Dynamic of Non-Autonomous Vector Infectious Disease Model with Cross Infection ()
1. Introduction
In real life, we are often confused by infectious diseases. Infectious diseases include humans, animals, plant infectious diseases, especially human infectious diseases, such as tuberculosis, AIDS/HIV, malaria, which are the top three single disease killers of health in the world. According to the World Health Organization statistics, in 2002 about 70 million people are infected with AIDS, causing around 20 million deaths. In recent years, each year more than 560 million people infected with AIDS [1] [2]. The control of infectious diseases spread has aroused great interest of the people and many mathematical models are established (see [3] [4] [5] [6] ) to understand the mechanism of disease transmission, and to prevent or slow down the transmission of infectious diseases. In order to effectively control the spread of infectious diseases, we often introduce three control strategies in the model: cohort immunization, time-dependent pulse vaccination, and state-dependent vaccination. The first strategy details a continuous vaccination effort of susceptible individuals, while the second and third strategies involve vaccinating a significant fraction of the susceptible population in a short period of time [7].
In recent years, some mathematical models incorporating treatment have been established and investigated by many researchers [8] - [15]. Infectious diseases are the most important biosecurity issues, and every country should pay attention and strive to have a maximal capacity treatment for diseases. Therefore, it is vital to describe the limited capacity for treatment [16]. In [12], Wang and Ruan proposed the constant treatment function of diseases in an SIR epidemic model. According to this model when people get sick and must be hospitalized but there are limited beds in hospitals, or there is not enough medicine for treatments, should be considered and simulated the limited resources for the treatment of patients. Wang [13] researched the piecewise linear treatment function. The model is assumed that treatment rate is proportional to the number of infectives below the capacity and is a constant when the number of infectives is greater than the capacity. In [13], Wang adopted a constant treatment, which is suitable for the case of a large number of infectives. Zhang and Liu [17] introduced a continuously differentiable treatment function, which describes the saturation phenomenon of the limited medical resources. Zhang and Kang [15] proposed discontinuous treatment function in an SEIR epidemic model, which describes that the treatment rate has at most a finite number of jump discontinuities in every compact interval.
Some infectious diseases are transmitted by vector, such as Malaria, Dengue and West Nile virus, which spread by Mosquitoes. The maintenance and resurgence of vector-borne diseases are related to ecological changes that favor increased vector densities or vector-host interactions, among other factors [18]. However, travel and transport are a major factor in the spread of vector-borne diseases, we have reasons to believe that the spatial movement of humans may be important for the epidemiology of vector-borne diseases. Every year there are more than 1 billion cases and over 1 million deaths from vector-borne diseases such as malaria, dengue, schistosomiasis, human African try-panosomiasis, leishmaniasis, Chagas disease, yellow fever, Japanese encephalitis and onchocerciasis [19]. So the vector-borne is a very important part of the transmission of epidemic diseases.
The structure of this paper is organized as follows. Section 2 presents the vector-borne diseases model. And positivity and boundary of the model (1.1) are studied. In Section 3 and 4, we deal with the existence and permanence of model (1.1). In Section 5, we had a brief discussion.
2. Definitions and Preliminaries
Based on [20], we get the following vector infectious disease model:
(2.1)
with initial value
(2.2)
where the variables
,
,
and
represent susceptible host, infected host, susceptible vector and infection vector, respectively.
represents the input rate of susceptible hosts,
(
) means effective contact rate.
and
represents the natural mortality of the host and the vector, respectively.
represents the birth rate of the newborn vectors, A represents the effective bite rate of the vector.
Assumption 2.1
1) Functions
and
are positive, bounded and continuous on
.
2) There exist constants
such that
In what follows, we denote
and
the solution of
(2.3)
the solution of
(2.4)
with initial value
,
,
,
.
Proposition 2.2
1) There exist constants
and
, which are independent from the chioce of initial value
, such that
(2.5)
2) There exist constants
and
, which are independent from the chioce of initial value
, such that
(2.6)
3) The solution
of system (1.1) with initial value (2.2) exists, uniformly bounded and
for all
.
For
and
we define
and
(2.7)
where
and
are solutions of system (1.1). In Sections 3 and 4 we use the following lemma in order to investigate the longtime behavior of system (2.1).
Lemma 2.3 If there exist positive contants
and
such that
for all
, then there exists
such that either
for all
or
for all
.
Proof. Suppose that there does not exist
such that
for all
or
for all
hold. Then there necessarily exists
such that
and
. Hence we have
(2.8)
and
(2.9)
Substituting (2.8) into (2.9) we have
From 3) of Proposition 2.2, we have
, which is a contradiction.
3. Extinction of Infectious Population
In this section, we obtain conditions for the extinction of infectious population of system (2.1). The definition of the extinction is as follows:
Definition 3.1. We say that the infectious population of system (2.1) is extinct if
From system (2.1), it’s easy to prove that if one of the above equalities hold, then the other one is certainly hold. We give one of the main results of this paper.
Theorem 3.2. If there exist positive constants
and
such that
(3.1)
(3.2)
and
for all
, then the infectious population of system (2.1) is extinct.
Proof. From Lemma 2.3, we only have to consider the following two cases.
1)
for all
.
2)
for all
.
First we consider the case 1). From the second equation of system (2.1), we have
Hence, we obtain
(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 have
. Then it follows from
for all
that
.
Next we consider the case 2). From the fourth equation of system (2.1), we have
(3.5)
Hence we have
(3.6)
From (3.2) we see that there exist constants
and
such that
(3.7)
for all
. From (3.6) and (3.7), we have
. Then it follows from
for all
that
.
4. Permanence of Infectious Population
In this section, we get sufficient conditions for the permanence of infectious population of system (2.1). The definition of the permanence is as follows:
Definition 4.1. We say that the infectious population of system (2.1) is permanent if there exist positive constants
and
, which are independent from the choice of initial value satisfying (2.2), such that
We give one of the main results of this paper.
Theorem 4.2. If there exist positive constants
and
such that
(4.1)
(4.2)
and
for all
, then the infectious population of system (2.1) is permanent.
Before we give the Proof of Theorem 4.2, we introduce the following lemma.
Lemma 4.3. If there exist positive constants
and
such that (4.1), (4.2) and
hold for all
, then
for all
, where
is given as in lemma 2.3.
Proof. From Lemma 2.3 we have only two cases to discuss,
for all
or
for all
. Suppose that
for all
. Then
for all
. It follows from the last equation of system (2.1) that
for all
. Hence, we obtain
(4.3)
for all
. From the equality (4.2), we see that there exist constants
and
such that
(4.4)
for all
. For convenience, we choose
satisfying
. Then the inequality (4.3) holds for
, it follows from (4.4) that
. This contradicts with the boundedness of
, stated in 2) of Proposition 2.2. Thus we have
for all
.
Using Lemma 4.4 we prove Theorem 4.2.
Proof (Proof of Theorem 4.2). For simplicity, let
,
,
, and
, where
is a constant. From the inequality (2.7) and (2.8), we see that for any
, there exists
such that
(4.5)
(4.6)
for all
. The inequality (4.1) and (4.2) implies that for sufficient small
, there exists
such that
(4.7)
(4.8)
for all
. We define
From (4.6) and (4.8), we see that for positive constants
and
there exist small
such that
(4.9)
(4.10)
hold for all
. From 2) of Assumption 2.1,
can be chosen sufficiently small satisfying
(4.11)
hold for all
.
First we claim that
.
In fact, if it is not true, then there exists
such that
(4.12)
for all
. Suppose that
for all
. Then, from (4.5) and (4.12) we have
for all
. Thus, from (4.11), we have
, which contradicts with 2) of Proposition 2.2. Therefore we see that there exists
such that
. Suppose that there exists an
such that
. Then, we see that there necessarily exists an
such that
and
for all
. Let n be an integer such that
. Then from (4.11), we have
which is a contradiction. Therefore, we see that
(4.13)
for all
. Now, from lemma 4.4, there exists
such that
for all
. Then
for all
. Hence, we have
It follows from (4.9) that
and this contradicts with the boundedness of
, stated in 2) of Proposition 2.2. Thus, we see that our claim
is true.
Next, we prove
where
is a constant given in the following lines. For the following convenience, we let
be the least common multiple of
and
. If we define
Then we have two cases to discuss, namely 1)
and 2)
. Firstly, we discuss the case 1). We set
such that
, then there exist
such that
for all
. Then, from inequalities (4.9), (4.11)-(4.12) and 2) of Assumption 2.1, we see that there exist constants
, which is an integral multiple of
, and
such that
(4.14)
(4.15)
(4.16)
for all
and
. Let
be an integer multiple of
satisfying
(4.17)
where
. Since we have proved
. There are only two possibilities as follows:
1)
for all
.
2)
oscillates about
for large
. In case 1), we have
. In case 2), there necessarily exist two constants
such that
Suppose that
. Then, from (1.1) we have
(4.18)
Hence, we obtain
(4.19)
for all
. Suppose that
. Then, from (4.18), we have
for all
. Now, we are in a position to show that
for all
. Suppose that
for all
. Then, from (4.14), we have
which is a contradiction. Therefore, there exists an
such that
. Then, as is in the proof of
, we can show that
(4.20)
for all
. From (4.18), we have
(4.21)
for all
. Thus, from (4.10), (4.20), (4.21), we have
for all
. Hence, from (4.16), we obtain
(4.22)
Now we suppose that there exists a
such that
,
and
for all
. Then there exists
such that
. Note that from Lemma 4.4. without loss of generality, we can assume that
is so large that
for all
. Then, from (4.20), we have
for all
. Thus, from (4.15) and (4.22), we have
Thus, from (4.20), we have
which contradicts with (4.17). Finally, if
, we let
be the integral multiple of
satisfying
(4.23)
Then, repeating the above steps, we have
Thus, from (4.20), we have
which is contradictive with (4.23). Therefore,
for all
, which implies
.
Since
, the infectious population of system (1.1) is permanent.
5. Discussion
In the paper, we have extended the epidemic models of vector-borne disease with direct mode of transmission presented in [20]. A non-autonomous vector infectious disease model that conforms to the actual environment has been established, which combines the spread of epidemics with changes in the natural environment and fully reflects the characteristics of the spread of epidemics that change over time. There are relatively few popular articles on the establishment of non-autonomous mathematical models, so the non-autonomous vector infectious disease models are even rarer. Therefore, our research has a certain theoretical value and application value.
Acknowledgements
The work was supported by the Science Fund of Education Department of Jiangxi Province (171373, 171374, 181361).