# The Dynamics of a Stochastic SIR Epidemic Model with Nonlinear Incidence and Vertical Transmission

Authors: Guihua Li; Yuanhang Liu
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

In this study, we build a stochastic SIR epidemic model with vertical infection and nonlinear incidence. The influence of the fluctuation of disease transmission parameters and state variables on the dynamic behaviors of the system is the focus of our study. Through the theoretical analysis, we obtain that there exists a unique global positive solution for any positive initial value. A thresholdR0s is given. When R0s<1, the diseases can be extincted with probability one. When R0s>1, we construct a stochastic Lyapunov function to prove that the system exists an ergodic stationary distribution, which means that the disease will persist. Then, we obtain the conditions that the solution of the stochastic model fluctuates widely near the equilibria of the corresponding deterministic model. Finally, the correctness of the results is verified by numerical simulation. It is further found that the fluctuation of disease transmission parameters and infected individuals with the environment can reduce the threshold of disease outbreak, while the fluctuation of susceptible and recovered individuals has a little effect on the dynamic behavior of the system. Therefore, we can make the disease extinct by adjusting the appropriate random disturbance.


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## Body

## 1. Introduction

At the beginning of 2020, a sudden epidemic (COVID-19) has disrupted people’s normal life. In order to curb the spread of the epidemic, the state has taken measures such as closing cities and delaying the opening of schools to protect people’s lives to the maximum extent. So far, the disease has not been completely controlled. Emerging infectious diseases have brought fear and inconvenience to people’s lives and have a great impact on the global economy. It is very important to study the spread of infectious diseases.Dynamic modeling is an important method to study the spread of infectious diseases. In the classical epidemic model, such as the SIR model, the total population is generally divided into three categories, the number of susceptible individuals bySt, the number of infectious individuals by It, and the number of permanently immune individuals by Rt. The model is as follows.(1)dSdt=A+bSt−βStItfI−dSt+b1−pIt,dIdt=bpIt+βStItfI−d+μ+γIt,dRdt=bRt+γIt−dRt,where A is the recruitment rate of susceptible corresponding to immigration, b is the birth rate, d is the nature death rate, μ is the disease induced mortality rate, γ is the rate of recovery from infection, a positive constant p0≤p≤1 is the proportion of infection in the offspring of infected mothers, 1−p is the proportion of susceptible individuals in the offspring of infected mothers, and β is the transmission coefficient between compartments St and It. fI is continuously differentiable function and assume f0=1 and f′I≥0.For the theoretical research of this model (1), there are a lot of literatures; interested readers can refer to the literatures [1–5]. The classical ODE model can reflect the spread of disease to some extent, but in real life, the spread of infectious diseases is also affected by many random factors (as shown in [6]), for example, the unpredictability of person-to-person contact, which means that there is not necessarily uniform contact between individuals. In other words, the process of disease transmission is inevitably affected by random factors. In this way, the stochastic differential model is more suitable than the deterministic model.There are many literatures on the stochastic differential model of infectious diseases [7–20]. Here, we mainly introduce the literature of stochastic differential models related to the SIR epidemic model in detail. In literature [8], Aadil and Omari built the SIRS stochastic differential model with parameter perturbation, vaccination of recruitment susceptible, and nonlinear incidence rate. They found that large enough random disturbances can suppress outbreaks. In 2015, Liu and Chen [7] considered a model of literature [8] and observed that the solution of the system fluctuates near the disease-free equilibrium under suitable conditions. Zhou et al. [16] studied an SIR model with the bilinear infection rate and stochastic perturbation of parameter and state variables and got the conditions of survival and stationary distribution.Since the contact between populations and the population itself are affected by many random factors in the environment, we will introduce stochastic white noise perturbations into system (1) by two different approaches.First of all, inspired by the literature [21], we consider a discrete time Markov chain. For a fixed time increment Δt>0, we define the process XΔtt=X1Δtt,X2Δtt,X3ΔttT (for convenience, we replace S,I,R by X1,X2,X3) for t=0,Δt,2Δt,…, where XΔt0=S0,I0,R0∈R+3 is a deterministic initial value. Let ViΔtkk=0∞i=1,2,3 be three sequences of random variables. Suppose that these variables are jointly independent and that within each sequence the variables are identically distributed, such that(2)EViΔtk=0,EViΔtk2=σi2Δt,i=1,2,3,where the parameter σi≥0i=1,2,3 represents the intensity of the environmental white noise. We assume that XΔt grows within that time period according to the deterministic system (1) and the random amount V1ΔtkX1ΔtkΔt,V2ΔtkX2ΔtkΔt,V3ΔtkX3ΔtkΔtT. Specifically, for k=0,1,…, we set(3)X1Δtk+1Δt=X1ΔtkΔt+V1ΔtkX1ΔtkΔt+A−d−bX1ΔtkΔt+b1−pX2ΔtkΔt−βX1ΔtkΔtX2ΔtkΔtfX2ΔtkΔtΔt,X2Δtk+1Δt=X2ΔtkΔt+V2ΔtkX2ΔtkΔt+bpX2ΔtkΔt−d+μ+γX2ΔtkΔt+βX1ΔtkΔtX2ΔtkΔtfX2ΔtkΔtΔt,X3Δtk+1Δt=X3ΔtkΔt+V3ΔtkX3ΔtkΔt+γX2ΔtkΔt−d−bX3ΔtkΔtΔt.Then, we will show thatXΔtt converges weakly to a diffusion process as Δt⟶0. To determine the drift coefficients of the diffusion, first, let pΔtx,dy denote the transition probabilities of the homogeneous Markov chain XΔtkΔtk=0∞, that is,(4)pΔtx,A=PXΔtk+1Δt∈A|XΔtkΔt=x,for all x=s,i,r∈R3+ and any Borel set A∈R3+. Combining with (2), we have for all x=s,i,r,(5)1Δt∫y1−spΔtx,dy=EA−d−bs−βsifi+b1−pi+sΔtV1Δtk=A−d−bs−βsifi+b1−pi,(6)1Δt∫y2−ipΔtx,dy=Ebpi+βsifi−d+μ+γi+iΔtV2Δtk=bpi+βsifi−d+μ+γi,(7)1Δt∫y3−rpΔtx,dy=Eγi−d−br+rΔtV3Δtk=γi−d−br.To determine the diffusion coefficients, we consider the moments(8)gijΔtz=1Δt∫yi−ziyj−zjpΔtx,dy,i,j=1,2,3Using (2), we obtain(9)g11Δtx−σ12s2=1ΔtEΔtA−d−bs−βsifi+b1−pi+sV1Δtk2−σ12s2=ΔtA−d−bs−βsifi+b1−pi2.Therefore,(10)limΔt⟶0+supx≤Kg11Δtx−σ12s2=0,for all 0<K<∞. Similarly, one can obtain that(11)limΔt⟶0+supx≤Kg22Δtx−σ22i2=0,limΔt⟶0+supx≤Kg33Δtx−σ32r2=0,(12)limΔt⟶0+supx≤Kg12Δtx=0,limΔt⟶0+supx≤Kg23Δtx=0,limΔt⟶0+supx≤Kg31Δtx=0,for all 0<K<∞.Assuming thatEViΔtk4=oΔt for i=1,2,3, one can deduce that for all 0<K<∞,(13)limΔt⟶0+supx≤K1Δt∫y−x3pΔtx,dy=0.Finally, we will extend the definition ofXΔtt to all t≥0 by setting XΔtt=XΔtkΔt for t∈kΔt,k+1Δt. According to Theorem 7.1 and Lemma 8.2 of [22], from (5)–(13), we can conclude that as Δt⟶0, XΔtt converge weakly to the solution Xt=St,It,Rt of the following SDE:(14)dS=A−d−bSt−βStItfIt+b1−pItdt+σ1StdB1t,dI=bpIt+βStItfIt−d+μ+γItdt+σ2ItdB2t,dR=γIt−d−bRtdt+σ3RtdB3t,where Bit,i=1,2,3 is a standard Brownian motion.Then, according to [23], we replace the parameter β by β+σ4dB4t. The stochastic SIR epidemic model with vertical transmission and nonlinear incidence is as follows.(15)dS=A−d−bSt−βStItfIt+b1−pItdt+σ1StdB1t−σ4StItfItdB4t,dI=bpIt+βStItfIt−d+μ+γItdt+σ2ItdB2t+σ4StItfItdB4t,dR=γIt−d−bRtdt+σ3RtdB3t,where B4t is a standard Brownian motion and parameter σ4≥0 represents the intensity of the environmental white noise.The structure of this study is as follows: In Section2, we study the existence and uniqueness of the positive solution of system (15). In Section 3, we give sufficient conditions for the extinction of the disease. In Section 4, we prove the existence of the stationary distribution of system (15) under some conditions by using the appropriate Lyapunov function. In Section 5, we discuss the solution of system (15) spirals around the disease-free equilibrium and endemic equilibrium of deterministic system (15) under proper conditions. Finally, the correctness of the theoretical results is verified by numerical simulation.In this study, letΩ,ℱ,ℱtt≥0,P be a complete probability space with a filtration, which satisfies the usual conditions (i.e., ℱt=∩s>tℱs, while ℱ0 contains P− null sets). Denote R3+=x,y,z∈R3:x>0,y≥0,z≥0.Thed-dimensional stochastic differential equations have the following form [24]:(16)dxt=fxt,tdt+gxt,tdBt,fort≥t0,with initial value xt0=x0∈R3. Bt is a d-dimensional standard Brownian motion which is defined in the above complete probability space.The differential operatorℒ of system (15) is defined by(17)ℒ=∂∂t+∑i=1dfix,t∂∂xi+12∑i,j=1dgx,tgTx,tij∂2∂xi∂xj.Ifℒ acts on a function V∈C2,1Rd×t0,+∞;R+, then(18)ℒVx,t=Vtx,t+Vxx,tfx,t+12tracegTx,tVxxx,tgx,t,where Vt=∂V/∂t,Vx=∂V/∂x1,…,∂V/∂xd, and Vxx=∂2V/∂xi∂xjd×d. By Itô’s formula [24],(19)dVx,t=ℒVx,tdt+Vxx,tgx,tdBt.


## 2. Existence and Uniqueness of the Global Positive Solution

In order to analyze the dynamic behavior of the system, we first need to discuss whether the solution of the system is nonnegative and global existence? In stochastic differential equations, if their coefficients satisfy local Lipschitz conditions and linear growth conditions, system (15) has a global positive solution [24]. However, the coefficients of system (15) do not satisfy linear growth conditions, which may lead to diseases exploded at a finite time. Next, we will show that the solution of system (15) does not blow up in finite time, so that the solution of system (15) will be global.Theorem 1.
For any given initial conditionS0,I0,R0∈R3+, there is a unique positive solution St,It,Rt to system (13) on t≥0, and the solution remains in R3+ with probability one, namely, St,It,Rt∈R3+ for all t≥0 almost surely.Proof.
Since the coefficients of system (15) are locally Lipschitz continuous, for any given initial value S0,I0,R0∈R3+, there is a unique local solution St,It,Rt on t∈0,τe, where τe is the explosion time. Next, we will verify that this solution is global, i.e., τe=+∞ a.s. First, let ξ0 be sufficiently large, such that S0,I0,R0 all lie within the interval 1/ξ0,ξ0. For each integer ξ≥ξ0, define the stopping time as [24](20)τξ=inft∈0,τe:St≤0orIt≤0orRt≤0.
Without loss of generality, let inf∅=+∞ (∅ is the empty set); we have, τξ≤τe. If we prove that τξ=+∞, a.s., then τe=+∞ and St,It,Rt∈R3+, a.s. for all t≥0. If the statement is not true, then there is a positive constant T, such that(21)Pτξ<T>0.
Define aC2 function V:R3+⟶R by(22)VSt,It,Rt=lnStItRt.
Forω∈τξ<T and all t∈0,τξ, by Itô’s formula, one can verify that(23)dVSt,It,Rt=ASt−βItfIt−d−b+b1−pItSt+βStfIt+bp−d+μ+γ+γItRt−d−b−12σ12+σ22+σ32−12σ42I2tf2It+σ42S2tf2Itdt+σ1dB1t−σ4ItfItdB4t+σ2dB2t+σ4StfItdB4t+σ3dB3t≥−βIt−2d−b−d+μ+γ−12σ42S2t−12σ12+σ22+σ32−12σ42I2tdt+σ1dB1t+σ2dB2t−σ4ItfItdB4t+σ4StfItdB4t+σ3dB3t.
Usingf0=1 and f′I≥0, the inequality (23) can be reduced to(24)dVSt,It,Rt≥HSt,It,Rtdt+σ1dB1t+σ2dB2t−σ4ItfItdB4t+σ4StfItdB4t+σ3dB3t,here,(25)HSt,It,Rt=−βIt−2d−b−d−μ−γ−12σ12+σ22+σ32−12σ42I2t−12σ42S2t.
Integrating both sides of the inequality (24) from 0 to t yields(26)VSt,It,Rt≥VS0,I0,R0+∫0tHSr,Ir,Rrdr+σ1B1t+σ2B2t+σ4∫0tSr−IrfIrdB4r+σ3dB3t.
Note that some components ofSτξ,Iτξ,Rτξ equal 0; therefore,(27)limt⟶τξVSt,It,Rt=−∞.
Lettingt⟶τξ in (26), it leads to the contradiction:(28)−∞≥VS0,I0,R0+∫0τξHSr,Ir,Rrdr+σ1B1τξ+σ2B2τξ+σ4∫0τξSr−IrfIrdB4r+σ3dB3τξ>−∞.
Thus,τξ=+∞ a.s. The proof is completed.


## 3. Extinction of the Disease

For system (1), using the notations in [25], we can deduce the basic reproductive number R0.First, we have two vectorsℱ and V to represent the new infection term and remaining transfer terms, respectively:(29)ℱ=βSIfI00,V=d+μ+γI−bpI−bR−γI+dR−A−bS+βSIfI+dS−b1−pI.The infected compartment isI; then,(30)F=βAd−b,V=d+μ+γ−bp.Hence, the reproduction number is given by(31)ρFV−1,R0=ρFV−1=βAd−bd+μ+γ−bp,where d>0 and 0≤p≤1. If R0≤1, the disease will go to extinction. If R0>0, the disease will persist. For system (15), we focus on whether diseases can be extincted by regulating the parameters of system (15). For the sake of simplicity, denote xt=1/t∫0txsds.Lemma 1.
LetSt,It,Rt be the solution of system (15) with initial value S0,I0,R0∈R+3, then(32)limt⟶∞Stt=0,limt⟶∞Itt=0,limt⟶∞Rtt=0,a.s.,(33)limt⟶∞∫0tSudB1ut=0,limt⟶∞∫0tIudB2ut=0,a.s.,limt⟶∞∫0tRudB3ut=0,limt⟶∞∫0tSu/fIudB4ut=0,a.s.,moreover,(34)limt⟶∞supSt+It+Rt≤Ad−b,a.s.Proof.
Using the method of Lemma1 in [26], it is easy to prove the conclusion (32) and (33). Then, we prove that conclusion (34) holds. From system (15), we obtain(35)dSt+It+Rt=A−d−bSt+It+Rt−μItdt+σ1StdB1t+σ2ItdB2t+σ3RtdB3t.
Integrating both sides of (35) from 0 to t, then dividing by t yields(36)St+It+Rtt−S0+I0+R0t=A−d−bSt+It+Rt−μIt+σ1∫0tSudB1ut+σ2∫0tIudB2ut+σ3∫0tRudB3ut.
Taking the limit of both sides of (36) and using (32) and (33), we have(37)limt⟶∞supSt+It+Rt≤Ad−b,a.s.
The proof of the lemma is completed.
Define(38)R0s=1d+μ+γ−bpAβd−b−12σ22−12σ42Ad−b2=R0−σ22+σ42A/d−b22d+μ+γ−bp.
It is obvious thatR0s≤R0.Theorem 2.
LetSt,It,Rt be the solution of system (15) with initial value S0,I0,R0∈R+3 and suppose that one of the following conditions holds:(39)AR0s<1andβ>Aσ42d−b,Bβ≤Aσ42d−bandβ22σ42−d+μ+γ−bp−12σ22<0,σ4≠0,then, It will tend to zero exponentially with probability one, i.e.,(40)limt⟶∞suplogItt≤d+μ+γ−bpR0s−1<0,a.s.,ifAholds,(41)limt⟶∞suplogItt≤β22σ42−d+μ+γ−bp−12σ22<0,a.s.,ifBholds.
In addition, we also have(42)limt⟶∞St=Ad−b,a.s.,limt⟶∞suplogRtt≤−mind−b+12σ32,k,a.s.,where k is a positive constant.Proof.
Using Itô’s formula, we have(43)logItt=logI0t+βStfIt−d+μ+γ−bp−12σ22−12σ42S2tf2It+σ2B2tt+σ4∫0tSu/fIudB4ut≤logI0t+βStfIt−d+μ+γ−bp−12σ22−12σ42StfIt2+σ2B2tt+σ4∫0tSu/fIudB4ut=logI0t−12σ42StfIt−βσ422+β22σ42−d+μ+γ−bp−12σ22+σ2B2tt+σ4∫0tSu/fIudB4ut.
Taking integrate both sides of (43) from 0 to t, then dividing by t, we yield(44)logItt=logI0t+βStfIt−d+μ+γ−bp−12σ22−12σ42S2tf2It+σ2B2tt+σ4∫0tSu/fIudB4ut≤logI0t+βStfIt−d+μ+γ−bp−12σ22−12σ42StfIt2+σ2B2tt+σ4∫0tSu/fIudB4ut=logI0t−12σ42StfIt−βσ422+β22σ42−d+μ+γ−bp−12σ22+σ2B2tt+σ4∫0tSu/fIudB4ut.
By the strong law of large numbers for martingales [24] and Lemma (32), we have(45)limt⟶∞B2tt=0,a.s.limt⟶∞∫0tSu/fIudB4ut=0,a.s.
If condition (A) holds, (44) becomes(46)logItt≤logI0t−12σ42A2d−b2+βAd−b−d+μ+γ−bp−12σ22+σ2B2tt+σ4∫0tSu/fIudB4ut,and using (45), we obtain(47)limt⟶∞suplogItt≤βAd−b−d+μ+γ−bp−12σ22−12σ42Ad−b2=d+μ+γ−bpR0s−1,<0.
If condition (B) holds, (44) becomes(48)logItt≥logI0t+β22σ42−d+μ+γ−bp−12σ22+σ2B2tt+σ4∫0tSu/fIudB4ut.
Furthermore, using (45), we can get(49)limt⟶∞suplogItt≤β22σ42−d+μ+γ−bp−12σ22<0,a.s.
So far, we proved (40) and (41). From these two limits, we can get that there is a constant k>0, such that for almost all ω∈Ω, there exists a T0=T0ω>0, when t>T0, It,ω≤e−kt. Without loss of generality, assume that It,ω≤e−kt for all t≥0, and this implies(50)limt⟶∞It=0,a.s.,limt⟶∞It=0,a.s.
Solving the third equation of system (15) and denoting ht=d−b+1/2σ32t−σ3B3t, we have(51)Rt=e−htR0+∫0tehuγIudu≤e−htR0+e−ht∫0tehuγe−kudu=F1+F2.
It is obvious that(52)limt⟶∞logF1t=−d−b−12σ32,limt⟶∞logF2t=−k.
Therefore, using (51), we have(53)limt⟶∞suplogRtt≤−mind−b+12σ32,k,a.s.
Furthermore, from system (15), we can obtain(54)St+Itt−S0+I0t=A−d−bSt−d+μ+γ−bIt+σ1∫0tSudB1ut+σ2∫0tIudB2ut.
Consequently,(55)St=Ad−b−d+μ+γ−bd−bIt+ψt,where(56)ψt=−1d−bSt+Itt−S0+I0t+σ1∫0tSudB1ut+σ2∫0tIudB2ut.
According to Lemma1, we get(57)limt⟶∞ψt=0,a.s.
Combining (50), (55), and (57), we can see that(58)limt⟶∞St=Ad−b,a.s.
The proof is completed.Remark 1.
In condition (B) of Theorem 2, if σ4=0, then we can get a degenerate result of condition (A):(59)R0s=1d+μ+γ−bpAβd−b−12σ22=R0−σ222d+μ+γ−bp<1.
Thus, we have the following corollary to supplement Theorem2.Corollary 1.
LetSt,It,Rt be the solution of system (15) with the initial value S0,I0,R0∈R+3. Assume that σ4=0. If condition (59) holds, then It, Rt almost surely exponentially converge to zero and St in the mean almost surely converges to A/d−b.


## 4. Existence of the Stationary Distribution

For the deterministic model (1), if R0>1, there was an unique globally asymptotically stable endemic equilibrium which implies the disease will be persistent. But for system (15), although there is no endemic equilibrium, we also expect to know the trend of the positive solution of the system. The trend of positive solution can be explained by the stationary distribution. If we can prove that system (15) exists a stationary distribution, it can show that the disease persists for a long time. Then, we will give a main theorem that there exists a stationary distribution for system (15) according to a well-known result from Khasminskii [27].LetXt be a regular time-homogeneous Markov process in Rd described by the stochastic differential equation:(60)dXt=hxdt+∑r=1kgrxdBrt,where hx=h1x,h2x,…,hdx, grx=gr1x,gr2x,…,grdx, and Brtr=1,2,…,k are the standard Brownian motions defined on some probability space Ω,ℱ,ℱtt≥0,P.The diffusion matrix for equation (60) is defined as follows:(61)Ax=aijx,aijx=∑r=1kgrixgrjx.Lemma 2 (See [27]).
Suppose that there exists a regular boundary domainU⊂Rd, satisfy the following properties.(i)
In the domainU and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix Ax is bounded away from zero(ii)
Ifx∈Rd∖U, the mean time τ at which a path issuing from x reaches the set U is finite, and supx∈KExτ<∞ for every compact subset K⊂Rd, then(62)PxlimT⟶∞1T∫0Tfxtdt=∫Rdfxπdx=1,where fx is a function integrable with respect to the measure π.Remark 2.
To validate (i), it suffices to prove that there is a positive numberG, such that ∑i,j=1daijxξiξj≥Gξ2 for all x∈U and ξ∈Rd [28, 29]. To validate (ii), it suffices to prove that there exists a nonnegative C2 function V and a neighborhood U, such that for some c>0, ℒVx<−c, x∈Rd∖U [30].Theorem 3.
Suppose thatR0s>1. If the conditions(63)0<θ<mink1S∗2,k2I∗2,k3R∗2,ki>0,i=1,2,3,hold, then there exists a stationary distribution and ergodic property for system (15). Here, S∗,I∗,R∗ is the unique endemic equilibrium of system (1), and(64)k1=d−b−σ12−2d+μ+γ−2bfI∗σ42I∗β,k2=d+μ+γ2−b−σ22,k3=d−b−γ2−σ32,θ=σ12+2d+μ+γ−2bfI∗σ42I∗βS∗2+σ22I∗I∗+2d+μ+γ−2bfI∗2β+σ32R∗2.Proof.
WhenR0s>1, which means that R0>1, there has an epidemic equilibrium E∗S∗,I∗,R∗ in system (1) and S∗,I∗,R∗ are satisfied with(65)A=d−bS∗+βS∗I∗fI∗−b1−pI∗,βS∗I∗fI∗=d+μ+γ−bpI∗,γI∗=d−bR∗.
Next, build a functionV:R+3⟶R+ by(66)VSt,It,Rt=V1St,It,Rt+2d+μ+γ−2bfI∗βV2St,It,Rt+V3St,It,Rt,where(67)V1St,It,Rt=12St−S∗+It−I∗2,V2St,It,Rt=It−I∗−I∗lnItI∗,V3St,It,Rt=12Rt−R∗2.
By computing and usinga+b2≤2a2+b2, f′I≥0, and ab≤a2+b2/2, we obtain(68)ℒV1=St−S∗+It−I∗A−d−bSt−d+μ+γ−bIt+12σ12S2t+12σ22I2t=St−S∗+It−I∗−d−bSt−S∗−d+μ+γ−bIt−I∗+σ122St−S∗+S∗2+σ222It−I∗+I∗2≤−d−b−σ12St−S∗2−2d+μ+γ−2bSt−S∗It−I∗ℒV1=St−S∗+It−I∗A−d−bSt−d+μ+γ−bIt+12σ12S2t+12σ22I2t=St−S∗+It−I∗−d−bSt−S∗−d+μ+γ−bIt−I∗+σ122St−S∗+S∗2+σ222It−I∗+I∗2≤−d−b−σ12St−S∗2−2d+μ+γ−2bSt−S∗It−I∗−d+μ+γ−b−σ22It−I∗2+σ12S∗2+σ22I∗2,ℒV2=It−I∗βStfIt−d+μ+γ−bp+12σ22I∗+12σ42I∗S2tf2It=It−I∗βStfIt−βS∗fI∗+12σ22I∗+12σ42I∗S2tf2It=−It−I∗fIt−fI∗βStfItfI∗+βSt−S∗It−I∗fI∗+12σ22I∗+12σ42I∗St−S∗+S∗2f2It≤βSt−S∗It−I∗fI∗+12σ22I∗+σ42I∗St−S∗2+σ42I∗S∗2,ℒV3=Rt−R∗γIt−γI∗+d−bR∗−d−bRt+12σ32Rt2=Rt−R∗γIt−I∗−d−bRt−R∗+12σ32Rt−R∗+R∗2≤γIt−I∗Rt−R∗−d−b−σ32Rt−R∗2+σ32R∗2≤γ2It−I∗2−d−b−σ32−γ2Rt−R∗2+σ32R∗2.
Substituting (68) into(69)ℒVt=ℒV1t+2d+μ+γ−2bfI∗βℒV2t+ℒV3t,leads to(70)ℒVt≤−k1St−S∗2−k2It−I∗2−k3Rt−R∗2+θ,where(71)k1=d−b−σ12−2d+μ+γ−2bfI∗σ42I∗β,k2=d+μ+γ2−b−σ22,k3=d−b−γ2−σ32,θ=σ12+2d+μ+γ−2bfI∗σ42I∗βS∗2+σ22I∗I∗+2d+μ+γ−2bfI∗2β+σ32R∗2.
If(72)0<θ<mink1S∗2,k2I∗2,k3R∗2,ki>0,i=1,2,3,holds, then(73)U=St,It,Rt|k1St−S∗2+k2It−I∗2+k3Rt−R∗2≤θ,lies entirely in R3+. Namely, the compact set U⊂R3+. Thus, there is a constant c>0, such that for any x=St,It,Rt∈R3+∖U, we have(74)k1St−S∗2+k2It−I∗2+k3Rt−R∗2≥θ+c.
Finally, we obtain(75)ℒVSt,It,Rt≤−c,St,It,Rt∈R3+∖U.
We verify that condition (ii) of Lemma2 holds.
Then, we will prove that condition (i) holds in Lemma (60). The diffusion matrix associated to system (15) is(76)A=σ12S2+σ42S2I2f2I−σ42S2I2f2I0−σ42S2I2f2Iσ22I2+σ42S2I2f2I000σ32R2.
Let a constantG=minσ1S2,σ2I2,σ3R2>0, for any S,I,R∈U¯, and ξ1,ξ2,ξ3∈R3+. We can calculate that(77)∑i,j=1daijξiξj=σ12S2ξ12+σ22I2ξ22+σ32R2ξ32+σ42S2I2f2Iξ1−ξ22≥minσ12S2,σ22I2,σ33R2ξ12+ξ22+ξ32=Gξ2.
This shows that condition (i) of Lemma2 holds. Therefore, there has a unique stationary distribution and the ergodic property of system (15).
The proof is completed.


## 5. Asymptotic Behavior around the Equilibria of System (1)

In the section, we will study the asymptotic behavior of the solution of system (15) around the disease-free equilibrium and endemic equilibrium under random perturbations.Theorem 4.
LetSt,It,Rt be the solution of system (15) with initial value S0,I0,R0∈R+3. Suppose R0≤1, if σ12<d−b, σ22<2d+μ−bp+γ, and σ32<2d−b−γ hold, then(78)limt⟶∞sup1tE∫0tSu−Ad−b2+I2u+R2udu≤σ12A2λd−b2,where λ=mind−b−σ12,d+μ+γ/2−bp−σ22/2,d−b−γ/2−σ32/2.Proof.
Now, define a functionV:R+3⟶R+ by(79)Vt=V1t+2d+μ+γ−b1+pβV2t+V3t,where(80)V1t=12St−Ad−b+It2,V2t=It,V3t=12R2t.
By calculating, we obtain(81)ℒV1t=St−Ad−b+ItA−d−bSt−d+μ+γ−bpIt+σ12S2t+σ22I2t2=St−Ad−b+It−d−bSt−Ad−b−d+μ+γ−bpIt+σ12St−A/d−b+A/d−b2+σ22I2t2≥−d−b−σ12St−Ad−b2−2d+μ+γ−b1+pSt−Ad−bIt−d+μ+γ−bp−σ222I2t+σ12A2d−b2,ℒV2t=βStfIt−Ad−b+Ad−bIt−d+μ+γ−bpIt≥βSt−Ad−bIt+βAd−b−d+μ+γ−bpIt≥βSt−Ad−bIt,ℒV3t=γRtIt−d−bR2t+σ32R2t2≤−d−b−γ2−σ322R2t+γ2I2t.
Thus,(82)ℒVt=ℒV1t+2d+μ+γ−b1+pβℒV2t+ℒV3tγ−d−b−σ12St−Ad−b2−d+μ+γ2−bp−σ222I2t−d−b−γ2−σ322R2t+σ12A2d−b2.
We calculate the derivative ofVt along the trajectories of system (15):(83)dVt=dV1t+2d+μ+γ−b1+pβdV2t+dV3t=ℒVtdt+St−Ad−b+Itσ1StdB1t−σ4StItfItdB4t+σ2ItdB2t+σ4StItfItdB4t+σ3R2tdB3t.
Integrating (83) from 0 to t on both sides, then taking expectation, we obtain(84)EVt−EV0≤E∫0t−d−b−σ12Su−Ad−b2−d+μ+γ2−bp−σ222I2u−d−b−γ2−σ322R2udu+σ12A2d−b2t,which implies that(85)limt⟶∞sup1tE∫0td−b−σ12Su−Ad−b2+d+μ+γ2−bp−σ222I2u+d−b−γ2−σ322R2udu≤σ12A2d−b2.
Denoteλ=mind−b−σ12,d+μ+γ/2−bp−σ22/2,d−b−γ/2−σ32/2. Finally, from (85), we have(86)limt⟶∞sup1tE∫0tSu−Ad−b2+I2u+R2udu≤σ12A2λd−b2.
The proof is completed.Corollary 2.
From Theorem4, when σi=0,i=1,2,3,4, by using the method of [31], we obtain(87)ℒVt≤−d−bSt−Ad−b2−d+μ+γ2−bpI2t−d−b−γ2R2t≤0.
Thus, whenR0≤1 and d−b>0, 2d+μ−bp+γ>0, and 2d−b−γ>0 hold, the disease-free equilibrium A/d−b,0,0 of system (1) is globally asymptotically stable.Remark 3.
In system (15), assume σi=0,i=1,2,3,4. Now, if σ1=0, we find that(88)ℒVt≤−d−bSt−Ad−b2−d+μ+γ2−bp−σ222I2t−d−b−γ2−σ322R2t,which shows that ℒVt is negative definite under the conditions σ22<2d+μ−bp+γ and σ32<2d−b−γ. So, we have the following corollary.Corollary 3.
Suppose thatR0≤1 and σ1=0. The solution of system (15) is stochastically asymptotically stable in the large [24] as σ22<2d+μ−bp+γ and σ32<2d−b−γ.Theorem 5.
LetSt,It,Rt be the solution of system (15) with the initial value S0,I0,R0∈R+3. Suppose R0s>1. Then, we have(89)limt⟶∞sup1tE∫0tk1Su−S∗2+k2Iu−I∗2+k3Ru−R∗2du≤θ.
Here,S∗,I∗,R∗ is the unique endemic equilibrium of system (1), and k1,k2,k3 and θ are the same as Theorem 3.Proof.
IfR0s>1, it is obvious that R0>1. Applying Itô’s formula to (66), we have(90)dVSt,It,Rt=dV1St,It,Rt+2d+μ+γ−2bfI∗βdV2St,It,Rt+dV3St,It,Rt=ℒVSt,It,Rt+St−S∗+It−I∗σ1StdB1t+σ2ItdB2t+It−I∗σ2dB2t+σ4StfItdB4t+Rt−R∗Rtσ3dB3t.
Integrating (90) from 0 to t on both sides and from (70), we obtain(91)VSt,It,Rt≥VS0,I0,R0+∫0t−k1Su−S∗2−k2Iu−I∗2−k3Ru−R∗2du+θt+Mt,where(92)Mt=∫0tSu−S∗+Iu−I∗σ1SudB1u+σ2IudB2u+∫0tIu−I∗σ2dB2u+σ4SufIudB4u+∫0tRu−R∗Ruσ3dB3u,which is a continuous local martingale; then, taking expectation on both sides of (91) yields(93)EVSt,It,Rt≥EVS0,I0,R0+E∫0t−k1Su−S∗2−k2Iu−I∗2−k3Ru−R∗2du+θt.
Finally, we obtain(94)limt⟶∞sup1tE∫0tk1Su−S∗2+k2Iu−I∗2+k3Ru−R∗2du≤θ.
This completes the proof.Corollary 4.
From Theorem5, when σi=0,i=1,2,3,4, by using the method of [31], we obtain(95)ℒVt≤−d−bSt−S∗2−d+μ+γ2−bIt−I∗2−d−b−γ2Rt−R∗2≤0.
Thus, whenR0s>1 and d−b>0, 2d+μ−b+γ>0, and 2d−b−γ>0 hold, the endemic equilibrium S∗,I∗,R∗ of system (1) is globally asymptotically stable.Remark 4.
From Theorems4 and 5, it can be seen that the solution of system (15) will oscillate around the disease-free equilibrium and endemic equilibrium of system (1) under some conditions, and the disturbance intensity is proportional to the white noise intensity. From a biological point of view, the solution of system (15) fluctuates most of the time around the disease-free equilibrium and endemic equilibrium of system (1) due to the small magnitude of the random disturbance.


## 6. Conclusions and Numerical Simulations

In this study, we are mainly concerned about the influence of stochastic factors on the behavior of the epidemic SIR model with a nonlinear incidence and vertical transmission. Here, the stochastic factors mainly include the disease transmission coefficient and state variables affected by white noise. System (15) has a unique global positive solution starting from the positive initial value. Furthermore, we give a threshold value R0s to distinguish the persistence and extinction of diseases, which differs from the basic reproductive number R0 corresponding system (1) by a noise term. When R0s<1, the conditions for extinction of disease are obtained in system (15). By constructing a stochastic Lyapunov function, we prove that an ergodic stationary distribution exists in system (15) when R0s>1, which means that the disease will persist. Finally, we study the asymptotic properties of solution of system (15) around disease-free equilibrium and endemic equilibrium of the deterministic model (1) under certain conditions and under the existence of stationary distribution, respectively. From the biological point of view, the interference of environmental white noise may have certain influence on the stability of the biological system: the ability of population to adapt to the environment is limited. If the intensity of white noise in the environment is small enough, the stability of the population will not be damaged. If the intensity of white noise is large in the environment, it may lead to the extinction of species.In the following part, we will give some examples to verify the theoretical results by numerical simulation. The method of numerical simulation is shown in reference [32]. For convenience, take fI=1+mI.Example 1.
Fix the parametersA=0.43,β=0.9,d=0.6,μ=0.1,γ=0.2,b=0.1,p=0.2,σ1=0.1,σ2=0.5,σ3=0.1,σ4=1,m=1 in system (15). It is easy to calculate(96)R0s=1d+μ+γ−bpAβd−b−12σ22−12σ42Ad−b2=0.3173<1,and 0.9=β>Aσ42/d−b=0.86. Obviously, this set of parameters satisfies condition (A) of Theorem 2. From Figure 1(a), it can be seen that the state variable St fluctuates around A/d−b=0.86, and It,Rt tend to zero with the increase of time. If we adjust β=0.8, other parameters remain unchanged. It is obvious that 0.8=β<Aσ42/d−b=0.86 and(97)β22σ42−d+μ+γ−bp−12σ22=−0.6850<0
Condition (B) of Theorem 2 is satisfied. As shown in Figure 1(b), St,It,Rt fluctuate around A/d−b=0.86,0,0 with the increase of time, respectively. This proves that the conclusion of Theorem 2 is correct. Furthermore, take σ4=0 and other parameters keep the same with the first set of parameters. Under this set of parameters, R0s=0.7375<1. From Figure 1(c), it can be seen that the result of Corollary 1 is true.Figure 1
The subgraphs (a)–(c) denote the numerical simulation of conditions (A) and (B) of Theorem 2 and Corollary 1, respectively.
(a)(b)(c)Example 2.
TakeA=0.55,β=0.9,d=0.6,μ=0.1,γ=0.2,b=0.1,p=0.2,σ1=0.1,σ2=0.5,σ3=0.1,σ4=1,m=1. Then, we draw the graph of the state variable St,It,Rt of system (1) and system (15) with the increase of time. Under the parameters, R0s<1<R0. When R0>1, there is a stable endemic equilibrium for system (1). When R0s<1, according to Theorem 2, St,It,Rt fluctuate around A/d−b=1.1,0,0 with the increase of time. From Figure 2, we can find that adding random perturbations to the environment can suppress the disease outbreaks.Figure 2
The subgraphs (a) and (b) denote the numerical simulation of the stochastic system (15) and deterministic system (1), respectively. R0s<1<R0 shows that large white noise can lead to disease extinction, though the deterministic system is persistent.
(a)(b)Example 3.
Choose the parametersA=2,β=0.9,d=0.6,μ=0.1,γ=0.2,b=0.1,p=0.2,σ1=σ2=σ3=0.1,σ4=0.01,m=1. We can calculate that R0≈4.09>1,R0s=4.0843>1, 0.776=θ<mink1S∗2,k2I∗2,k3R∗2=0.0858. When R0>1, there is a stable positive equilibrium for system (1). When R0s>1, there exists a stationary distribution for system (15) according to Theorem 3. From Figures 3(a)and 3(b), all the conclusions are explained. Furthermore, we draw the histogram of Figure 3(a) and find that St,It,Rt obey normal distribution, as shown in Figure 4.Figure 3
The subgraphs (a) and (b) denote the numerical simulation of the stochastic system (15) with R0s>1 and deterministic system (1) with R0>1, respectively. This shows that the disease will be permanent in a long time.
(a)(b)Figure 4
The histogram (a)–(c) of solution of the stochastic system (15) with R0s>1. System (15) exists a unique stationary distribution.
(a)(b)(c)Example 4.
FixA=0.43,β=0.9,d=0.6,μ=0.1,γ=0.2,b=0.1,p=0.2,σ2=0.5,σ3=0.1,σ4=1,m=1. Take σ1=0.2,0, respectively. From Figures 5(a) and 5(b), it is shown that solution of system (15) is stochastically asymptotically stable in the large.
In the study, we only consider the white noise. In real life, the environmental impact may be not only white noise but also color noise, so can we consider both of them in the model? In addition, it may be more reasonable to consider the spatial factors and time delay when building the model. We leave these issues for further investigation and look forward to resolution in the future.Figure 5
The subgraphs (a) and (b) are the numerical simulations of the stochastic system (15) with σ1=0.2, and σ1=0, respectively. Subgraph (b) shows that the solution of system (15) is stochastically asymptotically stable in the large.
(a)(b)



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