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Research article

Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate

  • Received: 12 February 2023 Revised: 24 March 2023 Accepted: 02 April 2023 Published: 23 April 2023
  • A time-delayed model of malaria transmission with asymptomatic infections and standard incidence rate is presented and its basic reproduction number R0 is calculated. We focus on the global dynamics of the model with respect to R0. If and only if R0>1, the model exists a unique malaria-infected equilibrium E, whereas it always possesses the malaria-free equilibrium E0. We first prove the local stability of the equilibria E0 and E by using proof by contradiction and the properties of complex modulus. Secondly, by utilizing the Lyapunov functional method and the limiting system of the model with some novel details, we show that the equilibrium E0 is globally asymptotically stable (GAS) when R0<1, globally attractive (GA) when R0=1 and unstable when R0>1; the equilibrium E is GAS if and only if R0>1. In particular, in order to obtain global attractivity of the equilibrium E, we demonstrate the weak persistence of the system for R0>1. Our results imply that malaria will gradually disappear if R01 and persistently exist if R0>1.

    Citation: Songbai Guo, Xin Yang, Zuohuan Zheng. Global dynamics of a time-delayed malaria model with asymptomatic infections and standard incidence rate[J]. Electronic Research Archive, 2023, 31(6): 3534-3551. doi: 10.3934/era.2023179

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  • A time-delayed model of malaria transmission with asymptomatic infections and standard incidence rate is presented and its basic reproduction number R0 is calculated. We focus on the global dynamics of the model with respect to R0. If and only if R0>1, the model exists a unique malaria-infected equilibrium E, whereas it always possesses the malaria-free equilibrium E0. We first prove the local stability of the equilibria E0 and E by using proof by contradiction and the properties of complex modulus. Secondly, by utilizing the Lyapunov functional method and the limiting system of the model with some novel details, we show that the equilibrium E0 is globally asymptotically stable (GAS) when R0<1, globally attractive (GA) when R0=1 and unstable when R0>1; the equilibrium E is GAS if and only if R0>1. In particular, in order to obtain global attractivity of the equilibrium E, we demonstrate the weak persistence of the system for R0>1. Our results imply that malaria will gradually disappear if R01 and persistently exist if R0>1.



    Malaria is one of the world's most significant infectious diseases [1]. Malaria is a life-threatening disease caused by parasites that is usually transmitted to persons through the bites of female Anopheles mosquitoes [2]. Malaria gives rise to great pressure for the global prevention and control of infectious diseases [3]. World Health Organization reported [2] that there were an estimated 247 million malaria cases, including 619,000 deaths worldwide in 2021, and the majority of cases and deaths occurred in sub-Saharan Africa. The African region accounted for a disproportionate share of the global malaria burdens [4,5]. In 2021, the African region was home to 95% of global malaria cases and 96% of global malaria deaths, and children under 5 years old accounted for about 80% of all malaria deaths there [2]. There are 5 kinds of parasite species that cause malaria in humans, and two of these species P. falciparum and P. vivax pose the greatest threat [6]. The first malaria symptoms such as headache, fever and chills usually appear 10–15 days after the bite of a malaria mosquito and may be mild and difficult to be recognized as malaria, which implies that malaria exists the incubation period [2]. It was reported that asymptomatic infections were more prevalent in sub-Saharan Africa, where an estimated 24 million people had asymptomatic malaria infections [7]. Thus asymptomatic infections can occur during malaria transmission.

    Asymptomatic infected people have no clinical symptoms, but they are contagious and the impact of asymptomatic infections on malaria transmission is enormous [8,9]. Bousema et al. [10] pointed out that asymptomatic carriers contributed to sustained transmission of malaria in local populations, and there was substantial evidence that an increase in the number of asymptomatic carriers at specific time intervals affected the dynamics of malaria transmission. Laishram et al. [4] concluded that asymptomatic malaria infections was a challenge for malaria control programs.

    Since the emergence of malaria, scholars at home and abroad have been studying the pathogenesis and transmission dynamics of malaria. In all research methods, mathematical modeling is undoubtedly one of the most intuitive and effective methods. Many researchers have studied the dynamic evolution of malaria transmission by applying some mathematical models of malaria. In 1911, Ross [11] put forward a basic ordinary differential equations (ODEs) malaria model. Afterwards, MacDonald [12] extended Ross's model, and gave first the definition of the basic reproduction number. The extended Ross's model was said to be the Ross-Macdonald model. Subsequently, the Ross-Macdonald model has been extended to higher dimensions and more factors affecting malaria transmission have been taken into account (see, e.g., [1,6,13]). For example, Kingsolver [14] extended the Ross-Macdonald model and explained the greater attraction of infectious humans to mosquitoes in 1987. Safan and Ghazi [1] developed a 4D ODEs malaria transmission model with standard incidence rates, and analyzed the dynamic properties of equilibria of the malaria model. In 2020, Aguilar and Gutierrez [6] established a high-dimensional ODEs malaria model with asymptomatic carriers and standard incidence rate, and dealt with local dynamics of the disease-free equilibrium of the malaria model.

    Over the years, considering the incubation period of malaria, lots of researchers established some time-delayed malaria models (see, e.g., [3,5,15,16,17,18]). For instance, in 2008, Ruan et al. [5] first established a class of Ross-Macdonald model with two time delays, and investigated the stability of equilibria of the model and the impact of time delays on the basic reproduction number. In 2019, Ding et al. [3] proposed a malaria model with time delay, and investigated the global stability of the uninfected equilibrium of the model as well as its uniform persistence. For the moment, there are few theoretical analysis of the model of malaria with standard incidence rate. Recently, Guo et al. [13] established a malaria transmission model with time delay and standard incidence rate, and they studied the global dynamic properties of equilibria of the model. Based on this, we extend and improve the model in [13], namely, we establish a malaria transmission model with asymptomatic infections, standard incidence rate and time delay, and then study the global dynamic properties of equilibria of the malaria model.

    The remainder of this paper is organized as follows. In Section 2, we put forward a time-delayed dynamic model of malaria with asymptomatic infections and standard incidence rate, and prove the well-posedness as well as dissipativeness of the system. In Section 3, we obtain the existence conditions of malaria-free and malaria-infected equilibria of the system, and verify the local dynamic properties of equilibria in terms of the basic reproduction number R0. In Section 4, to obtain the global dynamic property of the malaria-infected equilibrium for R0>1, we acquire the weak persistence of the system through some analysis techniques. In Section 5, by utilizing the Lyapunov functional method and the limiting system of the model combining stability of partial variables, we obtain the global stability results of malaria-free and malaria-infected equilibria in terms of R0, respectively.

    In order to delve into the details of malaria transmission, we develop a time-delayed model with asymptomatic infections and standard incidence rate. The population is classified into four compartments, which are denoted by Sh: susceptible individuals, Ah: asymptomatic infected individuals, Ih: symptomatic infected individuals, Rh: recovered individuals, respectively. The mosquitoes are classified into two compartments, which are denoted by Sm: susceptible mosquitoes and Im: infected mosquitoes, respectively. Then the model of malaria transmission is proposed as follows:

    {˙Sm(t)=λmβ1Sm(t)Ah(t)Nm(t)β2Sm(t)Ih(t)Nm(t)μmSm(t),˙Im(t)=β1Sm(t)Ah(t)Nm(t)+β2Sm(t)Ih(t)Nm(t)μmIm(t),˙Sh(t)=λhβhSh(t)Im(t)Nm(t)μhSh(t),˙Ah(t)=pβhSh(tτ)Im(tτ)Nm(tτ)(μh+γa)Ah(t),˙Ih(t)=(1p)βhSh(tτ)Im(tτ)Nm(tτ)(μh+γi)Ih(t),˙Rh(t)=γaAh(t)+γiIh(t)μhRh(t), (2.1)

    where Nm(t)=Sm(t)+Im(t). Here, time delay τ0, and all other parameters of system (2.1) are assumed to be positive and p(0,1). The description of parameters are listed in Table 1.

    Table 1.  Descriptions of parameters in the model.
    Parameter Description
    μm The natural death rate of mosquitoes
    μh The natural death rate of humans
    λm The natural birth rate of mosquitoes
    λh The natural birth rate of humans
    β1 The infection rate of susceptible mosquitoes biting asymptomatic individuals
    β2 The infection rate of susceptible mosquitoes biting symptomatic individuals
    βh The infection rate of infected mosquitoes biting susceptible individuals
    τ The incubation period of malaria
    γa The recovery rate of asymptomatic infected individuals
    γi The recovery rate of symptomatic infected individuals
    p The transition probability of asymptomatic infected individuals

     | Show Table
    DownLoad: CSV

    The phase space of system (2.1) is

    C+={ϕ=(ϕ1,ϕ2,ϕ3,ϕ4,ϕ5,ϕ6)TC=C([τ,0],R6+):ϕ1(θ)+ϕ2(θ)>0,θ[τ,0]},

    where C is the Banach space of continuous functions mapping from [τ,0] to R6+ with R+=[0,) and the supremum norm. In the following, the well-posedness as well as dissipativeness of system (2.1) will be investigated in C+.

    Theorem 2.1. The solution u(t)=(Sm(t),Im(t),Sh(t),Ah(t),Ih(t),Rh(t))T of system (2.1) with any ϕC+ exists uniquely, and is non-negative and ultimately bounded on R+. In particular, (Sm(t),Sh(t))T0 on (0,), and C+ is positively invariant for system (2.1).

    Proof. In view of the basic theory of delay differential equations (DDEs) [19,20], the solution u(t) of system (2.1) with any ϕC+ is unique on its maximum interval [0,Tϕ) of existence. Firstly, we will prove that the solution u(t) is non-negative on [0,Tϕ). According to the continuous dependence of solutions of DDEs on parameters [19,20], then for any b(0,Tϕ) and a sufficiently small ε>0, the solution u(t,ε)=(u1(t,ε),u2(t,ε),u3(t,ε),u4(t,ε),u5(t,ε),u6(t,ε))T through ϕ of the following model:

    {˙Sm(t)=λmβ1Sm(t)Ah(t)Nm(t)β2Sm(t)Ih(t)Nm(t)μmSm(t),˙Im(t)=β1Sm(t)Ah(t)Nm(t)+β2Sm(t)Ih(t)Nm(t)μmIm(t)+ε,˙Sh(t)=λhβhSh(t)Im(t)Nm(t)μhSh(t),˙Ah(t)=pβhSh(tτ)Im(tτ)Nm(tτ)(μh+γa)Ah(t)+ε,˙Ih(t)=(1p)βhSh(tτ)Im(tτ)Nm(tτ)(μh+γi)Ih(t)+ε,˙Rh(t)=γaAh(t)+γiIh(t)μhRh(t)+ε, (2.2)

    uniformly exists on [0,b]. Consequently, we claim u(t,ε)0 on [0,b). It is clear that ˙ui(0,ε)>0,iI6={1,2,3,4,5,6} whenever ui(0,ε)=0. Next, we prove the claim by contradiction. Suppose that there exists ˉt(0,b) such that ui(ˉt,ε)=0 for some iI6 and u(t,ε)0 for t(0,ˉt), where

    ˉt=min1i6{ti}, ti=sup{t(0,b):ui(x,ε)>0,x(0,t]}.

    As a result, it holds

    ˙ui(ˉt,ε)0. (2.3)

    Since

    u(ˉt,ε)0, Nm(ˉt)=Nm(ˉt,ε)=λm+εμm(1eμmˉt)+Nm(0,ε)eμmˉt>0,

    it follows from (2.2) that ˙ui(ˉt,ε)>0, which yields a contradiction to (2.3). Thus, we have u(t,ε)0 for t(0,b).

    Letting ε0+ gives that u(t,0)=u(t)0 for any t[0,b). Note that b(0,Tϕ) is chosen arbitrarily, so that u(t)0 on [0,Tϕ). It is obvious that Tϕ>τ. Therefore, from system (2.1), we have that for any tτ,

    ˙Sh(tτ)+˙Ah(t)+˙Ih(t)+˙Rh(t)=λhμh(Sh(tτ)+Ah(t)+Ih(t)+Rh(t)),˙Sm(t)+˙Im(t)=λmμm(Sm(t)+Im(t)).

    As a consequence, by the comparison principle, we can obtain that u(t) is bounded. Accordingly, from the continuation theorem of solutions of DDEs [19], it follows Tϕ=. Consequently, we have

    limt(Sh(tτ)+Ah(t)+Ih(t)+Rh(t))=λh/μh,limtNm(t)=λm/μm. (2.4)

    Therefore, the solution u(t) with any ϕC+ uniquely exists, and is non-negative and ultimately bounded on R+. Moreover, it is not difficult to get that (Sm(t),Sh(t))T0 on R+{0}, and C+ is a positive invariant set for system (2.1).

    To begin with, it follows easily the malaria-free equilibrium E0=(S0m,0,S0h,0,0,0)T, where S0m=λm/μm and S0h=λh/μh. By using the similar method in [21,22], we can calculate the basic reproduction number

    R0=pβhλhβ1μhλm(μh+γa)+(1p)βhλhβ2μhλm(μh+γi).

    To get a malaria-infected equilibrium (i.e., positive equilibrium) E=(Sm,Im,Sh,Ah,Ih,Rh)T, we have the following lemma.

    Lemma 3.1. System (2.1) exists a unique E0 when and only when R0>1.

    Proof. First of all, the malaria-infected equilibrium equations can be obtained as follows:

    {0=λmβ1SmAh+β2SmIhSm+ImμmSm,0=β1SmAh+β2SmIhSm+ImμmIm,0=λhβhShImSm+ImμhSh,0=pβhShImSm+Im(μh+γa)Ah,0=(1p)βhShImSm+Im(μh+γi)Ih,0=γaAh+γiIhμhRh. (3.1)

    Note that Sm+Im=λm/μm, it follows from (3.1) that

    {Im=β1SmAh+β2SmIhλm,Sh=λhλmμhλm+μmβhIm,Ah=pβhλhμmIm(μh+γa)(λmμh+μmβhIm),Ih=(1p)βhλhμmIm(μh+γi)(λmμh+μmβhIm). (3.2)

    Substituting the third and the fourth equations in (3.2) and Sm=λm/μmIm into the first equation in (3.2), there holds

    Im=λmμhImμhμm(Im)2μhλm+μmβhImR20. (3.3)

    In consequence, (3.3) possesses a unique positive root

    Im=λmμh(R201)μm(μhR20+βh)>0

    if and only if R0>1. Thus, we can conclude that E is a unique malaria-infected equilibrium of system (2.1) if and only if R0>1, where E satisfies

    {Sm=λm(μh+βh)μm(μhR20+βh),Im=λmμh(R201)μhμmR20+μmβh,Sh=λh(μhR20+βh)(μh+βh)μhR20,Ah=pβhλh(R201)(μh+βh)(μh+γa)R20,Ih=(1p)βhλh(R201)(μh+βh)(μh+γi)R20,Rh=γaAh+γiIhμh. (3.4)

    Next, by adopting similar techniques in [23,24,25], we will discuss the local dynamic properties of the malaria-free equilibrium E0 and the the malaria-infected equilibrium E with respect to R0. First of all, for the local stability of the equilibrium E0, we have the theorem as follows.

    Theorem 3.1. For any τ0, the malaria-free equilibrium E0 is locally asymptotically stable (LAS) when R0<1, and unstable when R0>1.

    Proof. With some calculations, the characteristic equation of the linear system of system (2.1) at E0 can be obtained as follows:

    |λ+μm00β1β200λ+μm0β1β200βhλhμmλmμhλ+μh0000pβhλhμmλmμheλτ0λ+(μh+γa)000(1p)βhλhμmλmμheλτ00λ+(μh+γi)0000γaγiλ+μh|=(λ+μm)(λ+μh)2H(λ)=0, (3.5)

    where

    H(λ)=(λ+μm)(λ+μh+γa)(λ+μh+γi)[pβ1βhλhμmλmμh(λ+μh+γi)+(1p)β2βhλhμmλmμh(λ+μh+γa)]eλτ. (3.6)

    Clearly, Eq (3.5) possesses three negative real roots: μh (double) and μm. The other roots of Eq (3.5) satisty H(λ)=0. Next, we will prove that any root λ of H(λ)=0 has negative real part. Suppose, by contradiction, λ has the nonegative real part. Then it follows from H(λ)=0 that

    λ+μm=pβ1βhλhμmλmμh(λ+μh+γa)eλτ+(1p)β2βhλhμmλmμh(λ+μh+γi)eλτ. (3.7)

    Taking the modulus of both sides in (3.7), we have

    |λ+μm|μm

    and

    |[pβ1βhλhμmλmμh(λ+μh+γa)+(1p)β2βhλhμmλmμh(λ+μh+γi)]eλτ||pβ1βhλhμmλmμh(λ+μh+γa)|+|(1p)β2βhλhμmλmμh(λ+μh+γi)|pβ1βhλhμmλmμh(μh+γa)+(1p)β2βhλhμmλmμh(μh+γi)=μmR20<μm

    for R0<1 and τ0, which leads to a contradiction. Therefore, the real part of each root of the Eq (3.5) is negative. Accordingly, E0 is LAS for R0<1 and τ0.

    Now, we prove that the E0 is unstable for R0>1 and τ0 by the zero theorem. Clearly, for R0>1 and τ0, we can get

    H(0)=μm(μh+γa)(μh+γi)(1R20)<0,limλH(λ)=.

    According to the zero theorem, there must exsit a positive real root in Eq (3.6). Thus, E0 is unstable for R0>1 and τ0.

    For the local stability of the equilibrium E, we can obtain the theorem as follows.

    Theorem 3.2. For any τ0, the malaria-infected equilibrium E is LAS if and only if R0>1.

    Proof. By Lemma 3.1, we just require to demonstrate the sufficiency. Let

    G=(R201)2μmμ2h(μh+βh)(μhR20+βh), H=βhλhμm(R201)(μh+βh)λmR20, K=μh+βhμhR20+βh,T=βhμh(R201)μhR20+βh, M=βhλhμmμhλmR20, J=μmμh(R201)μhR20+βh.

    With direct calculation, the characteristic equation of the linear system of system (2.1) at E can be got as follows:

    λ+G+μmJ0β1Kβ2K0Gλ+J+μm0β1Kβ2K0HMλ+T+μh000pHeλτpMeλτpTeλτλ+(μh+γa)00(1p)Heλτ(p1)Meλτ(p1)Teλτ0λ+(μh+γi)0000γaγiλ+μh=(λ+μh)(λ+μm)g(λ)=0, (3.8)

    where

    g(λ)=(λ+μh+γi)(λ+μh+γa)(λ+G+J+μm)(λ+T+μh)[(1p)β2(λ+μh+γa)+pβ1(λ+μh+γi)](λ+μh)βhλhμmμhλmR20eλτ.

    Clearly, Eq (3.8) has two negative roots: μh and μm. The other roots of Eq (3.8) satisfy g(λ)=0. Then, we will prove that any root λ of g(λ)=0 has negative real part by contradiction. Assume that λ has the non-negative real part. By g(λ)=0, we can get

    λ+T+μhλ+μh=[(1p)β2(λ+G+J+μm)(λ+μh+γi)+pβ1(λ+G+J+μm)(λ+μh+γa)]βhλhμmμhλmR20eλτ. (3.9)

    Taking the modulus of both sides in (3.9), for R0>1 any τ0, it follows

    |λ+T+μhλ+μh|>1

    and

    |[(1p)β2(λ+G+J+μm)(λ+μh+γi)+pβ1(λ+G+J+μm)(λ+μh+γa)]βhλhμmμhλmR20eλτ|[|(1p)β2(λ+G+J+μm)(λ+μh+γi)|+|pβ1(λ+G+J+μm)(λ+μh+γa)|]|βhλhμmμhλmR20eλτ|[(1p)β2βhλhλm(G+J+μm)μh(μh+γi)+pβ1βhλhλm(G+J+μm)μh(μh+γa)]μmR20<[(1p)β2βhλhλmμmμh(μh+γi)+pβ1βhλhλmμmμh(μh+γa)]μmR20=1.

    Obviously, this is a contradiction. Hence, the real part of each root of the Eq (3.8) is negative for R0>1 and τ0, which ensures the local stability of the equilibrium E.

    Generally, to obtain the global stability of the equilibrium E, we need to prove the strong persistence or uniform persistence of system (2.1). However, we study the weak persistence of system (2.1), which can ensure the global stability of the equilibrium E. Of course, the weak persistence of system (2.1) is more accessible than its strong or uniform persistence. Now, we define

    ϝ={ϕC+:ϕ2(0)>0},

    and let

    u(t)=(Sm(t),Im(t),Sh(t),Ah(t),Ih(t),Rh(t))T

    be the solution of system (2.1) with any ϕϝ. It follows easily that ϝ is a positive invariant set of system (2.1), and u(t)0 for t>0. Hence, we discuss the weak persistence of system (2.1) in ϝ.

    According to [26], system (2.1) is said to be weakly persistent if

    lim suptϱ(t)>0,ϱ=Sm,Im,Sh,Ah,Ih,Rh.

    We define ut=(Smt,Imt,Sht,Aht,Iht,Rht)TC+ to be ut(θ)=u(t+θ), θ[τ,0] for t0, and ut is the solution of system (2.1) with ϕ. Inspired by the work in [13], we study the weak persistence of system (2.1). First, we have the following lemma.

    Lemma 4.1. Assume that R0>1, θ(0,1) and lim suptIm(t)θIm. Then

    lim inftSm(t)ˉSmλmθμmImμm=λm[μhR20(1θ)+βh+θμh]μm(μhR20+βh)>Sm,lim inftSh(t)ˉShλhθβhIm/S0m+μh=S0hθβh(μh+βh)/μh(μhR20+βh)+1>Sh.

    Proof. It follows from (2.4) that

    lim inftSm(t)=lim inft(Nm(t)Im(t))=S0mlim suptIm(t)S0mθIm=ˉSm.

    For any ϵ>1, there can be found ϱ=ϱ(ϕ,ϵ)0 such that

    Im(t)Nm(t)ϵθImS0m,t>ϱ,

    and then

    ˙Sh(t)=λhβhSh(t)Im(t)Nm(t)μhSh(t)>λh(ϵθβhImS0m+μh)Sh(t).

    Consequently,

    lim inftSh(t)λhϵθβhIm/S0m+μh.

    Letting ϵ1+, it holds

    lim inftSh(t)ˉSh.

    The malaria-infected equilibrium equations imply that

    ˉSm=λm(μhR20(1θ)+βh+θμh)μm(μhR20+βh)>Sm, ˉSh=S0hθβh(μh+βh)/μh(μhR20+βh)+1>Sh.

    Theorem 4.1. Let R0>1. Then lim suptIm(t)Im.

    Proof. We will use the proof by contradiction to verify this result. Provided that lim suptIm(t)<Im. Whereupon, one can find a θ(0,1) such that lim suptIm(t)θIm. Using Lemma 4.1, we can get that there is an ϵ0>0 such that for any ϵ(0,ϵ0),

    ˉShS0m+ϵ>ShS0m,ˉSmS0m+ϵ>SmS0m. (4.1)

    Thanks to Lemma 4.1, it follows that for any ϵ(0,ϵ0), there can be found TT(ϵ,ϕ)>0 such that

    Sh(t)Nm(t)>ˉShS0m+ϵ,Sm(t)Nm(t)>ˉSmS0m+ϵ,Sm(t)>Sm,tT.

    Now, we define the functional on ϝ as follows,

    L(ϕ)=S0mϕ2(0)+β1Smμh+γaϕ4(0)+β2Smμh+γiϕ5(0)+λmR20SmS0h0τϕ3(θ)ϕ2(θ)ϕ1(θ)+ϕ2(θ)dθ.

    Obviously, L(ut) is bounded since L is continous on ϝ. Then for tT, the derivative of L along the solution ut is given by

    ˙L(ut)(λmR20SmSh(t)S0hNm(t)λm)Im(t)>(λmR20SmˉShS0h(S0m+ϵ)λm)Im(t).

    Denote

    ˉIm=minθ[τ,0]Im(T+τ+θ),
    c=min{ˉIm,(μh+γa)(S0m+ϵ)Ah(T+τ)pβhˉSh,(μh+γi)(S0m+ϵ)Ih(T+τ)(1p)βhˉSh}.

    Next, we will prove that Im(t)c for tT. If not, there exists a T00 such that Im(t)c for t[T,T+τ+T0], Im(T+τ+T0)=c and ˙Im(T+τ+T0)0. Then it follows that for t[T,T+τ+T0],

    ˙Ah(t)=pβhSh(tτ)Im(tτ)Nm(tτ)(μh+γa)Ah(t)pβhˉShcS0m+ϵ(μh+γa)Ah(t). (4.2)

    It easily follows from Eq (4.2) that for t[T,T+τ+T0],

    Ah(t)pβhˉShc(S0m+ϵ)(μh+γa)+(Ah(T)pβhˉShc(S0m+ϵ)(μh+γa))eμTμtpβhˉShc(S0m+ϵ)(μh+γa).

    Analogously, one can get

    Ih(t)(1p)βhˉShc(S0m+ϵ)(μh+γi),

    for t[T,T+τ+T0]. By R20=S0hS0m/ShSm and (4.1), we have

    λmR20SmˉShS0h(S0m+ϵ)λm>λmR20SmShS0hS0mλm=0.

    Accordingly, we conclude that

    ˙Im(T+τ+T0)=(β1Sm(T+τ+T0)Ah(T+τ+T0)+β2Sm(T+τ+T0)Ih(T+τ+T0)Nm(T+τ+T0)λmIm(T+τ+T0))>β1ˉSmpβhˉShc(S0m+ϵ)2(μh+γa)+β2ˉSm(1p)βhˉShc(S0m+ϵ)2(μh+γi)λmc=c(λmR20ˉSmˉShS0h(S0m+ϵ)2λm)>c(λmR20SmShS0hS0mλm)=0.

    Clearly, this contradicts ˙Im(T+τ+T0)0. As a result, Im(T)c for tT. Hence, for tτ,

    ˙L(ut)(λmR20ˉSmˉShS0h(S0m+ϵ)λm)c>0,

    which hints L(ut) as t. Accordingly, this contradicts the boundedness of L(ut).

    According to Theorem 4.1, we have the following result.

    Corollary 4.1. If R0>1, then for any τ0, system (2.1) is weakly persistent.

    We will study the global asymptotic stability of the equilibria E0 and E with respect to R0. For this purpose, we get from (2.4) the following limiting system of system (2.1):

    {˙Sm(t)=λmβ1Sm(t)Ah(t)+β2Sm(t)Ih(t)S0mμmSm(t),˙Im(t)=β1Sm(t)Ah(t)+β2Sm(t)Ih(t)S0mμmIm(t),˙Sh(t)=λhβhSh(t)Im(t)S0mμhSh(t),˙Ah(t)=pβhSh(tτ)Im(tτ)S0m(μh+γa)Ah(t),˙Ih(t)=(1p)βhSh(tτ)Im(tτ)S0m(μh+γi)Ih(t),˙Rh(t)=γaAh(t)+γiIh(t)μhRh(t). (5.1)

    Adopting a similar argument as in the proof of Theorem 2.1, it follows that the solution

    z(t)=(Sm(t),Im(t),Sh(t),Ah(t),Ih(t),Rh(t))T

    of system (5.1) through any φ=(φ1,φ2,φ3,φ4,φ5,φ6)TC+ uniquely exists, and is non-negative and ultimately bounded on [0,). Setting

    zt(θ)=z(t+θ),θ[τ,0]

    gives that zt=(Smt,Imt,Sht,Aht,Iht,Rht)TC+ is also the solution of system (5.1) through φ for t0. We can find easily that E and E0 are also the equilibria of system (5.1), and C+ is a positive invariant set of system (5.1). By the way, (Sm(t),Sh(t))T0 for t>0. Define H(v)=v1lnv,v>0. Thereupon, for the global dynamic property of the equilibrium E0 of system (2.1), we have the theorem as follows.

    Theorem 5.1. For any τ0, the malaria-free equilibrium E0 is GAS when R0<1 and GA when R0=1 in C+.

    Proof. By Theorem 3.1, it follows that for R0<1, E0 is LAS. Thus, we only need to prove that for R01, E0 is GA. Let ut be the solution of system (2.1) with any ϕC+ and zt be the solution of system (5.1) though any φC+. Let ω(ϕ) be the ω-limit set of ϕ with respect to system (2.1). In order to prove the global attractivity of E0, we just need to show that ω(ϕ)={E0}. By Theorem 2.1, we know that ut is bounded on C+. Hence, it follows from (2.4) that ω(ϕ) is a compact set, and is also a subset of C+.

    Let us define the following functional V on L1={φC+:φ1(0)>0,φ3(0)>0}C+

    V(φ)=V1(φ(0))+μhλmλh0τφ3(θ)φ2(θ)dθ, (5.2)

    where

    V1(φ(0))=(S0m)2H(φ1(0)S0m)+S0mφ2(0)+S0mλmβhH(φ3(0)S0h)+β1S0mμh+γaφ4(0)+β2S0mμh+γiφ5(0).

    Obviously, V1 is continuous on L1. Since ztL1 for t1, the derivative of V along zt (t1) is given by

    ˙V(zt)=λmμm(1S0mSm(t))(λmβ1Sm(t)Ah(t)+β2Sm(t)Ih(t)S0mμmSm(t))+λmμm(β1Sm(t)Ah(t)+β2Sm(t)Ih(t)S0mμmIm(t))+λmhS0h(1S0hSh(t))(λhhSh(t)Im(t)μhSh(t))+β1S0mμh+γa(phSh(tτ)Im(tτ)(μh+γa)Ah(t))+β2S0mμh+γi((1p)hSh(tτ)Im(tτ)(μh+γi)Ih(t))+μhλmλh(Sh(t)Im(t)Sh(tτ)Im(tτ))=λm(Sm(t)S0m)2Sm(t)μhλm(Sh(t)S0h)2hS0hSh(t)+λmμhλhSh(tτ)Im(tτ)(R201)0, (5.3)

    where h=βhμm/λm. Considering (5.2) and (5.3), we can conclude that both Sh(t) and Sm(t) are persistent. In other words, there exists a σ=σ(φ)>0 such that lim inftSh(t)>σ and lim inftSm(t)>σ. As a result, ω(φ)L1, where ω(φ) is the ω-limit set of φ with respect to system (5.1). It is evident that V is a Lyapunov functional on {zt:t1}L1. Then it follows from [27,Corollry 2.1] that ˙V(ψ)=0, ψω(φ).

    Assume that zt is the solution of system (5.1) through any ψω(φ). Then the invariance of ω(φ) gives that ztω(φ) for tR. According to (5.3), we have Sm(t)=S0m and Sh(t)=S0h for tR. From system (5.1) and the invariance of ω(φ), it follows that Im(t)=Ah(t)=Ih(t)=Rh(t)=0 for tR. Thus for R01, it holds that ω(φ)={E0}, which implies that Ws(E0)=C+, where Ws(E0) is the stable set of E0 with respect to system (5.1).

    Now, we prove that the equilibrium E0 of system (5.1) is uniformly stable for R01 by using the similar approach in [28,29]. Observe that the first five equations of system (5.1) can constitute an independent subsystem

    {˙Sm(t)=λmβ1Sm(t)Ah(t)+β2Sm(t)Ih(t)S0mμmSm(t),˙Im(t)=β1Sm(t)Ah(t)+β2Sm(t)Ih(t)S0mμmIm(t),˙Sh(t)=λhβhSh(t)Im(t)S0mμhSh(t),˙Ah(t)=pβhSh(tτ)Im(tτ)S0m(μh+γa)Ah(t),˙Ih(t)=(1p)βhSh(tτ)Im(tτ)S0m(μh+γi)Ih(t). (5.4)

    It is not difficult to find that

    C+={ξ=(ξ1,ξ2,ξ3,ξ4,ξ5)TC([τ,0],R5+):ξ1(θ)+ξ2(θ)>0,θ[τ,0]}

    is a positive invariant set with respect to system (5.4). Clearly, system (5.4) has a malaria-free equilibrium X0=(S0m,0,S0h,0,0)T. According to (5.2), (5.3), [27,Corollary 3.3] hints that X0 is uniformly stable. Define

    k:=min{μhγa,μhγi,1}.

    By the definition of uniform stability of X0, it follows that for any ϵ>0, there is δ2ϵ/3 such that for any ξC+ and ξX0<δ, there holds

    XtX0<ϵk3, t0,

    where Xt is the solution of system (5.4) with ξ. Then considering the sixth equation of system (5.1), we can get

    Rh(t)=ϕ6(0)eμht+γaeμhtt0Ah(s)eμhsds+γieμhtt0Ih(s)eμhsds. (5.5)

    Consequently, for any ϕC+ and ϕE0<δ, it follows that for any t0,

    Rh(t)<2ϵ3eμht+γaμhγaϵ31eμhtμh+γiμhγiϵ31eμhtμh=2ϵ3,

    and then

    utE0XtX0+Rht<ϵk3+2ϵ3ϵ.

    Thus, the equilibrium E0 is uniformly stable for system (5.1).

    Next, we claim that ω(ϕ)={E0} for R01. We first have E0ω(ϕ) since ω(ϕ)C+=Ws(E0). Assume that there exists ψω(ϕ) such that ψE0. Let α(ψ) be the α-limit set of ψ for system (5.1). Then it follows from the invariance and the compactness of ω(ϕ) that α(ψ)ω(ϕ). The invariance of α(ψ) and the stable set C+ of E0 yield that E0α(ψ). Obviously, this contradicts to the stability of E0 for system (5.1). Therefore, ω(ϕ)={E0}.

    Remark 5.1. In fact, the stability of the malaria-free equilibrium E0 of system (5.1) can be acquired for R0=1 in the proof of Theorem 5.1. But using the proof of Theorem 3.1, we can not obtain the stability of the equilibrium E0 for R0=1.

    For the global dynamic property of the equilibrium E of system (2.1), we can draw the following theorem.

    Theorem 5.2. For any τ0, the malaria-infected equilibrium E is globally asymptotically stable if and only if R0>1 in ϝ.

    Proof. From Lemma 3.1 and Theorem 3.2, we just require to prove that E is GA for R0>1. Let ut be the solution of system (2.1) with any ϕϝ and zt be the solution of system (5.1) through any φϝ. We can obtain that ϝ is positively invariant for system (5.1), and z(t)0 for t0. In order to show that E is GA, we only need to show that ω(ϕ)={E}. It follows from Theorem 2.1 that ut is bounded on ϝ. Thus, it holds that ω(ϕ) is compact.

    Let us define a functional V on L2={φC+:φi(0)>0,i=1,2,3,4,5}ϝ as follows

    V(φ)=V2(φ(0))+λmIm0τH(φ3(θ)φ2(θ)ShIm)dθ, (5.6)

    where

    V2(φ(0))=λmμmSmH(φ1(0)Sm)+λmμmImH(φ2(0)Im)+S0mλmβhH(φ3(0)Sh)+β1Smμh+γaH(φ4(0)Ah)+β2Smμh+γiH(φ5(0)Ih).

    Clearly, V2 is continuous on L2. In as much as ztL2 for tτ+1, the derivative of the functional V along zt in tτ+1 is given by

    ˙V(zt)=λmμm(1SmSm(t))(λmβ1Sm(t)Ah(t)+β2Sm(t)Ih(t)S0mμmSm(t))+λmμm(1ImIm(t))(β1Sm(t)Ah(t)+β2Sm(t)Ih(t)S0mμmIm(t))+λmhSh(1ShSh(t))(λhhSh(t)Im(t)μhSh(t))+β1Sm(μh+γa)(1AhAh(t))(phSh(tτ)Im(tτ)(μh+γa)Ah(t))+β2Sm(μh+γi)(1IhIh(t))((1p)hSh(tτ)Im(tτ)(μh+γi)Ih(t))+λmShSh(t)Im(t)λmShSh(tτ)Im(tτ)+λmImlnSh(tτ)Im(tτ)Sh(t)Im(t).

    where h=βhμm/λm. Further, it follows from the equilibrium equations that

    ˙V(zt)=λm(Sm(t)Sm)2Sm(t)λmImH(SmSm(t))β1SmAhH(Sm(t)Ah(t)ImSmAhIm(t))β2SmIhH(Sm(t)Ih(t)ImSmIhIm(t))λmμh(Sh(t)Sh)2hSh(t)ShλmImH(ShSh(t))β1SmAhH(Sh(tτ)Im(tτ)AhShImAh(t))β2SmIhH(Sh(tτ)Im(tτ)IhShImIh(t))0. (5.7)

    By (5.6) and (5.7), it follows that ω(φ)L2. It is clear that V is a Lyapunov functional on {zt:tτ+1}L2. As a consequence, [27,Corollary2.1] implies that ˙V(ψ)=0 for any ψω(φ).

    Let zt be the solution of system (5.1) for any ψω(φ). Then the invariance of ω(φ) indicates that ztω(φ) for any tR. Thus, from (5.7), it follows that for any tR,

    Sm(t)=Sm, Sh(t)=Sh, Ah(t)Im=AhIm(t), Ih(t)Im=IhIm(t). (5.8)

    By (5.8) and the third equation of system (5.1), we get that for any tR,Im(t)=Im, Ah(t)=Ah and Ih(t)=Ih. Consequently, by the invariance of ω(φ) and system (5.1), it holds that Rh(t)=Rh for any tR. Therefore, it follows that z0=ψ=E, and then ω(φ)={E}, which implies Ws(E)=ϝ, where Ws(E) is the stable set of E with respect to system (5.1).

    Now, we prove that the equilibrium E of system (5.1) is uniformly stable by using the similar argument in [28,29]. Note that system (5.4) has a unique malaria-infected equilibrium X=(Sm,Im,Sh,Ah,Ih). It follows from (5.6), (5.7) and [27,Corollary 3.3] that X is uniformly stable. By the definition of uniform stability of X, it follows that for any ϵ>0, there is δ2ϵ/3 such that for any ξC+ and ξX<δ, there holds

    XtX<ϵk3, t0,

    where Xt is the solution of system (5.4) through ξ. Hence, for any ϕL2 and ϕE<δ, it follow from (5.5) that

    Rh(t)Rh<2ϵ3eμht+γaμhγaϵ31eμhtμh+γiμhγiϵ31eμhtμh=2ϵ3

    for any t0, where

    Rh=Rheμht+γaeμhtt0Aheμhsds+γieμhtt0Iheμhsds

    is used. Thus, we have

    utEXtX+RhtRh<ϵk3+2ϵ3ϵ,

    which gives that the equilibrium E is uniformly stable with respect to system (5.1).

    Next, we claim that ω(ϕ)={E}. From Theorem 4.1, it follows that ω(ϕ)ϝ, which gives that Eω(ϕ). Assume that there is ψω(ϕ) such that ψE. Then the invariance and the compactness of ω(ϕ) implies that α(ψ)ω(ϕ). By Theorem 3.2, we have that Eα(ψ). Obviously, this contradicts to the stability of E with respect to system (5.1). Therefore, ω(ϕ)={E}.

    Remark 5.2. Indeed, the proof of Theorem 5.2 can be simplified, i.e., the stability of system (5.1) is not a must, because Eω(ϕ) and Theorem 3.2 can imply that ω(ϕ)={E}. But if we use [30,Theorem 4.1] to prove that ω(ϕ)={E}, then the stability of system (5.1) is required.

    This work is partially supported by the National NSF of China (No. 11901027), the Major Program of the National NSF of China (No. 12090014), the State Key Program of the National NSF of China (No. 12031020), the China Postdoctoral Science Foundation (No. 2021M703426), the Pyramid Talent Training Project of BUCEA (No. JDYC20200327), and the BUCEA Post Graduate Innovation Project (No. PG2022143).

    The authors declare there is no conflict of interest.



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