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 Open Access
Random periodic solution for a stochastic SIS epidemic model with constant population size
 Dianli Zhao^{1}Email author,
 Sanling Yuan^{1} and
 Haidong Liu^{2}
https://doi.org/10.1186/s1366201815114
© The Author(s) 2018
 Received: 26 September 2017
 Accepted: 1 February 2018
 Published: 20 February 2018
Abstract
In this paper, a stochastic susceptibleinfectedsusceptible (SIS) epidemic model with periodic coefficients is formulated. Under the assumption that the total population is fixed by N, an analogue of the threshold \({R_{0}^{T}}\) is identified. If \({R_{0}^{T} > 1}\), the model is proved to admit at least one random periodic solution which is nontrivial and located in \((0,N)\times(0,N)\). Further, the conditions for persistence and extinction of the disease are also established, where a threshold is given in the case that the noise is small. Comparing with the threshold of the autonomous SIS model, it is generalized to its averaged value in one period. The random periodic solution is illuminated by computer simulations.
Keywords
 Stochastic SIS epidemic model
 Random periodic solution
 Constant population size
 Persistence
 Extinction
MSC
 47D07
 60J60
 60H10
 92D30
1 Introduction
However, in the real word, many infectious diseases of humans, such as measles, mumps, rubella, chickenpox, diphtheria, pertussis, and influenza, fluctuate over time with seasonal variation [15]. This implies that the corresponding mathematical models may have the periodic solutions. Therefore, it is important to investigate the periodic dynamics of epidemic models. For more about the periodic properties of the epidemic model, one can see [16–19] and the references cited therein. At the same time, one can easily find that some diseases mentioned above always do not have significant effects on the total population size. Consequently, in this paper, the total population is assumed to be a positive constant, denoted by N.

What is the condition for the existence of a random periodic and positive solution of this model?

Under what conditions will the microorganism survive or will be washed out?

Is there a threshold which more or less helps to determine the survival of the microorganism?
For simplicity, we denote \({x^{*}} = \mathop{\sup} _{t \ge0} \{ {x ( t )} \}\) and \({x_{*}} = \mathop{\inf} _{t \ge 0} \{ {x ( t )} \}\) for a function \(x(t)\) defined on \([0,\infty)\). \(R_{0}^{T} = \frac{{\frac{1}{T}\int_{0}^{T} { [ {\beta ( t )N  \frac{{{\sigma^{2}} ( t ){N^{2}}}}{2}} ]\,dt} }}{{\frac{1}{T}\int_{0}^{T} { [ {\mu ( t ) + \delta ( t )} ]\,dt} }}\). One can easily check that \(\Gamma = \{ { ( {S,x} ) \in R_{+} ^{2}:S + x = N} \}\) is the positive invariant set of model (4), which is a crucial property for the proof of a periodic solution. In view of the biological meanings, we assume that the coefficients of model (4) are continuous, positive, bounded, and Tperiodic functions on \([0,\infty)\). Then \({\mu_{*}} > 0\).
The existence of the uniquely positive solution can be proved by following the standard procedure in [12], so we omit it. In the following, we mainly focus on finding the suitable condition for the existence of a random periodic solution, persistence, and extinction of (4). Main contributions of this paper are as follows.
Theorem 1.1
If \({R_{0}^{T} > 1}\) holds, then model (4) has at least one random positive Tperiodic solution in Γ.
The proof is located in Section 2. Here, we mainly illuminate Theorem 1.1 with an example.
Example 1
Remark 1
The following theorems concern the persistence in mean and extinction of model (4).
Theorem 1.2
Theorem 1.3
 (A)
\(\mathop{\sup} _{t \ge0} ( {{\sigma^{2}} ( t )N  \mu ( t )} ) \le0\) and \(R_{0}^{T} < 1\),
 (B)
\(\frac{1}{T}\int_{0}^{T} {{ [ {\frac{{{{ [ {N\beta ( s )  \theta ( {\mu ( s ) + \delta ( s )} )} ]}^{2}}}}{{2{N^{2}}{\sigma^{2}} ( s )}}  ( {1  \theta} ) ( {\mu ( s ) + \delta ( s )} )} ]}\,ds< 0}\) holds for any constant \(\theta\in[0,1)\),
Remark 2
From Theorems 1.2 and 1.3, under the assumption: \(\mathop{\sup} _{t \ge0} ( {{\sigma^{2}} ( t )N  \mu ( t )} ) \le0\), the disease is persistent if \({R_{0}^{T} > 1}\), while it goes extinct if \({R_{0}^{T} < 1}\). So we consider \({R_{0}^{T} }\) as the threshold of the stochastic model (4).
2 Proofs
First, we introduce some results concerning the periodic Markov process.
Definition 2.1
(see [20])
A stochastic process \(\xi(t) = \xi(t, \omega ), t\in R\), is said to be periodic with period T if, for every finite sequence of numbers \(t_{1}, t_{2}, \ldots, t_{n}\), the joint distribution of random variables \(\xi(t_{1} + h), \ldots, \xi(t_{n} + h)\) is independent of h, where \(h = kT\ (k = \pm1,\pm2, \ldots)\).
Lemma 2.2
(see [20])
Proof of Theorem 1.1
Remark 4
The proofs of Theorem 1.2 and (A) in Theorem 1.3 are similar to those in [12], and hence are omitted.
Proof of (B) in Theorem 1.3
3 Concluding remarks

When \({R_{0}^{T}}<1\), the disease will go extinct with probability 1 under extra mild conditions.

When \({R_{0}^{T}}>1\), the disease will be persistent in mean.
Let \(\sigma\equiv0\), we have \(R^{T} = \frac{{\frac{1}{T}\int_{0}^{T} { [ {\beta ( t )N } ]\,dt} }}{{\frac{1}{T}\int _{0}^{T} { [ {\mu ( t ) + \delta ( t )} ]\,dt} }}\), which is the threshold of the corresponding deterministic SIS model. Clearly, \({R_{0}^{T}} < R^{T}\), this means that the disease may go extinct due to the noises, while the deterministic SIS model predicts its survival. (B) in Theorem 1.3 shows that large noise can lead the disease to die out. In general, the noise has negative effects on persistence of the disease.
Comparing with the autonomous SIS model [12, 13], the threshold \({R_{0}^{T}}\) (see [13]) is replaced by its averaged value in one period, and thus the result is generalized.

When \({R_{0}^{T}}<1\), the disease modeled by (4) will almost surely go extinct without any extra condition.
Declarations
Acknowledgements
This work was supported by NSFC (No. 11671260, 11501148) and Shanghai Leading Academic Discipline Project (No. XTKX2015).
Authors’ contributions
All authors contributed equally to the writing of this paper. The authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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