# Evolutionary Dynamics of Stochastic SEIR Models with Migration and Human Awareness in Complex Networks

Authors: Yue Zhang; Yuxuan Li
Journal: Complexity
 (2020)

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## Abstract

In this paper, a stochastic SEIR (Susceptible-Exposed-Infected-Removed) epidemic dynamic model with migration and human awareness in complex networks is constructed. The awareness is described by an exponential function. The existence of global positive solutions for the stochastic system in complex networks is obtained. The sufficient conditions are presented for the extinction and persistence of the disease. Under the conditions of disease persistence, the distance between the stochastic solution and the local disease equilibrium of the corresponding deterministic system is estimated in the time sense. Some numerical experiments are also presented to illustrate the theoretical results. Although the awareness introduced in the model cannot affect the extinction of the disease, the scale of the disease will eventually decrease as human awareness increases.


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## Body

## 1. Introduction

Infectious diseases have been threatening human health and affecting people’s life. Many researchers use mathematical models to study the spread and control strategies of infectious diseases. Some current compartment models were proposed by Kermack and McKendrick (see [1–3]). Since then, many researchers have proposed other models of infectious diseases based on these models.It is usually assumed that the probability of contact between all different individuals is consistent in early classical compartment models. However, there is obvious heterogeneity in human behavior and interpersonal communication in society because there might exist some members who could spread the disease to many other members. Therefore, it is more suitable that the disease transmission is modeled over complex networks.Furthermore, most catenary systems have scale-free properties of several orders of magnitude. For example, Pastor-Satorras and Vespignani studied SIS (Susceptible-Infected-Susceptible) models on scale-free networks by mean-field approximation (see [4]). In order to comprehend the spread tendency of infectious diseases, more and more researchers concerned about the stability and asymptotic behavior of epidemic models on complex networks. For example, Wei et al. researched the global stability of the endemic equilibrium for a network-based SIS epidemic model with the nonmonotone incidence rate, a SIS epidemic model with feedback mechanism on networks and a SIQRS epidemic model on complex networks in papers [5–7], respectively. Li found a threshold value for the transmission rate of a network-based SIS epidemic model with the nonmonotone incidence rate (see [8]). Yang et al. obtained the basic reproduction numbers of the SIS model and SIR model with infection age to investigate the disease transmission on complex networks (see [9, 10]). These results provide good theoretical help for the prevention and control of infectious diseases.In addition, there are some diseases with incubation period. Humans can be divided into susceptible, infectious, and recovered population and undiagnosed infectious population. Undiagnosed infectious population are individuals with the disease who pass them on to susceptible population, such as measles and pertussis. The impact of undiagnosed populations on the spread of infectious diseases is also important. The susceptible-exposed-infective-recovered (SEIR) model is suited to model this kind of diseases in the incubation period. So, some SEIR epidemic models on the scale-free networks are presented. Using the analytical method, the threshold values on the stability of disease-free equilibrium and the persistence of the disease have been given in [11]. By researching the SEIR epidemic models on complex network, Zhu et al. found that epidemic propagation depends equally on the infection rate and the time of latent and infected periods, especially on the nature of the contacts among individuals (see [12]). Liu and Li discussed a new epidemic SEIR model with discrete delay on complex network in paper [13]. Moreover, there are many researchers discussed rumor spreading problems by SIER propagation model on heterogeneous network (see [14, 15]).In real communities, susceptible people will be vigilant in the presence of infectious diseases, thus reducing the probability of contact with suspected infected people. As the number of patients increases, susceptible people become more vigilant. As a result, susceptible people will raise awareness of staying away from disease. The awareness reveals “washing hands,” “staying at home,” and “increasing the distance between people.” The improvement of this awareness does not only lower the incidence of the disease but, in some cases, can also prevent the outbreak of the disease. Funk et al. investigated how the spread of awareness prompted by a first-hand contact with the disease affects the spread of the disease (see [16]). Agaba et al. provided information about potential spread of disease in a population (see [17]). Recently, there are also some research results about epidemic models considering awareness progress on complex networks. Yuan et al. considered an epidemic disease model about the effect of awareness programs on complex networks and obtained the basic reproduction number in [18]. Wu and Fu studied a discrete-time SIS model with awareness interactions on degree-uncorrelated networks. Their findings explored the effects of various types of awareness on epidemic spreading and addressed their roles in the epidemic control (see [19]). She et al. presented a theoretical analysis of the SIS epidemic spread from the perspective of bipartite network and risk aversion (see [20]).In real life, biological populations are inevitably affected by external uncertainties, such as absolute humidity, temperature, and precipitation. In order to be more practical, it is necessary to introduce the randomness into the deterministic model. The importance of stochasticity in the epidemic dynamics has been acknowledged for a long time. There are many research results in stochastic epidemic models. For instance, Lahrouzand Omari researched a stochastic SIRS epidemic model with general incidence rate in a population of varying size and obtained sufficient conditions for the extinction and the existence of a unique stationary distribution (see [21]). Li et al. studied a stochastic SIRS epidemic model with nonlinear incidence rate and varying population size and obtained sufficient conditions for extinction and persistence of the disease (see [22]). Ji and Jiang established a threshold condition under stochastic perturbation for persistence or extinction of the disease and global dynamical behavior for stochastic SIR epidemic models (see [23, 24]).Especially, there are also some results in stochastic SEIR epidemic model. For example, Zhang and Wang established stochastic SEIR models with jumps, which are used to describe the wide spread of the infectious diseases due to the medical negligence (see [25]). Witbooi proved that there is almost sure exponential stability of the disease-free equilibrium for an SEIR epidemic model with independent stochastic perturbations (see [26]). Liu et al. researched the asymptotic behaviors of the disease-free equilibrium and the endemic equilibrium for a stochastic delayed SEIR epidemic model with nonlinear incidence (see [27]). Liu et al. proposed a two-group stochastic SEIR model with infinite delay and obtained the sufficient conditions for asymptotic stability of endemic equilibrium in [28]. Han et al. researched the dynamics behaviors of a nonlinear stochastic SEIR epidemic system with varying population size in [29]. In recent years, there are several results for some stochastic epidemic models over complex networks, for example, Krause et al. established a stochastic epidemic metapopulation model and solved the stochastic model numerically. Furthermore, authors designed some control approaches to control the spread of a stochastic SIS epidemic over spatial networks (see [30]). Cao and Jin studied the epidemic threshold and ergodicity of an SIS model in switched networks (see [31]). Nevertheless, the research results about stochastic SEIR epidemic models over complex networks are seldom.Inspired by the abovementioned factors, we will establish a stochastic SEIR models with human awareness in complex networks in this paper, which consider the effects of undiagnosed populations, human awareness, and external uncertainties on the spread of infectious diseases. The rest of this paper is as follows. The stochastic SEIR models with migration and human awareness in complex networks are established in Section2. In Section 3, we discuss the existence of global positive solutions for the stochastic system in complex networks. In Sections 4 and 5, we study the persistence and extinction of the disease. In Section 6, we introduce numerical simulations to illustrate the main results. Finally, we close the paper with the summary of main results.


## 2. The Model with Migration and Human Awareness

In this section, we will establish a stochastic SEIR model on complex networks. This model will reflect the impact of susceptible, infected, recovered, and undiagnosed populations on the spread of infectious diseases. Furthermore, the effects of migration and human awareness on the spread of disease will also be considered on complex networks, and suppose that the network is inaccessible to people who have been diagnosed with the disease. The basic SEIR model on complex networks is as follows:(1)dSkt=1−αtΔ−λkSktΘt−bSktdt,dEkt=αtΔ+λkSktΘt−μ+b+γ1Ektdt,dIkt=μEkt−γ2+bIktdt,dRkt=γ1Ekt+γ2Ikt−bRktdt,where(2)Θt=1k∑k=1nkpkEkt+Ikt,k=∑k=1nkpk,limt⟶+∞αt=α,and Sk, Ek, Ik, and Rk denote the number of the population that are susceptible, undiagnosed infectious (also known as “exposed,” which have the disease and can pass it on to a susceptible person), infectious and recovered with immunity at node k, respectively. The parameter λ is the probability of exposure each time a susceptible person comes into contact with a pathogen carrier. b is for the death and emigration of the susceptible, exposed, infectious, and recovered. Δ is a growing and migrating population of susceptible and undiagnosed infectious individuals. Θt is a probability of a randomly selected link pointing to an undiagnosed or infected individual. αt∈0,1 is the proportion of these undiagnosed infectious individuals over time t, and αm≤αt≤αM. μ is the conversion rate from Ek to Ik. γ1 and γ2 represent the response rate from Ek and Ik to Rk, respectively. In addition, all parameters of model (1) are assumed to be positive constants.Throughout this paper, we letΩ,ℱ,ℙ be a complete probability space with a filtration ℱtt≥0 satisfying the usual conditions (i.e., it is right continuous and increasing while ℱ0 contains all ℙ-null sets), and ℝ+4n=x∈ℝ4n:xi>0,i=1,2,…,4n.Consider that the awareness of the population increases with the size of the infected population during an epidemic, and the rate of increase in awareness gradually decreases. Therefore, this awareness is described bye−cΘt. Supposing that Δ=b, then we obtain the following model:(3)dSkt=1−αtb−λkSktΘte−cΘt−bSktdt,dEkt=αtb+λkSktΘte−cΘt−μ+b+γ1Ektdt,dIkt=μEkt−γ2+bIktdt,dRkt=γ1Ekt+γ2Ikt−bRktdt.Assume that the transmission rateλ and the conversion rate μ in system (3) are perturbed by some random environmental effects; then, we have(4)λ=λ¯+σ1ξ1t,μ=μ¯+σ2ξ2t,where ξ1t and ξ2t are the white noise with mean zero and variance one. λ¯ and μ¯ represents the average of λ and μ in the stochastic setting, respectively. σ1,σ2>0 denotes the white noise intensity. Rewrite λ¯ and μ¯ as λ and μ for convenience. Noting that ξitdt=dBit,i=1,2, where Bit,i=1,2 represents the standard Brownian motion. Then, we obtain the following stochastic epidemic SEIR model:(5)dSkt=1−αtb−λkSktΘte−cΘt−bSktdt−σ1kSktΘte−cΘtdB1,dEkt=bαt+λkSktΘte−cΘt−μ+b+γ1Ektdt+σ1kSktΘte−cΘtdB1−σ2EktdB2,dIkt=μEkt−γ2+bIktdt+σ2EktdB2,dRkt=γ1Ekt+γ2Ikt−bRktdt.DenotingNkt is the size of total population with k−nodes, let Nkt:=Skt+Ekt+Ikt+Rkt and Nk0=1.


## 3. Existence of the Global and Positive Solution

To investigate the dynamical behavior of system (5), the first concern is whether the solution is global. Moreover, for an epidemic dynamics model, whether the value is positive is also required. The following theorem shows that the solution of system (5) is global and positive.Theorem 1.
For any initial valueSk0,Ek0,Ik0,Rk0∈ℝ+4,k=1,…,n and Nk0=1, there is a unique solution Skt,Ekt,Ikt,Rkt,k=1,…,n of system (5) on t≥0, and the solution will remain in ℝ+4 with probability 1.Proof.
Define the stopping timeτ+ by(6)τ+=inft∈0,+∞:Sk≤0orEk≤0orIk≤0orRk≤0,k=1,…,n.
To show this solution is positive globally in the first octant, we only need to show thatτ+=+∞ a.s. Assume that there exist T>0 and ε∈0,1 such that ℙτ+<T>ε. Let Ω˜=ω∈Ω:τ+<T, where ω represents a sample point of the sample space Ω. Define a C2−functionV:ℝ+4n⟶ℝ as follows:(7)VX=∑k=1nlnSk+lnEk+lnIk+lnRk.
Applying Itô’s formula, we obtain(8)dVX=∑k=1n1−αtbSk−λkΘe−cΘ−b−12σ12kΘe−cΘ2dt+αtbEk+λkSkΘe−cΘEk−μ+b+γ1−12σ12kSkΘe−cΘEk2−12σ22dt+μEkIk−γ2+b−12σ22EkIk2dt+μEkRk−γ1EkRk−bdt−σ1kΘe−cΘdB1+σ1kSkΘe−cΘEkdB1−σ2dB2+σ2EkIkdB2,≥∑k=1nKSk,Ek,Ik,Rkdt−σ1kΘe−cΘdB1+σ1kSkΘe−cΘEkdB1−σ2dB2+σ2EkIkdB2,where(9)KSk,Ek,Ik,Rk=−λkΘe−cΘ−12σ12kΘe−cΘ2−12σ12kSkΘe−cΘEk2−12σ22−12σ22EkIk2−γ1EkRk−μ+γ1+γ2+4b.
Integrating both sides of (8) from 0 to τ+∧t, we obtain(10)Vτ+∧t≥VX0+∑k=1n∫0τ+∧tKSks,Eks,Iks,Rksds−∫0τ+∧t1−SkEkσ1kΘe−cΘdB1−∫0τ+∧tσ21−EkIkdB2.
Lett⟶+∞ in both sides of the above inequality, and we obtain from the definition of τ+ that(11)−∞>−∞,which contradicts our assumption, and the proof of Theorem 1 is complete.Lemma 1.
For any initial valueX0=S10,E10,I10,R10,…,Sn0,En0,In0,Rn0 in ℝ+4n and Nk0=Sk0+Ek0+Ik0+Rk0=1,k=1,2,…,n, the solution of system (5) has the property that(12)Nkt=Skt+Ekt+Ikt+Rkt=1,for all ω∈Ω.Proof.
The proof is straightforward. Summing up the four equations in (5), we can obtain(13)dNkdt=0.
By Theorem1, we obtain the next assertion that(14)0≤Skt,Ekt,Ikt,Rkt≤1.


## 4. Persistence of the Disease

Now let us consider the persistence of disease, which corresponds to the stochastic strong persistence in the mean defined by Zhao et al. [32]. The following theorem shows that the disease will be almost surely persistent in the time mean sense.Lemma 2.
For any initial valueSk0,Ek0,Ik0,Rk0∈Sk,Ek,Ik,Rk∈ℝ+40≤Sk,Ek,Ik,Rk≤1,k=1,…,n and Nk0=1, ∃T>1, when t>T, the solution Ekt of system (5) has the property that(15)liminft⟶+∞1t∫0tEksds≥αmb.Proof.
From the second equation of system (5), we obtain(16)dEkt≥bαm−μ+b+γ1Ektdt+σ1kSktte−cΘtdB1−σ2EktdB2.
Considering the actual situation of parameterμ,b,andγ1, integrating the above inequality via t along 0,t, and then dividing both sides by t, we have(17)Ekt−Ek0t≥bαm−μ+b+γ11t∫0tEkds+1t∫0tσ1kSkΘe−cΘdB1s−1t∫0tσ2EkdB2s.
From the strong law of large numbers for local martingales, we obtain(18)liminft⟶+∞1t∫0tEksds≥αmb.
The proof of the lemma is complete.Theorem 2.
For any initial valueSk0,Ek0,Ik0,Rk0∈Sk,Ek,Ik,Rk∈ℝ+40≤Sk,Ek,Ik,Rk≤1,k=1,…,n and Nk0=1, if α>0, then the solution Skt,Ekt,Ikt,Rkt,k=1,…,n of system (5) has the property that(19)liminft⟶+∞1t∫0tSksds≥1−αMbλk+b,a.s.,liminft⟶+∞1t∫0tEksds≥2λk2λk−σ12k2Q1αmb−αMb−λkαmb,a.s.,liminft⟶+∞1t∫0tIksds≥μγ2+b2λk2λk−σ12k2Q1αmb−αMb−λkαmb,a.s.,liminft⟶+∞1t∫0tRksds≥μγ2+γ1γ2+γ1bγ2b+b22λk2λk−σ12k2Q1αmb−αMb−λkαmb,a.s.,where Qx=−1/2σ12k2x2+λkx+bαm−μ+b+γ1−1/2σ22.Proof.
LetSkt,Ekt,Ikt,Rkt be a solution of system (5) with any initial value Sk0,Ek0,Ik0,Rk0∈ℝ+4 and Nk0=1. By Theorem 1, it holds that 0<limt⟶+∞Θ≤1 for all ω∈Ω.This means that there exists a Tω=Tωε>0 such that Θω,t≤1+ε for all t>Tω, any ω∈Θ and any sufficiently small ε>0. From the first equation of system (5), we derive that, for all t>Tω,(20)dSkt≥1−αMb−λkSk1+ε−bSkdt−σ1kSkΘe−cΘdB1=1−αMb−λk1+ε+bSkdt−σ1kSkΘe−cΘdB1,where ω is omitted. For convenience and without losing generality, we assume that the above inequality holds for all t>0. Integrating the above inequality and then dividing both sides by t, we obtain(21)Skt−Sk0t≥1−αMb−λk1+ε+b∫0tSksds−1t∫0tσ1kSkΘe−cΘdB1s.
Fromlimt⟶+∞Skt≤1 for all ω∈Ω and strong law of large numbers for local martingales, we obtain(22)limt⟶+∞Skt−Sk0t+1t∫0tσ1kSkΘe−cΘdB1s=0,a.s.,together with (21), which implies(23)liminft⟶+∞1t∫0tSksds≥1−αMbλk1+ε+b,a.s.
Because of the arbitrariness ofε, we have(24)liminft⟶+∞1t∫0tSksds≥1−αMbλk+b,a.s.
This is the required first assertion.
Then, we will prove the second assertion.
By Itô’s formula, it can be seen from the second equation of system (5) that(25)dlnEkt≥bαm+λkSkΘe−cΘEk−μ+b+γ1−12σ12k2σ1SkΘe−cΘEk2−12σ22dt+σ1kSkΘe−cΘEkdB1t−σ2dB2t=QSkΘe−cΘEkdt+σ1kSkΘe−cΘEkdB1t−σ2dB2t.
Integrating both sides yields(26)lnEkt≥lnEk0+∫0tQSkΘe−cΘEkds+∫0tσ1kSkΘe−cΘEkdB1s−∫0tσ2dB2s.
According to Lemma2, supposing that Ek≥αmb, then we compute(27)QSkΘe−cΘEk−Q1αmb=12σ12k21αmb2−SkΘe−cΘEk2−λk1αmb−SkΘe−cΘEk.
By Lemma1 and Theorem 1, we assume that Skt≤1,Ekt≤1,Ikt≤1 for all t>0, which yields from (27) that(28)QSkΘe−cΘEk=Q1αmb−λk−12σ12k21αmb+SkΘe−cΘEk1αmb−SkΘe−cΘEk≥Q1αmb−λk−12σ12k21αmb−SkΘe−cΘEk.
Substituting this inequality into (26), we obtain(29)lnEkt≥lnEk0+Q1αmbt−λk−12σ12k2∫0t1αmb−SkΘe−cΘEkds+∫0tσ1kSkΘe−cΘEkdB1s−∫0tσ2dB2s.
From the first equation of system (5), we have(30)dSkt≥1−αtb−λkSkΘe−cΘEk−bSkdt−σ1kSkΘe−cΘdB1=1−αtb−λkαmb+λk1αmb−SkΘe−cΘEk−bSkdt−σ1kSkΘe−cΘdB1≥1−αMb−λkαmb+λk1αmb−SkΘe−cΘEk−b−Ektdt−σ1kSkΘe−cΘdB1=−αMb−λkαmb+λk1αmb−SkΘe−cΘEkb−Ektdt−σ1kSkΘe−cΘdB1.
Then, we obtain(31)λk∫0t1αmb−SkΘe−cΘEkds≤Skt−Sk0+λkαmbt+αMbt+∫0tEksds−∫0tσ1kSkΘe−cΘdB1s.
Substituting inequality (31) into (29), we obtain(32)lnEkt≥Q1αmbt−2λk−σ12k22λkαMbt+λkαmbt+∫0tEksds+Zt,where(33)Zt=lnEk0−2λk−σ12k22λkSkt−Sk0−∫0tσ1kSkΘe−cΘdB1s+∫0tσ1kSkΘe−cΘEkdB1s−∫0tσ2dB2s.
By Theorem1, Lemma 1 and the strong law of large numbers for local martingales, we have(34)limt⟶+∞Ztt=0,a.s.,limsupt⟶+∞lnEktt≤0.
Adding with (32), they can yield the following inequation:(35)liminft⟶+∞1t∫0tEksds≥2λk2λk−σ12k2Q1αmb−αMb−λkαmba.s.
This is the required second assertion.
Then, we shall prove the third assertion. Integrating the third equation of system (5) and dividing both sides by t, we obtain(36)Ikt−Ik0t=μ1t∫0tEksds−γ2+b1t∫0tIksds+1t∫0tσ2EksdB2s.
Adding with (35), they can yield the following inequation:(37)liminft⟶+∞1t∫0tIksds≥μγ2+b2λk2λk−σ12k2Q1αmb−αMb−λkαmb,a.s.
This is the required third assertion. Finally, similar to the proof of the third assertion, we can obtain(38)liminft⟶+∞1t∫0tRksds≥μγ2+γ1γ2+γ1bγ2b+b22λk2λk−σ12k2Q1αmb−αMb−λkαmb,a.s.
This completes the proof of Theorem2.
Recall thatlimt⟶+∞αt=α, which means that we have αt−α<ε when t≥T, ∃T>0, for any arbitrarily small number ε>0. Then, we consider the following simplified version of system (5):(39)dSkt=1−αb−λkSktΘte−cΘt−bSktdt−σ1kSktΘte−cΘtdB1,dEkt=bα+λkSktΘte−cΘt−μ+b+γ1Ektdt+σ1kSktΘte−cΘtdB1−σ2EktdB2,dIkt=μEkt−γ1+bIktdt+σ2EktdB2,where γ1=γ2.Theorem 3.
Suppose thatSkt,Ekt,Ikt,k=1,…,n is the solution of system (39) for any initial value Sk0,Ek0,Ik0∈Sk,Ek,Ik∈ℝ+30≤Sk,Ek,Ik≤1,k=1,…,n and Nk0=1. If Θ≤Θ∗ and Θ∗e−cΘ∗≤Θe−cΘ, then we have(40)limsupt⟶+∞1tE∫0tγ1+bΘ−Θ∗2Θ+1k∑k=1nkpkb+λkΘ∗e−cΘ∗Sk−Sk∗2Skds≤η,where Sk∗,Ek∗,Ik∗,k=1,…,n is the unique endemic equilibrium E∗ of the corresponding deterministic equation of system (39), and(41)η=1k∑k=1nkpk12σ12kΘ∗2+σ22Θ∗21αb2+12σ12k2Sk∗2.Proof.
For any arbitrarily small numberε>0, there exist a T>0, when t≥T, we have αt−α<ε. The corresponding deterministic version of system (39) has a unique endemic equilibrium E∗ satisfying(42)1−αb−λkSk∗Θ∗e−cΘ∗−bSk∗=0,bα+λkSk∗Θ∗e−cΘ∗−μ+b+γ1E∗=0,μEk∗−γ1+bIk∗=0.
DefineC2 functions as follows:(43)Vt=V1t+V2t,V1=Θ−Θ∗−Θ∗lnΘΘ∗,V2=1k∑k=1nkpkSk−Sk∗−Sk∗lnSk∗Sk.
Applying Itô’s formula, we have(44)LV1=1k∑k=1nkpkΘ−Θ∗ΘλkSkΘe−cΘ−λkSk∗Θ∗e−cΘ∗−γ1+bEk+Ik−Ek∗−Ik∗+12σ12kSkΘ∗e−cΘ2+σ22Θ∗2EkΘ2=−γ1+bΘ−Θ∗2Θ+1k∑k=1nkpkΘ−Θ∗ΘλkSkΘe−cΘ−λkSk∗Θ∗e−cΘ∗+12σ12kSkΘ∗e−cΘ2+σ22Θ∗2EkΘ2≤−γ1+bΘ−Θ∗2Θ+1k∑k=1nkpkΘ−Θ∗ΘλkSkΘe−cΘ−λkSk∗Θ∗e−cΘ∗+12σ12kΘ∗2+σ22Θ∗21αb2,(45)LV2=1k∑k=1nkpkSk−Sk∗SkλkSk∗Θ∗e−cΘ∗+bSk∗−λkSkΘe−cΘ−bSk+12σ12k2Sk∗2Θe−cΘ2=1k∑k=1nkpkSk−Sk∗SkbSk∗−bSk+λkSk∗Θ∗e−cΘ∗−λkSkΘe−cΘ−λkSkΘ∗e−cΘ∗+λkSkΘ∗e−cΘ∗+12σ12k2Sk∗2Θe−cΘ2=1k∑k=1nkpk−b+λkΘ∗e−cΘ∗Sk−Sk∗2Sk+Sk−Sk∗Sk−λkSkΘe−cΘ+λkSkΘ∗e−cΘ∗+12σ12k2Sk∗2Θe−cΘ2≤1k∑k=1nkpk−b+λkΘ∗e−cΘ∗Sk−Sk∗2Sk+Sk−Sk∗Sk−λkSkΘe−cΘ+λkSkΘ∗e−cΘ∗+12σ12k2Sk∗2.
Combining (44) with (45), this implies(46)LV≤−γ1+bΘ−Θ∗2Θ−1k∑k=1nkpkb+λkΘ∗e−cΘSk−Sk∗2Sk+1k∑k=1nkpkλkΘΘSkΘ∗e−cΘ∗−e−cΘ+Sk∗Θ∗e−cΘ∗−Θe−cΘΘ∗−Θ+12σ12kΘ∗2+σ22Θ∗21αb2+12σ12k2Sk∗2≤−γ1+bΘ−Θ∗2Θ−1k∑k=1nkpkb+λkΘ∗e−cΘ∗Sk−Sk∗2Sk+1k∑k=1nkpk12σ12kΘ∗2+σ22Θ∗21αb2+12σ12k2Sk∗2.
Based on Theorem 6 in [33], it can be obtain that(47)limsupt⟶+∞1tE∫0tγ1+bΘ−Θ∗2Θ+1k∑k=1nkpkb+λkΘ∗e−cΘ∗Sk−Sk∗2Skds≤η.
The proof of the theorem is complete.Remark 1.
From this result, the distance between the stochastic solution and the local disease equilibrium of the corresponding deterministic system can be estimated in the time sense under the conditions of disease persistence.η is related to the strength of the two white noise environment. Θ corresponds to the size of the infected person. When an endemic disease is formed, the scale of the final disease will fluctuate in the vicinity of the equilibrium point of the determining system under the influence of the environment. The greater the noise intensity, the more violent the fluctuation.


## 5. Extinction of a Disease

Finally, let us consider the case ofα=0, which assumes that no undiagnosed infection can enter people. As people migrate, the disease will persist as long as there are undiagnosed infections. α=0 will be an ideal case before migration is controlled. In this case, system (5) can be written as follows:(48)dSkt=b−λkSktΘte−cΘt−bSktdt−σ1kSktΘte−cΘtdB1,dEkt=λkSktΘte−cΘt−μ+b+γ1Ektdt+σ1kSktΘte−cΘtdB1−σ2EktdB2,dIkt=μEkt−γ2+bIktdt+σ2EktdB2,dRkt=γ1Ekt+γ2Ikt−bRktdt.At this point, we can discuss the conditions for disease extinction in system (48). The following theorem will show the conditions for disease extinction.Lemma 3 (see [34, 35]).
LetH be a given separable real-valued Hilbert space, Bt:t∈ℝ+ be a canonical cylindrical Wiener process defined on H, δ>1, and τh and νh be two sequences of positive numbers. Then, for almost all ω∈Ω, there exists a random integer h0ω such that, for all h≥h0,(49)∫0tfsdBs≤12νh∫0tfs2ds+δlnhνh,0≤t≤τh.Theorem 4.
For any initial valueSk0,Ek0,Ik0,Rk0∈Sk,Ek,Ik,Rk∈ℝ+40≤Sk,Ek,Ik,Rk≤1,k=1,…,n, and Nk0=1, the solution Ekt of system (48) has the property that(50)limsupt⟶+∞lnEktt≤λ22σ12−μ+b+γ1−12σ22,a.s.Ifλ2−2μ+2b+2γ1+σ22σ12<0, then Ekt tends to zero exponentially almost surely.Proof.
By using Itô’s formula, we have(51)lnEkt=lnEk0+∫0t−12σ12kSkΘe−cΘEk2+λkSkΘe−cΘEk−μ+b+γ1−12σ22ds+∫0tσ1kSkΘe−cΘEkdB1s−∫0tσ2dB2s.
Choosingδ>1, νh>0, and τh=h in Lemma 3, then for almost all ω∈Ω, there exists a random integer h0ω such that, for all h≥h0,(52)∫0tσ1kSkΘe−cΘEkdB1s≤12νh∫0tσ12kSkΘe−cΘEk2ds+δlnhνh,0≤t≤h.
Substituting (52) into (51), we obtain that(53)lnEkt≤lnEk0+∫0t−121−νkσ12kSkΘe−cΘEk2+λkSkΘe−cΘEk−μ+b+γ1−12σ22ds+δlnhνh−∫0tσ2dB2s.
Noting that(54)−121−νkσ12kSkΘe−cΘEk2+λkSkΘe−cΘEk≤λ221−νhσ12,it follows that, for all t∈0,h and h>h0,(55)lnEkt≤lnEk0+∫0tλ221−νhσ12−μ+b+γ1−12σ22ds+δlnhνh−∫0tσ2dB2s,dividing both sides by t; let h⟶+∞ and t⟶, and applying the strong law of large numbers, we have(56)limsupt⟶+∞lnEktt≤λ221−νhσ12−μ+b+γ1−12σ22,a.s.
Finally, by sendingνh⟶0, we obtain the desired inequality of Theorem 4.


## 6. Numerical Simulations

In this section, we will assume that all parameters are given in appropriate units.Example 1.
Consider system (48) with initial condition Sk0=0.7,Ek0=0.2,Ik0=0.1, and Rk0=0 under different system parameters:(57)αt=0,μ=0.3,b=0.05,λ=0.1,γ2=0.2,c=1,γ1=0.25,n=50,σ1=0.2,σ2=0.2.
Based on Theorem4, we can conclude that the solution of (48) obeys(58)limsupt⟶+∞lnEktt≤−0.0396,a.s.
The computer simulation in Figure1 supports this result clearly, illustrating the extinction of the disease.Figure 1
Under the initial conditions given in Example1, the time evolution of the average density of each state in a complex network.
(a)(b)(c)(d)Example 2.
Consider system (48) with initial condition Sk0=0.6,Ek0=0.2,Ik0=0.2, and Rk0=0 under different system parameters:(59)αt=0.3,μ=0.3,b=0.05,λ=0.1,γ2=0.2,c=1,γ1=0.25,n=50,σ1=0.25,σ2=0.2.
Figure2 shows the time plots of each state mean in the complex network in the stochastic model and the deterministic model, respectively.Figure 2
Under the initial conditions given in Example2, the time evolution of each state mean in the complex network in the stochastic model and the deterministic model, respectively.
(a)(b)(c)(d)Example 3.
Consider system (48) with initial condition Sk0=0.6,Ek0=0.2,Ik0=0.2, and Rk0=0 under different system parameters:(60)αt=0.3,μ=0.1,b=0.1,λ=0.2,γ2=0.2,γ1=0.2,n=50,σ1=0.1,σ2=0.1.
Figure3 shows the time plots of each state mean in the stochastic model with different values of c, respectively. The results show that as the value of c increases, the scale of the disease decreases.Figure 3
Under the initial conditions given in Example3, the time evolution of each state mean in the stochastic model with c=1,5,15, respectively.
(a)(b)(c)(d)


## 7. Conclusion

When infectious diseases are prevalent, healthy people, especially susceptible people, will be alert to the disease and thus reduce their contact with such people with disease or suspected disease. We take this property into account in the model. Moreover, in the real world, there are various random factors that have a significant impact on the rate of infection and the rate of diagnosis of patients with the disease, so it is necessary to consider this randomness in the model. In this paper, we have considered a stochastic SEIR epidemic model with the external variability in the transmission rateλ and the conversion rate μ. Combining analytical results with numerical simulations, we discussed the effects of these environmental noises on the transmission dynamics of epidemics. When outsiders enter the network, if they have undiagnosed infections, the disease persists. If there is no infected person in the alien population, then when λ2−2μ+2b+2γ1+σ22σ12<0, the disease dies out almost surely. Through numerical simulation, we found that the awareness introduced in the model cannot affect the extinction of the disease, but the scale of the disease will eventually decrease as the increase of human awareness, which may contribute to the control of the disease.



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*Source: 3768083-2020-05-28.xml*