Schistosomiasis (also known as bilharzia) is a disease caused by parasitic worms of the Schistosoma type  . Schistosomiasis affects almost 210 million people worldwide  , and an estimated 12,000 to 200,000 people die from it each year   . The disease is most commonly found in Africa, Asia and South America  . Around 700 million people, in more than 70 countries, live in areas where the disease is common   . Schistosomiasis is second only to malaria, as a parasitic disease with the greatest economic impact  .
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 (  -  , 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 (  -  , 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  . 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  . 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:
with initial value
Here , and denote the size of susceptible, exposed, infectious population at time t, respectively. is the growth rate of population, is the natural death rate of the population, is the rate of the efficient contact, and are the recovery rates of infectious population and exposed population, respectively, is the disease-related death rate and 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.
In this section, system (1.1) satisfies the following assumptions:
(H1): The functions are nonnegative, bounded and continuous on and .
(H2): There exist positive constants such that
Adding all the equations of model (1.1), then we have
Let be the total population in system (1.1) with the initial value . We denote by the solution of
with initial value (1.2), and denote by the solution of
with initial value (1.2). Then
By  , we have the following result:
Lemma 2.1. Suppose that assumptions (H1) and (H2) hold. Then:
(i) there exist positive constants and , such that
(ii) the solution of system (1.1) with the initial value (1.2) exists, is uniformly bounded and for all . For the solution of system (1.1), we define
for , . 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 and such that for all , then there exists such that or for all .
Proof: Suppose that there does not exist such that or for all . So we have
Substituting (2.5) into (2.6), we have
From Lemma 2.1, we have and , so , which is a contradiction with for all . 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 , and such that
and for all , then infectious population in system (1.1) is extinct. i.e.
Proof: From Lemma 2.2, we consider the following two cases:
(i) for all ;
(ii) for all .
First, we consider the cases (i). From the second equation of system (1.1), we have
So we have
for all . From (3.1), we see that there exist constants and such that
for all . From (3.3) and (3.4), we obtain . Therefore, it follows from , that . Now we consider the case (ii). From for all and the third equation of (1.1), we have
Then the following expression
for all hold. Hence, by (3.2), there exist and such that
for . From (3.5) and (3.6), we have
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 , and such that
and for all , then 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 , and such that (4.1), (4.2) and hold for all . Then there exists so that for all .
Proof: From Lemma 2.2, we consider the following two cases:
(i) for all ;
(ii) for all .
Suppose for all , then we have for all . From the third equation of system (1.1), we have
So we obtain
for all . From the inequality (4.2), there exist positive constants and such that
for all . So the inequality (4.3) holds for all . Then , which contradicts with the boundedness of in Lemma 2.1. Now, we prove Theorem 4.1 by using Lemma 4.1.
Proof: For simplicity, let , , where is a constant. In fact, Lemma 2.1 implies that for any sufficiently small , there exists such that
for all . The inequality (4.1) implies that for any sufficiently small , there exists such that
for all . We define
Thus, by (4.5) and (4.6), for any sufficiently small and , there exist very small , such that
for all , where . Lemma 2.1 implies that for any sufficiently small , we have
for all . First, we prove
In fact, if it is not true, there exists such that
for all . If for all , then from (4.5) and (4.6), we have
for all . It follows from inequality (4.9) that . This contradicts with the boundedness of solution. Hence, there exists an such that . In the following we prove
for all . If it is not true, there exists an such that
Hence, there necessarily exists an such that and for all . Let be an integer such that . By (4.9), we obtain
This contradicts with . Hence, (4.11) is valid. By Lemma 4.1, there exists such that for all . Therefore, by (4.10) and (4.11), we have for all , then
By (4.7) we obtain . This contradicts with Lemma 2.1 ( is uniformly bounded). Hence, is true.
Next, we prove
where is a constant given in the following lines. By inequality (4.7), (4.8), (4.9) and Lemma 2.1, there exist , , such that and , we obtain
Let be a constant satisfying
where , .
Because we have proved , there are only two possibilities as follows:
(i) There exists , then as , we obtain ;
(ii) oscillates about for all large t.
In case (i), we have . In case (ii), there necessarily exist such that
Suppose that . Then
for all . Suppose that . Then
for all . Now we only prove for all . If for all . By the second equation of system (1.1) and inequality (4.13), we have
which is contradiction. Hence, there exists an such that . We obtain that for ,
By inequality (4.16), then for
Therefore, by the second equation of system (1.1) and inequalities (4.8), (4.17), (4.18), we obtain that
for all . By (4.14), we have
Now, we suppose there exists such that , then and for all . By Lemma 4.1, we assume that is so large that for all . Hence, by (4.8), we further have
for all . By (4.12) and (4.19), we have
Thus, by (4.17), we have
This contradicts with (4.15). Hence, for all , which implies . 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 , , , , , , , . Then, from system (1.1), we have . We easily verify that assumptions (H1) and (H2) hold.We choose and . Then we have
for all . From Theorem 3.1, we see that the infectious population of system (1.1) is extinct, see Figure 1.
Fix , , , , , , , . We choose and . Then we have
for all . From Theorem 4.1, we see that the infectious population of system (1.1) is permanent, see Figure 2.
In this paper we obtain new sufficient conditions for the permanence and extinction of system (1.1). We prove that our conditions give the threshold-type result by the basic reproduction number given as in (3.1) when every parameter is given as a constant parameter. Thus our result is an extension result of the threshold-type result in the autonomous system. Our results may contribute to predicting the disease dynamics, such as permanence and extinction of the infectious population, when the phenomena are modeled as a nonautonomous system.
Figure 1. The trajectories of deterministic system (1.1) with , , . (a) Time series diagram of susceptible population, (b) Time series diagram of exposed population, (c) Time series diagram of infectious population, (d) phase diagram of three populations (susceptible, exposed, infectious), respectively.
Figure 2. The trajectories of deterministic system (1.1) with , , . The meaning of (a) ~ (d) is similar to Figure 1.
In Section 5, we provide numerical examples to illustrate the validity of our results. In those examples we show that conditions in Theorems 4.1 for the permanence and extinction of infectious population of system (1.1) are not satisfied. One may argue that our conditions for the permanence and extinction may not sharp.
It is still an open problem that if the basic reproduction number for (1.1) works as a threshold parameter to determine the permanence and extinction of infectious population like in the autonomous system.
The research has been partially supported by the Natural Science Foundation of China (No. 11561004).
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