A Class of Nonautonomous Schistosomiasis Transmission Model with Incubation Period

A nonautonomous schistosomiasis model with latent period and saturated incidence is investigated. Further, we study the long-time behavior of the epidemic model. The weaker sufficient conditions for the permanence and extinction of infectious population of the model are obtained by constructing some auxiliary functions. Numerical simulations show agreement with the theoretical results.


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 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.

Preliminaries
In this section, system (1.1) satisfies the following assumptions: (H 1 ): The functions From Lemma 2.1, we have 0 p > and ( ) 0 ≥ .The proof is completed.

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 (H 1 ) and (H 2 ) hold.If there exist First, we consider the cases (i).From the second equation of system (1.1), we have for all 2 t T ≥ .From (3.1), we see that there exist constants 1 0 for all 3 t T > .From (3.3) and (3.4), we obtain ( ) Then the following expression

Permanence of Infectious Population
In this section, we obtain the sufficient conditions for the permanence of infectious population.
From the third equation of system (1.1), we have .
for all 2 t T ≥ .From the inequality (4.2), there exist positive constants 0 In fact, Lemma 2.1 implies that for any sufficiently small 0 >  , there ex- for all t T ≥ .The inequality (4.1) implies that for any sufficiently small 0 η > , there exists for all . Lemma 2.1 implies that for any sufficiently small 2 0 >  , we have t T ≥ , then from (4.5) and (4.6), we have ( ) for all 3 t T ≥ .It follows from inequality (4.9) that ( ) This contradicts with the boundedness of solution.Hence, there exists an E s <  .In the following we prove for all 1 t s ≥ .If it is not true, there exists an 2 1 Hence, there necessarily exists an ( ) , t s s ∈ .Let 0 n ≥ be an integer such that ( )  .By (4.9), we obtain ( ) This contradicts with ( )  where 1 0 I > is a constant given in the following lines.By inequality (4.7), (4.8), (4.9) and Lemma 2.1, there exist ( ) (ii) ( )  for all large t.
which is contradiction.Hence, there exists an

Conclusions
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.
Y. Liu et al.  works as a threshold parameter to determine the permanence and extinction of infectious population like in the autonomous system.