Letter Special Issues

Similarity of epidemic spreading and information network connectivity mechanisms demonstrated by analysis of two probabilistic models

  • The modelling of epidemic spreading is essential in understanding the mechanisms of outbreaks and pandemics. Many models for different kinds of spreading have been proposed throughout the history of modelling, each suited for a specific scenario and parameters. On the other hand, models of information networks provide important tools for the analysis of the performance and reliability of such networks. We have previously presented a model for simulating the spreading of infectious disease throughout a social network and another one for simulating the connectivity of data traffic in an information network. We argue that these models are similar in that they produce equivalent results with appropriate parameters when run on the same network. We explain this by reasoning that the manners in which the models carry out their calculations, although devised from different assumptions, turn out to be equivalent. We also show empirical results of applying the models to calculate the spread of contagion and information connectivity on two complex networks suitable for the models. Based on the results, we calculate centrality metrics reflecting the outcome of the application, highlighting its important properties. We note that the centrality values obtained by running the epidemic model and the connectivity model turn out to be mutually equivalent, as predicted by their similar fashions of calculation. As the models were independently developed for their own applications, the equivalence in their calculation can not be explained by the models purposefully built similarly. Thus, not only are the two apparently completely separate areas of interest analysable with a single model but there appear to be inherent similarities in the mechanisms of epidemic spreading and determining network connectivity.

    Citation: Into Almiala, Vesa Kuikka. Similarity of epidemic spreading and information network connectivity mechanisms demonstrated by analysis of two probabilistic models[J]. AIMS Biophysics, 2023, 10(2): 173-183. doi: 10.3934/biophy.2023011

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  • The modelling of epidemic spreading is essential in understanding the mechanisms of outbreaks and pandemics. Many models for different kinds of spreading have been proposed throughout the history of modelling, each suited for a specific scenario and parameters. On the other hand, models of information networks provide important tools for the analysis of the performance and reliability of such networks. We have previously presented a model for simulating the spreading of infectious disease throughout a social network and another one for simulating the connectivity of data traffic in an information network. We argue that these models are similar in that they produce equivalent results with appropriate parameters when run on the same network. We explain this by reasoning that the manners in which the models carry out their calculations, although devised from different assumptions, turn out to be equivalent. We also show empirical results of applying the models to calculate the spread of contagion and information connectivity on two complex networks suitable for the models. Based on the results, we calculate centrality metrics reflecting the outcome of the application, highlighting its important properties. We note that the centrality values obtained by running the epidemic model and the connectivity model turn out to be mutually equivalent, as predicted by their similar fashions of calculation. As the models were independently developed for their own applications, the equivalence in their calculation can not be explained by the models purposefully built similarly. Thus, not only are the two apparently completely separate areas of interest analysable with a single model but there appear to be inherent similarities in the mechanisms of epidemic spreading and determining network connectivity.



    Zika virus disease, referred to as Zika, is a mosquito-borne infectious disease induced by Zika virus originally found in rhesus monkeys in the jungle of Uganda in 1947 [1] and afterward isolated from humans in Uganda and Tanzania in 1952 [2]. In the following decades, only a few cases from Africa and Southeast Asia were reported sporadically [3] until 2007 when Zika broke out on Yap Island in Micronesia in the western Pacific Ocean [4]. In early 2015, researchers detected Zika virus infection in Brazil [5]. The virus spread rapidly to Northern Europe, Australia, the United States, Canada [6,7,8,9], and then to Japan, China, India, and other countries [10,11,12], causing great harm to human health. At present, Zika is still prevalent at a low level in Central and South America. From January 1 to April 30, 2022, a total of 6171 suspected cases of Zika had been reported in Brazil, of which 541 cases were confirmed [13]. Zika remains an essential global public health challenge.

    The reason why Zika spreads fast and widely is mainly its multiple transmission channels. The virus is primarily propagated to mankind via the biting of infected Aedes aegypti and Aedes albopictus [14]. Meanwhile, it can be spread among humans through heterosexual or homosexual sexual contact [15,16]. In addition, it can also be spread from infected female mosquitoes to their descendants vertically [17] and from the water contaminated by the urine of the infected person to the mosquitoes in the aquatic stage [18]. The latency of the virus in the human body is generally 3–14 days [14]. The majority of infected people are asymptomatic, and only a quarter are believed to develop slight symptoms such as fever, erythema, conjunctivitis, and arthrodynia, with only a handful of documented fatalities [14]. Although the mortality of Zika virus disease is meager, it is believed that Zika infection during the gestational period is one of the causes of microcephaly and other congenital malformations in developing fetuses and newborns [19]. Zika infection is also a trigger factor for Guillain-Barre syndrome, myelitis, and neuropathy, especially in adults and older children [20]. Unfortunately, there is still no allowable vaccination or antivirus drug for the virus.

    As we all know, mathematical modeling is an effective and indispensable tool for a better understanding of population dynamics and epidemics [21]. Using this tool, many scholars have conducted rich and detailed research on the transmission of Zika disease, see [22,23,24,25,26,27] and their references. For example, Gao et al. [22] proposed an ordinary differential equation (ODE) model to examine the influences of media transmission and sexual transmission on the propagation of Zika disease and carried out a sensitive analysis of basic reproduction number. Agusto et al. [23] established an ODE model of Zika virus, including human vertical transmission, the birth of babies with microcephaly, and asymptomatic infection, and studied the dynamic behavior of the model. Considering the limitation of medical resources during the outbreak of Zika, Zhao et al. built an ODE model of Zika to investigate the effect of medical resources on the spread of Zika [24]. In addition, due to the impact of spatial differentiation and spatial mobility of human and vector populations on the dynamics of vector-borne diseases, some reaction-diffusion models for describing the spatial transmission of Zika virus have been developed accordingly, see [28,29,30] and references therein. However, the transmission of Zika virus is also influenced by temperature, wind, rain, fire, and other random environmental factors, which are ignored by the deterministic model. Using a stochastic differential equation (SDE) model to describe the epidemic dynamics can better reflect the actual phenomenon to some extent. The extinction and persistence of the epidemic driven by random noise have been studied in some works of literature, see [31,32,33,34]. Nevertheless, to our knowledge, there are few documents on the dynamic analysis of infectious diseases considering both random factors and spatial diffusion. Therefore, this paper intends to explore the permanence and extinction of Zika disease described by a random model with spatial diffusion, which includes human-mosquito transmission, human-human sexual transmission, and vertical transmission of mosquitoes, to fill this gap.

    The outbreak and prevalence of Zika have brought enormous economic burdens and health losses to the local people and government. Therefore, from the perspective of epidemiology and social economics, it is an essential and meaningful problem to formulate the optimal control strategy for Zika virus, that is, to achieve the greatest limitation of the disease with the least cost. There have been some studies on the optimal control problem of Zika. For instance, the literature [35] introduced vaccination as a control variable (although the vaccine has not been publicly available yet) and characterized the most economical and effective vaccination strategy in a reaction-diffusion model of Zika virus by utilizing the optimal control theory. The control variables (the prevention through mosquito nets, the treatment of infection patients, and the spraying of insecticides) were selected into the ODE model by authors in [36,37] to establish an optimal control problem of Zika virus. Their numerical simulation results suggested that it might be more beneficial to eliminate Zika virus infection if all three control measures are considered. This paper also plans to adopt three control variables, namely, personal protection, treatment of infected humans (here we use saturation treatment function due to limited medical resources), and reduction in the number of mosquitoes, and draw them into the SRDM to generate a stochastic control model of Zika virus.

    The rest of this paper is arranged as follows. In Section 2, we present the model description of Zika virus and prove the existence, uniqueness, and boundedness of the global positive solution of this model. In Sections 3 and 4, we discuss the conditions of disease extinction as well as the existence conditions of the stationary distribution. A stochastic optimal control problem is proposed and the expressions of the optimal controls are acquired in Section 5. Some numerical simulations are performed in Section 6 to declare and supplement the theoretical contents. At last, a summary is made in Section 7.

    According to the transmission mechanism of Zika virus, we plot the flow chart in Figure 1. In the flow chart, Sh(t,x), Eh(t,x), Ih(t,x), and Rh(t,x) are the number of susceptible, exposed, infected, and recovered human population at time t and location x, respectively. The total number of humans is Nh=Sh+Eh+Ih+Rh. The number of susceptible, exposed, and infected mosquitoes at time t and position x, are recorded as Sm(t,x), Em(t,x), and Im(t,x), respectively. Thus, Nm=Sm+Em+Im is the total amount of mosquitoes. Since only female mosquitoes suck blood and transmit diseases, the mosquitoes in this paper only refer to female mosquitoes.

    Figure 1.  Flow chart of the Zika model. Here (1)=bm(t,x)βhkhEmShNh,(2)=bm(t,x)βhImShNh,(3)=βhhkhhEhShNh, (4)=βhhIhShNh,(5)=kθμmEm+θμmIm, (6)=bh(t,x)βmkmEhSmNm,(7)=bh(t,x)βmIhSmNm.

    A susceptible human may be infected with Zika virus via the bite of an exposed or infected mosquito at a rate λmh(t,x)=bm(t,x)βh(khEm+Im)/Nh or through sexual contact with an exposed or infected partner at a rate λhh(t,x)=βhh(khhEh+Ih)/Nh, where bm(t,x)=b(t,x)/Nm, b(t,x)=αhNhαmNmαhNh+αmNm is the total number of bites per day at position x [38,39], and so bm(t,x) is the average number of bites per mosquito per day at position x. Susceptible mosquitoes move to the exposed class after biting exposed or infected humans at a rate λhm(t,x)=bh(t,x)βm(kmEh+Ih)/Nm, here bh(t,x)=b(t,x)/Nh is the average number of bites per day for an infectious person at position x. This paper considers the infectivity of humans and mosquitoes during the exposure period and the modification parameters 0<kh,khh,km<1 measure the reduction in transmissibility during the exposure period relative to the infection period.

    Zika virus can also be spread vertically from infected mothers to newborns and from infected female mosquitoes to their offspring [17,40]. This paper only deals with vertical transmission in mosquitoes, neglecting vertical transmission in humans because Zika has a very short transmission period compared to the human lifespan [26]. We assume that kθμmEm+θμmIm of the mosquito's offspring will be infected, and thus enter Em class, where μm is the average birth rate of mosquitoes, θ is the proportion of congenital infections in the progeny of infectious female mosquitoes, and 0<k<1 is also a modification parameter.

    Since the symptoms of Zika are slight and rarely fatal, we ignore the human mortality caused by the disease. And because of the short lifespan of mosquitoes, we assume that infected mosquitoes will not recover until natural death and that these mosquitoes will not die from Zika.

    Based on the above description and the flow chart in Figure 1, and taking into account the move of humans and mosquitoes, we establish the following reaction-diffusion system for Zika virus

    {Sht=d1ΔSh+Λhλmh(t,x)Shλhh(t,x)ShdhSh,Eht=d2ΔEh+λmh(t,x)Sh+λhh(t,x)Sh(ξh+dh)Eh,Iht=d3ΔIh+ξhEh(γ+dh)Ih,Rht=d4ΔRh+γIhdhRh,Smt=d5ΔSm+ΛmkθμmEmθμmImλhm(t,x)SmdmSm,Emt=d6ΔEm+kθμmEm+θμmIm+λhm(t,x)Sm(ξm+dm)Em,Imt=d7ΔIm+ξmEmdmIm, (2.1)

    for t>0,xQ, with the boundary conditions

    nSh=nEh=nIh=nRh=nSm=nEm=nIm=0,t>0,xQ,

    and initial conditions

    (Sh(0,x),Eh(0,x),Ih(0,x),Rh(0,x),Sm(0,x),Em(0,x),Im(0,x))
    =(S0h(x),E0h(x),I0h(x),R0h(x),S0m(x),E0m(x),I0m(x)),xQ.

    here Q is a bounded region possessing smooth boundary Q and n is the outward unit normal vector on Q. d1,d2,d3, and d4 represent the diffusion coefficients of susceptible, exposed, infected, and recovered human population, respectively, and d5,d6, and d7 denote the diffusion coefficients of susceptible, exposed, and infected mosquitoes population, respectively, di>0,i=1,2,,7. The meanings of the remaining parameters of model (2.1) are explained in Table 1.

    Table 1.  Parameters in model (2.1).
    Parameter Meaning Value or Range Source of data
    Λh Recruitment rate of the human population (per day) 30 [41]
    Λm Recruitment rate of the mosquitoes (per day) 2000 Assumed
    βh Probability of Zika virus spreading from an infected mosquito to a susceptible human 0.1–0.75 [42]
    βhh Transmission rate from infected humans to susceptible humans (per day) 0.001–0.1 [22]
    βm Probability of Zika virus spreading from an infected human to a susceptible mosquito 0.3–0.75 [43]
    αh The maximum number of bites that a susceptible human will tolerate being bitten (per day) 0.1–50 [39]
    αm Number of times a mosquito would bite a human (per day) 0.19–0.39 [39]
    ξh Average incubation rate of humans (per day) 0.14–0.25 [39]
    ξm Average incubation rate of mosquitoes (per day) 0.07–0.14 [39]
    dh Natural mortality rate of humans (per day) 3.65×105
    4.98×105
    [26]
    dm Natural mortality rate of mosquitoes (per day) 0.029–0.25 [26]
    γ Recovery rate of infected humans (per day) 0.07–0.3 [26]
    μm Natural birth rate of mosquitoes (per day) 0.029–0.25 [26]
    θ Proportion of congenital infections in the progeny of infectious female mosquitoes 0–0.004 [26]
    k,kh,khh,km Modification parameters 0.4, 0.1, 0.01, 0.1 [25]

     | Show Table
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    Because parameters in the infectious disease model are often subject to environmental noise and exhibit random fluctuations to a certain extent, this paper intends to build a stochastic Zika model by perturbing the natural death rates dh and dm for humans and mosquitoes with white noise. In other words, we will replace dh and dm in model (2.1) with dhσ1˙B1(t) and dmσ2˙B2(t), respectively, where B1(t) and B2(t) are independent standard Brownian motions in the complete probability space (Ω,F,P) with a filtration {Ft}t0, which is increasing, right continuous, and satisfies that F0 involves all P-null sets. σ1>0 and σ2>0 are the intensities of the noise. Then the corresponding stochastic system of model (2.1) has the following form

    {dSh=[d1ΔSh+Λhλmh(t,x)Shλhh(t,x)ShdhSh]dt+σ1ShdB1,dEh=[d2ΔEh+λmh(t,x)Sh+λhh(t,x)Sh(ξh+dh)Eh]dt+σ1EhdB1,dIh=[d3ΔIh+ξhEh(γ+dh)Ih]dt+σ1IhdB1,dRh=[d4ΔRh+γIhdhRh]dt+σ1RhdB1,dSm=[d5ΔSm+Λmθμm(kEm+Im)λhm(t,x)SmdmSm]dt+σ2SmdB2,dEm=[d6ΔEm+θμm(kEm+Im)+λhm(t,x)Sm(ξm+dm)Em]dt+σ2EmdB2,dIm=[d7ΔIm+ξmEmdmIm]dt+σ2ImdB2, (2.2)

    for t>0,xQ, with the boundary conditions

    nSh=nEh=nIh=nRh=nSm=nEm=nIm=0,t>0,xQ,

    and initial conditions

    (Sh(0,x),Eh(0,x),Ih(0,x),Rh(0,x),Sm(0,x),Em(0,x),Im(0,x))
    =(S0h(x),E0h(x),I0h(x),R0h(x),S0m(x),E0m(x),I0m(x)),xQ.

    Let H=H1(Q)={φ|φL2(Q),φxiL2(Q) is generalized partial derivative, i = 1, 2, 3}. H is a Sobolev space and HL2(Q)H, where H=H1(Q) is the dual space of H. and are the norms of H and L2(Q), respectively. φ2=φ2+φ2, and there exists a positive constant c such that φcφ. , indicates the dual product of H and H. The norm of Euclidean space is denoted by ||. H=H7. Denote H+={φ|φL2(Q;(0,)),φxiL2(Q),i=1,2,3}, H+=(H+)7. In addition, Rl+={(x1,x2,,xl)Rl:xi>0,i=1,2,,l}, R+=[0,). L2F([0,T]×Q;Rl) is a set of square integrable and Ft-adapted stochastic processes. The indicative function of set A is denoted by χA. ab=min{a,b}, ab=max{a,b}. φx represents the partial derivative of φ to x. B(t,x) is sometimes abbreviated to B for convenience without causing confusion.

    Theorem 2.1. For any initial value X(0,x)=(S0h(x),E0h(x),I0h(x),R0h(x),S0m(x),E0m(x),I0m(x))H+, stochastic Zika system (2.2) has a unique global positive solution X(t,x)=(Sh(t,x),Eh(t,x),Ih(t,x),Rh(t,x),Sm(t,x),Em(t,x),Im(t,x))H+ on t0. Moreover, there is a positive constant C0 such that

    Q[Sh(t,x)+Eh(t,x)+Ih(t,x)+Rh(t,x)+Sm(t,x)+Em(t,x)+Im(t,x)]dxC0a.s.

    Theorem 2.1 is an important fundamental theorem, which gives the existence, uniqueness, and boundness of the positive solution of system (2.2), and its proof is shown in Appendix A. The following theorem discusses the pth moment boundedness of system (2.2) and its proof can be found in Appendix B.

    Theorem 2.2. For any p>0, we have

    Esup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p)C,

    where C is a constant related to p,T, and the original condition and the parameters of system (2.2).

    In this section, we will discuss the conditions for almost surely exponential extinction of Zika disease.

    In general, consider an l-dimensional stochastic reaction-diffusion system by

    dυ(t,x)=(2xυ(t,x)+f(t,x,υ(t,x)))dt+g(t,x,υ(t,x))dB(t),t>t0,xQ, (3.1)

    with boundary condition υ(t,x)n=0(t>t0,xQ) and initial condition υ(t0,x)=υ0(x)(xQ).

    We give the definition of the almost surely exponential stability of system (3.1) [44].

    Definition 3.1. The trivial solution of system (3.1) is said to be almost surely exponentially stable if

    lim supt1t|log|υ(t,x;t0,υ0)||Q<0a.s.

    for all υ0Rl+, where |log|υ(,x)||Q:=Qlog|υ(,x)|dx.

    Next, the almost surely exponential extinction of Zika disease will be given in the following theorem.

    Theorem 3.2. For any starting value X(0,x)H+ of system (2.2), if

    d2d3d6d70, (3.2)

    and

    σ21σ22>8[(ξh+dhξh)2(ξm+dmξm)2](αmβh+βhh+θμm+αhβm), (3.3)

    then

    lim supt1t|log(Eh(t,x)+ξh+dhξhIh(t,x)+Em(t,x)+ξm+dmξmIm(t,x))|Q<0a.s.

    Proof. Let V(t,x)=log(Eh(t,x)+ξh+dhξhIh(t,x)+Em(t,x)+ξm+dmξmIm(t,x)), then

    |V(t,x)|Q=QV(t,x)dx=Qlog(Eh(t,x)+ξh+dhξhIh(t,x)+Em(t,x)+ξm+dmξmIm(t,x))dx.

    Let A(t,x)=(Eh(t,x)+ξh+dhξhIh(t,x)+Em(t,x)+ξm+dmξmIm(t,x)). By Itˆo's formula, we have

    d|V(t,x)|Q=QdV(t,x)dx=Qdlog(Eh(t,x)+ξh+dhξhIh(t,x)+Em(t,x)+ξm+dmξmIm(t,x))dx=Q{[1A(t,x)[d2ΔEh(t,x)+λmh(t,x)Sh(t,x)+λhh(t,x)Sh(t,x)(ξh+dh)Eh(t,x)]+ξh+dhξhA(t,x)[d3ΔIh(t,x)+ξhEh(t,x)(γ+dh)Ih(t,x)]+1A(t,x)[d6ΔEm(t,x)+θμm(kEm(t,x)+Im(t,x))+λhm(t,x)Sm(t,x)(ξm+dm)Em(t,x)]+ξm+dmξmA(t,x)[d7ΔIm(t,x)+ξmEm(t,x)dmIm(t,x)]12σ21E2h(t,x)+σ21I2h(t,x)(ξh+dhξh)2+σ22E2m(t,x)+σ22I2m(t,x)(ξm+dmξm)2(Eh(t,x)+ξh+dhξhIh(t,x)+Em(t,x)+ξm+dmξmIm(t,x))2]dt+(σ1Eh(t,x)A(t,x)+(ξh+dh)σ1Ih(t,x)ξhA(t,x))dB1(t)+(σ2Em(t,x)A(t,x)+(ξm+dm)σ2Im(t,x)ξmA(t,x))dB2(t)}dxQ{d2ΔEh(t,x)+d3Δ(ξh+dhξhIh(t,x))+d6ΔEm(t,x)+d7Δ(ξm+dmξmIm(t,x))Eh(t,x)+ξh+dhξhIh(t,x)+Em(t,x)+ξm+dmξmIm(t,x)+αmβh(khEm(t,x)+Im(t,x))Em(t,x)+ξm+dmξmIm(t,x)+βhh(khhEh(t,x)+Ih(t,x))Eh(t,x)+ξh+dhξhIh(t,x)+θμm(kEm(t,x)+Im(t,x))Em(t,x)+ξm+dmξmIm(t,x)+αhβm(kmEh(t,x)+Ih(t,x))Eh(t,x)+ξh+dhξhIh(t,x)(σ21σ21(ξh+dhξh)2σ22σ22(ξm+dmξm)2)(E2h+I2h+E2m+I2m)8((ξh+dhξh)2(ξm+dmξm)2)(E2h+I2h+E2m+I2m)+(σ1Eh(t,x)A(t,x)+(ξh+dh)σ1Ih(t,x)ξhA(t,x))dB1(t)+(σ2Em(t,x)A(t,x)+(ξm+dm)σ2Im(t,x)ξmA(t,x))dB2(t)}dxQ{(d2d3d6d7)|ΔA(t,x)|A(t,x)+αmβh+βhh+θμm+αhβmσ21σ228[(ξh+dhξh)2(ξm+dmξm)2]+(σ1Eh(t,x)A(t,x)+(ξh+dh)σ1Ih(t,x)ξhA(t,x))dB1(t)+(σ2Em(t,x)A(t,x)+(ξm+dm)σ2Im(t,x)ξmA(t,x))dB2(t)}dx.

    Integrating the above inequality from 0 to t, we get

    |V(t,x)|Q=|V(0,x)|Q+t0Q(d2d3d6d7)|ΔA(s,x)|A(s,x)dxds==+t0Q(αmβh+βhh+θμm+αhβmσ21σ228[(ξh+dhξh)2(ξm+dmξm)2])dxds==+t0Q(σ1Eh(s,x)A(s,x)+(ξh+dh)σ1Ih(s,x)ξhA(s,x))dxdB1(s)==+t0Q(σ2Em(s,x)A(s,x)+(ξm+dm)σ2Im(s,x)ξmA(s,x))dxdB2(s)=|V(0,x)|Q+t0Q(d2d3d6d7)|ΔA(s,x)|A(s,x)dxds==+(αmβh+βhh+θμm+αhβmσ21σ228[(ξh+dhξh)2(ξm+dmξm)2])|Q|t+M1(t)+M2(t), (3.4)

    where

    M1(t)=t0Q(σ1Eh(s,x)A(s,x)+(ξh+dh)σ1Ih(s,x)ξhA(s,x))dxdB1(s),
    M2(t)=t0Q(σ2Em(s,x)A(s,x)+(ξm+dm)σ2Im(s,x)ξmA(s,x))dxdB2(s).

    In addition, the quadratic variations of M1 and M2 are respectively

    M1,M1t=t0(Q(σ1Eh(s,x)A(s,x)+(ξh+dh)σ1Ih(s,x)ξhA(s,x))dx)2ds,
    M2,M2t=t0(Q(σ2Em(s,x)A(s,x)+(ξm+dm)σ2Im(s,x)ξmA(s,x))dx)2ds.

    Therefore

    lim suptM1,M1tt=lim supt1tt0(Q(σ1Eh(s,x)A(s,x)+(ξh+dh)σ1Ih(s,x)ξhA(s,x))dx)2ds(2σ1|Q|)2<a.s.,
    lim suptM2,M2tt=lim supt1tt0(Q(σ2Em(s,x)A(s,x)+(ξm+dm)σ2Im(s,x)ξmA(s,x))dx)2ds(2σ2|Q|)2<a.s.

    Thus, martingale's strong law of large numbers yields

    lim suptM1(t)t=0a.s.andlim suptM2(t)t=0a.s.

    Together with (3.2)–(3.4), , we obtain

    lim supt1t|log(Eh(t,x)+ξh+dhξhIh(t,x)+Em(t,x)+ξm+dmξmIm(t,x))|Q(αmβh+βhh+θμm+αhβmσ21σ228[(ξh+dhξh)2(ξm+dmξm)2])|Q|<0,

    which shows that

    limtEh(t,x)=limtIh(t,x)=limtEm(t,x)=limtIm(t,x)=0a.s.

    This completes the proof.

    Conditions (3.2) and (3.3) of Theorem 3.2 suggest that Zika virus will become exponentially extinct when the diffusion coefficients of infected people and mosquitoes are very small, that is, they hardly move, and the intensities of environmental noise are relatively large. There is no doubt that such conditions are very harsh. In what follows we will talk about the stationary distribution of system (2.2), which means the persistence of Zika disease.

    First of all, we give the definition of the stationary distribution of system (2.2) [45]. Let P(H) represent the space of all probability measures on (H,B(H)), here B(H) denotes the Borel σ-algebra on H. Cb(H) is the set of all bounded and continuous real-valued functions on H.

    Definition 4.1. A stationary distribution of the positive solution X(t,x), t0, of system (2.2) is defined as a probability measure πP(H) which satisfies

    π(g)=π(Ptg),t0,

    here π(g):=Hg(ϕ)π(dϕ), Ptg(ϕ):=Eg(|X(t,x,ϕ)|Q), and gCb(H).

    For π1,π2P(H), a measure on P(H) is defined by

    d(π1,π2)=supgN|Hg(ϕ1)π1(dϕ1)Hg(ϕ2)π2(dϕ2)|,

    where N:={g:HR,|g(ϕ1)g(ϕ2)|ϕ1ϕ2 for any ϕ1,ϕ2H and |g()|1}. P(H) is complete under the measure d(,) by [46], and then we can get an important lemma as follows, which provides an assertion for the existence of stationary distribution [45].

    Lemma 4.1. Assuming that for arbitrary bounded subset O of H+, p>1,

    (i)limtsupϕ1,ϕ2OEX(t,x,ϕ1)X(t,x,ϕ2)p=0;

    (ii)supt0supϕOEX(t,x,ϕ)p<.

    Then, X(t,x,ϕ),t0, has a stationary distribution for initial data ϕH+.

    Applying Lemma 4.1, we can obtain the conditions for the existence of the steady-state distribution of system (2.2).

    Theorem 4.2. Assume there are constants p>1, ϑ>0, and 0<ci<1(i=1,2,,7) such that

    2βphh+pβhhkhh+7(p1)+αph^A1+^A2+12p(p1)σ21<pdh+pˇB1, (4.1)

    and

    (θμm)p(1+kp)+pkθμm+ξpm+5(p1)+αpm^A3+12p(p1)σ22<pdm+pˇB2, (4.2)

    where ^A1=2βpmβph(1+kph), ^A2=ξphγp, ^A3=2βphβpm(1+kpm), ˇB1=c1d1c2d2c3d3c4d4, ˇB2=c5d5c6d6c7d7, then process X(t,x),t0, of system (2.2) has a unique stationary distribution πP(H).

    Proof. To illustrate the existence of steady-state distribution of system (2.2), we need to prove that the conditions (i) and (ii) in Lemma 4.1 hold. Since Theorem 2.2 implies that (ii) is true, we only need to verify (i). To this end, for p>1, ϑ>0, let

    Φ(t,x,ϕ1,ϕ2)=eϑt(y1p+y2p+y3p+y4p+y5p+y6p+y7p),

    where

    y1=y1(t,x,ϕ1,ϕ2)=Sh(t,x,ϕ1)Sh(t,x,ϕ2):=Sh1Sh2,y2=y2(t,x,ϕ1,ϕ2)=Eh(t,x,ϕ1)Eh(t,x,ϕ2):=Eh1Eh2,y3=y3(t,x,ϕ1,ϕ2)=Ih(t,x,ϕ1)Ih(t,x,ϕ2):=Ih1Ih2,y4=y4(t,x,ϕ1,ϕ2)=Rh(t,x,ϕ1)Rh(t,x,ϕ2):=Rh1Rh2,y5=y5(t,x,ϕ1,ϕ2)=Sm(t,x,ϕ1)Sm(t,x,ϕ2):=Sm1Sm2,y6=y6(t,x,ϕ1,ϕ2)=Em(t,x,ϕ1)Em(t,x,ϕ2):=Em1Em2,y7=y7(t,x,ϕ1,ϕ2)=Im(t,x,ϕ1)Im(t,x,ϕ2):=Im1Im2.

    Applying Itˆo's formula, we deduce

    dΦ(t,x,ϕ1,ϕ2)=ϑΦ(t,x,ϕ1,ϕ2)dt+eϑtd(y1p+y2p+y3p+y4p+y5p+y6p+y7p). (4.3)

    For simplicity, we assume that Nh(t,x,ϕ1)=Nh(t,x,ϕ2)=Nh(t,x) and Nm(t,x,ϕ1)=Nm(t,x,ϕ2)=Nm(t,x). Thus denote λmh1(t,x)=bm(t,x)βh(khEm1+Im1)/Nh=αhαmαhNh+αmNmβh(khEm1+Im1), λmh2(t,x)=bm(t,x)βh(khEm2+Im2)/Nh=αhαmαhNh+αmNmβh(khEm2+Im2), λhh1(t,x)=βhh(khhEh1+Ih1)/Nh, and λhh2(t,x)=βhh(khhEh2+Ih2)/Nh. By Itˆo's formula and embedding theorem,

    dy1p=py1p2y1,d1Δy1(λmh1(t,x)Sh1λmh2(t,x)Sh2)(λhh1(t,x)Sh1λhh2(t,x)Sh2)dhy1dt+12p(p1)y1p4y1,σ1y12dt+py1p2y1,σ1y1dB1=[pd1y1p2y12py1p2y1,αhαmβhαhNh+αmNm(khEm1Sh1khEm2Sh2+Im1Sh1Im2Sh2)py1p2y1,βhhNh(khhEh1Sh1khhEh2Sh2+Ih1Sh1Ih2Sh2)pdhy1p+12p(p1)σ21y1p]dt+pσ1y1pdB1[pc1d1y1p+pαmβhkhy1p1y6+pαmβhy1p1y7+pβhhkhhy1p1y2+pβhhy1p1y3pdhy1p+12p(p1)σ21y1p]dt+pσ1y1pdB1[(pc1d1+4(p1)pdh+12p(p1)σ21)y1p+(αmβhkh)py6p+(αmβh)py7p+(βhhkhh)py2p+βphhy3p]dt+pσ1y1pdB1, (4.4)

    here c1<1 is a constant and the last inequality sign takes advantage of the Young inequality. Similarly,

    dy2p=py2p2y2,d2Δy2+λmh1(t,x)Sh1λmh2(t,x)Sh2+λhh1(t,x)Sh1λhh2(t,x)Sh2(ξh+dh)y2dt+12p(p1)σ21y2pdt+pσ1y2pdB1[(pc2d2+7(p1)+pβhhkhhp(ξh+dh)+12p(p1)σ21)y2p+((αhβhkh)p+(αhβh)p+(βhhkhh)p+βphh)y1p+βphhy3p+(αmβhkh)py6p+(αmβh)py7p]dt+pσ1y2pdB1, (4.5)
    dy3p=py3p2y3,d3Δy3+ξhy2(γ+dh)y3dt+12p(p1)σ21y3pdt+pσ1y3pdB1[(pc3d3+p1p(γ+dh)+12p(p1)σ21)y3p+ξphy2p]dt+pσ1y3pdB1, (4.6)
    dy4p=py4p2y4,d4Δy4+γy3dhy4dt+12p(p1)σ21y4pdt+pσ1y4pdB1[(pc4d4+p1pdh+12p(p1)σ21)y4p+γpy3p]dt+pσ1y4pdB1, (4.7)
    dy5p=py5p2y5,d5Δy5kθμmy6θμmy7(λhm1(t,x)Sm1λhm2(t,x)Sm2)dmy5dt+12p(p1)σ22y5pdt+pσ2y5pdB2[(pc5d5+4(p1)pdm+12p(p1)σ22)y5p+(kθμm)py6p+(θμm)py7p+(αhβmkm)py2p+(αhβm)py3p]dt+pσ2y5pdB2, (4.8)
    dy6p=py6p2y6,d6Δy6+kθμmy6+θμmy7+λhm1(t,x)Sm1λhm2(t,x)Sm2(ξm+dm)y6dt+12p(p1)σ22y6pdt+pσ2y6pdB2[(pc6d6+5(p1)+pkθμmp(ξm+dm)+12p(p1)σ22)y6p+(kθμm)py7p+((αmβmkm)p+(αmβm)p)y5p+(αhβmkm)py2p+(αhβm)py3p]dt+pσ2y6pdB2, (4.9)
    dy7p=py7p2y7,d7Δy7+ξmy6dmy7dt+12p(p1)σ22y7|pdt+pσ2y7pdB2[(pc7d7+p1pdm+12p(p1)σ22)y7p+ξpmy6p]dt+pσ2y7pdB2. (4.10)

    Substituting (4.4)–(4.10) into (4.3), integrating the two sides of (4.3), and seeking mathematical expectation, then

    EΦ(t,x,ϕ1,ϕ2)EΦ(0,x,ϕ1,ϕ2)+Et0ϑΦ(s,x,ϕ1,ϕ2)ds+Et0eϑsC7(y1p+y2p+y3p+y4p+y5p+y6p+y7p)ds=EΦ(0,x,ϕ1,ϕ2)+Et0(ϑ+C7)Φ(s,x,ϕ1,ϕ2)ds,

    where

    C7=max{pc1d1+4(p1)pdh+12p(p1)σ21+(αhβh)p(kph+1)+βphh(kphh+1),pc2d2+7(p1)+pβhhkhhp(ξh+dh)+12p(p1)σ21+(βhhkhh)p+ξph+2(αhβmkm)p,pc3d3+p1p(γ+dh)+12p(p1)σ21+2βphh+γp+2(αhβm)p,pc4d4+p1pdh+12p(p1)σ21,pc5d5+4(p1)pdm+12p(p1)σ22+(αmβmkm)p+(αmβm)p,pc6d6+5(p1)+pkθμmp(ξm+dm)+ξpm
    +12p(p1)σ22+2(αmβhkh)p+(kθμm)p,pc7d7+p1pdm+12p(p1)σ22+(θμm)p(1+kp)+2(αmβh)p},

    and ϑ+C7>0. Next, we take the supϕ1,ϕ2O and use the Gronwall inequality to get

    supϕ1,ϕ2OEΦ(t,x,ϕ1,ϕ2)supϕ1,ϕ2OEΦ(0,x,ϕ1,ϕ2)e(ϑ+C7)t,

    i.e.,

    supϕ1,ϕ2OE(y1p+y2p+y3p+y4p+y5p+y6p+y7p)supϕ1,ϕ2OEΦ(0,x,ϕ1,ϕ2)eC7t. (4.11)

    According to (4.1) and (4.2), C7<0. Therefore,

    limtsupϕ1,ϕ2OE(y1p+y2p+y3p+y4p+y5p+y6p+y7p)=0.

    Thus, the condition (i) of Lemma 4.1 is proved. Let us now explain the uniqueness of steady-state distribution of system (2.2).

    Suppose πP(H) is another steady-state distribution for X(t,x),t0, of system (2.2). Clb(H) is a bounded and Lipschitz continuous function family on H. Then by the definition of stationary distribution, the Hölder inequality, and (4.11), for gClb(H), we can derive that

    |π(g)π(g)|H×H|Ptg(ϕ1)Ptg(ϕ2)|π(dϕ1)π(dϕ2)C8e12C7t,t0, (4.12)

    here C8>0 is a constant. Whereupon, the uniqueness of stationary distribution can be obtained by setting t in (4.12) when C7<0. The proof is completed.

    From (4.1) and (4.2) of Theorem 4.2, we find that Zika disease will be persistent when the intensities of environmental noise are low, while the diffusion coefficients of humans and mosquitoes are relatively large, which is the opposite of exponential extinction.

    The objective of this section is to illustrate that anti-Zika control strategies can be implemented while minimizing the cost of implementing these measures. So we formulate a stochastic optimal control problem by introducing three control variables into system (2.2). The control u1(t,x) denotes the level of personal protective efforts among the population, so the correlative infectivity is decreased by the factor (1u1(t,x)). The control u2(t,x) represents the level of treatment for infected people. We choose saturated treatment rate function cu2Ih1+αIh with treatment rate c>0 and saturation coefficient α0 due to the limited medical resources (medical staff, medicines, hospital beds, etc.), where cα is the largest medical resource provided per unit of time. The control u3(t,x) indicates the level of insecticides used to kill mosquitoes in mosquito breeding grounds, which increases the mosquito mortality rate from dm to dm+c0u3 with killing efficacy c0. In this thesis, 0ui1(i=1,2,3) means that there is no effort (i.e., no control) when the control is zero, and the maximum control is put when the control is one. Let u=(u1,u2,u3). Thus the stochastic control system for Zika disease will be written as

    {dSh=[d1ΔSh+Λh(1u1)λmh(t,x)Sh(1u1)λhh(t,x)ShdhSh]dt+σ1ShdB1:=z1(t,x,X,u)dt+σ1ShdB1,dEh=[d2ΔEh+(1u1)λmh(t,x)Sh+(1u1)λhh(t,x)Sh(ξh+dh)Eh]dt+σ1EhdB1:=z2(t,x,X,u)dt+σ1EhdB1,dIh=[d3ΔIh+ξhEh(γ+dh)Ihcu2Ih1+αIh]dt+σ1IhdB1:=z3(t,x,X,u)dt+σ1IhdB1,dRh=[d4ΔRh+γIhdhRh+cu2Ih1+αIh]dt+σ1RhdB1:=z4(t,x,X,u)dt+σ1RhdB1,dSm=[d5ΔSm+Λmθμm(kEm+Im)λhm(t,x)Sm(dm+c0u3)Sm]dt+σ2SmdB2:=z5(t,x,X,u)dt+σ2SmdB2,dEm=[d6ΔEm+θμm(kEm+Im)+λhm(t,x)Sm(ξm+dm+c0u3)Em]dt+σ2EmdB2:=z6(t,x,X,u)dt+σ2EmdB2,dIm=[d7ΔIm+ξmEm(dm+c0u3)Im]dt+σ2ImdB2:=z7(t,x,X,u)dt+σ2ImdB2, (5.1)

    for t>0,xQ, with the boundary conditions

    nSh=nEh=nIh=nRh=nSm=nEm=nIm=0,t>0,xQ,

    and initial conditions

    (Sh(0,x),Eh(0,x),Ih(0,x),Rh(0,x),Sm(0,x),Em(0,x),Im(0,x))
    =(S0h(x),E0h(x),I0h(x),R0h(x),S0m(x),E0m(x),I0m(x)),xQ.

    Our optimal control study aims at minimizing the number of exposed and infected people, the total number of mosquitoes, and the cost of executing the control in time interval [0,T] and region Q. In order to realize this goal, an objective functional is defined as

    J(u)=E{T0Qf(X(t,x),u(t,x))dxdt+Qφ(X(T,x))dx}, (5.2)

    with

    f(X,u)=a1Eh+a2Ih+a3Nm+b1u1Sh+b2u2Ih+b3u3Nm+123j=1cju2j,

    where a1,a2, and a3 are positive coefficients of weight of the exposed, infected human and the total mosquito populations, respectively, b1,b2, and b3 are positive coefficients of weigh for the linear costs of personal protection, the treatment for infected people, and mosquito control, respectively, c1,c2, and c3 are positive coefficients of weight for the quadratic costs, respectively. A function of X(t,x) at the terminal time T is denoted by φ(X(T,x)). The next task is to find the optimal control ˉu=(ˉu1,ˉu2,ˉu3) such that

    J(ˉu)=minuUJ(u),

    here U is the admissible control set as follows

    U={(u(t,x)|ui(t,x)[0,1] is {Ft}t0adapted,t[0,T],xQ,i=1,2,3}. (5.3)

    Similar to Theorem 2.1, the existence, uniqueness, and boundness of the positive solution of system (5.1) can also be verified. Further, we can obtain the boundedness and convexity of fi(t,x,X,u)(i=1,2,,7) and the compactness of U, and then the existence of optimal control ˉu can be shown according to Theorem 3.1 in [47].

    Next, we will makes use of the Pontryagin maximum principle [48] to obtain the optimal control. Denote λ(t,x)=(λ1(t,x),λ2(t,x),,λ7(t,x)), μ(t,x)=(μ1(t,x),μ2(t,x),,μ7(t,x)). ˉX=(ˉX1,ˉX2,ˉX3,ˉX4,ˉX5,ˉX6,ˉX7)=(ˉSh,ˉEh,ˉIh,ˉRh,ˉSm,ˉEm,ˉIm) is the optimal state variable of system (5.1) corresponding to the optimal control ˉu. Then by the stochastic maximum principle, there exists a pair of processes (λ(t,x),μ(t,x))L2F([0,T]×Q;R7)×L2F([0,T]×Q;R7) that satisfy the following SDE

    {dλ1(t,x)=g1(ˉX(t,x),ˉu(t,x),λ(t,x),μ(t,x))dt+μ1(t,x)dB1(t),dλ2(t,x)=g2(ˉX(t,x),ˉu(t,x),λ(t,x),μ(t,x))dt+μ2(t,x)dB1(t),dλ3(t,x)=g3(ˉX(t,x),ˉu(t,x),λ(t,x),μ(t,x))dt+μ3(t,x)dB1(t),dλ4(t,x)=g4(ˉX(t,x),ˉu(t,x),λ(t,x),μ(t,x))dt+μ4(t,x)dB1(t),dλ5(t,x)=g5(ˉX(t,x),ˉu(t,x),λ(t,x),μ(t,x))dt+μ5(t,x)dB2(t),dλ6(t,x)=g6(ˉX(t,x),ˉu(t,x),λ(t,x),μ(t,x))dt+μ6(t,x)dB2(t),dλ7(t,x)=g7(ˉX(t,x),ˉu(t,x),λ(t,x),μ(t,x))dt+μ7(t,x)dB2(t),λi(T,x)=φXi(ˉX(T,x)), (5.4)

    where

    g1(ˉX,ˉu,λ,μ)=d1Δλ1[(1ˉu1)ˉλmhαmˉNm+αh(ˉEh+ˉIh+ˉRh)αmˉNm+αhˉNh+(1ˉu1)ˉλhhˉEh+ˉIh+ˉRhˉNh+dh]λ1+[(1ˉu1)ˉλmhαmˉNm+αh(ˉEh+ˉIh+ˉRh)αmˉNm+αhˉNh+(1ˉu1)ˉλhhˉEh+ˉIh+ˉRhˉNh]λ2+[ˉλhmαhαmˉNm+αhˉNhˉSm]λ5[ˉλhmαhαmˉNm+αhˉNhˉSm]λ6+σ1μ1+b1u1,g2(ˉX,ˉu,λ,μ)=d2Δλ2+[(1ˉu1)ˉλmhαhˉShαmˉNm+αhˉNh(1ˉu1)βhhkhhˉSh(ˉSh+ˉIh+ˉRh)ˉIhˉShˉN2h]λ1[(1ˉu1)ˉλmhαhˉShαmˉNm+αhˉNh(1ˉu1)βhhkhhˉSh(ˉSh+ˉIh+ˉRh)ˉIhˉShˉN2h+ξh+dh]λ2+ξhλ3[αmαhβmkmˉSmαhˉλhmˉSmαmˉNm+αhˉNh]λ5+[αmαhβmkmˉSmαhˉλhmˉSmαmˉNm+αhˉNh]λ6+σ1μ2+a1,
    g3(ˉX,ˉu,λ,μ)=d3Δλ3+[(1ˉu1)ˉλmhαhˉShαmˉNm+αhˉNh(1ˉu1)βhhˉSh(ˉSh+ˉEh+ˉRh)khhˉEhˉShˉN2h]λ1[(1ˉu1)ˉλmhαhˉShαmˉNm+αhˉNh(1ˉu1)βhhˉSh(ˉSh+ˉEh+ˉRh)khhˉEhˉShˉN2h]λ2[γ+dh+cu2(1+αIh)2]λ3+[γ+cu2(1+αIh)2]λ4[αmαhβmˉSmαhˉλhmˉSmαmˉNm+αhˉNh]λ5+[αmαhβmˉSmαhˉλhmˉSmαmˉNm+αhˉNh]λ6+σ1μ3+a2+b2u2,g4(ˉX,ˉu,λ,μ)=d4Δλ4+[(1ˉu1)ˉλmhαhˉShαmˉNm+αhˉNh+(1ˉu1)ˉλhhˉShˉNh]λ1[(1ˉu1)ˉλmhαhˉShαmˉNm+αhˉNh+(1ˉu1)ˉλhhˉShˉNh]λ2dhλ4+[αhˉλhmˉSmαmˉNm+αhˉNh]λ5[αhˉλhmˉSmαmˉNm+αhˉNh]λ6+σ1μ4,g5(ˉX,ˉu,λ,μ)=d5Δλ5+[(1ˉu1)ˉλmhαmˉShαmˉNm+αhˉNh]λ1[(1ˉu1)ˉλmhαmˉShαmˉNm+αhˉNh]λ2[ˉλhmαm(ˉEm+ˉIm)+αhˉNhαmˉNm+αhˉNh+dm+c0u3]λ5+[ˉλhmαm(ˉEm+ˉIm)+αhˉNhαmˉNm+αhˉNh]λ6+σ2μ5+a3+b3u3,g6(ˉX,ˉu,λ,μ)=d6Δλ6[(1ˉu1)αmαhβhkhˉShˉλmhˉShαmˉNm+αhˉNh]λ1+[(1ˉu1)αmαhβhkhˉShˉλmhˉShαmˉNm+αhˉNh]λ2+[αmˉλhmˉSmαmˉNm+αhˉNhkθμm]λ5+[kθμmαmˉλhmˉSmαmˉNm+αhˉNh(ξm+dm+c0u3)]λ6+ξmλ7+σ2μ6+a3+b3u3,g7(ˉX,ˉu,λ,μ)=d7Δλ7[(1ˉu1)αmαhβhˉShˉλmhˉShαmˉNm+αhˉNh]λ1+[(1ˉu1)αmαhβhˉShˉλmhˉShαmˉNm+αhˉNh]λ2+[αmˉλhmˉSmαmˉNm+αhˉNhθμm]λ5[αmˉλhmˉSmαmˉNm+αhˉNhθμm]λ6(dm+c0u3)λ7+σ2μ7+a3+b3u3,

    and ˉNh=ˉSh+ˉEh+ˉIh+ˉRh, ˉNm=ˉSm+ˉEm+ˉIm, ˉλmh=αmαhβh(khˉEm+ˉIm)αmˉNm+αhˉNh, ˉλhh=βhh(khhˉEh+ˉIh)ˉNh, ˉλhm=αmαhβm(kmˉEh+ˉIh)αmˉNm+αhˉNh.

    Define the following Hamilton function:

    H(t,X,u,λ,μ)=7i=1zi(t,x,X,u),λi+σ1Sh,μ1+σ1Eh,μ2+σ1Ih,μ3+σ1Rh,μ4+σ2Sm,μ5+σ2Em,μ6+σ2Im,μ7+Qf(X(t,x),u(t,x))dx=Q(7i=1zi(t,x,X,u)λi+σ1Shμ1+σ1Ehμ2+σ1Ihμ3+σ1Rhμ4+σ2Smμ5+σ2Emμ6+σ2Imμ7+f(X(t,x),u(t,x)))dx, (5.5)

    for (t,X,u,λ,μ)[0,T]×H+×U×R7×R7. So, according to the maximum condition of stochastic maximum principle, from H(t,X,u,λ,μ)uj=0 and 0uj1,j=1,2,3, we get the following conclusion:

    Theorem 5.1. Under objective functional (5.2), the expressions for the optimal controls of system (5.1) are

    ˉu1=min{max{1c1[(ˉλmh+ˉλhh)(λ2λ1)ˉShb1ˉSh],0},1},ˉu2=min{max{1c2[cIh1+αIh(λ3λ4)b2ˉIh],0},1},ˉu3=min{max{1c3[c0(ˉSmλ5+ˉEmλ6+ˉImλ7)b3ˉNm],0},1}.

    In this part, some numerical simulations will be conducted to illustrate our theoretical results more intuitively. We can write the discrete form (13.1) of the state Eq (2.2) shown in Appendix C using Milstein's method [49]. The initial conditions of model (2.2) and model (5.1) are selected as S0h(x)=750000+200cosπx30,E0h(x)=150+10cosπx30,I0h(x)=10+5cosπx30,R0h(x)=0,S0m(x)=80000+100cosπx30,E0m(x)=100+10cosπx30,I0m(x)=10+2cosπx30,x(0,100).

    We choose parameters βh=0.1,βhh=0.001,βm=0.3,αh=0.1,αm=0.19,ξh=0.14,ξm=0.14,dh=0.000039,dm=0.03,γ=0.14,μm=0.029,θ=0.001,d1=0.012,d2=d3=0.0002,d4=d5=0.008,d6=d7=0.0001,σ1=σ2=0.59. Please refer to Table 1 for other parameter values. Through calculation, conditions (3.2) and (3.3) for the almost surely exponential extinction of Zika disease in Theorem 3.2 are established at this time. Under these parameter values, we draw the evolutions of Eh,Ih,Em, and Im as shown in Figure 2. Here, the left is the spatio-temporal graphs, the right is the relevant projection graphs, where the curves of different colors indicate the variations of the population in different regions over time. Obviously, Figure 2 verifies the conclusion of Theorem 3.2.

    Figure 2.  The spatio-temporal graphs and corresponding projection graphs of Eh,Ih,Em, and Im in model (2.2).

    It can be seen from the above, to make Zika disease almost surely exponentially extinct, on the one hand, the intensities of noise should be higher; on the other hand, the diffusion coefficients of the infected people and mosquitoes should be very small, which can guide us on how to eliminate Zika disease faster.

    In this subsection, the existence of the stationary distribution of system (2.2) will be numerically simulated. We take βh=0.23,βhh=0.10,βm=0.33,αh=0.1,αm=0.26,ξh=0.2,ξm=0.1,σ1=0.1,σ2=0.1,d1=0.6,d2=0.6,d3=0.55,d4=0.58,d5=0.3,d6=0.3,d7=0.25,p=1.01,ci=0.96(i=1,2,,7). Other parameter values are the same as those in Figure 2. At this time, conditions (4.1) and (4.2) in Theorem 4.2 describing the existence of steady-state distribution hold. The trajectories of the solution of system (2.2) are presented in Figures 3 and 4, whose left side is the spatio-temporal graphs and the right side is the corresponding projection graphs, which indicate that the system will achieve a steady state over time t. Figures 5 and 6 show the evolutions of the solution of the system when spatial variable x=10 and their corresponding histograms, from which we can see that the system has a stationary distribution.

    Figure 3.  The spatio-temporal graphs and corresponding projection graphs of Sh,Eh,Ih, and Rh in model (2.2).
    Figure 4.  The spatio-temporal graphs and corresponding projection graphs of Sm,Em, and Im in model (2.2).
    Figure 5.  The trajectories of Sh(t,10),Eh(t,10),Ih(t,10), and Rh(t,10), as well as the corresponding histograms.
    Figure 6.  The trajectories of Sm(t,10),Em(t,10), and Im(t,10), as well as the corresponding histograms.

    1) The impact of noise on disease

    We choose the same parameters as in Figure 2 and take the values of noise as σ1=σ2=0, σ1=σ2=0.05, σ1=σ2=0.3, σ1=σ2=0.8. Figure 7 describes the variations of Ih(t,x) and Im(t,x) under different noise intensities when x=10. We can observe that a smaller noise intensity has a slight fluctuation in the number of infected people and mosquitoes, however, as the noise intensity enhances, the number of infected persons and mosquitoes decreases significantly. Therefore, we can consider random noise as a control strategy, such as human treatment and mosquito repellent spraying, to achieve the control of Zika disease.

    Figure 7.  The trajectories of Ih(t,10) and Im(t,10) under different noise intensities.

    2) The impact of diffusion coefficients on disease

    We set diffusion intensity 1 to d1=0.6,d2=d3=0.0002,d4=0.5,d5=0.3,d6=d7=0.0001, diffusion intensity 2 to d1=0.6,d2=d3=0.1,d4=0.58,d5=0.3,d6=0.18,d7=0.1, diffusion intensity 3 to d1=d2=0.6,d3=0.55,d4=0.58,d5=d6=0.3,d7=0.25, σ1=σ2=0.01, and the other parameter values are the same as those in Figure 2. The impacts of different diffusion strengths on infected people and mosquitoes are given in Figure 8, from which we find that with the increasing movement of infected people and mosquitoes, the number of infected people and mosquitoes also increases, indicating that controlling the movement of infected people and mosquitoes can reduce the risk of Zika disease.

    Figure 8.  The trajectories of Ih(t,10) and Im(t,10) under different diffusion intensities.

    Some numerical results of optimal controls will be presented in this subsection. We choose c=0.5, α=0.1, c0=0.5 in (5.1), a1=0.25, a2=0.35, a3=0.33, b1=0.1, b2=0.2, b3=0.2, c1=44, c2=40, c3=50 in (5.2), and μi(t,x)=2.0(i=1,2,,7) in (5.4). The remaining parameter values are consistent with those in Figure 2.

    Figure 9 shows the space-time diagrams of optimal controls. Figure 10 is the time evolutions of optimal controls when space variable x=10. From these two figures, we can see that the levels of human control (individual protection u1 and medical treatment of the infected people u2) are very high in the early and middle stages of the disease, but they are low in the later stages; however, the control level of mosquitoes (u3) has maintained the maximum for a long time, which means that the strength of control for mosquitoes has exceeded that for humans. Figure 11 demonstrates the trajectories of Ih(t,x) and Im(t,x) for x=10 under the four conditions of no control, only controlling humans, only controlling mosquitoes, and controlling both humans and mosquitoes. We observe that the effects on the disease of implementing three control variables and no control are very significant. In addition, the control variables u1 and u2 have great influence on the changes of infected people, but have little influence on infected mosquitoes. However, u3 has a great impact on both people and mosquitoes. Therefore, to sum up, reducing the number of mosquitoes is the primary factor to control Zika disease and personal protection and treatment of the infected humans are also two indispensable measures.

    Figure 9.  The space-time diagrams of optimal controls u1(t,x),u2(t,x), and u3(t,x).
    Figure 10.  The trajectories of optimal controls u1(t,10),u2(t,10), and u3(t,10).
    Figure 11.  The trajectories of Ih(t,10) and Im(t,10) with and without control.

    This paper presents a stochastic Zika disease model with spatial diffusion, which includes human-mosquito transmission, human-human sex transmission, and vertical transmission of mosquitoes, and studies the dynamic behavior and optimal control of the model. Firstly, we give the conditions for almost surely exponential extinction of Zika disease, and the result signifies that the Zika disease will disappear when the diffusion coefficients of infected people and mosquitoes are very small and the fluctuations of environmental noise are relatively large. Secondly, we prove the sufficient conditions for the existence and uniqueness of the steady-state distribution representing the persistence of the disease, and research suggests that when the strengths of environmental noise are low and the diffusion coefficients of humans and mosquitoes are relatively large, Zika disease will continue to exist, which is contrary to the situation of disease extinction. In addition, numerical simulations have shown that increasing the intensity of random noise or decreasing the movement of infected people and mosquitoes can lessen the occurrence of Zika disease. Finally, we take three control variables, namely, individual protection, medical treatment of the infected people, and insecticides for spraying mosquitoes, into the model, and derive the expressions of optimal controls according to the Pontryagin maximum principle. Numerical simulations show that individual protection and treatment of infected persons are very effective for human beings, but reducing the number of mosquitoes is still the most important measure to control Zika.

    The experiments demonstrate that the growth, survival, propagation, biting rate, transmission, and infection probability of Aedes aegypti and Aedes albopictus are closely related to the temperature, which is an essential factor affecting the dynamics of the spread of mosquito-borne diseases [50]. Thus, incorporating seasonality, establishing a stochastic periodic system, and studying the dynamics and control of Zika disease are our next exploration directions.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    We would like to thank the editor and the reviewers for their help, which greatly improve this paper. The research is funded by the National Natural Science Foundation of China under grant number 11971013.

    The authors declare there is no conflict of interest.

    Proof. Because of the local Lipschitz continuity of the coefficients of system (2.2), for any given initial function (S0h(x),E0h(x),I0h(x),R0h(x),S0m(x),E0m(x),I0m(x))H+, there is a unique local solution (Sh(t,x),Eh(t,x),Ih(t,x),Rh(t,x),Sm(t,x),Em(t,x),Im(t,x)) in t[0,τe),xQ, here τe is the moment of explosion. To validate the local solution is also global, we only need to prove that τe=a.s. Let k0 be large enough such that every component of (|S0h(x)|Q,|E0h(x)|Q,|I0h(x)|Q,|R0h(x)|Q,|S0m(x)|Q,|E0m(x)|Q,|I0m(x)|Q) is in the interval (1k0,k0]. Then for every integer kk0, define a stopping time

    τk=inf{t[0,τe)|min{|Sh(t,x)|Q,|Eh(t,x)|Q,|Ih(t,x)|Q,|Rh(t,x)|Q,|Sm(t,x)|Q,|Em(t,x)|Q,|Im(t,x)|Q}1kormax{|Sh(t,x)|Q,|Eh(t,x)|Q,|Ih(t,x)|Q,|Rh(t,x)|Q,|Sm(t,x)|Q,|Em(t,x)|Q,|Im(t,x)|Q}k}.

    We set inf= ( denotes the empty set usually) throughout this paper. Apparently, τk is increasing constantly as k. Let τ=limkτk, accordingly ττea.s. As long as we can verify that τ=a.s., then τe=a.s. This means that (Sh(t,x),Eh(t,x),Ih(t,x),Rh(t,x),Sm(t,x),Em(t,x),Im(t,x))H+a.s. for all t0. Before showing that τ=a.s., let us prove the boundedness of solution for every k when t[0,τk).

    Let

    N(t):=Q{Sh(t,x)+Eh(t,x)+Ih(t,x)+Rh(t,x)+Sm(t,x)+Em(t,x)+Im(t,x)}dx,

    then

    ddtN(t)==Q{tSh(t,x)+tEh(t,x)+tIh(t,x)+tRh(t,x)+tSm(t,x)+tEm(t,x)+tIm(t,x)}dx==Q{d1ΔSh(t,x)+d2ΔEh(t,x)+d3ΔIh(t,x)+d4ΔRh(t,x)+d5ΔSm(t,x)+d6ΔEm(t,x)==+d7ΔIm(t,x)+Λhdh(Sh(t,x)+Eh(t,x)+Ih(t,x)+Rh(t,x))+Λmdm(Sm(t,x)==+Em(t,x)+Im(t,x))+σ1Sh˙B1(t)+σ1Eh˙B1(t)+σ1Ih˙B1(t)+σ1Rh˙B1(t)==+σ2Sm˙B2(t)+σ2Em˙B2(t)+σ2Im˙B2(t)}dx==d1QnSh(t,x)dx+d2QnEh(t,x)dx+d3QnIh(t,x)dx+d4QnRh(t,x)dx==+d5QnSm(t,x)dx+d6QnEm(t,x)dx+d7QnIm(t,x)dx+Q(Λh+Λm)dx==+Q(dh(Sh(t,x)+Eh(t,x)+Ih(t,x)+Rh(t,x))dm(Sm(t,x)+Em(t,x)+Im(t,x)))dx==+Qσ1(Sh(t,x)+Eh(t,x)+Ih(t,x)+Rh(t,x))˙B1(t)dx==+Qσ2(Sm(t,x)+Em(t,x)+Im(t,x))˙B2(t)dx=(Λh+Λm)|Q|νN(t)+Qσ1(Sh(t,x)+Eh(t,x)+Ih(t,x)+Rh(t,x))˙B1(t)dx==+Qσ2(Sm(t,x)+Em(t,x)+Im(t,x))˙B2(t)dx,

    here |Q| stands for the volume of Q, ν=dhdm.

    Consider the following SDE

    {dZ(t)=[(Λh+Λm)|Q|νZ(t)]dt+Qσ1(Sh(t,x)+Eh(t,x)+Ih(t,x)+Rh(t,x))dxdB1(t)+Qσ2(Sm(t,x)+Em(t,x)+Im(t,x))dxdB2(t),Z(0)=N(0). (A.1)

    By the constant variation method, the solution of equation (A.1) can be obtained as

    Z(t)=(Λh+Λm)|Q|ν+(Z(0)(Λh+Λm)|Q|ν)eνt+M(t),

    where M(t)=t0eν(ts)Qσ1(Sh(s,x)+Eh(s,x)+Ih(s,x)+Rh(s,x))dxdB1(s)+t0eν(ts)Qσ2(Sm(s,x)+Em(s,x)+Im(s,x))dxdB2(s) is a continuous local martingale with M(0)=0a.s. Combining the stochastic comparison theorem, we can get that there is a constant C0>0 such that N(t)Z(t)C0a.s. That is, for each k, when t[0,τk),

    Q{Sh(t,x)+Eh(t,x)+Ih(t,x)+Rh(t,x)+Sm(t,x)+Em(t,x)+Im(t,x)}dxC0a.s. (A.2)

    Next, we continue to prove that τ=a.s. For any T>0, define V(t)=Sh,Sh+Eh,Eh+Ih,Ih+Rh,Rh+Sm,Sm+Em,Em+Im,Im, t[0,τkT). Using the Itˆo formula, we have

    dV(t)=dSh,Sh+dEh,Eh+dIh,Ih+dRh,Rh+dSm,Sm+dEm,Em+dIm,Im=[2Sh,d1ΔSh+Λhλmh(t,x)Shλhh(t,x)ShdhSh+σ21Sh2]dt+2Sh,σ1ShdB1(t)+[2Eh,d2ΔEh+λmh(t,x)Sh+λhh(t,x)Sh(ξh+dh)Eh+σ21Eh2]dt+2Eh,σ1EhdB1(t)+[2Ih,d3ΔIh+ξhEh(γ+dh)Ih+σ21Ih2]dt+2Ih,σ1IhdB1(t)+[2Rh,d4ΔRh+γIhdhRh+σ21Rh2]dt+2Rh,σ1RhdB1(t)+[2Sm,d5ΔSm+Λmθμm(kEm+Im)λhm(t,x)SmdmSm+σ22Sm2]dt+[2Em,d6ΔEm+θμm(kEm+Im)+λhm(t,x)Sm(ξm+dm)Em+σ22Em2]dt+[2Im,d7ΔIm+ξmEmdmIm+σ22Im2]dt+2Sm,σ2SmdB2(t)+2Em,σ2EmdB2(t)+2Im,σ2ImdB2(t).

    Integrating the two ends of the above equation from 0 to τkT, taking the expectation, and using (A.2) and the fundamental inequality, yield

    EV(τkT)V(0))=2EτkT0[d1Sh,ΔSh+Sh,Λh+d2Eh,ΔEh+Eh,λmh(t,x)Sh+Eh,λhh(t,x)Sh==+d3Ih,ΔIh+ξhIh,Eh+d4Rh,ΔRh+γRh,ΔIh+d5Sm,ΔSm+Sm,Λm==+d6Em,ΔEm+θμmEm,ΔkEm+Im+Em,λhm(t,x)Sm+d7Im,ΔIm+ξmIm,Em==+12σ21Sh2+12σ21Eh2+12σ21Ih2+12σ21Rh2+12σ22Sm2+12σ22Em2+12σ22Im2]dt=2EτkT0[d1Sh2+CΛhd2Eh2+αmβh(2Eh2+Em2+Im2)+2βhhEh2==+βhhIh2d3Ih2+ξhEh2+ξhIh2d4Rh2+γRh2+γIh2d5Sm2==+CΛmd6Em2+2θμmEm2+θμmIm2+αhβm(2Em2+Eh2+Ih2)d7Im2==+ξmIm2+ξmEm2+12σ21Sh2+12σ21Eh2+12σ21Ih2+12σ21Rh2+12σ22Sm2==+12σ22Em2+12σ22Im2]dt.

    Thereby,

    EV(τkT)V(0)+2C(Λh+Λm)T+EτkT0[σ21Sh2+(4αmβh+4βhh+2ξh+2αhβm+σ21)Eh2+(2βhh+2ξh+2γ+2αhβm+σ21)Ih2+(2γ+σ21)Rh2+σ22Sm2+(2αmβh+4θμm+4αhβm+2ξm+σ22)Em2+(2αmβh+2θμm+2ξm+σ22)Im2]dtC1+C2EτkT0(Sh2+Eh2+Ih2+Rh2+Sm2+Em2+Im2)dt=C1+C2T0EV(τkt)dt,

    where

    C1=V(0)+2C(Λh+Λm)T=S0h2+E0h2+I0h2+R0h2+S0m2+E0m2+I0m2+2C(Λh+Λm)T,C2=max{4αmβh+4βhh+2ξh+2αhβm+σ21,2βhh+2ξh+2γ+2αhβm+σ21,2αmβh+4θμm+4αhβm+2ξm+σ22}.

    By taking advantage of the Gronwall inequality, we get

    EV(τkT)C1eC2T. (A.3)

    Denote ϱk=infX(t,x)k,0<t<V(t) for kk0. Obviously, ϱk (k). Combine (A.3) to get C1eC2TEV(τkT)=E[V(τk)χ{τkT}]ϱkP(τkT). Setting k, then P(τkT)=0. Thence P(τ>T)=1. The proof is completed.

    Proof. Define V(t)=Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p. First, consider p2. Making use of the Itˆo formula, we obtain

    dV(t)=(pShp2Sh,d1ΔSh+Λhλmh(t,x)Shλhh(t,x)ShdhSh+pEhp2Eh,d2ΔEh+λmh(t,x)Sh+λhh(t,x)Sh(ξh+dh)Eh+pIhp2Ih,d3ΔIh+ξhEh(γ+dh)Ih+pRhp2Rh,d4ΔRh+γIhdhRh+pSmp2Sm,d5ΔSm+Λmθμm(kEm+Im)λhm(t,x)SmdmSm+pEmp2Em,d6ΔEm+θμm(kEm+Im)+λhm(t,x)Sm(ξm+dm)Em+pImp2Im,d7ΔIm+ξmEmdmIm+12p(p1)σ21(Shp+Ehp+Ihp+Rhp)+12p(p1)σ22(Smp+Emp+Imp))dt+pσ1(Shp+Ehp+Ihp+Rhp)dB1(t)+pσ2(Smp+Emp+Imp)dB2(t).

    Integrating the two sides of the above equation and taking the supremum and expectation, we can get

    Esup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p)E(S0h(x)p+E0h(x)p+I0h(x)p+R0h(x)p+S0m(x)p+E0m(x)p+I0m(x)p)+Esup0tTt0[pSh(s,x)p2Sh,Λh+pEh(s,x)p2Eh,λmh(s,x)Sh+λhh(s,x)Sh+pIh(s,x)p2Ih,ξhEh+pRh(s,x)p2Rh,γIh+pSm(s,x)p2Sm,Λm+pEm(s,x)p2Em,θμm(kEm+Im)+λhm(s,x)Sm+pIm(s,x)p2Im,ξmEm+12p(p1)σ21(Sh(s,x)p+Eh(s,x)p+Ih(s,x)p+Rh(s,x)p)+12p(p1)σ22(Sm(s,x)p+Em(s,x)p+Im(s,x)p)]ds+Esup0tT|t0pσ1(Sh(s,x)p+Eh(s,x)p+Ih(s,x)p+Rh(s,x)p)dB1(s)|+Esup0tT|t0pσ2(Sm(s,x)p+Em(s,x)p+Im(s,x)p)dB2(s)|.

    Using the Young inequality as well as the Burkholder-Davis-Gundy inequality, we can further see that

    Esup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p)=E(S0h(x)p+E0h(x)p+I0h(x)p+R0h(x)p+S0m(x)p+E0m(x)p+I0m(x)p)==+Esup0tTt0[Λph|Q|p2+Λpm|Q|p2+(p1+βphh+12p(p1)σ21)Sh(s,x)p+(3(p1)==+ξph+αphβpmkpm+12p(p1)σ21)Eh(s,x)p+(p1+γp+αphβpm+12p(p1)σ21)Ih(s,x)p==+(p1+12p(p1)σ21)Rh(s,x)p+(p1+12p(p1)σ22)Sm(s,x)p+(αpmβphkph+pθμmk==+3(p1)+ξpm+12p(p1)σ22)Em(s,x)p+(αpmβph+θpμpm+p1+12p(p1)σ22)Im(s,x)p]ds==+12Esup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p==+Im(t,x)p)+16p2σ21Esup0tTt0(Sh(s,x)p+Eh(s,x)p+Ih(s,x)p+Rh(s,x)p)ds==+16p2σ22Esup0tTt0(Sm(s,x)p+Em(s,x)p+Im(s,x)p)ds.

    After organizing, we have

    Esup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p)C3+Esup0tTt02C4(Sh(s,x)p+Eh(s,x)p+Ih(s,x)p+Rh(s,x)p+Sm(s,x)p+Em(s,x)p+Im(s,x)p)ds,

    where

    C3=2E(S0h(x)p+E0h(x)p+I0h(x)p+R0h(x)p+S0m(x)p+E0m(x)p+I0m(x)p)+2(Λph+Λpm)|Q|p2T,C4=max{p1+βphh+12p(p1)σ21+16p2σ21,3(p1)+ξph+αphβpmkpm+12p(p1)σ21+16p2σ21,p1+γp+αphβpm+12p(p1)σ21+16p2σ21,αpmβphkph+pθμmk+3(p1)+ξpm+12p(p1)σ22+16p2σ22,αpmβph+θpμpm+p1+12p(p1)σ22+16p2σ22}.

    The Gronwall inequality implies

    Esup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p)C3exp{2C4T}:=C5(p).

    Next suppose 0<p<2. By the Hölder inequality, we have

    Esup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p)(E|sup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p)|2p)p2=(Esup0tT(Sh(t,x)p+Eh(t,x)p+Ih(t,x)p+Rh(t,x)p+Sm(t,x)p+Em(t,x)p+Im(t,x)p)2p)p2(Esup0tT72pp(Sh(t,x)2+Eh(t,x)2+Ih(t,x)2+Rh(t,x)2+Sm(t,x)2+Em(t,x)2+Im(t,x)2))p2(72p1C5(2))p2:=C6,

    here the second inequality sign makes use of the fundamental inequality |x1+x2++x7|r7r1(|x1|r+|x2|r++|x7|r),r>1. This completes the proof.

    The discrete version of model (2.2) is as follows

    {Sh(i+1,j)=Sh(i,j)+[d1Sh(i,j+1)2Sh(i,j)+Sh(i,j1)(x)2+Λhλmh(i,j)Sh(i,j)λhh(i,j)Sh(i,j)dhSh(i,j)]t+σ1Sh(i,j)ζ1it+12σ21S2h(i,j)(ζ21i1)t,Eh(i+1,j)=Eh(i,j)+[d2Eh(i,j+1)2Eh(i,j)+Eh(i,j1)(x)2+λmh(i,j)Sh(i,j)+λhh(i,j)Sh(i,j)(ξh+dh)Eh(i,j)]t+σ1Eh(i,j)ζ1it+12σ21E2h(i,j)(ζ21i1)t,Ih(i+1,j)=Ih(i,j)+[d3Ih(i,j+1)2Ih(i,j)+Ih(i,j1)(x)2+ξhEh(i,j)(γ+dh)Ih(i,j)]t+σ1Ih(i,j)ζ1it+12σ21I2h(i,j)(ζ21i1)t,Rh(i+1,j)=Rh(i,j)+[d4Rh(i,j+1)2Rh(i,j)+Rh(i,j1)(x)2+γIh(i,j)dhRh(i,j)]t+σ1Rh(i,j)ζ1it+12σ21R2h(i,j)(ζ21i1)t,Sm(i+1,j)=Sm(i,j)+[d5Sm(i,j+1)2Sm(i,j)+Sm(i,j1)(x)2+Λmθμm(kEm(i,j)+Im(i,j))λhm(i,j)Sm(i,j)dmSm(i,j)]t+σ2Sm(i,j)ζ2it+12σ22S2m(i,j)(ζ22i1)t,Em(i+1,j)=Em(i,j)+[d6Em(i,j+1)2Em(i,j)+Em(i,j1)(x)2+θμm(kEm(i,j)+Im(i,j))+λhm(i,j)Sm(i,j)(ξm+dm)Em(i,j)]t+σ2Em(i,j)ζ2it+12σ22E2m(i,j)(ζ22i1)t,Im(i+1,j)=Im(i,j)+[d7Im(i,j+1)2Im(i,j)+Im(i,j1)(x)2+ξmEm(i,j)dmIm(i,j)]t+σ2Im(i,j)ζ2it+12σ22I2m(i,j)(ζ22i1)t, (A.4)

    where ζ1i,ζ2i(i=1,2,) are independent standard normal variables.



    Conflict of interest



    All authors declare no conflicts of interest.

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