A Class of Nonautonomous Schistosomiasis Transmission Model with Incubation Period ()
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:
(1.1)
with initial value
(1.2)
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.
2. Preliminaries
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
(2.1)
with initial value (1.2), and denote by
the solution of
(2.2)
with initial value (1.2). Then
By [22] , we have the following result:
Lemma 2.1. Suppose that assumptions (H1) and (H2) hold. Then:
(i) there exist positive constants
and
, such that
(2.3)
(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
and
(2.4)
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
(2.5)
and
(2.6)
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
(3.1)
(3.2)
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
(3.3)
for all
. From (3.1), we see that there exist constants
and
such that
(3.4)
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
(3.5)
for all
hold. Hence, by (3.2), there exist
and
such that
(3.6)
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
(4.1)
(4.2)
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
(4.3)
for all
. From the inequality (4.2), there exist positive constants
and
such that
(4.4)
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
(4.5)
for all
. The inequality (4.1) implies that for any sufficiently small
, there exists
such that
(4.6)
for all
. We define
Thus, by (4.5) and (4.6), for any sufficiently small
and
, there exist very small
,
such that
(4.7)
(4.8)
for all
, where
. Lemma 2.1 implies that for any sufficiently small
, we have
(4.9)
for all
. First, we prove
In fact, if it is not true, there exists
such that
(4.10)
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
(4.11)
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
We obtain
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
(4.12)
(4.13)
(4.14)
Let
be a constant satisfying
(4.15)
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
(4.16)
which implies
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
,
(4.17)
By inequality (4.16), then for
(4.18)
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
(4.19)
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
and
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
and
for all
. From Theorem 4.1, we see that the infectious population of system (1.1) is permanent, see Figure 2.
6. 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.
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.
Acknowledgements
The research has been partially supported by the Natural Science Foundation of China (No. 11561004).