Research article

Global dynamics of an impulsive vector-borne disease model with time delays


  • Received: 14 October 2023 Revised: 12 November 2023 Accepted: 14 November 2023 Published: 21 November 2023
  • In this paper, we investigate a time-delayed vector-borne disease model with impulsive culling of the vector. The basic reproduction number R0 of our model is first introduced by the theory recently established in [1]. Then the threshold dynamics in terms of R0 are further developed. In particular, we show that if R0<1, then the disease will go extinct; if R0>1, then the disease will persist. The main mathematical approach is based on the uniform persistent theory for discrete-time semiflows on some appropriate Banach space. Finally, we carry out simulations to illustrate the analytic results and test the parametric sensitivity on R0.

    Citation: Rong Ming, Xiao Yu. Global dynamics of an impulsive vector-borne disease model with time delays[J]. Mathematical Biosciences and Engineering, 2023, 20(12): 20939-20958. doi: 10.3934/mbe.2023926

    Related Papers:

    [1] Rundong Zhao, Qiming Liu, Huazong Zhang . Dynamical behaviors of a vector-borne diseases model with two time delays on bipartite networks. Mathematical Biosciences and Engineering, 2021, 18(4): 3073-3091. doi: 10.3934/mbe.2021154
    [2] Yan-Xia Dang, Zhi-Peng Qiu, Xue-Zhi Li, Maia Martcheva . Global dynamics of a vector-host epidemic model with age of infection. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1159-1186. doi: 10.3934/mbe.2017060
    [3] Xia Wang, Yuming Chen . An age-structured vector-borne disease model with horizontal transmission in the host. Mathematical Biosciences and Engineering, 2018, 15(5): 1099-1116. doi: 10.3934/mbe.2018049
    [4] Hongyong Zhao, Yangyang Shi, Xuebing Zhang . Dynamic analysis of a malaria reaction-diffusion model with periodic delays and vector bias. Mathematical Biosciences and Engineering, 2022, 19(3): 2538-2574. doi: 10.3934/mbe.2022117
    [5] Fahad Al Basir, Yasuhiro Takeuchi, Santanu Ray . Dynamics of a delayed plant disease model with Beddington-DeAngelis disease transmission. Mathematical Biosciences and Engineering, 2021, 18(1): 583-599. doi: 10.3934/mbe.2021032
    [6] Rocio Caja Rivera, Shakir Bilal, Edwin Michael . The relation between host competence and vector-feeding preference in a multi-host model: Chagas and Cutaneous Leishmaniasis. Mathematical Biosciences and Engineering, 2020, 17(5): 5561-5583. doi: 10.3934/mbe.2020299
    [7] Xinli Hu, Yansheng Liu, Jianhong Wu . Culling structured hosts to eradicate vector-borne diseases. Mathematical Biosciences and Engineering, 2009, 6(2): 301-319. doi: 10.3934/mbe.2009.6.301
    [8] Ling Xue, Caterina Scoglio . Network-level reproduction number and extinction threshold for vector-borne diseases. Mathematical Biosciences and Engineering, 2015, 12(3): 565-584. doi: 10.3934/mbe.2015.12.565
    [9] Hai-Feng Huo, Fan Wu, Hong Xiang . On threshold dynamics for periodic and time-delayed impulsive systems and application to a periodic disease model. Mathematical Biosciences and Engineering, 2022, 19(1): 836-854. doi: 10.3934/mbe.2022038
    [10] Lin Zhao, Zhi-Cheng Wang, Liang Zhang . Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1535-1563. doi: 10.3934/mbe.2017080
  • In this paper, we investigate a time-delayed vector-borne disease model with impulsive culling of the vector. The basic reproduction number R0 of our model is first introduced by the theory recently established in [1]. Then the threshold dynamics in terms of R0 are further developed. In particular, we show that if R0<1, then the disease will go extinct; if R0>1, then the disease will persist. The main mathematical approach is based on the uniform persistent theory for discrete-time semiflows on some appropriate Banach space. Finally, we carry out simulations to illustrate the analytic results and test the parametric sensitivity on R0.



    Vector-borne disease involves a vector (an organism) that transmits infectious pathogens from the infected host to the uninfected host. For example, the malaria parasites are transmitted through the bite of infected female mosquitoes, spreading the disease from mosquitoes to humans and back to mosquitoes [2]. West Nile virus (WNv) is another vector-borne disease caused by Flavivirus and transmitted primarily by mosquitoes (Culex species) to vertebrate hosts, such as humans, birds and horses [3]. Those vector-borne diseases have caused a significant threat to public health as well as wildlife worldwide.

    Mathematical models give insights to predict the spread of vector-borne diseases, and to test control strategies. Impulsive (delayed) differential systems are widely used in population biology and epidemiology to study the various factors: birth pulses [4,5], effectiveness of vaccination [6,7,8,9,10], and elimination of vector insects [11,12], see also [13,14] for stochastic effects. To investigate the control strategies against vector-borne diseases, Gourley et al. [11] used two time-delayed models with impulses to evaluate the effectiveness of age-structured culling strategies. Yang et al. [10] studied the effect of impulsive controls in multiple patch models with age-structure. Xu and Xiao [9] considered the impacts of periodic impulsive culling of mosquitoes for control of WNv transmission. Based on [9], the incubation period of mosquitoes was furthered incorporated in [15].

    To describe the disease transmission between vector and host as well as the strategy of periodically culling the vector, we use a set of impulsive delayed differential equations to incorporate the extrinsic incubation period (EIP) of vector and host. By the theory recently developed in [1], the basic reproduction ratio R0 of the model is first established. The global dynamics of the model are further investigated in terms of R0. The uniform persistence theory based on dynamical systems is used extensively in population biology and epidemiology models, see for instance [16,17,18,19] and references therein. However, it seems like this dynamical system approach is rarely adapted to time-delayed impulsive systems to study the threshold dynamics. There are two possible issues: first, it is not an easy task to find an appropriate phase space for time-delayed impulsive systems so that the evolution operators are well-defined, which was recently solved by [1] for a large class of time-delayed impulsive systems; second, the solutions of the time-delayed impulsive systems may not be continuous in t, and hence, such systems might not generate continuous periodic semiflows (see [20]), which brings trouble for direct application of the uniform persistence theory in [21] to investigate the sharp permanence of infectious compartments when R0>1. Meanwhile, the analytic method developed in [1,6,7] for the uniform persistence of a time-delayed impulsive system is quite useful for a form like Susceptible-Exposed-Infected-Removed (SEIR), but it would not be easy to apply to the vector-borne model (including multiple infected compartments). We refer to [9,15] for attempts in this direction. Combing the ideas in [1] and [21], we establish the sharp persistence of infectious compartments, by using the uniform persistence theory of discrete-time semiflows on some appropriate phase space. It is worth pointing out that our current approach would be easily applied to the models where the period of model coefficients is the same as that of fixed impulsive moments.

    The rest of this paper is organized as follows: In Section 2, we formulate a time-delayed impulsive differential model including several factors: EIP and the strategy of periodically culling the vector, EIP and vertical transmission of host. In Section 3, we first introduce the basic reproduction number R0 for the model and then study the threshold dynamics in terms of R0. In Section 4, we use numerical simulations to test the differences with and without culling, and further explore the impacts of parameters such as time delays and vertical transmission rate on R0. A brief discussion then concludes the paper.

    Inspired by the WNv infection process in [9,15,18], we consider an impulsive delay differential equation model to describe the periodic culling of the vector with two differnt EIPs. Let Sv(t), Iv(t), Sh(t), Eh(t) and Ih(t) be the the total numbers of the susceptible adult vector, infected adult vector, susceptible host, exposed host and infected host at time t, respectively. The total number of the adult vector is given by Nh(t)=Sh(t)+Eh(t)+Ih(t). Let Λv and Λh be the recruitment rate of the vector and (susceptible) host, respectively, μv and μh be the natural death rate of vector and host, and dh be the disease-induced death rate of the host. Similar to [7], we suppose that the birth rate of the vector equals its natural death rate. Vertical transmission of the virus in the host population is incorporated by a fraction rvt(0,1). For the virus transmission, we assume that the susceptible host becomes infectious given contact with the infected vector, and the susceptible vector can receive the infection by biting a virus-carrying host. By similar arguments to those in [2,17,19], we simply assume that the numbers of newly occurred infectious host and newly occurred infected vector per unit time at time t are given by

    bβIh(t)Nh(t)Sv(t) and cβSh(t)Nh(t)Iv(t),

    respectively, where β is the average biting rate of the vector, b and c are the transmission probabilities of the virus from vector to host and from host to vector, respectively. However, the newly infected vector and infected host need to survive the EIP to become infectious. We denote the finite constants τ1 and τ2 to represent the length of the EIP in the vector and host, respectively. The probability that the vector and host survive the EIP is eμvτ1 and eμhτ2, then of those vectors and hosts infected τ1 and τ2 unit times ago, only the proportions

    bβIh(tτ1)Nh(tτ1)Sv(tτ1)eμvτ1 and cβSh(tτ2)Nh(tτ2)Iv(tτ2)eμhτ2

    are infectious at time t. Throughout this paper, we suppose that culling occurs at certain particular times, and denote T to be the period of culling. We also assume that spraying reduces both the susceptible and infected vector, p[0,+) is the culling intensity of those killed, where p=0 means that there is no culling.

    The transmission of cross-infection between vector and host is shown in Figure 1. Consequently, our model takes the form:

    {dSv(t)dt=ΛvbβIh(t)Nh(t)Sv(t)μvSv(t),dIv(t)dt=bβIh(tτ1)Nh(tτ1)Sv(tτ1)eμvτ1μvIv(t),dSh(t)dt=ΛhcβSh(t)Nh(t)Ih(t)μhSh(t)+(1rvt)μhIh(t),dEh(t)dt=cβSh(t)Nh(t)Ih(t)cβSh(tτ2)Nh(tτ2)Iv(tτ2)eμhτ2μhEh(t),dIh(t)dt=cβSh(tτ2)Nh(tτ2)Iv(tτ2)eμhτ2(μh+dhrvtμh)Ih(t),}a.e.tnT,nN,Sv(t+)=11+pSv(t),Iv(t+)=11+pIv(t),Sh(t+)=Sh(t),Eh(t+)=Eh(t),Ih(t+)=Ih(t),}t=nT,nN. (2.1)

    All the parameters in model (2.1) are positive. In view of the biological meaning of τ2, we impose the following compatibility condition:

    ϕ4(0)=0τ2cβϕ2(s)ϕ3(s)5i=3ϕi(s)eμhsds. (2.2)
    Figure 1.  Chain of infection.

    To investigate the long time behavior of system (2.1) from the point view of dynamical systems, we first introduce a few notations related to the phase space (see also [1,22]). Given two constants a<b, let

    PC([a,b],Rm):={ϕ:[a,b]Rm|ϕ(t)=ϕ(t),t(a,b],ϕ(t+) exists for t[a,b) and ϕ(t+)=ϕ(t) for all but at most a finite number of points t[a,b)},PC([a,b),Rm):={ϕ:[a,b)Rm|ϕ(t)=ϕ(t),t(a,b),ϕ(t+) exists for t[a,b) and ϕ(t+)=ϕ(t) for all but at most a finite number of points t[a,b)},

    where ϕ(t+)=limst+ϕ(s) and ϕ(t)=limstϕ(s). Then for any r>0, PC([r,0],Rm) is a Banach space with the norm r given by ϕr=suprθ0|ϕ(θ)|. Set ˆτ=max{τ1,τ2}>0, ˜J=[ˆτ,0){tn=nT:nZ}(possibly empty) and J=[ˆτ,0)˜J. Denote

    PCJ:={ϕ:[ˆτ,0]R|ϕ(t)=ϕ(t),t(ˆτ,0],ϕ(t+) exists for t[ˆτ,0) and ϕ(t+)=ϕ(t) for tJ},

    and PC+J:={ϕPCJ:ϕ(t)0,t[ˆτ,0]}. It then follows that PCJ is a Banach space endowed with the norm ˆτ, and PC+J is a closed cone of PCJ, which induces a partial ordering on PCJ. Clearly, PCJPC([ˆτ,0],R). For any u=(u1,u2,...,u5)PC([ˆτ,η),R5) with η>0, define utPC([ˆτ,0],R5) by

    ut(θ)=(u1(t+θ),u2(t+θ),...,u5(t+θ)),θ[ˆτ,0],t[0,η).

    Define

    X:={ϕ=(ϕ1,ϕ2,...,ϕ5)PC+J×PC+J×PC+J×PC+J×PC+J:5i=3ϕi(θ)>0,θ[ˆτ,0]},

    and

    Dϵ:={ϕX:5i=3ϕi(θ)>ϵ,θ[ˆτ,0],ϕ4(0)=0τ2cβϕ2(s)ϕ3(s)5i=3ϕi(s)eμhsds} (2.3)

    for any ϵ(0,Λhμh+dh). Now we are ready to state the well-posed result on the solutions of system (2.1).

    Theorem 2.1. For any ϕDϵ, system (2.1) exists a unique non-negative solution u(t,ϕ) on [0,+) with the initial value u0=ϕ, unTDϵ for any nN. Moreover, ut+T(ϕ)=ut(uT(ϕ)) for all t0 and Φ:=uT admits a global attractor in Dϵ.

    Proof. For convenience, we denote

    (u1(t),u2(t),u3(t),u4(t),u5(t))=(Sv(t),Iv(t),Sh(t),Eh(t),Ih(t)).

    For any t0,tnT,nN,ϕDϵ, define

    f(t,ϕ):=(f1(ϕ),f2(ϕ),f3(ϕ),f4(ϕ),f5(ϕ)),

    where

    f1(ϕ)=Λvbβϕ5(0)5i=3ϕi(0)ϕ1(0)μvϕ1(0),f2(ϕ)=bβϕ5(τ1)5i=3ϕi(τ1)ϕ1(τ1)eμvτ1μvϕ2(0),f3(ϕ)=Λhcβϕ3(0)5i=3ϕi(0)ϕ3(0)μhϕ5(0)+(1rvt)μhϕ5(0),f4(ϕ)=cβϕ3(0)5i=3ϕi(0)ϕ5(0)cβϕ3(τ2)5i=3ϕi(τ2)ϕ5(τ2)eμhτ2μhϕ4(0),f5(ϕ)=cβϕ3(τ2)5i=3ϕi(τ2)ϕ5(τ2)eμhτ2(μh+dhrvtμh)ϕ5(0).

    It is easy to check that f(t,ϕ) satisfies Lipschitz condition for ϕ on any compact subset of R+×X. It follows from [22, Theorems 3.1 and 3.3] that system (2.1) admits a unique solution u(t,ϕ) satisfying 5i=3ui(t,ϕ)>ϵ on the maximal existence interval [0,tϕ) with u0=ϕDϵ and tϕ(0,).

    Next we prove the solution of system (2.1) is non-negative on [0,tϕ). Given ϕDϵ and t[0,T), we see that fi(t,ϕ):=fi(ϕ)0 provided i{1,2,3,5} and ϕi(0)=0. It follows from [23, Theorem 5.2.1] and its proof that for ui(t,ϕ)0,i{1,2,3,5}, t[0,T)[0,tϕ). If T<tϕ, then by the formulas of system (2.1) at the impulsive points, we have ui(t,ϕ)0,i{1,2,3,5},t[0,T]. It then follows from the method of steps that ui(t,ϕ)0,i{1,2,3,5},t[0,tϕ). Meanwhile, by the uniqueness of solutions of system (2.1) and the compatibility condition (2.2), we obtain that

    u4(t)=ttτ2cβu2(s)u3(s)5i=3ui(s)eμh(ts)ds. (2.4)

    Therefore, u4(t,ϕ)0 for t[0,tϕ).

    Notice that the total host population (Nh(t):=5i=3ui(t)) satisfies

    dNh(t)dt=ΛhμhNh(t)(dhμh)Ih(t)Λh(μh+dh)Nh(t), (2.5)

    which implies Nh(t) is an upper solution of

    x(t)=Λh(μh+dh)x(t). (2.6)

    Denote x(t;Nh(0)) as the unique solution of (2.6) with x(0)=Nh(0), then Nh(t)x(t;Nh(0)) for any t[0,tϕ). This further yields that Nh(t)x(t;Nh(0))ϵ for any t[0,tϕ) provided Nh(0)ϵ with ϵ(0,Λhμh+dh). Furthermore,

    dNh(t)dtΛhmin{μh,dh}Nh(t), (2.7)

    which implies that if ϵ5i=3ϕi(0)Λhmin{μh,dh}, then ϵ5i=3ui(t)Λhmin{μh,dh} for any t[0,tϕ).

    Since ϵNh(t) and u5(t)Nh(t) for any t[0,tϕ), by comparison, we have

    u1(t)Λvμvu1,t>0,u2(t)bβu1(tτ1)eμvτ1μvu2,t>0,

    which yields that,

    lim suptu1(t)Λvμv,lim suptu2(t)bβΛveμvτ1μ2v and lim suptNh(t)Λhmin{μh,dh}.

    Therefore, the solutions of system (2.1) with initial data in Dϵ exist globally on [0,) and are ultimately bounded. By similar argument to that in [1, Lemma 4] (see also [24, Page 192]), we infer that for each n>ˆτT, unT is compact. It then follows from [21, Theorem 1.1.3] that UT:DϵDϵ admits a global attractor. Moreover, since if ut(ϕ) is a solution of (2.1), then ut+T(ϕ) is also a solution (2.1). By the uniqueness of the solution, we then have ut+T(ϕ)=ut(uT(ϕ)).

    In this section, we first introduce the basic reproduction number R0 for system (2.1) and then investigate its threshold dynamics in terms of R0.

    The basic reproduction number R0 is defined as the expected number of secondary infections produced by a typical infectious individual in a completely susceptible population. We will use the theory developed in [1] to give the definition of R0. In order to obtain the disease-free periodic solution of system (2.1), we recall a lemma (see [6, Lemma 1] for detail).

    Lemma 3.1. Consider the following impulsive differential equation:

    {dudt=abu(t),tnT,nN,u(t+)=(1θ)u(t),t=nT,nN, (3.1)

    where a,b>0, θ(0,1). Then system (3.1) admits a unique positive periodic solution ˆue(t):=ab+(uab)eb(tnT), nT<t<(n+1)T, which is globally asymptotically stable. Here u=a(1θ)(1ebT)b(1(1θ)ebT).

    Letting Iv=Ih=0 in system (2.1), we then get the following disease-free system:

    {dSv(t)dt=ΛvμvSv(t),dSh(t)dt=ΛhμhSh(t),}a.e.tnT,nN,Sv(t+)=11+pSv(t),Sh(t+)=Sh(t),}t=nT,nN. (3.2)

    By Lemma 3.1, system (3.2) exists a unique positive T-periodic solution ˆu(t):=(ˆSv(t),ˆSh), with ˆSh=Λhμh and

    ˆSv(t)=(1peμv(tnT)1+peμvT)Λvμv,nT<t(n+1)T,nN, (3.3)

    which is globally asymptotically stable.

    Let E=PC([ˆτ,0],R2), E+={ϕE:ϕ(t)0,t[ˆτ,0]}, and

    XT={v:RR2|v is continuous in (tn,tn+1),tn=nT,nZ,v(tn),v(t+n) exist v(tn)=v(tn),v(t+T)=v(t) for tR, and v(t+n+1)=v(t+n)},

    with the norm vXT=supt[0,T]|v(t)|. It then follows that XT is a Banach space. Linearizing system (2.1) at its disease-free periodic solution E0(t)=(ˆSv(t),0,ˆSh(t),0,0), we obtain the following linear system for the infectious compartments:

    {dIv(t)dt=μvIv(t)+bβeμvτ1μhΛhˆSv(tτ1)Ih(tτ1),dIh(t)dt=(μh+dhrvtμh)Ih(t)+cβeμhτ2Iv(tτ2),}tnT,nN,Iv(t+)=11+pIv(t),Ih(t+)=Ih(t),}t=nT,nN. (3.4)

    Define F(t):ER2, where

    F(t)(ϕ1ϕ2):=(bβeμvτ1μhΛhˆSv(tτ1)ϕ2(τ1)cβeμhτ2ϕ1(τ2)+rvtμhϕ2(0)),tR,ϕ=(ϕ1,ϕ2)E.

    Set Y(t,s),ts as the Cauchy matrix [24, Section 1.2] of

    {du1(t)dt=μvu1(t),du2(t)dt=(μh+dh)u2(t),}tnT,nN,u1(t+)=11+pu1(t),u2(t+)=u2(t),}t=nT,nN, (3.5)

    that is,

    Y(t,s)=eV(ts)k:tk[s,t)diag (11+p,1),

    where each tk denotes the impulsive point on [s,t) and V=(μv00μh+dh). It is easy to check that F(t) and Y(t,s) satisfy the following properties: (a) for each tR, F(t) is a positive operator; (b) the matrix V is cooperative and r(Y(T,0))<1, where r(Y(T,0) is the spectral radius of Y(T,0).

    Consequently, the linear impulse periodic differential system (3.5) could be rewritten as:

    {du(t)dt=F(t)utVu(t), a.e. t>0,ttn,u(t+)=Pu(t),t=tn,nN, (3.6)

    where P= diag (11+p,1).

    After the above settings, we can apply the theory and method in [1] to define basic the reproduction number for system (2.1). Let v(t) be the initial distribution of infected individuals with period T. For ts, then F(ts)vts is the distribution of newly infected individuals at time ts. It is produced by infected individuals introduced during the time interval [tsˆτ,ts], which means it is the distribution of newly infected at time ts and still infected at time t. Define the linear operator L on XT:

    [Lv](t):=0Y(t,ts)F(ts)v(ts+)ds,tR,vXT.

    We define the basic reproduction number as

    R0:=r(L),

    where r(L) is the spectral radius of the linear operator L. Denote

    EJ=PCJ×PCJ,E+J=PC+J×PC+J.

    For any ϕEJ, let w(t,ϕ)=(w1(t,ϕ),w2(t,ϕ)) be the unique solution of (3.4) with the initial value w0=ϕ, and

    wt(ϕ)(θ)=(w1t(ϕ)(θ),w2t(ϕ)(θ))=(w1(t+θ,ϕ),w2(t+θ,ϕ)),θ[ˆτ,0].

    Let Q(t):EJE=PC([ˆτ,0],R2) be the solution map of (3.4), then

    Q(t)ϕ=(w1t(ϕ),w2t(ϕ)),t0,ϕEJ.

    In view of the definition of PCJ, we infer that Q:=Q(T) is the operator from EJ to EJ (in other words, they have the same number of discontinuous points), and set r(Q) to be the spectral radius of Q on EJ. By [1, Theorem 1], we have the following observation:

    Lemma 3.2. R01 has the same sign as r(Q)1.

    Let

    Z:=PCτ2×PCτ1,Z+:=PC+τ2×PC+τ1,

    where PCτi is understood as PCJ with ˆτ replaced by τi, and then (Z,Z+) is an ordered Banach space. For any ψZ+, system (3.4) admits a unique non-negative solution z(t,ψ) with the intial value z0=ψ. For all t0, we have zt(ψ)=(z1t(ψ),Z2t(ψ)), for all θi[τi,0], and then Zit(ψ)(θi)=zi(t+θi,ψ), i=1,2. For any given t0, let ˆQ(t) be the solution map of system (3.4) on Z, that is,

    ˆQ(t)ψ=zt(ψ),ψZ.

    We remark that z1tPC([τ2,0],R) and z2tPCτ1 for all t0 as z2(t,ψ) is continuous for all t0 even if ψ is piecewise continuous. Therefore, ˆQ(T) is a map from Z to Z. Next, we will illustrate that ˆQ(t) is eventually strongly positive on Z+.

    Lemma 3.3. ˆQ(t)φ0 for any t>3ˆτ provided φ=(φ1,φ2)Z+{0}.

    Proof. For convenience, denote

    (z1(t),z2(t)):=(z1(t,φ),z2(t,φ)).

    By the similar argument to that in Theorem 2.1, we obtain that zi(t)0 for any t0, i=1,2.

    In the case that φ1>0. We argue that there exists ˆt[0,ˆτ] such that z2(ˆt)>0. Assume, by contradiction, that z2(t)=0 for all t[0,ˆτ], that is, z2(t)=0 for all t[0,ˆτ]. Then we deduce from the second equation of system (3.4) that:

    z1(tτ2)=0,t[0,ˆτ].

    This yields φ1(t)0 for any t[τ2,0], which is a contradiction. Moreover, from the second equation of system (3.4) and z1(t)0 for all tτ2, we obtain

    z2(t)z2(ˆt)e(μh+dhrvtμh)(tˆt)>0 for all tˆt.

    Then by the integral form of the first equation of system (3.4), we have

    z1(t)=T(t,0)z1(0)+bβeμvτ1μhΛht0T(t,s)ˆSv(sτ1)z2(sτ1)ds,

    where

    T(t,s)=eμv(ts)k:kT[s,t)11+p.

    Since ˆt[0,ˆτ], it follows that z1(t)>0 for all t>2ˆτ. The result as follows.

    In the case that φ2>0. We have

    z1(t)bβeμvτ1μhΛht0T(t,s)ˆSv(sτ1)z2(sτ1)ds>0

    provided t>ˆτ. Then by the integral form of the second equation of system (3.4), we have

    z2(t)=e(μh+dbrvtμh)tz2(0)+cβeμhτ2t0e(μh+dhrvtμh)(ts)z1(sτ2)ds,cβeμhτ2t0e(μh+dbrvtμh)(ts)z1(sτ2)ds>0,

    whenever t2ˆτ. Therefore, ˆQ(t)φ0 for t>3ˆτ.

    Denote r(ˆQ) as the spectral radius of ˆQ:=ˆQ(T). One might perform the same argument as in [16, Lemma 3.8] to obtain r(Q)=r(ˆQ). Moreover, we have the following observation:

    Lemma 3.4. Let μ=lnr(Q)T. Then there exists a positive T-periodic function ˜v(t)=(˜v1(t),˜v2(t)) such that eμtˆv(t) is a solution of system (3.4) with the feasible domain either Z+ or E+J for any t0.

    Proof. Resembling the arguments in [1, Lemma 4], we infer that for each t>ˆτ, Q(t) and ˆQ(t) are compact on EJ and Z, respectively.

    In the case that the feasible domain is Z+, fix an integer n0>0 such that n0T>3ˆτ. It follows that ˆQn0=ˆQ(n0T) is compact and strongly positively on Z. By [25, Lemma 3.1], we obtain that there is a strongly positive eigenvector φ=(φ1,φ2) such that ˆQ(φ)=r(ˆQ)φ. Let v(t,φ) be the solution of system (3.4) with the initial value

    (v1(θ1),v2(θ2))=(φ1(θ1),φ2(θ2)),(θ1,θ2)×[τ2,0]×[τ1,0].

    Since φ0, it is easy to see that vt(φ)0 for any t0. Let ˆv(t)=eμtv(t,φ). Then ˆv(t)0 satisfies

    {dˆv1(t)dt=μvˆv1(t)+bβeμvτ1μhΛhˆSv(tτ1)ˆv2(tτ1)μˆv1,dˆv2(t)dt=cβeμhτ2ˆv1(tτ2)(μh+dhrvtμh)ˆv2(t)μˆv2,}tnT,nN,ˆv1(t+)=11+pˆv1(t),ˆv2(t+)=ˆv2(t),}t=nT,nN. (3.7)

    For θ2[τ1,0] and θ1[τ2,0], we have

    ˆvi(θi)=eμθivi(θi,φ)=eμθiφi(θi),ˆvi(T+θi)=eμ(T+θi)vi(T+θi,φ)=eμ(T+θi)[ˆQ(φ)]i=eμ(T+θi)r(ˆQ)φi(θi)=eμθiφi(θi),

    This implies ˆvi(θi)=ˆvi(T+θi), i=1,2. Therefore, ˆv(t) is a T-periodic solution satisfying ˜vi(θi)=eμθiφi(θi), i=1,2.

    In the case that the feasible domain is E+J, since eμtˆv(t) given in the above is a function for tR. We denote ˆϕ=(^ϕ1,^ϕ2) as:

    ˆϕi(θ)=eμθˆvi(θ),θ[ˆτ,0],i=1,2,ˆϕi(θ+)=eμθˆvi(θ+),whenever θJ.

    By the uniqueness of solutions, we have that w(t,ˆφ):=eμtˆv(t) satisfies (3.4) for all t0 with the initial value w0=ˆϕE+J.

    We are now in a position to prove a threshold-type result of system (2.1) in terms of R0.

    Theorem 3.5. If R0<1 and μhdh, then the disease-free periodic solution E0(t)=(ˆSv(t),0, ˆSh(t),0,0) is globally attractive for system (2.1) with respect to Dϵ.

    Proof. By the first and sixth equations of system (2.1), we have

    {dSv(t)dtΛvμvSv(t),tnT,nN,Sv(t+)=11+pSv(t),t=nT,nN.

    Consider the following auxiliary system

    {dx(t)dt=Λvμvx(t),tnT,nN,x(t+)=11+px(t),t=nT,nN.

    According to Lemma 3.1 and the comparison theorem on impulsive differential equations, we have Sv(t,a)x(t,a) with the same initial condition Sv(0)=x(0)=a>0, t0 and |x(t)ˆSv(t)|0 as t with ˆSv(t) given in (3.3). Then for any δ(0,Λhμh), there exists ˆt1>0 such that

    Sv(t)x(t)ˆSv(t)+δ,tˆt1. (3.8)

    When μhdh, in view of system (2.1), we have

    dNh(t)dt=ΛhμhNh(t)+(μhdh)Ih(t)ΛhμhNh(t).

    It then follows that for the above δ, there exists ˆt2>ˆt1 such that

    Nh(t)Λhμhδ,t>ˆt2.

    Thus, we have

    {dIv(t)dtbβeμvτ1μhΛhμhδ(ˆSv(tτ1)+δ)Ih(tτ1)μvIv(t),dIh(t)dt(μh+dhrvtμh)Ih(t)+cβIv(tτ2),} a.e. tˆt2+ˆτ,tnT,Iv(t+)=11+pIv(t),Ih(t+)=Ih(t),}tˆt2+ˆτ,t=nT.

    We consider the following impulsive differential equations with parameter δ:

    {dy1(t)dt=bβeμvτ1μhΛhμhδ(ˆSv(tτ1)+δ)y2(tτ1)μvy1(t),dy2(t)dt=(μh+dhrvtμh)y2(t)+cβy1(tτ2),}tnT,nN,y1(t+)=11+py1(t),y2(t+)=y2(t),}t=nT,nN. (3.9)

    From Lemma 3.2, we know that R0<1 if and only if r(Q)<1. Let Qδ be the Poincaré map of system (3.9). Observe that limδ0+r(Qδ)=r(Q)<1 (as r(Qδ) is non-decreasing and upper semi-continuous in δ0 but close to 0 (see [26,27] for example), we can fix a small positive number δ such that r(Qδ)<1. By Lemma 3.4, there is a positive T-periodic function ˆvδ(t)=(ˆvδ1(t),ˆvδ2(t)) such that eμδtˆvδ(t) is a positive solution of system (3.9), where μδ=lnr(Qδ)T<0. Choose a positive constant Kδ such that:

    (Iv(t),Ih(t))Kδeμδtˆvδ(t),t[ˆt2,ˆt2+ˆτ].

    Applying the comparison principle for cooperative impulsive delay differential systems [28, Lemma 2.2], we obtain that

    (Iv(t),Ih(t))Kδeμδtˆvδ(t),t[ˆt2,+). (3.10)

    Letting t in (3.10), we have (Iv(t),Ih(t))(0,0). It then follows from the fourth equation or (2.4) that Eh(t)0 as t. In view of the theories of asymptotically periodic semiflows and internally chain transitive sets [21], we further deduce from the third equation of system (2.1) that

    limtSh(t)=Λhμh.

    Then for any given δ1(0,δ), there exists ˆt3>ˆt2+ˆτ such that

    Ih(t)δ1,Sh(t)Λhμhδ1,tˆt3,

    and hence,

    dSv(t)dtΛv(bβμhδ1Λhμhδ1+μv)Sv(t),tˆt3.

    For this inequality, we consider the following impulsive differential equation:

    {dSv(t)dt=Λv(bβμhδ1Λhμhδ1+μv)Sv(t),tnT,nN,Sv(t+)=11+pSv(t),t=nT,nN. (3.11)

    By Lemma 3.1, system (3.11) admits a globally stable T-periodic solution, denoted as ˆSδ1v(t), which depends continuously on δ1. It follows that for the above δ, there exists a ˆt4>ˆt3 such that

    Sv(t)ˆSδ1v(t)δ,tˆt4. (3.12)

    This, together with (3.8), yields for any δ1(0,δ)

    ˆSδ1v(t)δSv(t)ˆSv(t)+δ,tˆt4.

    Letting δ10+, we obtain for any small δ>0, that there holds

    ˆSv(t)δSv(t)ˆSv(t)+δ,tˆt4, (3.13)

    which implies that limt(Sv(t)ˆSv(t))=0.

    The result follows. In the remainder of this section, we investigate the uniform persistence of system (2.1).

    Theorem 3.6. Assume that (Sv(t,ϕ),Iv(t,ϕ),Sh(t,ϕ),Eh(t,ϕ),Ih(t,ϕ)) is the unique solution of system (2.1) through ϕX0:={(ϕ1,ϕ2,ϕ3,ϕ4,ϕ5)Dϵ:ϕ2(0)>0,ϕ5(0)>0}. If R0>1, then there exists ρ>0 such that lim inftIi(t,ϕ)ρ,i=v,h.

    Proof. Let X0:={ϕDϵ:ϕ2(0)=0 or ϕ5(0)=0}. For any ϕX0, it is easy to check that

    Ii(t,ϕ)>0,t>0,i=v,h,

    that is, Φn(X0)X0 for any nN with Φ:=uT given in Theorem 2.1 having a global attractor in Dϵ. Define

    M={ϕX0:Φn(ϕ)X0 for any nN},

    let ω(ψ) be the omega limit set of the forward orbit Γ={Φnψ:nN}, and

    M={(ˆSv(),0,ˆSh,0,0}.

    Claim 1: For any ψM, ω(ψ)=M, which is locally stable in M.

    For any given ψM, it follows from the definition of M that for each nN, there holds Iv(nT,ψ)=0 or Ih(nT,ψ)=0. Consequently, we infer that either Iv(t,ψ)0 or Ih(t,ψ)0 for any t0 (as if there exists t00 such that Ii0(t0,ψ)0 for some i0{v,h}, then Ii0(t,ψ)>0 for all tt0).

    In the case that Iv(t,ψ)0 for each t0, it follows from the second equation of system (2.1) that Ih(tτ1)Sv(tτ1)=0 for any t0. By the first and sixth equations of (2.1), we have

    Sv(t)Λv(μv+bβ)Sv(t),t>0,tnT and Sv(t+)=11+pSv(t),t=nT,nN.

    This implies there exists ˉt1>0, such that Sv(t)>0 for all t>ˉt1+τ1, and hence, Ih(t)=0 and Eh(t)=0 for all t0. Since disease-free system (3.2) admits a globally stable T periodic solution (ˆS(),ˆSh), we obtain that ω(ψ)=M.

    In the case that Ih(t,ψ)0 for each t0, it follows from the fifth equation of system (2.1) that Sh(tτ2)Iv(tτ2)=0 for any t0. By the third equation of (2.1), we have Sh(t)Λh(μv+cβ)Sh(t),t>0. This implies there exists ˉt2>0, such that Sh(t)>0 for all t>ˉt2+τ2, and hence, Iv(t)=0 and Eh(t)=0 for all t0. For the same reason, we obtain that ω(ψ)=M. Therefore, Claim 1 is valid.

    For σ(0,min{mint(0,T]ˆSv(t),ˆSh}), let Qσ(t) be the solution map of the following system and Qσ=Qσ(T) be the associated Poincaré map

    {dIσv(t)dt=bβeμvτ1ˆSv(tτ1)σˆSh+3σIσh(tτ1)μvIσv(t),dIσh(t)dt=cβeμhτ2ˆShσˆSh+3σIσv(tτ2)(μh+dhrvtμh)Iσh(t),}tnT,nN,Iσv(t+)=11+pIσv(t),Iσh(t+)=Iσh(t),}t=nT,nN. (3.14)

    For σ close to 0+, by the comparison principle, it follows that the spectral radius r(Qσ) is non-increasing in σ, which implies limσ0+r(Qσ) exists and limσ0+r(Qσ)r(Q). Note that r(Qσ) is upper semi-continuous in σ0 (see [26,27]), which further yields lim supσ0+r(Qσ)r(Q), and hence, limσ0+r(Qσ)=r(Q)>1. Thus, we can fix a sufficiently small number σ>0 such that r(Qσ0)>1. By Lemma 3.4, there exists a positive T-periodic function ˆvσ=(ˆvσ1,ˆvσ2) such that eμσtˆvσ(t) is a positive solution of system (3.14), where μσ=lnr(Qσ)T>0.

    By the continuous dependence of the solution (see [20]) on the initial value, we have for the above chosen σ>0, and there exists σ(0,σ) such that ut(ϕ)ut(M)<σ for any t[0,T] provided ϕM<σ. Next we prove the following claim:

    Claim 2: For all ϕX0, there holds lim supnΦnϕMσ. Assume, by contradiction, that lim supnΦnˆϕM<σ for some ˆϕX0. Then there exists n11 such that ΦnˆϕM<σ for any nn1. For any tn1T, letting t=nT+t with n=[tT] and t[0,T), by Theorem 2.1, we have

    ut(ˆϕ)M=ut(ˆϕ)ut(M)=ut(Φnˆϕ)ut(M)<σ. (3.15)

    It then follows that for any t>n1Tˆτ, 0<Ii(t)<σ,i=v,h, Sv(t)>ˆSv(t)σ, ˆShσ<Sh(t)Nh(t)<ˆSh+3σ. And hence, Sv(tτ1)Nh(tτ1)ˆSv(tτ1)σˆSh+3σ>0 and Sh(tτ2)Nh(tτ2)ˆShσˆSh+3σ>0 for any t>n1T. By the comparison principle for cooperative systems, it follows that Iv(t,ˆϕ) and Ih(t,ˆϕ) in system (2.1) satisfy

    {dIv(t)dtbβeμvτ1ˆSv(tτ1)σˆSh+3σIh(tτ1)μvIv(t),dIh(t)dtcβeμhτ2ˆShσˆSh+3σIv(tτ2)(μh+dhrvtμh)Ih(t),}t>n1T,tnT,Iv(t+)=11+pIv(t),Ih(t+)=Ih(t),}t=nT,nn1. (3.16)

    Recall that Keμσtˆvσ(t) is a positive solution of (3.14) for any K>0. Choose Kσ>0 such that

    (Iv(t,ˆϕ),Ih(t,ˆϕ))Kσeμσtˆvσ(t),t[n1T,n1T+ˆτ].

    Then the comparison theorem for cooperative impulsive delay differential systems implies that

    (Iv(t,ˆϕ),Ih(t,ˆϕ))Kσeμσtˆvσ(t),tn1T+ˆτ.

    Since μσ>0, we get Iv(t,ˆϕ),Ih(t,ˆϕ) as t, which leads to a contradiction.

    The above claims indicate that M cannot form a cycle for Φ in Dϵ and Ws(M)X0=, where Ws(M) is the stable set of M for Φ. Now we define a continuous distance function p:DϵR+ by p(ϕ)=min{ϕ2(0),ϕ5(0)}. In view of Claim 1, we have Ws(M)p1(0,)=. Now by [21, Lemma 1.2.1 and Theorem 1.3.2] and Theorem 2.1, it then follows that Φ is uniformly persistent with respect to (X0,X0,p), that is, there exists ρ1>0 such that

    lim infnp(Φnϕ)=lim infnmin{Iv(nT,ϕ),Ih(nT,ϕ)}ρ1>0,ϕX0.

    Consequently, there exists n21, such that

    min{Iv(nT),Ih(nT)}ρ12,nn2,nN.

    Now for any tn2T, letting t=nT+ˉt with n=[tT] and ˉt[0,T), we obtain from the equation of Iv that

    Iv(t)μvIv(t),t(nT,(n+1)T].

    Therefore,

    Iv(t)11+pIv(nT)eμvT=ρ1eμvT2(1+p).

    Similarly, we see from Ih(t)(μh+dh)Ih(t),tnT that

    Ih(t)Ih(nT)e(μh+dh)T=ρ1e(μh+dh)T2.

    Now set ρ=min{ρ1eμvT2(1+p),ρ1e(μh+dh)T2}>0, we see that lim inftIi(t)ρ,i=v,h.

    In this section, we perform illustrative numerical simulations to verify theoretical results and explore the influences of key model parameters on the disease transmission. Note that the following simulations were based on the WNv disease transmission process studied in [9,15], where vector and host are mosquito and bird, respectively, and some of parameters were chosen only for the test of parameter sensitivity on the basic reproduction number.

    Below, we take day as the time unit. Following the work in [9], we choose the baseline parameters as Λv=100,Λh=2.1,μv=0.05,μh=0.001,β=0.16,c=0.6,dh=0.005. Since the EIP of vector (mosquito) is about 8–12 days, and the EIP of the host (bird) is about 3–5 days, we set τ1=8,τ2=3 unless stated otherwise.

    Similar to [15], choose T=10,p=0.6 while R0=1.1490. Figure 2(a) shows that the number of infected vectors tends to the periodical oscillation when culling occurs, whereas it tends to a equilibrium without culling. In addition, we see from Figure 2(a) that culling significantly reduces the amount of infections of the vector compared to the case of no culling. Choose T=10,p=2 while R0=0.7142. As shown in the Figure 3(b), the number of infected hosts that are not culled will continue to increase, and after the action of culling, the number of infected hosts will decrease, and the disease will eventually die out.

    Figure 2.  While R0>1, comparison of the long-term behavior of infectious vector (the left plot) and host (the right plot) in different scenarios: culling and without culling.
    Figure 3.  While R0<1, comparison of the long-term behavior of infectious vector (the left plot) and host (the right plot) in different scenarios: culling and without culling.

    Next, we use PRCCs (partial rank correlation coefficients) to obtain the sensitivity analysis of R0. We take Λv,Λh,μv,μh,β,b,c,p,dh and rvt as the input variables and the value of R0 as the output variable. Figure 4 illustrates that parameters Λh,μv,μh,p and dh are negatively correlated with R0 and the others are positively correlated. We also see that R0 is more sensitive to μh,β,b,c and p. Thus, the corresponding control measures should be taken for these sensitive parameters. For instance, to reduce mosquito-bird contact and further control the disease spread, we could burn repellent plants in the bird habitat or at their water sources.

    Figure 4.  Sensitivity analysis of R0.

    In Figure 5, we investigate the joint effects of β,p on R0. It shows that when the biting rate β is large, it requires very strong culling effect. In Figure 6 we also choose different rvt=0.01,r=0.05,r=0.1, while β=0.05, and get R0 increasing in rvt (compare (a)–(c)). Both Figures 5 and 6 indicate that when all parameters are equal, R0 is decreasing in p. All the results are consistent with the conclusion in sensitivity analysis.

    Figure 5.  The contour plot of R0 with respect to p and β with rvt=0.001.
    Figure 6.  The curve of R0 with respect to τ1 with β=0.05. The vertical transmission rate rvt for subplots (a), (b) and (c) is 0.1, 0.2 and 0.3, respectively.

    Finally, we examine the impact of the EIP on R0. We let τ1 vary from 0 to 10 and τ2 vary from 0 to 5. Figure 7 describes the dependence of R0 on τ1 and τ2 for three different frequencies of culling (i.e. 1T). As we see from this figure, R0 is decreasing with respect to τ1 as well as τ2 and it increases as T increases for fixed τ2 and τ1. This suggests that culling during the EIP and prolonging the EIP would be beneficial for disease control, while infrequent culling might be counterproductive.

    Figure 7.  The curve of R0 with respect to τ1 (the left panel) and τ2 (the right panel) for different culling intervals.

    In this paper, we have formulated an impulsive vector-borne disease model with time delays to investigate the joint effects of the EIP and impulsive intervention. The basic reproduction number R0 is first derived by the theory in [1], which serves as a threshold value to determine the extinction and uniform persistence of the disease. Unlike most existing works [1,6,9,15], we utilize the dynamical system approach to show the sharp uniform persistence as R0>1. As is well known, the uniform persistence theory often gives rise to the existence of a positive periodic solution (see [21, Theorem 1.3.10]). As a complement, we could also show the existence of a positive periodic solution in [1,9,15] via our strategy. However, we emphasize that since our feasible domain Dϵ in (2.3) is non-convex, we could not directly utilize it to verify the existence of the positive periodic solution, and we will leave the general existence and its stability for further investigation.

    In addition, there are several possible extensions of this work. For example, EIP is often sensitive to temperature, which would lead to study of the time-varying delay (especially time-periodic delays as in [16]), whose periods might be different from the fixed impulsive moments, and so it requires a more careful and delicate deviation of the theory in [1,21]. At the moment, we are working on a time-periodic impulsive model with constant delays, where the impulsive moments are different from the real-time periods. In a different work, one could elaborate our model by incorporating the stochastic effects, and perform the stability analysis of the model system using some recent results in [29,30,31].

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

    We are grateful to anonymous reviewers for their careful reading and valuable suggestions which led to an improvement of our original manuscript. This research was supported by the NSF of China (Nos. 12001205 and 12371169).

    The authors declare there is no conflict of interest.



    [1] Z. Bai, X. Q. Zhao, Basic reproduction ratios for periodic and time-delayed compartmental models with impulses, J. Math. Biol., 80 (2020), 1095–1117. https://doi.org/10.1007/s00285-019-01452-2 doi: 10.1007/s00285-019-01452-2
    [2] S. Ruan, D. Xiao, J. C. Beier, On the delayed Ross-Macdonald model for malaria transmission, Bull. Math. Biol., 70 (2008), 1098–1114. https://doi.org/10.1007/s11538-007-9292-z doi: 10.1007/s11538-007-9292-z
    [3] D. Nash, F. Mostashari, A. Fine, J. Miller, D. O'Leary, K. Murray, et al., The outbreak of West Nile virus infection in the New York City area in 1999, N. Engl. J. Med., 344 (2001), 1807–1814. https://doi.org/10.1056/NEJM200106143442401 doi: 10.1056/NEJM200106143442401
    [4] S. Tang, L. Chen, Density-dependent birth rate, birth pulses and their population dynamic consequences, J. Math. Biol., 44 (2002), 185–199. https://doi.org/10.1007/s002850100121 doi: 10.1007/s002850100121
    [5] Z. Bai, Y. Lou, X. Q. Zhao, A delayed succession model with diffusion for the impact of diapause on population growth, SIAM J. Appl. Math., 80 (2020), 1493–1519. https://doi.org/10.1137/19M1236448 doi: 10.1137/19M1236448
    [6] S. Gao, L. Chen, Z. Teng, Impulsive vaccination of an SEIRS model with time delay and varying total population size, Bull. Math. Biol., 69 (2007), 731–745. https://doi.org/10.1007/s11538-006-9149-x doi: 10.1007/s11538-006-9149-x
    [7] X. Meng, L. Chen, H. Cheng, Two profitless delays for the SEIRS epidemic disease model with nonlinear incidence and pulse vaccination, Appl. Math. Comput., 186 (2007), 516–529. https://doi.org/10.1016/j.amc.2006.07.124 doi: 10.1016/j.amc.2006.07.124
    [8] Y. P. Yang, Y. Xiao, Threshold dynamics for compartmental epidemic models with impulses, Nonlinear Anal. Real World Appl., 13 (2012), 224–234. https://doi.org/10.1016/j.nonrwa.2011.07.028 doi: 10.1016/j.nonrwa.2011.07.028
    [9] X. Xu, Y. Xiao, R. A. Cheke, Models of impulsive culling of vector to interrupt transmission of West Nile virus to host, Appl. Math. Model., 39 (2015), 3549–3568. https://doi.org/10.1016/j.apm.2014.10.072 doi: 10.1016/j.apm.2014.10.072
    [10] Z. Yang, C. Huang, X. Zou, Effect of impulsive controls in a model system for age-structured population over a patchy environment, J. Math. Biol., 76 (2018), 1387–1419. https://doi.org/10.1007/s00285-017-1172-z doi: 10.1007/s00285-017-1172-z
    [11] S. A. Gourley, R. Liu, J. Wu, Eradicating vector-borne diseases via age-structured culling, J. Math. Biol., 54 (2007), 309–335. https://doi.org/10.1007/s00285-006-0050-x doi: 10.1007/s00285-006-0050-x
    [12] A. J. Terry, Impulsive culling of a structured population on two patches. J. Math. Biol., 61 (2010), 843–875. https://doi.org/10.1007/s00285-009-0325-0 doi: 10.1007/s00285-009-0325-0
    [13] S. P. Rajasekar, M. Pitchaimani, Q. Zhu, Probing a stochastic epidemic hepatitis C virus model with a chronically infected treated population, Acta Math. Sci., 42 (2022), 2087–2112. https://doi.org/10.1007/s10473-022-0521-1 doi: 10.1007/s10473-022-0521-1
    [14] Y. Zhao, L. Wang, Practical exponential stability of impulsive stochastic food chain system with time-varying delays, Mathematics, 11 (2023), 147. https://doi.org/10.3390/math11010147 doi: 10.3390/math11010147
    [15] Y. Han, Z. Bai, Threshold dynamics of a West Nile virus model with impulsive culling and incubation period., Discrete Contin. Dyn. Syst. B, 27 (2022), 4515–4529. https://doi.org/10.3934/dcdsb.2021239 doi: 10.3934/dcdsb.2021239
    [16] Y. Lou, X. Q. Zhao, A theoretical approach to understanding population dynamics with seasonal developmental durations, J. Nonlinear Sci., 27 (2017), 573–603. https://doi.org/10.1007/s00332-016-9344-3 doi: 10.1007/s00332-016-9344-3
    [17] X. Wang, X. Q. Zhao, A periodic vector-bias malaria model with incubation period, SIAM J. Appl. Math., 77 (2017), 181–201. https://doi.org/10.1137/15M1046277 doi: 10.1137/15M1046277
    [18] F. B. Wang, R. Wu, X. Q. Zhao, A West Nile virus transmission model with periodic incubation periods, SIAM J. Appl. Dyn. Syst., 18 (2019), 1498–1535. https://doi.org/10.1137/18M123616 doi: 10.1137/18M123616
    [19] F. B. Wang, R. Wu, X. Yu, Threshold dynamics of a bat-borne rabies model with periodic incubation periods, Nonlinear Anal. Real World Appl., 61 (2021), 103340. https://doi.org/10.1016/j.nonrwa.2021.103340 doi: 10.1016/j.nonrwa.2021.103340
    [20] X. Liu, G. Ballinger, Continuous dependence on initial values for impulsive delay differential equations, Appl. Math. Lett., 17 (2004), 483–490. https://doi.org/10.1016/S0893-9659(04)90094-8 doi: 10.1016/S0893-9659(04)90094-8
    [21] X. Q. Zhao, Dynamical Systems in Population Biology, 2nd edition, CMS Books in Mathematics, Springer, 2017. https://doi.org/10.1007/978-3-319-56433-3
    [22] G. Ballinger, X. Liu, Existence, uniqueness and boundedness results for impulsive delay differential equations, Appl. Anal., 74 (2000), 71–93. https://doi.org/10.1080/00036810008840804 doi: 10.1080/00036810008840804
    [23] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, American Mathematical Society, Providence, RI, 1995. https://doi.org/10.1090/surv/041
    [24] D. Bainov, P. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman Harlow (1993), New York. https://doi.org/10.1201/9780203751206
    [25] X. Liang, X. Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications, Commun. Pure Appl. Math., 60 (2007), 1–40. https://doi.org/10.1002/cpa.20154 doi: 10.1002/cpa.20154
    [26] L. Burlando, Monotonicity of spectral radius for positive operators on ordered Banach spaces, Arch. Math., 56 (1991), 49–57. https://doi.org/10.1007/BF01190081 doi: 10.1007/BF01190081
    [27] T. Kato, Perturbation Theory for Linear Operators, Springer, 1976. https://doi.org/10.1007/978-3-642-66282-9
    [28] X. Liu, X. Shen, Y. Zhang, A comparison principle and stability for large-scale impulsive delay differential systems, ANZIAM J., 47 (2005), 203–235. https://doi.org/10.1017/S1446181100009998 doi: 10.1017/S1446181100009998
    [29] W. Hu, Q. Zhu, H. R. Karimi, Some improved Razumikhin stability criteria for impulsive stochastic delay differential systems, IEEE Trans. Autom. Control, 64 (2019), 5207–5213. https://doi.org/10.1109/TAC.2019.2911182 doi: 10.1109/TAC.2019.2911182
    [30] W. Hu, Q. Zhu, Stability criteria for impulsive stochastic functional differential systems with distributed delay dependent impulsive effects, IEEE Trans. Syst. Man Cybern. Syst., 51 (2021), 2027–2032. https://doi.org/10.1109/TSMC.2019.2905007 doi: 10.1109/TSMC.2019.2905007
    [31] M. Xia, L. Liu, J. Fang, Y. Zhang, Stability analysis for a class of stochastic differential equations with impulses, Mathematics, 11 (2023), 1541. https://doi.org/10.3390/math11061541 doi: 10.3390/math11061541
  • This article has been cited by:

    1. Nabeela Anwar, Aqsa Ghaffar, Muhammad Asif Zahoor Raja, Iftikhar Ahmad, Muhammad Shoaib, Adiqa Kausar Kiani, Stochastic supervised neuro-architecture design for analyzing vector-borne plant virus epidemics with latency and incubation effects, 2024, 139, 2190-5444, 10.1140/epjp/s13360-024-05883-9
  • Reader Comments
  • © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(1326) PDF downloads(73) Cited by(1)

Figures and Tables

Figures(7)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog