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The persistence and extinction of a stochastic SIS epidemic model with Logistic growth
- Jiamin Liu^{1},
- Lijuan Chen^{1} and
- Fengying Wei^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1528-8
© The Author(s) 2018
- Received: 8 August 2016
- Accepted: 13 November 2017
- Published: 23 February 2018
Abstract
The dynamical properties of a stochastic susceptible-infected epidemic model with Logistic growth are investigated in this paper. We show that the stochastic model admits a nonnegative solution by using the Lyapunov function method. We then obtain that the infected individuals are persistent under some simple conditions. As a consequence, a simple sufficient condition that guarantees the extinction of the infected individuals is presented with a couple of illustrative examples.
Keywords
- Extinction
- Persistence
- Stochastic SIS model
- Logistic growth
MSC
- 60H10
- 37H10
1 Introduction
Throughout this paper, we will work on the complete probability space \((\Omega, \{\mathcal{F}_{t}\}_{t\geq 0},P)\) with its filtration \(\{\mathcal{F}_{t}\}_{t\geq 0}\) satisfying the usual conditions (i.e., it is right continuous and \(\mathcal{F}_{0}\) contains all P-null sets). We will investigate the dynamical properties of stochastic SIS model from several aspects: the result that stochastic model (9) admits a unique positive solution will be studied in the next section. The sufficient conditions of the persistence for the infected individuals would be derived. Further, we still find a simple condition to reach the extinction for the infected individuals. As a consequence, several illustrative examples are carried out to support the main results of this paper.
2 Existence and uniqueness of positive solution
In this section, we first show that the solution of system (9) is positive and global. Our proof is motivated by the work of Mao et al. [13].
Theorem 1
There exists a unique solution \((S(t),I(t))\) of system (9) on \(t\geq 0\) for any initial value \((S(0),I(0))\in \mathbb{R}^{2}_{+}\), and the solution will remain in \(\mathbb{R}^{2}_{+}\) with probability 1, namely \((S(t),I(t))\in \mathbb{R}^{2}_{+} \) for all \(t\geq 0\) almost surely.
Proof
3 Persistence in the mean
Lemma 1
([13], Strong law of large numbers)
Theorem 2
Proof
Example 1
4 Extinction
In the previous section, we have investigated the persistence of the solution to model (9). In this section, we shall prove that the density of the infected individuals will be driven to extinction with a negative exponential power under some simple assumptions.
Theorem 3
Proof
Example 2
5 Conclusion
The dynamical properties of the stochastic SIS model with Logistic growth are paid more attention to in this paper. According to the approach shown in many recent literature works, we still construct a \(C^{2}\)-function to show that the stochastic SIS epidemic model admits a unique positive global solution. Based on the general assumption of this paper, the total population is separated into two compartments: one is the susceptible, another is the infected. We also assume that the transmission rate β is perturbed by a white noise. The two indicators \(\tilde{R}_{0}\) and \(\breve{R}_{0}\) are kind of thresholds of this paper: when \(\tilde{R}_{0}> 1\), under some extra conditions, the density of the infected individuals keeps persistent; when \(\breve{R}_{0}<1\) holds or (36) is valid, the density of the infected individuals declines to zero in a long run. Several illustrative examples support the main results of this paper.
Declarations
Acknowledgements
The authors would like to thank Zhanshuai Miao for his discussion and the anonymous referees for their good comments. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11201075, 11601085), Natural Science Foundation of Fujian Province of China (Grant No. 2016J01015, 2017J01400).
Authors' contributions
The main idea of this paper was proposed by JL. JL prepared the manuscript initially, LC and FW performed all the steps of the proofs in this research. All authors read and approved the final manuscript.
Competing interests
We claim that none of the authors have any competing interests in the manuscript.
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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