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ABSTRACT
In this paper, we consider a stochastic SIRS epidemic model with seasonal variation and saturated incidence. Firstly, we obtain the threshold of stochastic system which determines whether the epidemic occurs or not. Secondly, we prove that there is a non-trivial positive periodic solution if .
1. Introduction
Understanding the periodic behaviour of epidemic dynamical system is of paramount importance in applications. This is because many childhood diseases such as measles, chickenpox, and mumps, have been found to be endemic and to exhibit regular oscillatory levels of incidence in large populations [Citation1,Citation3,Citation4,Citation10].
In order to predict sustained oscillatory, many authors take the effect of seasonal variation and stochasticity into account, see, for example [Citation2,Citation6]. However, in Ref. [Citation2,Citation6] the authors prove the the existence of positive periodic solution for a stochastic epidemic model only by numerical methods, but not by theoretical methods. Recently there are several literatures, in which the existence of stochastic periodic solution can be showed analytically (e.g. [Citation8,Citation9,Citation12]). In Ref. [Citation12] the authors considered a periodic stochastic SIR epidemic model with pulse vaccination. Under some conditions a unique positive periodic disease-free solution was obtained. But the existence of non-trivial positive periodic solution could not be obtained by Wang et al. [Citation12]. Lin et al. [Citation8] later filled this gap. They provided the sufficient conditions for the existence of non-trivial periodic solution. Further, Liu et al. [Citation9] considered positive periodic solution for a stochastic non-autonomous SIR epidemic model with logistic growth.
In this paper we continue to do some work in this direction. We consider a stochastic SIRS epidemic model with seasonal variation and saturated incidence, which takes the form of
(1)
(1) where
,
and
denote the number of susceptible, infected and removed individuals at time t respectively. Parameter functions Λ, β, ϵ, γ, μ, δ,
are positive, non-constant and continuous functions of period ω on
;
are independent standard Brownian motions, defined on a complete probability space
with a filtration
satisfying the usual conditions (see [Citation11]).
In fact, model (Equation1(1)
(1) ) with constant coefficients and
has been considered by Yang et al. [Citation14]. They presented the sufficient conditions for the ergodicity and extinction of (Equation1
(1)
(1) ).
This paper is organized as follows. In Section 2, we present some auxiliary results concerning the existence of a periodic Markov process. In Section 3, we obtain the threshold for the epidemic to occur. The existence of non-trivial positive periodic solution is obtained in Section 4.
2. Preliminary
To begin with, we introduce some notations. If is an integrable function on
, define
If
is a bounded function on
, define
Next we present some auxiliary results concerning the existence of a periodic Markov process which will be used in the proof of our main result. The author may also refer to [Citation7] for details.
Definition 2.1
A stochastic process is said to be periodic with period θ if for every finite sequence of numbers
the joint distribution of random variables
is independent of h, where
.
Remark 2.1
It is showed in [Citation7] that a Markov process is θ-periodic if and only if its transition probability function is θ-periodic and the function
satisfies the equation
.
Consider the following equation
(2)
(2)
Lemma 1
Suppose that the coefficient of (Equation2(2)
(2) ) are θ-periodic in t and satisfy condition:
in every cylinder
where B is a constant; and suppose further that there exists a function
in
which is θ-periodic in t, and satisfies the following conditions
(3)
(3) and
(4)
(4) where the operator
is given by
Then there exists a solution of (Equation2
(2)
(2) ) which is a θ-periodic Markov process.
Remark 2.2
According to the proof of Lemma 1, linear growth condition is only used to guarantee the existence and uniqueness of the solution of (Equation2(2)
(2) ).
3. Extinction and persistence of the disease
According to the similar arguments in [Citation5], we know that system (Equation1(1)
(1) ) has a unique global positive solution for any
. In the following result we determine the threshold for the disease to occur.
Theorem 3.1
Assume . When
and
the disease I will persist in the sense that
where
and
are the unique positive ω-periodic solution of the equation
and
respectively.
Remark 3.1
In fact, we have that
From the equations which
and
satisfy, it follows that
. This means
. Hence,
on
.
Proof.
By using the similar arguments as in Zhao and Jiang [Citation15], we can get that if
then
(5)
(5) Applying Itô formula, we have
This together with Equation (Equation5
(5)
(5) ) implies
(6)
(6) By Itô formula, we have
That is,
(7)
(7) where
Obviously, Equation (Equation6
(6)
(6) ) implies that
Let
and
In view of Equation (Equation7
(7)
(7) ), we have
(8)
(8) and
(9)
(9) where
.
From Equations (Equation1(1)
(1) ) and (Equation5
(5)
(5) ), the following holds:
which together with
implies that
According to Lemma 17 in [Citation13] and Lemma A.2 in [Citation15], it follows from Equations (Equation8
(8)
(8) ) and (Equation9
(9)
(9) ) that
Theorem 3.2
Assume . The disease will become extinct exponentially almost surely when
and
.
Proof.
It follows from Equation (Equation7(7)
(7) ) that
Noting
a.s., we get
which completes the proof.
4. Existence of ω-periodic solution
Theorem 4.1
Assume . If
then there exists a ω-periodic solution for system (Equation1
(1)
(1) ).
Proof.
Since for any system (Equation1
(1)
(1) ) has a unique global positive solution, we take
as the whole space. It is obvious that the coefficients of system (Equation1
(1)
(1) ) satisfy the local Lipschitz condition. According to Lemma 1 and Remark 2.2, in order to prove Theorem 4.1 it suffices to find a
-function
and a closed set
such that Equations (Equation3
(3)
(3) ) and (Equation4
(4)
(4) ) hold.
Take and r>0 such that
(10)
(10) where the function
and
are given in Equations (Equation11
(11)
(11) ) and (Equation12
(12)
(12) ), respectively. Define the auxiliary function
by
where
and
are given in Theorem 3.1,
is a function defined on
satisfying
and
. It is clear that
is a ω-periodic function on
. Hence
is ω-periodic in t and satisfies Equation (Equation3
(3)
(3) ).
Next we will find a closed set such that
Denote
Then
In the following, for simplicity we always use Λ to denote the function , the other parameter functions are the same. Direct calculation implies that
and
Hence,
where
(11)
(11)
(12)
(12) In view of Equation (Equation10
(10)
(10) ), we can obtain
Take small
such that
(13)
(13) where
.
For , in view of Equation (Equation10
(10)
(10) ) we have
and
Taking
small enough, it follows that
(14)
(14) where
.
For , we obtain
Choosing
small enough, it is obvious that
(15)
(15) where
is defined by
Noting , we obtain
. Combining Equations (Equation13
(13)
(13) ) – (Equation15
(15)
(15) ), it follows
The proof is complete.
5. Conclusion
In this paper, we consider a stochastic SIRS epidemic model with seasonal variation. Denote According to Theorems 3.1 and 3.2, imposing some restrictions on the intensities of white noises, the disease dies out if
; whereas if
the disease will persist. According to Theorem 4.1 we show that model (Equation1
(1)
(1) ) has at least one ω-periodic solution when
.
Disclosure statement
No potential conflict of interest was reported by the authors.
ORCID
Yuguo Lin http://orcid.org/0000-0003-2754-6698
Additional information
Funding
References
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