1. Introduction

Schistosomiasis (also known as bilharzia) is a disease caused by parasitic worms of the Schistosoma type [1] . Schistosomiasis affects almost 210 million people worldwide [2] , and an estimated 12,000 to 200,000 people die from it each year [3] [4] . The disease is most commonly found in Africa, Asia and South America [5] . Around 700 million people, in more than 70 countries, live in areas where the disease is common [4] [6] . Schistosomiasis is second only to malaria, as a parasitic disease with the greatest economic impact [7] .

Mathematical modeling has become an important tool in analyzing the spread and control of infectious diseases. In recent years, many schostosomiasis models have been proposed and studied ( [8] - [13] , etc.). These models provide a detailed exposition on how to describe, analyze, and predict epidemics of schistosomiasis for the ultimate purposes of developing control strategies and tactics for schistosomiasis transmission.

Many diseases incubate inside the hosts for a period of time before the hosts become infectious. Using a compartmental approach, one may assume that a susceptible individual first goes through an incubation period (and is said to become exposed or in the class E) after infection, before becoming infectious. The resulting models are of SEIR or SEIRS types, respectively, depending on whether the acquired immunity is permanent or otherwise.

In the aforementioned framework, their coefficients are considered as constants, which are approximated by average values. However, we note that ecosystems in the real world often appear the nonautonomous phenomenon. Recently many nonautonomous epidemic systems have been studied ( [14] - [20] , etc.). In fact, natural factors, such as seasonal changes in moisture and temperature, affect the abundance and activity of the intermediate snail host, Oncomelania hupensis, and the transmission dynamics of schistosomiasis are in a constant state of flux [21] . Moreover, there are many social factors related to human behaviors accounting for the change of schistosomiasis incidence, such as marked changes of contact rates caused by daily production activities [22] . This illustrates that the transmission of schistosomiasis shows seasonal behavior. In order to describe this kind of phenomenon, in the model, the parameters of the system should be functions of time. As far as we know, the research work on the nonautonomous schistosomiasis models is very few. Therefore, it is necessary to study nonautonomous schistosomiasis models.

In this paper, we assume large intermediate host population and thus ignore snail dynamics. Motivated by the above description, we develop a class of nonautonomous schistosomiasis transmission model with incubation period:

$\{\begin{array}{l}\frac{\text{d}S\left(t\right)}{\text{d}t}=\Lambda \left(t\right)-\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha S\left(t\right)}-\mu \left(t\right)S\left(t\right)+\delta \left(t\right)I\left(t\right)+\gamma \left(t\right)E\left(t\right),\\ \frac{\text{d}E\left(t\right)}{\text{d}t}=\frac{\beta \left(t\right)S\left(t\right)I\left(t\right)}{1+\alpha S\left(t\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right),\\ \frac{\text{d}I\left(t\right)}{\text{d}t}=\epsilon \left(t\right)E\left(t\right)-\left(\mu \left(t\right)+\delta \left(t\right)+d\left(t\right)\right)I\left(t\right).\end{array}$ (1.1)

with initial value

$S\left(0\right)>0,E\left(0\right)>0,I\left(0\right)>0.$ (1.2)

Here $S\left(t\right)$ , $E\left(t\right)$ and $I\left(t\right)$ denote the size of susceptible, exposed, infectious population at time t, respectively. $\Lambda \left(t\right)$ is the growth rate of population, $\mu \left(t\right)$ is the natural death rate of the population, $\beta \left(t\right)$ is the rate of the efficient contact, $\delta \left(t\right)$ and $\gamma \left(t\right)$ are the recovery rates of infectious population and exposed population, respectively, $d\left(t\right)$ is the disease-related death rate and $\epsilon \left(t\right)$ is the rate of developing infectivity at time t.

The organization of this paper is as follows. In the next section, we present preliminaries setting and propositions, which we use to analyze the long-time behavior of system (1.1) in the following sections. In Section 3, we establish the extinction of the disease of system (1.1). In Section 4, we will discuss the permanence of the infectious population. Our results are verified by numerical simulations in Section 5.

2. Preliminaries

In this section, system (1.1) satisfies the following assumptions:

(H₁): The functions $\Lambda \left(t\right)\mathrm{,}\delta \left(t\right)\mathrm{,}\mu \left(t\right)\mathrm{,}\delta \left(t\right)\mathrm{,}\gamma \left(t\right)\mathrm{,}\epsilon \left(t\right)\mathrm{,}d\left(t\right)$ are nonnegative, bounded and continuous on $\left[\mathrm{0,}+\infty \right)$ and $\beta \left(0\right)>0$ .

(H₂): There exist positive constants ${\omega }_i>0\left(i=1,2,3\right)$ such that

$\underset{t\to +\infty }{\lim \inf }{\displaystyle {\int }_t^{t+{\omega }_2}}\beta \left(s\right)\text{d}s>0,$

$\underset{t\to +\infty }{\lim \inf }{\displaystyle {\int }_t^{t+{\omega }_2}}\mu \left(s\right)\text{d}s>0,$

$\underset{t\to +\infty }{\lim \inf }{\displaystyle {\int }_t^{t+{\omega }_2}}\Lambda \left(s\right)\text{d}s>0.$

Adding all the equations of model (1.1), then we have

$\begin{array}{c}\Lambda \left(t\right)-\mu \left(t\right)N\left(t\right)\ge \frac{\text{d}N\left(t\right)}{\text{d}t}=\Lambda \left(t\right)-\mu \left(t\right)N\left(t\right)-d\left(t\right)I\left(t\right)\\ \ge \Lambda \left(t\right)-\left(\mu \left(t\right)+d\left(t\right)\right)N(t)\end{array}$

Let $N\left(t\right)=S\left(t\right)+E\left(t\right)+I\left(t\right)$ be the total population in system (1.1) with the initial value $N\left(0\right)=S\left(0\right)+E\left(0\right)+I\left(0\right)$ . We denote by $N^{\mathrm{<mo>\; *\; </mo>}}\left(t\right)$ the solution of

$\frac{\text{d}{N}^{*}\left(t\right)}{\text{d}t}=\Lambda \left(t\right)-\mu \left(t\right)N^{*}\left(t\right)$ (2.1)

with initial value (1.2), and denote by $N_{\ast }\left(t\right)$ the solution of

$\frac{\text{d}{N}_{\mathrm{<mo>\; *\; </mo>}}\left(t\right)}{\text{d}t}=\Lambda \left(t\right)-\left(\mu \left(t\right)+d\left(t\right)\right)N_{\mathrm{<mo>\; *\; </mo>}}\left(t\right)$ (2.2)

with initial value (1.2). Then

$N_{\mathrm{<mo>\; *\; </mo>}}\left(t\right)\le S\left(t\right)+E\left(t\right)+I\left(t\right)\le N^{\mathrm{<mo>\; *\; </mo>}}\left(t\right)\mathrm{.}$

By [22] , we have the following result:

Lemma 2.1. Suppose that assumptions (H₁) and (H₂) hold. Then:

(i) there exist positive constants $m>0$ and $M>0$ , such that

$\begin{array}{c}0<m\le \underset{t\to +\infty }{\lim \inf }{N}_{*}\left(t\right)\le \underset{t\to +\infty }{\lim \sup }{N}_{*}\left(t\right)\\ \le \underset{t\to +\infty }{\lim \inf }{N}^{*}\left(t\right)\le \underset{t\to +\infty }{\lim \sup }{N}^{*}\left(t\right)\le M<+\infty .\end{array}$ (2.3)

(ii) the solution $\left(S\left(t\right)\mathrm{,}E\left(t\right)\mathrm{,}I\left(t\right)\right)$ of system (1.1) with the initial value (1.2) exists, is uniformly bounded and $S\left(t\right)>0,E\left(t\right)>0,I\left(t\right)>0$ for all $t>0$ . For the solution $\left(S\left(t\right)\mathrm{,}E\left(t\right)\mathrm{,}I\left(t\right)\right)$ of system (1.1), we define

$G\left(p,t\right):=\frac{p\beta \left(t\right)S\left(t\right)}{1+\alpha S\left(t\right)}+\delta \left(t\right)+d\left(t\right)-\gamma \left(t\right)-\left(1+\frac{1}{p}\right)\epsilon (t)$

and

$W\left(p,t\right):=pE\left(t\right)-I\left(t\right).$ (2.4)

for $p>0$ , $t>0$ . In Sections 3 and 4 we use the following lemma in order to investigate the longtime behavior of system (1.1).

Lemma 2.2. If there exist positive constants $p>0$ and $T_1>0$ such that $G\left(p,t\right)<0$ for all $t\ge T_1$ , then there exists $T_2\ge T_1$ such that $W\left(p,t\right)>0$ or $W\left(p\mathrm{,}t\right)\le 0$ for all $t\ge T_2$ .

Proof: Suppose that there does not exist $T_2\ge T_1$ such that $W\left(p,t\right)>0$ or $W\left(p\mathrm{,}t\right)\le 0$ for all $t\ge T_2$ . So we have

$pE\left(s\right)=I\left(s\right)$ (2.5)

and

$\begin{array}{c}\frac{\text{d}W\left(p\mathrm{,}s\right)}{\text{d}t}=p\left\{\frac{\beta \left(s\right)S\left(s\right)I\left(s\right)}{1+\alpha S\left(s\right)}-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)E\left(s\right)\right\}\\ -\left\{\epsilon \left(s\right)E\left(s\right)-\left(\mu \left(s\right)+\delta \left(s\right)+d\left(s\right)\right)I\left(s\right)\right\}\\ =I\left(s\right)\left\{\frac{p\beta \left(s\right)S\left(s\right)}{1+\alpha S\left(s\right)}+\left(\mu \left(s\right)+\delta \left(s\right)+d\left(s\right)\right)\right\}\\ -pE\left(s\right)\left\{\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)+\frac{\epsilon \left(s\right)}{p}\right\}\\ >0.\end{array}$ (2.6)

Substituting (2.5) into (2.6), we have

$\begin{array}{c}0<pE\left(s\right)\left\{\frac{p\beta \left(s\right)S\left(s\right)}{1+\alpha S\left(s\right)}+\delta \left(s\right)+d\left(s\right)-\gamma \left(s\right)-\left(1+\frac{1}{p}\right)\epsilon \left(s\right)\right\}\\ =pE\left(s\right)G\left(p\mathrm{,}s\right)\mathrm{.}\end{array}$

From Lemma 2.1, we have $p>0$ and $E\left(s\right)>0$ , so $G\left(p,s\right)>0$ , which is a contradiction with $G\left(p,t\right)<0$ for all $t\ge T_1$ . The proof is completed.

3. Extinction of Infectious Population

In this section, we obtain conditions for focus on the extinction of the infectious population of system (1.1).

Theorem 3.1 Suppose that assumptions (H₁) and (H₂) hold. If there exist $\lambda >0$ , $p>0$ and $T_1>0$ such that

$R_1\left(\lambda ,p\right):=\underset{t\to +\infty }{\lim \sup }{\displaystyle {\int }_t^{t+\lambda }}\left\{p\frac{\beta \left(s\right)}{1+\alpha N^{*}\left(s\right)}{N}^{*}\left(s\right)-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)\right\}\text{d}s<0,$ (3.1)

$R_1^{\mathrm{<mo>\; *\; </mo>}}\left(\lambda \mathrm{,}p\right)\mathrm{<mo>\; :\; </mo>}=\underset{t\to +\infty }{\lim \sup }{\displaystyle {\int }_t^{t+\lambda }}\left\{\epsilon \left(s\right)\frac{1}{p}-\left(\mu \left(s\right)+\delta \left(s\right)+d\left(s\right)\right)\right\}\text{d}s<0$ (3.2)

and $G\left(p,t\right)<0$ for all $t\ge T_1$ , then infectious population $I\left(t\right)$ in system (1.1) is extinct. i.e.

$\underset{t\to +\infty }{\lim }I\left(t\right)=0.$

Proof: From Lemma 2.2, we consider the following two cases:

(i) $pE\left(t\right)>I\left(t\right)$ for all $t\ge T_2$ ;

(ii) $pE\left(t\right)\le I\left(t\right)$ for all $t\ge T_2$ .

First, we consider the cases (i). From the second equation of system (1.1), we have

$\begin{array}{c}\frac{\text{d}E\left(t\right)}{\text{d}t}=\frac{\beta \left(t\right)I\left(t\right)}{1+\alpha \left(N\left(t\right)-E\left(t\right)-I\left(t\right)\right)}\left(N\left(t\right)-E\left(t\right)-I\left(t\right)\right)\\ -\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ <\frac{p\beta \left(t\right)E\left(t\right)}{1+\alpha \left(N^{*}\left(t\right)-E\left(t\right)-I\left(t\right)\right)}\left(N^{*}\left(t\right)-E\left(t\right)-I\left(t\right)\right)\\ -\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ <\frac{p\beta \left(t\right)E\left(t\right)}{1+\alpha N^{*}\left(t\right)}{N}^{*}\left(t\right)-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ =E\left(t\right)\left\{\frac{p\beta \left(t\right)}{1+\alpha N^{*}\left(t\right)}{N}^{*}\left(t\right)-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)\right\}.\end{array}$

So we have

$E\left(t\right)<E\left(T_2\right)\exp \left({\displaystyle {\int }_{T_2}^t}\left\{p\frac{\beta \left(s\right)}{1+\alpha N^{*}\left(s\right)}{N}^{*}\left(s\right)-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)\right\}\text{d}s\right)$ (3.3)

for all $t\ge T_2$ . From (3.1), we see that there exist constants ${\delta }_1>0$ and $T_3>T_2$ such that

${\displaystyle {\int }_t^{t+\lambda }}\left\{p\frac{\beta \left(s\right)}{1+\alpha N^{\mathrm{<mo>\; *\; </mo>}}\left(s\right)}{N}^{\mathrm{<mo>\; *\; </mo>}}\left(s\right)-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)\right\}\text{d}s<-{\delta }_1$ (3.4)

for all $t>T_3$ . From (3.3) and (3.4), we obtain ${\lim }_{t\to +\infty }E\left(t\right)=0$ . Therefore, it follows from $pE\left(t\right)>I\left(t\right)$ , that ${\lim }_{t\to +\infty }I\left(t\right)=0$ . Now we consider the case (ii). From $E\left(t\right)\le \frac{I\left(t\right)}{p}$ for all $t\ge T_2$ and the third equation of (1.1), we have

$\frac{\text{d}I\left(t\right)}{\text{d}t}\le I\left(t\right)\left\{\epsilon \left(t\right)\frac{1}{p}-\left(\mu \left(t\right)+\delta \left(t\right)+d\left(t\right)\right)\right\}\mathrm{.}$

Then the following expression

$I\left(t\right)\le I\left(T_2\right)\exp \left({\displaystyle {\int }_{T_2}^t}\left\{\frac{\epsilon \left(s\right)}{p}-\mu \left(s\right)-\delta \left(s\right)-d\left(s\right)\right\}\text{d}s\right)$ (3.5)

for all $t>T_2$ hold. Hence, by (3.2), there exist ${\delta }_2>0$ and $T_4>T_2$ such that

${\displaystyle {\int }_{t+\lambda }^t}\left\{\epsilon \left(s\right)\frac{1}{p}-\left(\mu \left(s\right)+\delta \left(s\right)+d\left(s\right)\right)\right\}\text{d}s<-{\delta }_2,$ (3.6)

for $t\ge T_4$ . From (3.5) and (3.6), we have

$\underset{t\to +\infty }{\lim }I\left(t\right)=0.$

4. Permanence of Infectious Population

In this section, we obtain the sufficient conditions for the permanence of infectious population.

Theorem 4.1. Suppose that assumptions (H₁) and (H₂) hold. If there are $\lambda >0$ , $p>0$ and $T_1>0$ such that

$R_2\left(\lambda ,p\right):=\underset{t\to +\infty }{\lim \inf }{\displaystyle {\int }_t^{t+\lambda }}\left\{p\frac{\beta \left(s\right)}{1+\alpha N_{*}\left(s\right)}{N}_{*}\left(s\right)-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)\right\}\text{d}s>0,$ (4.1)

$R_2^{*}\left(\lambda ,p\right):=\underset{t\to +\infty }{\lim \inf }{\displaystyle {\int }_t^{t+\lambda }}\left\{\epsilon \left(s\right)\frac{1}{p}-\left(\mu \left(s\right)+\delta \left(s\right)+d\left(s\right)\right)\right\}\text{d}s>0$ (4.2)

and $G\left(p,t\right)<0$ for all $t\ge T_1$ , then $I\left(t\right)$ in system (1.1) is permanent.

Before we give the proof of Theorem 4.1, we first prove the following lemma.

Lemma 4.1. If there exist constants $\lambda >0$ , $p>0$ and $T_1>0$ such that (4.1), (4.2) and $G\left(p\mathrm{,}t\right)>0$ hold for all $t\ge T_1$ . Then there exists $T_2>T_1$ so that $W\left(p\mathrm{,}t\right)\le 0$ for all $t\ge T_2$ .

Proof: From Lemma 2.2, we consider the following two cases:

(i) $W\left(p\mathrm{,}t\right)>0$ for all $t\ge T_2$ ;

(ii) $W\left(p\mathrm{,}t\right)\le 0$ for all $t\ge T_2$ .

Suppose $W\left(p\mathrm{,}t\right)>0$ for all $t\ge T_2$ , then we have $E\left(t\right)>\frac{I\left(t\right)}{p}$ for all $t\ge T_2$ . From the third equation of system (1.1), we have

$\begin{array}{c}\frac{\text{d}I\left(t\right)}{\text{d}t}>\epsilon \left(t\right)\frac{1}{p}I\left(t\right)-\left(\mu \left(t\right)+\delta \left(t\right)+d\left(t\right)\right)I\left(t\right)\\ =I\left(t\right)\left\{\epsilon \left(t\right)\frac{1}{p}-\left(\mu \left(t\right)+\delta \left(t\right)+d\left(t\right)\right)\right\}.\end{array}$

So we obtain

$I\left(t\right)>I\left(T_2\right)\exp \left({\displaystyle {\int }_{T_2}^t}\left\{\frac{\epsilon \left(s\right)}{p}-\mu \left(s\right)-\delta \left(s\right)-d\left(s\right)\right\}\text{d}s\right)$ (4.3)

for all $t\ge T_2$ . From the inequality (4.2), there exist positive constants $\eta >0$ and $T>0$ such that

${\displaystyle {\int }_{t+\lambda }^t}\left\{\epsilon \left(s\right)\frac{1}{p}-\left(\mu \left(s\right)+\delta \left(s\right)+d\left(s\right)\right)\right\}\text{d}s>\eta$ (4.4)

for all $t\ge T$ . So the inequality (4.3) holds for all $t\ge \max \left\{T_2,T\right\}$ . Then ${\lim }_{t\to +\infty }I\left(t\right)=+\infty$ , which contradicts with the boundedness of $I\left(t\right)$ in Lemma 2.1. Now, we prove Theorem 4.1 by using Lemma 4.1.

Proof: For simplicity, let $m_{ϵ}:=m-ϵ$ , $M_{ϵ}:=M+ϵ$ , where $ϵ>0$ is a constant. In fact, Lemma 2.1 implies that for any sufficiently small $ϵ>0$ , there exists $T>0$ such that

$m_{ϵ}<N_{*}\left(t\right)\le N^{*}\left(t\right)<M_{ϵ}$ (4.5)

for all $t\ge T$ . The inequality (4.1) implies that for any sufficiently small $\eta >0$ , there exists $T_1>T$ such that

${\displaystyle {\int }_t^{t+\lambda }}\left\{\frac{p\beta \left(s\right)N_{*}\left(s\right)}{1+\alpha N_{*}\left(s\right)}-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)\right\}\text{d}s>\eta$ (4.6)

for all $t\ge T_1$ . We define

${\beta }^{+}:=\underset{t\ge 0}{\sup }\beta \left(t\right),{\mu }^{+}:=\underset{t\ge 0}{\sup }\mu \left(t\right),{\delta }^{+}:=\underset{t\ge 0}{\sup }\delta \left(t\right),$

$d^{+}:=\underset{t\ge 0}{\sup }d\left(t\right),{\epsilon }^{+}:=\underset{t\ge 0}{\sup }\epsilon \left(t\right),{\gamma }^{+}:=\underset{t\ge 0}{\sup }\gamma \left(t\right).$

Thus, by (4.5) and (4.6), for any sufficiently small ${\eta }_1<\eta$ and $T_2>T_1$ , there exist very small ${ϵ}_i$ , $i\in \left\{\mathrm{1,2,3}\right\}$ such that

${\displaystyle {\int }_t^{t+\lambda }}\left\{\frac{\beta \left(s\right)}{1+\alpha N_{*}\left(s\right)}\left(N_{*}\left(s\right)-{ϵ}_1-k{ϵ}_2\right)p-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)\right\}\text{d}s>{\eta }_1,$ (4.7)

$N_{*}\left(t\right)-{ϵ}_1-k{ϵ}_2>m_{ϵ}$ (4.8)

for all $t\ge T_2$ , where $k:=1+\frac{{\beta }^{+}}{1+\alpha M_{ϵ}}{M}_{ϵ}{\omega }_2$ . Lemma 2.1 implies that for any sufficiently small ${ϵ}_2>0$ , we have

${\displaystyle {\int }_{t_1}^{t_1+{\omega }_2}}\left\{\frac{\beta \left(s\right)M_{ϵ}{ϵ}_2}{1+\alpha M_{ϵ}}-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right){ϵ}_1\right\}\text{d}s<-{\eta }_1$ (4.9)

for all $t\ge T_2$ . First, we prove

$\underset{t\to +\infty }{\lim \sup }I\left(t\right)>{ϵ}_2.$

In fact, if it is not true, there exists $T_3\ge T_2$ such that

$I\left(t\right)\le {ϵ}_2$ (4.10)

for all $t\ge T_3$ . If $E\left(t\right)\ge {ϵ}_1$ for all $t\ge T_3$ , then from (4.5) and (4.6), we have

$\begin{array}{l}E\left(t\right)\\ =E\left(T_3\right)+{\displaystyle {\int }_{T_3}^t}\left\{\frac{\beta \left(s\right)I\left(s\right)}{1+\alpha S\left(s\right)}\left(N\left(s\right)-E\left(s\right)-I\left(s\right)\right)-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)E\left(s\right)\right\}\text{d}s\\ \le E\left(T_3\right)+{\displaystyle {\int }_{T_3}^t}\left\{\frac{\beta \left(s\right)}{1+\alpha M_{ϵ}}{M}_{ϵ}{ϵ}_2-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right){ϵ}_1\right\}\text{d}s\end{array}$

for all $t\ge T_3$ . It follows from inequality (4.9) that ${\lim }_{t\to +\infty }E\left(t\right)=-\infty$ . This contradicts with the boundedness of solution. Hence, there exists an $s_1\ge T_3$ such that $E\left(s_1\right)<{ϵ}_1$ . In the following we prove

$E\left(t\right)\le {ϵ}_1+\frac{{\beta }^{+}}{1+\alpha M_{ϵ}}{M}_{ϵ}{\omega }_2{ϵ}_2\mathrm{,}$ (4.11)

for all $t\ge s_1$ . If it is not true, there exists an $s_2>s_1$ such that

$E\left(s_2\right)>{ϵ}_1+\frac{{\beta }^{+}}{1+\alpha M_{ϵ}}{M}_{ϵ}{\omega }_2{ϵ}_2.$

Hence, there necessarily exists an $s_3\in \left(s_1\mathrm{,}{s}_2\right)$ such that $E\left(s_3\right)={ϵ}_1$ and $E\left(t\right)>{ϵ}_1$ for all $t\in \left(s_3\mathrm{,}{s}_2\right)$ . Let $n\ge 0$ be an integer such that $s_2\in \left[s_3+n{\omega }_2\mathrm{,}{s}_3+\left(n+1\right){\omega }_2\right]$ . By (4.9), we obtain

$\begin{array}{l}E\left(s_2\right)\\ =E\left(s_3\right)+{\displaystyle {\int }_{s_2}^{s_3}}\left\{\frac{\beta \left(s\right)I\left(s\right)}{1+\alpha S\left(s\right)}\left(N^{*}\left(s\right)-E\left(s\right)-I\left(s\right)\right)-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)E\left(s\right)\right\}\text{d}s\\ <{ϵ}_1+\left\{{\displaystyle {\int }_{s_3}^{s_3+n{\omega }_2}}+{\displaystyle {\int }_{s_3+n{\omega }_2}^{s_2}}\right\}\left\{\frac{\beta \left(s\right)M_{ϵ}{ϵ}_2}{1+\alpha M_{ϵ}}-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right){ϵ}_1\right\}\text{d}s\\ <{ϵ}_1+{\displaystyle {\int }_{s_3+n{\omega }_2}^{s_2}}\frac{\beta \left(s\right)}{1+\alpha M_{ϵ}}{M}_{ϵ}{ϵ}_2\text{d}s\\ <{ϵ}_1+\frac{{\beta }^{+}}{1+\alpha M_{ϵ}}{M}_{ϵ}{\omega }_2{ϵ}_2,\end{array}$

This contradicts with $E\left(s_2\right)>{ϵ}_1+\frac{{\beta }^{+}}{1+\alpha M_{ϵ}}{M}_{ϵ}{\omega }_2{ϵ}_2$ . Hence, (4.11) is valid. By Lemma 4.1, there exists $T_4\ge s_1$ such that $W\left(p,t\right)=pE\left(t\right)-I\left(t\right)\le 0$ for all $t\ge T_4$ . Therefore, by (4.10) and (4.11), we have $E\left(t\right)+I\left(t\right)\le {ϵ}_1+k{ϵ}_2$ for all $t\ge T_4$ , then

$\begin{array}{c}\frac{\text{d}E\left(t\right)}{\text{d}t}=\frac{\beta \left(t\right)I\left(t\right)\left(N\left(t\right)-E\left(t\right)-I\left(t\right)\right)}{1+\alpha \left(N\left(t\right)-E\left(t\right)-I\left(t\right)\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ \ge \frac{\beta \left(t\right)I\left(t\right)\left(N_{*}\left(t\right)-E\left(t\right)-I\left(t\right)\right)}{1+\alpha \left(N_{*}\left(t\right)-E\left(t\right)-I\left(t\right)\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ \ge \frac{\beta \left(t\right)E\left(t\right)\left(N_{*}\left(t\right)-E\left(t\right)-I\left(t\right)\right)p}{1+\alpha N_{*}\left(t\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ \ge E\left(t\right)\left\{\frac{\beta \left(t\right)\left(N_{*}\left(t\right)-E\left(t\right)-I\left(t\right)\right)p}{1+\alpha N_{*}\left(t\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)\right\}\\ \ge E\left(t\right)\left\{\frac{\beta \left(t\right)\left(N_{*}\left(t\right)-{ϵ}_1-k{ϵ}_2\right)p}{1+\alpha N_{*}\left(t\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)\right\}.\end{array}$

We obtain

$E\left(t\right)\ge E\left(T_4\right)\exp \left({\displaystyle {\int }_{T_4}^t}\left\{\frac{\beta \left(s\right)\left(N_{*}\left(s\right)-{ϵ}_1-k{ϵ}_2\right)p}{1+\alpha N_{*}\left(s\right)}-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)\right\}\text{d}s\right).$

By (4.7) we obtain ${\lim }_{t\to +\infty }E\left(t\right)=+\infty$ . This contradicts with Lemma 2.1 ( $E\left(t\right)$ is uniformly bounded). Hence, $\lim {\sup }_{t\to +\infty }I\left(t\right)>{ϵ}_2$ is true.

Next, we prove

$\underset{t\to +\infty }{\lim \inf }I\left(t\right)\ge I_1\mathrm{,}$

where $I_1>0$ is a constant given in the following lines. By inequality (4.7), (4.8), (4.9) and Lemma 2.1, there exist ${\tilde{T}}_3\left(\ge T_2\right)$ , ${\lambda }_2>0$ , ${\eta }_2>0$ such that ${\lambda }_3\ge {\lambda }_2$ and $t\ge {\tilde{T}}_3$ , we obtain

${\displaystyle {\int }_t^{t+{\lambda }_3}}\left\{\frac{\beta \left(s\right)}{1+\alpha N_{*}\left(s\right)}\left(N_{*}\left(s\right)-{ϵ}_1-k{ϵ}_2\right)p-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right)\right\}\text{d}s>{\eta }_2,$ (4.12)

${\displaystyle {\int }_{t_1}^{t_1+{\lambda }_3}}\left\{\frac{\beta \left(s\right)M_{ϵ}{ϵ}_2}{1+\alpha M_{ϵ}}-\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right){ϵ}_1\right\}\text{d}s<-M_{ϵ},$ (4.13)

${\displaystyle {\int }_{t_1}^{t_1+{\lambda }_3}}\beta \left(s\right)\text{d}s>{\eta }_2.$ (4.14)

Let $C>0$ be a constant satisfying

${\text{e}}^{-\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right){\lambda }_2}\frac{m_{ϵ}{\nu }_2{\eta }_2{\text{e}}^{n^{*}{\eta }_2}}{1+\alpha M_{ϵ}}>{ϵ}_1+\frac{{\beta }^{+}{M}_{ϵ}{\omega }_2{ϵ}_2}{1+\alpha M_{ϵ}}$ (4.15)

where ${\nu }_2={ϵ}_2{\text{e}}^{-\left({\mu }^{+}+{\delta }^{+}+d^{+}\right)2{\lambda }_2}$ , $n^{*}\le \frac{C}{{\lambda }_2}$ .

Because we have proved $\lim {\sup }_{t\to +\infty }I\left(t\right)>{ϵ}_2$ , there are only two possibilities as follows:

(i) There exists ${\tilde{T}}_4\ge {\tilde{T}}_3$ , then as $t\ge {\tilde{T}}_4$ , we obtain $I\left(t\right)\ge {ϵ}_2$ ;

(ii) $I\left(t\right)$ oscillates about ${ϵ}_2$ for all large t.

In case (i), we have $\lim {\inf }_{t\to +\infty }I\left(t\right)\ge {ϵ}_2=:I_1$ . In case (ii), there necessarily exist $t_1\mathrm{,}{t}_2\ge {\tilde{T}}_3\left(t_2\ge t_1\right)$ such that

$\{\begin{array}{l}I\left(t_1\right)=I\left(t_2\right)={ϵ}_2,\\ I\left(t\right)<{ϵ}_2,t\in \left(t_1,t_2\right).\end{array}$

Suppose that $t_2-t_1\le C+2{\lambda }_2$ . Then

$\frac{\text{d}I\left(t\right)}{\text{d}t}\ge -\left({\mu }^{+}+{\delta }^{+}+d^{+}\right)I\left(t\right),$ (4.16)

which implies

$\begin{array}{c}I\left(t\right)\ge I\left(t_1\right)\exp \left({\displaystyle {\int }_{t_1}^t}-\left({\mu }^{+}+{\delta }^{+}+d^{+}\right)\text{d}s\right)\\ \ge {ϵ}_2{\text{e}}^{-\left({\mu }^{+}+{\delta }^{+}+d^{+}\right)\left(C+2{\lambda }_2\right)}:=I_1.\end{array}$

for all $t\in \left(t_1\mathrm{,}{t}_2\right)$ . Suppose that $t_2-t_1>C+2{\lambda }_2$ . Then

$I\left(t\right)\ge {ϵ}_2{\text{e}}^{-\left({\mu }^{+}+{\delta }^{+}+d^{+}\right)\left(C+2{\lambda }_2\right)}=I_1$

for all $t\in \left(t_1\mathrm{,}{t}_1+C+2{\lambda }_2\right)$ . Now we only prove $I\left(t\right)\ge I_1$ for all $t\in \left[t_1+C+2{\lambda }_2\mathrm{,}{t}_2\right)$ . If $E\left(t\right)\ge {ϵ}_1$ for all $t\in \left[t_1\mathrm{,}{t}_1+{\lambda }_2\right]$ . By the second equation of system (1.1) and inequality (4.13), we have

$\begin{array}{c}E\left(t_1+{\lambda }_2\right)\le E\left(t_1\right)+{\displaystyle {\int }_{t_1}^{t_1+{\lambda }_2}\left\{\frac{\beta \left(s\right)M_{ϵ}}{1+\alpha M_{ϵ}}{ϵ}_2+\left(\mu \left(s\right)+\epsilon \left(s\right)+\gamma \left(s\right)\right){ϵ}_1\right\}\text{d}s}\\ <M_{ϵ}-M_{ϵ}=0,\end{array}$

which is contradiction. Hence, there exists an $s_4\in \left[t_1\mathrm{,}{t}_1+{\lambda }_2\right]$ such that $E\left(s_4\right)<{ϵ}_1$ . We obtain that for $t\ge s_4$ ,

$E\left(t\right)\le {ϵ}_1+\frac{{\beta }^{+}}{1+\alpha M_{ϵ}}{M}_{ϵ}{\omega }_2{ϵ}_2\mathrm{.}$ (4.17)

By inequality (4.16), then for $t\in \left[t_1\mathrm{,}{t}_1+2{\lambda }_2\right]$

$I\left(t\right)\ge {\nu }_2={ϵ}_2{\text{e}}^{-\left({\mu }^{+}+{\delta }^{+}+d^{+}\right)2{\lambda }_2}.$ (4.18)

Therefore, by the second equation of system (1.1) and inequalities (4.8), (4.17), (4.18), we obtain that

$\begin{array}{c}\frac{\text{d}E\left(t\right)}{\text{d}t}=\frac{\beta \left(t\right)I\left(t\right)\left(N\left(t\right)-E\left(t\right)-I\left(t\right)\right)}{1+\alpha \left(N\left(t\right)-E\left(t\right)-I\left(t\right)\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ \ge \frac{\beta \left(t\right)I\left(t\right)}{1+\alpha M_{ϵ}}\left(N_{*}\left(t\right)-E\left(t\right)-I\left(t\right)\right)-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ \ge \frac{\beta \left(t\right)}{1+\alpha M_{ϵ}}{m}_{ϵ}-\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right)E\left(t\right).\end{array}$

for all $t\in \left[t_1+{\lambda }_2\mathrm{,}{t}_1+2{\lambda }_2\right]$ . By (4.14), we have

$\begin{array}{c}E\left(t\right)\ge {\text{e}}^{-\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right)\left(t_1+2{\lambda }_2\right)}\{E\left(t_1+{\lambda }_2\right){\text{e}}^{\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right)\left(t_1+{\lambda }_2\right)}\\ +{\displaystyle {\int }_{t_1+{\lambda }_2}^{t_1+2{\lambda }_2}}\frac{\beta \left(s\right)}{1+\alpha M_{ϵ}}{m}_{ϵ}{\nu }_2{\text{e}}^{\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right)s}\text{d}s\}\\ \ge {\text{e}}^{-\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right)\left(t_1+2{\lambda }_2\right)}{\displaystyle {\int }_{t_1+{\lambda }_2}^{t_1+2{\lambda }_2}}\frac{\beta \left(s\right)}{1+\alpha M_{ϵ}}{m}_{ϵ}{\nu }_2{\text{e}}^{\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right)s}\text{d}s\\ \ge {\text{e}}^{-\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right){\lambda }_2}\frac{1}{1+\alpha M_{ϵ}}{m}_{ϵ}{\nu }_2{\eta }_2\mathrm{.}\end{array}$ (4.19)

Now, we suppose there exists $t_0>0$ such that $t_0\in \left(t_1+C+2{\lambda }_2\mathrm{,}{t}_2\right)$ , then $I\left(t_0\right)=I_1$ and $I\left(t\right)\ge I_1$ for all $t\in \left[t_1\mathrm{,}{t}_0\right]$ . By Lemma 4.1, we assume that $t_1$ is so large that $W\left(p,t\right)=pE\left(t\right)-I\left(t\right)\le 0$ for all $t\ge t_1+2{\lambda }_2$ . Hence, by (4.8), we further have

$\begin{array}{c}\frac{\text{d}E\left(t\right)}{\text{d}t}=\frac{\beta \left(t\right)I\left(t\right)\left(N\left(t\right)-E\left(t\right)-I\left(t\right)\right)}{1+\alpha \left(N\left(t\right)-E\left(t\right)-I\left(t\right)\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)E\left(t\right)\\ \ge E\left(t\right)\left\{\frac{\beta \left(t\right)\left(N_{*}\left(t\right)-{ϵ}_1-k{ϵ}_2\right)p}{1+\alpha N_{*}\left(t\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)\right\}\end{array}$

for all $t\in \left(t_1+2{\lambda }_2\mathrm{,}{t}_2\right)$ . By (4.12) and (4.19), we have

$\begin{array}{l}E\left(t\right)\\ =E\left(t_1+2{\lambda }_2\right)\exp \left({\displaystyle {\int }_{t_1+2{\lambda }_2}^{t_0}}\left\{\frac{\beta \left(t\right)\left(N_{*}\left(t\right)-{ϵ}_1-k{ϵ}_2\right)p}{1+\alpha N_{*}\left(t\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)\right\}\text{d}s\right)\\ =E\left(t_1+2{\lambda }_2\right)\exp (\{{\displaystyle {\int }_{t_1+2{\lambda }_2}^{t_1+2{\lambda }_2+{\lambda }_2}}+{\displaystyle {\int }_{t_1+2{\lambda }_2+{\lambda }_2}^{t_1+2{\lambda }_2+2{\lambda }_2}}+\cdots \begin{array}{c}{}^{}\\ \end{array}\\ +{\displaystyle {\int }_{t_1+2{\lambda }_2+(n^{*}-1){\lambda }_2}^{t_0}}\}\left\{\frac{\beta \left(t\right)\left(N_{*}\left(t\right)-{ϵ}_1-k{ϵ}_2\right)p}{1+\alpha N_{*}\left(t\right)}-\left(\mu \left(t\right)+\epsilon \left(t\right)+\gamma \left(t\right)\right)\right\}\text{d}s)\\ \ge {\text{e}}^{-\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right){\lambda }_2}\frac{1}{1+\alpha M_{ϵ}}{m}_{ϵ}{\nu }_2{\eta }_2{\text{e}}^{n^{*}{\eta }_2}.\end{array}$

Thus, by (4.17), we have

${\epsilon }_1+\frac{{\beta }^{+}{M}_{ϵ}{\omega }_2{ϵ}_2}{1+\alpha M_{ϵ}}\ge {\text{e}}^{-\left({\mu }^{+}+{\epsilon }^{+}+{\gamma }^{+}\right){\lambda }_2}\frac{m_{ϵ}{\nu }_2{\eta }_2{\text{e}}^{n^{*}{\eta }_2}}{1+\alpha M_{ϵ}},$

This contradicts with (4.15). Hence, $I\left(t\right)\ge I_1$ for all $t\in \left[t_1+C+2{\lambda }_2\mathrm{,}{t}_2\right]$ , which implies $\lim {\inf }_{t\to +\infty }I\left(t\right)\ge I_1$ . Thus, the infectious population of system (1.1) is permanent.

5. Numerical Simulations

Numerical verification of the results is necessary for completeness of the analytical study. In Sections 3 and 4, we focused our attention on the dynamic analysis of system (1.1). In the present section, numerical simulations are carried out to illustrate the analytical results of system (1.1) by means of the software Matlab.

In order to testify the validity of our results, in system (1.1), fix $\Lambda =1$ , $\beta \left(t\right)=0.3+0.1\cos t$ , $\alpha =0.3$ , $\mu =0.1$ , $\delta =0.1$ , $\gamma =0.8$ , $ϵ=0.2$ , $d=0.5$ . Then, from system (1.1), we have ${\lim }_{t\to +\infty }{N}^{*}\left(t\right)=1$ . We easily verify that assumptions (H1) and (H2) hold.We choose $\lambda =1$ and $p=2$ . Then we have

$\begin{array}{l}{R}_1\left(\lambda ,p\right):={\displaystyle {\int }_0^1}\left\{2\frac{0.3+0.1\cos t}{1+0.3}-\left(0.1+0.2+0.8\right)\right\}\text{d}t\approx -0.84<0,\\ R_1^{*}\left(\lambda ,p\right):={\displaystyle {\int }_0^1}\left\{0.2\frac{1}{2}-\left(0.1+0.1+0.5\right)\right\}\text{d}t=-0.7<0.\end{array}$