Asymptotic analysis of a vector-borne disease model with the age of infection

Abstract

Vector-borne infectious diseases may involve both horizontal transmission between hosts and transmission from infected vectors to susceptible hosts. In this paper, we incorporate these two transmission modes into a vector-borne disease model that includes general nonlinear incidence rates and the age of infection for both hosts and vectors. We show the existence, uniqueness, nonnegativity, and boundedness of solutions for the model. We study the existence and local stability of steady states, which is shown to be determined by the basic reproduction number. By showing the existence of a global compact attractor and uniform persistence of the system, we establish the threshold dynamics using the Fluctuation Lemma and the approach of Lyapunov functionals. When the basic reproduction number is less than one, the disease-free steady state is globally asymptotically stable and otherwise the disease will be established when there is initial infection force for the hosts. We also study a model with the standard incidence rate and discuss the effect of different incidence rates on the disease dynamics.

Keywords:

• Vector-borne disease
• age of infection
• horizontal transmission
• threshold dynamics
• global stability

1. Introduction

Vectors are living organisms that can transmit infectious diseases between humans (horizontal transmission) or from animals to humans. Many of these vectors are bloodsucking insects, which ingest disease-producing microorganisms during a blood meal from an infected host (human or animal) and later inject these microorganisms into a new host during their subsequent blood meal [59]. Mosquitoes are the best known disease vector. Other vectors include ticks, flies, sandflies, fleas, and some freshwater aquatic snails. Vector-borne diseases are infections caused by pathogens and parasites in human populations which are transmitted by the bite of infected vectors. They have been a substantial threat to human health [9]. According to the WHO [59], vector-borne diseases account for a significant proportion of infectious diseases that affect humans. Every year there are more than one billion cases and over one million deaths from vector-borne diseases such as malaria, West Nile virus, leishmaniasis, dengue (the arboviral human disease), human African trypanosomiasis, Chagas disease, and so on. Therefore, it is imperative to understand the transmission dynamics of vector-borne diseases and, based on these results, design appropriate control and prevention measures.

Mathematical modeling is an efficient way to investigate infectious diseases dynamics. Over the past few decades, mathematical models have made significant contribution to the understanding of the transmission dynamics of vector-borne diseases. Many models are described by ordinary differential equations (see [2,14,16,18,23,31,39,43,46,48,49,52,53] and references therein). A fundamental assumption in these models is that individuals in each epidemiological class are homogeneous. For example, infectious individuals have the same infectivity during their infectious period. Although it may provide a reasonable approximation to the biological process that is modeled, this is not always the case. To describe such phenomena as variable infectivity in the infectious period, continuous age structures have been introduced into mathematical models described by ordinary differential equations.

There is a rich literature on SIR (susceptible-infectious-recovered) epidemic models and their extensions with continuous age structures. Many diseases can simply spread because of the existence of biting vectors, which is important in the transmission dynamics of vector-borne diseases. The age at which a vector becomes infectious can affect the number of secondary infectious individuals. When infection occurs at the beginning of the vector's life, it makes a higher number of bites. From the point of biological modeling, Novoseltsev et al. [44] provided an extension and refinement of the vectorial capacity paradigm by introducing an age-structured extension to the model. They assumed that insect vectors may have age-dependent mortality in the model, which was shown to greatly influence the pathogen transmission dynamics. Bellan [3] developed a simple model to assess how relaxing the classical assumption of constant mortality rate affects the predicted effectiveness of anti-vectorial interventions. Comparing with a more realistic age-dependent model, he concluded that models with constant mortality overestimated the sensitivity of disease transmission to interventions that reduce mosquito survival. Pigeault et al. [45] carried out experiments by using the avian malaria parasite (Plasmodium relictum) and its natural vector in the field, the mosquito (Culex pipiens). Their results indicated that older insects are often more resistant to infections than younger ones. These results suggested that structural and functional alterations in mosquito physiology with age may be more important than immunity in determining the probability of a Plasmodium infection in old mosquitoes. In view of mathematical modeling, age structure in vector-borne transmission models is not in itself new, although vector-borne age-structured models are predominantly focused on the age in the host population rather than in the vector population [13,26,36,54,55,57]. In [13], Dang et al. proposed a vector-borne disease model that includes both incubation age of exposed hosts and infection age of infectious hosts. By using the method of Lyapunov functionals, threshold dynamics is established. Inaba and Sekine [26] developed a mathematical model for Chagas disease with infection-age-dependent infectivity. Under the assumption that the population size of vectors is constant, the local and global stability of steady states were investigated. In particular, it was shown that bifurcation of endemic steady states can occur even when the basic reproduction number is less than one. In [55], Vogt-Geisse et al. first obtained a partial differential equation model, which is reduced to an ordinary differential equation model with multiple age groups coupled with age since infection. The theoretical results may help generate insights into effective control measures by targeting age groups in an optimal way.

In addition to the transmission from vectors to hosts, vector-borne diseases may also involve horizontal transmission between hosts. Horizontal transmission is the spread of viruses from one individual to another, usually through contact with bodily excretions or fluids, such as sputum or blood, that contain pathogens. The horizontal transmission risk may not be high in general, but it cannot be neglected [26]. For example, for Chagas disease, the prevalence rate among blood donors in Bolivia reached the level beyond 10% in 1993-1994, but in other countries of Central and South America, prevalence rates were at most several percent [50]. Some models have been developed for vector-borne diseases with horizontal transmission [1,15,20,31–33]. They are described by ordinary differential equations. Wei et al. [58] also used a delay differential equation system to study vector-borne diseases.

It is well-known that an important factor affecting the transmission dynamics is the incidence rate, which depends on the population behavior and the infectivity of the disease. Mathematically, incidence rate is the number of new infections per unit time. In the above mentioned references, the incidence rates are either bilinear or standard. Experiments and field work have indicated that nonlinear incidences may be more reasonable. Liu et al. [35] showed that epidemiological models with nonlinear incidence rates exhibit much richer dynamics than those with bilinear incidence rates. By describing how the biology of host-parasite association could determine the functional form of the transmission rate, Georgescu and Hsieh [21] and Korobeinikov et al. [29] formulated a variety of models with incidence rates having the form $f\left(S\right(t\left)\right)g\left(I\right(t\left)\right)$, where $f\left(S\right(t\left)\right)$ is the contact rate function dependent on susceptible class S and $g\left(I\right(t\left)\right)$ represents the force of infection by infectious class I. Thereafter, Korobeinikov et al. [27,28] used a very general form $f(S,I)$ for the incidence rate. For more other nonlinear incidence rates, we refer readers to [5,10,11,17,34,44,56].

Motivated by the above studies, in this paper we will further the study on vector-borne diseases by incorporating age structures into both infectious hosts and infectious vectors and employing general nonlinear incidence rates. This paper is different from a previous study [57] in that we include two routes of transmission of the infectious disease. We evaluate the influence of the horizontal transmission and exposed classes on disease dynamics. We also compare the results under different incidence rates. The model is formulated and some preliminary results are presented in Section 2. In Section 3, we study the existence and local stability of steady states. Using the Fluctuation Lemma and Lyapunov functional approaches, we study the asymptotic dynamics of the model in Section 4. We show that if the basic reproduction number is less than one, then the disease-free steady state is globally asymptotically stable and hence the infection can be eradicated; if the basic reproduction number is larger than one, the infectious steady state is globally asymptotically stable. If the total population of hosts or vectors is not constant, then the standard incidence rate does not satisfy the condition for the stability of the infectious steady state. As a result, in Section 5, for variable populations of both hosts and vectors, we study a model with the standard incidence rates. We extend some results to this model but the stability of the infectious steady state is challenging due to the complexity of the transcendental characteristic equation. The paper concludes with a brief discussion.

2. The model and preliminary results

The host population $N_h$ in our model includes four classes, namely the susceptible $S_h$, exposed $E_h$, infectious $I_h$, and recovered $R_h$. We assume that the recovered hosts have acquired permanent immunity and hence in the sequel there is no need to include its dynamics in the model. The vector population $N_v$ has three classes: susceptible $S_v$, exposed $E_v$, and infectious $I_v$. It is assumed that there are no recovered vectors. Letting a be the age of infection, we have $i_h(t,a)$ and $i_v(t,a)$ as infectious hosts and vectors with age a at time t, respectively. The total numbers of infectious hosts and infectious vectors at time t are $I_h\left(t\right)={\int }_0^{\mathrm{\infty }}{i}_h(t,a)\mathrm{d}a$ and $I_v\left(t\right)={\int }_0^{\mathrm{\infty }}{i}_v(t,a)\mathrm{d}a$, respectively.

Let the natural death rate of the host be ${\mu }_h$ and the recruitment rate of hosts be ${\lambda }_h$. We assume that all the recruited are susceptible. Infectious hosts can infect susceptible hosts at the rate $\varphi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)$ and infectious vectors can also infect them at the rate $\psi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_v(a\left)i_v\right(t,a\left)\mathrm{d}a\right)$, where ${\text{β}}_h\left(a\right)$ denotes the age-dependent horizontal transmission rate of the disease from infectious hosts to susceptible hosts and ${\text{β}}_v\left(a\right)$ is the transmission rate from infectious vectors to susceptible hosts. Note that ${\text{β}}_h\left(a\right)$ or ${\text{β}}_v\left(a\right)$ lumps all the transmission-related parameters such as the contact or biting rate and the probability of successful transmission. The transition rate from exposed hosts to infectious hosts is ${\alpha }_h$. The parameter ${\gamma }_h\left(a\right)$ is the per capita recovery rate of infectious hosts and ${\rho }_h\left(a\right)$ is the disease-induced death rate of infectious hosts.

Unlike many of the existing models, we do not assume the population size of vectors to be constant. We let ${\lambda }_v$ be the birth rate of vectors and ${\mu }_v$ be the natural death rate. For susceptible vectors, the force of infection is $f\left(S_v\right(t),{\int }_0^{\mathrm{\infty }}{k}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)$, where $k_h\left(a\right)$ is the transmission rate from infectious hosts to susceptible vectors due to biting. The transition rate from exposed vectors to infectious vectors is ${\alpha }_v$ and the infection-related death rate of infectious vectors is ${\rho }_v\left(a\right)$.

Based on the above assumptions, we develop the following model (1) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)\\ & -\psi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_v(a\left)i_v\right(t,a\left)\mathrm{d}a\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)+\psi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_v(a\left)i_v\right(t,a\left)\mathrm{d}a\right)\\ & -({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-({\mu }_h+{\gamma }_h(a)+{\rho }_h(a\left)\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),{\int }_0^{\mathrm{\infty }}{k}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),{\int }_0^{\mathrm{\infty }}{k}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-({\mu }_v+{\rho }_v(a\left)\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(1) Here ${\mathbb{R}}_{+}=[0,\mathrm{\infty })$ and $L_{+}^1$ is the positive cone of $L^1(0,\mathrm{\infty })$.

Denote $\begin{array}{rl}{J}_h\left(t\right)& ={\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a,\\ J_v\left(t\right)& ={\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a,\\ J\left(t\right)& ={\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a,\\ {\delta }_h\left(a\right)& ={\mu }_h+{\gamma }_h\left(a\right)+{\rho }_h\left(a\right),\\ {\delta }_v\left(a\right)& ={\mu }_v+{\rho }_v\left(a\right).\end{array}$ Clearly, ${\delta }_h\left(a\right)\ge {\mu }_h$ and ${\delta }_v\left(a\right)\ge {\mu }_v$ for all $a\in {\mathbb{R}}_{+}$. Using the new notations, model (1(1) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)\\ & -\psi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_v(a\left)i_v\right(t,a\left)\mathrm{d}a\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)+\psi \left(S_h\right(t),{\int }_0^{\mathrm{\infty }}{\text{β}}_v(a\left)i_v\right(t,a\left)\mathrm{d}a\right)\\ & -({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-({\mu }_h+{\gamma }_h(a)+{\rho }_h(a\left)\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),{\int }_0^{\mathrm{\infty }}{k}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),{\int }_0^{\mathrm{\infty }}{k}_h(a\left)i_h\right(t,a\left)\mathrm{d}a\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-({\mu }_v+{\rho }_v(a\left)\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(1) ) can be rewritten as (2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) For model (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ), we assume $i_h(0,0)=i_{h0}\left(0\right)$ and $i_v(0,0)=i_{v0}\left(0\right)$. Thus (3) $i_{h0}\left(0\right)={\alpha }_h{E}_{h0}\text{and}{i}_{v0}\left(0\right)={\alpha }_v{E}_{v0}.$(3)

We also assume the following for the parameters and the nonlinear incidence functions in (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ).

(C1)	${\text{β}}_h,{\text{β}}_v,{\gamma }_h,{\rho }_h,{\rho }_v,k_h\in L_{+}^{\mathrm{\infty }}(0,\mathrm{\infty })$, where $L_{+}^{\mathrm{\infty }}(0,\mathrm{\infty })$ is the positive cone of the Banach space $L^{\mathrm{\infty }}(0,\mathrm{\infty })$. The functions ${\text{β}}_h,{\text{β}}_v$, and $k_h$ are uniformly continuous and bounded. Moreover, at least one of ${\text{β}}_h,{\text{β}}_v$ and $k_h$ cannot be the zero function. Otherwise, there is no transmission of the disease.
(C2)	$\varphi (u,v),\psi (u,v)$ and $f(u,v)$ are differentiable with $\frac{\mathrm{\partial }\varphi (u,v)}{\mathrm{\partial }u}>0,\frac{\mathrm{\partial }\psi (u,v)}{\mathrm{\partial }u}>0,\frac{\mathrm{\partial }f(u,v)}{\mathrm{\partial }u}>0$ for u>0 and v>0, and $\frac{\mathrm{\partial }\varphi (u,v)}{\mathrm{\partial }v}>0,\frac{\mathrm{\partial }\psi (u,v)}{\mathrm{\partial }v}>0,\frac{\mathrm{\partial }f(u,v)}{\mathrm{\partial }v}>0$ for u>0 and $v\ge 0$. Furthermore, $\varphi (u,0)=\varphi (0,v)=\psi (u,0)=\psi (0,v)=f(u,0)=f(0,v)=0$ for all $u,v\in {\mathbb{R}}_{+}$.
(C3)	$\frac{\mathrm{\partial }\varphi (u,v)}{\mathrm{\partial }v},\frac{\mathrm{\partial }\psi (u,v)}{\mathrm{\partial }v}$ and $\frac{\mathrm{\partial }f(u,v)}{\mathrm{\partial }v}$ are continuous at $(S_h^0,0),(S_h^0,0)$ and $(S_v^0,0)$, respectively, where $S_h^0=\frac{{\lambda }_h}{{\mu }_h}$ and $S_v^0=\frac{{\lambda }_v}{{\mu }_v}$. Moreover, $\frac{{\mathrm{\partial }}^2\varphi (u,v)}{\mathrm{\partial }{v}^2}\le 0,\frac{{\mathrm{\partial }}^2\psi (u,v)}{\mathrm{\partial }{v}^2}\le 0$, and $\frac{{\mathrm{\partial }}^{2}f(u,v)}{\mathrm{\partial }{v}^2}\le 0$ for all $u,v>0$.
(C4)	$\varphi (u,v),\psi (u,v)$ and $f(u,v)$ are all locally Lipschitz continuous, that is, given C>0, there exist positive values $L_u,L_v,K_u,K_v,M_u$ and $M_v$ such that $\begin{array}{rl}& |\varphi (u,v)-\varphi (\tilde{u},v)|\le L_u|u-\tilde{u}|,|\varphi (u,v)-\varphi (u,\tilde{v})|\le L_v|v-\tilde{v}|,\\ & |\psi (u,v)-\psi (\tilde{u},v)|\le K_u|u-\tilde{u}|,|\psi (u,v)-\psi (u,\tilde{v})|\le K_v|v-\tilde{v}|,\\ & |f(u,v)-f(\tilde{u},v)|\le M_u|u-\tilde{u}|,|f(u,v)-f(u,\tilde{v})|\le M_v|v-\tilde{v}|\end{array}$for any $u,\tilde{u},v,\tilde{v}$ in the interval $[0,C]$.

For positive u and v, we know that $\varphi (u,v)>0,\psi (u,v)>0$ and $f(u,v)>0$. From assumption $\left({\mathbf{C}}_2\right)$, these functions are also strictly increasing in terms of one independent variable when the other is fixed and positive. $\left({\mathbf{C}}_3\right)$ implies that the incidence rates are concave down for the first fixed variable.

The state space of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) is $X_{+}={\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times L_{+}^1(0,\mathrm{\infty })\times {\mathbb{R}}_{+}\times {\mathbb{R}}_{+}\times L_{+}^1(0,\mathrm{\infty }),$ the positive cone of the Banach space $\mathbb{R}\times \mathbb{R}\times L^1(0,\mathrm{\infty })\times \mathbb{R}\times \mathbb{R}\times L^1(0,\mathrm{\infty })$ equipped with the norm $\parallel (S_h,E_h,i_h,S_v,E_v,i_v)\parallel =|S_h|+|E_h|+\parallel i_h{\parallel }_1+|S_v|+|E_v|+\parallel i_v{\parallel }_1$ for $(S_h,E_h,i_h,S_v,E_v,i_v)\in \mathbb{R}\times \mathbb{R}\times L^1(0,\mathrm{\infty })\times \mathbb{R}\times \mathbb{R}\times L^1(0,\mathrm{\infty })$.

It follows from [6,25,37] that system (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) has a unique continuous solution in $X_{+}$ if the initial value $(S_{h0},E_{h0},i_{h0},S_{v0},E_{v0},i_{v0})\in X_{+}$ satisfies (3(3) $i_{h0}\left(0\right)={\alpha }_h{E}_{h0}\text{and}{i}_{v0}\left(0\right)={\alpha }_v{E}_{v0}.$(3) ). This leads to a solution semiflow $\mathrm{\Phi }:{\mathbb{R}}_{+}\times X_{+}\to X_{+}$ defined by $\mathrm{\Phi }(t,(S_{h0},E_{h0},i_{h0},S_{v0},E_{v0},i_{v0}\left)\right)=\left(S_h\right(t),E_h(t),i_h(t,\cdot ),S_v(t),E_v(t),i_v(t,\cdot \left)\right),$ where $t\in {\mathbb{R}}_{+}$ and $\left(S_h\right(t),E_h(t),i_h(t,a),S_v(t),E_v(t),i_v(t,a\left)\right)$ is the unique solution of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) through the initial value $(S_{h0},E_{h0},i_{h0},S_{v0},E_{v0},i_{v0})\in X_{+}$.

Let $\begin{array}{rl}{G}_h\left(t\right)& =S_h\left(t\right)+E_h\left(t\right)+{\int }_0^{\mathrm{\infty }}{i}_h(t,a)\mathrm{d}a,\\ N_v\left(t\right)& =S_v\left(t\right)+E_v\left(t\right)+{\int }_0^{\mathrm{\infty }}{i}_v(t,a)\mathrm{d}a.\end{array}$ It follows from (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) that (4) $\begin{array}{rl}\frac{\mathrm{d}{G}_h\left(t\right)}{\mathrm{d}t}& \le {\lambda }_h-{\mu }_h{S}_h\left(t\right)-{\mu }_h{E}_h\left(t\right)-{\int }_0^{\mathrm{\infty }}{\delta }_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & \le {\lambda }_h-{\mu }_h{G}_h\left(t\right)\end{array}$(4) and (5) $\frac{\mathrm{d}{N}_v\left(t\right)}{\mathrm{d}t}\le {\lambda }_v-{\mu }_v{N}_v\left(t\right),$(5) which respectively imply that $\underset{t\to \mathrm{\infty }}{lim sup}{G}_h\left(t\right)\le \frac{{\lambda }_h}{{\mu }_h}\text{and}\underset{t\to \mathrm{\infty }}{lim sup}{N}_v\left(t\right)\le \frac{{\lambda }_v}{{\mu }_v}.$ Furthermore, one can easily see from (4(4) $\begin{array}{rl}\frac{\mathrm{d}{G}_h\left(t\right)}{\mathrm{d}t}& \le {\lambda }_h-{\mu }_h{S}_h\left(t\right)-{\mu }_h{E}_h\left(t\right)-{\int }_0^{\mathrm{\infty }}{\delta }_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & \le {\lambda }_h-{\mu }_h{G}_h\left(t\right)\end{array}$(4) ) and (5(5) $\frac{\mathrm{d}{N}_v\left(t\right)}{\mathrm{d}t}\le {\lambda }_v-{\mu }_v{N}_v\left(t\right),$(5) ) that the set $\mathrm{\Omega }=\left\{(S_h,E_h,i_h,S_v,E_v,i_v)\in X_{+}\left|\begin{array}{l}{S}_h+E_h+\parallel i_h{\parallel }_1\le \frac{{\lambda }_h}{{\mu }_h},\\ S_v+E_v+\parallel i_v{\parallel }_1\le \frac{{\lambda }_v}{{\mu }_v}\end{array}\right.\right\}$ is a positively invariant and attracting subset for system (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ).

3. The existence and local stability of steady states

Obviously, system (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) always has the disease-free steady state $E^0=(S_h^0,0,0,S_v^0,0,0)$. Recall that $S_h^0=\frac{{\lambda }_h}{{\mu }_h}$ and $S_v^0=\frac{{\lambda }_v}{{\mu }_v}$.

For the convenience of notation, we denote $\begin{array}{rl}{\xi }_h& ={\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right){\sigma }_h\left(a\right)\mathrm{d}a,\\ {\xi }_v& ={\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right){\sigma }_v\left(a\right)\mathrm{d}a,\\ \eta & ={\int }_0^{\mathrm{\infty }}{k}_h\left(a\right){\sigma }_h\left(a\right)\mathrm{d}a,\\ {\sigma }_h\left(a\right)& =\exp \left(-{\int }_0^a{\delta }_h\left(s\right)\mathrm{d}s\right)\text{for }a\in {\mathbb{R}}_{+}\text{,}\\ {\sigma }_v\left(a\right)& =\exp \left(-{\int }_0^a{\delta }_v\left(s\right)\mathrm{d}s\right)\text{for }a\in {\mathbb{R}}_{+}\text{.}\end{array}$ It turns out that the structure of steady states of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) is determined by the basic reproduction number ${\mathcal{R}}_0$, which is given by ${\mathcal{R}}_0=\frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}\cdot \frac{{\xi }_h{\alpha }_h}{{\alpha }_h+{\mu }_h}+\frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\cdot \frac{{\alpha }_h\eta }{{\alpha }_h+{\mu }_h}\cdot \frac{{\xi }_v{\alpha }_v}{{\alpha }_v+{\mu }_v}.$ The expression of ${\mathcal{R}}_0$ is derived in the coming discussion of the existence of steady states and the local stability of the disease-free steady state. ${\mathcal{R}}_0$ is the number of infectious hosts generated by the introduction of an infectious host into a population consisting of only susceptible hosts, which agrees with the biological interpretation of the basic reproduction number. The first term in ${\mathcal{R}}_0$ is the number generated by the horizontal transmission while the second term is the number generated through vectors.

Let $E^{\ast }=(S_h^{\ast },E_h^{\ast },i_h^{\ast },S_v^{\ast },E_v^{\ast },i_v^{\ast })\in X_{+}$ (in fact in Ω) be a steady state of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ), that is, (6a) $\begin{array}{rl}0& ={\lambda }_h-{\mu }_h{S}_h^{\ast }-\varphi (S_h^{\ast },J_h^{\ast })-\psi (S_h^{\ast },J_v^{\ast }),\end{array}$(6a) (6b) $\begin{array}{rl}0& =\varphi (S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast })-({\alpha }_h+{\mu }_h)E_h^{\ast },\end{array}$(6b) (6c) $\begin{array}{rl}\frac{\mathrm{d}{i}_h^{\ast }\left(a\right)}{da}& =-{\delta }_h\left(a\right)i_h^{\ast }\left(a\right),\end{array}$(6c) (6d) $\begin{array}{rl}0& ={\lambda }_v-f(S_v^{\ast },J^{\ast })-{\mu }_v{S}_v^{\ast },\end{array}$(6d) (6e) $\begin{array}{rl}f(S_v^{\ast },J^{\ast })& =({\alpha }_v+{\mu }_v)E_v^{\ast },\end{array}$(6e) (6f) $\begin{array}{rl}\frac{\mathrm{d}{i}_v^{\ast }\left(a\right)}{da}& =-{\delta }_v\left(a\right)i_v^{\ast }\left(a\right),\end{array}$(6f) (6g) $\begin{array}{rl}{J}_h^{\ast }& ={\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a={\xi }_h{i}_h^{\ast }\left(0\right)={\xi }_h{\alpha }_h{E}_h^{\ast },\end{array}$(6g) (6h) $\begin{array}{rl}{J}_v^{\ast }& ={\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v^{\ast }\left(a\right)\mathrm{d}a={\xi }_v{i}_v^{\ast }\left(0\right)={\xi }_v{\alpha }_v{E}_v^{\ast },\end{array}$(6h) (6i) $\begin{array}{rl}{J}^{\ast }& ={\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a=\eta i_h^{\ast }\left(0\right)=\eta {\alpha }_h{E}_h^{\ast }.\end{array}$(6i) From (6a(6a) $\begin{array}{rl}0& ={\lambda }_h-{\mu }_h{S}_h^{\ast }-\varphi (S_h^{\ast },J_h^{\ast })-\psi (S_h^{\ast },J_v^{\ast }),\end{array}$(6a) ) and (6b(6b) $\begin{array}{rl}0& =\varphi (S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast })-({\alpha }_h+{\mu }_h)E_h^{\ast },\end{array}$(6b) ), we have (7) $S_h^{\ast }=\frac{{\lambda }_h-({\alpha }_h+{\mu }_h)E_h^{\ast }}{{\mu }_h}.$(7) By applying the implicit function theorem, we see from (6d(6d) $\begin{array}{rl}0& ={\lambda }_v-f(S_v^{\ast },J^{\ast })-{\mu }_v{S}_v^{\ast },\end{array}$(6d) ) and (6i(6i) $\begin{array}{rl}{J}^{\ast }& ={\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a=\eta i_h^{\ast }\left(0\right)=\eta {\alpha }_h{E}_h^{\ast }.\end{array}$(6i) ) that there exists a function p such that $S_v^{\ast }=p\left(E_h^{\ast }\right)$, where $p\left(0\right)=S_v^0$ and $p^{\prime }\left(E_h^{\ast }\right)=-\frac{\eta {\alpha }_h\frac{\mathrm{\partial }f\left(p\right(E_h^{\ast }),\eta {\alpha }_h{E}_h^{\ast })}{\mathrm{\partial }J}}{{\mu }_v+\frac{\mathrm{\partial }f\left(p\right(E_h^{\ast }),\eta {\alpha }_h{E}_h^{\ast })}{\mathrm{\partial }{S}_v}}.$ This, combined with (6e(6e) $\begin{array}{rl}f(S_v^{\ast },J^{\ast })& =({\alpha }_v+{\mu }_v)E_v^{\ast },\end{array}$(6e) ) and (6i(6i) $\begin{array}{rl}{J}^{\ast }& ={\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a=\eta i_h^{\ast }\left(0\right)=\eta {\alpha }_h{E}_h^{\ast }.\end{array}$(6i) ), gives (8) $E_v^{\ast }=\frac{f(S_v^{\ast },J^{\ast })}{{\alpha }_v+{\mu }_v}=\frac{f\left(p\right(E_h^{\ast }),\eta {\alpha }_h{E}_h^{\ast })}{{\alpha }_v+{\mu }_v}.$(8) It follows from (6b(6b) $\begin{array}{rl}0& =\varphi (S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast })-({\alpha }_h+{\mu }_h)E_h^{\ast },\end{array}$(6b) ), (6g(6g) $\begin{array}{rl}{J}_h^{\ast }& ={\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a={\xi }_h{i}_h^{\ast }\left(0\right)={\xi }_h{\alpha }_h{E}_h^{\ast },\end{array}$(6g) ), (6h(6h) $\begin{array}{rl}{J}_v^{\ast }& ={\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v^{\ast }\left(a\right)\mathrm{d}a={\xi }_v{i}_v^{\ast }\left(0\right)={\xi }_v{\alpha }_v{E}_v^{\ast },\end{array}$(6h) ), (7(7) $S_h^{\ast }=\frac{{\lambda }_h-({\alpha }_h+{\mu }_h)E_h^{\ast }}{{\mu }_h}.$(7) ), and (8(8) $E_v^{\ast }=\frac{f(S_v^{\ast },J^{\ast })}{{\alpha }_v+{\mu }_v}=\frac{f\left(p\right(E_h^{\ast }),\eta {\alpha }_h{E}_h^{\ast })}{{\alpha }_v+{\mu }_v}.$(8) ) that $E_h^{\ast }$ is a nonnegative zero of $H\left(x\right)$, where $\begin{array}{rl}H\left(x\right)& =\varphi \left(\frac{{\lambda }_h-({\alpha }_h+{\mu }_h)x}{{\mu }_h},{\xi }_h{\alpha }_{h}x\right)\\ & +\psi \left(\frac{{\lambda }_h-({\alpha }_h+{\mu }_h)x}{{\mu }_h},{\xi }_v{\alpha }_v\frac{f\left(p\right(x),\eta {\alpha }_{h}x)}{{\alpha }_v+{\mu }_v}\right)-({\alpha }_h+{\mu }_h)x.\end{array}$ Obviously, $H\left(0\right)=0$, which gives the disease-free steady state. Moreover, from the above relations, one can easily see that if $E_h^{\ast }\ne 0$ then none of $i_h^{\ast },E_v^{\ast }$, and $i_v^{\ast }$ is zero. This means that a steady state rather than the disease-free one is an infectious steady state. In the following, we discuss their existence.

First, we assume ${\mathcal{R}}_0\le 1$. Then for any x>0, it follows from (${\mathbf{C}}_2$) and (${\mathbf{C}}_3$) that $\begin{array}{rl}H\left(x\right)& \le \left(\frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}{\xi }_h{\alpha }_h+\frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}\cdot \frac{{\xi }_v{\alpha }_v}{{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\cdot \eta {\alpha }_h-({\alpha }_h+{\mu }_h)\right)x\\ & =({\alpha }_h+{\mu }_h)({\mathcal{R}}_0-1)x\\ & \le 0.\end{array}$ This implies that there is no infectious steady state when ${\mathcal{R}}_0\le 1$.

Next, we suppose ${\mathcal{R}}_0>1$. Note that $H\left(0\right)=0,H\left(\frac{{\lambda }_h}{{\alpha }_h+{\mu }_h}\right)=-{\lambda }_h<0$, and $\begin{array}{rl}{H}^{\mathrm{\prime }}\left(0\right)& =\frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}{\xi }_h{\alpha }_h+\frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}\cdot \frac{{\xi }_v{\alpha }_v}{{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\cdot \eta {\alpha }_h-({\alpha }_h+{\mu }_h)\\ & =({\alpha }_h+{\mu }_h)({\mathcal{R}}_0-1)\\ & >0.\end{array}$ It follows that $H\left(x\right)>0$ if x>0 is sufficiently small. By the Intermediate Value Theorem, $H\left(x\right)$ has at least one positive zero and hence there is at least one infectious steady state. We claim that (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) has a unique infectious steady state. Otherwise, suppose that there exist at least two infectious steady states, say $E^{\ast }=(S_h^{\ast },E_h^{\ast },i_h^{\ast },S_v^{\ast },E_v^{\ast },i_v^{\ast })$ and $\overline{E}=({\overline{S}}_h,{\overline{E}}_h,{\overline{i}}_h,{\overline{S}}_v,{\overline{E}}_v,{\overline{i}}_v)$. Without loss of generality, we can assume that ${\overline{E}}_h>E_h^{\ast }$. This implies ${\overline{S}}_v=p\left({\overline{E}}_h\right)<p\left(E_h^{\ast }\right)=S_v^{\ast }$ since p is decreasing. Let $b=\frac{{\overline{E}}_h}{E_h^{\ast }}$, which is larger than 1. We have $\begin{array}{r}{E}_v^{\ast }=\frac{f(S_v^{\ast },J^{\ast })}{{\alpha }_v+{\mu }_v}>\frac{f({\overline{S}}_v,J^{\ast })}{{\alpha }_v+{\mu }_v}=\frac{f({\overline{S}}_v,\frac{1}{b}\overline{J})}{{\alpha }_v+{\mu }_v}>\frac{1}{b}\frac{f({\overline{S}}_v,\overline{J})}{{\alpha }_v+{\mu }_v}=\frac{1}{b}{\overline{E}}_v.\end{array}$ The first inequality comes from the monotonicity of f in the first independent variable while the second inequality comes from the concavity of f with respect to the second independent variable. Similarly, we have $\begin{array}{rl}{E}_h^{\ast }& =\frac{\varphi (S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast })}{{\alpha }_h+{\mu }_h}\\ & >\frac{\varphi ({\overline{S}}_h,{\alpha }_h{\xi }_h{E}_h^{\ast })+\psi ({\overline{S}}_h,{\alpha }_v{\xi }_v{E}_v^{\ast })}{{\alpha }_h+{\mu }_h}\\ & >\frac{\varphi ({\overline{S}}_h,\frac{1}{b}{\alpha }_h{\xi }_h{\overline{E}}_h)+\psi ({\overline{S}}_h,\frac{1}{b}{\alpha }_v{\xi }_v{\overline{E}}_v)}{{\alpha }_h+{\mu }_h}\\ & \ge \frac{1}{b}\cdot \frac{\varphi ({\overline{S}}_h,{\xi }_h{\alpha }_h{\overline{E}}_h)+\psi ({\overline{S}}_h,{\xi }_v{\alpha }_v{\overline{E}}_v)}{{\alpha }_h+{\mu }_h}\\ & =\frac{1}{b}{\overline{E}}_h,\end{array}$ which is a contradiction to $E_h^{\ast }=\frac{1}{b}{\overline{E}}_h$. This proves the uniqueness of infectious steady states.

The following theorem summarizes the conditions in terms of the basic reproduction number on the existence of steady states.

Theorem 3.1

1. If ${\mathcal{R}}_0\le 1,$ then system (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) only has the disease-free steady state $E^0$.

2. If ${\mathcal{R}}_0>1,$ then besides $E^0,$ system (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) also has a unique infectious steady state, denoted by $E^{\ast }$.

Now, we investigate the local stability of steady states. Let $\overline{E}=({\overline{S}}_h,{\overline{E}}_h,{\overline{i}}_h,{\overline{S}}_v,{\overline{E}}_v,{\overline{i}}_v)$ be a steady state of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ). Linearizing (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) at the steady state leads to (9) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& =-({\mu }_h+\mathrm{\Lambda })S_h\left(t\right)-\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & -\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a,\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\mathrm{\Lambda }{S}_h\left(t\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right)+\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & +\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a,\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& =-\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a-({\mu }_v+\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v})S_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a+\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v}{S}_v\left(t\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\end{array}$(9) where $\mathrm{\Lambda }=\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{S}_h}+\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{S}_h}.$ Letting $S_h\left(t\right)=x_0{e}^{\tau t},E_h\left(t\right)=y_0{e}^{\tau t},i_h(t,a)=z_0\left(a\right)e^{\tau t}$, and $S_v\left(t\right)=u_0{e}^{\tau t},E_v\left(t\right)=v_0{e}^{\tau t},i_v(t,a)=w_0\left(a\right)e^{\tau t}$, and substituting them into (9(9) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& =-({\mu }_h+\mathrm{\Lambda })S_h\left(t\right)-\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & -\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a,\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\mathrm{\Lambda }{S}_h\left(t\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right)+\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & +\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a,\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& =-\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a-({\mu }_v+\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v})S_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a+\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v}{S}_v\left(t\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\end{array}$(9) ), from the third equation and the seventh equation, we can get $z_0\left(a\right)=z_0\left(0\right){\sigma }_h\left(a\right)e^{-\tau a}$ and $w_0\left(a\right)=w_0\left(0\right){\sigma }_v\left(a\right)e^{-\tau a}$. Then system (9(9) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& =-({\mu }_h+\mathrm{\Lambda })S_h\left(t\right)-\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & -\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a,\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\mathrm{\Lambda }{S}_h\left(t\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right)+\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & +\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a,\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& =-\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a-({\mu }_v+\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v})S_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a+\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v}{S}_v\left(t\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\end{array}$(9) ) becomes $\begin{array}{rl}(\tau +{\mu }_h+\mathrm{\Lambda })x_0+\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}\widehat{{\xi }_h}\left(\tau \right)z_0\left(0\right)+\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}\widehat{{\xi }_v}\left(\tau \right)w_0\left(0\right)& =0,\\ -\mathrm{\Lambda }{x}_0+(\tau +{\alpha }_h+{\mu }_h)y_0-\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}\widehat{{\xi }_h}\left(\tau \right)z_0\left(0\right)-\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}\widehat{{\xi }_v}\left(\tau \right)w_0\left(0\right)& =0,\\ {\alpha }_h{y}_0-z_0\left(0\right)& =0,\\ \frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}\hat{\eta }\left(\tau \right)z_0\left(0\right)+(\tau +{\mu }_v+\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v})u_0& =0,\\ -\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}\hat{\eta }\left(\tau \right)z_0\left(0\right)-\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v}{u}_0+(\tau +{\alpha }_v+{\mu }_v)v_0& =0,\\ {\alpha }_v{v}_0-w_0\left(0\right)& =0,\end{array}$ where $\begin{array}{rl}{\hat{\xi }}_h\left(\tau \right)& ={\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right){\sigma }_h\left(a\right)e^{-\tau a}\mathrm{d}a,\\ {\hat{\xi }}_v\left(\tau \right)& ={\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right){\sigma }_v\left(a\right)e^{-\tau a}\mathrm{d}a,\\ \hat{\eta }\left(\tau \right)& ={\int }_0^{\mathrm{\infty }}{k}_h\left(a\right){\sigma }_h\left(a\right)e^{-\tau a}\mathrm{d}a.\end{array}$ This leads to the characteristic equation

$\mathrm{\Delta }\left(\overline{E}\right)=\left|\begin{array}{cccccc}\tau +{\mu }_h+\mathrm{\Lambda }& 0& \frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}{\hat{\xi }}_h\left(\tau \right)& 0& 0& \frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}{\hat{\xi }}_v\left(\tau \right)\\ -\mathrm{\Lambda }& \tau +{\alpha }_h+{\mu }_h& -\frac{\mathrm{\partial }\varphi ({\overline{S}}_h,{\overline{J}}_h)}{\mathrm{\partial }{J}_h}{\hat{\xi }}_h\left(\tau \right)& 0& 0& -\frac{\mathrm{\partial }\psi ({\overline{S}}_h,{\overline{J}}_v)}{\mathrm{\partial }{J}_v}{\hat{\xi }}_v\left(\tau \right)\\ 0& {\alpha }_h& -1& 0& 0& 0\\ 0& 0& \frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}\hat{\eta }\left(\tau \right)& \tau +{\mu }_v+\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v}& 0& 0\\ 0& 0& -\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }J}\hat{\eta }\left(\tau \right)& -\frac{\mathrm{\partial }f({\overline{S}}_v,\overline{J})}{\mathrm{\partial }{S}_v}& \tau +{\alpha }_v+{\mu }_v& 0\\ 0& 0& 0& 0& {\alpha }_v& -1\end{array}\right|=0.$The steady state $\overline{E}$ is locally asymptotically stable if all eigenvalues of the characteristic equation have negative real parts and it is unstable if at least one eigenvalue has a positive real part. For more detail, we refer the readers to Martcheva and Thieme [40].

Theorem 3.2

1. When ${\mathcal{R}}_0<1,$ the infection-free steady state $E^0$ of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) is locally asymptotically stable. When ${\mathcal{R}}_0>1,$ it is unstable.

2. The infectious steady state $E^{\ast }$ of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) is locally asymptotically stable when ${\mathcal{R}}_0>1$.

Proof.

(i) Note that $\frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{S}_h}=\frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{S}_h}=\frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }{S}_v}=0$ by $\left({\mathbf{C}}_2\right)$. The characteristic equation at $E^0$ is $\mathrm{\Delta }\left(E^0\right)=(\tau +{\mu }_h)(\tau +{\mu }_v)(\tau +{\alpha }_h+{\mu }_h)(\tau +{\alpha }_v+{\mu }_v)g_1\left(\tau \right)=0,$ where $\begin{array}{rl}{g}_1\left(\tau \right)& =1-\frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}\cdot \frac{{\hat{\xi }}_h\left(\tau \right){\alpha }_h}{\tau +{\alpha }_h+{\mu }_h}-\frac{{\alpha }_h\hat{\eta }\left(\tau \right)}{\tau +{\alpha }_h+{\mu }_h}\\ & \cdot \frac{{\alpha }_v{\hat{\xi }}_v\left(\tau \right)}{\tau +{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}.\end{array}$ The roots of $g_1\left(\tau \right)=0$ determines the stability of $E^0$. If ${\mathcal{R}}_0>1$, then $g_1\left(0\right)=1-{\mathcal{R}}_0<0$. We also have $\underset{\tau \to \mathrm{\infty }}{lim}{g}_1\left(\tau \right)=1>0$. Therefore, the equation $g_1\left(\tau \right)=0$ admits a positive root, which shows that E⁰ is unstable. Next, we claim that the root of $g_1\left(\tau \right)=0$ has negative real parts when ${\mathcal{R}}_0<1$. Prove by contradiction. If τ₀ is a root with non-negative real part, then from the definition of g₁ we have $\begin{array}{rl}1& =\left|\frac{{\alpha }_h{\hat{\xi }}_h\left({\tau }_0\right)}{{\tau }_0+{\alpha }_h+{\mu }_h}\cdot \frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}+\frac{{\alpha }_h\hat{\eta }\left({\tau }_0\right)}{{\tau }_0+{\alpha }_h+{\mu }_h}\cdot \frac{{\alpha }_v{\xi }_v\left({\tau }_0\right)}{{\tau }_0+{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\right|\\ & \le \frac{{\alpha }_h{\xi }_h}{{\alpha }_h+{\mu }_h}\cdot \frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}+\frac{{\alpha }_h\eta }{{\alpha }_h+{\mu }_h}\cdot \frac{{\alpha }_v{\xi }_v}{{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\\ & ={\mathcal{R}}_0,\end{array}$ which leads to a contradiction. This proves the claim and hence E⁰ is locally asymptotically stable when ${\mathcal{R}}_0<1$.

(ii) When ${\mathcal{R}}_0>1$, the characteristic equation at the infectious steady state $E^{\ast }$ is (10) $\begin{array}{rl}\mathrm{\Delta }\left(E^{\ast }\right)& =(\tau +{\mu }_h+{\mathrm{\Lambda }}^{\ast })(\tau +{\alpha }_h+{\mu }_h)\left(\tau +{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v}\right)(\tau +{\alpha }_v+{\mu }_v)\\ & -{\alpha }_h(\tau +{\mu }_h)\frac{\mathrm{\partial }\varphi (S_h^{\ast },J_h^{\ast })}{\mathrm{\partial }{J}_h}{\hat{\xi }}_h\left(\tau \right)\left(\tau +{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v}\right)(\tau +{\alpha }_v+{\mu }_v)\\ & -{\alpha }_h(\tau +{\mu }_h){\alpha }_v{\hat{\xi }}_v\left(\tau \right)\frac{\mathrm{\partial }\psi (S_h^{\ast },J_v^{\ast })}{\mathrm{\partial }{J}_v}(\tau +{\mu }_v)\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }J}\hat{\eta }\left(\tau \right)\\ & =0,\end{array}$(10) where ${\mathrm{\Lambda }}^{\ast }=\frac{\mathrm{\partial }\varphi (S_h^{\ast },J_h^{\ast })}{\mathrm{\partial }{S}_h}+\frac{\mathrm{\partial }\psi (S_h^{\ast },J_v^{\ast })}{\mathrm{\partial }{S}_h}$. We now prove that all roots of (10(10) $\begin{array}{rl}\mathrm{\Delta }\left(E^{\ast }\right)& =(\tau +{\mu }_h+{\mathrm{\Lambda }}^{\ast })(\tau +{\alpha }_h+{\mu }_h)\left(\tau +{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v}\right)(\tau +{\alpha }_v+{\mu }_v)\\ & -{\alpha }_h(\tau +{\mu }_h)\frac{\mathrm{\partial }\varphi (S_h^{\ast },J_h^{\ast })}{\mathrm{\partial }{J}_h}{\hat{\xi }}_h\left(\tau \right)\left(\tau +{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v}\right)(\tau +{\alpha }_v+{\mu }_v)\\ & -{\alpha }_h(\tau +{\mu }_h){\alpha }_v{\hat{\xi }}_v\left(\tau \right)\frac{\mathrm{\partial }\psi (S_h^{\ast },J_v^{\ast })}{\mathrm{\partial }{J}_v}(\tau +{\mu }_v)\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }J}\hat{\eta }\left(\tau \right)\\ & =0,\end{array}$(10) ) have negative real parts with contradictive arguments. If ${\tau }^0$ is a solution with nonnegative real part, then we have (11) $\begin{array}{rl}1& =\left|\frac{{\alpha }_h({\tau }^0+{\mu }_h)\frac{\mathrm{\partial }\varphi (S_h^{\ast },J_h^{\ast })}{\mathrm{\partial }{J}_h}{\hat{\xi }}_h\left({\tau }^0\right)({\tau }^0+{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v})({\tau }^0+{\alpha }_v+{\mu }_v)}{({\tau }^0+{\mu }_h+{\mathrm{\Lambda }}^{\ast })({\tau }^0+{\alpha }_h+{\mu }_h)({\tau }^0+{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v})({\tau }^0+{\alpha }_v+{\mu }_v)}\right.\\ & +\left.\frac{{\alpha }_h({\tau }^0+{\mu }_h){\alpha }_v{\hat{\xi }}_v\left({\tau }^0\right)\frac{\mathrm{\partial }\psi (S_h^{\ast },J_v^{\ast })}{\mathrm{\partial }{J}_v}({\tau }^0+{\mu }_v)\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }J}\hat{\eta }\left({\tau }^0\right)}{({\tau }^0+{\mu }_h+{\mathrm{\Lambda }}^{\ast })({\tau }^0+{\alpha }_h+{\mu }_h)({\tau }^0+{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v})({\tau }^0+{\alpha }_v+{\mu }_v)}\right|\\ & <\frac{{\alpha }_h{\xi }_h}{{\alpha }_h+{\mu }_h}\cdot \frac{\mathrm{\partial }\varphi (S_h^{\ast },J_h^{\ast })}{\mathrm{\partial }{J}_h}+\frac{{\alpha }_h\eta }{{\alpha }_h+{\mu }_h}\cdot \frac{{\alpha }_v{\xi }_v}{{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }\psi (S_h^{\ast },J_v^{\ast })}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }J}.\end{array}$(11) However, since $i_v^{\ast }\left(0\right)={\alpha }_v{E}_v^{\ast }=\frac{{\alpha }_v}{{\alpha }_v+{\mu }_v}f\left(S_v^{\ast },{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a\right)=\frac{{\alpha }_v}{{\alpha }_v+{\mu }_v}f\left(p\right(E_h^{\ast }),\eta {\alpha }_h{E}_h^{\ast })$ and $i_h^{\ast }\left(0\right)={\alpha }_h{E}_h^{\ast }=\frac{{\alpha }_h}{{\alpha }_h+{\mu }_h}\left(\varphi \left(S_h^{\ast },{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a\right)+\psi \left(S_h^{\ast },{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v^{\ast }\left(a\right)\mathrm{d}a\right)\right),$ we have $\begin{array}{rl}1& =\frac{1}{{\alpha }_h+{\mu }_h}\left(\frac{\varphi (S_h^{\ast },{\int }_0^{\mathrm{\infty }}{\text{β}}_h(a\left)i_h^{\ast }\right(a\left)\mathrm{d}a\right)}{E_h^{\ast }}+\frac{\psi (S_h^{\ast },{\int }_0^{\mathrm{\infty }}{\text{β}}_v(a\left)i_v^{\ast }\right(a\left)\mathrm{d}a\right)}{E_h^{\ast }}\right)\\ & =\frac{1}{{\alpha }_h+{\mu }_h}\frac{\varphi (S_h^{\ast },{\xi }_h{\alpha }_h{E}_h^{\ast })}{E_h^{\ast }}+\frac{1}{{\alpha }_h+{\mu }_h}\frac{\psi (S_h^{\ast },{\xi }_v{\alpha }_v{E}_v^{\ast })}{E_h^{\ast }}\\ & =\frac{1}{({\alpha }_h+{\mu }_h)E_h^{\ast }}\varphi (S_h^{\ast },{\xi }_h{\alpha }_h{E}_h^{\ast })+\frac{1}{({\alpha }_h+{\mu }_h)E_h^{\ast }}\psi \left(S_h^{\ast },{\xi }_v{\alpha }_v\cdot \frac{f\left(p\right(E_h^{\ast }),\eta {\alpha }_h{E}_h^{\ast })}{{\alpha }_v+{\mu }_v}\right)\\ & \ge \frac{{\xi }_h{\alpha }_h}{{\alpha }_h+{\mu }_h}\cdot \frac{\mathrm{\partial }\varphi (S_h^{\ast },J_h^{\ast })}{\mathrm{\partial }{J}_h}+\frac{\eta {\alpha }_h}{{\alpha }_h+{\mu }_h}\cdot \frac{\mathrm{\partial }\psi (S_h^{\ast },J_v^{\ast })}{\mathrm{\partial }{J}_v}\cdot \frac{{\xi }_v{\alpha }_v}{{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }J},\end{array}$ which is a contradiction to (11(11) $\begin{array}{rl}1& =\left|\frac{{\alpha }_h({\tau }^0+{\mu }_h)\frac{\mathrm{\partial }\varphi (S_h^{\ast },J_h^{\ast })}{\mathrm{\partial }{J}_h}{\hat{\xi }}_h\left({\tau }^0\right)({\tau }^0+{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v})({\tau }^0+{\alpha }_v+{\mu }_v)}{({\tau }^0+{\mu }_h+{\mathrm{\Lambda }}^{\ast })({\tau }^0+{\alpha }_h+{\mu }_h)({\tau }^0+{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v})({\tau }^0+{\alpha }_v+{\mu }_v)}\right.\\ & +\left.\frac{{\alpha }_h({\tau }^0+{\mu }_h){\alpha }_v{\hat{\xi }}_v\left({\tau }^0\right)\frac{\mathrm{\partial }\psi (S_h^{\ast },J_v^{\ast })}{\mathrm{\partial }{J}_v}({\tau }^0+{\mu }_v)\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }J}\hat{\eta }\left({\tau }^0\right)}{({\tau }^0+{\mu }_h+{\mathrm{\Lambda }}^{\ast })({\tau }^0+{\alpha }_h+{\mu }_h)({\tau }^0+{\mu }_v+\frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }{S}_v})({\tau }^0+{\alpha }_v+{\mu }_v)}\right|\\ & <\frac{{\alpha }_h{\xi }_h}{{\alpha }_h+{\mu }_h}\cdot \frac{\mathrm{\partial }\varphi (S_h^{\ast },J_h^{\ast })}{\mathrm{\partial }{J}_h}+\frac{{\alpha }_h\eta }{{\alpha }_h+{\mu }_h}\cdot \frac{{\alpha }_v{\xi }_v}{{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }\psi (S_h^{\ast },J_v^{\ast })}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f(S_v^{\ast },J^{\ast })}{\mathrm{\partial }J}.\end{array}$(11) ). This completes the proof.

4. Threshold dynamics

We first establish the global stability of the disease-free steady state $E^0$ of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) using the Fluctuation Lemma. Let $u_{\mathrm{\infty }}=\underset{t\to \mathrm{\infty }}{lim inf}u\left(t\right)$ and $u^{\mathrm{\infty }}=\underset{t\to \mathrm{\infty }}{lim sup}u\left(t\right).$

Lemma 4.1

Fluctuation Lemma [24]

Let $u:{\mathbb{R}}_{+}\to \mathbb{R}$ be bounded and continuously differentiable. Then there exist two sequences $\left\{s_n\right\}$ and $\left\{t_n\right\}$ such that $s_n\to \mathrm{\infty },t_n\to \mathrm{\infty },u\left(s_n\right)\to u_{\mathrm{\infty }},u^{\mathrm{\prime }}\left(s_n\right)\to 0,u\left(t_n\right)\to u^{\mathrm{\infty }}$, and $u^{\mathrm{\prime }}\left(t_n\right)\to 0$ as $n\to \mathrm{\infty }$.

Lemma 4.2

[25]

Suppose $k\in L_{+}^1(0,\mathrm{\infty })$ and B is a bounded function from ${\mathbb{R}}_{+}$ to $\mathbb{R}$. Then $\underset{t\to \mathrm{\infty }}{lim sup}{\int }_0^{t}k\left(\theta \right)B(t-\theta )\mathrm{d}\theta \le B^{\mathrm{\infty }}\parallel k{\parallel }_1.$

Theorem 4.3

The disease-free steady state $E^0$ of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) is globally asymptotically stable when ${\mathcal{R}}_0<1$.

Proof.

On the basis of the local stability, it suffices to prove that $E^0$ is globally attractive.

We first get the expressions of $i_h(t,a)$ and $i_v(t,a)$. Consider $\begin{array}{rl}\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a)\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right)\\ i_h(0,0)& =i_{h0}\end{array}$ and $\begin{array}{rl}\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a)\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right)\\ i_v(0,0)& =i_{v0}\end{array}$ Solving them by the method of characteristic lines gives the solutions (12) $i_h(t,a)=\left\{\begin{array}{ll}{\sigma }_h\left(a\right){\alpha }_h{E}_h(t-a),& 0\le a\le t\\ \frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}{i}_{h0}(a-t),& a>t>0\end{array}\right.$(12) and $i_v(t,a)=\left\{\begin{array}{ll}{\sigma }_v\left(a\right){\alpha }_v{E}_v(t-a),& 0\le a\le t,\\ \frac{{\sigma }_v\left(a\right)}{{\sigma }_v(a-t)}{i}_{v0}(a-t),& a>t>0.\end{array}\right.$

Next, we prove that $E_h^{\mathrm{\infty }}=E_v^{\mathrm{\infty }}=0$. From the second and sixth equations of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ), we see that (13) $E_h^{\mathrm{\infty }}\le \frac{\left(\varphi \right(S_h,J_h){)}^{\mathrm{\infty }}+(\psi (S_h,J_v){)}^{\mathrm{\infty }}}{{\alpha }_h+{\mu }_h}\text{and}{E}_v^{\mathrm{\infty }}\le \frac{\left(f\right(S_v,J){)}^{\mathrm{\infty }}}{{\alpha }_v+{\mu }_v}.$(13) By $\left({\mathbf{C}}_3\right)$ and (12(12) $i_h(t,a)=\left\{\begin{array}{ll}{\sigma }_h\left(a\right){\alpha }_h{E}_h(t-a),& 0\le a\le t\\ \frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}{i}_{h0}(a-t),& a>t>0\end{array}\right.$(12) ), we have $\begin{array}{rl}\varphi \left(S_h\right(t),J_h(t\left)\right)& \le \frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}{J}_h\left(t\right)\\ & =\frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & =\frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}\left({\int }_0^t{\text{β}}_h\left(a\right){\alpha }_h{E}_h(t-a){\sigma }_h\left(a\right)\mathrm{d}a\right.\\ & \left.+{\int }_t^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_{h0}(a-t)\frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}\mathrm{d}a\right)\\ & \le \frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}\left({\int }_0^t{\text{β}}_h\left(a\right){\alpha }_h{E}_h(t-a){\sigma }_h\left(a\right)\mathrm{d}a+e^{-{\mu }_{h}t}\parallel {\text{β}}_h{\parallel }_{\mathrm{\infty }}\parallel i_{h0}{\parallel }_1\right).\end{array}$ It follows from Lemma 4.2 that (14) $\left(\varphi \right(S_h,J_h){)}^{\mathrm{\infty }}\le \frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}{\xi }_h{\alpha }_h{E}_h^{\mathrm{\infty }}.$(14) Similarly, (15) $\left(\psi \right(S_h,J_v){)}^{\mathrm{\infty }}\le \frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}{\xi }_v{\alpha }_v{E}_v^{\mathrm{\infty }}\text{and}(f(S_v,J){)}^{\mathrm{\infty }}\le \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\eta {\alpha }_h{E}_h^{\mathrm{\infty }}.$(15) Therefore, it follows from (13(13) $E_h^{\mathrm{\infty }}\le \frac{\left(\varphi \right(S_h,J_h){)}^{\mathrm{\infty }}+(\psi (S_h,J_v){)}^{\mathrm{\infty }}}{{\alpha }_h+{\mu }_h}\text{and}{E}_v^{\mathrm{\infty }}\le \frac{\left(f\right(S_v,J){)}^{\mathrm{\infty }}}{{\alpha }_v+{\mu }_v}.$(13) )–(15(15) $\left(\psi \right(S_h,J_v){)}^{\mathrm{\infty }}\le \frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}{\xi }_v{\alpha }_v{E}_v^{\mathrm{\infty }}\text{and}(f(S_v,J){)}^{\mathrm{\infty }}\le \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\eta {\alpha }_h{E}_h^{\mathrm{\infty }}.$(15) ) that $E_h^{\mathrm{\infty }}\le \frac{{\alpha }_h{E}_h^{\mathrm{\infty }}}{{\alpha }_h+{\mu }_h}\left({\xi }_h\cdot \frac{\mathrm{\partial }\varphi \left(S_h^0.0\right)}{\mathrm{\partial }{J}_h}+\frac{{\xi }_v{\alpha }_v}{{\alpha }_v+{\mu }_v}\cdot \frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}\cdot \eta \cdot \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\right)={\mathcal{R}}_0{E}_h^{\mathrm{\infty }},$ which implies that $E_h^{\mathrm{\infty }}=0$ because ${\mathcal{R}}_0<1$. Then $E_v^{\mathrm{\infty }}=0$ follows immediately from the second inequalities in both (13(13) $E_h^{\mathrm{\infty }}\le \frac{\left(\varphi \right(S_h,J_h){)}^{\mathrm{\infty }}+(\psi (S_h,J_v){)}^{\mathrm{\infty }}}{{\alpha }_h+{\mu }_h}\text{and}{E}_v^{\mathrm{\infty }}\le \frac{\left(f\right(S_v,J){)}^{\mathrm{\infty }}}{{\alpha }_v+{\mu }_v}.$(13) ) and (15(15) $\left(\psi \right(S_h,J_v){)}^{\mathrm{\infty }}\le \frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}{\xi }_v{\alpha }_v{E}_v^{\mathrm{\infty }}\text{and}(f(S_v,J){)}^{\mathrm{\infty }}\le \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\eta {\alpha }_h{E}_h^{\mathrm{\infty }}.$(15) ).

Similarly as in the proof of (14(14) $\left(\varphi \right(S_h,J_h){)}^{\mathrm{\infty }}\le \frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}{\xi }_h{\alpha }_h{E}_h^{\mathrm{\infty }}.$(14) ), we can get $(\parallel i_h(t,\cdot ){\parallel }_1{)}^{\mathrm{\infty }}=(\parallel i_v(t,\cdot ){\parallel }_1{)}^{\mathrm{\infty }}=0$ and hence $\underset{t\to \mathrm{\infty }}{lim}\parallel i_h(t,\cdot ){\parallel }_1=\underset{t\to \mathrm{\infty }}{lim}\parallel i_v(t,\cdot ){\parallel }_1=0$.

Finally, we show $\underset{t\to \mathrm{\infty }}{lim}{S}_h\left(t\right)=S_h^0$ and $\underset{t\to \mathrm{\infty }}{lim}{S}_v\left(t\right)=S_v^0$. We only give the proof of $\underset{t\to \mathrm{\infty }}{lim}{S}_h\left(t\right)=S_h^0$ as that for the other is similar. Recall that $S_h^{\mathrm{\infty }}\le S_h^0$. It suffices to show $(S_h{)}_{\mathrm{\infty }}\ge S_h^0$. By Lemma 4.1, there exists a sequence $\left\{t_n\right\}$ such that $t_n\to \mathrm{\infty },S_h\left(t_n\right)\to (S_h{)}_{\mathrm{\infty }}$, and $\frac{\mathrm{d}{S}_h\left(t_n\right)}{\mathrm{d}t}\to 0$ as $n\to \mathrm{\infty }$. Note that $\underset{t\to \mathrm{\infty }}{lim}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a=0\text{and}\underset{t\to \mathrm{\infty }}{lim}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a=0$ because $\underset{t\to \mathrm{\infty }}{lim}\parallel i_h(t,\cdot ){\parallel }_1=\underset{t\to \mathrm{\infty }}{lim}\parallel i_v(t,\cdot ){\parallel }_1=0$. It follows from $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t_n\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t_n\right)-\varphi \left(S_h\left(t_n\right),{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t_n,a)\mathrm{d}a\right)\\ & -\psi \left(S_h\left(t_n\right),{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t_n,a)\mathrm{d}a\right)\end{array}$ by letting $n\to \mathrm{\infty }$ that $0={\lambda }_h-{\mu }_h(S_h{)}_{\mathrm{\infty }}$. Hence $(S_h{)}_{\mathrm{\infty }}=\frac{{\lambda }_h}{{\mu }_h}=S_h^0$.

In summary, we have shown that $\underset{t\to \mathrm{\infty }}{lim}\left(S_h\right(t),E_h(t),i_h(t,\cdot ),S_v(t),E_v(t),i_v(t,\cdot \left)\right)=E^0.$ This completes the proof.

In the remaining of this section, we establish the attractivity of the infectious steady state $E^{\ast }$ by constructing suitable Lyapunov functionals. For this purpose, we need the persistence of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ).

We start with the existence of a compact attractor $\mathcal{A}$ that attracts all points in $X_{+}$. The existence is established by applying the following result.

Lemma 4.4

[22]

If $\mathrm{\Phi }\left(t\right):X_{+}\to X_{+},t\in {\mathbb{R}}_{+}$ is asymptotically smooth, point dissipative, and orbits of bounded sets are bounded, then there exists a global attractor.

From Smith [51], if each forward invariant bounded closed set is attracted by a non-empty compact set, then the semiflow is asymptotically smooth.

Lemma 4.5

$\mathrm{\Phi }\left(t\right)={\mathrm{\Phi }}_1\left(t\right)+{\mathrm{\Phi }}_2\left(t\right)$ is asymptotically smooth if the following are satisfied: $\left(1\right){\mathrm{\Phi }}_2\left(t\right)$ is completely continuous;$\left(2\right)$ There exists a continuous function $\tilde{u}$ such that $\tilde{u}(t,\tilde{r})\to 0$ as $t\to \mathrm{\infty }$ and $\tilde{u}(t,\tilde{r})\ge \parallel {\mathrm{\Phi }}_1\left(t\right)x\parallel$ when $\parallel x\parallel <\tilde{r}$.

Theorem 4.6

If ${\mathcal{R}}_0>1,$ then there exists a global compact attractor $\mathcal{A}$ for the solution semiflow Φ of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) in $X_{+}$.

Proof.

It follows from (4(4) $\begin{array}{rl}\frac{\mathrm{d}{G}_h\left(t\right)}{\mathrm{d}t}& \le {\lambda }_h-{\mu }_h{S}_h\left(t\right)-{\mu }_h{E}_h\left(t\right)-{\int }_0^{\mathrm{\infty }}{\delta }_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & \le {\lambda }_h-{\mu }_h{G}_h\left(t\right)\end{array}$(4) ) and (5(5) $\frac{\mathrm{d}{N}_v\left(t\right)}{\mathrm{d}t}\le {\lambda }_v-{\mu }_v{N}_v\left(t\right),$(5) ) that Φ is point dissipative and orbits of bounded sets are bounded. In order to apply Lemma 4.4, we only need to prove that Φ is asymptotically smooth. For any $t\in {\mathbb{R}}_{+}$ and $x=(S_{h0},E_{h0},i_{h0},S_{v0},E_{v0},i_{v0})\in X_{+}$, let ${\mathrm{\Phi }}_1(t,x)=(0,0,{\hat{i}}_h(t,\cdot ),0,0,{\hat{i}}_v(t,\cdot \left)\right)$ and ${\mathrm{\Phi }}_2(t,x)=\left(S_h\right(t),E_h(t),{\tilde{i}}_h(t,\cdot ),S_v(t),E_v(t),{\tilde{i}}_v(t,\cdot \left)\right),$ where (16) $\begin{array}{rl}{\hat{i}}_h(t,a)& =\left\{\begin{array}{ll}0,& 0\le a\le t,\\ \frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}{i}_{h0}(a-t),& 0<t<a,\end{array}\right.\end{array}$(16) (17) $\begin{array}{rl}{\hat{i}}_v(t,a)& =\left\{\begin{array}{ll}0,& 0\le a\le t,\\ \frac{{\sigma }_v\left(a\right)}{{\sigma }_v(a-t)}{i}_{v0}(a-t),& 0<t<a,\end{array}\right.\end{array}$(17) and $\begin{array}{rl}{\tilde{i}}_h(t,a)& =i_h(t,a)-{\hat{i}}_h(t,a)=\left\{\begin{array}{ll}{\sigma }_h\left(a\right){\alpha }_h{E}_h(t-a),& 0\le a\le t,\\ 0,& 0<t<a,\end{array}\right.\\ {\tilde{i}}_v(t,a)& =i_v(t,a)-{\hat{i}}_v(t,a)=\left\{\begin{array}{ll}{\sigma }_v\left(a\right){\alpha }_v{E}_v(t-a),& 0\le a\le t,\\ 0,& 0<t<a.\end{array}\right.\end{array}$ Then $\mathrm{\Phi }={\mathrm{\Phi }}_1+{\mathrm{\Phi }}_2$. It is easy to see that ${\hat{i}}_h,{\hat{i}}_v,{\tilde{i}}_h$, and ${\tilde{i}}_v$ are nonnegative. Using (16(16) $\begin{array}{rl}{\hat{i}}_h(t,a)& =\left\{\begin{array}{ll}0,& 0\le a\le t,\\ \frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}{i}_{h0}(a-t),& 0<t<a,\end{array}\right.\end{array}$(16) ) and (17(17) $\begin{array}{rl}{\hat{i}}_v(t,a)& =\left\{\begin{array}{ll}0,& 0\le a\le t,\\ \frac{{\sigma }_v\left(a\right)}{{\sigma }_v(a-t)}{i}_{v0}(a-t),& 0<t<a,\end{array}\right.\end{array}$(17) ), we get $\begin{array}{rl}\parallel {\mathrm{\Phi }}_1(t,x)\parallel & =\parallel {\hat{i}}_h(t,\cdot ){\parallel }_1+\parallel {\hat{i}}_v(t,\cdot ){\parallel }_1\\ & ={\int }_t^{\mathrm{\infty }}{i}_{h0}(a-t)\frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}\mathrm{d}a+{\int }_t^{\mathrm{\infty }}{i}_{v0}(a-t)\frac{{\sigma }_v\left(a\right)}{{\sigma }_v(a-t)}\mathrm{d}a\\ & ={\int }_0^{\mathrm{\infty }}{i}_{h0}\left(a\right)\frac{{\sigma }_h(a+t)}{{\sigma }_h\left(a\right)}\mathrm{d}a+{\int }_0^{\mathrm{\infty }}{i}_{v0}\left(a\right)\frac{{\sigma }_v(a+t)}{{\sigma }_v\left(a\right)}\mathrm{d}a\\ & \le e^{-{\mu }_{h}t}\parallel i_{h0}{\parallel }_1+e^{-{\mu }_{v}t}\parallel i_{v0}{\parallel }_1\\ & \le e^{-\mu t}\parallel x\parallel ,\end{array}$ where $\mu =min\{{\mu }_h,{\mu }_v\}$. Thus, the second condition in Lemma 4.5 regarding ${\mathrm{\Phi }}_1$ is satisfied.

Now, we show ${\mathrm{\Phi }}_2$ is completely continuous, i.e., for a positive t and a bounded $\mathcal{B}\subseteq X_{+}$, the set $\left\{{\mathrm{\Phi }}_2\right(t,x)|x=(S_{h0},E_{h0},i_{h0},S_{v0},E_{v0},i_{v0})\in \mathcal{B}\},$ denoted by ${\mathcal{B}}_t$, is precompact. As in Chen et al. [12], we only need to prove that ${\mathcal{B}}_t(i_h,i_v)=\{\left({\tilde{i}}_h\right(t,\cdot ),{\tilde{i}}_v(t,\cdot \left)\right)|\left(S_h\right(t),E_h(t),{\tilde{i}}_h(t,\cdot ),S_v,E_v,{\tilde{i}}_v(t,\cdot \left)\right)\in {\mathcal{B}}_t\}$ is precompact by employing the Fréchet-Kolmogrov Theorem. We only verify the second condition (as the others are trivial to verify) that ${\mathcal{B}}_t(i_h,i_v)$ is uniformly continuous or (18) $\underset{k\to 0^{+}}{lim}\parallel {\tilde{i}}_h(t,\cdot )-{\tilde{i}}_h(t,\cdot +k){\parallel }_1=0=\underset{k\to 0^{+}}{lim}\parallel {\tilde{i}}_v(t,\cdot )-{\tilde{i}}_v(t,\cdot +k){\parallel }_1$(18) uniformly in ${\mathcal{B}}_t(i_h,i_v)$.

Note that, by (4(4) $\begin{array}{rl}\frac{\mathrm{d}{G}_h\left(t\right)}{\mathrm{d}t}& \le {\lambda }_h-{\mu }_h{S}_h\left(t\right)-{\mu }_h{E}_h\left(t\right)-{\int }_0^{\mathrm{\infty }}{\delta }_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & \le {\lambda }_h-{\mu }_h{G}_h\left(t\right)\end{array}$(4) ) and (5(5) $\frac{\mathrm{d}{N}_v\left(t\right)}{\mathrm{d}t}\le {\lambda }_v-{\mu }_v{N}_v\left(t\right),$(5) ) again, there exists C>0 such that $G_h\left(t\right)\le C$ and $N_v\left(t\right)\le C$ for all $t\ge 0$ and $x\in \mathcal{B}$.

We only show the first part of (18(18) $\underset{k\to 0^{+}}{lim}\parallel {\tilde{i}}_h(t,\cdot )-{\tilde{i}}_h(t,\cdot +k){\parallel }_1=0=\underset{k\to 0^{+}}{lim}\parallel {\tilde{i}}_v(t,\cdot )-{\tilde{i}}_v(t,\cdot +k){\parallel }_1$(18) ) holds as the proof for the second one is similar. Clearly, it holds when t = 0 since ${\tilde{i}}_h(0,\cdot )=0$. Without losing generality, let t be positive. When $k\to 0^{+}$, we let k be in the interval $(0,t)$. It follows that (19) $\begin{array}{rl}\parallel {\tilde{i}}_h(t,\cdot )-{\tilde{i}}_h(t,\cdot +k){\parallel }_1& ={\int }_0^{\mathrm{\infty }}|{\tilde{i}}_h(t,a)-{\tilde{i}}_h(t,a+k)|\mathrm{d}a\\ & ={\int }_0^{t-k}|{\alpha }_h{E}_h(t-a-k){\sigma }_h(a+k)-{\alpha }_h{E}_h(t-a){\sigma }_h\left(a\right)|\mathrm{d}a\\ & +{\int }_{t-k}^t{\alpha }_h{E}_h(t-a){\sigma }_h\left(a\right)\mathrm{d}a\\ & \le {\int }_0^{t-k}{\alpha }_h{E}_h(t-a-k)|{\sigma }_h(a+k)-{\sigma }_h\left(a\right)|\mathrm{d}a\\ & +{\int }_0^{t-k}{\alpha }_h|E_h(t-a-k)-E_h(t-a)|{\sigma }_h\left(a\right)\mathrm{d}a+Ck{\alpha }_h.\end{array}$(19) Recall that ${\sigma }_h\left(a\right)=e^{-{\int }_0^a({\mu }_h+{\rho }_h(s)+{\gamma }_h(s\left)\right)\mathrm{d}s}$ is a monotone decreasing function of a. It is clear that $\begin{array}{rl}{\sigma }_h\left(a\right)-{\sigma }_h(a+k)& ={\sigma }_h\left(a\right)\left(1-\frac{{\sigma }_h(a+k)}{{\sigma }_h\left(a\right)}\right)\\ & \le e^{-{\mu }_{h}a}\left(1-e^{-{\int }_a^{a+k}{\delta }_h\left(s\right)\mathrm{d}s}\right)\\ & \le e^{-{\mu }_{h}a}{\int }_a^{a+k}{\delta }_h\left(s\right)\mathrm{d}s\\ & \le e^{-{\mu }_{h}a}k\parallel {\delta }_h{\parallel }_{\mathrm{\infty }}.\end{array}$ Moreover, $\begin{array}{rl}\left|\frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}\right|& \le \varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)+({\alpha }_h+{\mu }_h)C\\ & \le \varphi (C,J_h(t\left)\right)+\psi (C,J_v(t\left)\right)+({\alpha }_h+{\mu }_h)C\\ & \le \frac{\mathrm{\partial }\varphi (C,0)}{\mathrm{\partial }{J}_h}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & +\frac{\mathrm{\partial }\psi (C,0)}{\mathrm{\partial }{J}_v}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a+({\alpha }_h+{\mu }_h)C\\ & \le \left(\frac{\mathrm{\partial }\varphi (C,0)}{\mathrm{\partial }{J}_h}\parallel {\text{β}}_h{\parallel }_{\mathrm{\infty }}+\frac{\mathrm{\partial }\psi (C,0)}{\mathrm{\partial }Jv}\parallel {\text{β}}_v{\parallel }_{\mathrm{\infty }}+{\alpha }_h+{\mu }_h\right)C\\ & \triangleq L_E.\end{array}$ Therefore, it follows from (19(19) $\begin{array}{rl}\parallel {\tilde{i}}_h(t,\cdot )-{\tilde{i}}_h(t,\cdot +k){\parallel }_1& ={\int }_0^{\mathrm{\infty }}|{\tilde{i}}_h(t,a)-{\tilde{i}}_h(t,a+k)|\mathrm{d}a\\ & ={\int }_0^{t-k}|{\alpha }_h{E}_h(t-a-k){\sigma }_h(a+k)-{\alpha }_h{E}_h(t-a){\sigma }_h\left(a\right)|\mathrm{d}a\\ & +{\int }_{t-k}^t{\alpha }_h{E}_h(t-a){\sigma }_h\left(a\right)\mathrm{d}a\\ & \le {\int }_0^{t-k}{\alpha }_h{E}_h(t-a-k)|{\sigma }_h(a+k)-{\sigma }_h\left(a\right)|\mathrm{d}a\\ & +{\int }_0^{t-k}{\alpha }_h|E_h(t-a-k)-E_h(t-a)|{\sigma }_h\left(a\right)\mathrm{d}a+Ck{\alpha }_h.\end{array}$(19) ) that $\parallel {\tilde{i}}_h(t,\cdot )-{\tilde{i}}_h(t,\cdot +k){\parallel }_1\le {\alpha }_{h}C\frac{k\parallel {\delta }_h{\parallel }_{\mathrm{\infty }}}{{\mu }_h}+{\alpha }_h{L}_{E}kt+Ck{\alpha }_h.$ This completes the proof.

Clearly, the global attractor $\mathcal{A}\subset \mathrm{\Omega }$. Since it is invariant, it can only contain points with a total trajectory through it. From the reference [51], a total trajectory $x:\mathbb{R}\to X_{+}$ of Φ requires $\mathrm{\Phi }(s,x(t\left)\right)=x(s+t)$ for $t\in \mathbb{R}$ and $s\in {\mathbb{R}}_{+}$. Thus $i_h(t,a)=i_h(t-a,0){\sigma }_h\left(a\right)$ and $i_v(t,a)=i_v(t-a,0){\sigma }_v\left(a\right)$ for a>0. The α-limit of a total trajectory $x\left(t\right)$ that passes through the initial value $x\left(0\right)=x_0$ is $\alpha \left(x_0\right)=\bigcap _{t\le 0}\overline{\bigcup _{s\le t}\left\{x\right(s\left)\right\}}.$ We will apply Theorem 5.2 in [51] to establish the uniform persistence. For this purpose, we define $\rho :X_{+}\to {\mathbb{R}}_{+}$ by $\rho (S_h,E_h,i_h,S_v,E_v,i_v)={\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h\left(a\right)\mathrm{d}a+{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v\left(a\right)\mathrm{d}a$ for $(S_h,E_h,i_h,S_v,E_v,i_v)\in X_{+}$. Clearly ρ is continuous and not identically zero. A biological interpretation of ρ is the initial infection force for the host. Let $X_0=\{x_0=(S_{h0},E_{h0},i_{h0},S_{v0},E_{v0},i_{v0})\in X_{+}|\rho (\mathrm{\Phi }(t,x_0))=0\text{ for all }t\ge 0\}.$ We say that Φ is uniformly weakly ρ-persistent if there exists $\gamma >0$ such that $\underset{t\to \mathrm{\infty }}{lim sup}\rho \left(\mathrm{\Phi }\right(t,x_0\left)\right)>\gamma \text{whenever }\rho \left(x_0\right)>0$ and is uniformly ρ-persistent if we can replace lim sup by lim inf above (see [51, pp. 126]). Note that $\rho \left(\mathrm{\Phi }\right(t,x_0\left)\right)=J_h\left(t\right)+J_v\left(t\right)$ for all $t\ge 0$ and $x_0\in X_{+}$.

To prove the uniform weak ρ-persistence, we need the following result. Though the result has been used in the literature, to the best of our knowledge, it has been explicitly stated and proved recently for the first time by Bentout et al. [4].

Lemma 4.7

Let $x_0\in X_{+}$ such that $\rho \left(x_0\right)>0$. Then there exists a $T\ge 0$ such that $\rho \left(\mathrm{\Phi }\right(t,x_0\left)\right)>0$ for $t\ge T$.

Proof.

It is easy to see that $S_h\left(t\right)>0$ and $S_v\left(t\right)>0$ for all t>0. Also by (4) and (5(5) $\frac{\mathrm{d}{N}_v\left(t\right)}{\mathrm{d}t}\le {\lambda }_v-{\mu }_v{N}_v\left(t\right),$(5) ), we know that $\left\{\mathrm{\Phi }\right(t,x_0)|t\ge 0\}$ is bounded. Let C>0 such that $\parallel \mathrm{\Phi }(t,x_0)\parallel \le C$ for $t\ge 0$. Then for $t\ge 0$, using the Lipschitz continuity on ϕ and ψ, we have $\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}\ge {\lambda }_h-({\mu }_h+L_u+K_u)S_h\left(t\right),$ which implies that $\underset{t\to \mathrm{\infty }}{lim inf}{S}_h\left(t\right)\ge \frac{{\lambda }_h}{{\mu }_h+L_u+K_u}$. Similarly, $\underset{t\to \mathrm{\infty }}{lim inf}{S}_v\left(t\right)\ge \frac{{\lambda }_v}{{\mu }_v+M_u}$. Therefore, without loss of generality (we can make a translation), we can assume that $\rho \left(x_0\right)>0$, and there exist $\zeta >0$ and $\gamma >0$ such that $S_h\left(t\right)\ge \zeta ,S_v\left(t\right)\ge \zeta ,J_h\left(t\right)\le \gamma ,J_v\left(t\right)\le \gamma$, and $J\left(t\right)\le \gamma$ for $t\ge 0$.

Now, using (12(12) $i_h(t,a)=\left\{\begin{array}{ll}{\sigma }_h\left(a\right){\alpha }_h{E}_h(t-a),& 0\le a\le t\\ \frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}{i}_{h0}(a-t),& a>t>0\end{array}\right.$(12) ) and the monotonicity and concavity of ϕ and ψ, we get $\begin{array}{rl}{J}_h\left(t\right)& ={\int }_t^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a+{\int }_0^t{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & ={\int }_t^{\mathrm{\infty }}{\text{β}}_h\left(a\right)\frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}{i}_{h0}(a-t)\mathrm{d}a+{\int }_0^t{\text{β}}_h\left(a\right){\alpha }_h{\sigma }_h\left(a\right)E_h(t-a)\mathrm{d}a\\ & ={\int }_0^{\mathrm{\infty }}{\text{β}}_h(t+a)\frac{{\sigma }_h(a+t)}{{\sigma }_h\left(a\right)}{i}_{h0}\left(a\right)\mathrm{d}a+{\int }_0^t{\alpha }_h{\text{β}}_h\left(a\right){\sigma }_h\left(a\right)[e^{-({\alpha }_h+{\mu }_h)(t-a)}{E}_h\left(0\right)\\ & \left.+{\int }_0^{t-a}{e}^{-({\alpha }_h+{\mu }_h)(t-a-s)}\left[\varphi \right(S_h\left(s\right),J_h\left(s\right))+\psi (S_h\left(s\right),J_v\left(s\right)\left)\right]\mathrm{d}s\right]\mathrm{d}a\\ & +{\int }_t^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\\ & \ge {\int }_0^t{\int }_0^{t-a}{\alpha }_h{\text{β}}_h\left(a\right){\sigma }_h\left(a\right)e^{-({\alpha }_h+{\mu }_h)(t-a-s)}\\ & \times \left[\frac{\mathrm{\partial }\varphi (\zeta ,\gamma )}{\mathrm{\partial }{J}_h}{J}_h\left(s\right)+\frac{\mathrm{\partial }\psi (\zeta ,\gamma )}{\mathrm{\partial }{J}_v}{J}_v\left(s\right)\right]\mathrm{d}s\mathrm{d}a+R_h\left(t\right),\end{array}$ where $R_h\left(t\right)={\int }_0^{\mathrm{\infty }}{\text{β}}_h(t+a)\frac{{\sigma }_h(a+t)}{{\sigma }_h\left(a\right)}{i}_{h0}\left(a\right)\mathrm{d}a$. Changing the order of integration gives us $J_h\left(t\right)\ge {\alpha }_{h}min\left\{\frac{\mathrm{\partial }\varphi (\zeta ,\gamma )}{\mathrm{\partial }{J}_h},\frac{\mathrm{\partial }\psi (\zeta ,\gamma )}{\mathrm{\partial }{J}_v}\right\}{\int }_0^t\left[J_h\right(s)+J_v(s\left)\right]{\kappa }_h(t-s)\mathrm{d}s+R_h\left(t\right),$ where ${\kappa }_h\left(t\right)={\int }_0^t{\text{β}}_h\left(a\right){\sigma }_h\left(a\right)e^{-({\alpha }_h+{\mu }_h)(t-a)}\mathrm{d}a$. With a similar but tedious computation, we can get $J_v\left(t\right)\ge {\alpha }_h{\alpha }_v\frac{\mathrm{\partial }f(\zeta ,\gamma )}{\mathrm{\partial }J}min\left\{\frac{\mathrm{\partial }\varphi (\zeta ,\gamma )}{\mathrm{\partial }{J}_h},\frac{\mathrm{\partial }\psi (\zeta ,\gamma )}{\mathrm{\partial }{J}_v}\right\}{\int }_0^t\left[J_h\right(s)+J_v(s\left)\right]{\kappa }_v(t-s)\mathrm{d}s+R_v\left(t\right),$ where ${\kappa }_v\left(t\right)={\int }_0^t{\int }_0^{t-a}{\int }_{\tau }^{t-a}{\text{β}}_v\left(a\right){\sigma }_v\left(a\right)k_h\left(\tau \right){\sigma }_h\left(\tau \right)e^{-({\alpha }_v+{\mu }_v)(t-a-\eta )}{e}^{-({\alpha }_h+{\mu }_h)(\eta -\tau )}\mathrm{d}\eta \mathrm{d}\tau \mathrm{d}a$ and $R_v\left(t\right)={\int }_0^{\mathrm{\infty }}{\text{β}}_v(t+a)\frac{{\sigma }_v(t+a)}{{\sigma }_v\left(a\right)}{i}_{v0}\left(a\right)\mathrm{d}a.$ In summary, we have obtained, $\rho \left(\mathrm{\Phi }\right(t,x_0\left)\right)\ge {\int }_0^t\rho \left(\mathrm{\Phi }\right(s,x_0\left)\right)\kappa (t-s)\mathrm{d}s+R\left(t\right),$ where $\kappa \left(t\right)={\alpha }_{h}min\left\{\frac{\mathrm{\partial }\varphi (\zeta ,\gamma )}{\mathrm{\partial }{J}_h},\frac{\mathrm{\partial }\psi (\zeta ,\gamma )}{\mathrm{\partial }{J}_v}\right\}\left[{\kappa }_h\left(t\right)+{\alpha }_v\frac{\mathrm{\partial }f(\zeta ,\gamma )}{\mathrm{\partial }J}{\kappa }_v\left(t\right)\right]$ and $R\left(t\right)=R_h\left(t\right)+R_v\left(t\right)$. Note that $R\left(0\right)=\rho \left(x_0\right)>0$ and $R\left(t\right)$ is locally integrable. Moreover, by $\left({\mathbf{C}}_1\right)$, we know that κ is not zero a.e. By Corollary B.6 [51], there exists $T\ge 0$ (dependent only on κ and not on R) such that $\rho \left(\mathrm{\Phi }\right(t,x_0\left)\right)>0$ for $t\ge T$.

Theorem 4.8

Suppose ${\mathcal{R}}_0>1$. Then model (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) is uniformly weakly ρ-persistent.

Proof.

By contradiction, we suppose that (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) is not uniformly weakly ρ-persistent. Then for any $\epsilon >0$, there is an $x^{\epsilon }=(S_{h0}^{\epsilon },E_{h0}^{\epsilon },i_{h0}^{\epsilon },S_{v0}^{\epsilon },E_{v0}^{\epsilon },i_{v0}^{\epsilon })\in X_{+}$ such that $\rho \left(x^{\epsilon }\right)>0$ and $\underset{t\to \mathrm{\infty }}{lim sup}\rho \left(\mathrm{\Phi }\right(t,x^{\epsilon }\left)\right)\le \epsilon .$ Since ${\mathcal{R}}_0>1$, we can choose a positive ${\epsilon }_0$ such that $\begin{array}{rl}{\epsilon }_1& =\frac{{\lambda }_h-(L_v+K_v){\epsilon }_0}{{\mu }_h}-{\epsilon }_0>0,\\ {\epsilon }_2& =\frac{{\alpha }_h\eta (L_v+K_v){\epsilon }_0}{{\alpha }_h+{\mu }_h}+{\epsilon }_0,\\ {\epsilon }_3& =\frac{{\lambda }_v-M_v{\epsilon }_2}{{\mu }_v}-{\epsilon }_0>0,\end{array}$ and (20) $\begin{array}{rl}& \frac{\mathrm{\partial }\varphi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_h}\cdot \frac{{\alpha }_h\widehat{{\text{β}}_h{\sigma }_h}\left({\epsilon }_0\right)}{{\alpha }_h+{\mu }_h+{\epsilon }_0}+\frac{\mathrm{\partial }\psi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f({\epsilon }_3,{\epsilon }_2)}{\mathrm{\partial }J}\\ & \cdot \frac{{\alpha }_h\widehat{k_h{\sigma }_h}}{\alpha +h+{\mu }_h+{\epsilon }_0}\cdot \frac{{\alpha }_v\widehat{{\text{β}}_v{\sigma }_v}\left({\epsilon }_0\right)}{{\alpha }_v+{\mu }_v+{\epsilon }_0}>1.\end{array}$(20) Here and in the remaining of the proof, the notation $\hat{\cdot }$ represents the Laplace transform. Given this ${\epsilon }_0$, we can find an $x^{\frac{{\epsilon }_0}{2}}\in X_{+}$ with $\rho \left(x^{\frac{{\epsilon }_0}{2}}\right)>0$ such that $\underset{t\to \mathrm{\infty }}{lim sup}\rho \left(\mathrm{\Phi }\right(t,x^{\frac{{\epsilon }_0}{2}}\left)\right)\le \frac{{\epsilon }_0}{2}$. For simplicity of notation, we write x for $x^{\frac{{\epsilon }_0}{2}}$. In the following, we will first establish some estimates on $S_h,S_v$, and J, and then use Laplace transform to obtain a contradiction.

From $\underset{t\to \mathrm{\infty }}{lim sup}\rho \left(\mathrm{\Phi }\right(t,x\left)\right)\le \frac{{\epsilon }_0}{2}$, we can find $t_0>0$ such that when $t\ge t_0,\rho \left(\mathrm{\Phi }\right(t,x\left)\right)<{\epsilon }_0$. We further assume that $t_0$ is 0 since we can replace x with $\mathrm{\Phi }(t_0,x)$. By the first equation of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ), $\left({\mathbf{C}}_2\right)$, and $\left({\mathbf{C}}_4\right)$, we have $\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}\ge {\lambda }_h-(L_v+K_v){\epsilon }_0-{\mu }_h{S}_h\left(t\right)\text{for }t\in {\mathbb{R}}_{+}\text{,}$ which implies that $\underset{t\to \mathrm{\infty }}{lim inf}{S}_h\left(t\right)\ge \frac{{\lambda }_h-(L_v+K_v){\epsilon }_0}{{\mu }_h}$. Without losing generality, we let $S_h\left(t\right)\ge {\epsilon }_1$. Similarly, from the second equation of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ), we have $\underset{t\to \mathrm{\infty }}{lim sup}{E}_h\left(t\right)\le \frac{(L_v+K_v){\epsilon }_0}{{\alpha }_h+{\mu }_h}$. This, combined with $\begin{array}{rl}J\left(t\right)& ={\int }_0^t{k}_h\left(a\right){\sigma }_h\left(a\right){\alpha }_h{E}_h(t-a)\mathrm{d}a+{\int }_t^{\mathrm{\infty }}{k}_h\left(a\right)\frac{{\sigma }_h\left(a\right)}{{\sigma }_h(a-t)}{i}_{h0}(a-t)\mathrm{d}a\\ & \le {\int }_0^t{k}_h\left(a\right){\sigma }_h\left(a\right){\alpha }_h{E}_h(t-a)\mathrm{d}a+e^{-{\mu }_{h}t}{\int }_t^{\mathrm{\infty }}{k}_h\left(a\right)i_{h0}(a-t)\mathrm{d}a\\ & \le {\int }_0^t{k}_h\left(a\right){\sigma }_h\left(a\right){\alpha }_h{E}_h(t-a)\mathrm{d}a+e^{-{\mu }_{h}t}\parallel k_h{\parallel }_{\mathrm{\infty }}\parallel i_{h0}{\parallel }_1\end{array}$ and Lemma 4.2, gives $\underset{t\to \mathrm{\infty }}{lim sup}J\left(t\right)\le \frac{{\alpha }_h\eta (L_v+K_v){\epsilon }_0}{{\alpha }_h+{\mu }_h}$. Again, we let $J\left(t\right)\le {\epsilon }_2$. Using the similar argument as those for obtaining $S_h\left(t\right)\ge {\epsilon }_1$, we can get $S_v\left(t\right)\ge {\epsilon }_3$ for $t\in {\mathbb{R}}_{+}$.

Now, from the second equation of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ), $\left({\mathbf{C}}_2\right)$, and $\left({\mathbf{C}}_3\right)$, we get $\frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}\ge \frac{\mathrm{\partial }\varphi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_h}{J}_h\left(t\right)+\frac{\mathrm{\partial }\psi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_v}{J}_v\left(t\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right)\text{for }t\in {\mathbb{R}}_{+}\text{.}$ We take Laplace transforms to get $\lambda {\hat{E}}_h\left(\lambda \right)-E_h\left(0\right)\ge \frac{\mathrm{\partial }\varphi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_h}{\hat{J}}_h\left(\lambda \right)+\frac{\mathrm{\partial }\psi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_v}{\hat{J}}_v\left(\lambda \right)-({\alpha }_h+{\mu }_h){\hat{E}}_h\left(\lambda \right)\text{for }\lambda \in {\mathbb{R}}_{+}\text{,}$ or (21) ${\hat{E}}_h\left(\lambda \right)\ge \frac{\frac{\mathrm{\partial }\varphi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_h}{\hat{J}}_h\left(\lambda \right)}{{\alpha }_h+{\mu }_h+\lambda }+\frac{\frac{\mathrm{\partial }\psi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_v}{\hat{J}}_v\left(\lambda \right)}{{\alpha }_h+{\mu }_h+\lambda }\text{for }\lambda \in {\mathbb{R}}_{+}$(21) since $E_h\left(0\right)\ge 0$. By Lemma 4.7, ${\hat{J}}_h\left(\lambda \right)+{\hat{J}}_v\left(\lambda \right)>0$ for $\lambda \in {\mathbb{R}}_{+}$, which combined with (21(21) ${\hat{E}}_h\left(\lambda \right)\ge \frac{\frac{\mathrm{\partial }\varphi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_h}{\hat{J}}_h\left(\lambda \right)}{{\alpha }_h+{\mu }_h+\lambda }+\frac{\frac{\mathrm{\partial }\psi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_v}{\hat{J}}_v\left(\lambda \right)}{{\alpha }_h+{\mu }_h+\lambda }\text{for }\lambda \in {\mathbb{R}}_{+}$(21) ) implies that ${\hat{E}}_h\left(\lambda \right)\ne 0$ for any $\lambda \in {\mathbb{R}}_{+}$. Similarly, (22) ${\hat{E}}_v\left(\lambda \right)\ge \frac{\frac{\mathrm{\partial }f({\epsilon }_3,{\epsilon }_2)}{\mathrm{\partial }J}}{{\alpha }_v+{\mu }_v+\lambda }\hat{J}\left(\lambda \right)\text{for }\lambda \in {\mathbb{R}}_{+}\text{.}$(22) Moreover, from $J_h\left(t\right)={\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a\ge {\int }_0^t{\text{β}}_h\left(a\right){\sigma }_h\left(a\right){\alpha }_h{E}_h(t-a)\mathrm{d}a\text{for }t\in {\mathbb{R}}_{+},$ we obtain (23) ${\hat{J}}_h\left(\lambda \right)\ge {\alpha }_h\widehat{{\text{β}}_h{\sigma }_h}\left(\lambda \right){\hat{E}}_h\left(\lambda \right)\text{for {}\lambda \in {\mathbb{R}}_{+}\text{}.}$(23) Similarly, for $\lambda \in {\mathbb{R}}_{+}$, (24) $\begin{array}{rl}{\hat{J}}_v\left(\lambda \right)& \ge {\alpha }_v\widehat{{\text{β}}_v{\sigma }_v}\left(\lambda \right){\hat{E}}_v\left(\lambda \right),\end{array}$(24) (25) $\begin{array}{rl}\hat{J}\left(\lambda \right)& \ge {\alpha }_h\widehat{k_h{\sigma }_h}\left(\lambda \right){\hat{E}}_h\left(\lambda \right).\end{array}$(25) It follows from (21(21) ${\hat{E}}_h\left(\lambda \right)\ge \frac{\frac{\mathrm{\partial }\varphi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_h}{\hat{J}}_h\left(\lambda \right)}{{\alpha }_h+{\mu }_h+\lambda }+\frac{\frac{\mathrm{\partial }\psi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_v}{\hat{J}}_v\left(\lambda \right)}{{\alpha }_h+{\mu }_h+\lambda }\text{for }\lambda \in {\mathbb{R}}_{+}$(21) )–(25(25) $\begin{array}{rl}\hat{J}\left(\lambda \right)& \ge {\alpha }_h\widehat{k_h{\sigma }_h}\left(\lambda \right){\hat{E}}_h\left(\lambda \right).\end{array}$(25) ) that $\begin{array}{rl}{\hat{E}}_h\left(\lambda \right)& \ge \left(\frac{\mathrm{\partial }\varphi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_h}\cdot \frac{{\alpha }_h\widehat{{\text{β}}_h{\sigma }_h}\left(\lambda \right)}{{\alpha }_h+{\mu }_h+\lambda }\right.\\ & \left.+\frac{\mathrm{\partial }\psi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f({\epsilon }_3,{\epsilon }_2)}{\mathrm{\partial }J}\cdot \frac{{\alpha }_h\widehat{k_h{\sigma }_h}\left(\lambda \right)}{{\alpha }_h+{\mu }_h+\lambda }\cdot \frac{{\alpha }_v\widehat{{\text{β}}_v{\sigma }_v}\left(\lambda \right)}{{\alpha }_v+{\mu }_v+\lambda }\right){\hat{E}}_h\left(\lambda \right),\end{array}$ which is impossible at $\lambda ={\epsilon }_0$ because of (20(20) $\begin{array}{rl}& \frac{\mathrm{\partial }\varphi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_h}\cdot \frac{{\alpha }_h\widehat{{\text{β}}_h{\sigma }_h}\left({\epsilon }_0\right)}{{\alpha }_h+{\mu }_h+{\epsilon }_0}+\frac{\mathrm{\partial }\psi ({\epsilon }_1,{\epsilon }_0)}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f({\epsilon }_3,{\epsilon }_2)}{\mathrm{\partial }J}\\ & \cdot \frac{{\alpha }_h\widehat{k_h{\sigma }_h}}{\alpha +h+{\mu }_h+{\epsilon }_0}\cdot \frac{{\alpha }_v\widehat{{\text{β}}_v{\sigma }_v}\left({\epsilon }_0\right)}{{\alpha }_v+{\mu }_v+{\epsilon }_0}>1.\end{array}$(20) ) and ${\hat{E}}_h\left({\epsilon }_0\right)>0$. This completes the proof.

In order to apply Theorem 5.2 [51], it leaves to check hypothesis (H1) [51, page 125]. This is done by the following two lemmas.

Lemma 4.9

Let $x\left(t\right)=\left(S_h\right(t),E_h(t),i_h(t,\cdot ),S_v(t),E_v(t),i_v(t,\cdot \left)\right)$ be a total trajectory of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) in $\mathcal{A}$. If $\rho \left(x\right(t\left)\right)=0$ for all $t\le 0$ then $\rho \left(x\right(t\left)\right)=0$ for all t>0.

Proof.

Clearly, for $t\le 0$, $\frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}=-(\alpha +{\mu }_h)E_h\left(t\right).$ Since $x\left(t\right)\in \mathcal{A}\subset \mathrm{\Omega }$, we must have $E_h\left(0\right)=0$ (and hence $E_h\left(t\right)=0$ for $t\le 0$) otherwise $\underset{t\to -\mathrm{\infty }}{lim}{E}_h\left(t\right)=\mathrm{\infty }$. This implies that, for $t\le 0$, $i_h(t,a)=i_h(t-a,0){\sigma }_h\left(a\right)={\alpha }_h{E}_h(t-a){\sigma }_h\left(a\right)=0$ and hence $J\left(t\right)=0$. Using similar arguments as above we can get $E_v\left(t\right)=0$ and $i_v(t,a)=0$ for $t\le 0$. Then, for $t\le 0$, we have $\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}={\lambda }_h-{\mu }_h{S}_h\left(t\right)\text{and}\frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}={\lambda }_v-{\mu }_v{S}_v\left(t\right).$ Using $x\left(t\right)\in \mathcal{A}$ again, we see that $S_h\left(t\right)=S_h^0$ and $S_v\left(t\right)=S_v^0$ for $t\le 0$. By the uniqueness of solutions, we get $x\left(t\right)\equiv E^0$. In particular, we have that $J_h\left(t\right)+J_v\left(t\right)=0$ for t>0.

Lemma 4.10

For a total trajectory $x\left(t\right)=\left(S_h\right(t),E_h(t),i_h(t,\cdot ),S_v(t),E_v(t),i_v(t,\cdot \left)\right)$ in $\mathcal{A},$ both $S_h\left(t\right)$ and $S_v\left(t\right)$ are positive and either $\rho \left(x\right(t\left)\right)\equiv 0$ or $\rho \left(x\right(t\left)\right)$ is positive on $\mathbb{R}$.

Proof.

For the first part, suppose by way of contradiction, there exists some $t_0\in \mathbb{R}$ such that $S_h\left(t_0\right)=0$. Then $\frac{\mathrm{d}{S}_h\left(t_0\right)}{\mathrm{d}t}={\lambda }_h>0$. This implies that $S_h\left(t\right)<0$ for $t<t_0$ sufficiently small, a contradiction to $x\left(t\right)\in \mathcal{A}\subset \mathrm{\Omega }$. Thus $S_h\left(t\right)$ is positive. Similarly, we can show that $S_v\left(t\right)$ is positive.

Now, suppose $\rho \left(x\right(t\left)\right)\not\equiv 0$. We claim that $\rho \left(x\right(t\left)\right)>0$ for all $t\in \mathbb{R}$. Otherwise, with the assistance of Lemma 4.9, there exists a sequence $\left\{t_n\right\}$ such that $t_n\to -\mathrm{\infty }$ as $n\to \mathrm{\infty }$ and $\rho \left(x\right(t\left)\right)>0$ for all n. For every n, define $x_n:\mathbb{R}\to \mathcal{A}$ by $x_n\left(t\right)=x(t+t_n)$. Then using similar arguments as those in the proof of Lemma 4.7, there exists a $T\ge 0$ (independent of n) such that $\rho \left(x_n\right(t\left)\right)>0$ for $t\ge T$ and all n. This is a contradiction and hence the proof is completed.

By now we have verified all the assumptions of Theorem 5.2 [51]. Therefore, we have the following result.

Theorem 4.11

Suppose ${\mathcal{R}}_0>1$. Then there exists a global compact attractor ${\mathcal{A}}_1$ in $X_{+}\setminus X_0$ and the semi-flow Φ is uniformly ρ-persistent.

Corollary 4.12

Suppose ${\mathcal{R}}_0>1$. Then there is an ${\epsilon }_0$ such that $S_h\left(t\right),E_h\left(t\right),i_h(t,0),S_v\left(t\right),E_v\left(t\right),i_v(t,0)\ge {\epsilon }_0$ for all $t\in \mathbb{R},$ where $\left(S_h\right(t),E_h(t),i_h(t,\cdot ),S_v(t),E_v(t),i_v(t,\cdot \left)\right)$ is any total trajectory in ${\mathcal{A}}_1$.

Proof.

It follows from the first equation of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) and $\left({\mathbf{C}}_4\right)$ that $\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}\ge {\lambda }_h-{\mu }_h{S}_h\left(t\right)-(L_u+K_u)S_h\left(t\right),$ which implies $\underset{t\to \mathrm{\infty }}{lim inf}{S}_h\left(t\right)\ge \frac{{\lambda }_h}{{\mu }_h+L_u+K_u}\triangleq {\epsilon }_{h1}$. Similarly, $\underset{t\to \mathrm{\infty }}{lim inf}{S}_v\left(t\right)\ge \frac{{\lambda }_v}{{\mu }_v+M_u}\triangleq {\epsilon }_{v1}$. By invariance, $S_h\left(t\right)\ge {\epsilon }_{h1}$ and $S_v\left(t\right)\ge {\epsilon }_{v1}$.

From Theorem 4.11, we know there is positive $\tilde{\epsilon }$ such that ${\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a+{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a\ge \tilde{\epsilon }$ By the concavity in $\left({\mathbf{C}}_3\right)$, we have $\begin{array}{rl}\frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& \ge \varphi ({\epsilon }_{h1},J_h(t\left)\right)+\psi ({\epsilon }_{h1},J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right)\\ & \ge \frac{\mathrm{\partial }\varphi ({\epsilon }_{h1},\frac{\parallel {\text{β}}_h{\parallel }_{\mathrm{\infty }}{\lambda }_h}{{\mu }_h})}{\mathrm{\partial }{J}_h}{J}_h\left(t\right)+\frac{\mathrm{\partial }\psi ({\epsilon }_{h1},\frac{\parallel {\text{β}}_v{\parallel }_{\mathrm{\infty }}{\lambda }_v}{{\mu }_v})}{\mathrm{\partial }{J}_v}{J}_v\left(t\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right)\\ & \ge A\tilde{\epsilon }-({\alpha }_h+{\mu }_h)E_h\left(t\right),\end{array}$ which implies $\underset{t\to \mathrm{\infty }}{lim inf}{E}_h\left(t\right)\ge \frac{A\tilde{\epsilon }}{{\alpha }_h+{\mu }_h}\triangleq {\epsilon }_{h2},$ where $A=min\left\{\frac{\mathrm{\partial }\varphi ({\epsilon }_{h1},\frac{\parallel {\text{β}}_h{\parallel }_{\mathrm{\infty }}{\lambda }_h}{{\mu }_h})}{\mathrm{\partial }{J}_h},\frac{\mathrm{\partial }\psi ({\epsilon }_{h1},\frac{\parallel {\text{β}}_v{\parallel }_{\mathrm{\infty }}{\lambda }_v}{{\mu }_v})}{\mathrm{\partial }{J}_v}\right\}.$

By invariance again, $E_h\left(t\right)\ge {\epsilon }_{h2}$ for $t\in \mathbb{R}$. It follows that $i_h(t,0)={\alpha }_h{E}_h\left(t\right)\ge {\alpha }_h{\epsilon }_{h2}\triangleq {\epsilon }_{h3}\text{for {}t\in \mathbb{R}\text{}.}$ Thirdly, since $J\left(t\right)={\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a={\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t-a,0){\sigma }_h\left(a\right)\mathrm{d}a\ge \eta {\epsilon }_{h3}$ for $t\in \mathbb{R}$, it follows from $\frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}\ge f({\epsilon }_{v1},\eta {\epsilon }_{h3})-({\alpha }_v+{\mu }_v)E_v\left(t\right)$ for $t\in \mathbb{R}$ that $\underset{t\to \mathrm{\infty }}{lim inf}{E}_v\left(t\right)\ge \frac{f({\epsilon }_{v1},\eta {\epsilon }_{h3})}{{\alpha }_v+{\mu }_v}\triangleq {\epsilon }_{v2}$. Again by invariance, $E_v\left(t\right)\ge {\epsilon }_{v2}$ for $t\in \mathbb{R}$ and hence $i_v(t,0)={\alpha }_v{E}_v\left(t\right)\ge {\alpha }_v{\epsilon }_{v2}\triangleq {\epsilon }_{v3}\text{for {}t\in \mathbb{R}\text{}.}$ Letting ${\epsilon }_0=min\{{\epsilon }_{h1},{\epsilon }_{h2},{\epsilon }_{h3},{\epsilon }_{v1},{\epsilon }_{v2},{\epsilon }_{v3}\}$, we immediately complete the proof.

Now, we are ready to establish the global stability of the infectious steady state $E^{\ast }$ under the following additional assumption on the nonlinear incidences.

(C5)

For $S_h>0$ and $S_v>0$, $\begin{array}{rl}& \left\{\begin{array}{ll}\frac{J_h}{J_h^{\ast }}\le \frac{S_h^{\ast }\varphi (S_h,J_h)}{S_h\varphi (S_h^{\ast },J_h^{\ast })}\le 1,& 0<J_h\le J_h^{\ast },\\ 1\le \frac{S_h^{\ast }\varphi (S_h,J_h)}{S_h\varphi (S_h^{\ast },J_h^{\ast })}\le \frac{J_h}{J_h^{\ast }},& 0<J_h^{\ast }\le J_h,\end{array}\right.\\ & \left\{\begin{array}{ll}\frac{J_v}{J_v^{\ast }}\le \frac{S_h^{\ast }\psi (S_h,J_v)}{S_h\psi (S_h^{\ast },J_v^{\ast })}\le 1,& 0<J_v\le J_v^{\ast },\\ 1\le \frac{S_h^{\ast }\psi (S_h,J_v)}{S_h\psi (S_h^{\ast },J_v^{\ast })}\le \frac{J_v}{J_v^{\ast }},& 0<J_v^{\ast }\le J_v,\end{array}\right.\\ & \left\{\begin{array}{ll}\frac{J}{J^{\ast }}\le \frac{S_v^{\ast }f(S_v,J)}{S_{v}f(S_v^{\ast },J^{\ast })}\le 1,& 0<J\le J^{\ast },\\ 1\le \frac{S_v^{\ast }f(S_v,J)}{S_{v}f(S_v^{\ast },J^{\ast })}\le \frac{J}{J^{\ast }},& 0<J^{\ast }\le J.\end{array}\right.\end{array}$

Theorem 4.13

Under the conditions ${\mathcal{R}}_0>1$ and $\left({\mathbf{C}}_1\right)$–$\left({\mathbf{C}}_5\right),$ the infectious steady state $E^{\ast }$ of (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ) is globally asymptotically stable in $X_{+}\setminus X_0$.

Proof.

Since $E^{\ast }$ is locally asymptotically stable, we only need to prove that ${\mathcal{A}}_1=\left\{E^{\ast }\right\}$. The approach is the technique of Lyapunov functionals. Let $x\left(t\right)=\left(S_h\right(t),E_h(t),i_h(t,\cdot ),S_v(t),E_v(t),i_v(t,\cdot \left)\right)$ be a total trajectory in ${\mathcal{A}}_1$. From Corollary 4.12, we can find an ${\tilde{\epsilon }}_0$ ensuring $g\left(x\right)\in [0,{\tilde{\epsilon }}_0]$, where x can be any of $\frac{S_h\left(t\right)}{S_h^{\ast }},\frac{E_h\left(t\right)}{E_h^{\ast }},\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)},\frac{S_v\left(t\right)}{S_v^{\ast }},\frac{E_v\left(t\right)}{E_v^{\ast }}$ and $\frac{i_v(t,a)}{i_v^{\ast }\left(a\right)}$. Here $g:(0,\mathrm{\infty })\ni x\mapsto x-1-\ln x\in \mathbb{R}$. Clearly, g is nonnegative and $g\left(x\right)=0$ if and only if x = 1.

Define a Lyapunov functional by $\begin{array}{rl}L\left(t\right)& =L\left(S_h\right(t),E_h(t),i_h(t,\cdot ),S_v(t),E_v(t),i_v(t,\cdot \left)\right)\\ & =L_1\left(t\right)+L_2\left(t\right)+L_3\left(t\right)+L_4\left(t\right)+L_5\left(t\right),\end{array}$ where $\begin{array}{rl}{L}_1\left(t\right)& =S_h^{\ast }g\left(\frac{S_h\left(t\right)}{S_h^{\ast }}\right)+E_h^{\ast }g\left(\frac{E_h\left(t\right)}{E_h^{\ast }}\right),\\ L_2\left(t\right)& =\frac{\psi (S_h^{\ast },J_v^{\ast })}{f(S_v^{\ast },J^{\ast })}\left(S_v^{\ast }g\left(\frac{S_v\left(t\right)}{S_v^{\ast }}\right)+E_v^{\ast }g\left(\frac{E_v\left(t\right)}{E_v^{\ast }}\right)\right),\\ L_3\left(t\right)& =\frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}\eta \left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\mathrm{d}a,\\ L_4\left(t\right)& =\frac{\varphi (S_h^{\ast },J_h^{\ast })}{{\xi }_h{\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}{\xi }_h\left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\mathrm{d}a,\\ L_5\left(t\right)& =\frac{\psi (S_h^{\ast },J_v^{\ast })}{{\xi }_v{\alpha }_v{E}_v^{\ast }}{\int }_0^{\mathrm{\infty }}{\xi }_v\left(a\right)i_v^{\ast }\left(a\right)g\left(\frac{i_v(t,a)}{i_v^{\ast }\left(a\right)}\right)\mathrm{d}a,\end{array}$ with $\begin{array}{rl}{\xi }_h\left(a\right)& ={\int }_a^{\mathrm{\infty }}{\text{β}}_h\left(\theta \right)e^{-{\int }_a^{\theta }{\delta }_h\left(s\right)\mathrm{d}s}\mathrm{d}\theta ,\\ {\xi }_v\left(a\right)& ={\int }_a^{\mathrm{\infty }}{\text{β}}_v\left(\theta \right)e^{-{\int }_a^{\theta }{\delta }_v\left(s\right)\mathrm{d}s}\mathrm{d}\theta ,\\ \eta \left(a\right)& ={\int }_a^{\mathrm{\infty }}{k}_h\left(\theta \right)e^{-{\int }_a^{\theta }{\delta }_h\left(s\right)\mathrm{d}s}\mathrm{d}\theta .\end{array}$ Then $L\left(t\right)$ is bounded on the solution $x\left(t\right)$. In the following, we calculate the time derivatives of $L_i(i=1,2,\dots ,5)$.

With the assistance of ${\lambda }_h=\varphi (S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast })+{\mu }_h{S}_h^{\ast }$ and ${\alpha }_h+{\mu }_h=\frac{\varphi (S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast })}{E_h^{\ast }}$, we have $\begin{array}{rl}\frac{\mathrm{d}{L}_1\left(t\right)}{\mathrm{d}t}& =\left(1-\frac{S_h^{\ast }}{S_h\left(t\right)}\right)({\lambda }_h-\varphi (S_h\left(t\right),J_h\left(t\right))-\psi (S_h\left(t\right),J_v\left(t\right))-{\mu }_h{S}_h(t\left)\right)\\ & +\left(1-\frac{E_h^{\ast }}{E_h\left(t\right)}\right)\left(\varphi \right(S_h\left(t\right),J_h\left(t\right))+\psi (S_h\left(t\right),J_v\left(t\right))-({\alpha }_h+{\mu }_h\left)E_h\right(t\left)\right)\\ & =\left(1-\frac{S_h^{\ast }}{S_h\left(t\right)}\right)({\mu }_h{S}_h^{\ast }-{\mu }_h{S}_h(t\left)\right)+\left(1-\frac{S_h^{\ast }}{S_h\left(t\right)}\right)\left(\varphi \right(S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast }\left)\right)\\ & +\frac{S_h^{\ast }}{S_h\left(t\right)}\left(\varphi \right(S_h\left(t\right),J_h\left(t\right))+\psi (S_h\left(t\right),J_v\left(t\right)\left)\right)-\frac{E_h\left(t\right)}{E_h^{\ast }}\left(\varphi \right(S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast }\left)\right)\\ & -\frac{E_h^{\ast }}{E_h\left(t\right)}\left(\varphi \right(S_h\left(t\right),J_h\left(t\right))+\psi (S_h\left(t\right),J_v\left(t\right)\left)\right)+\varphi (S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast })\\ & ={\mu }_h{S}_h^{\ast }\left(2-\frac{S_h^{\ast }}{S_h\left(t\right)}-\frac{S_h\left(t\right)}{S_h^{\ast }}\right)-\left(\varphi \right(S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast }\left)\right)\left(\frac{S_h^{\ast }}{S_h\left(t\right)}-1-\ln \frac{S_h^{\ast }}{S_h\left(t\right)}\right)\\ & +\varphi (S_h^{\ast },J_h^{\ast })\left(\frac{S_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{S_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}-1-\ln \frac{S_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{S_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}\right)\\ & +\psi (S_h^{\ast },J_v^{\ast })\left(\frac{S_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{S_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}-1-\ln \frac{S_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{S_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}\right)\\ & -\left(\varphi \right(S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast }\left)\right)\left(\frac{E_h\left(t\right)}{E_h^{\ast }}-1-\ln \frac{E_h\left(t\right)}{E_h^{\ast }}\right)\\ & -\varphi (S_h^{\ast },J_h^{\ast })\left(\frac{E_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{E_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}-1-\ln \frac{E_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{E_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}\right)\\ & -\psi (S_h^{\ast },J_v^{\ast })\left(\frac{E_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{E_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}-1-\ln \frac{E_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{E_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}\right)\\ & ={\mu }_h{S}_h^{\ast }\left(2-\frac{S_h^{\ast }}{S_h\left(t\right)}-\frac{S_h\left(t\right)}{S_h^{\ast }}\right)-\left(\varphi \right(S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast }\left)\right)g\left(\frac{S_h^{\ast }}{S_h\left(t\right)}\right)\\ & +\varphi (S_h^{\ast },J_h^{\ast })g\left(\frac{S_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{S_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}\right)+\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{S_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{S_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}\right)\\ & -\left(\varphi \right(S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast }\left)\right)g\left(\frac{E_h\left(t\right)}{E_h^{\ast }}\right)-\varphi (S_h^{\ast },J_h^{\ast })g\left(\frac{E_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{E_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}\right)\\ & -\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{E_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{E_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}\right).\end{array}$ Similarly, $\begin{array}{rl}\frac{\mathrm{d}{L}_2\left(t\right)}{\mathrm{d}t}& =\frac{\psi (S_h^{\ast },J_v^{\ast })}{f(S_v^{\ast },J^{\ast })}{\mu }_v{S}_v^{\ast }\left(2-\frac{S_v^{\ast }}{S_v\left(t\right)}-\frac{S_v\left(t\right)}{S_v^{\ast }}\right)-\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{S_v^{\ast }}{S_v\left(t\right)}\right)\\ & +\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{S_v^{\ast }f\left(S_v\right(t),J(t\left)\right)}{S_v\left(t\right)f(S_v^{\ast },J^{\ast })}\right)-\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{E_v\left(t\right)}{E_v^{\ast }}\right)\\ & -\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{E_v^{\ast }f\left(S_v\right(t),J(t\left)\right)}{E_v\left(t\right)f(S_v^{\ast },J^{\ast })}\right).\end{array}$

Next, $\begin{array}{rl}\frac{\mathrm{d}{L}_3\left(t\right)}{\mathrm{d}t}& =\frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}\eta \left(a\right)\left(1-\frac{i_h^{\ast }\left(a\right)}{i_h(t,a)}\right)\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}\mathrm{d}a\\ & =\frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}\eta \left(a\right)\left(1-\frac{i_h^{\ast }\left(a\right)}{i_h(t,a)}\right)\left(-\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}-{\delta }_h\left(a\right)i_h(t,a)\right)\mathrm{d}a.\end{array}$ Note that $\begin{array}{rl}{i}_h^{\ast }\left(a\right)\frac{\mathrm{\partial }}{\mathrm{\partial }a}g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)& =i_h^{\ast }\left(a\right)\frac{\mathrm{\partial }}{\mathrm{\partial }a}g\left(\frac{i_h(t-a,0)}{i_h^{\ast }\left(0\right)}\right)\\ & =-i_h^{\ast }\left(a\right)\frac{\mathrm{\partial }}{\mathrm{\partial }t}g\left(\frac{i_h(t-a,0)}{i_h^{\ast }\left(0\right)}\right)\\ & =-i_h^{\ast }\left(a\right)\left(1-\frac{i_h^{\ast }\left(0\right)}{i_h(t-a,0)}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }t}\frac{i_h(t-a,0)}{i_h^{\ast }\left(0\right)}\\ & =-i_h^{\ast }\left(a\right)\left(1-\frac{i_h^{\ast }\left(a\right)}{i_h(t,a)}\right)\frac{\mathrm{\partial }}{\mathrm{\partial }t}\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\\ & =\left(1-\frac{i_h^{\ast }\left(a\right)}{i_h(t,a)}\right)\left(\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}+{\delta }_h\left(a\right)i_h(t,a)\right).\end{array}$ Using $\eta \left(0\right)=\eta ,i_h^{\ast }\left(0\right)={\alpha }_h{E}_h^{\ast }$, and $i_h(t,0)={\alpha }_h{E}_h\left(t\right)$, we get $\begin{array}{rl}\frac{\mathrm{d}{L}_3\left(t\right)}{\mathrm{d}t}& =-\frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}\eta \left(a\right)i_h^{\ast }\left(a\right)\frac{\mathrm{\partial }}{\mathrm{\partial }a}\left(g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\right)\mathrm{d}a\\ & =-\frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}\eta \left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right){|}_{a=0}^{a=\mathrm{\infty }}\\ & +\frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\left({\eta }^{\mathrm{\prime }}\right(a)-{\delta }_h(a\left)\eta \right(a\left)\right)i_h^{\ast }\left(a\right)\mathrm{d}a\\ & =\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{E_h\left(t\right)}{E_h^{\ast }}\right)-\frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\mathrm{d}a.\end{array}$ Similarly, we obtain $\frac{\mathrm{d}{L}_4\left(t\right)}{\mathrm{d}t}=\varphi (S_h^{\ast },J_h^{\ast })g\left(\frac{E_h\left(t\right)}{E_h^{\ast }}\right)-\frac{\varphi (S_h^{\ast },J_h^{\ast })}{{\xi }_h{\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\mathrm{d}a,$ and $\frac{\mathrm{d}{L}_5\left(t\right)}{\mathrm{d}t}=\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{E_v\left(t\right)}{E_v^{\ast }}\right)-\frac{\psi (S_h^{\ast },J_v^{\ast })}{{\xi }_v{\alpha }_v{E}_v^{\ast }}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v^{\ast }\left(a\right)g\left(\frac{i_v(t,a)}{i_v^{\ast }\left(a\right)}\right)\mathrm{d}a.$ To sum up, we have achieved that $\begin{array}{rl}\frac{\mathrm{d}L\left(t\right)}{\mathrm{d}t}& ={\mu }_h{S}_h^{\ast }\left(2-\frac{S_h^{\ast }}{S_h\left(t\right)}-\frac{S_h\left(t\right)}{S_h^{\ast }}\right)-\left(\varphi \right(S_h^{\ast },J_h^{\ast })+\psi (S_h^{\ast },J_v^{\ast }\left)\right)g\left(\frac{S_h^{\ast }}{S_h\left(t\right)}\right)\\ & +\varphi (S_h^{\ast },J_h^{\ast })g\left(\frac{S_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{S_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}\right)+\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{S_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{S_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}\right)\\ & -\varphi (S_h^{\ast },J_h^{\ast })g\left(\frac{E_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{E_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}\right)-\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{E_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{E_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}\right)\\ & -\frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\mathrm{d}a\\ & -\frac{\varphi (S_h^{\ast },J_h^{\ast })}{{\xi }_h{\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\mathrm{d}a\\ & -\frac{\psi (S_h^{\ast },J_v^{\ast })}{{\xi }_v{\alpha }_v{E}_v^{\ast }}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v^{\ast }\left(a\right)g\left(\frac{i_v(t,a)}{i_v^{\ast }\left(a\right)}\right)\mathrm{d}a\\ & +\frac{\psi (S_h^{\ast },J_v^{\ast })}{f(S_v^{\ast },J^{\ast })}{\mu }_v{S}_v^{\ast }\left(2-\frac{S_v^{\ast }}{S_v\left(t\right)}-\frac{S_v\left(t\right)}{S_v^{\ast }}\right)\\ & -\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{S_v^{\ast }}{S_v\left(t\right)}\right)+\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{S_v^{\ast }f\left(S_v\right(t),J(t\left)\right)}{S_v\left(t\right)f(S_v^{\ast },J^{\ast })}\right)\\ & -\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{E_v^{\ast }f\left(S_v\right(t),J(t\left)\right)}{E_v\left(t\right)f(S_v^{\ast },J^{\ast })}\right).\end{array}$ It follows from the monotonicity of $g,\left({\mathbf{C}}_5\right)$, and the Jensen Inequality that $\begin{array}{rl}& \varphi (S_h^{\ast },J_h^{\ast })g\left(\frac{S_h^{\ast }\varphi \left(S_h\right(t),J_h(t\left)\right)}{S_h\left(t\right)\varphi (S_h^{\ast },J_h^{\ast })}\right)\\ & \le \varphi (S_h^{\ast },J_h^{\ast })g\left(\frac{J_h\left(t\right)}{J_h^{\ast }}\right)\\ & =\varphi (S_h^{\ast },J_h^{\ast })g\left(\frac{{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a}{{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a}\right)\\ & =\frac{\varphi (S_h^{\ast },J_h^{\ast })}{{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\mathrm{d}a}{{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)\mathrm{d}a}\right)\mathrm{d}a\\ & \le \frac{\varphi (S_h^{\ast },J_h^{\ast })}{{\xi }_h{\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\mathrm{d}a,\end{array}$ and similarly $\begin{array}{rl}\psi (S_h^{\ast },J_v^{\ast })g\left(\frac{S_h^{\ast }\psi \left(S_h\right(t),J_v(t\left)\right)}{S_h\left(t\right)\psi (S_h^{\ast },J_v^{\ast })}\right)& \le \frac{\psi (S_h^{\ast },J_v^{\ast })}{{\xi }_v{\alpha }_v{E}_v^{\ast }}{\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v^{\ast }\left(a\right)g\left(\frac{i_v(t,a)}{i_v^{\ast }\left(a\right)}\right)\mathrm{d}a,\\ \psi (S_h^{\ast },J_v^{\ast })g\left(\frac{S_v^{\ast }f\left(S_v\right(t),J(t\left)\right)}{S_v\left(t\right)f(S_v^{\ast },J^{\ast })}\right)& \le \frac{\psi (S_h^{\ast },J_v^{\ast })}{\eta {\alpha }_h{E}_h^{\ast }}{\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h^{\ast }\left(a\right)g\left(\frac{i_h(t,a)}{i_h^{\ast }\left(a\right)}\right)\mathrm{d}a.\end{array}$ Therefore, $\frac{\mathrm{d}L\left(t\right)}{\mathrm{d}t}\le 0$. Because the Lyapunov functional L is bounded on $x\left(t\right)$, we conclude that the alpha limit of $x(\cdot )$ is in the largest invariant subset $\mathcal{M}$ in $\{\frac{\mathrm{d}L\left(t\right)}{\mathrm{d}t}=0\}$. However, $\mathcal{M}$ is exactly $\left\{E^{\ast }\right\}$. Therefore, $L\left(x\right(t\left)\right)\le L\left(E^{\ast }\right)$ and ${\mathcal{A}}_1=\left\{E^{\ast }\right\}$. This completes the proof.

5. A model with standard incidence rates

If the total populations of hosts $N_h\left(t\right)$ and vectors $N_v\left(t\right)$ are not constants, then the standard incidence rate does not satisfy the condition $\left({\mathbf{C}}_5\right)$. In this section, we study the following system with standard incidence rates (26) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}-\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)},\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}+\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)}-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{R}_h\left(t\right)}{\mathrm{d}t}& ={\int }_0^{\mathrm{\infty }}{\gamma }_h\left(a\right)i_h(t,a)\mathrm{d}a-{\mu }_h{R}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ R_h\left(0\right)& =R_{h0}\in {\mathbb{R}}_{+},\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(26) The variables and parameters in the above model are the same as those in system (2(2) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\varphi \left(S_h\right(t),J_h(t\left)\right)-\psi \left(S_h\right(t),J_v(t\left)\right),\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\varphi \left(S_h\right(t),J_h(t\left)\right)+\psi \left(S_h\right(t),J_v(t\left)\right)-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-f\left(S_v\right(t),J(t\left)\right)-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =f\left(S_v\right(t),J(t\left)\right)-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(2) ). The total host population is $N_h\left(t\right)=S_h\left(t\right)+E_h\left(t\right)+{\int }_0^{\mathrm{\infty }}{i}_h(t,a)\mathrm{d}a+R_h\left(t\right)$.

System (26(26) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}-\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)},\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}+\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)}-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{R}_h\left(t\right)}{\mathrm{d}t}& ={\int }_0^{\mathrm{\infty }}{\gamma }_h\left(a\right)i_h(t,a)\mathrm{d}a-{\mu }_h{R}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ R_h\left(0\right)& =R_{h0}\in {\mathbb{R}}_{+},\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(26) ) always has the disease-free steady state $\widetilde{E^0}=(\widetilde{S_h^0},0,0,0,\widetilde{S_v^0},0,0)$, where $\widetilde{S_h^0}=S_h^0=\frac{{\lambda }_h}{{\mu }_h}$ and $\widetilde{S_v^0}=S_v^0=\frac{{\lambda }_v}{{\mu }_v}$. The basic reproduction number ${\tilde{R}}_0$ is give by ${\tilde{R}}_0=\frac{{\xi }_h{\alpha }_h}{{\alpha }_h+{\mu }_h}+\frac{{\alpha }_h\eta }{{\alpha }_h+{\mu }_h}\frac{{\xi }_v{\alpha }_v}{{\alpha }_v+{\mu }_v}\frac{S_v^0}{N_h^0},$ where $N_h^0=S_h^0$.

If $\widetilde{E^{\ast }}=(\widetilde{S_h^{\ast }},\widetilde{E_h^{\ast }},\widetilde{i_h^{\ast }},\widetilde{R_h^{\ast }},\widetilde{S_v^{\ast }},\widetilde{E_v^{\ast }},\widetilde{i_v^{\ast }})$ is an infectious steady state, then $\begin{array}{rl}\widetilde{S_h^{\ast }}& =\frac{{\lambda }_h-({\alpha }_h+{\mu }_h)\widetilde{E_h^{\ast }}}{{\mu }_h},\widetilde{R_h^{\ast }}=\frac{\gamma {\alpha }_h\widetilde{E_h^{\ast }}}{{\mu }_h},\\ \gamma & ={\int }_0^{\mathrm{\infty }}{\gamma }_h\left(a\right){\sigma }_h\left(a\right)\mathrm{d}a,\sigma ={\int }_0^{\mathrm{\infty }}{\sigma }_h\left(a\right)\mathrm{d}a,\\ \widetilde{i_h^{\ast }}\left(a\right)& ={\sigma }_h\left(a\right){\alpha }_h\widetilde{E_h^{\ast }},\\ \widetilde{S_v^{\ast }}& =\frac{{\lambda }_v}{{\mu }_v+\frac{\eta {\alpha }_h\widetilde{E_h^{\ast }}}{\widetilde{N_h^{\ast }}}},\widetilde{N_h^{\ast }}=\frac{{\lambda }_h}{{\mu }_h}+\frac{\gamma -1+\sigma {\mu }_h}{{\mu }_h}{\alpha }_h\widetilde{E_h^{\ast }},\\ \widetilde{E_v^{\ast }}& =\frac{\widetilde{S_v^{\ast }}\widetilde{J^{\ast }}}{\widetilde{N_h^{\ast }}({\alpha }_v+{\mu }_v)}=\frac{{\lambda }_v}{{\mu }_v+\frac{\eta {\alpha }_h\widetilde{E_h^{\ast }}}{\widetilde{N_h^{\ast }}}}\frac{\eta {\alpha }_h\widetilde{E_h^{\ast }}}{({\alpha }_v+{\mu }_v)\widetilde{N_h^{\ast }}},\\ \widetilde{i_v^{\ast }}\left(a\right)& ={\sigma }_v\left(a\right){\alpha }_v\widetilde{E_v^{\ast }},\end{array}$ and $\widetilde{E_h^{\ast }}$ is a positive root of the following quadratic equation (27) $H\left(\widetilde{E_h^{\ast }}\right)=m{\widetilde{E_h^{\ast }}}^2+n\widetilde{E_h^{\ast }}+q=0,$(27) where $\begin{array}{rl}m& =-({\alpha }_h+{\mu }_h){\mu }_h{\mu }_v(m_1^2+\left(\frac{{\xi }_h{\alpha }_h}{{\mu }_h}+\frac{\eta {\alpha }_h}{{\mu }_v}\right)m_1+\frac{{\xi }_h{\alpha }_h}{{\mu }_h}\cdot \frac{\eta {\alpha }_h}{{\mu }_v}),\\ m_1& =(\frac{\gamma -1+\sigma {\mu }_h}{{\mu }_h}){\alpha }_h,\\ n& ={\lambda }_h{\xi }_h{\alpha }_h^2\eta -({\alpha }_h+{\mu }_h)\frac{{\lambda }_h}{{\mu }_h}{\mu }_v\left({\xi }_h{\alpha }_h+\frac{{\xi }_v{\alpha }_v}{{\alpha }_v+{\mu }_v}\frac{\eta {\alpha }_h{S}_v^0}{N_h^0}+\eta {\alpha }_h\frac{{\mu }_h}{{\mu }_v}\right)\\ & +({\lambda }_h{\xi }_h{\alpha }_h{\mu }_v-2({\alpha }_h+{\mu }_h){\mu }_h{\mu }_v\frac{{\lambda }_h}{{\mu }_h})(\frac{\gamma -1+\sigma {\mu }_h}{{\mu }_h}){\alpha }_h,\\ q& =({\alpha }_h+{\mu }_h)\frac{{\lambda }_h^2}{{\mu }_h}{\mu }_v({\tilde{R}}_0-1).\end{array}$

Using the relationship between roots and coefficients of the quadratic equation $H\left(\widetilde{E_h^{\ast }}\right)=0$, given in  (27(27) $H\left(\widetilde{E_h^{\ast }}\right)=m{\widetilde{E_h^{\ast }}}^2+n\widetilde{E_h^{\ast }}+q=0,$(27) ), we list all the possibilities for the number of positive roots of $H\left(\widetilde{E_h^{\ast }}\right)=0$ in Table 1. From all the cases enumerated in Table 1, we have the following results: The system (26(26) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}-\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)},\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}+\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)}-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{R}_h\left(t\right)}{\mathrm{d}t}& ={\int }_0^{\mathrm{\infty }}{\gamma }_h\left(a\right)i_h(t,a)\mathrm{d}a-{\mu }_h{R}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ R_h\left(0\right)& =R_{h0}\in {\mathbb{R}}_{+},\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(26) ) has a unique infectious steady state $\widetilde{E_h^{\ast }}$ if either (i) ${\tilde{R}}_0>1,m<0$ or (ii) ${\tilde{R}}_0<1,m>0$ is satisfied. The system (26(26) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}-\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)},\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}+\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)}-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{R}_h\left(t\right)}{\mathrm{d}t}& ={\int }_0^{\mathrm{\infty }}{\gamma }_h\left(a\right)i_h(t,a)\mathrm{d}a-{\mu }_h{R}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ R_h\left(0\right)& =R_{h0}\in {\mathbb{R}}_{+},\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(26) ) could have two infectious steady states if either (iii) ${\tilde{R}}_0>1,m>0,n<0,n^2-4mq>0$ or (iv) ${\tilde{R}}_0<1,m<0,n>0,n^2-4mq>0$ is satisfied. Therefore, it is possible for the system with standard incidence rates to have backward bifurcation.

Table 1. Number of positive zeros of H given in  (27(27) $H\left(\widetilde{E_h^{\ast }}\right)=m{\widetilde{E_h^{\ast }}}^2+n\widetilde{E_h^{\ast }}+q=0,$(27) ).

Cases	m	n	$n^2-4mq$	Number of positive zeros of H
${\tilde{R}}_0>1$	−	$N{R}^{\mathrm{a}}$	$A{P}^{\mathrm{b}}$	1
	+	−	+	2
	+	+	+	0
	+	NR	−	0
${\tilde{R}}_0<1$	+	NR	AP	1
	−	+	+	2
	−	+	−	0
	−	−	NR	0

${}^{\mathrm{a}}$means no restriction on the sign; ${}^{\mathrm{b}}$means always positive.

Theorem 5.1

When ${\tilde{R}}_0<1,$ the disease-free steady state $\widetilde{E^0}$ of (26(26) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}-\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)},\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}+\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)}-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{R}_h\left(t\right)}{\mathrm{d}t}& ={\int }_0^{\mathrm{\infty }}{\gamma }_h\left(a\right)i_h(t,a)\mathrm{d}a-{\mu }_h{R}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ R_h\left(0\right)& =R_{h0}\in {\mathbb{R}}_{+},\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(26) ) is locally asymptotically stable. When ${\tilde{R}}_0>1,$ it is unstable.

Proof.

Similar to the proof of Theorem 3.2, the characteristic equation at $\widetilde{E^0}$ is $\mathrm{\Delta }\left(\widetilde{E^0}\right)=(\tau +{\mu }_h{)}^2(\tau +{\mu }_v\left)\right(\tau +{\alpha }_h+{\mu }_h\left)\right(\tau +{\alpha }_v+{\mu }_v\left)g_2\right(\tau )=0,$ where $g_2\left(\tau \right)=1-\frac{{\alpha }_h\widehat{{\xi }_h}\left(\tau \right)}{\tau +{\alpha }_h+{\mu }_h}-\frac{{\alpha }_h\widehat{{\xi }_v}\left(\tau \right){\alpha }_v\hat{\eta }\left(\tau \right)S_v^0}{(\tau +{\alpha }_h+{\mu }_h)(\tau +{\alpha }_v+{\mu }_v)N_h^0}.$ The roots of $g_2\left(\tau \right)=0$ determine the stability of $\widetilde{E^0}$. If ${\tilde{R}}_0>1$, then $g_2\left(0\right)=1-{\tilde{R}}_0<0,\underset{\tau \to \mathrm{\infty }}{lim}{g}_2\left(\tau \right)=1$. Therefore, $g_2\left(\tau \right)=0$ admits a positive root, which means that $\widetilde{E^0}$ is unstable. If ${\tilde{R}}_0<1$, we show that all roots of $g_2\left(\tau \right)=0$ have negative real parts. By contradiction, we suppose that ${\tau }_0$ is a root with non-negative real part. Then $\begin{array}{rl}1& =\left|\frac{{\alpha }_h\widehat{{\xi }_h}\left({\tau }_0\right)}{{\tau }_0+{\alpha }_h+{\mu }_h}+\frac{{\alpha }_h\widehat{{\xi }_v}\left({\tau }_0\right){\alpha }_v\hat{\eta }\left({\tau }_0\right)\widetilde{S_v^0}}{({\tau }_0+{\alpha }_h+{\mu }_h)({\tau }_0+{\alpha }_v+{\mu }_v)N_h^0}\right|\\ & \le \frac{{\alpha }_h{\xi }_h}{{\alpha }_h+{\mu }_h}+\frac{{\alpha }_h{\xi }_v{\alpha }_v\eta S_v^0}{({\alpha }_h+{\mu }_h)({\alpha }_v+{\mu }_v)N_h^0}\\ & ={\tilde{R}}_0,\end{array}$ which yields a contradiction. This completes the proof.

When an infectious steady state $\widetilde{E^{\ast }}$ exists, the characteristic equation evaluated at $\widetilde{E^{\ast }}$ is $\mathrm{\Delta }\left(E^{\ast }\right)=(\tau +{\mu }_h)({\tau }^4+a_1{\tau }^3+a_2{\tau }^2+a_3\tau +a_4)=0,$ where $\begin{array}{rl}{a}_1& ={\alpha }_v+2{\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}+2{\mu }_h+{\alpha }_h+A+{\mathrm{\Lambda }}_1-{\alpha }_{h}B,\\ a_2& =({\alpha }_v+{\mu }_v)\left({\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)+(2{\mu }_h+{\alpha }_h+A)\left({\alpha }_v+2{\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)\\ & +{\mu }_h({\mu }_h+{\alpha }_h+A)+\left(2{\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}+{\alpha }_v+{\alpha }_h+{\mu }_h\right){\mathrm{\Lambda }}_1\\ & -{\alpha }_{h}B\left(2{\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}+{\alpha }_v+{\mu }_h\right)+A{\alpha }_h\hat{\gamma }\left(\tau \right)-{\alpha }_h{\alpha }_{v}C{D}_2,\\ a_3& =(2{\mu }_h+{\alpha }_h+A)({\alpha }_v+{\mu }_v)\left({\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)\\ & +{\mu }_h({\mu }_h+{\alpha }_h+A)\left({\alpha }_v+2{\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)\\ & +{\mathrm{\Lambda }}_1\left({\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)({\alpha }_v+{\mu }_v+{\alpha }_h+{\mu }_h)+({\alpha }_h+{\mu }_h)({\alpha }_v+{\mu }_v){\mathrm{\Lambda }}_1\\ & -{\alpha }_{v}CD({\alpha }_h+{\mu }_h+{\mu }_v)+CD{\alpha }_v({\mu }_v+{\mu }_h)-{\alpha }_h{\alpha }_{v}C{D}_2({\mu }_v+{\mu }_h)\\ & -{\alpha }_{h}B\left(({\alpha }_v+{\mu }_v+{\mu }_h)({\mu }_v+\frac{J^{\ast }}{N_h^{\ast }})+{\mu }_h({\alpha }_v+{\mu }_v)\right)\\ & +A{\alpha }_h\hat{\gamma }\left(\tau \right)\left({\alpha }_v+2{\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)+{\alpha }_h{\alpha }_{v}CD\hat{\gamma }\left(\tau \right),\\ a_4& ={\mu }_h({\mu }_h+{\alpha }_h+A)({\alpha }_v+{\mu }_v)\left({\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)+CD{\alpha }_v{\mu }_h{\mu }_v\\ & +{\mathrm{\Lambda }}_1({\alpha }_h+{\mu }_h)({\alpha }_v+{\mu }_v)\left({\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)-{\alpha }_{v}CD({\alpha }_h+{\mu }_h){\mu }_v\\ & -{\alpha }_{h}B{\mu }_h({\alpha }_v+{\mu }_v)\left({\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)+A{\alpha }_h\hat{\gamma }\left(\tau \right)\left({\mu }_v+\frac{J^{\ast }}{N_h^{\ast }}\right)({\alpha }_v+{\mu }_v)\\ & +{\alpha }_h{\alpha }_{v}CD\hat{\gamma }\left(\tau \right){\mu }_v-{\alpha }_h{\alpha }_{v}C{D}_2{\mu }_h{\mu }_v,\end{array}$ and $\begin{array}{rl}A& =\frac{S_h^{\ast }(J_h^{\ast }+J_v^{\ast })}{{N_h^{\ast }}^2},\\ {\mathrm{\Lambda }}_1& =\frac{(N_h^{\ast }-S_h^{\ast })(J_h^{\ast }+J_v^{\ast })}{{N_h^{\ast }}^2},\\ B& =\frac{S_h^{\ast }{\hat{\xi }}_h\left(\tau \right)}{N_h^{\ast }}-\frac{S_h^{\ast }{J}_h^{\ast }\hat{\sigma }\left(\tau \right)}{{N_h^{\ast }}^2},\\ C& =\frac{S_h^{\ast }{\hat{\xi }}_v\left(\tau \right)}{N_h^{\ast }},\\ D& =\frac{S_v^{\ast }{J}^{\ast }}{{N_h^{\ast }}^2},\\ D_2& =\frac{S_h^{\ast }\hat{\eta }\left(\tau \right)}{N_h^{\ast }}-D\hat{\sigma }\left(\tau \right),\\ \hat{\gamma }\left(\tau \right)& ={\int }_0^{\mathrm{\infty }}{\gamma }_h\left(a\right){\sigma }_h\left(a\right)e^{-\tau a}\mathrm{d}a,\\ \hat{\sigma }\left(\tau \right)& ={\int }_0^{\mathrm{\infty }}{\sigma }_h\left(a\right)e^{-\tau a}\mathrm{d}a.\end{array}$${\hat{\xi }}_h\left(\tau \right),{\hat{\xi }}_v\left(\tau \right)$, and $\hat{\eta }\left(\tau \right)$ are the same as previous definitions. The analysis of the distribution of the zeros of this complicated transcendental equation is challenging and remains to be further investigated.

6. Conclusion and discussion

Infection age is an important factor in studying the transmission of vector-borne diseases such as malaria. We included the age of infection for both infectious hosts and vectors in a mathematical model that studies both horizontal host transmission and direct vector transmission. Because of the extra disease-induced death, the total population of hosts or vectors does not converge to a constant. We assumed that incidences of horizontal transmission, transmission from vectors to hosts, and transmission from hosts to vectors are all in general nonlinear forms. Because the model is an infinite dimensional system consisting of coupled ordinary differential equations and partial differential equations, the analysis is not trivial. Using the Fluctuation Lemma, the approach of Lyapunov functionals, and some techniques in [38,42], we have established the threshold dynamics, which are completely determined by the basic reproduction number ${\mathcal{R}}_0$. Specifically, the disease-free steady state is globally asymptotically stable if ${\mathcal{R}}_0<1$ while the infectious steady state is globally asymptotically stable if ${\mathcal{R}}_0>1$. In other words, if ${\mathcal{R}}_0>1$ and there is initial infection force for the hosts then the solution of the system converges to the infectious steady state.

From the expression of ${\mathcal{R}}_0$, ${\mathcal{R}}_0=\frac{\mathrm{\partial }\varphi (S_h^0,0)}{\mathrm{\partial }{J}_h}\cdot \frac{{\xi }_h{\alpha }_h}{{\alpha }_h+{\mu }_h}+\frac{\mathrm{\partial }\psi (S_h^0,0)}{\mathrm{\partial }{J}_v}\cdot \frac{\mathrm{\partial }f(S_v^0,0)}{\mathrm{\partial }J}\cdot \frac{{\alpha }_h\eta }{{\alpha }_h+{\mu }_h}\cdot \frac{{\xi }_v{\alpha }_v}{{\alpha }_v+{\mu }_v},$ we see that ${\mathcal{R}}_0$ is the number of infectious hosts generated by the introduction of an infectious host into a population consisting of only susceptible hosts, which agrees with the biological interpretation of the basic reproduction number. The first term in ${\mathcal{R}}_0$ is the number generated by horizontal transmission while the second term is the number generated through vectors. Ignoring the horizontal transmission would underestimate the value of ${\mathcal{R}}_0$, which may result in suboptimal and unsuccessful treatment.

Transmission is a key process in the host-vector interaction. In many models, the transmission is described by the mass action law. Given the number of susceptible hosts $S_h\left(t\right)$ and number of infected hosts $I_h\left(t\right)$ (or vectors $I_v\left(t\right)$), the number of new infected hosts per unit of time caused by horizontal transmission is ${\text{β}}_h{S}_h\left(t\right)I_h\left(t\right)$ (or by vector transmission is ${\text{β}}_v{S}_h\left(t\right)I_v\left(t\right)$), where ${\text{β}}_h$ (or ${\text{β}}_v$) is the transmission rate. Transmission depends on the number of susceptible individuals with which an infectious individual might interact. This implies that the number of susceptible individuals in a ‘neighborhood’ of an infectious individual is important, rather than the total number of susceptible individuals in the population. McCallum et al. [41] suggested the use of mass action might not be accurate in some scenarios and that other functions could be used to describe the transmission. In our paper, using $\varphi (S_h,J_h)$ where $J_h={\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a$ as an example, the transmission can be used to describe the Holling Type II functional response $\frac{\text{β}{S}_h{J}_h}{1+\alpha J_h}$ [47] and $k{S}_h\ln (1+\frac{\text{β}{J}_h}{k})$ [41]. These functions, as well as the bilinear mass action term, are special cases of our nonlinear incidence rates satisfying (${\mathbf{C}}_1$)–(${\mathbf{C}}_5$).

Another popular form used to describe infection is the standard incidence rate. When the total population is not constant, the standard incidence rate does not belong to our previous class of incidence rates. Therefore, in Section 5, we studied the corresponding model with standard incidences. In this case, the horizontal transmission between hosts is $\frac{S_h\left(t\right){\int }_0^{\mathrm{\infty }}{\text{β}}_h\left(a\right)i_h(t,a)\mathrm{d}a}{N_h\left(t\right)}$, the transmission from vectors to hosts is $\frac{S_h\left(t\right){\int }_0^{\mathrm{\infty }}{\text{β}}_v\left(a\right)i_v(t,a)\mathrm{d}a}{N_h\left(t\right)}$, and the transmission to susceptible vectors because of biting infectious hosts is given by $\frac{S_v\left(t\right){\int }_0^{\mathrm{\infty }}{k}_h\left(a\right)i_h(t,a)\mathrm{d}a}{N_h\left(t\right)}$, where $N_h\left(t\right)=S_h\left(t\right)+E_h\left(t\right)+{\int }_0^{\mathrm{\infty }}{i}_h(t,a)\mathrm{d}a+R_h\left(t\right)$. For this model, we listed the conditions for the existence of one or more infectious steady states in Table 1. The disease-free steady sate always exists and when the basic reproduction number ${\tilde{R}}_0$ is less than 1 it is locally asymptotically stable. In addition to the disease-free steady state, one or two infectious steady states may also exist and be stable when ${\tilde{R}}_0<1$. Thus, backward bifurcation may take place. However, at this moment we are not able to carry out the stability analysis of the infectious steady state due to the complexity of the transcendental characteristic equation.

For models described by ordinary differential equations, lots of studies have indicated that standard incidence rate coupled with other factors can generate bifurcations. For example, Garba et al. [19] investigated a deterministic model with a standard incidence rate for the transmission dynamics of dengue disease. They showed that the use of standard incidence can generate the backward bifurcation phenomenon of dengue disease. Roop-O et al. [47] studied the effect of the choice of the incidence function for the occurrence of backward bifurcation in a malaria model. They found that three methods may eliminate the backward bifurcation: (i) reducing the reproductive number to below a sub-threshold, (ii) the rate of malaria-induced mortality in humans is zero, (iii) the incidence function takes a nonlinear incidence function rather than the standard incidence rate. In a recent paper by Cai et al. [8], the authors also found that the backward bifurcation can be generated in an ordinary differential equation malaria model by a standard incidence rate coupled with a nonzero disease-induced mortality rate or superinfection (i.e., asymptomatic individuals can be reinfected and move to the symptomatic class). Cai et al. [7] proposed delayed vector-host disease models with two time delays, one describing the incubation period in the vector population and the other representing the incubation period in the host population. They also assumed that the total populations of both host $N_h$ and vector $N_v$ are constants. Then the standard incidence rate can be reduced to the bilinear incidence rate, which satisfies our conditions $\left({\mathbf{C}}_1\right)$–$\left({\mathbf{C}}_5\right)$. Their analytic results revealed that the global dynamics of such vector-host disease models with time delays are completely determined by the basic reproduction number by constructing suitable Liapunov functionals. Kribs-Zaleta and Martcheva [30] considered partial differential equation models for a disease with acute and chronic infective stages, in which the standard incidence rate was also used. Their models for SIRS and SIS disease cycles without vectors exhibit backward bifurcations under certain conditions.

Besides a more detailed description of the dynamics of model (26(26) $\begin{array}{rl}\frac{\mathrm{d}{S}_h\left(t\right)}{\mathrm{d}t}& ={\lambda }_h-{\mu }_h{S}_h\left(t\right)-\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}-\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)},\\ \frac{\mathrm{d}{E}_h\left(t\right)}{\mathrm{d}t}& =\frac{S_h\left(t\right)J_h\left(t\right)}{N_h\left(t\right)}+\frac{S_h\left(t\right)J_v\left(t\right)}{N_h\left(t\right)}-({\alpha }_h+{\mu }_h)E_h\left(t\right),\\ \frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_h(t,a)}{\mathrm{\partial }a}& =-{\delta }_h\left(a\right)i_h(t,a),\\ i_h(t,0)& ={\alpha }_h{E}_h\left(t\right),\\ \frac{\mathrm{d}{R}_h\left(t\right)}{\mathrm{d}t}& ={\int }_0^{\mathrm{\infty }}{\gamma }_h\left(a\right)i_h(t,a)\mathrm{d}a-{\mu }_h{R}_h\left(t\right),\\ \frac{\mathrm{d}{S}_v\left(t\right)}{\mathrm{d}t}& ={\lambda }_v-\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-{\mu }_v{S}_v\left(t\right),\\ \frac{\mathrm{d}{E}_v\left(t\right)}{\mathrm{d}t}& =\frac{S_v\left(t\right)J\left(t\right)}{N_h\left(t\right)}-({\alpha }_v+{\mu }_v)E_v\left(t\right),\\ \frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }t}+\frac{\mathrm{\partial }{i}_v(t,a)}{\mathrm{\partial }a}& =-{\delta }_v\left(a\right)i_v(t,a),\\ i_v(t,0)& ={\alpha }_v{E}_v\left(t\right),\\ S_h\left(0\right)& =S_{h0}\in {\mathbb{R}}_{+},E_h\left(0\right)=E_{h0}\in {\mathbb{R}}_{+},i_h(0,\cdot )=i_{h0}\in L_{+}^1(0,\mathrm{\infty }),\\ R_h\left(0\right)& =R_{h0}\in {\mathbb{R}}_{+},\\ S_v\left(0\right)& =S_{v0}\in {\mathbb{R}}_{+},E_v\left(0\right)=E_{v0}\in {\mathbb{R}}_{+},i_v(0,\cdot )=i_{v0}\in L_{+}^1(0,\mathrm{\infty }).\end{array}$(26) ), future work may include models with the physiological heterogeneity in susceptibility. Highly susceptible individuals tend to acquire infection first, while resistant individuals acquire infection later and at a slower rate. This may yield a nonlinear relation between time and number of new infections acquired [41]. Therefore, different transmission modes should be evaluated in studying vector-borne disease dynamics by mathematical models.

Acknowledgments

The authors would like to thank the editors and the anonymous reviewers for their very valuable comments and suggestions on improving the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Wang was supported by NSFC (No. 11771374), the Nanhu Scholar Program for Young Scholars of Xinyang Normal University. Chen was supported by NSERC of Canada. Martcheva and Rong were supported by the NSF grant DMS-1515661 and DMS-1758290, respectively.