Global dynamics of an age-structured malaria model with prevention

1.   Introduction

Malaria, an infectious disease caused by the malaria parasite, is one of the most severe public health problems in the world. About half of the world population live in areas at risk of malaria transmission. According to the World Malaria Report $2016$ ^[1], in $2015$, there were $212$ million new clinical cases of the disease and $429,000$ deaths. The majority of malaria cases and deaths are concentrated in Africa. Malaria transmission between people mainly involves malaria parasites infecting successively two types of hosts: humans and female Anopheles mosquitoes. In humans, the parasites (sporozoites) grow and multiply first in the liver cells. After a period of time, the parasites are found in the red cells of the blood. The blood stage parasites are those that cause the symptoms of malaria. In the blood, the parasites grow inside the red cells and then destroy them, releasing more daughter parasites that continue the cycle by invading other red cells. A proportion of the blood stage parasites develop into the gametocytes at some point of this process. When gametocytes are ingested by a female Anopheles mosquito during a blood meal, they start another different cycle of growth and multiplication in the mosquito. After 10-18 days, the parasites (sporozoites) are found in salivary glands of the mosquito. When the anopheles mosquito takes a blood meal on another human, the sporozoites are injected with the mosquito's saliva and start another human infection when they parasitize the liver cells. Thus, the mosquito carries the disease from one human to another. Differently from the human host, the mosquito does not suffer from the presence of the parasites ^[2].

Mathematical modeling of malaria transmission has always provided useful insights into malaria transmission mechanism since the pioneer work of Ross ^[3] and Macdonald ^[4], it has also become an important tool in understanding the dynamics of malaria transmission. Over the years, the Ross-Macdonald model has been extended by many researchers ^{[5,6,7,8,9,10,11,12,13]}. Ruan et al. ^[7] introduced modified Ross-Macdonald model to include two discrete time delays which represent incubation periods of parasites within the human and the mosquito. They verified that extending the incubation periods in either humans or mosquitos were beneficial to the malaria control. Cai et al. ^[8] investigated the effect of distributed delays on the vector-host disease dynamics, they showed that incubation periods can play significant role in affecting the disease transmission. Gao et al. ^[12] presented a multi-patch malaria model to study the impact of mobility of vector and host populations on malaria transmission. By using analysis and numerical simulation, they found that human movement was a critical factor in the spatial spread of malaria around the world. Li et al. ^[13] investigated two malaria models with relapse. They separated the dynamics into the fast time dynamics and the slow time dynamics and showed the full dynamics were determined by the slow systems. Other models about epidemic model or age-structured can be found in ^{[14,15,16,17,18,19,20,21,22,23,24,35]}.

Prevention and treatment may be the effective way to reduce malaria transmission. In most malaria-endemic countries, four interventions—case management (diagnosis and treatment), insecticide-treated nets (ITNs), intermittent preventive treatment of malaria in pregnant women (IPTp) and indoor residual spraying (IRS) —make up the essential package of malaria interventions. These measures are considered to have made a major contribution to the reduction in malaria burden since 2000. However, global investment for malaria has barely changed in recent years, which has affected malaria control to a certain extent. In 2015, 43% of the population of sub-Saharan Africa were not covered by ITNs or IRS, 69% of pregnant women did not receive three doses of IPTp and 36% of children with fever were not taken for care ^[1]. Prevention of malaria may be more cost-effective than treatment of the disease in the long run. In March 2010, the newest intervention recommended by the World Health Organization is intermittent preventive treatment during infancy (IPTi). Studies show that IPTi can significantly reduce clinical malaria and anemia in the first year of life, as well as hospital admissions associated with malaria infection or for any cause ^[1]. Intermittent preventive treatment of malaria in pregnant women (IPTp) is also a malaria intervention. WHO has recommended that pregnant women be given at least three doses during each pregnancy ^[1]. Vaccination is also a means of prevention. RTS, S(developed by PATH Malaria Vaccine Initiative (MVI) and GlaxoSmithKline (GSK) with support from the Bill and Melinda Gates Foundation) is the most recently developed recombinant vaccine. Phase Ⅲ clinical trial indicated that RTS, S reduced the number of cases among young children by almost 50 percent and among infants by around 25 percent. But, overall efficacy seem to wane with time ^[25]. According to this, we introduce age structure in the prevention period of susceptible population in our model.

Motivated by the above, in this paper, we not only introduce age structure to explain the prevention period of susceptible population but also involve age structures to account for the latent periods in humans and mosquitoes. We show that there exists a compact global attractor and obtain a sufficient condition for uniform persistence of the solution semiflow. We also identify the basic reproduction number $\mathcal{R}_{0}$ and show that $\mathcal{R}_{0}$ completely determines the global dynamics of our model.

This paper is organized as follows. In Section 2, we introduce the malaria model and present some basic properties. In Section 3, we define the basic reproductive number and prove the local stability of the disease-free equilibrium and the unique endemic equilibrium. In Section 4, we present the uniform persistence and prove the global stability of the disease-free equilibrium and the unique endemic equilibrium. In Section 5, we perform some numerical simulations. In Section 6, we give a brief discussion.

2.   Malaria model and basic properties

2.1.   The model

We divide the total human population at time $t$ into five mutually-exclusive subetaoups: susceptible individuals $S_{h}(t)$, protected individuals $P(t, a)$, where the parameter $a$ denotes the preventive age of the susceptible individuals, exposed individuals $E_{h}(t, \theta)$, where $\theta$ denotes the latent age of the exposed individuals, infective individuals $I_{h}(t)$ and removed individuals $R_{h}(t)$. We assume that the removed individuals are given special protection, they will no longer be involved in the transmission process. We divide the total vector population at time $t$ into three mutually-exclusive subetaoups: susceptible vectors $S_{m}(t)$, exposed vectors $E_{m}(t, \tau)$, where the parameter $\tau$ denotes the latent age of the exposed vectors, infective vectors $I_{m}(t)$. The flow among those subetaoups is shown in the following flowchart (Figure 1).

By the flowchart (Figure 1) and noting that the removed individuals are decoupled, we can formulate the malaria model as follows:

with the following initial and boundary conditions

where $P_{0}(a), E_{h0}(\theta), E_{m0}(\tau)\in L_{+}^{1}(0, +\infty)$ and $s_{h0}, i_{h0}, r_{h0}, s_{m0}, i_{m0}\in \mathbb{R}_{+}.$ The meanings of the parameters in $(2.1)$ are explained in Table 1.

Throughout this paper, we make the following assumptions and notations.

$(A1):$ $\Lambda_{i}, \mu_{i}, \beta_{i}, \delta, \alpha > 0, (i = 1, 2), \ \psi\geq0;$

$(A2):$ $\gamma_{i}\in L_{+}^{\infty}(0, +\infty) \ (i = 1, 2, 3)$ with essential upper bounds $\bar{ \gamma_{i}} > 0$, respectively;

$(A3):$ $P(0, 0) = \psi S_{h}(0), \ E_{h}(0, 0) = b\beta_{1}S_{h}(0)I_{m}(0), \ E_{m}(0, 0) = b\beta_{2}S_{m}(0)I_{h}(0).$

For $a, \theta, \tau \geq 0$, we denote

  $k_{1}(a) = e^{-\int_{0}^{a}(\mu_{1}+\gamma_{1}(s))da},$    $\mathscr{K}_{1} = \int_{0}^{+\infty}\gamma_{1}(a)k_{1}(a)da$,     $\phi(a) = \psi\gamma_{1}(a)k_{1}(a)$,

  $k_{2}(\theta) = e^{-\int_{0}^{\theta}(\mu_{1}+\gamma_{2}(s))d\theta},$    $\mathscr{K}_{2} = \int_{0}^{+\infty}\gamma_{2}(\theta)k_{2}(\theta)d\theta$,

  $k_{3}(\tau) = e^{-\int_{0}^{\tau}(\mu_{2}+\gamma_{3}(s))d\tau},$    $\mathscr{K}_{3} = \int_{0}^{+\infty}\gamma_{3}(\tau)k_{3}(\tau)d\tau$,

  $\mathscr{Y} = \mathbb{R}_{+}^{4}\times (Ł_{+}^{1}(0, +\infty))^{3}$ with norm

2.2.   Well-posedness

Base on the above assumptions, we can verify the local existence of unique and nonnegative solution of the model $(2.1)$ with the nonnegative initial conditions (see Webb ^[26] and Iannelli ^[27]), thus obtain the following proposition.

Proposition 2.1. Let $x_{0}\in \mathscr{Y},$ then there exists $\epsilon > 0$ and neighborhood $B_{0}\subset \mathscr{Y}$ with $x_{0}\in B_{0}$ such that there exists a unique continuous function, $\Phi : [0, \epsilon]\times B_{0}\rightarrow \mathscr{Y}$, where $\Phi(t, x_{0})$ is the solution to $(2.1)$ with $\Phi(0, x_{0}) = x_{0}.$

For $t\in[0, \epsilon],$

and setting $\mu = min\{\mu_{1}, \mu_{2}\},$ we deduce that $\parallel\Phi(t, x_{0})\parallel_{\mathscr{Y}}$ satisfies the following inequality:

therefore

which yields

Boundedness is a direct consequence of nonnegativity of solutions. Then we have following proposition.

Proposition 2.2. Let $x_{0}\in \mathscr{Y},$ then there exists a unique continuous semiflow, $\Phi : \mathbb{R}_{+}\times \mathscr{Y}\rightarrow \mathscr{Y}$, where $\Phi(t, x_{0})$ is the solution to $(2.1)$ with $\Phi(0, x_{0}) = x_{0},$ and $(2.3), (2.4)$ are satisfied for $t\in\mathbb{R}_{+}.$ The following set is positively invariant for system $(2.1)$

From $Proposition 2.2.$ and $(2.3)$, we obtain the following proposition.

Proposition 2.3. (1) The solution of $(2.1)$, $\Phi(t, \cdot)$, is point dissipative and $\Omega$ attracts all points in $\mathscr{Y}$;

(2) Let $B \subset \mathscr{Y}$ be bounded, then $\Phi(t, B)$ is bounded;

(3) If $x_{0}\in \mathscr{Y}$ and $\parallel x_{0}\parallel_{\mathscr{Y}}\leq A$ for some $A\geq\frac{\Lambda_{1}+\Lambda_{2}}{\mu},$ then $S_{i}(t), \ I_{i}(t), \parallel P(t, \cdot)\parallel_{L_{+}^{1}}, \parallel E_{i}(t, \cdot)\parallel_{L_{+}^{1}}\leq A, (i = h, m).$

2.3.   Asymptotic smoothness

Integrating the equations for $P, E_{h}, E_{m}$ in $(2.1)$ along the characteristic lines, $t-a$ = const., $t-\theta$ = cost., $t-\tau$ = const., respectively, we have

Continuous semiflow $\{\Phi(t, \cdot)\}_{t\geq0}$ is said to be asymptotically smooth, if each positively invariant bounded closed set is attracted by a nonempty compact set.

We will use the following two lemmas ^[28] to prove the asymptotic smoothness of the semiflow.

Lemma 2.1. The semiflow $\Phi: \mathbb{R}_{+}\times \mathscr{Y}\rightarrow \mathscr{Y}$ is asymptotically smooth if there are maps $U_{1}, U_{2}:\mathbb{R}_{+}\times \mathscr{Y}\rightarrow \mathscr{Y}$ such that $\Phi(t, x) = U_{1}(t, x) + U_{2}(t, x)$, and the following hold for any bounded closed set $\mathscr{A}\subset \mathscr{Y}$ that is forward invariant under $\Phi$:

(1) $\lim\limits_{t\rightarrow +\infty} diam U_{2}(t, \mathscr{A}) = 0$;

(2) there exists $t_{\mathscr{A}}\geq0$ such that $U_{1}(t, \mathscr{A})$ has compact closure for each $t\geq t_{\mathscr{A}}$.

Since $\mathscr{Y}$ is an infinite dimensional space, infinite dimensional space $L_{+}^{1}(0, +\infty)$ is a component of $\mathscr{Y}$. For infinite dimensional space, we cannot deduce precompactness only from boundedness. We apply following lemma.

Lemma 2.2. Let $\mathscr{B}$ be a bounded subset of $L_{+}^{1}(0, +\infty)$. Then $\mathscr{B}$ has compact closure if and only if the following conditions hold:

($i$) $\sup\limits_{f\in\mathscr{B}}\int_{0}^{+\infty}\mid f(s)\mid ds < +\infty$;

($ii$) $\lim\limits_{h\rightarrow +\infty}\int_{h}^{+\infty}\mid f(s)\mid ds = 0$ uniformly in $f\in \mathscr{B}$;

($iii$) $\lim\limits_{h\rightarrow 0^{+}}\int_{0}^{+\infty}\mid f(s+h)-f(s)\mid ds = 0$ uniformly in $f\in \mathscr{B}$;

($iv$) $\lim\limits_{h\rightarrow 0^{+}}\int_{0}^{h}\mid f(s)\mid ds = 0$ uniformly in $f\in \mathscr{B}$.

We are now ready to prove a result on the semiflow $\Phi$ generated by system $(2.1)$ is asymptotically smooth.

Theorem 2.3. The semiflow $\Phi$ generated by system $(2.1)$ is asymptotically smooth.

The proof of this theorem is in the appendix section.

Combining Proposition $2.3$ and $\Phi$ is asymptotically smooth, as well as the existence theory of global attractors, the following result is immediate from Theorem $2.6$ in ^[29] and Theorem $2.4$ in ^[30].

Theorem 2.4. The semiflow $\Phi$ has a global attractor $\mathcal{A}$ in $\mathscr{Y},$ which attracts any bounded subset of $\mathscr{Y}$.

3.   Equilibria and their local stability

3.1.   Existence of equilibria

System $(2.1)$ always has the disease free equilibrium $E_{0} = (S_{h}^{0}, S_{m}^{0}, 0, 0, P^{0}(a), 0_{L^{1}}, 0_{L^{1}})$, where

Define the basic reproduction number by

$\mathscr{R}_{0}$ represents the average number of secondary infectious human cases produced by a primary infectious human case that is introduced into two entirely susceptible populations: humans and female anopheles mosquitoes. The biological relevance of the threshold $\mathscr{R}_{0}$ can be interpreted as follows. There are

susceptible people and $\frac{\Lambda_{2}}{\mu_{2}}$ susceptible female anopheles mosquitoes. A primary infectious human case has a removal rate $\mu_{1}+\alpha+\delta$, the average infectious period is

During this time, the average number of mosquito bites from the susceptible mosquitoes is

so that the average number of infected but not infectious mosquitoes from the infectious human case will be

Then

represents the total number of infectious female anopheles mosquitoes produced by infected but not infectious mosquitoes. The infectious mosquitoes have a removal rate $\mu_{2}$, the average infectious period is $\frac{1}{\mu_{2}}$. During this time, the average number of mosquito bites from the susceptible mosquitoes is $\frac{b}{\mu_{2}}$, so that the average number of infected but not infectious human cases from the infectious mosquitoes will be

Then

represents the average number of infectious human cases produced by infected but not infectious human cases. Therefore, the average number of secondary infectious human cases from a primary infectious human case is

which is $\mathscr{R}_{0}$.

Next, we investigate the endemic equilibria of system $(2.1)$. Any endemic equilibrium $(S_{h}^{*}, S_{m}^{*}, I_{h}^{*}, I_{m}^{*}, P^{*}(a), E_{h}^{*}(\theta), E_{m}^{*}(\tau))$ of system $(2.1)$ should satisfy the following equations:

From $(3.1)$ we can easily find that if $\mathscr{R}_{0} > 1,$ system $(2.1)$ has a unique endemic equilibrium $E^{*} = (S_{h}^{*}, S_{m}^{*}, I_{h}^{*}, I_{m}^{*}, P^{*}(a), E_{h}^{*}(\theta), E_{m}^{*}(\tau)),$ where

Summarizing the discussions above, we have the following theorem.

Theorem 3.1. System $(2.1)$ always has the disease free equilibrium $E_{0}$. Moreover, apart from $E_{0}$, if $\mathscr{R}_{0} > 1$, system $(2.1)$ has a unique endemic equilibrium $E^{*}.$

3.2.   Local stability of the equilibria

Now we consider the linearized system of $(2.1)$ at an equilibrium

Let $\overline{S_{h}}(t) = S_{h}(t)-\widetilde{S_{h}}, \ \overline{S_{m}}(t) = S_{m}(t)-\widetilde{S_{m}}, \ \overline{I_{h}}(t) = I_{h}(t)-\widetilde{I_{h}}, \ \overline{I_{m}}(t) = I_{m}(t)-\widetilde{I_{m}}, \ \overline{P}(a, t) = P(a, t)-\widetilde{P}(a), \ \overline{E_{h}}(t, \theta) = E_{m}(t, \theta)-\widetilde{E_{h}}(\theta), \ \overline{E_{m}}(t, \tau)) = E_{m}(t, \tau)-\widetilde{E_{m}}(\tau),$ then removing the bar, we obtain the following linearized system:

with the following initial and boundary conditions

Let

For $(3.2)$, let $S_{h}(t) = S_{h}^{0}e^{\lambda t}, \ S_{m}(t) = S_{m}^{0}e^{\lambda t}, \ I_{h}(t) = I_{h}^{0}e^{\lambda t}, \ I_{m}(t) = I_{m}^{0}e^{\lambda t}, \ P(t, a) = P^{0}(a)e^{\lambda t}, E_{h}(t, \theta) = E_{h}^{0}(\theta)e^{\lambda t}, E_{m}(t, \tau) = E_{m}^{0}(\tau)e^{\lambda t},$ we have

with initial conditions

We obtain from the system $(3.3)$ that

On substituting the above two equations into the second and third equations of system $(3.4)$, and by simple calculation, we have

and

We derive that

We obtain the characteristic equation of model $(2.1)$ at an equilibrium $\widetilde{E}$ as follows:

where $f(\lambda) = \dfrac{b^{2}\beta_{1}\beta_{2}\mathscr{K}_{2}(\lambda)\mathscr{K}_{3}(\lambda)\widetilde {S_{h}}\widetilde {S_{m}} (\lambda+\mu_{1}+\psi(1-\mathscr{K}_{1}(\lambda))}{(\lambda+\mu_{1}+\alpha+\delta)(\lambda+\mu_{2}+b\beta_{2}\widetilde {I_{h}}) (\lambda+\mu_{1}+\psi(1-\mathscr{K}_{1}(\lambda))+b\beta_{1}\widetilde {I_{m}})}.$

Theorem 3.2. (Local stability)

($i$) The disease-free equilibrium $E_{0}$ of system $(2.1)$ is locally stable if $\mathscr{R}_{0} < 1$ and unstable if $\mathscr{R}_{0} > 1$.

($ii$) The endemic equilibrium $E^{*}$ of system $(2.1)$ is locally stable if $\mathscr{R}_{0} > 1$.

Proof. First consider the local stability of the disease-free steady state $E_{0}$.

Clearly, we have $f(0) = \mathscr{R}_{0}$, $f'(\lambda) < 0$ and $\lim\limits_{\lambda \rightarrow \infty} f(\lambda) = 0$. Hence, if $\mathscr{R}_{0} > 1$, then $f(\lambda) = 1$ has a unique positive real root. Thus, if $\mathscr{R}_{0} > 1$, the disease-free steady state $E_{0}$ is unstable.

If $\mathscr{R}_{0} < 1$, the disease-free equilibrium $E_{0}$ is locally stable. Otherwise, $f(\lambda_{0}) = 1$ has at least one root $\lambda_{0} = a_{1}+ib_{1}$ satisfying $a_{1}\geq 0.$ But

Hence, if $\mathscr{R}_{0} < 1$, all roots of $f(\lambda) = 1$ have negative real parts, then the disease-free equilibrium $E_{0}$ is locally stable.

Next, we consider the local stability of the endemic equilibrium $E^{*}$.

If $\mathscr{R}_{0} > 1$, the endemic equilibrium $E^{*}$ is locally stable. Otherwise, $f(\lambda^{*}) = 1$ has at least one root $\lambda^{*} = a_{2}+ib_{2}$ satisfying $a_{2}\geq 0.$ But

Notice, $S_{h}^{*} S_{m}^{*} = \dfrac{(\mu_{1}+\alpha+\delta)\mu_{2}}{b^{2}\beta_{1}\beta_{2}\mathscr{K}_{2}\mathscr{K}_{3}}$, we know $\mid f(\lambda^{*})\mid < 1,$ Hence, if $\mathscr{R}_{0} > 1$, all roots of $f(\lambda) = 1$ have negative real parts, then the endemic equilibrium $E^{*}$ is locally stable. This completes the proof.

4.   Uniform persistence and global stability

4.1.   Uniform persistence

In this subsection, our purpose is to show that system $(2.1)$ is uniformly persistent when $\mathscr{R}_{0} > 1$. Define

let $\partial M_{0} = \mathscr{Y}\setminus M_{0}$, then we have $\mathscr{Y} = M_{0}\cup \partial M_{0}.$

Theorem 4.1. The sets $M_{0}$ and $\partial M_{0}$ are forward invariant under the semiflow $\Phi(t, \cdot)$. Also, the disease-free equilibrium $E_{0}$ of system $(2.1)$ is globally asymptotically stable for the semiflow $\Phi(t, \cdot)$ restricted to $\partial M_{0}$.

The proof of this theorem is in the appendix section. Theorem 4.1 states that if the number of people and mosquitoes initially infected by plasmodium belongs to $\partial M_{0}$, then malaria will eventually die out. But if the number of people and mosquitoes initially infected by plasmodium belongs to $M_{0}$, then whether malaria is eventually extinct, we need to verify by the following Theorem 4.2, Theorem 4.3 and Theorem 4.5.

Theorem 4.2. If $\mathscr{R}_{0} > 1,$ then semiflow $\{\Phi(t, \cdot)\}_{t\geq0}$ generated by system $(2.1)$ is uniformly persistent with respect to the decomposition $(M_{0}, \ \partial M_{0})$. Moreover, there is a compact subset $\mathcal{A}_{0}\subset M_{0}$, which is a global attractor for $\{\Phi(t, \cdot)\}_{t\geq0}$ in $M_{0}$.

Details of this proof are in appendix section. Theorem 4.2 shows that if $\mathscr{R}_{0}$ is greater than 1, when the number of people and mosquitoes initially infected by plasmodium belongs to $M_{0}$, malaria will always exist at a certain scale.

4.2.   Global stablility

Let $x \in \mathcal{A}$, we can find a complete orbit $\{\Phi(t, \cdot)\}_{t\in\mathbb{R}}$ through $x$ in $\mathcal{A}$. By similar analytical method used in [^[31], subsection $3.2$], system $(2.1)$ can be written as

Theorem 4.3. The disease-free equilibrium $E_{0}$ of system $(2.1)$ is globally asymptotically stable if $\mathcal{R}_{0} < 1$.

The proof of this theorem is in the appendix section. Theorem 4.3 shows that if $\mathcal{R}_{0}$ is less than 1, malaria will be extinct regardless of whether the number of people and mosquitoes initially infected by plasmodium belongs to $M_{0}$ or $\partial M_{0}$.

If $\mathcal{R}_{0} > 1$, we have know that the system $(2.1)$ is uniformly persistent and has a global attractor $\mathcal{A}_{0}$ in $M_{0}$, meanwhile, let $(S_{h}(0), \ S_{m}(0), \ I_{h}(0), \ I_{m}(0), \ P(0, a), \ E_{h}(0, \theta), \ E_{m}(0, \tau))\in \mathcal{A}_{0}$, system $(2.1)$ can be written as $(4.1)$ (replace $\mathcal{A}$ with $\mathcal{A}_{0}$). In order to study the global stability of $E^{*}$, we first prove the following lemma.

Lemma 4.4. There exist $\epsilon, M > 0$ such that all solutions in $\mathcal{A}_{0}$ for $t\in \mathbb{R}$, the following inequalities are satisfied

Proof. Let $(S_{h}(t), S_{m}(t), I_{h}(t), I_{m}(t), P(t, a), E_{h}(t, \theta), E_{m}(t, \tau))\in \mathcal{A}_{0}$.

First we prove that $S_{h}(t) > 0, \ S_{m}(t) > 0$ for $t\in \mathbb{R}$. We assume that there exists $t_{1}\in \mathbb{R}$ such that $S_{m}(t_{1}) = 0$. From $(4.1)$ we have $\dfrac{dS_{m}(t_{1})}{dt}\geq\Lambda_{1} > 0$, then $\exists \ \eta > 0$ such that $S_{m}(t_{1}-\eta) < 0$. This contradicts the $\mathcal{A}_{0} \subset M_{0}$. Thus, $S_{m}(t) > 0$ for $t\in \mathbb{R}$. Similarly, we can get $S_{h}(t) > 0$ for $t\in \mathbb{R}$.

Next we prove that $I_{h}(t) > 0, \ I_{m}(t) > 0$ for $t\in \mathbb{R}$. First we assume that there exists $t_{0}\in \mathbb{R}$ such that $I_{h}(t_{0}) = 0$ and $I_{m}(t_{0}) = 0$, then according to $I_{h}(t), \ I_{m}(t)$ equations in $(4.1)$, we can deduce that $I_{h}(t) = 0, \ I_{m}(t) = 0$ for $t\leq t_{0}.$ Next form $(4.1)$ we know $\int_{0}^{+\infty}E_{h}(t, \theta)d\theta = \int_{0}^{+\infty}k_{2}(\theta)b\beta_{1}S_{h}(t-\theta)I_{m}(t-\theta))d\theta = 0, \ \int_{0}^{+\infty}E_{m}(t, \tau)d\tau = \int_{0}^{+\infty}k_{3}(\tau)b\beta_{2}S_{m}(t-\tau)I_{h}(t-\tau))d\tau = 0$ for all $t\leq t_{0}$, then obviously, $\int_{0}^{+\infty}\gamma_{2}(\theta+s_{1})E_{h}(t, \theta)d\theta = 0, \ \int_{0}^{+\infty}\gamma_{3}(\tau+s_{2})E_{m}(t, \tau)d\tau = 0$ for all $s_{1}, \ s_{2}\geq 0$. This contradicts the $\mathcal{A}_{0} \subset M_{0}$. On the other hand, we assume that there exists $t_{0}\in \mathbb{R}$ such that one of $I_{h}(t_{0}), \ I_{m}(t_{0})$ is positive, so without loss of generality, we assume that $I_{h}(t_{0}) = 0, \ I_{m}(t_{0}) > 0$. Similarly, we have $I_{h}(t) = 0, \ \int_{0}^{+\infty}\gamma_{3}(\tau+s)E_{m}(t, \tau)d\tau = 0$ for all $s\geq 0, t\leq t_{0}.$ Then $\dfrac{dI_{m}(t)}{dt} = -\mu_{2}I_{m}(t)$ for all $t\leq t_{0}$, we deduce that $I_{m}(t_{0}) = I_{m}(t)e^{-\mu(t_{0}-t)}$, obviously, $I_{m}(t)\rightarrow +\infty$ as $t\rightarrow -\infty$. This contradicts the compactness of $\mathcal{A}_{0}$. Hence, $I_{h}(t) > 0, \ I_{m}(t) > 0$ for all $t\in \mathbb{R}$. In addition, from $(4.1)$, we deduce that $P(t, a) > 0, E_{h}(t, \theta) > 0, \ E_{m}(t, \tau) > 0$ for $t\in \mathbb{R}, a, \theta, \tau \in \mathbb{R}_{+}$.

According to the compactness of $\mathcal{A}_{0}$, we know there exist $\epsilon, M > 0$ such that

The proof is complete.

Theorem 4.5. If $\mathcal{R}_{0} > 1$, then endemic equilibrium $E^{*}$ of system $(2.1)$ is globally asymptotically stable in $M_{0}$.

Details of this proof are in appendix section. Theorem 4.5 shows that if $\mathcal{R}_{0}$ is greater than 1, when the number of people and mosquitoes initially infected by plasmodium belongs to $M_{0}$, the number of people and mosquitoes will eventually stabilize at $E^{*}$.

5.   Simulations for the malaria model

In this section, we present some numerical simulations to confirm the above theoretical results (i.e., Theorems $4.3$ and $4.5$). Some of the parameters are taken from^[8] $b = 0.2, \beta_{1} = 0.015, \delta = 0.05, \mu_{1} = 0.00004$. Other parameters are estimated. we estimate other parameters as follows $\Lambda_{1} = 0.2, \Lambda_{2} = 20, \beta_{2} = 0.015, \mu_{2} = 0.03, \alpha = 0.001$. Moreover, we assume that the prevention wane rate and the removal rate from latent class take the form

where $i = 1, 2, 3$, $t_{1} = 12, t_{2} = 7, t_{3} = 10$ (time units are days), $\gamma_{1} = 0.001, \gamma_{2} = 0.11, \gamma_{3} = 0.1$. By solving $\mathcal{R}_{0} = 1$, we obtain that $\psi_{0} = 0.4872.$ Obviously, $\mathcal{R}_{0}$ decreases as the degree of prevention $\psi$ increases. Thus, first, we let $\psi = 0.9 > \psi_{0}$, we can compute $\mathcal{R}_{0} = 0.5414 < 1$, then the disease-free equilibrium $E_{0} = (0.2426,666.667, 0, 0, 0.2183k_{1}(a), 0, 0)$ is globally stable, as depicted in Fig. 2, Fig. 3. Next, we let $\psi = 0.3 < \psi_{0}$, we can compute $\mathcal{R}_{0} = 1.6239 > 1$, then the endemic equilibrium $E^{*} = (0.5025,594.5913, 1.2122, 41.0726, 0.1507k_{1}(a), 0.0619k_{2}(\theta), 2.1623k_{3}(\tau))$ is globally stable, as depicted in Fig. 4, Fig. 5. From the numerical simulations we can conjecture that increasing $\psi$, the rate of prevention of the susceptible people, can eliminate malaria. This is consistent with the report ^[1] of the WHO that prevention is considered to have made a major contribution to the reduction in malaria burden since 2000.

6.   Discussion and conclusion

Malaria is a dangerous and sometimes fatal disease caused by malaria parasite that commonly infect female anopheles mosquitoes which take a blood meal on humans. It is one of the greatest challenges in the field of global public health. The cost of malaria to individuals, families, communities, nations is huge. This issue may become more serious due to lack of prevention. In our paper, we considered an age-structured malaria epidemic model with prevention period of susceptible people and latent period of two types of infected hosts. The basic reproduction number $\mathcal{R}_{0}$ of our model $(2.1)$ has been found by the definition. Based on regional division of state space and selection of appropriate Lyapunov functions, we proved that disease-free equilibrium $E_{0}$ is globally asymptotically stable if $\mathcal{R}_{0} < 1$, while endemic equilibrium $E^{*}$ is globally asymptotically stable in $M_{0}$ if $\mathcal{R}_{0} > 1$.

We are easy to see that $\mathcal{R}_{0}$ decreases as the rate of prevention $\psi$ increases, that is, increasing the degree of prevention can lead to malaria extinct. Thus, the prevention is very important for the control of malaria. If we don't take into account the prevention period and latent period of two types of infected hosts, and according to the dynamics of the ODE model in ^[32], we can easily know that the basic reproduction number $\mathcal{R}_{0}$ is reduced to $\overline{\mathcal{R}}_{0} = \frac{b^{2}\beta_{1}\beta_{2}\Lambda_{1}\Lambda_{2}}{(\mu_{1}+\alpha+\delta)\mu_{1}\mu_{2}^{2}}$, in addition, notice that $\mathscr{K}_{1} < 1$, $\mathscr{K}_{2} < 1$, $\mathscr{K}_{3} < 1$, we can find that $\mathcal{R}_{0} < \overline{\mathcal{R}}_{0}$, so considering the prevention period and latent period of two types of infected hosts can reduce the basic reproduction number, which are beneficial to understanding the dynamics of malaria control.

Acknowledgments

This work is supported by the NNSF of China(11861044and11661050), the NSF of Gansu Province (148RJZA024), and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology. We are grateful to the anonymous referees and the editors for their valuable comments and suggestions which improved the quality of the paper.

Conflict of interest

All authors declare no conflicts of interest in this paper.

Appendix

Proof of Theorem 2.3

Proof. We first decompose $\Phi$ into the following two parts: $U_{1}, U_{2}$ defined respectively by

where

for $x = (S_{h}(0), S_{m}(0), I_{h}(0), I_{m}(0), P_{0}(a), E_{h0}(\theta), E_{m0}(\tau)),$ clearly, we have $\Phi(t, x) = U_{1}(t, x)+U_{2}(t, x).$

Let $\mathscr{A}\subset \mathscr{Y}$, $r$ is a constant greater than $\frac{\Lambda_{1}+\Lambda_{2}}{\mu}$, for each $x \in \mathscr{A}$, we have $\parallel x \parallel_{\mathscr{Y}}\leq r.$

Thus, $\lim\limits_{t\rightarrow +\infty} diam \ U_{2}(t, \mathscr{A}) = 0$. In the following we show that $U_{1}(t, \mathscr{A})$ has compact closure for each $t\geq 0$.

From Proposition $2.3$, we know that $S_{i}(t), I_{i}(t)\ (i = h, m)$ remain in the compact set $[0, r]$ for all $t\geq 0$. Next, we will show that $\widetilde{P}(t, a), \widetilde{E_{h}}(t, \theta)\ and \ \widetilde{E_{m}}(t, \tau)$ remain in a precompact subset of $L_{1}^{+}(0, +\infty)$ which is independent of $x$.

It is easy to show that

Therefore, the conditions $(i)$, $(ii)$ and $(iv)$ of Lemma $2.2$ are satisfied. Now, we only to check the condition $(iii)$ of Lemma $2.2$.

Notice that $\mid\dfrac{dI_{h}(t)}{dt}\mid\leq\bar{\gamma}_{2}r+(\mu_{1}+\alpha+\delta)r$, $\mid\dfrac{dS_{m}(t)}{dt}\mid\leq\Lambda_{1}+(\mu_{1}+\varphi)r+b\beta_{1}r^{2}+\bar{\gamma_{1}}r$, we have

Then

Hence,

Thus, condition $(iii)$ of Lemma $2.2$ holds, then we can get that $\widetilde{E_{m}}(t, \tau)$ satisfies the conditions of Lemma $2.2$. In a similar way, $\widetilde{P}(t, a), \ \widetilde{E_{h}}(t, \theta)$ also satisfy the conditions of Lemma $2.2$. Therefore, we obtain $U_{1}(t, \mathscr{A})$ has compact closure for each $t\geq 0$, Using Lemma $2.1$, we know semiflow $\Phi$ is asymptotically smooth. This completes the proof.

Proof of Theorem 4.1

Proof. First we prove $M_{0}$ is forward invariant under the semiflow $\Phi(t, \cdot)$. Let $\Phi(0, x_{0})\in M_{0}$, if $I_{h}(0) > 0$ or $I_{m}(0) > 0$, From $(2.1)$, it's easy to know $I_{h}(t)\geq I_{h}(0)e^{-(\mu_{1}+\alpha+\delta)t} > 0$ or $I_{m}(t)\geq I_{m}(0)e^{-\mu_{2}t} > 0$, then $M_{0}$ is forward invariant. Otherwise, if $I_{h}(0) = I_{m}(0) = 0$, then without loss of generality, $\exists\ t_{1}\in\mathbb{R}_{+}$ such that $\int_{0}^{+\infty}\gamma_{2}(\theta+t_{1})E_{h}(0, \theta)d\theta > 0$ $(\ x_{0}\in M_{0})$. $\forall \ t\in[0, t_{1}]$, $\exists\ s = t_{1}-t\geq 0$ such that

If $\exists \ t_{2}\in(0, t_{1}]$ such that $I_{h}(t_{2}) > 0$, then $I_{h}(t) > 0, \ \forall t > t_{2},$ otherwise, since

we know $I_{h}(t) > 0$ for all $t > t_{1}$. Hence, $\Phi(t, M_{0})\subset M_{0}.$

This completes the fact that $\Phi(t, M_{0})\subset M_{0},$ i.e. $M_{0}$ is forward invariant under the semiflow $\Phi(t, \cdot)$.

Next, we will prove $\partial M_{0}$ is forward invariant under the semiflow $\Phi(t, \cdot)$. Let $\Phi(0, x_{0})\in \partial M_{0},$ we consider the following system

Since $S_{h}(t), \ S_{m}(t)\leq \Delta$, where $\Delta = max\{\dfrac{\Lambda_{1}+\Lambda_{2}}{\mu}, \parallel x_{0}\parallel_{\mathscr{Y}}\},$ it follows that

where

with the following initial and boundary conditions

By use of Volterra formulation, we have

Substituting $(6.5)$ into the first two equations of $(6.3)$ yield

where

Since

due to $\Phi(0, x_{0})\in \partial M_{0},$ then $F_{1}(t)\equiv0$ for $t\geq 0$. In a similar way, we have $F_{2}(t)\equiv0$ for $t\geq 0$. Accordingly, $(6.6)$ has a unique solution $\hat{I}_{i}(t)\equiv 0, (i = h, m)$ for $t\geq 0.$

From $(6.5)$, we know that $\hat{E}_{i}(t, s) = 0, (i = h, m)$ for $0\leq s\leq t.$ Thus, for all $u\geq0$,

Similarly, we have $\parallel \gamma_{3}(\tau+u)\hat{E}_{m}(t, \tau)\parallel_{L_{+}^{1}} = 0.$

By using $(6.2)$ we can obtain that $I_{h}(t) = 0, \ I_{m}(t) = 0, \parallel \gamma_{2}(\theta+u_{1})E_{h}(t, \theta)\parallel_{L_{+}^{1}} = 0, \parallel\gamma_{3}(\tau+u_{2}) E_{m}(t, \tau)\parallel_{L_{+}^{1}} = 0$ for $t, u_{1}, u_{2}\geq0$, and $E_{h}(t, \theta), \ E_{m}(t, \tau)\rightarrow 0$ as $t\rightarrow +\infty.$ Thus, $\partial M_{0}$ is forward invariant under the semiflow $\Phi(t, \cdot)$.

Finally, we prove the disease-free equilibrium $E_{0}$ of system $(2.1)$ is globally asymptotically stable for the semiflow $\Phi(t, \cdot)$ restricted to $\partial M_{0}$. In $\partial M_{0}$, system $(2.1)$ can be divided into the following two subsystems

Obviously, $\lim\limits_{t\rightarrow+\infty}S_{m}(t) = \frac{\Lambda_{2}}{\mu_{2}}.$ $(6.7)$ has unique equilibrium $(S_{h}^{0}, \ P^{0}(a)) = (\dfrac{\Lambda_{1}}{\mu_{1}+\psi(1-\mathscr{K}_{1})}, \psi S_{h}^{0}k_{1}(a)),$ and $S_{h}(t) > 0, \ \int_{0}^{+\infty}P(t, a)da > 0$ for $t > 0$. Similar to the analysis of the section $2$, $(6.7)$ has a global attractor $\mathcal{A}_{1}$ of bounded sets in $(0, +\infty)\times (L_{+}^{1}(0, +\infty)\setminus 0_{L^{1}}).$

Let $x \in \mathcal{A}_{1}$, we can find a complete orbit $\{\Phi(t, \cdot)\}_{t\in\mathbb{R}}$ through $x$ in $\mathcal{A}_{1}$. By similar analytical method used in [^[31], subsection $3.2$], system $(6.7)$ can be written as

Meanwhile, we easily find that $(6.9)$ can be written as

Using similar to the proof of lemma $4.2$^[33], we can get that any solution in $\mathcal{A}_{1}$ is satisfied that $S_{h}(t) > 0$ for $t\in \mathbb{R}$. Once the stability of system $(6.10)$ is obtained, in combination with $(6.9)$, we can obtain the stability of system $(6.7)$. In the following, we will define Lyapunov functionals on $\mathcal{A}_{1}$. Define

It is not difficult to find that $V_{0}(t)$ is positive-definite with $S_{h}^{0}$ as its global minimum point. Since compactness of $\mathcal{A}_{1}$, we can easily deduce $V_{0}(t)$ is bounded on $\mathcal{A}_{1}$. Calculating the derivative of $V_{0}(t)$ along the solution of system $(6.10)$, we obtain have

In addition, we have noted that $\Lambda_{1} = (\mu_{1}+\psi)S_{h}^{0}-\int_{0}^{+\infty}\phi(a)S_{h}^{0}da$, it follows that

Notice that $\mathscr{K}_{1} < 1$, then $\dfrac{dV_{0}}{dt}\leq0$ holds. Furthermore, the strict equality holds only if $S_{h}(t) = S_{h}^{0}$. Consequently, $S_{h}^{0}$ of $(6.10)$ is globally asymptotically stable. In combination with $(6.9)$, we have $(S_{h}^{0}, \ P^{0}(a))$ of $(6.7)$ is globally asymptotically stable. Hence, the equilibrium $E_{0}$ is globally asymptotically stable restricted to $\partial M_{0}$. The proof of Theorem $4.1$ is complete.

Proof of Theorem 4.2

Proof. Since the disease-free equilibrium $E_{0}$ is globally asymptotically stable restricted to $\partial M_{0}$, applying Theorem $4.2$ in ^[34], we need only to prove

where $W_{s}(E_{0}) = \{x\in\mathscr{Y}\ :\lim\limits_{t\rightarrow +\infty}\Phi(t, x) = E_{0}\}.$ By way of contradiction, we assume that there exists a $x_{0}\in M_{0}$ such that $x_{0}\in W_{s}(E_{0}).$ Then we can find a list of $\{x_{n}\}\subset M_{0}$ such that

Denote $\Phi(t, x_{n}) = (S_{hn}(t), S_{mn}(t), I_{hn}(t), I_{mn}(t), P_{n}(t, \cdot), E_{hn}(t, \cdot), E_{mn}(t, \cdot)).$ Then for all $t\geq0$, we have

and $\Phi(t, x_{n}) \subset M_{0}.$ Similar to the first part of theorem $4.1$ proof, we know that there exists $t_{0}\geq0$ such that $I_{hn}(t) > 0$ for $t\geq t_{0}$ or $I_{mn}(t) > 0$ for $t\geq t_{0}$, so without loss of generality, we can take $t_{0} = 0$ and $I_{hn}(0) > 0$. Since $\mathcal{R}_{0} > 1,$ we can choose large enough $n$ such that $S_{h}^{0} > \frac{1}{n}, \ S_{m}^{0} > \frac{1}{n}$ and

Now we construct the following system

Using similar analysis of Sect.2, we can get existence, uniqueness, and nonnegative of solution to system $(6.11)$. By the comparison principle, we know

By use of Volterra formulation, we have

Substituting $(6.13)$ into the first two equations of $(6.11)$ yield

We can claim that at least one of $\hat{I}_{h}(t), \hat{I}_{m}(t)$ is unbounded. Otherwise, we can use Laplace transform in the first two inequations of $(6.14)$ yield

where $\mathcal{L}[\hat{I}_{i}](\lambda) = \int_{0}^{+\infty}e^{-\lambda t}\hat{I}_{i}(t)dt, \ (i = h, m)$ for $(\lambda > 0)$, $\mathcal{L}[u_{1}](\lambda) = \int_{0}^{\infty} b\beta_{1}(S_{h}^{0}-\frac{1}{n})\gamma_{2}(\theta)k_{2}(\theta)e^{-\lambda \theta}d\theta$ and $\mathcal{L}[u_{2}](\lambda) = \int_{0}^{\infty} b\beta_{2}(S_{m}^{0}-\frac{1}{n})\gamma_{3}(\tau)k_{3}(\tau)e^{-\lambda \tau}d\tau$ for $(\lambda\geq0)$.

After a simple calculation, we obtain

$\mathcal{L}[u_{1}](\lambda)\rightarrow \mathcal{L}[u_{1}](0), \mathcal{L}[u_{2}](\lambda)\rightarrow \mathcal{L}[u_{2}](0)$ as $\lambda\rightarrow 0$ by the Dominated Convergence Theorem.

Since

then there exists $\varepsilon > 0$ such that

for all $\lambda\in[0, \varepsilon)$. According to $(6.16)$, we have $\mathcal{L}[\hat{I}_{h}](\lambda) < 0$ for all $\lambda\in(0, \varepsilon)$. But this contradicts the nonnegative of $\hat{I}_{h}(t), \ (t\geq0)$. Hence, at least one of $\hat{I}_{h}(t), \hat{I}_{m}(t)$ is unbounded. Since $I_{in}(t)\geq\hat{I}_{i}(t), (i = h, m)$, we get that at least one of $I_{hn}(t), I_{mn}(t)$ is unbounded. This contradicts the proposition $2.3$. Therefore, $W_{s}(E_{0})\cap M_{0} = \emptyset$. By Theorem $4.2$ ^[34], we get that semiflow $\{\Phi(t, \cdot)\}_{t\geq0}$ generated by system $(2.1)$ is uniformly persistent. By Theorem $3.7$ ^[29], we get that there exists a compact subset $\mathcal{A}_{0}\subset M_{0}$ which is a global attractor for $\{\Phi(t, \cdot)\}_{t\geq0}$ in $M_{0}$.

Proof of Theorem 4.3

Proof. Using similar to the proof of lemma $4.2$^[33], we can get that any solution in $\mathcal{A}$ is satisfied that $S_{h}(t), \ S_{m}(t) > 0$ for $t\in \mathbb{R}$.

Let $g(x) = x-\ln x-1$, note that $g(x)$ is non-negative and continuous on $(0, +\infty)$ with a unique root at $x = 1$. Next, we construct the following Lyapunov function $L = L_{1}+L_{2}+L_{3}$ on the global attractor $\mathcal{A}$, since compactness of $\mathcal{A}$, we can easily deduce $L$ is bounded on $\mathcal{A}$, where

Calculating the derivative of $L_{1}$, $L_{2}$, $L_{3}$ along solutions of system $(4.1)$, respectively. In the process of calculate the time derivative of $L_{1}$, we used $\Lambda_{2} = \mu_{2}S_{m}^{0}$, $\Lambda_{1} = (\mu_{1}+\psi)S_{h}^{0}-\psi\mathscr{K}_{1}S_{h}^{0}$, $\int_{0}^{+\infty}\phi(a)\int_{0}^{a}g(\dfrac{S_{h}(t-r)}{S_{h}^{0}})drda = \int_{0}^{+\infty}\phi(a)\int_{t-a}^{t}g(\dfrac{S_{h}(r)}{S_{h}^{0}})drda$ and $\int_{0}^{+\infty}\phi(a)da = \psi\mathscr{K}_{1}.$ We can deduce

Therefore,

Notice that $\mathscr{K}_{1} < 1$, if $\mathcal{R}_{0} < 1$, then $\frac{dL}{dt}\leq0$ holds. Let $T$ is the largest invariant subset of $\{\frac{dL}{dt}\mid_{(4.1)} = 0\}$, the equality holds only if $S_{m}(t) = S_{m}^{0}, \ S_{h}(t) = S_{h}^{0}, \ I_{h}(t) = 0$. In $T$, $S_{m}(t) = S_{m}^{0}$ for all $t$, then we have $\frac{dS_{h}(t)}{dt}\equiv0$. Combining this with $(4.1)$, it follows that $I_{m}(t) = 0$ for all $t$, moreover, $P(t, a) = \psi S_{h}^{0}k_{1}(a), \ E_{m}(t, \tau) = 0, \ E_{h}(t, \theta) = 0$ for all $t\in \mathbb{R}$, $a, \ \theta, \ \tau \in \mathbb{R}_{+}$. Hence, $T = \{E_{0}\}$. Assume that there exists $x\in\mathcal{A}\setminus T$ and we can find $\Phi(t, x)\subset \mathcal{A}$ through $x$ at $t = 0$, with alpha limit set $\alpha (x)$. Since $x\neq E_{0}$, we can deduce that $t\rightarrow L(\Phi(t, x))$ is a non-increasing and bounded function, then $L$ is a constant on $\alpha (x)$. Since $\alpha (x)$ is invariant in $\mathcal{A}$, we know $\alpha (x) = T$. From Theorem $3.2$ we know that the disease free equilibrium $E_{0}$ is locally asymptotically stable, which implies $x = E_{0}$, this contradicts the $x\neq E_{0}$. Hence, $\mathcal{A} = \{E_{0}\}$. This proves that $E_{0}$ is globally asymptotically stable. This completes the proof.

Proof of Theorem 4.5

Proof. We define the following Lyapunov function $V = V_{1}+V_{2}+V_{3}+V_{4}+V_{5}+V_{6}$ on the global attractor $\mathcal{A}_{0}$, using lemma $4.4$, we can easily deduce $V$ is bounded on $\mathcal{A}_{0}$, where

Calculating the derivative of $V$ along a solution in $\mathcal{A}_{0}$. In the process of calculate the time derivative of $V_{1}$, we notice that $\Lambda_{1} = b\beta_{1}S_{h}^{*}I_{m}^{*}+(\mu_{1}+\psi)S_{h}^{*}-\int_{0}^{+\infty}\phi(a)S_{h}^{*}da$.

By using

we have

Recalling that $I_{h}^{*} = \dfrac{\mathscr{K}_{2}E_{h}^{*}(0)}{\mu_{1}+\alpha+\delta}$, then

Thus,

Notice that $\Lambda_{2} = b\beta_{2}S_{m}^{*}I_{h}^{*}+\mu_{2}S_{m}^{*}$, we can get

Notice that $\dfrac{\partial }{\partial\tau}g(\dfrac{E_{m}(t, \tau)}{E_{m}^{*}(\tau)}) = \dfrac{1}{E_{m}^{*}(\tau)}(1-\dfrac{E_{m}^{*}(\tau)}{E_{m}(t, \tau)})(\frac{\partial E_{m}}{\partial \tau}+(\mu_{2}+\gamma_{3}(\tau))E_{m})$, we have

Notice that $I_{m}^{*} = \dfrac{\mathscr{K}_{3}E_{m}^{*}(0)}{\mu_{2}}$, we have

Thus,

By using $\dfrac{\int_{0}^{+\infty}\gamma_{2}(\theta)E_{h}^{*}(\theta)d\theta}{\mathscr{K}_{2}E_{h}^{*}(0)} = 1$, $\dfrac{\int_{0}^{+\infty}\gamma_{3}(\tau)E_{m}^{*}(\tau)d\tau}{\mathscr{K}_{3}E_{m}^{*}(0)} = 1$, we can get

It follows from the non-negativity of $g$, we know that $\dfrac{dV}{dt}\leq0$, and

Let $T^{*}$ is the largest invariant subset of $\{\frac{dV}{dt}\mid_{(4.1)} = 0\}$, then we have $\frac{dS_{h}(t)}{dt}\equiv0, \ \frac{dS_{m}(t)}{dt}\equiv0$. Combining this with $(4.1)$, we can obtain that $I_{h}(t) = I_{h}^{*}$ for all $t$, $P(t, a) = \psi S_{h}^{*}k_{1}(a).$ Further, $I_{m}(t) = I_{m}^{*}$ for all $t$, in combining with $(6.17)$, we can get $E_{h}(t, \theta) = E_{h}^{*}(\theta), \ E_{m}(t, \tau) = E_{m}^{*}(\tau)$. Hence, $T^{*} = \{E^{*}\}$. According to the proof of the theorem $4.3$, we can easily obtain that $\mathcal{A}_{0} = \{E^{*}\}$. This proves that $E^{*}$ is globally asymptotically stable. The proof is complete.