# Asymptotic Behavior of Multigroup SEIR Model with Nonlinear Incidence Rates under Stochastic Perturbations

Authors: Feng Wang; Shan Wang; Youhua Peng
Journal: Discrete Dynamics in Nature and Society
 (2020)

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## Abstract

In this paper, the asymptotic behavior of a multigroup SEIR model with stochastic perturbations and nonlinear incidence rate functions is studied. First, the existence and uniqueness of the solution to the model we discuss are given. Then, the global asymptotical stability in probability of the model withR0<1 is established by constructing Lyapunov functions. Next, we prove that the disease can die out exponentially under certain stochastic perturbation while it is persistent in the deterministic case when R0>1. Finally, several examples and numerical simulations are provided to illustrate the dynamic behavior of the model and verify our analytical results.


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## Body

## 1. Introduction

The history of human beings is full of struggle against diseases which cause great disaster to humans. At present, many countries and people around the world are suffering from the COVID-19, which has seriously affected people’s lives and brought huge losses to the economy. Epidemiology is the subject to study the spread of diseases and formulate the strategies and measures for controlling and eliminating diseases. Mathematical modeling has been widely used in epidemiology to depict the mechanism of disease transmission and study the behavior of disease. One of the classic epidemic models is the SIR model which divides the host population into three parts, the susceptible, the infective, and the removed, and records their sizes bySt, It, and Rt at time t, respectively. However, many diseases do not break out immediately, and there will be a latent period of time, so SEIR models with latent period have been widely studied. In SEIR models, the size of the exposed individuals is labeled by Et at time t.Many models have considered the case of only one group; however, groups in different communities, regions, or with different cultural backgrounds have various lifestyles, dietary habits, and so on, which will make the disease have different ways of transmission. Therefore, considering different contact patterns, transmission, or geographic distributions, it is more reasonable to divide the host population into several subgroups and study the disease interactions among different subgroups. This is known as the multigroup model. One of the earliest works on the multigroup disease model was done by Lajmanovich and Yorke [1], who discussed a class of SIS multigroup models for the transmission of gonorrhea and used Lyapunov functions to prove the stability of the unique endemic equilibrium. Since then, there has been a great quantity of literature on the multigroup model, such as [2–8].In the classic SEIR models, the incidence function takes the bilinear form. A premise for this form is that the host population is homogeneously mixed, and everyone has the same possibility to be infected when the infectives are introduced to the group. In real life, however, the population may not be homogeneously mixed, and the immunity of each person may be different such that the chances of being infected are disparate, so extending bilinear incidence to nonlinear functions can conform to the actual situation better. Many scholars have studied the epidemic models with nonlinear incidence rate, such as [4, 8–11] and the reference therein. Also, many scholars investigated the epidemic models with time delays, such as [12, 13]. In [4], the authors discussed the global stability of the multigroup epidemic model with nonlinear incidence rates of the form fkjSk,Ij, which satisfies the following assumptions:(i)
(H1) for0<Sk≤Sk0, it has 0<limIj⟶0+fkjSk,Ij/Ij=CkjSk, where Sk0 is the positive solution of certain function.(ii)
(H2)fkjSk,Ij≤CkjSkIj for any Ij>0.(iii)
(H3)CkjSk≤CkjSk0, for 0<Sk≤Sk0.This research intends to study this general form of incidence function and assumes further thatCkjSk/Sk≤K, where for K is a positive constant. The above incidence rate functions fkjSk,Ij include some special cases which can be seen in some literature, for example,(1)fkjSk,Ij=SkIj,fkjSk,Ij=SkqIj,q≥1,fkjSk,Ij=SkIj1+αIj2,fkjSk,Ij=SkIjφIj.The multigroup SEIR model with above incidence functions can be obtained:(2)dSkdt=Λk−∑j=1nβkjfkjSk,Ij−dkSSk,dEkdt=∑j=1nβkjfkjSk,Ij−ϵk+dkEEk,dIkdt=ϵkEk−αk+dkI+γkIk,dRkdt=γkIk−dkRRk.What the parameters mean can be summarized in the following list:Λk: the influx of individuals in the kth group.βkj: the transmission rate between Sk and Ij.dkS,dkE,dkI,anddkR: the natural death rate of S, E, I, and R in the kth group, respectively.ϵk: the rate of becoming infectious in the kth group.αk: the death rate caused by disease in the kth group.γk: the cure rate in the kth group.The parameters above are all nonnegative. In particular, whenβkj=0, it means that there is no disease transmission between Sk and Ij. The matrix B=βkjn×n reflects the transmission mechanism of disease among different subgroups built in the model. In this paper, we assume that the matrix B is irreducible.Since thatRk,k=1,2,…,n, do not appear in the first three equations of model (2) but only in the fourth equation, their properties and behaviors can be solved easily if Ik,k=1,2,…,n, are known; they can be omitted when analyzed. Therefore, the model can be simplified into the following form:(3)dSkdt=Λk−∑j=1nβkjfkjSk,Ij−dkSSk,dEkdt=∑j=1nβkjfkjSk,Ij−ϵk+dkEEk,dIkdt=ϵkEk−αk+dkI+γkIk.In the epidemic models, the basic reproduction numberR0, which represents the number of second generations produced by a single infected individual, plays an important role in the spread of disease for the long time. According to [4, 14], R0=ρM0, where M0=βkjϵkCkjSk0/αk+dkI+γkϵk+dkEn×n, Sk0=Λk/dkS, and ρ is the spectral radius of the matrix M0. If R0<1, there is only disease-free equilibrium P0, where P0=Λ1/d1S,0,0,…,Λn/dnS,0,0. When R0>1, then P0 is unstable, and the model has an endemic equilibrium P∗ which means the disease will be persistent. In this situation, our concern is whether there is a way to exterminate the disease.The reality is filled with randomness, and the epidemic models are often influenced by random environments. For example, there are a lot of natural disasters in reality, such as storm and earthquake. If these randomnesses happen, the parameters and the transmission mechanism in the model are likely to be affected. Thus, the deterministic model has some limitations to fully describe transmission of disease. Many scholars have studied the epidemic model with stochastic perturbations depicted by Brownian motion, and a lot of literature studies have been published; we refer the readers to [5, 7, 10, 12, 13, 15–17]. In [18–20], the authors studied the SIR or SIRS model with Markovian switching, and they gave some conditions on extinction or ergodicity of the model.Influenced by the work of predecessors, we use the similar method of Dalal et al. and Witbooi [21, 22] to construct stochastic perturbations, that is, we replace the parameters dkE and dkI by dkE−σ1kdBk and dkI−σ2kdWk, where the stochastic perturbations Bk and Wk are independent standard Brownian motions. The reason that not all parameters but only some of them are disturbed by stochastic perturbations may be the uncertainty of stochastic factors and the change of behavior of the infected.For all we know, the papers that discuss asymptotic behaviors of stochastic multigroup SEIR models with nonlinear incidence rate functions are relatively few. In this paper, we will study the following stochastic multigroup SEIR model:(4)S˙k=Λk−∑j=1nβkjfkjSk,Ij−dkSSkdt,E˙k=∑j=1nβkjfkjSk,Ij−ϵk+dkEEkdt+σ1kEkdBkt,I˙k=ϵkEk−αk+dkI+γkIkdt+σ2kIkdWkt,where σik, i=1,2, are the intensities of stochastic perturbation.Because the incidence rate functionsfkjSk,Ij are general and can be of different types in one model, which increase the difficulty of research, we will overcome it by some inequality techniques. This paper is organized as follows. Section 2 presents some background knowledge and lemmas which will be used afterwards. In Section 3, we prove that there is a unique positive solution to the model for any initial value. Section 4 proves that the disease-free equilibrium is globally asymptotically stable in probability when R0<1 by constructing Lyapunov functions. In Section 5, the disease will die out exponentially under certain stochastic perturbations when R0>1, and in Section 6, we provide some numerical simulations of the model to verify our analytical results.


## 2. Preliminaries

Throughout the paper, unless otherwise specified,Ω,ℱtt≥0,P denotes a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is right continuous, and ℱ0 contains all P-null sets). Denote(5)ℝ+n=x∈Rn:xi>0for all1≤i≤n.In general, letX be a regular homogeneous Markov process in Rn; consider the stochastic differential equation(6)dXt=bXtdt+∑k=1dσkXtdBkt,with initial value Xt0=x0∈Rn and Bkt,1≤k≤d, are standard Brownian motions. Define the differential operator ℒ associated with the above equation by(7)ℒ=∑k=1nbkx∂∂xk+12∑i=1n∑j=1nσTxσxij∂2∂xi∂xj.Ifℒ acts on a function V∈C2,1El×ℝ+;ℝ+, then by Ito^ formula,(8)dVX,t=ℒVX,tdt+∑r=1dVxX,tσrXtdBrt.where(9)ℒVX,t=VtX,t+∑k=1nbkx∂V∂xk+12∑i=1n∑j=1nσTxσxij∂2V∂xi∂xj.Next, we introduce some definitions about stability and lemmas which will be used latter. Assume thatb0=0 and σk0=0,k=1,2,…,d; then, Xt≡0 is the trivial solution to (6).Definition 1.
The trivial solution is called to be(i)
Stable in probability if for anyϵ>0 and the solution Xt,x0 with initial value X0=x0, then(10)limx0⟶0ℙsupt≥0Xt,x0≥ϵ=0.(ii)
Globally asymptotically stable in probability if it is stable in probability, and for anyx0∈ℝn,(11)ℙlimt⟶∞Xt,x0=0=1.Lemma 1 (cf. [23]).
If there is a positive definite functionVt,x∈C2ℝn with an infinitesimal upper limit such that the function ℒV is negative definite, then the trivial solution is globally asymptotically stable in probability.Lemma 2 (Perron–Frobenius).
IfA=aijn×n is irreducible and nonnegative, then the spectral radius ρA of A is a single eigenvalue, and there is a positive eigenvector ω=ω1,ω2,…,ωn corresponding to ρA of A. Moreover, ρA satisfies the inequality(12)mini∑jaij≤ρA≤maxi∑jaij.Remark 1.
From our previous description in Introduction, we know thatR0=ρM0<1 will lead to the extinction of disease in deterministic model (3). Combining the expression of R0 with the estimation of ρM0 in (12), we can infer that if transmission rate βkj decreases, ρM0 will become smaller, which provides the possibility of eliminating disease. A very important way to reduce βkj is to isolate people at home and restrict them from going out. This measure is being taken in many countries to combat COVID-19.Lemma 3 (cf. [24]).
LetM=Mtt≥0 be a real-valued continuous local martingale vanishing at t=0. Then,(13)limt⟶∞M,Mt=∞a.s.⟹limt⟶∞MtM,Mt=0a.s.lim supt⟶∞M,Mtt<∞a.s.⟹limt⟶∞Mtt=0 a.s..


## 3. The Existence and Uniqueness of the Solution to Model (4)

The first question we concern is whether the system has a solution or not. In this section, we prove that the system has a global and positive solution for any initial value.Theorem 1.
Given any initial valueS10,E10,I10⋯Sn0,En0,In0∈ℝ+3n, then model (4) has a unique solution on t≥0, and the solution will remain in ℝ+3n with probability one, that is, S1t,E1t,I1t⋯Snt,Ent,Int∈ℝ+3n for t≥0 almost surely.Proof.
Since the coefficients of the model are locally Lipschitz continuous, there is a unique local solutionS1t,E1t,I1t⋯Snt,Ent,Int on t∈0,τe, where τe is the explosion time (cf. [24]). In order to illustrate the solution is global, we only need to prove τe=∞. Assume c0 is sufficiently large so that S10,E10,I10⋯Sn0,En0,In0 lie within the interval 1/c0,c0. For c≥c0, define the stopping time(14)τc=inft∈0,τe,min1≤k≤nSkt,Ekt,Ikt≤1c,ormax1≤k≤nSkt,Ekt,Ikt≥c.
We setinf∅=∞ (where ∅ denotes the empty set). Clearly, τc is increasing as c⟶∞. Let τ∞=limc⟶∞τc, and τ∞≤τe a.s. If we can prove τ∞=∞ a.s., then equality τe=∞ holds true, and the conclusion can be obtained. If the assertion is false, then there exist two constants T>0 and ϵ∈0,1 such that(15)ℙτ∞≤T>ϵ.
Hence, there exists a positive integerc1≥c0 such that(16)ℙτc≤T>ϵ,for allc≥c1.
Then, we define a functionV:R+3n⟶R by(17)VSk,Ek,Ik≔∑k=1nSk−ak−aklnSkak+Ek−1−lnEk+Ik−1−lnIk,where ak,k=1,2,⋯n, are constants which will be determined later. Using Ito^’s formula, we can get(18)dV=∑k=1n1−akSkΛk−∑j=1nβkjfkjSk,Ij−dkSSkdt+12∑k=1nσ1k2+σ2k2+∑k=1n1−1Ek∑j=1nβkjfkjSk,Ij−ϵk+dkEEkdt+σ1kEkdBkt+∑k=1n1−1IkϵkEk−αk+dkI+γkIkdt+σ2kIkdWktdt=ℒVdt+∑k=1nσ1kEk−1dBkt+σ2kIk−1dWkt,where(19)ℒV=∑k=1nΛk−dkSSk−akSkΛk+akSk∑j=1nβkjfkjSk,Ij+akdkS−dkEEk−1Ek∑j=1nβkjfkjSk,Ij−αk+dkI+γkIk−ϵkEkIk+ϵk+dkE+αk+dkI+γk+12∑k=1nσ1k2+σ2k2≤∑k=1nΛk+ak∑j=1nβkjKIj+akdkS−αk+dkI+γkIk+ϵk+dkE+αk+dkI+γk+12∑k=1nσ1k2+σ2k2.
Notice that(20)∑k=1n∑j=1nakβkjKIj−∑k=1nαk+dkI+γkIk=∑j=1n∑k=1nakβkjKIj−∑j=1nαj+djI+γjIj=∑j=1n∑k=1nKakβkj−αj+djI+γkIj.
We choose appropriate numbersak,1≤k≤n, such that ∑k=1nKakβkj−αj+djI+γk=0; then, ℒV≤M, where M is a positive constant. Therefore,(21)dV≤Mdt+∑k=1nσ1kEk−1dBkt+σ2kIk−1dWkt.
Integrate both sides of (21) from 0 to τc∧T and take expectation; then,(22)EVSkτc∧T,Ekτc∧T,Ikτc∧T≤VSk0,Ek0,Ik0+E∫0τc∧TMdt≤VSk0,Ek0,Ik0+MT.
SetΩc=τc≤T; we have ℙΩc≥ϵ. Notice that, for every ω∈Ωc, there exists at least one of Sτc,ω,Eτc,ω,Iτc,ω which equals either c or 1/c. Therefore,(23)VSkτc∧T,Ekτc∧T,Ikτc∧T≥min1≤k≤nc−ak−aklncak,1c−ak−akln1akc∧c−1−lnc∧1c−1−ln1c.
Combining (22) with (23), we can obtain that(24)VSk0,Ek0,Ik0+MT≥E1ΩcωVSkτc∧T,Ekτc∧T,Ikτc∧T≥ϵmin1≤k≤nc−ak−aklncak,1c−ak−akln1akc∧c−1−lnc∧1c−1−ln1c,where 1Ωmω is the indicator function of Ωm. Letting m⟶∞ leads to the contradiction that ∞>VSk0,Ek0,Ik0+MT=∞. So, τe=∞ a.s. The proof is completed.Corollary 1.
ForSk, k=1,2⋯n, in model (4), there exists a set of Mk such that Sk≤Mk. Furthermore, the set Γ=Sk:Sk>0,Sk≤Λk/dkS is the invariant set, that is to say, if the initial value Sk0∈Γ, then Skt∈Γ, for t≥0 almost surely.Proof.
For the first equation of model (4), we have dSk≤Λk−dkSSkdt. By the method of variation of constants, we get that(25)Skt≤ΛkdkS+Sk0−ΛkdkSe−dkSt.
IfSk0≤Λk/dkS, then Skt≤Λk/dkS. If Sk0>Λk/dkS, then Skt≤Sk0. Let Mk=maxΛk/dkS,Sk0. The proof is complete.
The assumptionSk0≤Λk/dkS will be used in the rest of the paper.


## 4. The Behavior of the Model withR0<1

In the deterministic SEIR model,P0 is the disease-free equilibrium, and it is globally stable which means that the disease will die out with any initial value when R0<1. In this section, we will discuss the asymptotic behavior of the stochastic model with R0<1.Theorem 2.
LetS1t,E1t,I1t,…,Snt,Ent,Int be the solution to model (4) with the initial valueinitial valueS10,E10,I10,…,Sn0,En0,In0∈ℝ+3n. If B=βkjn×n is irreducible and R0=ρM0<1, then P0 is the unique equilibrium of model (4), and it is globally asymptotically stable in probability.Proof.
According to the assumption,B is irreducible and nonnegative; then, by Lemma 2, M0 has a single eigenvalue ρM0 and a positive eigenvector ω=ω1,ω2,…,ωn corresponding to ρM0 such that(26)ω1,ω1,…,ωnM0=ω1,ω1,…,ωnρM0.
LetV1=∑k=1n1/2akΛk/dkS−Sk2 and V2=∑k=1nckEk+ϵk+dkE/ϵkIk, where ck=ωkϵk/dkE+ϵkαk+dkI+γk and ak will be determined later. Using Ito^’s formula, we can obtain that(27)ℒV1=−∑k=1nakΛkdkS−SkΛk−∑j=1nβkjfkjSk,Ij−dkSSk≤−∑k=1nakdkSΛkdkS−Sk2+∑k=1n∑j=1nakβkjCkjSkIjSk−ΛkdkS≤−∑k=1nakdkSΛkdkS−Sk2+ϵ2∑k=1n∑j=1nakβkjCkjSk0ΛkdkS−Sk2+12ϵ∑k=1n∑j=1nakβkjCkjSk0Ij2=−∑k=1nakdkS−ϵ2∑j=1nβkjCkjSk0ΛkdkS−Sk2+12ϵ∑k=1n∑j=1nakβkjCkjSk0Ij2.
Here, the second inequality holds true because of the inequalityab≤ϵ/2a2+1/2ϵb2. Similarly, we use Ito^’s formula to V2 to get(28)ℒV2=∑k=1nck∑j=1nβkjfkjSk,Ij−ϵk+dkEαk+dkI+γkϵkIk≤∑j=1n∑k=1nωkϵkβkjCkjSk0ϵk+dkEαk+dkI+γkIj−∑k=1nωkIk=ρ0−1∑k=1nωkIk.
Define Lyapunov functionVt by Vt=V1t+V2t, and according to Theorem 1, Vt is positive definite; then,(29)ℒV≤−∑k=1nakdkS−ϵ2∑j=1nβkjCkjSk0ΛkdkS−Sk2+12ϵ∑k=1n∑j=1nakβkjCkjSk0Ij2+ρ0−1∑k=1nωkIk.
We can choose smallϵ such that dkS−ϵ/2∑j=1nβkjCkjSk0>0, ak are chosen to be sufficiently small, and because of R0<1, we have ℒV<0. Hence, applying Lemma 1, we arrive at the desired assertion. The proof is completed.


## 5. The Influence of Large Noise on Disease

In this section, we will discuss the influence of large noises on disease whenR0>1. Before we give the theorem, an inequality is presented first.Lemma 4.
Forak, bk, ck, dk, k=1,2⋯,n, the following inequality holds true:(30)∑k=1nakbk+ckdk2≤∑k=1nak2+∑k=1nck2∑k=1nbk2+∑k=1ndk2.Proof.
We prove it by transforming it into an inner product in spaceℝn. Let a≔a1,a2,…,anT∈ℝn, and the vectors b, c, and d are defined in a similar way. Then,(31)aTb+cTd2=a,b+c,d2≤a2b2+c2d2+2abcd≤a2b2+c2d2+a2c2+b2d2≤a2+c2b2+d2.
The proof is completed.Theorem 3.
IfB=βkj1≤k,j≤n is irreducible, then we have(32)max1≤k≤nlimsupt⟶∞1tlnEkt,limsupt⟶∞1tlnIkt≤R0−1max1≤k≤nαk+dkI+γk−12∑i=1n1/σ1k2+1/σ2k2,a.s.Proof.
We define aC2 function VEk,Ik by(33)V=∑k=1nckEk+dkE+ϵkϵkIk,where ck=ωkϵk/dkE+ϵkαk+dkI+γk. By calculation, we can get that(34)dV=∑k=1n∑j=1nckβkjfkjSk,Ij−∑k=1nωkIkdt+∑k=1nckσ1kEkdBkt+∑k=1nckσ2kdkE+ϵkϵkIkdWkt.
Using Ito’s formula, we arrive at(35)dlnV=1V∑k=1n∑j=1nckβkjfkjSk,Ij−∑k=1nωkIkdt−12V2∑k=1nck2σ1k2Ek2+σ2k2dkE+ϵk2ϵk2Ik2dt+1V∑k=1nckσ1kEkdBkt+1V∑k=1nckσ2kdkE+ϵkϵkIkdWkt.≤1V∑k=1n∑j=1nckβkjCkjSk0Ij−∑k=1nωkIkdt−12V2∑k=1nck2σ1k2Ek2+σ2k2dkE+ϵk2ϵk2Ik2dt+1V∑k=1nckσ1kEkdBkt+1V∑k=1nckσ2kdkE+ϵkϵkIkdWkt.≕V1tdt+V2tdt+V3t+V4t.
ForV1t, from the expression of eigenvector of R0, i.e., ω1,ω1⋯ωnM0=R0ω1,ω1⋯ωn, we obtain that(36)V1t=1VR0−1∑k=1nωkIk≤R0−1∑k=1nckdkE+ϵk/ϵkIk∑k=1nωkIk≤max1≤k≤nαk+dkI+γkR0−1.
According to the expression ofV, utilizing Lemma 4 yields(37)V2=∑k=1nckσ1kEk1σ1k+ckσ2kdkE+ϵkIkϵk1σ2k2≤∑k=1nck2σ1k2Ek2+σ2k2dkE+ϵk2Ik2ϵk2∑k=1n1σ1k2+1σ2k2.
Hence,V2t satisfies the inequality(38)V2t≤12∑k=1n1/σ1k2+1/σ2k2.
Because(39)lim supt⟶∞1t∫0t∑k=1nck2σ1k2Ek2V2dt<∞,applying Lemma 3 to V3t yields(40)lim supt⟶∞1t∫0t∑k=1nV−1ckσ1kEkdBkt=0.
V
4
t can be done in the same way. Therefore,(41)lim supt⟶∞ln Vtt≤max1≤k≤ndkI+γkR0−1−12∑k=1n1/σ1k2+1/σ2k2.
Since(42)1tln∑k=1nckEk≤1tlnmax1≤k≤ncknmax1≤k≤nEk≤1tlnmax1≤k≤nckn+lnmax1≤k≤nEk,1tln∑k=1nckEk≥1tlnmin1≤k≤nckmax1≤k≤nEk≥1tlnmin1≤k≤nck+lnmax1≤k≤nEk,taking the upper limit yields(43)lim supt⟶∞1tln∑k=1nckEk=max1≤k≤nlim supt⟶∞1tlnEk.
Making use of the same method, we can obtain that(44)lim supt⟶∞1tln∑k=1nckdkE+ϵkϵkIk=max1≤k≤nlim supt⟶∞1tlnIk.
Combining (43) and (44) yields(45)lim supt⟶∞1tlnVt≥max1≤k≤nlim supt⟶∞1tlnEk,lim supt⟶∞1tlnIk.
Along with (41), we arrive at the desired assertion. The proof is complete.Corollary 2.
For the solution to model (4), Ekt and Ikt, k=1,2⋯n, decay exponentially to zero almost surely if(46)R0−1max1≤k≤nαk+dkI+γk<12∑i=1n1/σ1k2+1/σ2k2.Remark 2.
From (46), we know that the right side of the inequality increases with the increase of σ1k and σ2k; therefore, the inequality above holds true for certain αk, dkI, γk, and sufficiently large σ1k and σ2k even if R0>1, which makes the disease extinct. It reflects that stochastic perturbations play an important role in disease control. Compared with the deterministic model in [4], the SEIR model with stochastic perturbations can show more properties and different behaviors.Remark 3.
We can see from many literature studies that the incidence function of the multigroup SEIR model is single one, such asSktIjt in [5, 7] and SktIjt/1+αkIjt in [9]. These may have some limitations and cannot reflect the actual situation well. Incidence functions fkjSkt,Ijt in this paper can be expressed in different forms, which can better describe the reality of life. We will provide different examples to illustrate the results in Section 6.


## 6. Examples and Numerical Simulations

In this section, we give some simulations of model (4) to confirm the analytical results above. By using Milstein’s higher-order method [25], we obtain the corresponding discretization equation:(47)Si,k+1=Si,k+Λi−∑j=1nβijfkjSi,k,Ij,k−diSSi,kΔt,Ei,k+1=Ei,k+∑j=1nβijfkjSi,k,Ij,k−ϵi+diEEi,kΔt+σ1iEi,kηi,kΔt+12σ1i2Ei,kηi,k2Δt−Δt,Ii,k+1=Ii,k+ϵiEi,k−αi+diI+γiIi,kΔt+σ2iIi,kρi,kΔt+12σ2i2Ii,kρi,k2Δt−Δt,where ηik,ρik are Gaussian random variables which follow the distribution N0,1. Let n=2, i.e., we consider the interactions of diseases in two groups.First, we give an example to verify Theorem2.Example 1.
Assume thatfkj=SkIj/1+2Ij2. We choose Λ1=0.3,Λ2=0.4;ϵ1=0.5,ϵ2=0.6;γ1=0.4,γ2=0.3;α1=0.1,α2=0.07;β11=0.02,β12=0.05,β21=0.04,β22=0.02,d1S=0.05,d2S=0.1,d1E=0.08,d2E=0.12,d1I=0.1,d2I=0.15 such that R0=0.4835<1, which satisfies the condition of Theorem 2. Moreover, let σ11=1,σ12=0.5,σ21=0.8,andσ22=1.5. Its trajectory is shown in Figure 1.
From Figure1(a), we can see that the diseases are extinct when stochastic perturbations are absent. From Figures 1(b) and 1(c), we can see the diseases in two groups are globally asymptotically stable.
Now, we move forward to verify Theorem3. We will present two examples to illustrate the two cases of incidence functions. In Example 2, we give the same incidence function for two groups, and in Example 3, two different incidence functions are presented.Figure 1
The trajectories withR0<1 and initial value S10=5,E10=2.7,I10=2.3;S20=2,E20=2.1,I20=1: (a) the trajectory without stochastic perturbation; (b, c) the trajectories with stochastic perturbations where parameters are shown in Example 1.
(a)
(b)
(c)Example 2.
Assume thatfkj=SkIj/1+2Ij2. We choose Λ1=0.3,Λ2=0.4;ϵ1=0.5,ϵ2=0.6;γ1=0.4,γ2=0.3;α1=0.1,α2=0.07;β11=0.1,β12=0.05,β21=0.12,β22=0.2,d1S=0.05,d2S=0.1,d1E=0.08,d2E=0.12,d1I=0.1,d2I=0.15 such that R0=1.685>1, which satisfies the condition of Theorem 3. Moreover, let σ11=1,σ12=0.5,σ21=0.8,σ22=1.5 so that R0−1max1≤k≤nαk+dkI+γk<1/2∑i=1n1/σ1k2+1/σ2k2 is satisfied. Its trajectory is shown in Figure 2. From Figure 2(a), we can see that the diseases are persistent because of R0>1 when stochastic perturbation is absent. We can see in Figures 2(b) and 2(c) that the diseases in two groups die out under certain stochastic perturbations and the exposed are the same results.Figure 2
The trajectories withR0>1: (a) the trajectory without stochastic perturbation; (b, c) the trajectories with stochastic perturbations where parameters are shown in Example 2.
(a)
(b)
(c)Example 3.
Assume thatf1j=S12Ij, f2j=SkIj/1+2Ij2, j=1,2, such that(48)M0=β11ϵ1Λ1/d1S2α1+d1I+γ1ϵ1+d1Eβ12ϵ1Λ1/d1S2α1+d1I+γ1ϵ1+d1Eβ21ϵ2Λ2/d2Sα2+d2I+γ2ϵ2+d2Eβ22ϵ2Λ2/d2Sα2+d2I+γ2ϵ2+d2E.
We chooseΛ1=0.3,Λ2=0.4;ϵ1=0.5,ϵ2=0.6;γ1=0.4,γ2=0.3;α1=0.1,α2=0.07;β11=0.04,β12=0.15,β21=0.1,β22=0.05,d1S=0.05,d2S=0.1,d1E=0.08,d2E=0.12,d1I=0.1,d2I=0.15 so that R0=3.61>1 can be obtained. Moreover, let σ11=4,σ12=4.5,σ21=2.8,andσ22=5; then, the conditions in Theorem 3 are satisfied. Its trajectory is shown in Figure 3. From Figure 3(a), we can see that the diseases are persistent because R0>1 without stochastic perturbation. We can see in Figures 3(b) and 3(c) that the exposed and infected in two groups die out under certain stochastic perturbations, which conform to the results of Theorem 3.Figure 3
The trajectories withR0>1 and two different incidence functions: (a) the trajectory without stochastic perturbation; (b, c) the trajectories with stochastic perturbations where parameters are shown in Example 3.
(a)
(b)
(c)



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*Source: 9367879-2020-05-05.xml*