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Convolutional neural network based obstacle detection for unmanned surface vehicle

  • Unmanned surface vehicles (USV) is the development trend of future ships, and it will be widely used in various kinds of marine tasks. Obstacle avoidance is one key technology for autonomous navigation of USV. Convolutional neural network based obstacle classification and detection method is applied to USV visual images in environment sensing task. To solve the problem of low detection and classification accuracy of obstacles in the visual inspection of USV, a bidirectional feature pyramid networks is proposed combining hybrid network architecture of ResNet and improved DenseNet. The proposed method can further enhance the detection and classification some types of obstacles by using the underlying multi-layer detail features and high-level strong semantic features in the network architecture. The detection and classification performance of the proposed method is evaluated on a self built dataset. Ablation experiments and performance tests on open datasets are also employed. The experimental results show that the proposed algorithm has best performance for obstacles detection, and it is more suitable for autonomous navigation of USV.

    Citation: Liyong Ma, Wei Xie, Haibin Huang. Convolutional neural network based obstacle detection for unmanned surface vehicle[J]. Mathematical Biosciences and Engineering, 2020, 17(1): 845-861. doi: 10.3934/mbe.2020045

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  • Unmanned surface vehicles (USV) is the development trend of future ships, and it will be widely used in various kinds of marine tasks. Obstacle avoidance is one key technology for autonomous navigation of USV. Convolutional neural network based obstacle classification and detection method is applied to USV visual images in environment sensing task. To solve the problem of low detection and classification accuracy of obstacles in the visual inspection of USV, a bidirectional feature pyramid networks is proposed combining hybrid network architecture of ResNet and improved DenseNet. The proposed method can further enhance the detection and classification some types of obstacles by using the underlying multi-layer detail features and high-level strong semantic features in the network architecture. The detection and classification performance of the proposed method is evaluated on a self built dataset. Ablation experiments and performance tests on open datasets are also employed. The experimental results show that the proposed algorithm has best performance for obstacles detection, and it is more suitable for autonomous navigation of USV.


    Avian influenza is an animal infectious disease caused by the transmission of influenza A viruses. Influenza A viruses are divided into subtypes according to two proteins on the surface of the virus: Hemagglutinin (HA) and neuraminidase (NA) [1]. Most avian influenza viruses infect only certain species and do not infect humans. However, a few avian influenza viruses have crossed the species barrier to infect humans and even kill them, such as H5N1, H7N1, H7N2, H7N3, H7N7, H9N2 and H7N9. Among them, H5N1 is a highly pathogenic avian influenza virus, which was first detected in human in Hong Kong in 1997. After that, humans infection with avian influenza have occurred from time to time. As of December 2019, the global cumulative number of cases of human infection with H5N1 avian influenza arrives 861, with 455 deaths. Unlike H5N1, H7N9 is classified as a low pathogenicity avian influenza virus [2]. In March 2013, there was the first case of human infection with the H7N9 avian influenza virus in Shanghai, China. In the following weeks, this virus spread to several provinces and municipalities in mainland China. As of May 2017, H7N9 has resulted in 1263 human cases in China, of whom 459 died, with a mortality rate of nearly 37%. The frequent outbreak of avian influenza in the world not only brings a serious threat to human health, but also causes psychological panic and huge social impact, and brings a huge blow to the national economy. Therefore, it has been important to understand the dynamical behavior of avian influenza and to predict what may occur. Mathematical modeling has been a useful tool to describe the dynamical behavior of avian influenza and to obtain a better understanding of transmission mechanisms. Recently, many avian influenza models have been built from different perspectives (see [2,3,4,5,6,7,8,9,10,11,12] and references therein).

    As we all know, there exist time delays during the spread of avian influenza, which can be used to describe not only the infection period of avian influenza virus in poultry (human) population, but also the incubation period of avian influenza in poultry (human) population and the immune period of recovered human to avian influenza. Therefore, the time delays should be considered such that the avian influenza models are more realistic. Generally speaking, delayed differential equations exhibit more complex dynamical behavior than differential equations without delay because time delay can make a stable equilibrium position to be unstable [13,14,15,16]. Consequently, it is of great interest to describe the transmission mechanism of avian influenza by introducing time delay into the models. For example, Liu et al. [7] and Kang et al. [12] established avian influenza models with different time delays in the poultry and human populations by considering the incubation periods of avian influenza virus and the survival probabilities of infected poultry and humans. By considering the existence of intracellular delay between initial infection of a cell and the release of new virus particles, Samanta [17] established a non-autonomous ordinary differential equation with distributed delay to characterize the spread of avian influenza between poultry and humans. These surveys imply that the research of time delay on avian influenza is a meaningful issue and is still open for study.

    On the other hand, many existing literatures only focus on the deterministic avian influenza models that do not consider the impact of environmental noise. However, in the real world, the spread of avian influenza is often affected by the variations of environmental factors, such as humidity, temperature and so on [18,19]. Due to the fluctuations in the environment, an actual avian influenza system would not remain in a stable state, which would interfere with this stable state by acting directly on the density or indirectly affecting the parameter values. Therefore, it is of great significance to reveal the impact of environmental noise on avian influenza model by using stochastic model, so as to obtain more real benefits and accurately predict the future dynamics of avian influenza. To better understand the transmission dynamics of avian influenza, some authors have introduced stochastic perturbations into the deterministic models [20,21,22]. Zhang et al. [20] constructed a stochastic avian-human influenza model with logistic growth for avian population, and discussed the dynamical behavior of this model. Further, Zhang et al. [21] investigated a stochastic avian-human influenza epidemic model with psychological effect in human population and saturation effect within avian population. On the basis of the deterministic model established by Iwami et al. [3], Zhang et al. [22] established the corresponding stochastic model by introducing density disturbance. All the papers mentioned above only focused on the extinction and persistence of stochastic avian influenza models. However, to the best of our knowledge, there is no results related to the asymptotic behavior of stochastic avian influenza model around the equilibria of the corresponding deterministic model.

    Motivated by the above discussions, in this paper, we investigate the asymptotic behavior of a stochastic delayed avian influenza model with saturated incidence rate. This work differs from existing results [7,12,17,20,21,22] in that (a) time delays and white noise are taken into account to describe the latency period of avian influenza virus in both poultry and human population and the environmental fluctuations; (b) asymptotic behavior of a stochastic delayed avian influenza model is studied. Overview of the rest of the article is as follows: In section 3, we show that there exists a unique global positive solution of system (2.3) with the given initial value (2.4). In section 4, we prove that the solution of system (2.3) is going around E0 under certain conditions. Further, we derive that the solution of system (2.3) is going around E under certain conditions in section 5. In section 6, some numerical examples are introduced to illustrate the effectiveness of theoretic results. Finally, some conclusions are given in section 7.

    Although the avian influenza virus spreads between wild birds and poultry, and between poultry and humans, we will only consider the transmission dynamics of avian influenza between poultry and humans because poultry is the main source of infection. Moreover, we assume that the virus is not spread between humans and mutate. We denote the total population of poultry and humans at time t by Na(t) and Nh(t), respectively. When the susceptible poultry contact with the infected poultry closely, there is usually no quick way to detect whether they are infected or the detection cost is too high, which makes it impossible to distinguish whether the close contacts of poultry are infected with the avian influenza virus. Therefore, the poultry population is divided into three sub-populations depending on the state of the disease: susceptible poultry Sa(t), exposed poultry Ea(t) and infected poultry Ia(t). The total poultry population at time t is denoted by Na(t)=Sa(t)+Ea(t)+Ia(t). The human population is divided into three sub-populations, which are susceptible human Sh(t), infected human with avian influenza Ia(t) and recovered human from avian influenza Rh(t). The total population of human at time t is given by Nh(t)=Sh(t)+Ih(t)+Rh(t).

    The reason why we do not consider the exposed class for human population is that the close contacts of human beings are usually isolated and tested to determine whether they are infected with the avian influenza virus. The poultry in Ea either shows symptoms after incubation period and move to Ia, or always stays in Ea until natural death. The number of susceptible poultry (human) is increased by new recruitment, but decreases by natural death and infection (moving to class Ia (Ih)). The number of infected poultry (human) is increased by the infection of susceptible poultry (human) and reduced through natural and disease-related death. In addition, the number of infected humans is also reduced by recovery from the disease (moving to class Rh). Based on the above discussions, we obtain the schematic diagram of our model (see Figure 1).

    Figure 1.  Schematic diagram of the model (2.1).

    The corresponding avian influenza model can be represented by the following equations:

    {dSa(t)dt=ΛaμaSa(t)βaSa(t)Ia(t)1+α1Ia(t),dEa(t)dt=βaeμaτaSa(tτa)Ia(tτa)1+α1Ia(tτa)(μa+γa)Ea(t),dIa(t)dt=γaEa(t)(μa+δa)Ia(t),dSh(t)dt=ΛhμhSh(t)βhSh(t)Ia(t)1+α2Ia(t),dIh(t)dt=βheμhτhSh(tτh)Ia(tτh)1+α2Ia(tτh)(μh+δh+θh)Ih(t),dRh(t)dt=θhIh(t)μhRh(t). (2.1)

    All parameters in model (2.1) are assumed non-negative and described in Table 1.

    Table 1.  Parameters description in the model (2.1).
    Parameter Description
    Λa new recruitment of the poultry populations
    Λh new recruitment of the human population
    βa the transmission rate from infective poultry to susceptible poultry
    βh the transmission rate from infective poultry to susceptible human
    μa the natural death rate of poultry populations
    μh the natural death rate of human populations
    δa the disease-related death rate of poultry populations
    δh the disease-related death rate of humans populations
    γa the transfer rate of exposed poultry to infected poultry
    θh the recovery rate of the infective human
    αi(i=1,2) parameters that measure the inhibitory effect

     | Show Table
    DownLoad: CSV

    Because the removed human populations Rh(t) has no effect on the dynamics of the first five equations, system (2.1) can be decoupled to the following system:

    {dSa(t)dt=ΛaμaSa(t)βaSa(t)Ia(t)1+α1Ia(t),dEa(t)dt=βaeμaτaSa(tτa)Ia(tτa)1+α1Ia(tτa)(μa+γa)Ea(t),dIa(t)dt=γaEa(t)(μa+δa)Ia(t),dSh(t)dt=ΛhμhSh(t)βhSh(t)Ia(t)1+α2Ia(t),dIh(t)dt=βheμhτhSh(tτh)Ia(tτh)1+α2Ia(tτh)(μh+δh+θh)Ih(t). (2.2)

    A realistic avian influenza system would not remain in this stable state due to environmental fluctuations. In this paper, we will reveal how the environmental white noise affects the spread of avian influenza through investigating the dynamics of a stochastic delayed avian influenza model with saturated incidence rate. Taking the same approach as the literatures [23,24], we assume that the environmental white noise is directly proportional to the variables Sa(t), Ea(t), Ia(t), Sh(t) and Ih(t), respectively. Then, corresponding to system (2.2), the stochastic avian influenza model with time delay is of the following form

    {dSa(t)=(ΛaμaSa(t)βaSa(t)Ia(t)1+α1Ia(t))dt+σ1Sa(t)dB1(t),dEa(t)=(βaeμaτaSa(tτa)Ia(tτa)1+α1Ia(tτa)(μa+γa)Ea(t))dt+σ2Ea(t)dB2(t),dIa(t)=(γaEa(t)(μa+δa)Ia(t))dt+σ3Ia(t)dB3(t),dSh(t)=(ΛhμhSh(t)βhSh(t)Ia(t)1+α2Ia(t))dt+σ4Sh(t)dB4(t),dIh(t)=(βheμhτhSh(tτh)Ia(tτh)1+α2Ia(tτh)(μh+δh+θh)Ih(t))dt+σ5Ih(t)dB5(t), (2.3)

    in which Bi(t)(i=1,2,,5) are mutually independent standard Brownian motions defined on a complete probability space (Ω,F,P) with a filtration{Ft}t0 satisfying the usual conditions (i.e., it is increasing and right continuous while F0 contains all P-null sets), σi(i=1,2,,5) denote the intensities of the white noises. The initial value of system (2.3) are

    {Sa(θ)=φ1(θ),Ea(θ)=φ2(θ),Ia(θ)=φ3(θ),Sh(θ)=φ4(θ),Ih(θ)=φ5(θ),φi(θ)C([τ,0],R5+),i=1,2,3,4,5,τ=max{τa,τh}, (2.4)

    where C is the Banach space C([τ,0];R5+) of continuous functions mapping the interval [τ,0] into R5+, and R5+={x=(x1,x2,x3,x4,x5):xi>0,i=1,2,3,4,5}. By a biological meaning, we assume that φi(0)>0(i=1,2,3,4,5).

    In this section, we prove that the solution of system (2.3) is global and positive for any initial value (2.4).

    Theorem 1. For any initial value (2.4), system (2.3) has a unique positive solution (Sa(t),Ea(t),Ia(t),Sh(t),Ih(t)) on t0 and the solution will remain in R5+ with probability one, in other words, (Sa(t),Ea(t),Ia(t),Sh(t),Ih(t))R5+ for all t0 almost surely.

    Proof. Since the coefficients of system (2.3) satisfy the local Lipschitz conditions, then for any initial value (2.4), there exists a unique local solution (Sa(t),Ea(t),Ia(t),Sh(t),Ih(t)) on t[τ,τe), where τe is the explosive time. To show this solution is global, we only need to show that τe= a.s. To this end, let k01 be sufficiently large such that (Sa(θ),Ea(θ),Ia(θ),Sh(θ),Ih(θ))(θ[τ,0]) all lie within the interval [1k0,k0]. For each integer kk0, define the stopping time as

    τk=inf{t[0,τe):Sa(t)(1k,k) or Ea(t)(1k,k) or Ia(t)(1k,k) or Sh(t)(1k,k) or Ih(t)(1k,k)}.

    We set inf=. Obviously, τk increasing when k. Let τ=limkτk, where ττe a.s. If we can verify τ= a.s., then τe= and (Sa(t),Ea(t),Ia(t),Sh(t),Ih(t))R5+ a.s. for all t0. That is to say, to complete the proof we only need to show that τ= a.s. If this assertion is not true, then there is a pair of constants T>0 and ε(0,1) such that

    P{τT}>ε.

    There exists an integer k1k0 such that

    P{τkT}ε for all kk1. (3.1)

    Define a C2-function V: R5+R+ by

    V(Sa,Ea,Ia,Sh,Ih)=eμaτa(SaaalnSaa)+(Ea1lnEa)+(Ia1lnIa)+βaeμaτattτaSa(s)Ia(s)1+α1Ia(s)ds+eμhτh(ShbblnShb)+(Ih1lnIh)+βheμhτhttτhSh(s)Ia(s)1+α2Ia(s)ds,

    in which a and b are positive constants to be determined later. The nonnegativity of this function can be derived from x1lnx0 for any x>0. Applying the Itô's formula to V, we get

    dV=eμaτa(1aSa)dSa+eμaτaa2S2a(dSa)2+(11Ea)dEa+a2E2a(dEa)2+(11Ia)dIa+a2I2a(dIa)2+βaeμaτaSaIa1+α1IaβaeμaτaSa(tτa)Ia(tτa)1+α1Ia(tτa)+eμhτh(1bSh)dSh+eμhτhb2S2h(dSh)2+(11Ih)dIh+12I2h(dIh)2+βheμhτhShIh1+α2IaβheμhτhSh(tτh)Ia(tτh)1+α2Ia(tτh)=LVdt+eμaτaσ1(Saa)dB1(t)+σ2(Ea1)dB2(t)+σ3(Ia1)dB3(t)+eμhτhσ4(Shb)dB4(t)+σ5(Ih1)dB5(t), (3.2)

    where

    LV=eμaτa(1aSa)(ΛaμaSa)(11Ea)(μa+γa)Ea+(11Ia)(γaEa(μa+δa)Ia)+eμhτh(1bSh)(ΛhμhSh)(11Ih)(μh+δh+θh)Ih+eμaτaaσ212+σ222+σ232+eμhτhbσ242+σ252eμaτaΛa+aμaeμaτa+aσ212eμaτa+2μa+δa+γa+12σ22+12σ23+eμhτhΛh+bμheμhτh+μh+δh+θh+bσ242eμhτh+12σ25+(aβaeμaτa+bβheμhτh(μa+δa))Ia.

    Choose a=μaeμaτaβa and b=δaeμhτhβh or a=δaeμaτaβa and b=μaeμhτhβh such that

    aβaeμaτa+bβheμhτh(μa+δa)=0.

    Then, we can get

    LV(Sa,Ea,Ia,Sh,Ih)eμaτaΛa+aμaeμaτa+eμhτhΛh+bμheμhτh+2μa+γa+δa+μh+δh+θh+aσ212eμaτa+bσ242eμhτh+12(σ22+σ23+σ25)=:K,

    where K is a positive constant. It thus follows from (3.2) that

    dV(Sa,Ea,Ia,Sh,Ih)Kdt+eμaτaσ1(Saa)dB1(t)+σ2(Ea1)dB2(t)+σ3(Ia1)dB3(t)+eμhτhσ4(Shb)dB4(t)+σ5(Ih1)dB5(t). (3.3)

    Integrating both sides of (3.3) from 0 to τkT=min{τk,T} and then taking the expectation results in

    EV(Sa(τkT),Ea(τkT),Ia(τkT),Sh(τkT),Ih(τkT))V(Sa(0),Ea(0),Ia(0),Sh(0),Ih(0))+KE(τkT)V(Sa(0),Ea(0),Ia(0),Sh(0),Ih(0))+KT. (3.4)

    Set Ωk={τkT} for kk1, and according to (3.1), we have P(Ωk)ε. For every ωΩk, there exists Sa(τk,ω) or Ea(τk,ω) or Ia(τk,ω) or Sh(τk,ω) or Ih(τk,ω) equals either k or 1k. Therefore, V(Sa(τk,ω),Ea(τk,ω),Ia(τk,ω),Sh(τk,ω),Ih(τk,ω)) is no less either k1lnk or 1k1ln1k or kaalnka or 1ka+alnak or kbblnkb or 1kb+blnbk.

    Therefore, we have

    V(Sa(τk,ω),Ea(τk,ω),Ia(τk,ω),Sh(τk,ω),Ih(τk,ω))(k1lnk)(1k1+lnk)(kaalnka)(1ka+alnak)(kbblnkb)(1kb+blnbk).

    It follows from (3.4) that

    V(Sa(0),Ea(0),Ia(0),Sh(0),Ih(0))+KTE[1ΩkV(Sa(τk,ω),Ea(τk,ω),Ia(τk,ω),Sh(τk,ω),Ih(τk,ω))]ε[(k1lnk)(1k1+lnk)(kaalnka)(1ka+alnak)(kbblnkb)(1kb+blnbk)],

    where 1Ωk denotes the indicator function of Ωk. Letting k, then

    >V(Sa(0),Ea(0),Ia(0),Sh(0),Ih(0))+KT=,

    which leads to the contradiction. This completes the proof.

    In this section, we will investigate the solution of system (2.3) around disease-free equilibrium E0 under certain conditions. It is worthwhile to mention that, if R0=βaγaΛaeμaτaμa(μa+δa)(μa+γa)<1, the deterministic system (2.2) is globally asymptotically stable around the unique disease-free equilibrium E0=(S0a,0,0,S0h,0)=(Λaμa,0,0,Λhμh,0), but E0 is not the equilibrium of the stochastic system (2.3). Thus, the result concerning the solution of stochastic system (2.3) around E0 is presented by the following theorem.

    Theorem 2. Let (Sa(t),Ea(t),Ia(t),Sh(t),Ih(t)) be the solution of system (2.3) with the initial value (2.4). If R0<1 and the following conditions hold

    σ21<μa,σ22<μa+γa,σ23<μa+δa,σ24<μh,σ25<μh+δh+θh,

    then,

    lim supt1tEt0(SaΛaμa)2dsσ21Λ2aμ2a(μaσ21),lim supt1tEt0(E2a+I2a)dsP1M1,lim supt1tEt0(ShΛhμh)2dsΛ2hμ2h(μhσ24)(σ24+βhα2),lim supt1tEt0I2hdsP2,

    where

    M1=min{μa+γaσ224,(μa+γaσ22)(μa+δaσ23)(μa+δa)4γ2a},P1=e2μaτaσ21Λ2aμ2a[1μaσ21(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)+1],P2=2e2μhτhΛhμ2h(μh+δh+θhσ25)[α2σ24+βhα2(μhσ24)(2μ2h+2μhδh+2μhθh+(δh+θh)22(μh+δh+θh)+σ24)+σ24].

    Proof. Since (S0a,0,0,S0h,0) is the disease-free equilibrium of system (2.2), then

    Λa=μaS0a,Λh=μhS0h.

    According to system (2.3), we can obtain that

    dSa(t)=[μa(SaΛaμa)βaSaIa1+α1Ia]dt+σ1SadB1(t)=[μa(SaΛaμa)βa(SaΛaμa)Ia1+α1IaβaΛaμaIa1+α1Ia]dt+σ1SadB1(t), (4.1)

    and

    d[Ea(t+τa)+μa+γaγaIa(t+τa)]=dEa(t+τa)+μa+γaγadIa(t+τa)[βaeμaτa(SaΛaμa)Ia1+α1Ia(μa+γa)(μa+δa)γaIa(t+τa)+βaeμaτaΛaμaIa]dt+σ2Ea(t+τa)dB2(t)+σ3(μa+γa)γaIa(t+τa)dB3(t)[βaeμaτa(SaΛaμa)Ia1+α1Ia+(μa+γa)(μa+δa)γa(Ia(t)Ia(t+τa))]dt+σ2Ea(t+τa)dB2(t)+σ3(μa+γa)γaIa(t+τa)dB3(t). (4.2)

    Let V1=12(SaΛaμa)2, then applying the Itô's formula to V1, together with (4.1), we have

    dV1=[(SaΛaμa)(μa(SaΛaμa)βa(SaΛaμa)Ia1+α1IaβaΛaμaIa1+α1Ia)+12σ21S2a]dt+σ1Sa(SaΛaμa)dB1(t)=[μa(SaΛaμa)2βa(SaΛaμa)2Ia1+α1IaβaΛaμa(SaΛaμa)Ia1+α1Ia+12σ21S2a]dt+σ1Sa(SaΛaμa)dB1(t)=:LV1dt+σ1Sa(SaΛaμa)dB1(t),

    where

    LV1μa(SaΛaμa)2βaΛaμa(SaΛaμa)Ia1+α1Ia+σ21(SaΛaμa)2+σ21Λ2aμ2a=(μaσ21)(SaΛaμa)2βaΛaμa(SaΛaμa)Ia1+α1Ia+σ21Λ2aμ2a. (4.3)

    Similarly, let V2=Ea(t+τa)+μa+γaγaIa(t+τa)+(μa+γa)(μa+δa)γat+τatIa(s)ds, it follows from (4.2) that

    dV2βaeμaτa(SaΛaμa)Ia1+α1Ia+σ2Ea(t+τa)dB2(t)+σ3(μa+γa)γaIa(t+τa)dB3(t).

    Define ˉV=eμaτaV1+ΛaμaV2, then

    dˉV[eμaτa(μaσ21)(SaΛaμa)2+eμaτaσ21Λ2aμ2a]dt+σ1Sa(SaΛaμa)dB1(t)+σ2Ea(t+τa)dB2(t)+σ3(μa+γa)γaIa(t+τa)dB3(t). (4.4)

    Integrating both sides of (4.4) from 0 to t and taking expectation, we get

    EˉV(t)EˉV(0)eμaτa(μaσ21)Et0(SaΛaμa)2ds+eμaτaσ21Λ2aμ2at.

    Therefore, we can obtain

    lim supt1tEt0(SaΛaμa)2dsσ21Λ2aμ2a(μaσ21).

    Similarly, we define

    V3=12[eμaτa(SaΛaμa)+Ea(t+τa)]2,

    then,

    LV3=e2μaτaμa(SaΛaμa)2eμaτa(2μa+γa)(SaΛaμa)Ea(t+τa)(μa+γa)E2a(t+τa)+12e2μaτaσ21S2a+12σ22E2a(t+τa)e2μaτaμa(SaΛaμa)2+μa+γa2E2a(t+τa)+(2μa+γa)2e2μaτa2(μa+γa)(SaΛaμa)2(μa+γa)E2a(t+τa)+e2μaτaσ21(SaΛaμa)2+e2μaτaσ21Λ2aμ2a+12σ22E2a(t+τa)=e2μaτa(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)(SaΛaμa)212(μa+γaσ22)E2a(t+τa)+e2μaτaσ21Λ2aμ2a.

    Let V4=V3+12(μa+γaσ22)t+τatE2a(s)ds, we get

    LV4e2μaτa(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)(SaΛaμa)212(μa+γaσ22)E2a+e2μaτaσ21Λ2aμ2a.

    Let V5=12I2a, the derivative of V5 can be calculated as

    LV5=γaEaIa(μa+δa)I2a+12σ23I2aμa+δa2I2a+γ2a2(μa+δa)E2a(μa+δa)I2a+12σ23I2a=γ2a2(μa+δa)E2a12(μa+δaσ23)I2a.

    The Young's inequality is used above. Let

    ˜V=V4+eμaτaμaσ21(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)ˉV+(μa+γaσ22)(μa+δa)2γ2aV5,

    which implies that

    L˜V12(μa+γaσ22)E2a+e2μaτaσ21Λ2aμ2a+e2μaτaσ21Λ2aμ2a(μaσ21)(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)+14(μa+γaσ22)E2a(μa+γaσ22)(μa+δaσ23)(μa+δa)4γ2aI2a=14(μa+γaσ22)E2a(μa+γaσ22)(μa+δaσ23)(μa+δa)4γ2aI2a+e2μaτaσ21Λ2aμ2a[1μaσ21(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)+1]. (4.5)

    Integrating both sides of (4.5) from 0 to t and then taking expectation yields

    E˜V(t)E˜V(0)14(μa+γaσ22)Et0E2a(s)ds(μa+γaσ22)(μa+δaσ23)(μa+δa)4γ2aEt0I2a(s)ds+e2μaτaσ21Λ2aμ2a[1μaσ21(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)+1]t.

    Consequently, we can obtain

    lim supt1tEt0(E2a(s)+I2a(s))dsP1M1,

    where M1 and P1 are defined in Theorem 2. Further, according to system (2.3), we have

    dSh(t)=[μh(ShΛhμh)βhShIa1+α2Ia]dt+σ4ShdB4(t)=[μh(ShΛhμh)(ShΛhμh)βhIa1+α2IaβhΛhIaμh(1+α2Ia)]dt+σ4ShdB4(t), (4.6)

    and

    dIh(t+τh)=[βheμhτhShIa1+α2Ia(μh+δh+θh)Ih(t+τh)]dt+σ5Ih(t+τh)dB5(t)[βheμhτhIa1+α2Ia(ShΛhμh)+βhΛheμhτhα2μh(μh+δh+θh)Ih(t+τh)]dt+σ5Ih(t+τh)dB5(t). (4.7)

    Let V6=12(ShΛhμh)2. Noting (4.6), we have

    LV6=μh(ShΛhμh)2βh(ShΛhμh)2Ia1+α2IaβhΛhμh(ShΛhμh)Ia1+α2Ia+12σ24S2hμh(ShΛhμh)2βhΛhμh(ShΛhμh)Ia1+α2Ia+σ24(ShΛhμh)2+σ24Λ2hμ2h=(μhσ24)(ShΛhμh)2βhΛhμh(ShΛhμh)Ia1+α2Ia+σ24Λ2hμ2h.

    Let V7=eμhτhV6+ΛhμhIh(t+τh), it follows from (4.7) that

    LV7eμhτh(μhσ24)(ShΛhμh)2+eμhτhσ24Λ2hμ2h+βhΛ2heμhτhα2μ2hΛhμh(μh+δh+θh)Ih(t+τh)eμhτh(μhσ24)(ShΛhμh)2+eμhτhΛ2hμ2h(σ24+βhα2). (4.8)

    Integrating both sides of (4.8) from 0 to t and then taking the expectation yields

    EV7(t)EV7(0)eμhτh(μhσ24)Et0(ShΛhμh)2ds+eμhτhΛ2hμ2h(σ24+βhα2)t,

    therefore, we can get

    lim supt1tEt0(ShΛhμh)2dsΛ2hμ2h(μhσ24)(σ24+βhα2).

    Let V8=12[eμhτh(ShΛhμh)+Ih(t+τh)]2, then

    LV8=(eμhτh(ShΛhμh)+Ih(t+τh))[eμhτh(ΛhμhSh)(μh+δh+θh)Ih(t+τh)]+12e2μhτhσ24S2h+12σ25I2h(t+τh)e2μhτhμh(ShΛhμh)2+(2μh+δh+θh)2e2μhτh2(μh+δh+θh)(ShΛhμh)2+μh+δh+θh2I2h(t+τh)(μh+δh+θh)I2h(t+τh)+e2μhτhσ24(ShΛhμh)2+e2μhτhσ24Λ2hμ2h+12σ25I2h(t+τh)=e2μhτh(2μ2h+2μhδh+2μhθh+(δh+θh)22(μh+δh+θh)+σ24)(ShΛhμh)212(μh+δh+θhσ25)I2h(t+τh)+e2μhτhσ24Λ2hμ2h.

    Defining

    V9=V8+eμhτhμhσ24(2μ2h+2μhδh+2μhθh+(δh+θh)22(μh+δh+θh)+σ24)V7+12(μh+δh+θhσ25)t+τhtI2h(s)ds,

    we get

    LV912(μh+δh+θhσ25)I2h+e2μhτhΛ2hμ2h[1μhσ24(σ24+βhα2)(2μ2h+2μhδh+2μhθh+(δh+θh)22(μh+δh+θh)+σ24)+σ24]. (4.9)

    Integrating both sides of (4.9) from 0 to t and taking expectation, we obtain

    EV9(t)EV9(0)12(μh+δh+θhσ25)Et0I2h(s)ds+e2μhτhΛ2hμ2h[1μhσ24(σ24+βhα2)(2μ2h+2μhδh+2μhθh+(δh+θh)22(μh+δh+θh)+σ24)+σ24]t.

    Consequently, we can obtain

    lim suptEt0I2h(s)dsP2,

    where P2 is defined in Theorem 2. This completes the proof.

    If R0>1, there exists an endemic equilibrium E=(Sa,Ea,Ia,Sh,Ih) of system (2.2), but it is not the equilibrium of system (2.3), where Sa=Λa(1+α1Ia)μa(1+α1Ia)+βaIa, Ea=βaΛaeμaτaIa(μa+γa)[μa(1+α1Ia)+βaIa], Ia=μa(R01)α1μa+βa, Sh=Λh(1+α2Ia)μh(1+α2Ia)+βhIa, Ea=βheμhτhShIa(μh+δh+θh)(1+α2Ia). In this section, we show that the solution of system (2.3) is going around E under certain conditions.

    Theorem 3. Let (Sa(t),Ea(t),Ia(t),Sh(t),Ih(t)) be the solution of system (2.3) with initial value (2.4). If R0>1 and the following conditions hold

    (i) σ21<μa,σ22<12(μa+γa),σ23<12(μa+δa),σ24<μh,σ25<μh+δh+θh;

    (ii) max(P3,P4,P5,P6)<d(E,E0),

    then

    lim supt1tEt0(SaSa)2dsP3,lim suptEt0[(Ea(s)Ea)2+(Ia(s)Ia)2]dsL1L2=:P4,lim suptEt0(ShSh)2dsP5,lim suptEt0(IhIh)2dsP6,

    where

    d(E,E0)=(SaΛaμa)2+(Ea)2+(Ia)2+(ShΛhμh)2+(Ih)2P3=1μaσ21[σ21(Sa)2+σ21SaL32μa+(eμaτaSa+L3μaeμaτa)(12σ22Ea+μa+γa2γaσ23Ia)],P4=L1L2,P5=σ24(Sh)2μhσ24,P6=σ24L24(μhσ24)(μh+δh+θhσ25)2+2σ25(Ih)2μh+δh+θhσ25,L1=eμaτaμaσ21(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)[σ21(Sa)2+σ21SaL32μa+(eμaτaSa+L3μaeμaτa)(12σ22Ea+12μa+γaγaσ23Ia)]+e2μaτaσ21(Sa)2+σ22(Ea)2+σ23(μa+δa)(μa+γa2σ22)2γ2a(Ia)2,L2=min{14(μa+γa2σ22),(μa+δa)(μa+γa2σ22)(μa+δa2σ23)4γ2a},L3=βaSaIa1+α1Ia,L4=βhShIa1+α2Ia.

    Proof. Since (Sa,Ea,Ia,Sh,Ih) is the interior equilibrium of system (2.2), then

    Λa=μaSa+βaSaIa1+α1Ia,(μa+γa)Ea=βaeμaτaSaIa1+α1Ia,IaEa=γaμa+δa,Λh=μhSh+βhShIa1+α2Ia,(μh+δh+θh)Ih=βheμhτhShIa1+α2Ia. (5.1)

    Define the Lyapunov function W1 as W1=SaSaSalnSaSa, from which we have

    dW1=(ΛaμaSaβaSaIa1+α1IaΛaSaSa+μaSa+βaSaIa1+α1Ia+12Saσ21)dt+σ1(SaSa)dB1(t)=[(μaSa+βaSaIa1+α1Ia)(2SaSaSaSa)+βaSaIa1+α1Ia(SaIa(1+α1Ia)SaIa(1+α1Ia)+SaSa+Ia(1+α1Ia)Ia(1+α1Ia)1)+12Saσ21]dt+σ1(SaSa)dB1(t)=LW1dt+σ1(SaSa)dB1(t),

    where

    LW1=(μa+βaIa1+α1Ia)(SaSa)2Saβa(SaSa)(Ia1+α1IaIa1+α1Ia)+12Saσ21. (5.2)

    Similarly, we can define W2 as

    W2=Ea(t+τa)EaEalnEa(t+τa)Ea+μa+γaγa(Ia(t+τa)IaIalnIa(t+τa)Ia).

    By using the Itô's formula, the derivative of W2 is calculated as follows

    LW2=(1EaEa(t+τa))(βaeμaτaSaIa1+α1Ia(μa+γa)Ea(t+τa))+μa+γaγa(1IaIa(t+τa))(γaEa(t+τa)(μa+δa)Ia(t+τa))+12σ22Ea+μa+γa2γaσ23Ia=βaeμaτaIa1+α1Ia(SaSa)(1+α1IaIaIa1+α1Ia1)+βaeμaτaSaIa1+α1Ia(SaSa1+α1IaSaIaSaIa1+α1IaEaEa(t+τa)+1+α1IaIaIa1+α1IaIa(t+τa)IaEa(t+τa)EaIaIa(t+τa))+12σ22Ea+12μa+γaγaσ23Ia. (5.3)

    Since x1lnx0 for x>0, the following estimate can be obtained

    1+α1IaSaIaSaIa1+α1IaEaEa(t+τa)1+ln(1+α1IaSaIaSaIa1+α1IaEaEa(t+τa))=1+lnSaSalnIa(t+τa)Ia+lnIa(1+α1Ia)Ia(1+α1Ia)lnEa(t+τa)EaIaIa(t+τa). (5.4)

    Substituting (5.4) into (5.3), we can get

    LW2βaeμaτaIa1+α1Ia(SaSa)(1+α1IaIaIa1+α1Ia1)+βaeμaτaSaIa1+α1Ia(SaSa1lnSaSa+lnIa(t+τa)IalnIa(1+α1Ia)Ia(1+α1Ia)+lnEa(t+τa)EaIaIa(t+τa)+1+α1IaIaIa1+α1IaIa(t+τa)IaEa(t+τa)EaIaIa(t+τa))+12σ22Ea+12μa+γaγaσ23Ia=βaeμaτaIa1+α1Ia(SaSa)(1+α1IaIaIa1+α1Ia1)+βaeμaτaSaIa1+α1Ia[(SaSalnSaSa)(Ia(t+τa)IalnIa(t+τa)Ia)+(Ia(1+α1Ia)Ia(1+α1Ia)lnIa(1+α1Ia)Ia(1+α1Ia))(Ea(t+τa)EaIaIa(t+τa)lnEa(t+τa)EaIaIa(t+τa))1]+12σ22Ea+12μa+γaγaσ23Ia. (5.5)

    Choose W3=W2+βaeμaτaSaIa1+α1Iat+τat(Ia(s)IalnIa(s)Ia1)ds. Therefore, LW3 can be obtained as follows by using (5.5):

    LW3βaeμaτaIa1+α1Ia(SaSa)(Ia(1+α1Ia)Ia(1+α1Ia)1)+βaeμaτaSaIa1+α1Ia[(SaSalnSaSa)(Ia(t+τa)IalnIa(t+τa)Ia)+(Ia(1+α1Ia)Ia(1+α1Ia)lnIa(1+α1Ia)Ia(1+α1Ia))(Ea(t+τa)EaIaIa(t+τa)lnEa(t+τa)EaIaIa(t+τa))1]+12σ22Ea+12μa+γaγaσ23Ia+βaeμaτaSaIa1+α1Ia(Ia(t+τa)IalnIa(t+τa)Ia1)βaeμaτaSaIa1+α1Ia(IaIalnIaIa1)βaeμaτaIa1+α1Ia(SaSa)(Ia(1+α1Ia)Ia(1+α1Ia)1)+βaeμaτaSaIa1+α1Ia[SaSa+SaSa1IaIa+Ia(1+α1Ia)Ia(1+α1Ia)+lnIa(1+α1Ia)Ia(1+α1Ia)IaIa1]+12σ22Ea+12μa+γaγaσ23Ia. (5.6)

    Noting that x1lnx0 holds for x>0, we also have

    IaIa+Ia(1+α1Ia)Ia(1+α1Ia)+lnIa(1+α1Ia)Ia(1+α1Ia)IaIaIaIa+Ia(1+α1Ia)Ia(1+α1Ia)+Ia(1+α1Ia)Ia(1+α1Ia)IaIa1Ia(1+α1Ia)Ia(1+α1Ia)IaIa(Ia(1+α1Ia)Ia(1+α1Ia)IaIa1)(Ia(1+α1Ia)Ia(1+α1Ia)1)=(1+α1Ia)(1+α1Ia)Ia(11+α1Ia11+α1Ia)(Ia1+α1IaIa1+α1Ia)<0, (5.7)

    substituting (5.7) into (5.6) and using SaSa+SaSa2=(SaSa)2SaSa, we know that

    LW3βaeμaτaIa1+α1Ia(SaSa)(Ia(1+α1Ia)Ia(1+α1Ia)1)+βaeμaτaIa1+α1Ia(SaSa)2Sa+12σ22Ea+12μa+γaγaσ23Ia. (5.8)

    Let W4=W1+1+α1IaβaeμaτaIa(μa+βaIa1+α1Ia)W3. Applying the Itô's formula, together with (5.2) and (5.8), derives that

    LW4=LW1+1+α1IaβaeμaτaIa(μa+βaIa1+α1Ia)LW3(μa+βaIa1+α1Ia)(SaSa)2Saβa(SaSa)(Ia1+α1IaIa1+α1Ia)+12σ21Sa+1+α1IaβaeμaτaIa(μa+βaIa1+α1Ia)[βaeμaτaIa1+α1Ia(SaSa)(Ia(1+α1Ia)Ia(1+α1Ia)1)+βaeμaτaIa1+α1Ia(SaSa)2Sa+12σ22Ea+12μa+γaγaσ23Ia]=(μa+βaIa1+α1Ia)(SaSa)(Ia(1+α1Ia)Ia(1+α1Ia)1)βa(SaSa)(Ia1+α1IaIa1+α1Ia)+12σ21Sa+1+α1IaβaeμaτaIa(μa+βaIa1+α1Ia)(12σ22Ea+12μa+γaγaσ23Ia)=(SaSa)(Ia1+α1IaIa1+α1Ia)[(μa+βaIa1+α1Ia)1+α1IaIaβa]+12σ21Sa+1+α1IaβaeμaτaIa(μa+βaIa1+α1Ia)(12σ22Ea+12μa+γaγaσ23Ia)=μa(1+α1Ia)Ia(SaSa)(Ia1+α1IaIa1+α1Ia)+12σ21Sa+1+α1IaβaeμaτaIa(μa+βaIa1+α1Ia)(12σ22Ea+12μa+γaγaσ23Ia). (5.9)

    Choose Lyapunov function W5 as W5=(SaSa)22, then its derivative is

    LW5=(SaSa)[ΛaμaSaβaSaIa1+α1Ia]+12σ21S2a=(SaSa)[μaSaμaSa+βaSaIa1+α1IaβaSaIa1+α1Ia]+12σ21S2a=μa(SaSa)2βaSa(SaSa)(Ia1+α1IaIa1+α1Ia)βa(SaSa)2Ia1+α1Ia+12σ21S2aμa(SaSa)2βaSa(SaSa)(Ia1+α1IaIa1+α1Ia)+σ21(SaSa)2+σ21(Sa)2=(μaσ21)(SaSa)2βaSa(SaSa)(Ia1+α1IaIa1+α1Ia)+σ21(Sa)2.

    Let ˉW=W5+βaSaIaμa(1+α1Ia)W4, one can derive that

    LˉW(μaσ21)(SaSa)2βaSa(SaSa)(Ia1+α1IaIa1+α1Ia)+σ21(Sa)2+βaSaIaμa(1+α1Ia)[μa(1+α1Ia)Ia(SaSa)(Ia1+α1IaIa1+α1Ia)+12σ21Sa+1+α1IaβaeμaτaIa(μa+βaIa1+α1Ia)(12σ22Ea+12μa+γaγaσ23Ia)]=(μaσ21)(SaSa)2+σ21(Sa)2+βaSaIa2μa(1+α1Ia)σ21Sa+(eμaτaSa+βaSaIaμaeμaτa(1+α1Ia))(12σ22Ea+12μa+γaγaσ23Ia). (5.10)

    Integrating both sides of (5.10) from 0 to t and then taking expectation yields

    EˉW(t)EˉW(0)(μaσ21)Et0(Sa(s)Sa)2ds+[σ21(Sa)2+βaSaIa2μa(1+α1Ia)σ21Sa+(eμaτaSa+βaSaIaμaeμaτa(1+α1Ia))(12σ22Ea+12μa+γaγaσ23Ia)]t.

    Then, we can get

    lim supt1tEt0(Sa(s)Sa)2dsP3,

    where P3 is defined in Theorem 3. Defining W6=12[eμaτa(SaSa)+Ea(t+τa)Ea]2, the use of Itô's formula yields that

    LW6=μae2μaτa(SaSa)2(μa+γa)(Ea(t+τa)Ea)2(2μa+γa)eμaτa(SaSa)(Ea(t+τa)Ea)+12e2μaτaσ21S2a+12σ22E2a(t+τa)μae2μaτa(SaSa)2(μa+γa)(Ea(t+τa)Ea)2+μa+γa2(Ea(t+τa)Ea)2+(2μa+γa)2e2μaτa2(μa+γa)(SaSa)2+e2μaτaσ21(SaSa)2+e2μaτaσ21(Sa)2+σ22(Ea(t+τa)Ea)2+σ22(Ea)2=e2μaτa(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)(SaSa)2(μa+γa2σ22)(Ea(t+τa)Ea)2+e2μaτaσ21(Sa)2+σ22(Ea)2.

    Let W7=W6+(μa+γa2σ22)t+τat(Ea(s)Ea)2ds and W8=12(IaIa)2. We have

    LW7e2μaτa(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)(SaSa)2(μa+γa2σ22)(EaEa)2+e2μaτaσ21(Sa)2+σ22(Ea)2, (5.11)

    and

    LW8=(IaIa)(γaEa(μa+δa)Ia)+12σ23I2a=γa(EaEa)(IaIa)(μa+δa)(IaIa)2+12σ23I2aμa+δa2(IaIa)2+γ2a2(μa+δa)(EaEa)2(μa+δa)(IaIa)2+σ23(IaIa)2+σ23(Ia)2=γ2a2(μa+δa)(EaEa)2(μa+δa2σ23)(IaIa)2+σ23(Ia)2. (5.12)

    Let ˜W=W7+eμaτaμaσ21(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)ˉW+(μa+δa)(μa+γa2σ22)2γ2aW8. Making use of (5.10), (5.11) and (5.12) yields that

    L˜W=LW7+eμaτaμaσ21(2μ2a+2μaγa+γ2a2(μa+γa)+σ21)LˉW+(μa+δa)(μa+γa2σ22)2γ2aLW814(μa+γa2σ22)(EaEa)2(μa+δa)(μa+γa2σ22)(μa+δa2σ23)4γ2a(IaIa)2+L1. (5.13)

    Integrating both sides of (5.13) from 0 to t and then taking expectation yields

    E˜W(t)E˜W(0)14(μa+γa2σ22)Et0(Ea(s)Ea)2ds(μa+δa)(μa+γa2σ22)(μa+δa2σ23)4γ2aEt0(Ia(s)Ia)2ds+L1t.

    Therefore, we can obtain

    lim suptEt0[(Ea(s)Ea)2+(Ia(s)Ia)2]dsL1L2=:P4,

    where L1,L2 have been defined in Theorem 3. Taking U1=12(ShSh)2, we have

    LU1=(ShSh)(ΛhμhShβhShIa1+α2Ia)+12σ24Sh=(ShSh)[μhShμhSh+βhShIa1+α2IaβhShIa1+α2Ia]+12σ24Sh=(μh+βhIa1+α2Ia)(ShSh)2+σ24(ShSh)2+σ24(Sh)2(μhσ24)(ShSh)2+σ24(Sh)2. (5.14)

    Integrating both sides of (5.14) from 0 to t and then taking expectation, we get

    EU1(t)EU1(0)(μhσ24)Et0(ShSh)2ds+σ24(Sh)2t.

    Therefore, we can obtain

    lim suptEt0(ShSh)2dsσ24(Sh)2μhσ24.

    Let U2=12[Ih(t+τh)Ih]2, we have

    LU2=(Ih(t+τh)Ih)[βhShIa1+α2Ia(μh+δh+θh)Ih(t+τh)]+12σ25I2h(t+τh)=βhIa1+α2Ia(Ih(t+τh)Ih)(ShSh)(μh+δh+θh)(Ih(t+τh)Ih)2+12σ25I2h(t+τh)β2h(Ia)22(1+α2Ia)2(μh+δh+θhσ25)(ShSh)2μh+δh+θhσ252(Ih(t+τh)Ih)2(μh+δh+θh)(Ih(t+τh)Ih)2+σ25(Ih(t+τh)Ih)2+σ25(Ih)2=β2h(Ia)2(ShSh)22(1+α2Ia)2(μh+δh+θhσ25)μh+δh+θhσ252(Ih(t+τh)Ih)2+σ25(Ih)2.

    Let ¯U=β2h(Ia)22(μhσ24)(1+α2Ia)2(μh+δh+θhσ25)U1+U2, then

    L¯U=μh+δh+θhσ252(Ih(t+τh)Ih)2+β2h(Ia)2σ24(Sh)22(μhσ24)(1+α2Ia)2(μh+δh+θhσ25)+σ25(Ih)2.

    Let U3=μh+δh+θhσ252t+τht(Ih(s)Ih)2ds, we obtain

    LU3=μh+δh+θhσ252[(Ih(t+τh)Ih)2+(IhIh)2].

    Let ˜U=¯U+U3, then,

    L˜U=μh+δh+θhσ252(Ih(t+τh)Ih)2+β2h(Ia)2σ24(Sh)22(μhσ24)(1+α2Ia)2(μh+δh+θhσ25)+σ25(Ih)2+μh+δh+θhσ252[(Ih(t+τh)Ih)2+(IhIh)2]=μh+δh+θhσ252(IhIh)2+β2h(Ia)2σ24(Sh)22(μhσ24)(1+α2Ia)2(μh+δh+θhσ25)+σ25(Ih)2. (5.15)

    Integrating both sides of (5.15) from 0 to t and then taking expectation, we have

    E˜U(t)E˜U(0)μh+δh+θhσ252Et0(IhIh)2ds+β2h(Ia)2σ24(Sh)22(μhσ24)(1+α2Ia)2(μh+δh+θhσ25)t+σ25(Ih)2t.

    Therefore, we can obtain

    lim suptEt0(IhIh)2dsP6,

    where P6 has been defined in Theorem 3. The proof is completed.

    This section is devoted to illustrating the theoretical results by numerical examples. The parameters of system (2.3) are selected as in Table 2, α1 and α2 are varying parameters that is taken value from 0.001 to 0.1, and σ1=0.01,σ2=σ3=σ5=0.04,σ4=0.008. The initial conditions of system (2.3) are Sa(θ)=3,000,000,Ea(θ)=1,000,Ia(θ)=10,Sh(θ)=1,000,Ih(θ)=5,θ[τ,0]. The Milstein method [25] is used to obtain the discrete form of system (2.3) as follows:

    {Sa(k+1)=Sa(k)+(ΛaμaSa(k)βaSa(k)Ia(k)1+α1Ia(k))Δt+σ1Sa(k)Δtξ1(k)+12σ21Sa(k)(ξ21(k)1)Δt,Ea(k+1)=Ea(k)+(βaeμaτaSa(kτaΔt)Ia(kτaΔt)1+α1Ia(kτaΔt)(μa+γa)Ea(k))Δt+σ2Ea(k)Δtξ2(k)+12σ22Ea(k)(ξ22(k)1)Δt,Ia(k+1)=Ia(k)+(γaEa(k)(μa+δa)Ia(k))Δt+σ3Ia(k)Δtξ3(k)+12σ23Ia(k)(ξ23(k)1)Δt,Sh(k+1)=Sh(k)+(ΛhμhSh(k)βhSh(k)Ia(k)1+α2Ia(k))Δt+σ4Sh(k)Δtξ4(k)+12σ24Sh(k)(ξ24(k)1)Δt,Ih(k+1)=Ih(k)+(βheμhτhSh(kτhΔt)Ia(kτhΔt)1+α2Ia(kτhΔt)(μh+δh+θh)Ih(k))Δt+σ5Ih(k)Δtξ5(k)+12σ25Ih(k)(ξ25(k)1)Δt, (6.1)
    Table 2.  Parameter values used in numerical simulations for model (2.3).
    Parameter Value Source of data
    Λa 30000 Assumed
    Λh μh×1000 Assumed
    βa (0.5---12.5)×106day1 [10]
    βh 3×104 [10]
    μa 1/100day1 [10]
    μh 200/(70×365)day1 Assumed
    δa 5day1 [10]
    δh 0.03day1 [10,11]
    γa 0.3day1 [11]
    θh 0.16day1 [11]
    τa 7 day Assumed
    τh 14 day Assumed

     | Show Table
    DownLoad: CSV

    where ξi(k)N(0,1)(i=1,,5;k=1,2,) are independent Gaussian random variables. Initially, we study the effect of R0, which, by Theorems 4.1 and 5.1, can govern the asymptotic behavior.

    Example 1 Effect of basic reproduction number R0.

    Choose different βa such that R0 take different values, which are shown in Table 3. Since σ21=104<μa=102,σ22=0.0016<12(μa+γa)=0.155,σ23=0.0016<12(μa+δa)=5.01,σ24=0.000064<μh=0.0078,σ25=0.0016<μh+δh+θh=0.1978, the condition (ⅰ) of Theorem 3 is satisfied. From Table 3, we see that for each R0, the inequality Pm<dE holds, which means the condition (ⅱ) of Theorem 3 is also satisfied. Thus, all the conclusions of Theorem 3 hold. It follows from Table 3 that the change of R0 can result in different values of E, which also illustrate the value of E is related to R0. By the discrete form of system (2.3), the numerical results under different R0 are presented by Figures 2 and 3 when R0>1, which show that the solution of system (2.3) goes around the endemic equilibrium E. The effectiveness of Theorem 3 is also indicated by these two figures. In addition, we can see from Figures 2 and 3 and Table 3 that the number of infected poultry and humans will reduce with the decrease of R0. On the other hand, in order to explore if the results of Theorem 3 hold, we enhance the intensity of perturbation as σ=(σ1,,σ5)=(0.02,0.08,0.08,0.016,0.08) (Case Ⅰ: condition (ⅰ) of Theorem 3 is satisfied but condition (ⅱ) is not satisfied), σ=(0.06,0.24,0.24,0.048,0.24) (Case Ⅱ: Both conditions (ⅰ) and (ⅱ) are not satisfied) and σ=(0.10,0.40,0.40,0.080,0.40) (Case Ⅲ: Both conditions (ⅰ) and (ⅱ) are not satisfied). The simulation results are presented in Figure 4, which are obtained by computing the average of 800 simulations. The equilibrium of corresponding deterministic model is E=(2.3931×106,1.8254×104,0.7812×103,227.3975,11.5855). From Figure 4, we see that the curves will move away from the equilibrium point E with the increasing of intensity of perturbation, which violate the conclusions of Theorem 3.

    Table 3.  Value of E(Sa,Ea,Ia,Sh,Ih) under different R0.
    R0 Sa(×106) Ea(×104) Ia(×103) Sh Ih Pm(×105) dE(×105)
    4.8269 0.7977 6.6239 2.8348 212.6795 11.8062 2.5226 22.033
    3.2823 1.1336 5.6137 2.4024 213.7049 11.7908 2.7378 18.673
    1.9308 1.7948 3.6249 1.5513 217.3720 11.7359 3.2294 12.057
    1.3515 2.3931 1.8254 0.7812 227.3975 11.5855 3.7266 6.072
    Pm=max(P3,P4,P5,P6), dE=d(E,E0).

     | Show Table
    DownLoad: CSV
    Figure 2.  The behavior of avian population under different R0>1.
    Figure 3.  The behavior of human population under different R0>1.
    Figure 4.  The trajectories of Ea,Ia and Ih for large perturbation when R0=1.3515>1.

    According to the values of σ1,,σ5 and the parameters values in Table 2, we easily verify the conditions of Theorem 2 are satisfied. Therefore, from Theorem 2 we know that the solution of system (2.3) will go around the disease-free equilibrium E0 when R0<1. The numerical simulation results of R0 are presented in Figures 5 and 6. These figures show Ea,Ia and Ih all go to zero when R0<1, which illustrate the effectiveness of the theoretical results in Theorem 2. Meanwhile, Figures 5 and 6 also show that the rate of Ea,Ia and Ih converges to zero is increasing with the decrease of R0. The conditions of Theorem 2 are only a sufficient ones, so we want to know whether the conclusions of Theorem 2 hold when the intensity of perturbation increase such that these conditions are not satisfied. Thus, we choose σ=(σ1,,σ5)=(0.02,0.08,0.08,0.016,0.08) (Case I), σ=(0.06,0.24,0.24,0.048,0.24) (Case II) and σ=(0.12,0.48,0.48,0.096,0.48) (Case III), and the simulation results are presented in Figure 7. Figure 7 shows Ea,Ia and Ih converge to zero for each cases, so the results of Theorem 2 also hold.

    Figure 5.  The behavior of infected avian population under different R0<1.
    Figure 6.  The behavior of infected human population under different R0<1.
    Figure 7.  The trajectories of Ea,Ia and Ih for large perturbation when R0=0.7723<1.

    Example 2 Effect of time delays τa and τh.

    In order to study the effect of time delays, we consider the average peak values of Ea, Ia and Ih, and the time of reaching average peak values by 300 simulation runs. The simulation results are shown in Figures 8 and 9. It follows from Figure 8 that the increase of time delay τa or τh can reduce the peak value of both infected poultry and human population. Meanwhile, from Figure 9, we know that the large time delay also lead to the delay of reaching peak value. Thus, we may conclude that time delays have significate influence for the spread of avian influenza. According to the practical meaning of τa and τh, related department can adopt some measures to increase the spread delay to suppress the outbreak of influenza, such as isolation. In addition, the adopting of those control measures will win time for taking drug control.

    Figure 8.  The average peak values of Ea, Ia and Ih under different τa and τh by 300 simulation runs.
    Figure 9.  The time of reaching average peak values of Ea, Ia and Ih by 300 simulation runs.

    Example 3 Effect of saturation constants α1 and α2.

    According to the analysis of Introduction, the saturation constants α1 and α2 are important parameters for avian influenza. We thus explore the effects of α1 and α2 in this example. In order to explore the effect of α1 under fixed α2, we run 1000 simulations and take their average values. The results are shown in Figure 10. It follows from Figure 10 that α1 can influence the rate of convergence to the equilibria of the poultry population, while it can not significantly influence the rate of convergence to the equilibria of the human population. In addition, we study the influence of α2 under fixed α1. The simulation results are presented in Figure 11, which implies that α2 can not change the rate of convergence to the equilibria of the poultry population. Figure 11 also means that α2 can not increase the rate of convergence to the equilibria of the human population, but it can evidently reduce the peak value of Ih(t). In summary, α1 and α2 have evidently influence to the spreading of avian influenza among both avian and human population.

    Figure 10.  The effect of α1 and α2 by 1000 simulation runs (for fixed α2).
    Figure 11.  The effect of α1 and α2 by 1000 simulation runs (for fixed α1).

    In this paper, we establish a stochastic delayed avian influenza model with saturated incidence rate. To begin with, we investigate the existence and uniqueness of the global positive solution to the system (2.3) with any positive initial value (2.4). Since there is no equilibrium point in the system (2.3) at this time, thus, the asymptotic behaviors of the disease-free equilibrium and the endemic equilibrium are given by constructing some suitable Lyapunov functions and applying the Young's inequality and Hölder's inequality. Theorem 2 shows that if R0<1, then the solution of system (2.3) is going around E0 while from Theorem 3, we obtain that if R0>1, then the solution of system (2.3) is going around E. Finally, some numerical examples are given to illustrate the accuracy of the theoretical results.

    There are some interesting issues deserve further investigations. On the one hand, we can formulate some more realistic but complex avian influenza models, such as considering the effects of Lévy jumps or impulsive perturbations on system (2.3). On the other hand, the coefficients in our model studied in this paper are all constants. If the coefficients are with Markov switching, how will the properties change? We leave these investigations as our future work.

    The research was supported by the National Natural Science Foundation of China (11661064), Ningxia Natural Science Foundation Project (2019AAC03069) and the Funds for Improving the International Education Capacity of Ningxia University (030900001921).

    The authors declare that they have no conflict of interest.



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