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

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. 'en, the global asymptotical stability in probability of the model with R0 < 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 themodel and verify our analytical results.


Introduction
e 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 by S(t), I(t), and R(t) 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 E(t) 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. erefore, 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. is 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][3][4][5][6][7][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][9][10][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 f kj (S k , I j ), which satisfies the following assumptions: is research intends to study this general form of incidence function and assumes further that (C kj (S k )/S k ) ≤ K, where for K is a positive constant. e above incidence rate functions f kj (S k , I j ) include some special cases which can be seen in some literature, for example, (1) e multigroup SEIR model with above incidence functions can be obtained: What the parameters mean can be summarized in the following list: Λ k : the influx of individuals in the kth group. β kj : the transmission rate between S k and I j . d S k , d E k , d I k , and d R k : the natural death rate of S, E, I, and R in the kth group, respectively. ϵ k : the rate of becoming infectious in the kth group. α k : the death rate caused by disease in the kth group. c k : the cure rate in the kth group. e parameters above are all nonnegative. In particular, when β kj � 0, it means that there is no disease transmission between S k and I j .
e matrix B � (β kj ) n×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 that R k , 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 I k , k � 1, 2, . . . , n, are known; they can be omitted when analyzed. erefore, the model can be simplified into the following form: In the epidemic models, the basic reproduction number R 0 , 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], , 0, 0). When R 0 > 1, then P 0 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. e 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. us, 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][16][17]. In [18][19][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 d E k and d I k by d E k − σ 1k dB k and d I k − σ 2k dW k , where the stochastic perturbations B k and W k are independent standard Brownian motions. e 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: 2 Discrete Dynamics in Nature and Society where σ ik , i � 1, 2, are the intensities of stochastic perturbation.
Because the incidence rate functions f kj (S k , I j ) 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.
is 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 R 0 < 1 by constructing Lyapunov functions. In Section 5, the disease will die out exponentially under certain stochastic perturbations when R 0 > 1, and in Section 6, we provide some numerical simulations of the model to verify our analytical results.

Preliminaries
roughout the paper, unless otherwise specified, (Ω, F t t ≥ 0 , P) denotes a complete probability space with a filtration F t t ≥ 0 satisfying the usual conditions (i.e., it is right continuous, and F 0 contains all P-null sets). Denote In general, let X be a regular homogeneous Markov process in R n ; consider the stochastic differential equation with initial value X(t 0 ) � x 0 ∈ R n and B k (t), 1 ≤ k ≤ d, are standard Brownian motions. Define the differential operator L associated with the above equation by If L acts on a function V ∈ C 2,1 (E l × R + ; R + ), then by Ito formula, Next, we introduce some definitions about stability and lemmas which will be used latter. Assume that b(0) � 0 and σ k (0) � 0, k � 1, 2, . . . , d; then, X(t) ≡ 0 is the trivial solution to (6).
e trivial solution is called to be (i) Stable in probability if for any ϵ > 0 and the solution (ii) Globally asymptotically stable in probability if it is stable in probability, and for any Lemma 1 (cf. [23]). If there is a positive definite function V(t, x) ∈ C 2 (R n ) 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). If A � (a ij ) n×n is irreducible and nonnegative, then the spectral radius ρ(A) of A is a single eigenvalue, and there is a positive eigenvector Remark 1. From our previous description in Introduction, we know that R 0 � ρ(M 0 ) < 1 will lead to the extinction of disease in deterministic model (3). Combining the expression of R 0 with the estimation of ρ(M 0 ) in (12), we can infer that if transmission rate β kj decreases, ρ(M 0 ) will become smaller, which provides the possibility of eliminating disease. A very important way to reduce β kj is to isolate people at home and restrict them from going out. is measure is being taken in many countries to combat COVID-19.
Lemma 3 (cf. [24]). Let M � M t t ≥ 0 be a real-valued continuous local martingale vanishing at t � 0. en, Discrete Dynamics in Nature and Society

The Existence and Uniqueness of the Solution to Model (4)
e 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. (4) has a unique solution on t ≥ 0, and the solution will remain in R 3n + with probability one, that is,

Theorem 1. Given any initial value
Proof. Since the coefficients of the model are locally Lipschitz continuous, there is a unique local solution where τ e is the explosion time (cf. [24]). In order to illustrate the solution is global, we only need to prove τ e � ∞. Assume c 0 is sufficiently large so that We set inf∅ � ∞ (where ∅ denotes the empty set). Clearly, τ c is increasing as c ⟶ ∞. Let τ ∞ � lim c⟶∞ τ c , and τ ∞ ≤ τ e a.s. If we can prove τ ∞ � ∞ a.s., then equality τ e � ∞ 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 Hence, there exists a positive integer c 1 ≥ c 0 such that en, we define a function V: R 3n where a k , k � 1, 2, · · · n, are constants which will be determined later. Using Ito's formula, we can get where Notice that 4 Discrete Dynamics in Nature and Society We choose appropriate numbers a k , 1 ≤ k ≤ n, such that n k�1 Ka k β kj − (α j + d I j + c k ) � 0; then, LV ≤ M, where M is a positive constant. erefore, Integrate both sides of (21) from 0 to τ c ∧ T and take expectation; then, Set Ω c � τ c ≤ T ; we have P(Ω c ) ≥ ϵ. Notice that, for every ω ∈ Ω c , there exists at least one of S(τ c , ω), E(τ c , ω), I(τ c , ω) which equals either c or 1/c. erefore, Combining (22) with (23), we can obtain that where 1 Ω m (ω) is the indicator function of Ω m . Letting m ⟶ ∞ leads to the contradiction that ∞ > V(S k (0), E k (0), I k (0)) + MT � ∞. So, τ e � ∞ a.s. e proof is completed. □ Corollary 1. For S k , k � 1, 2 · · · n, in model (4), there exists a set of M k such that S k ≤ M k . Furthermore, the set Γ � S k : S k > 0, S k ≤ (Λ k /d S k ) is the invariant set, that is to say, if the initial value S k (0) ∈ Γ, then S k (t) ∈ Γ, for t ≥ 0 almost surely.
Proof. For the first equation of model (4), we have dS k ≤ (Λ k − d S k S k )dt. By the method of variation of constants, we get that e assumption S k (0) ≤ (Λ k /d S k ) will be used in the rest of the paper. In the deterministic SEIR model, P 0 is the disease-free equilibrium, and it is globally stable which means that the disease will die out with any initial value when R 0 < 1. In this section, we will discuss the asymptotic behavior of the stochastic model with R 0 < 1.
Proof. According to the assumption, B is irreducible and nonnegative; then, by Lemma 2, M 0 has a single eigenvalue ρ(M 0 ) and a positive eigenvector ω � (ω 1 , ω 2 , . . . , ω n ) corresponding to ρ(M 0 ) such that Discrete Dynamics in Nature and Society and a k will be determined later. Using Ito's formula, we can obtain that Here, the second inequality holds true because of the inequality ab ≤ (ϵ/2)a 2 + (1/2ϵ)b 2 . Similarly, we use Ito's formula to V 2 to get (28) , and according to eorem 1, V(t) is positive definite; then, We can choose small ϵ such that d S k − (ϵ/2) n j�1 β kj C kj (S 0 k ) > 0, a k are chosen to be sufficiently small, and because of R 0 < 1, we have LV < 0. Hence, applying Lemma 1, we arrive at the desired assertion. e proof is completed.

The Influence of Large Noise on Disease
In this section, we will discuss the influence of large noises on disease when R 0 > 1. Before we give the theorem, an inequality is presented first.
, a.s. where . By calculation, we can get that Using Ito's formula, we arrive at .

Corollary 2.
For the solution to model (4), E k (t) and I k (t), k � 1, 2 · · · n, decay exponentially to zero almost surely if Remark 2. From (46), we know that the right side of the inequality increases with the increase of σ 1k and σ 2k ; therefore, the inequality above holds true for certain α k , d I k , c k , and sufficiently large σ 1k and σ 2k even if R 0 > 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 as S k (t)I j (t) in [5,7] and S k (t)I j (t)/(1 + α k I j (t)) in [9]. ese may have some limitations and cannot reflect the actual situation well. Incidence functions f kj (S k (t), I j (t)) 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.

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:

10
Discrete Dynamics in Nature and Society where η ik , ρ ik are Gaussian random variables which follow the distribution N(0, 1). Let n � 2, i.e., we consider the interactions of diseases in two groups. First, we give an example to verify eorem 2.
From Figure 1(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 eorem 3. 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.
Example 3. Assume that f 1j � S 2 1 I j , f 2j � S k I j /(1 + 2I 2 j ), j � 1, 2, such that M 0 � β 11 ϵ 1 Λ 1 /d S  Figure 3. From Figure 3(a), we can see that the diseases are persistent because R 0 > 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 eorem 3.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that they have no conflicts of interest.