ω0,ωϕ′T=ωTd,ωϕ′T−ε=0,where ωϕ satisfies (40) for t∈T−ε,T. If we take T adequately big, we can spread the function ωϕ:0,ε∩T−ε,T⟶Λ/μ+γ+m,Λ/μ−m to a C2 function ωϕ on the whole segment 0,T for which Λ−μ+γ·ωϕt<−μ+γm<ωϕ′t<μm<Λ−μωϕt for ε,T−ε. So, the function ωϕ satisfies (41) on [0, T].
As a result, a continuous functionφ can be determined from the first equation of (38), while two functions xϕ and zϕ can be found where these functions satisfy the other equations in (38). This finishes the proof.Lemma 5.
Assume thatR0s>1. For any density g, we get limt⟶∞∭ΠPtgx,y,zdxdydz=1.
whereΠ is obtained from (13).Proof.
Following the proof of Lemma 5.6, we substituteZt=St+Et+It. Then, model (3) can be rewritten as follows:(79)dSt=g1St,Zt,Itdt−σIt+αZt−St−ItStfItdBt,dEt=g2St,Zt,Itdt−σIt+αZt−St−ItStfItdBt,dIt=g3St,Zt,Itdt,where g1x,w,z,g2x,w,z, andg3x,w,z are introduced in (38). Since St,Et,It is a positive solution of model (5) with a probability of 1, and given g2, we have the following equation:(80)Λ−μ+γZt<dZtdt<Λ−μZt,t∈0,∞a.s.
Now, for almost everyw∈Ω, we can show that there exists t0=t0w for which(81)Λμ+γ<Ztw<dZtdt<Λ−μZt,fort>t0.
Actually, three cases exist.(a)
Z0∈Λ/μ+γ,Λ/μ: the conclusion is obvious from (45).(b)
Z0∈0,Λ/μ+γ: assume on the contrary that our claim is not true. Then, there would be Ω′∈Ω with Prob (Ω′) > 0 for which Z0∈0,Λ/μ+γ. From (44), Ztw is carefully expanding on [0,∞) for any w∈Ω′ . Consequently,(82)limt⟶∞Ztw=Λμ+γ,w∈Ω′.From (43), we get limt⟶∞Stw=limt⟶∞Itw=0,w∈Ω′ and consequently limt⟶∞Etw=Λ/μ+γ,w∈Ω′. Thus, we get using the Itô formula,(83)dlnEt=βαfItSt−μ+η−σ2α2St22f2Itdt+σαStfItdBt.Hence,(84)lnEt−lnE0t=1t∫0tβαfIsSs−μ+η−σ2α2Ss22f2Is+1t∫0tσαSsfIsdBs.Since1/t∫0tσ2α2Ss2/f2Isds≤σ2α2Λ2/μ2f20<+∞, and using the strong law of large numbers for martingales [20], we get the following:(85)limt⟶∞1t∫0tσαSsfIsdBs=0.Therefore, taking into consideration the continuity of the functionsSt,Et,It, and fIt, we obtain the following:(86)limt⟶∞1t∫0tβαSsfIs−μ+η−σ2α2Ss22f2Isds+1t∫0tσαSsfIsdBs=−μ+η.This contradicts the limitlimt⟶∞lnEt−lnE0/t=0, and thus, the claim is proved.(c)
Z0∈Λ/μ+γ,+∞: we use again a proof by contradiction with arguments similar to those in (b) to deduce that there is Ω′∈Ω with Prob (Ω′) > 0 for which(87)limt⟶∞Ztw=Λμ,w∈Ω′.
Using (44) and for any w∈Ω′, we get the following:(88)g2x,y,z=Λ−μ+δ+νZt+δEt+St+γRt+νSt+νEt,Ztw=e−tμ+δ+νZ0+∫0tesμ+δ+νΛ+δEt+St+γR+νSt+νEtds.Itw=e−tμ+δ+νR0+η∫0tesμ+δ+νZsw−Ssw−Itwds.
Hence,limt⟶∞Etw=0, limt⟶∞Stw=0, and limt⟶∞Itw=0 for any w∈Ω′. Hence,(89)limt⟶∞lnEt−lnE0t=limt⟶∞1t∫0tβαSsfIs−μ+η−σ2α2Ss22f2Isds+1t∫0tσαSsfIsdBs=βαΛμfI0−μ+η−σ2α2Λ22μ2f20=μ+η2μf0βαΛ2μ2f20μ+η−1>0,a.s.onΩ′.
This is contradictory to the assumption thatlimt⟶∞It=0 a.s. and the claim follows. Remark 7: from Lemmas 4 and 5, we realize that when the Fokker–Planck equation (11) has a stationary solution U∗, then sup U∗ = Π.Lemma 6.
Assume thatR0s > 1, the semigroup {P (t)} t ≥ 0 is either sweeping with respect to minimal sets or asymptotically stable.Proof.
By Lemma3, the operator family {P (t)} t ≥ 0 is a fundamental Markov semigroup with a constant kernel kt,x,y,z,x0,y0,z0 for t > 0. Then, the appropriation of (St,Et,It) possesses a density Ux,y,z,t, which fulfills (19). From Lemma 5, the semigroup {P (t)} t ≥ 0 can be restricted to the space L0 (Π). As indicated by Lemma 4, for each f ∈ D, we have the following:(90)∫0∞Ptfdt>0,a.s.
Thus, from Lemma1, the semigroup {P(t)}, t ≥ 0 is asymptotically stable or is sweeping with respect to minimal sets.


## 4.1. Preliminaries

Basic definitions and remarks on the Markov semigroups and their asymptotic characteristics [32–38] are given here to facilitate the demonstration of our results.


### 4.1.1. Markov Semigroups

Let∑=BX be the σ− algebra of the Borel subsets of X, and let m be the Lebesgue measure on X,Σ. For the space L1=L1X,Σ,m, let D=DX,Σ,m denote the subset of all density functions, i.e.,(9)D=g∈L1:g≥0,g=1,where the norm · is defined in L1. A linear operator P: L′⟶L′ is of the Markov type if PD⊂D.Letk: X×X⟶0,∞ be a measurable function that satisfies ∫Xkx,ymdx=1 for essentially all y∈X. The operator PgX=∫Xkx,ygymdy is thus an integral Markov operator, with a kernel k. Let Ptt≥0 be a family of the Markov-type operators that fulfills the conditions:(1)
P0=Id;(2)
Pt+s=PtPs for all s,t>0; and(3)
The functiont⟶Ptg is continuous for each g∈L′. Then, the operator family Ptt≥0 is called a Markov semigroup. This semigroup is called essential if the operator Pt is a vital Markov operator for every t>0. That is, a measurable function k:0,∞×X×X⟶0,∞ exists so that PtgX=∫Xkt,x,ygymdy for each g∈D.Key terms follow for the asymptotic analysis of Markov semigroups. Firstly, a densityg∗ is said to be invariant under the Markov semigroup Ptt≥0 if Ptg∗=g∗ for every t>0. Secondly, the Markov semigroup Ptt≥0 is asymptotically stable if an invariant density g∗ exists such that limt⟶∞Ptg−g∗=0 for any g∈D. If a differential equation system (e.g., a SDE model) generates the semigroup, then the asymptotic stability implies the convergence of all solutions starting from any density in D to the invariant density. Thirdly, a Markov semigroup Ptt≥0 is sweeping (or zero type) with respect to a set A∈Σ if, for each g∈D, limt⟶∞∫APtgXmdX=0.Remark 4.
A Markov semigroup that is sweeping with respect to limited measure sets possesses no invariant density [32, 34]. Thus, a positive kernel vital Markov semigroup with no invariant density can be non-sweeping with respect to smaller sets. Sweeping with respect to minimal sets is not identical to sweeping with respect to limited measure sets. While a Markov semigroup could be both repetitive and sweeping, it should be noted that dissipativity does not necessarily imply sweeping.
The next lemma characterizes Markov semigroups as asymptotically stable or sweeping [38].Lemma 1.
AssumePtt≥0 is an integral Markov semigroup having a continuous kernel kt,x,y for t>0 and that ∫Xkx,ymdX=1 for all y∈X. Assume for every density g∈D that ∫0∞PtgXdt>0. Then, this semigroup is either asymptotically stable or sweeping with respect to minimal sets.
The fact that a Markov semigroupPtt≥0 is asymptotically stable or sweeping from an adequately large family of sets (e.g., from every minimal set) is known as the Foguel alternative [33].


### 4.1.2. Fokker–Planck Equation

For anyA∈Σ, let Pt,x,y,z,A denote the progress likelihood work for the dissemination procedure St,It,Et, where(10)Rt=N−St−Et−It,pt,x,y,z,A=probSt,It,Et∈A.Assume thatSt,It,Et is a solution of (3) such that the distribution S0,I0,E0 is uniformly continuous and with density νx,y,z. Thus, St,It,Et has a density Ut,x,y,z that satisfies the Fokker–Planck equation [35, 37]:(11)∂U∂t=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z,where φx,y,z=x2y+αz2/f2y and f1x,y,z=Λ−μx−βy+αz/fyx+γN−x−y−z,(12)f2x,y,z=ηz−μ+ν+δy,f3x,y,z=βy+αzfyx−μ+ηz.Define the operatorPt by setting Ptνx,y,z=Ut,x,y,z for ν∈D. Because the operator pt is a contraction on D, it may be protracted to a contraction on L1. Thus, the operator family Ptt≥0 creates a Markov semigroup, whose infinitesimal generator A satisfies (12), i.e.,(13)ΑV=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z.The adjoint ofA is given by the following equation:(14)Α∗V=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z.


## 4.1.1. Markov Semigroups

Let∑=BX be the σ− algebra of the Borel subsets of X, and let m be the Lebesgue measure on X,Σ. For the space L1=L1X,Σ,m, let D=DX,Σ,m denote the subset of all density functions, i.e.,(9)D=g∈L1:g≥0,g=1,where the norm · is defined in L1. A linear operator P: L′⟶L′ is of the Markov type if PD⊂D.Letk: X×X⟶0,∞ be a measurable function that satisfies ∫Xkx,ymdx=1 for essentially all y∈X. The operator PgX=∫Xkx,ygymdy is thus an integral Markov operator, with a kernel k. Let Ptt≥0 be a family of the Markov-type operators that fulfills the conditions:(1)
P0=Id;(2)
Pt+s=PtPs for all s,t>0; and(3)
The functiont⟶Ptg is continuous for each g∈L′. Then, the operator family Ptt≥0 is called a Markov semigroup. This semigroup is called essential if the operator Pt is a vital Markov operator for every t>0. That is, a measurable function k:0,∞×X×X⟶0,∞ exists so that PtgX=∫Xkt,x,ygymdy for each g∈D.Key terms follow for the asymptotic analysis of Markov semigroups. Firstly, a densityg∗ is said to be invariant under the Markov semigroup Ptt≥0 if Ptg∗=g∗ for every t>0. Secondly, the Markov semigroup Ptt≥0 is asymptotically stable if an invariant density g∗ exists such that limt⟶∞Ptg−g∗=0 for any g∈D. If a differential equation system (e.g., a SDE model) generates the semigroup, then the asymptotic stability implies the convergence of all solutions starting from any density in D to the invariant density. Thirdly, a Markov semigroup Ptt≥0 is sweeping (or zero type) with respect to a set A∈Σ if, for each g∈D, limt⟶∞∫APtgXmdX=0.Remark 4.
A Markov semigroup that is sweeping with respect to limited measure sets possesses no invariant density [32, 34]. Thus, a positive kernel vital Markov semigroup with no invariant density can be non-sweeping with respect to smaller sets. Sweeping with respect to minimal sets is not identical to sweeping with respect to limited measure sets. While a Markov semigroup could be both repetitive and sweeping, it should be noted that dissipativity does not necessarily imply sweeping.
The next lemma characterizes Markov semigroups as asymptotically stable or sweeping [38].Lemma 1.
AssumePtt≥0 is an integral Markov semigroup having a continuous kernel kt,x,y for t>0 and that ∫Xkx,ymdX=1 for all y∈X. Assume for every density g∈D that ∫0∞PtgXdt>0. Then, this semigroup is either asymptotically stable or sweeping with respect to minimal sets.
The fact that a Markov semigroupPtt≥0 is asymptotically stable or sweeping from an adequately large family of sets (e.g., from every minimal set) is known as the Foguel alternative [33].


## 4.1.2. Fokker–Planck Equation

For anyA∈Σ, let Pt,x,y,z,A denote the progress likelihood work for the dissemination procedure St,It,Et, where(10)Rt=N−St−Et−It,pt,x,y,z,A=probSt,It,Et∈A.Assume thatSt,It,Et is a solution of (3) such that the distribution S0,I0,E0 is uniformly continuous and with density νx,y,z. Thus, St,It,Et has a density Ut,x,y,z that satisfies the Fokker–Planck equation [35, 37]:(11)∂U∂t=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z,where φx,y,z=x2y+αz2/f2y and f1x,y,z=Λ−μx−βy+αz/fyx+γN−x−y−z,(12)f2x,y,z=ηz−μ+ν+δy,f3x,y,z=βy+αzfyx−μ+ηz.Define the operatorPt by setting Ptνx,y,z=Ut,x,y,z for ν∈D. Because the operator pt is a contraction on D, it may be protracted to a contraction on L1. Thus, the operator family Ptt≥0 creates a Markov semigroup, whose infinitesimal generator A satisfies (12), i.e.,(13)ΑV=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z.The adjoint ofA is given by the following equation:(14)Α∗V=12σ2∂2φU∂x2−2∂2φU∂x∂y+∂2φU∂y2−∂f1U∂x−∂f2U∂y−∂f3U∂z.


## 4.2. Proofs of Theorems1 and 2

We give here rigorous proofs for the theoretical results of Section3 using the preliminaries.The deterministic SEIRS model (1) has two equilibrium states: one is the epidemic-free equilibrium E0=Λ/μ,0,0,0, which can be obtained for any parameter settings, while the other state is the endemic equilibrium E−∗=S∗,E∗,I∗,R∗, which is a positive solution of the following scheme:(15)Λ−μs−βI+αEfIS+γR=0,βI+αEfIS−μ+ηE=0,ηE−μI−νI−δI=0,νI−μ+γR=0.The endemic equilibrium terms, namely,S∗,E∗,I∗, and R∗, can be expressed as follows:(16)S∗=μI∗fI∗βI∗+αE∗,R∗=νμ+γ,E∗=μ+ν+δηI∗,and Λ−ημ2fI∗/βη+αμ+ν+δ+μγ/μ+γ−βμI∗=0.DefineFI=Λ−ημ2fI∗/βη+αμ+ν+δ+μγ/μ+γ−βμI∗. Based on the assumption (H1), the function FI is decreasing. Since (17)F0=Λ−μ2ηβη+αμ+ν+δf0=μ2ηβη+αμ+ν+δβη+αμ+ν+δΛμ2ηf0−1f0=μ2ηβη+αμ+ν+δR0−1f0.The equationFI=0 possesses a unique positive solution I∗ if R0>1. Therefore, a unique endemic equilibrium E−∗=S∗,E∗,I∗,R∗ exists for model (1).The next lemma demonstrates that the solutions for model (1) are limited, contained in a reduced set, and continuous for all t>0.Lemma 2.
Model (1) is decidedly invariant where pulls of each solution with initial conditions begin in its state space X. Also, every direction of model (1) will in the long run remain in a reduced subset of Γ.Proof.
Joining all conditions in (1) and considering Nt=St+Et+It+Rt, we have the following:(18)Λ−μ+δN≤dNdt=Λ−μN−δI≤Λ−μN.
Hereafter, by integrating (18), we obtain the following equation:(19)Λμ+δ+N0−Λμ+δe−μ+δt≤Nt≤Λμ+N0−Λμe−μt.
This concludes the proof of the lemma.Remark 5.
Lemma2 shows in particular that the dynamics of model (1) can be studied in the restricted set Γ obtained in (7).


### 4.2.1. Epidemic-Free Dynamics of Model (1)

Here, the global asymptotic stability of the epidemic-free equilibrium E0 is investigated. In particular, we prove Theorem1 (I).Proof.
Construct the following Lyapunov function:VS,E,I,R=1/2S−Λ/μ2+θ1E+θ2I,
whereθ1=Λ/μ and θ2=θ1/ημ+η1−R0−εifR0<10ifR0=1 for each adequately small ε>0.
Hence, the time derivative ofV for a solution of model (1) is as follows:(20)dVdt=S−ΛμdSdt+θ1βI+αEfIS−μ+ηE+θ2ηE−μ+v+δI=−µS−Λμ2−S−ΛμβSI+αEfI+γRS−Λμ+θ1βSI+αEfI−θ1μ+η−θ2ηE−θ2μ+v+δI≤−μ+βI+αEfIS−Λμ2+θ1Λβ−μfIθ1μ+η−θ2ημfII−θ2μ+v+δI,where the following is applied:(21)βSI−αEfI=βI−αEfIS−Λμ+ΛβI−αEμfI,γs−ΛμR≤0.
IfR0<1, and since fI=f0+f′0I+oI, we get the following:(22)θ1Λβ−µfIθ1μ+η−θ2η=θ1Λβ−µf0μ+η+μf0θ2η−µθ1µ+η−θ2ηf′0I+oI≤−θ1µf0μ+η1−R0+μf0θ2η−μf′0θ1μ+η−θ2ηI.
Moreover, sinceθ2=θ1/ημ+η1−R0−εifR0<10ifR0=1,
we have the following:(23)−θ1μf0μ+η1−R0+μf0θ2η=−μηf0ε,θ1μ+η−θ2η=R0θ1μ+η+εη.
Hence,dV/dt≤−μ+βI+αE/fIS−Λ/μ2−ηf0ε/fII−θ2μ+v+δI−R0θ1μ+η+εη/fIf′0I2. Note the nonnegativity of the functions S,E,I,andR. Also, note that the relationships in the right side of the last inequality are nonpositive; i.e., dV/dt≤0, if and only if dV/dt=0 Consequently, the best invariant S=Λ/μ,E=0,I=0,andR=0 set in S,E,I,R:dV/dt=0 is a singleton E0.
IfR0=1, then(24)dVdt≤−μ+βI+αEfIS−Λμ2−θ1μ+ηfII2f′0.
By LaSalle’s invariance principle [39, 40], the solutions of model (1) tend to M⊂S,E,I,R||S=Λ/μ,E=0,I=0,R=0, the biggest invariant subset of model (1). From the description of model (1), M=E0 is a singleton set. Thus, E0 is universally asymptotically constant on the set Γ if R0≤0.
WhenR0>1, the Jacobian of model (1) at E0 is given by the following equation:(25)JE0=−μ−Λαμf0−Λβμf0γ0−μ+η+βαΛμf0βΛμf000η−μ+ν+δ000ν−μ+γ,with eigenvalues −µ<0,−µ+v+δ<0,−μ+γ<0, and(26)−μ+η+βαΛμf0βαΛμf0η−μ+ν+δ=μ+ημ+v+δ−R0μη=μημ+ημ+v+δμη−R0.
Therefore, the epidemic-free equilibrium is perturbed ifR0>μ+ημ+v+δ/μη>1. This concludes the proof.


### 4.2.2. Endemic Dynamics of Model (1)

Here, the global asymptotic stability of the endemic equilibriumE∗ is addressed. In particular, Theorem 1 (II) is proved.Proof.
The Jacobian of (2) at E∗ is as follows:(27)JE∗=−μ−βI∗+αE∗fI∗βαS∗fI∗βS∗I∗1−I∗f2I∗γβI∗+αE∗fI∗−μ+η000η−μ−ν−δ000ν−μ+γ.
The characteristic polynomial of the JacobianJE∗ is as follows:(28)λ4+c1λ3+c2λ2+c3λ+c4=0,c1=4μ+η+v+δ+γ+βI∗+αE∗fI∗>0,c2=μ+v+δμ+γ+2μ+v+δ+γβI∗+αE∗fI∗+2µ+η+µ+ηµ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0,c3=μ+v+δμ+γβI∗+αE∗fI∗+2μ+η+2μ+v+δ+γμ+ημ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0,c4=μ+v+δμ+γβI∗+αE∗fI∗+2μ+η+μ+v+δ+μ+γμ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0.
It can be verified thatc1c2−c3>0,c3c1c2−c3−c12c4>0. Hence, the asymptotic stability of E∗ can be determined by exploiting the Routh–Hurwitz criterion.
Now, by proving thatS∗,E∗,andI∗ of model (1) are globally asymptotically stable, we will immediately prove the same type of stability for the endemic equilibrium of model (1).


## 4.2.1. Epidemic-Free Dynamics of Model (1)

Here, the global asymptotic stability of the epidemic-free equilibrium E0 is investigated. In particular, we prove Theorem1 (I).Proof.
Construct the following Lyapunov function:VS,E,I,R=1/2S−Λ/μ2+θ1E+θ2I,
whereθ1=Λ/μ and θ2=θ1/ημ+η1−R0−εifR0<10ifR0=1 for each adequately small ε>0.
Hence, the time derivative ofV for a solution of model (1) is as follows:(20)dVdt=S−ΛμdSdt+θ1βI+αEfIS−μ+ηE+θ2ηE−μ+v+δI=−µS−Λμ2−S−ΛμβSI+αEfI+γRS−Λμ+θ1βSI+αEfI−θ1μ+η−θ2ηE−θ2μ+v+δI≤−μ+βI+αEfIS−Λμ2+θ1Λβ−μfIθ1μ+η−θ2ημfII−θ2μ+v+δI,where the following is applied:(21)βSI−αEfI=βI−αEfIS−Λμ+ΛβI−αEμfI,γs−ΛμR≤0.
IfR0<1, and since fI=f0+f′0I+oI, we get the following:(22)θ1Λβ−µfIθ1μ+η−θ2η=θ1Λβ−µf0μ+η+μf0θ2η−µθ1µ+η−θ2ηf′0I+oI≤−θ1µf0μ+η1−R0+μf0θ2η−μf′0θ1μ+η−θ2ηI.
Moreover, sinceθ2=θ1/ημ+η1−R0−εifR0<10ifR0=1,
we have the following:(23)−θ1μf0μ+η1−R0+μf0θ2η=−μηf0ε,θ1μ+η−θ2η=R0θ1μ+η+εη.
Hence,dV/dt≤−μ+βI+αE/fIS−Λ/μ2−ηf0ε/fII−θ2μ+v+δI−R0θ1μ+η+εη/fIf′0I2. Note the nonnegativity of the functions S,E,I,andR. Also, note that the relationships in the right side of the last inequality are nonpositive; i.e., dV/dt≤0, if and only if dV/dt=0 Consequently, the best invariant S=Λ/μ,E=0,I=0,andR=0 set in S,E,I,R:dV/dt=0 is a singleton E0.
IfR0=1, then(24)dVdt≤−μ+βI+αEfIS−Λμ2−θ1μ+ηfII2f′0.
By LaSalle’s invariance principle [39, 40], the solutions of model (1) tend to M⊂S,E,I,R||S=Λ/μ,E=0,I=0,R=0, the biggest invariant subset of model (1). From the description of model (1), M=E0 is a singleton set. Thus, E0 is universally asymptotically constant on the set Γ if R0≤0.
WhenR0>1, the Jacobian of model (1) at E0 is given by the following equation:(25)JE0=−μ−Λαμf0−Λβμf0γ0−μ+η+βαΛμf0βΛμf000η−μ+ν+δ000ν−μ+γ,with eigenvalues −µ<0,−µ+v+δ<0,−μ+γ<0, and(26)−μ+η+βαΛμf0βαΛμf0η−μ+ν+δ=μ+ημ+v+δ−R0μη=μημ+ημ+v+δμη−R0.
Therefore, the epidemic-free equilibrium is perturbed ifR0>μ+ημ+v+δ/μη>1. This concludes the proof.


## 4.2.2. Endemic Dynamics of Model (1)

Here, the global asymptotic stability of the endemic equilibriumE∗ is addressed. In particular, Theorem 1 (II) is proved.Proof.
The Jacobian of (2) at E∗ is as follows:(27)JE∗=−μ−βI∗+αE∗fI∗βαS∗fI∗βS∗I∗1−I∗f2I∗γβI∗+αE∗fI∗−μ+η000η−μ−ν−δ000ν−μ+γ.
The characteristic polynomial of the JacobianJE∗ is as follows:(28)λ4+c1λ3+c2λ2+c3λ+c4=0,c1=4μ+η+v+δ+γ+βI∗+αE∗fI∗>0,c2=μ+v+δμ+γ+2μ+v+δ+γβI∗+αE∗fI∗+2µ+η+µ+ηµ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0,c3=μ+v+δμ+γβI∗+αE∗fI∗+2μ+η+2μ+v+δ+γμ+ημ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0,c4=μ+v+δμ+γβI∗+αE∗fI∗+2μ+η+μ+v+δ+μ+γμ+βI∗+αE∗fI∗−βαS∗fI∗βI∗+αE∗fI∗>0.
It can be verified thatc1c2−c3>0,c3c1c2−c3−c12c4>0. Hence, the asymptotic stability of E∗ can be determined by exploiting the Routh–Hurwitz criterion.
Now, by proving thatS∗,E∗,andI∗ of model (1) are globally asymptotically stable, we will immediately prove the same type of stability for the endemic equilibrium of model (1).


## 4.3. Proof of Theorem2

For proving Theorem2, we first prove the existence and uniqueness of a positive global solution for model (2).Theorem 3.
For some random initial solutionsS0,E0,I0,R0∈Γ, there is an nontrivial positive solution St,Et,It,Rt of model (2) for t≥0, which stays in X with a likelihood of 1.Proof.
LetS0,E0,I0,R0∈Γ. Adding up the three equations in model (2) and using Nt=St+Et+Rt+It, we have dNt=Λ−μN−δIdt.
Then, ifS0,E0,I0,R0∈X for all 0≤t1≤t almost surely (briefly a.s.), then we get Λ−μ+δNt1dt≤dNt1≤Λ−μNt1dta.s.
By integration, we obtainΛ/μ+δ≤Ns≤Λ/μ. Thus, St1,Et1,It1,Rt1∈0,Λ/μ for all t1∈0,t a.s. Because the model coefficients for (2) fulfill the neighborhood Lipschitz condition, an extraordinary nearby solution exists on t1∈0,τe, where τe is the blast time. In this manner, the unique nearby solution of model (2) is certain by Itô’s equation. Now, the global nature of this solution is shown, i.e., τe=∞ a.s. Let n0>0 be appropriately big so that S0,I0, and R0 lie inside the interval 1/n0,n0. For every integer n>n0, the stop times are obtained:(29)τn=inft∈0,τe:minSt,Et,It,Rt≤1normaxSt,Et,It,Rt≥n.
Set infϕ=∞ (∞ represents the empty set). τn grows as n⟶∞. Let τ∞=limn⟶∞τn. Then, τ∞≤τe a.s.
In the following, we demonstrate thatτ∞=∞. Assume on the contrary that this is not true. Thus, there exists a steady T>0 such that Prob τ∞≤t>ε for any ε∈0,1. As a result, a whole number n1≥n0 exists for which(30)Probτn≤T≥ε,n≥n1.
Describe the positiveC2 function V:D⟶ℝ++by the following equation:(31)VS,E,I,R=S−1−lnS+I−1−lnI+R−1−lnR+I−1−lnI.
IfSt,Et,It,Rt∈X, then by the Itô formulation, we obtain the following equation:(32)dV=1−1SΛ−μS−βIt+αEtfItSt+γRt+σ2It+αEt2St22f2Itdt+1−1EβIt+αEtfItSt−μ+ηE+σ2It+αEt2St2f2Itdt+1−1IηE−μ+ν+δIdt+1−1RνI−μ+γRdt−1−1SσIt+αEtfItSt+1−1IσIt+αEtfItStdBt=LVdt+σIt−St+αEtfItdBt,where(33)LV=Λ+4μ+γ+η+v+δ−μSt+Et+It+Rt+3σ2It+αEt2f2ItSt2−βIt+αEtEfItSt+βIt+αEtfIt+γR+γRs−ηIE−1Rν−ΛS−δI−γR−ηE<Λ+4μ+γ+η+ν+δ+3σ2ξ2+α2+2αf2ξ+2βξ+αfξ≔K.
Replacing this inequality in equation (32), we get the following equation:(34)dVSt,Et,It,Rt≤Kdt+σIt−StfItdBt,which implies that(35)∫0τn∧TdVSr,Er,Ir,Rr≤∫0τn∧TKdt+∫0τn∧TσIr−SrfIrdBr,where τnΛT=minτn,T. Evaluating the integrals of the last inequality gives the following equation:(36)EVSτn∧T,Iτn∧T,Eτn∧T,Rτn∧T≤VS0,E0,I0,R0+KT.
SetΩn=τn≤T. From (35), we have Prob Ωn≥ε. For each w∈Ωn, at least one exists among Sτnw,Eτnw,Iτnw, and Rτnw with a value of either n or 1/n. Hence,(37)VSτnw,Eτnw,Iτnw,Rτnw≥n−1−lnn∧1n−1−ln1n.
Next, from (34), we have the following:(38)VS0,E0,I0,R0≥Eχτnw·VSτnw,Eτnw,Iτnw,Rτnw+KT≥εn−1−lnn∧1n−1−ln1n,where χΩn is the characteristic function of Ωn.Asn⟶∞, the following contradiction is obtained:(39)∞>VS0,E0,I0,R0+KT=∞a.s.
Therefore,τ∞=∞, and the solution of model (2) shall not blast within a limited time with a probability of one. The proof is complete.Remark 6.
From Theorem1, the set is an almost surely positive invariant of the SDE (2). That is, for S0,E0,I0,R0∈Γ,(40)ProbSt,Et,It,Rt∈Γ=1∀t≥0.


### 4.3.1. Disease Extinction in the SDE Model

Here, Theorem2 (I) on the disease extinction in the stochastic model (3) will be proved.Proof.
Based on the Itô formulation,(41)dlnEt=φSt,Et,It,Rtdt+σαStfItdBt,where φ:R3+⟶R is given by the following equation:(42)φu,v=βαufν−μ+η+σ2α2u22f2v.
Hence,(43)lnEt=lnI0+∫0tφSs,Isds+∫0tσαSsfIsdBs.
SettingGt≔∫0tσαSs/fIsdBs, we have the following equation:(44)G,Gtt≔1t∫0tσ2α2Ss2f2IsdBs≤σ2α2Λ2μ2f20<+∞.
From the strong law of large numbers for martingales [38], we obtain limt⟶∞supGt/t=0a.s.
Based on (9), we have the following equation:(45)φSs,Is=βαStfIt−μ+η+σ2α2Ss22f2Is=−σ2α22StfIt−βσ2α2+β22σ2−µ+η≤β22σ2−µ+η.
It then follows from (29) that(46)lnEt≤lnI0+∫0tβ22σ2−μ+ηds+Gt.
If we divide both sides of (46) by t and let t⟶∞, we get the following equation:(47)limt⟶∞lnItt≤β22σ2−µ+ηa.s.
Now, consider the case whenσ2<βμf0/Λ. Thus,(48)φSs,Is=−σ2α22StfIt−βσ2α2+β22σ2−μ+η≤−σ2α22Λμf0−βσ2α2+β22σ2−µ+η=Λαβμf0−σ2α2Λ22μ2f20−µ+η=µ+η2μf0αΛβ−σ2α2Λ22μ2µ+ηf20−1=µ+ηR0s−1,where(49)R0s=2μf0αΛβ−σ2α2Λ22μ2µ+ηf20.
It then follows from (49) that(50)lnIt≤lnI0+μ+ηR0s−1t+Gt.
Therefore, from the last inequality and (10),(51)limt⟶∞suplnEtt≤µ+ηRs0−1<0a.s.
The reason is thatR0<1,R0s=2μf0αΛβ−σ2α2Λ2/2μ2µ+ηf20<μf0αΛβ/μ2ηf0=μαΛβ/μ2ηf0<1, and there exists a null set N1 for which Prob N1=0 and for any ω∉N1,(52)limt⟶∞supEtωt<−c.
Thus, for each adequately smallε>0, there is T1=T1ω for which(53)Etω≤e−c+εt,∀t≥T1.
From the 3rd equation of the stochastic model (3), for each ω∈Ω, if t≥T1ω,(54)Itω=e−μ+v+δt∫0teµ+v+δsIsds+I0≤I0e−μ+v+δt+ηe−μ+v+δt∫0teµ+v+δsIsds+νe−μ+v+δt∫T1teμ+v+δ−c+εsds.
Thus, for any(55)ω∉N1,limsup1tlnItω≤min−μ+v+δ,−c+εa.s.
Lettingε⟶0, we get limsup1/tlnItω≤min−μ+v+δ,−c,a.s. Correspondingly, there is a null set N2 such that Prob N2=0 and for each ω∉N2,(56)limsupItωt<−λ∼a.s.for a constant λ∼>0. Therefore, for each adequately small ε>0, there is T2=T2ω such that Itω≤e−λ∼+εt,∀t≥T2. Similarly, we have the following equation:(57)Rtω=e−μ+γtv∫0teµ+γsIsds+R0≤R0e−µ+γt+ve−μ+γt∫0T2eµ+γtIsds+ve−µ+γt∫T2teµ+γ−λ+ε∼sIsds.
It follows that for anyω∉N2,(58)limsup1tlnRtω≤min−μ+γ,−λ˜+εa.s.
Lettingε⟶0, we get linsup1/tlnRtω≤min−μ+γ,−λ˜,a.s. Likewise, a null set N3 exists so that Prob N3=0 and for all ω∉N3,(59)linsupRtωt<−λ˜,a.s.,for some constant −λ˜>0. Thus, for any adequately small ε>0, there exists T3=T3ω for which Rtω≤e−λ˜+εt,∀t≥T3. Finally, we consider St. In view of the above analysis, there exists the null set N=N1∪N2∪N3 and T=Tω=maxT1,T2,T3 for which
ProbN=0 and for all ω∉N,(60)dSt+Et+It+Rt=Λ−δIt−μSt+Et+It+Rtdt≥Λ−δe−λ˜+εt−μSt+Et+It+Rtdt,∀t≥T.
This implies(61)1t∫0tSt+Et+It+Rtds≥Λμ−δt∫0te−λ˜+εtds−φt,where (62)φt=1μSt+Et+It+Rtt−S0+E0+I0+R0tandlimt⟶∞φt=0a.s.
For a randomε, we have the following equation:(63)limt⟶∞inf1t∫0tSs+Is+Es+Rsds≥Λμa.s.
From Remark6, we deduce that(64)limt⟶∞1t∫0tSs+Is+Es+Rsds=Λμa.s.
Together with the aforementioned results, we getlimt⟶∞1/tSsds=Λ/μa.s. Hence, the proof is finished.


### 4.3.2. Stochastic Asymptotic Stability

In this subsection, we show that under mild additional conditions the solutions of model (2) converge to the endemic state a.s., and in particular, we prove Theorem 2 (II). We will initially demonstrate the asymptotic stability of the Markov semigroup by showing the existence of an invariant density for the semigroup.Lemma 3.
For each pointx0,y0,z0∈X and t>0, the progress likelihood work Pt,x0,y0,z0,A possesses a continuous density kt,x,y,z;x0,y0,z0.Proof.
For proving this lemma, we utilize the Hörmander hypothesis [41] on the presence of smooth densities of the change likelihood work for dispersion processes.
Let(65)a0S,E,I=Λ−µS−βI+αEfIS+γRβI+αEfIS−μE−ηEµE−μv+δI,and(66)a1S,E,I=−σI+αEfISσI+αEfIS0.
By straight computations, the Lie bracketa0,a1 is a vector field expressible as follows:(67)a1=a0,a1=−σA11f2IσA11f2I−SI+αEησfI,where(68)A11=−I2γ+EEαγ−Nαγ+Sαγ−Sη+Sαη−αΛ+Sαμ+IE−N+Eαγ−Λ+Sγ+δ+μfI+SI+αE−Eη+Iv+δ+ωf′I.
Seta3=a1,a2. The vector fields a1,a2,a3 are linearly independent on the space X. Hence, for all S,E,I∈X, a1,a2,anda3 span the space X. Based on the Hörmander theorem [42, 43], the transition probability function Pt,x0,y0,z0,A has a continuous density kt,x,y,z;x0,y0,z0 and k∈C∞0,∞×X×X. Next, the positivity of k is examined using sustenance theorems [41, 44].
Pick a fixed pointx0,y0,z0∈X and a function φ∈L20,T,ℝ. Note this system of integral equations:(69)xϕt=x0+∫0tf1xϕs,yϕs,zϕs−σϕxϕs+αyϕsfyϕsxϕsds,yϕt=y0+∫0tf2xϕs,yϕs,zϕsσϕxϕs+αyϕsfyϕsxϕsds,zϕt=z0+∫0tf3xϕs,yϕs,zϕsds,where f1x,y,z,f2x,y,z,andf3x,y,z are given in (11).
LetX=x,y,zTandX0=x0,y0,z0T. Let DX0;ϕ be the Fréchet derivative of the h⟶Xϕ+hT function from L20,T;ℝ to X. If for some ϕ∈L20,T;ℝ the derivative DX0;ϕ has a rank of 3, then kT,x,y,z;x0,y0,z0>0 for X=XϕT.Letψt=f′Xϕt+φg′Xϕt, where f′ and g′ are, respectively, the Jacobians of f=f1x,y,zf2x,y,zf3x,y,z and −σxy/fyσxy/fy0.
LetQt,t0, for 0≤t0≤t≤T, be a matrix function for which Qt0,t0=Id and ∂Qt,t0/∂t=ψtQt,t0. Then, DX0;ϕh=∫0TQT,sgshsds.Lemma 4.
For everyx0,y0,z0∈Πandx,y,z∈Π, there exists kT,x,y,z;x0,y0,z0>0, where Π is characterized as in (68).Proof.
Since a continuous control workϕ is considered, the inequality (35) could be supplemented by these differential equations:(70)x′ϕt=f1xϕs,yϕs,zϕs−σϕxϕs+αyϕsfyϕsxϕs,y′ϕt=f2xϕs,yϕs,zϕs+σϕxϕs+αyϕsfyϕsxϕs,z′ϕt=f3xϕs,yϕs,zϕs.
Firstly, the rank ofDX0;ϕ is shown to be 3. Let
ht=χT−ε,Ttfyϕt/χϕt+αyϕt,t∈0,T, where χ is a characteristic function. Since(71)QT,s=Id+ψTs−T+12ψ2Ts−T2+os−T2,we obtain DX0;φh=εV−1/2ε2ψTv+1/6ε3ψ2Tv+oε3, where ν=−σσ0, and(72)ψT.v=σ−x+y+αzβfy+μf2y+xy+αzβf′yf2y−ν+δ+μσβσx−y−αzfy−xy+αzf′yf2y.
Thus,v,ψTv,andψ2Tv are straightly autonomous and the subsidiary DX0;ϕ has a rank of 3.
Next, for anyX0∈Ωand X∈Ω, we demonstrate the existence of a control work ϕ and T > 0 for which Xφ0=X0andXφT=X. Set ωϕ=xϕ+yϕ+zϕ. Model (37) becomes(73)g1x,w,z=Λ−μx−βz+αw−x−zfzx+γN−ω−σφz+αω−z−zStf,g2x,w,z=Λ+μz−μz−vz−δz−γ+μw,g3x,w,z=ηw−x−z−μ+v+δz.
LetΠ0=x,w,z∈:0<x,z<Λ/μ,Λ/Λ+μ<w<Λ/μandx,z,<w. For any x0,w0,z0∈Π0and.x1,w1,z1∈Π0, it can be claimed that there exists a control function ϕandT>0 for which
xϕ0,wϕ0,zϕ0=x0,w0,z0andxϕT,wϕT,zϕT=x1,w1,z11. We create the function ϕ in the next steps. First of all, we determine a positive constant T and a differentiable function wϕ:0,T⟶Λ/μ+γ,Λ/μ, for which(74)wϕ0=w0,wϕ′T=w1,wϕ′0=g2x0,w0,z0=w0d,wϕ′T=g2x1,w1,z1=wTdandΛ−μ+γwϕt<wϕ′t<Λ−μwϕtfort∈0,T.
For achieving this, the domain of the functionwϕ is divided into 3 segments 0,ε,ε,Tε,andT−ε,T, where 0<ε<T/2. Let(75)η=12minω0−Λμ+γ,ω1−Λμ+γ,Λμ−ω0,Λμ−ω1.
Ifωϕ∈Λ/μ+γ+m,Λ/μ−m, then we have the following equation:(76)Λ−μ+γωϕt<−μ+γm<0andΛ−μωϕt>μm>0fort∈0,t.
Based on (41), a C2 function ωϕ:0,ε⟶Λ/μ+γ+m,Λ/μ−m can be obtained for which(77)ωϕ0=ω0,ωϕ′0=ω0d,ωϕ′ε=0,where ωϕ satisfies (40) for t∈0,t. Similarly, a C2 function ωϕ:T−ε,T⟶Λ/μ+γ+m,Λ/μ−m is constructed so that(78)ωϕT=ω0,ωϕ′T=ωTd,ωϕ′T−ε=0,where ωϕ satisfies (40) for t∈T−ε,T. If we take T adequately big, we can spread the function ωϕ:0,ε∩T−ε,T⟶Λ/μ+γ+m,Λ/μ−m to a C2 function ωϕ on the whole segment 0,T for which Λ−μ+γ·ωϕt<−μ+γm<ωϕ′t<μm<Λ−μωϕt for ε,T−ε. So, the function ωϕ satisfies (41) on [0, T].
As a result, a continuous functionφ can be determined from the first equation of (38), while two functions xϕ and zϕ can be found where these functions satisfy the other equations in (38). This finishes the proof.Lemma 5.
Assume thatR0s>1. For any density g, we get limt⟶∞∭ΠPtgx,y,zdxdydz=1.
whereΠ is obtained from (13).Proof.
Following the proof of Lemma 5.6, we substituteZt=St+Et+It. Then, model (3) can be rewritten as follows:(79)dSt=g1St,Zt,Itdt−σIt+αZt−St−ItStfItdBt,dEt=g2St,Zt,Itdt−σIt+αZt−St−ItStfItdBt,dIt=g3St,Zt,Itdt,where g1x,w,z,g2x,w,z, andg3x,w,z are introduced in (38). Since St,Et,It is a positive solution of model (5) with a probability of 1, and given g2, we have the following equation:(80)Λ−μ+γZt<dZtdt<Λ−μZt,t∈0,∞a.s.
Now, for almost everyw∈Ω, we can show that there exists t0=t0w for which(81)Λμ+γ<Ztw<dZtdt<Λ−μZt,fort>t0.
Actually, three cases exist.(a)
Z0∈Λ/μ+γ,Λ/μ: the conclusion is obvious from (45).(b)
Z0∈0,Λ/μ+γ: assume on the contrary that our claim is not true. Then, there would be Ω′∈Ω with Prob (Ω′) > 0 for which Z0∈0,Λ/μ+γ. From (44), Ztw is carefully expanding on [0,∞) for any w∈Ω′ . Consequently,(82)limt⟶∞Ztw=Λμ+γ,w∈Ω′.From (43), we get limt⟶∞Stw=limt⟶∞Itw=0,w∈Ω′ and consequently limt⟶∞Etw=Λ/μ+γ,w∈Ω′. Thus, we get using the Itô formula,(83)dlnEt=βαfItSt−μ+η−σ2α2St22f2Itdt+σαStfItdBt.Hence,(84)lnEt−lnE0t=1t∫0tβαfIsSs−μ+η−σ2α2Ss22f2Is+1t∫0tσαSsfIsdBs.Since1/t∫0tσ2α2Ss2/f2Isds≤σ2α2Λ2/μ2f20<+∞, and using the strong law of large numbers for martingales [20], we get the following:(85)limt⟶∞1t∫0tσαSsfIsdBs=0.Therefore, taking into consideration the continuity of the functionsSt,Et,It, and fIt, we obtain the following:(86)limt⟶∞1t∫0tβαSsfIs−μ+η−σ2α2Ss22f2Isds+1t∫0tσαSsfIsdBs=−μ+η.This contradicts the limitlimt⟶∞lnEt−lnE0/t=0, and thus, the claim is proved.(c)
Z0∈Λ/μ+γ,+∞: we use again a proof by contradiction with arguments similar to those in (b) to deduce that there is Ω′∈Ω with Prob (Ω′) > 0 for which(87)limt⟶∞Ztw=Λμ,w∈Ω′.
Using (44) and for any w∈Ω′, we get the following:(88)g2x,y,z=Λ−μ+δ+νZt+δEt+St+γRt+νSt+νEt,Ztw=e−tμ+δ+νZ0+∫0tesμ+δ+νΛ+δEt+St+γR+νSt+νEtds.Itw=e−tμ+δ+νR0+η∫0tesμ+δ+νZsw−Ssw−Itwds.
Hence,limt⟶∞Etw=0, limt⟶∞Stw=0, and limt⟶∞Itw=0 for any w∈Ω′. Hence,(89)limt⟶∞lnEt−lnE0t=limt⟶∞1t∫0tβαSsfIs−μ+η−σ2α2Ss22f2Isds+1t∫0tσαSsfIsdBs=βαΛμfI0−μ+η−σ2α2Λ22μ2f20=μ+η2μf0βαΛ2μ2f20μ+η−1>0,a.s.onΩ′.
This is contradictory to the assumption thatlimt⟶∞It=0 a.s. and the claim follows. Remark 7: from Lemmas 4 and 5, we realize that when the Fokker–Planck equation (11) has a stationary solution U∗, then sup U∗ = Π.Lemma 6.
Assume thatR0s > 1, the semigroup {P (t)} t ≥ 0 is either sweeping with respect to minimal sets or asymptotically stable.Proof.
By Lemma3, the operator family {P (t)} t ≥ 0 is a fundamental Markov semigroup with a constant kernel kt,x,y,z,x0,y0,z0 for t > 0. Then, the appropriation of (St,Et,It) possesses a density Ux,y,z,t, which fulfills (19). From Lemma 5, the semigroup {P (t)} t ≥ 0 can be restricted to the space L0 (Π). As indicated by Lemma 4, for each f ∈ D, we have the following:(90)∫0∞Ptfdt>0,a.s.
Thus, from Lemma1, the semigroup {P(t)}, t ≥ 0 is asymptotically stable or is sweeping with respect to minimal sets.


## 4.3.1. Disease Extinction in the SDE Model

Here, Theorem2 (I) on the disease extinction in the stochastic model (3) will be proved.Proof.
Based on the Itô formulation,(41)dlnEt=φSt,Et,It,Rtdt+σαStfItdBt,where φ:R3+⟶R is given by the following equation:(42)φu,v=βαufν−μ+η+σ2α2u22f2v.
Hence,(43)lnEt=lnI0+∫0tφSs,Isds+∫0tσαSsfIsdBs.
SettingGt≔∫0tσαSs/fIsdBs, we have the following equation:(44)G,Gtt≔1t∫0tσ2α2Ss2f2IsdBs≤σ2α2Λ2μ2f20<+∞.
From the strong law of large numbers for martingales [38], we obtain limt⟶∞supGt/t=0a.s.
Based on (9), we have the following equation:(45)φSs,Is=βαStfIt−μ+η+σ2α2Ss22f2Is=−σ2α22StfIt−βσ2α2+β22σ2−µ+η≤β22σ2−µ+η.
It then follows from (29) that(46)lnEt≤lnI0+∫0tβ22σ2−μ+ηds+Gt.
If we divide both sides of (46) by t and let t⟶∞, we get the following equation:(47)limt⟶∞lnItt≤β22σ2−µ+ηa.s.
Now, consider the case whenσ2<βμf0/Λ. Thus,(48)φSs,Is=−σ2α22StfIt−βσ2α2+β22σ2−μ+η≤−σ2α22Λμf0−βσ2α2+β22σ2−µ+η=Λαβμf0−σ2α2Λ22μ2f20−µ+η=µ+η2μf0αΛβ−σ2α2Λ22μ2µ+ηf20−1=µ+ηR0s−1,where(49)R0s=2μf0αΛβ−σ2α2Λ22μ2µ+ηf20.
It then follows from (49) that(50)lnIt≤lnI0+μ+ηR0s−1t+Gt.
Therefore, from the last inequality and (10),(51)limt⟶∞suplnEtt≤µ+ηRs0−1<0a.s.
The reason is thatR0<1,R0s=2μf0αΛβ−σ2α2Λ2/2μ2µ+ηf20<μf0αΛβ/μ2ηf0=μαΛβ/μ2ηf0<1, and there exists a null set N1 for which Prob N1=0 and for any ω∉N1,(52)limt⟶∞supEtωt<−c.
Thus, for each adequately smallε>0, there is T1=T1ω for which(53)Etω≤e−c+εt,∀t≥T1.
From the 3rd equation of the stochastic model (3), for each ω∈Ω, if t≥T1ω,(54)Itω=e−μ+v+δt∫0teµ+v+δsIsds+I0≤I0e−μ+v+δt+ηe−μ+v+δt∫0teµ+v+δsIsds+νe−μ+v+δt∫T1teμ+v+δ−c+εsds.
Thus, for any(55)ω∉N1,limsup1tlnItω≤min−μ+v+δ,−c+εa.s.
Lettingε⟶0, we get limsup1/tlnItω≤min−μ+v+δ,−c,a.s. Correspondingly, there is a null set N2 such that Prob N2=0 and for each ω∉N2,(56)limsupItωt<−λ∼a.s.for a constant λ∼>0. Therefore, for each adequately small ε>0, there is T2=T2ω such that Itω≤e−λ∼+εt,∀t≥T2. Similarly, we have the following equation:(57)Rtω=e−μ+γtv∫0teµ+γsIsds+R0≤R0e−µ+γt+ve−μ+γt∫0T2eµ+γtIsds+ve−µ+γt∫T2teµ+γ−λ+ε∼sIsds.
It follows that for anyω∉N2,(58)limsup1tlnRtω≤min−μ+γ,−λ˜+εa.s.
Lettingε⟶0, we get linsup1/tlnRtω≤min−μ+γ,−λ˜,a.s. Likewise, a null set N3 exists so that Prob N3=0 and for all ω∉N3,(59)linsupRtωt<−λ˜,a.s.,for some constant −λ˜>0. Thus, for any adequately small ε>0, there exists T3=T3ω for which Rtω≤e−λ˜+εt,∀t≥T3. Finally, we consider St. In view of the above analysis, there exists the null set N=N1∪N2∪N3 and T=Tω=maxT1,T2,T3 for which
ProbN=0 and for all ω∉N,(60)dSt+Et+It+Rt=Λ−δIt−μSt+Et+It+Rtdt≥Λ−δe−λ˜+εt−μSt+Et+It+Rtdt,∀t≥T.
This implies(61)1t∫0tSt+Et+It+Rtds≥Λμ−δt∫0te−λ˜+εtds−φt,where (62)φt=1μSt+Et+It+Rtt−S0+E0+I0+R0tandlimt⟶∞φt=0a.s.
For a randomε, we have the following equation:(63)limt⟶∞inf1t∫0tSs+Is+Es+Rsds≥Λμa.s.
From Remark6, we deduce that(64)limt⟶∞1t∫0tSs+Is+Es+Rsds=Λμa.s.
Together with the aforementioned results, we getlimt⟶∞1/tSsds=Λ/μa.s. Hence, the proof is finished.


## 4.3.2. Stochastic Asymptotic Stability

In this subsection, we show that under mild additional conditions the solutions of model (2) converge to the endemic state a.s., and in particular, we prove Theorem 2 (II). We will initially demonstrate the asymptotic stability of the Markov semigroup by showing the existence of an invariant density for the semigroup.Lemma 3.
For each pointx0,y0,z0∈X and t>0, the progress likelihood work Pt,x0,y0,z0,A possesses a continuous density kt,x,y,z;x0,y0,z0.Proof.
For proving this lemma, we utilize the Hörmander hypothesis [41] on the presence of smooth densities of the change likelihood work for dispersion processes.
Let(65)a0S,E,I=Λ−µS−βI+αEfIS+γRβI+αEfIS−μE−ηEµE−μv+δI,and(66)a1S,E,I=−σI+αEfISσI+αEfIS0.
By straight computations, the Lie bracketa0,a1 is a vector field expressible as follows:(67)a1=a0,a1=−σA11f2IσA11f2I−SI+αEησfI,where(68)A11=−I2γ+EEαγ−Nαγ+Sαγ−Sη+Sαη−αΛ+Sαμ+IE−N+Eαγ−Λ+Sγ+δ+μfI+SI+αE−Eη+Iv+δ+ωf′I.
Seta3=a1,a2. The vector fields a1,a2,a3 are linearly independent on the space X. Hence, for all S,E,I∈X, a1,a2,anda3 span the space X. Based on the Hörmander theorem [42, 43], the transition probability function Pt,x0,y0,z0,A has a continuous density kt,x,y,z;x0,y0,z0 and k∈C∞0,∞×X×X. Next, the positivity of k is examined using sustenance theorems [41, 44].
Pick a fixed pointx0,y0,z0∈X and a function φ∈L20,T,ℝ. Note this system of integral equations:(69)xϕt=x0+∫0tf1xϕs,yϕs,zϕs−σϕxϕs+αyϕsfyϕsxϕsds,yϕt=y0+∫0tf2xϕs,yϕs,zϕsσϕxϕs+αyϕsfyϕsxϕsds,zϕt=z0+∫0tf3xϕs,yϕs,zϕsds,where f1x,y,z,f2x,y,z,andf3x,y,z are given in (11).
LetX=x,y,zTandX0=x0,y0,z0T. Let DX0;ϕ be the Fréchet derivative of the h⟶Xϕ+hT function from L20,T;ℝ to X. If for some ϕ∈L20,T;ℝ the derivative DX0;ϕ has a rank of 3, then kT,x,y,z;x0,y0,z0>0 for X=XϕT.Letψt=f′Xϕt+φg′Xϕt, where f′ and g′ are, respectively, the Jacobians of f=f1x,y,zf2x,y,zf3x,y,z and −σxy/fyσxy/fy0.
LetQt,t0, for 0≤t0≤t≤T, be a matrix function for which Qt0,t0=Id and ∂Qt,t0/∂t=ψtQt,t0. Then, DX0;ϕh=∫0TQT,sgshsds.Lemma 4.
For everyx0,y0,z0∈Πandx,y,z∈Π, there exists kT,x,y,z;x0,y0,z0>0, where Π is characterized as in (68).Proof.
Since a continuous control workϕ is considered, the inequality (35) could be supplemented by these differential equations:(70)x′ϕt=f1xϕs,yϕs,zϕs−σϕxϕs+αyϕsfyϕsxϕs,y′ϕt=f2xϕs,yϕs,zϕs+σϕxϕs+αyϕsfyϕsxϕs,z′ϕt=f3xϕs,yϕs,zϕs.
Firstly, the rank ofDX0;ϕ is shown to be 3. Let
ht=χT−ε,Ttfyϕt/χϕt+αyϕt,t∈0,T, where χ is a characteristic function. Since(71)QT,s=Id+ψTs−T+12ψ2Ts−T2+os−T2,we obtain DX0;φh=εV−1/2ε2ψTv+1/6ε3ψ2Tv+oε3, where ν=−σσ0, and(72)ψT.v=σ−x+y+αzβfy+μf2y+xy+αzβf′yf2y−ν+δ+μσβσx−y−αzfy−xy+αzf′yf2y.
Thus,v,ψTv,andψ2Tv are straightly autonomous and the subsidiary DX0;ϕ has a rank of 3.
Next, for anyX0∈Ωand X∈Ω, we demonstrate the existence of a control work ϕ and T > 0 for which Xφ0=X0andXφT=X. Set ωϕ=xϕ+yϕ+zϕ. Model (37) becomes(73)g1x,w,z=Λ−μx−βz+αw−x−zfzx+γN−ω−σφz+αω−z−zStf,g2x,w,z=Λ+μz−μz−vz−δz−γ+μw,g3x,w,z=ηw−x−z−μ+v+δz.
LetΠ0=x,w,z∈:0<x,z<Λ/μ,Λ/Λ+μ<w<Λ/μandx,z,<w. For any x0,w0,z0∈Π0and.x1,w1,z1∈Π0, it can be claimed that there exists a control function ϕandT>0 for which
xϕ0,wϕ0,zϕ0=x0,w0,z0andxϕT,wϕT,zϕT=x1,w1,z11. We create the function ϕ in the next steps. First of all, we determine a positive constant T and a differentiable function wϕ:0,T⟶Λ/μ+γ,Λ/μ, for which(74)wϕ0=w0,wϕ′T=w1,wϕ′0=g2x0,w0,z0=w0d,wϕ′T=g2x1,w1,z1=wTdandΛ−μ+γwϕt<wϕ′t<Λ−μwϕtfort∈0,T.
For achieving this, the domain of the functionwϕ is divided into 3 segments 0,ε,ε,Tε,andT−ε,T, where 0<ε<T/2. Let(75)η=12minω0−Λμ+γ,ω1−Λμ+γ,Λμ−ω0,Λμ−ω1.
Ifωϕ∈Λ/μ+γ+m,Λ/μ−m, then we have the following equation:(76)Λ−μ+γωϕt<−μ+γm<0andΛ−μωϕt>μm>0fort∈0,t.
Based on (41), a C2 function ωϕ:0,ε⟶Λ/μ+γ+m,Λ/μ−m can be obtained for which(77)ωϕ0=ω0,ωϕ′0=ω0d,ωϕ′ε=0,where ωϕ satisfies (40) for t∈0,t. Similarly, a C2 function ωϕ:T−ε,T⟶Λ/μ+γ+m,Λ/μ−m is constructed so that(78)ωϕT=ω0,ωϕ′T=ωTd,ωϕ′T−ε=0,where ωϕ satisfies (40) for t∈T−ε,T. If we take T adequately big, we can spread the function ωϕ:0,ε∩T−ε,T⟶Λ/μ+γ+m,Λ/μ−m to a C2 function ωϕ on the whole segment 0,T for which Λ−μ+γ·ωϕt<−μ+γm<ωϕ′t<μm<Λ−μωϕt for ε,T−ε. So, the function ωϕ satisfies (41) on [0, T].
As a result, a continuous functionφ can be determined from the first equation of (38), while two functions xϕ and zϕ can be found where these functions satisfy the other equations in (38). This finishes the proof.Lemma 5.
Assume thatR0s>1. For any density g, we get limt⟶∞∭ΠPtgx,y,zdxdydz=1.
whereΠ is obtained from (13).Proof.
Following the proof of Lemma 5.6, we substituteZt=St+Et+It. Then, model (3) can be rewritten as follows:(79)dSt=g1St,Zt,Itdt−σIt+αZt−St−ItStfItdBt,dEt=g2St,Zt,Itdt−σIt+αZt−St−ItStfItdBt,dIt=g3St,Zt,Itdt,where g1x,w,z,g2x,w,z, andg3x,w,z are introduced in (38). Since St,Et,It is a positive solution of model (5) with a probability of 1, and given g2, we have the following equation:(80)Λ−μ+γZt<dZtdt<Λ−μZt,t∈0,∞a.s.
Now, for almost everyw∈Ω, we can show that there exists t0=t0w for which(81)Λμ+γ<Ztw<dZtdt<Λ−μZt,fort>t0.
Actually, three cases exist.(a)
Z0∈Λ/μ+γ,Λ/μ: the conclusion is obvious from (45).(b)
Z0∈0,Λ/μ+γ: assume on the contrary that our claim is not true. Then, there would be Ω′∈Ω with Prob (Ω′) > 0 for which Z0∈0,Λ/μ+γ. From (44), Ztw is carefully expanding on [0,∞) for any w∈Ω′ . Consequently,(82)limt⟶∞Ztw=Λμ+γ,w∈Ω′.From (43), we get limt⟶∞Stw=limt⟶∞Itw=0,w∈Ω′ and consequently limt⟶∞Etw=Λ/μ+γ,w∈Ω′. Thus, we get using the Itô formula,(83)dlnEt=βαfItSt−μ+η−σ2α2St22f2Itdt+σαStfItdBt.Hence,(84)lnEt−lnE0t=1t∫0tβαfIsSs−μ+η−σ2α2Ss22f2Is+1t∫0tσαSsfIsdBs.Since1/t∫0tσ2α2Ss2/f2Isds≤σ2α2Λ2/μ2f20<+∞, and using the strong law of large numbers for martingales [20], we get the following:(85)limt⟶∞1t∫0tσαSsfIsdBs=0.Therefore, taking into consideration the continuity of the functionsSt,Et,It, and fIt, we obtain the following:(86)limt⟶∞1t∫0tβαSsfIs−μ+η−σ2α2Ss22f2Isds+1t∫0tσαSsfIsdBs=−μ+η.This contradicts the limitlimt⟶∞lnEt−lnE0/t=0, and thus, the claim is proved.(c)
Z0∈Λ/μ+γ,+∞: we use again a proof by contradiction with arguments similar to those in (b) to deduce that there is Ω′∈Ω with Prob (Ω′) > 0 for which(87)limt⟶∞Ztw=Λμ,w∈Ω′.
Using (44) and for any w∈Ω′, we get the following:(88)g2x,y,z=Λ−μ+δ+νZt+δEt+St+γRt+νSt+νEt,Ztw=e−tμ+δ+νZ0+∫0tesμ+δ+νΛ+δEt+St+γR+νSt+νEtds.Itw=e−tμ+δ+νR0+η∫0tesμ+δ+νZsw−Ssw−Itwds.
Hence,limt⟶∞Etw=0, limt⟶∞Stw=0, and limt⟶∞Itw=0 for any w∈Ω′. Hence,(89)limt⟶∞lnEt−lnE0t=limt⟶∞1t∫0tβαSsfIs−μ+η−σ2α2Ss22f2Isds+1t∫0tσαSsfIsdBs=βαΛμfI0−μ+η−σ2α2Λ22μ2f20=μ+η2μf0βαΛ2μ2f20μ+η−1>0,a.s.onΩ′.
This is contradictory to the assumption thatlimt⟶∞It=0 a.s. and the claim follows. Remark 7: from Lemmas 4 and 5, we realize that when the Fokker–Planck equation (11) has a stationary solution U∗, then sup U∗ = Π.Lemma 6.
Assume thatR0s > 1, the semigroup {P (t)} t ≥ 0 is either sweeping with respect to minimal sets or asymptotically stable.Proof.
By Lemma3, the operator family {P (t)} t ≥ 0 is a fundamental Markov semigroup with a constant kernel kt,x,y,z,x0,y0,z0 for t > 0. Then, the appropriation of (St,Et,It) possesses a density Ux,y,z,t, which fulfills (19). From Lemma 5, the semigroup {P (t)} t ≥ 0 can be restricted to the space L0 (Π). As indicated by Lemma 4, for each f ∈ D, we have the following:(90)∫0∞Ptfdt>0,a.s.
Thus, from Lemma1, the semigroup {P(t)}, t ≥ 0 is asymptotically stable or is sweeping with respect to minimal sets.


## 5. Numerical Simulation Results

We demonstrate here the results of simulations of the deterministic and the stochastic models. These simulations clarify the effects of stochasticity on the epidemic dynamics. The simulations of the stochastic model are performed following the Milstein strategy [45]. We simulate the SDE solutions with fI=1+aI2. For the convenience of display, the simulation is set as 100 times 100 in the space-time range, the abscissa represents the time, and the ordinate represents the number of patients. The simulations can help us to investigate how the ecological perturbations and the harmfully idle periods influence the spread of epidemics. In particular, we consider the global characteristics of a general SDE model with infection forces for both the deterministic case (without infection forces) and the stochastic case (with infection forces). In the first set of simulations, the parameters of the stochastic model are set as follows: λ = 0.23, μ = 0.01, α = 0.36, β = 0.52, γ = 0.45, σ = 0.6, δ = 0.31, v = 0.13, η = 0.25, and a = 0.1 (see Figure 1).Figure 1
Temporal functions ofSt, Et, It, and Rt for the stochastic model.The results in Figure1 are based on a stochastic reproduction number of R0s = 2.8917, which is more than 1. We take the initial conditions to be (St,Et,It,Rt) = (0.9, 0.06, 0.04, 0). It is easy to see that the system is oscillating. Next, we study how environmental oscillations affect the spread of epidemics by reviewing the global dynamics of the general SEIRS model.We take the parameter values as follows:λ = 0.001, µ = 0.01, α = 0.35, β = 0.6, γ = 0.05, σ = 0.15, a = 5, δ = 0.05, ν = 0.6, η = 0.33, R0s = 1.5329 > 1, and R0 = 0.9877 < 1. The simulation results for the deterministic model are shown in Figure 2. It is easy to see that the deterministic system is stable. To understand the influence of the environmental noise on the system, we increase gradually the disturbance parameter σ = 0.15, 0.35, 0.55, and 0.75, while keeping the other parameters unchanged.Figure 2
Temporal functions ofSt, Et, It, and Rt for the deterministic model with initial values of (S0,E0,I0,R0) = (0.9, 0.06, 0.04, 0).The results are shown in Figures3–6. From these figures, we can conclude that increasing the intensity of the system disturbance gradually leads naturally to more disturbances of the relevant quantities. However, when the noise level is above a certain threshold, these quantities are severely disturbed at the beginning but then stabilize gradually.Figure 3
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values (S0,E0,I0,R0) = (0.9, 0.06, 0.04, 0), σ = 0.15, and R0s = 1.4286 > 1.Figure 4
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values (S0,E0,I0,R0) = (0.9, 0.06, 0.04, 0), σ = 0.35, and R0s = 1.4062 > 1.Figure 5
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values (S0,E0,I0,R0) = (0.9, 0.06, 0.04, 0), σ = 0.55, and R0s = 1.2035 > 1.Figure 6
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values S0,E0,I0,R0 = (0.9, 0.06, 0.04, 0), σ = 0.75, and R0s = 0.9108 < 1.Figures7–9 discuss the influence of the change in a variable on the system. The conclusion is that with the increase in α, the system has stronger disturbance and worse control ability. Therefore, the incubation period is an important variable in disease control. The existence of the incubation period will lead to the difficulty of disease control.Figure 7
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values (S0,E0,I0,R0) = (0.9, 0.06, 0.04, 0), α = 0.55, and R0s = 6.7825.Figure 8
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values (S0,E0,I0,R0) = (0.9, 0.06, 0.04, 0), α = 0.8, and R0s = 1.4329.Figure 9
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values S0,E0,I0,R0 = (0.9, 0.06, 0.04, 0), α = 1.02, and R0s = 67.5118.We computed the time series and confidence intervals of each variable, as shown in Figures10 and 11. From the simulations, we can see that the stability of the system is affected, and the fluctuation range is big. For this set of simulations, we set the parameters as follows: λ = 0.001, μ = 0.01, α = 0.75, β = 0.1, γ = 0.25, σ = 0.35, a = 0.001, δ = 0.05, ν = 0.1, and η = 0.33.Figure 10
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values (S0,E0,I0,R0) = (0.9, 0.06, 0.04, 0).Figure 11
Temporal functions ofSt, Et, It, and Rt for the stochastic model with initial values (S0,E0,I0,R0) = (0.9, 0.06, 0.04, 0) and confidence intervals.There are many variables in the system. We only discussed several representative variables in detail. In the actual disease control, we can discuss the influence of each variable on the system, so as to better control the spread of disease.Several groups of simulation results show that the conclusion of this study is correct. In the actual disease model control, we should pay attention to the types of diseases and fully consider the interference of random factors. The establishment of control variables in this study can provide basic theoretical basis and model reference for the simulation of subsequent infectious disease models.


## 6. Conclusions

Worldwide populations have been largely and negatively impacted by infectious disease outbreaks, which had detrimental effects socially and economically [5]. Individual responses go from maintaining a safe distance from infected people to wearing defensive covers, or taking immunizations. Intervention approaches seek to change human behavior, to decrease the contact rates of susceptible people [6]. Compared with other models, such as literature [21, 46], this model establishes a four-variable random infectious disease model, which adds the influence of incubation period, which is more in line with reality. At present, there are few studies on relevant theories and simulation.Natural infection forces affect the spread of epidemics. In this study, we investigated the components of a stochastic SEIRS model with a general contamination force. The stochastic effects were considered by incorporating a multiplicative background noise in the development conditions of both the susceptible and exposed populations.Our investigations uncover two important perspectives. Firstly, the generation numberRos can be used to control the stochastic elements of a SDE model based on the Markov semigroup assumptions. If Ros < 1, and with gentle additional conditions, the SDE framework has a disease-free solution set, which implies the eradication of the epidemic with a likelihood of 1. When Ros > 1, and again under mild additional conditions, the SDE framework has an endemic equilibrium. This prompts the stochastic persistence of the disease.The numberR0S is the main control variable of random infectious disease model control, which should be considered in practice. In addition, the change in initial value may also lead to uncontrollable results of the system, which brings greater challenges to infectious disease control.



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*Source: 4538045-2022-01-10.xml*