# On the Dynamics Behaviors of a Stochastic Echinococcosis Infection Model with Environmental Noise

Authors: Jianguo Sun; Huina Zhang; Daqing Jiang
Journal: Discrete Dynamics in Nature and Society
 (2021)

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## Abstract

This paper pays main attention to the dynamics behaviors of a stochastic echinococcosis infection model with environmental noise. The existence and uniqueness of the stochastic model is showed in this paper. We obtain the sufficient condition of the ergodic stationary distribution. What is more, the condition of extinction of the stochastic model is also given.


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## Body

## 1. Introduction

Echinococcosis is a chronic parasitic disease caused by the infection of echinococcosis larvae with human hosts and other mammals [1]. Echinococcus can parasitize many parts of the human body and infect many organs at the same time. At present, the cystic echinococcosis and vesicular echinococcosis are the main types of echinococcosis. Those areas, such as Eastern and Central Asia, Northern Africa, and Southern America, are mainly distributed by echinococcosis [2]. Every year, in the world, there are 18800 new cases of cystic echinococcosis, which causes about 18400 DALYs [3]. Echinococcosis infection is one of the most important public health problems in the world, seriously endangering human health and economic development.Echinococcosis has a wide range of epidemic and serious harm, and its spread has attracted extensive attention in the world [4–8]. The research on the epidemic area, influencing factors, and transmission mechanism of echinococcosis is a hot topic in recent decades [4–6, 9]. Based on the long-term epidemic situation of echinococcosis, scholars also used a deterministic model to explore the influence of host activity and individual behavior on the transmission of echinococcosis [9, 10].The criteria of global asymptotic stability of the traveling wave solutions of the echinococcosis model were given by Xu and Ai [11]. Wang et al. [12] gave the following model of echinococcosis and used the results to Xinjiang of China. The variables in the model are defined in Table 1, and the parameters are defined in Table 2.(1)dSD=A1−β1SDIL−d1SD+σIDdt,dID=β1SDIL−d1+σIDdt,dSL=A2−β2SLID−d2SLdt,dIL=β2SLID−d2ILdt,dSH=A3−β3SHID−d3SH+γIHdt,dEH=β3SHID−d3+ωEHdt,dIH=ωEH−d3+μ+γIHdt.Table 1
Variables in the model (adapted from [12]).
PopulationDefinitionSDtSusceptible dogs at timetIDtInfectious dogs at timetSLtSusceptible livestock at timetILtInfectious livestock at timetSHtSusceptible human at timetEHtExposed human at timetIHtInfectious human at timetTable 2
Parameters in the model (adapted from [12]).
ParameterDefinitionA1Annual recruitment rate of the dog populationd1Natural death rate of the dog populationσRecovery rate of transition from infected to noninfected dogsA2Annual recruitment rate of the livestock populationd2Natural death rate of the livestock populationA3Annual recruitment rate of the human populationd3Natural death rate of the human populationμDisease-related death rate of the human population1/ωIncubation period of infected individuals of the human populationγRecovery rate of the human populationβ1Infection rate of susceptible dogs by infectious livestockβ2Infection rate of susceptible livestock by infectious dogsβ3Infection rate of susceptible human by infectious dogsThe results showed that the dynamical properties of the model were completely determined by the basic reproduction numberR0R03=β1β2A1A2/d1d22d1+σ. That was, if R0<1, the disease-free equilibrium was globally asymptotically stable, and if R0>1, the model was permanent and the endemic equilibrium was globally asymptotically stable.Actually, in the real world, environmental noise is an inevitable important factor, so the epidemic model will be affected by environmental noise [13–20]. In this paper, we will consider a type of environmental noise (namely white noise) with the system (1). Then, the stochastic version corresponding to system (1) can be described by the following equations:(2)dSD=A1−β1SDIL−d1SD+σIDdt+σ1SDdB1t,dID=β1SDIL−d1+σIDdt+σ2IDdB2t,dSL=A2−β2SLID−d2SLdt+σ3SLdB3t,dIL=β2SLID−d2ILdt+σ4ILdB4t,dSH=A3−β3SHID−d3SH+γIHdt+σ5SHdB5tdEH=β3SHID−d3+ωEHdt+σ6EHdB6t,dIH=ωEH−d3+μ+γIHdt+σ7IHdB7t,where Bit stands for the mutually Brownian motions independently and Bi0=0; σi2>0 replaces the intensities of the environmental noise, i=1,2,3,4,5,6,7.In Section2 of this paper, we give the existence and uniqueness of global positive solutions of the infection model (2) by using the Lyapunov function. The result that the system (2) is ergodic stationary distribution on certain conditions is proved in Section 3. Section 4 gives the sufficient conditions for the existence of extinction of the system (2). We illustrate some simulations to demonstrate our results in Section 5.


## 2. Existence and Uniqueness of the Positive Solution

Defining a differential operatorL associated with equation (1) using the same methods as [15], there is positive and global solution of one system if the coefficients of the system satisfy local Lipschitz conditions. Actually, for the system (2), it may explode in finite time for not satisfying the linear growth condition. Here, we only use the Lyapunov function to proof the solution of the model (2) is positive and global.Denote(3)R+n=x∈Rn:xi>0,for all1≤i≤n,R¯+n=x∈Rn:xi≥0,for all1≤i≤n.We consider the general n-dimensional stochastic differential equation(4)dxt=fxt,tdt+gxt,tdBt,fort≥t0,with initial value xt0=x0∈Rn, where Bt denotes n-dimensional standard Brownian motions defined on the above probability space.Theorem 1.
There is a unique solutionSDt,IDt,SLt,ILt,SHt,EHt,IHt of system (2) on t≥0 for any initial value SD0,ID0,SL0,IL0,SH0,EH0,IH0∈R+7, and the solution will remain in R+7 with probability 1, namely, SDt,IDt,SLt,ILt,SHt,EHt,IHt∈R+7, for all t≥0 almost surely.Proof.
Using the same method as [17], when the local solution of the system (2) is global with the initial value(5)SD0,ID0,SL0,IL0,SH0,EH0,IH0∈R+7,we only need to prove SDt,IDt,SLt,ILt,SHt,EHt,IHt on t∈0,τe. Let n0≥0 be a large number and SD0,ID0,SL0,IL0,SH0,EH0,W0∈1/n0,n0. Defining the stopping time(6)τn=inft∈0,τe:minSDt,IDt,SLt,ILt,SHt,EHt,IHt≤1normaxSDt,IDt,SLt,ILt,SHt,EHt,IHt≥n,for n≥n0. We set infϕ=∞ (as usual ϕ denotes the empty set). According to the definition, τn is increasing as n⟶∞. Set τ∞=limn⟶∞τn, where τ∞≤τe a.s. If we can prove that τ∞=∞ a.s., then τe=∞ and SDt,IDt,SLt,ILt,SHt,EHt,IHt∈R+7 a.s., for all t≥0, and the formula τ∞=∞ a.s. is needed to show. Otherwise, there exists constants S>0 and ε∈0,1 such that(7)Pτ∞≤S>ε.
Hence, there is an integern1≥n0 such that(8)Pτ∞≤S≥ε,for alln≥n1.
Define aC2-function V:R+7⟶R¯+ by(9)V=SD−a−alnSDa+ID−b−blnIDb+SL−1−lnSL+IL−1−lnIL+SH−1−lnSH+EH−1−lnEH+IH−1−lnIH,where a,b are positive constants to be determined later. The nonnegativity of this function can be seen from u−1−logu≥0, ∀u>0. Let n≥n0 and S>0 be arbitrary. Using Itô’s formula, we obtain(10)dV=LVdt+σ1SD−adB1t+σ2ID−bdB2t+σ3SL−1dB3t+σ4IL−1dB4t+σ5SH−1dB5t+σ6EH−1dB6t+σ7IH−1dB7t,LV=A1−d1SD−aA1SD+aβ1IL+ad1−aσIDSD+12aσ12−bd1ID−bβ1SDILID+bd1+σ+12σ22+A2−d2SL−A2SL+β2ID+d2+12σ32−d2IL−β2SLIDIL+d2+12σ42+A3−d3SH−A3SH+β3ID+d3−γIHSH+12σ52−d3EH−β3SHIDEH+d3+ω+12σ62−d3+μIH−ωEHIH+d3+μ+γ+12σ72≤aβ1−d2IL+β2+β3−bd1ID+A1+A2+A3+a+bd1+2d2+3d3+12aσ12+bσ22+σ32+σ42+σ52+σ62+σ72+bσ+ω+μ+γ.
Choosing(11)a=d2β1,b=β2+β3d1,then we get(12)LV≤A1+A2+A3+a+bd1+2d2+3d3+12aσ12+bσ22+σ32+σ42+σ52+σ62+σ72+bσ+ω+μ+γ≔K.
The remained proof follows that in Mao et al.’s study [14].


## 3. Existence of Unique and Ergodic Stationary Distribution

Lemma 1 (see [14]).
The Markov processXt has a stationary distribution μ⋅ if there exists a bounded domain U⊂El with regular boundary Γ and(i)
(B.1) there is a positive numberM such that ∑i,j=1laijxξiξj≥Mξ2,x∈U,ξ∈Rl.(ii)
(B.2) there exists a nonnegativeC2 function V such that LV is negative for any El\U. Then,(13)PxlimT⟶∞1T∫0TfXtdt=∫Elfxμdx=1,for all x∈El, where f⋅ is a function integrable with respect to the measure μ.Define a parameter(14)R0s=β1β2A1A2d1+1/2σ12d1+σ+1/2σ22d2+1/2σ32d2+1/2σ42.Theorem 2.
Assume thatR0s>1, then there is a unique stationary distribution μ⋅ for system (2) and it has ergodic property.Proof.
(i)
Step 1: verify that (B.1) holds. Apparently, the diffusion matrix of system (2) is(15)ΛSD,ID,SL,IL,SH,EH,IH=σ12SD20000000σ22ID20000000σ32SL20000000σ42IL20000000σ52SH20000000σ62EH20000000σ72IH2.Besides, there isR=minSD,ID,SL,IL,SH,EH,IH∈R+7σ12SD2,σ22ID2,σ32SL2,σ42IL2,σ52SH2,σ62EH2,σ72IH2>0 such that(16)∑i,j=17λijSD,ID,SL,IL,SH,EH,IHξiξj=σ12SD2+σ22ID2+σ32SL2+σ42IL2+σ52SH2+σ62EH2+σ72IH2≥Rξ2,for allSD,ID,SL,IL,SH,EH,IH∈U¯,ξ∈R7, which implies condition (B.1) is satisfied.(ii)
Step 2: verify that (B.2) holds. Now, we will construct a nonnegativeC2-function V and a closed set U∈Σ such that(17)supSD,ID,SL,IL,SH,EH,IH∈R+7\ULVSD,ID,SL,IL,SH,EH,IH<0,which can assure that (B.2) is satisfied. For the convenience of calculation, let(18)S¯D=A1d1+1/2σ12,S¯L=A2d2+1/2σ32.
I¯D and I¯L satisfy the following equations:(19)β3S¯LI¯D=d2+12σ42I¯L.
Beyond that(20)S^D=SDS¯D,I^D=IDI¯D,S^L=SLS¯L,I^L=ILI¯L.
Considering aC2-function R+7⟶R(21)ΦSD,ID,SL,IL,SH,EH,IH=MV1+V2+V3,we assume that Φ˜ is the minimum value of Φ. Then, we define a nonnegative C2-function V,(22)V=Φ−Φ˜,where(23)V1=−I¯DlnID−c1S¯DlnSD−c2S¯LlnSL−c3I¯LlnIL+cIL,V2=−lnSD−lnSL−lnIL−lnSH−lnEH−lnIH,V3=1m+2SD+ID+SL+IL+SH+EH+IHm+2,and c,c1,c2,c3 are positive constants to be determined later; m>0 is a sufficiently small number such that(24)m<2d1∧d2∧d3σ12∨σ22∨σ32∨σ42∨σ52∨σ62∨σ72−1.
M is a sufficiently large number such that(25)AMI¯D+E≤−2,where(26)E=supSD,ID,SL,IL,SH,EH,IH∈R+7β1IL+β2+β3ID+Mcβ2εSLm+2+1−12ρSD+ID+SL+IL+SH+EH+IHm+2+B+C,where ε is a sufficiently small number such that(27)Mcβ2ε−12ρ≤0,and B,C will be defined in (36) and (38), respectively, then(28)L−lnSD=−A1SD+β1IL−σIDSD+d1+12σ12.
By applying the inequalityx−1≥lnxx>0, we obtain(29)L−S¯DlnSD=−A1S^D+β1S¯DIL−σS¯DIDSD+S¯Dd1+12σ12≤−A11S^D−1+β1S¯D≤A1lnS^D+β1S¯DIL.
From (25), we can obtain(30)I¯LI¯D=β2S¯Ld2+1/2σ42.
Then,(31)L−lnID=−β1SDILID+d1+σ+12σ22=−β1S¯DI¯LI¯DS^DI^LI^D+d1+σ+12σ22=−β1S¯DI¯LI¯DS^DI^LI^D−1−β1S¯DI¯LI¯D+d1+σ+12σ22≤−β1S¯Dβ2S¯Ld2+1/2σ42+d1+σ+12σ22−β1S¯DI¯LI¯DlnS^DI^LI^D=−d1+σ+12σ22β1S¯Dβ2S¯Ld1+σ+1/2σ22d2+1/2σ42−1−β1S¯DI¯LI¯DlnS^DI^LI^D=−R0s−1d1+σ+12σ22−β1S¯DI¯LI¯DlnS^DI^LI^D≔−A−β1S¯DI¯LI¯DlnS^DI^LI^D,L−I¯DlnID≤−AI¯D−β1S¯DI¯LlnS^DI^LI^D,L−S¯LlnSL=−A2S^L+β2S¯LID+S¯Ld2+12σ32≤A2lnS^L+β2S¯LID,L−I¯LlnIL=−β2S¯LI¯DS^LI^DI^L+I¯Ld2+12σ42=−β2S¯LI¯DS^LI^DI^L−1≤−β2S¯LI¯DlnS^LI^DI^L.
Then,(32)LV1≤−AI¯D+c1β1S¯D−cd2IL+c2β2S¯LID+cβ2SLID+c1A1−β1S¯DI¯LlnS^D+β1S¯DI¯L−c3β2S¯LI¯DlnI^D+c2A2−c3β2S¯LI¯DlnS^L+−β1S¯DI¯L+c3β2S¯LI¯DlnI^L.
Let(33)c1A1−β1S¯DI¯L=0,β1S¯DI¯L−c3β2S¯LI¯D=0,c2A2−c3β2S¯LI¯D=0,−β1S¯DI¯L+c3β2S¯LI¯D=0,then the unique solution of above equations is given as follows:(34)c1=β1S¯DI¯LA1,c2=β1S¯DI¯LA2,c3=β1S¯DI¯Lβ2S¯LI¯D.
Choosingc=c1β1S¯D/d2, then(35)LMV1≤−AMI¯D+Mc2β2S¯LID+Mcβ2SLID,(36)LV2=L−lnSD−lnSL−lnIL−lnSH−lnEH−lnIH=−A1SD+β1IL−σIDSD+d1+12σ12−A2SL+β2ID+d2+12σ32−β2SLIDIL+d2+12σ42−A3SH+β3ID−γIHSH+d3+12σ52−β3SHIDEH+d3+ω+12σ62−ωEHIH+d3+μ+γ+12σ72≤−A1SD−A2SL−β2SLIDIL−A3SH−β3SHIDEH−ωEHIH+β1IL+β2+β3ID+B,where(37)B=d1+2∑i=23di+12σ12+∑i=37σi2+ω+μ+γ,(38)LV3=L1m+2SD+ID+SL+IL+SH+EH+IHm+2=SD+ID+SL+IL+SH+EH+IHm+1A1+A2+A3−d1SD+ID−d2SL+IL−d3SH+EH−d3+μIH+m+12SD+ID+SL+IL+SH+EH+IHmσ12SD2+σ22ID2+σ32SL2+σ42IL2+σ52SH2+σ62EH2+σ72IH2≤A1+A2+A3SD+ID+SL+IL+SH+EH+IHm+1−d1∧d2∧d3SD+ID+SL+IL+SH+EH+IHm+2+m+12σ12∨σ22∨σ32∨σ42∨σ52∨σ62∨σ72SD+ID+SL+IL+SH+EH+IHm+2≤C−12ρSD+ID+SL+IL+SH+EH+IHm+2,where(39)C=supSD,ID,SL,IL,SH,EH,IH∈R+7A1+A2+A3SD+ID+SL+IL+SH+EH+IHm+1−12ρSD+ID+SL+IL+SH+EH+IHm+2,ρ=d1∧d2∧d3−m+12σ12∨σ22∨σ32∨σ42∨σ52∨σ62∨σ72>0.
Then, combining with (32), (33), and (35), we get the value of LV finally:(40)LV≤−AMI¯D+c2Mβ2S¯LID+Mcβ2SLID−A1SD−A2SL−β2SLIDIL−A3SH−β3SHIDEH−ωEHIH−12ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+β2+β3ID+B+C.
Then, defining a closed set(41)Uε=SD,ID,SL,IL,SH,EH,IH∈R+7:ε≤SD≤1ε,ε≤ID≤1ε,ε≤SL≤1ε,ε3≤IL≤1ε3,ε≤SH≤1ε,ε3≤EH≤1ε3,ε4≤IH≤1ε4,where ε is a sufficiently small positive number such that(42)−θε+D≤−1,(43)−AMI¯D+Mc2β2S¯Lε+E≤−1,(44)−θε+F≤−1,(45)−14ρ1εm+2+G≤−1,(46)−14ρ1ε3m+2+G≤−1,(47)−14ρ1ε4m+2+G≤−1,where θ=A1∧A2∧A3∧β2∧β3∧ω, D, E, F are positive constants which can be found from inequations. Divide R+7Uε to fourteen domains:(48)U1c=SD,ID,SL,IL,SH,EH,IH∈R+7:0<SD<ε,U2c=SD,ID,SL,IL,SH,EH,IH∈R+7:0<ID<ε,U3c=SD,ID,SL,IL,SH,EH,IH∈R+7:0<SL<ε,U4c=SD,ID,SL,IL,SH,EH,IH∈R+7:0<IL<ε3,SL>ε,ID>ε,U5c=SD,ID,SL,IL,SH,EH,IH∈R+7:0<SH<ε,U6c=SD,ID,SL,IL,SH,EH,IH∈R+7:0<EH<ε3,SH>ε,ID>ε,U7c=SD,ID,SL,IL,SH,EH,IH∈R+7:0<IH<ε4,EH>ε3,U8c=SD,ID,SL,IL,SH,EH,IH∈R+7:SD>1ε,U9c=SD,ID,SL,IL,SH,EH,IH∈R+7:ID>1ε,U10c=SD,ID,SL,IL,SH,EH,IH∈R+7:SL>1ε,U11c=SD,ID,SL,IL,SH,EH,IH∈R+7:IL>1ε3,U12c=SD,ID,SL,IL,SH,EH,IH∈R+7:SH>1ε,U13c=SD,ID,SL,IL,SH,EH,IH∈R+7:EH>1ε3,U14c=SD,ID,SL,IL,SH,EH,IH∈R+7:IH>1ε4.
Obviously,R+7Uε=∪1≤i≤14Uic. Next, we will show that LV≤−1 on R+7Uε, which is equivalent to verifying it on the above fourteen domains.Case 1.
For anySD,ID,SL,IL,SH,EH,IH∈U1c, in view of (39) and (41), thus(49)LV≤−A1SD−12ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−A1ε+D≤−θε+D≤−1,where(50)D=supSD,ID,SL,IL,SH,EH,IH∈R+7−12ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C.Case 2.
For anySD,ID,SL,IL,SH,EH,IH∈U2c, in view of (24), (39), and (42), thus(51)LV≤−AMI¯D+Mc2β2S¯LID+β1IL+β2+β3ID+Mcβ2SLID−12ρSD+ID+SL+IL+SH+EH+IHm+2+B+C≤−AMI¯D+Mc2β2S¯Lε+E≤−1.Case 3.
For anySD,ID,SL,IL,SH,EH,IH∈U3c, in view of (39) and (41), thus(52)LV≤−A2SL−12ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−A2ε+D≤−θε+F≤−1,where(53)F=supSD,ID,SL,IL,SH,EH,IH∈R+7β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2εIDm+2+1−12ρSD+ID+SL+IL+SH+EH+IHm+2+B+C.Case 4.
For anySD,ID,SL,IL,SH,EH,IH∈U4c, in view of (39) and (41), thus(54)LV≤−β2SLIDIL−12ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−β2ε+D≤−θε+D≤−1.Case 5.
For anySD,ID,SL,IL,SH,EH,IH∈U5c, in view of (39) and (41), thus(55)LV≤−A3SH−12ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−A3ε+D≤−θε+D≤−1.Case 6.
For anySD,ID,SL,IL,SH,EH,IH∈U6c, in view of (39) and (41), thus(56)LV≤−β3SHIDEH−12ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−β3ε+D≤−θε+D≤−1.Case 7.
For anySD,ID,SL,IL,SH,EH,IH∈U7c, in view of (39) and (41), thus(57)LV≤−ωEHIH−12ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−ωε+D≤−θε+D≤−1.Case 8.
For anySD,ID,SL,IL,SH,EH,IH∈U8c, in view of (39) and (43), thus(58)LV≤−14ρSD+ID+SL+IL+SH+EH+IHm+2−14ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−14ρSDm+2+G≤−14ρ1εm+2+G≤−1,where(59)G=supSD,ID,SL,IL,SH,EH,IH∈R+7−14ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C.Case 9.
For anySD,ID,SL,IL,SH,EH,IH∈U9c, in view of (39) and (43), thus(60)LV≤−14ρSD+ID+SL+IL+SH+EH+IHm+2−14ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−14ρIDm+2+G≤−14ρ1εm+2+G≤−1.Case 10.
For anySD,ID,SL,IL,SH,EH,IH∈U10c, in view of (39) and (43), thus(61)LV≤−14ρSD+ID+SL+IL+SH+EH+IHm+2−14ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−14ρSLm+2+G≤−14ρ1εm+2+G≤−1.Case 11.
For anySD,ID,SL,IL,SH,EH,IH∈U11c, in view of (39) and (44), thus(62)LV≤−14ρSD+ID+SL+IL+SH+EH+IHm+2−14ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−14ρILm+2+G≤−14ρ1ε3m+2+G≤−1.Case 12.
For anySD,ID,SL,IL,SH,EH,IH∈U12c, in view of (39) and (43), thus(63)LV≤−14ρSD+ID+SL+IL+SH+EH+IHm+2−14ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−14ρSHm+2+G≤−14ρ1εm+2+G≤−1.Case 13.
For anySD,ID,SL,IL,SH,EH,IH∈U13c, in view of (39) and (44), thus(64)LV≤−14ρSD+ID+SL+IL+SH+EH+IHm+2−14ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−14ρEHm+2+G≤−14ρ1ε3m+2+G≤−1.Case 14.
For anySD,ID,SL,IL,SH,EH,IH∈U14c, in view of (39) and (45), thus(65)LV≤−14ρSD+ID+SL+IL+SH+EH+IHm+2−14ρSD+ID+SL+IL+SH+EH+IHm+2+β1IL+Mc2β2S¯L+β2+β3ID+Mcβ2SLID+B+C≤−14ρIHm+2+G≤−14ρ1ε4m+2+G≤−1.
Through the discussion of the above ten cases, we can satisfy condition (B.2) of Lemma1. We complete the proof of Theorem 2.


## 4. Extinction

Define a parameter(66)R^0=β1SD0+β2∫0∞xπxdx+β3SH0+ω1/8σ22∧σ42∧σ62∧σ72,where(67)πx=Qx−2−2d2/σ32e−2/σ32A2/x,x∈0,∞.Theorem 3.
LetSD,ID,SL,IL,SH,EH,IH be the solution of system (2) with any initial value(68)SD0,ID0,SL0,IL0,SH0,EH0,IH0∈R+7.IfR^0<1, then the solution SD,ID,SL,IL,SH,EH,IH of system (2) satisfies(69)limsupt⟶∞lnID+IL+EH+IHt≤β1SD0+β2∫0∞xπxdx+β3SH0+ω−18σ22∧σ42∧σ62∧σ72<0a.s.Namely,(70)limsupt⟶∞ID=limsupt⟶∞IL=limsupt⟶∞EH=limsupt⟶∞IH=0.Also, the distribution ofSLt converges weakly to the measure which has the density(71)πx=Qx−2−2d2/σ32e−2/σ32A2/x,x∈0,∞,where Q is a constant such that ∫0∞πxdx=1.Proof.
Consider the following auxiliary logistic equation with random perturbation(72)dXt=A2−d2Xdt+σ3XdB3t,with the initial value X0=SL0>0. Setting(73)bx=A2−d2x,σx=σ3x,x∈0,∞,we compute that(74)∫buσ2udu=1σ32∫A2u2−d2udu=1σ32−A2x−d2lnx+Q.
Therefore,(75)e∫bu/σ2udu=eQx−d2/σ32e−1/σ32A2/x.
Clearly, we have(76)∫0∞1σ2xe∫1x2bτ/σ2τdτ=C∫0∞x−2−2d2/σ32e−2/σ32A2/xdx<∞.
Consequently, the condition of Theorem 1.16 in [14] follows clearly from above. Thus, system (72) has the ergodic property, and the invariant density is given by(77)πx=Qx−2−2d2/σ32e−2/σ32A2/xdx,x∈0,∞,where Q is a constant such that ∫0∞πxdx=1. From the ergodic theorem, it follows that(78)limt⟶∞1t∫0txsds=∫0∞xπxdxa.s.
LetXt be the solution of SDE (72) with the initial value X0=SL0>0, then we can see that(79)SLt≤Xta.s.
On the other hand, integrating from 0 tot and then dividing by t on both sides of (2) lead to(80)SDt−SD0t=A1−β1SDIL−d1SD+σID+σ1t∫0tSrdB1r,IDt−ID0t=β1SDIL−d1+σID+σ2t∫0tIDrdB2r,SHt−SH0t=A3−β3SHID−d3SH+γIH+σ5t∫0tSHrdB5r,EHt−EH0t=β3SHID−d3+ωEH+σ6t∫0tEHrdB6r,IHt−IH0t=ωEH−d3+μ+γIH+σ7t∫0tIHrdB7r.
We can obtain that(81)SDt−SD0t+IDt−ID0t=A1−d1SDt−d1IDt,then(82)SDt=A1−d1IDtd1+φ1t=A1d1−IDt+φ1t=SD0−IDt+φ1t,where(83)φ1t=−1d1SDt−SD0t+IDt−ID0t,SHt−SH0t+EHt−EH0t+IHt−IH0t=A3−d3SHt−d3EHt−d3+μIHt,then(84)SHt=SH0−EHt−d3+μd3+φ2t,where(85)φ2t=−1d3SHt−SH0t+EHt−EH0t+IHt−IH0t.
From [9], we can obtain that(86)limt⟶∞φit=0,a.s.i=1,2.
LetPt=ID+IL+EH+IH. Applying Itô’s formula, we can obtain that(87)dlnPt=1Pβ1SDIL−d1+σID+β2SLID−d2IL+β3SHID−d3+ωEH+ωEH−d3+μ+γIH−12P2σ22ID2+σ42IL2+σ62EH2+σ72IH2dt+1Pσ2IDdB2t+σ4ILdB4t+σ6EHdB6t+σ7IHdB7t≤β1SD+β2SL+β3SH+ω−18σ22∧σ42∧σ62∧σ72dt+σ2dB2t+σ4dB4t+σ6dB6t+σ7dB7t.
Integrating (87) from 0 to t and then dividing by t on both sides, one can see that(88)lnPtt−lnP0t≤β1SDt+β2t∫0tSLsds+β3SHt+ω−18σ22∧σ42∧σ62∧σ72+σ2tB2t+σ4tB4t+σ6tB6t+σ7tB7t.
As application of strong law of large numbers [13], one has(89)limt⟶∞Bitt=0,a.s.i=2,4,6,7.
Taking the superior limit on both sides of (88) and note that R^0<1 leads to(90)limsupt⟶∞lnPtt≤β1SD0+β2∫0∞xπxdx+β3SH0+ω−18σ22∧σ42∧σ62∧σ72<0a.s,which implies that(91)limsupt⟶∞ID=limsupt⟶∞IL=limsupt⟶∞EH=limsupt⟶∞IH=0a.s.
Thus, for any smallε>0, there exist t0 and a set Ωε⊂Ω such that PΩε>1−ε and β2SLID≤εSL for t≥t0 and ω∈Ωε. Now, from(92)A2−β2SLID−d2SLdt+σ3SLdB3t≤dSLt≤A2−d2SLdt+σ3SLdB3t,we can obtain that the process SLt converges to the measure π. We finish the proof.


## 5. Numerical Simulation

By using Milstein’s higher order method in [12], some numerical simulations are used to illustrate our results. We choose the initial values(93)SD0,ID0,SL0,IL0,SH0,EH0,IH0=0.9,0.5,1.2,0.6,1.2,0.3,0.3.The corresponding discretizing equations of model (2) are as follows:(94)SDk+1=A1−β1SDkILk−d1SDk+σIDkΔt+σ1SDkΔtξ1k+σ12SDk2Δtξ1k2−1,IDk+1=β1SDkILk−d1+σIDkΔt+σ2IDkΔtξ2k+σ22IDk2Δtξ2k2−1,SLk+1=A2−β2SLkIDk−d2SLkΔt+σ3SLkΔtξ3k+σ32SLk2Δtξ3k2−1,ILk+1=β2SLkIDk−d2ILkΔt+σ4ILkΔtξ4k+σ42ILk2Δtξ4k2−1,SHk+1=A3−β3SHkIDk−d3SHk+γIHkΔt+σ5SHkΔtξ5k+σ52SHk2Δtξ5k2−1,EHk+1=β3SHkIDk−d3+ωEHkΔt+σ6EHkΔtξ6k+σ62EHk2Δtξ6k2−1,IHk+1=ωEHk−d3+μ+γIHkΔt+σ7IHkΔtξ7k+σ72IHk2Δtξ7k2−1,where the time increment Δt is positive and ξik are the Gaussian random variables which follow the distribution N0,1,i=1,2,3,4,5,6,7.Example 1.
Firstly, we choose the values of system parameters to test the existence of ergodic stationary distribution of the system (94) σ12=10−4,σ22=3.6×10−3,σ32=10−2,σ42=4×10−4,σ52=10−4,σ62=4×10−4,σ72=2.5×10−3. For other values of the system parameters, see Table 3. Direct calculation leads to R0s=754.928752>1, where R0s is defined before Theorem 2. In other words, the conditions of Theorem 2 hold. In view of Theorem 2, there is an ergodic stationary distribution μ⋅ of system (2). Figures 1–3 illustrate this.Table 3
List of parameters.
ParameterValueParameterValueA10.9A21.0A30.7d10.02d20.01d30.05β10.08β20.02β30.08σ0.6μ0.03ω0.3γ0.8Figure 1
The red lines describe the solution of system (2); the green lines stand for the solution of the corresponding system (1). The right pictures are the histogram of the density function for SD,ID, and SL populations.Figure 2
The red lines describe the solution of system (2); the green lines stand for the solution of the corresponding system (1). The right pictures are the histogram of the probability density function for IL,SH populations.Figure 3
The red lines describe the solution of system (2); the green lines stand for the solution of the corresponding system (1). The right pictures are the histogram of the probability density function for EH,IH populations.Example 2.
In order to obtain the extinction of the infection, in (2), we choose the values of the system parameters as follows: σ12=10−2,σ22=9×10−2,σ32=10−2,σ42=1.6×10−1,σ52=4×10−4,σ62=9×10−2;σ72=1.6×10−1. For other values of the system parameters, see Table 4. By Theorem 3, ID,IL,EH,IH will tend to be zero exponentially with probability one. We give the simulations to support our results in Figure 4.Table 4
List of parameters.
ParameterValueParameterValueA10.5A20.6A30.5d10.1d20.08d30.05β10.02β20.02β30.02σ0.8μ0.03ω0.1γ0.8Figure 4
The red line, blue line, and magenta line representSD,SL, and SH, respectively; the green line, cyan line, yellow line, and black line represent ID,IL,EH, and IH, respectively.



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