- Research
- Open Access
Dynamics analysis of stochastic epidemic models with standard incidence
- Wencai Zhao1, 2Email authorView ORCID ID profile,
- Jinlei Liu1,
- Mengnan Chi1 and
- Feifei Bian1
https://doi.org/10.1186/s13662-019-1972-0
© The Author(s) 2019
- Received: 9 July 2018
- Accepted: 15 January 2019
- Published: 23 January 2019
Abstract
In this paper, two stochastic SIRS epidemic models with standard incidence were proposed and investigated. For the non-autonomous periodic model, the sufficient criteria for extinction of the disease are obtained firstly. Then we show that the stochastic system has at least one nontrivial positive T-periodic solution under some conditions. For the model that are both disturbed by the white noise and telephone noise, we construct a suitable Lyapunov functions to verify the existence of a unique ergodic stationary distribution. Meanwhile, the sufficient condition for the extinction of the disease is also established. Finally, examples are introduced to illustrate the theoretical analysis.
Keywords
- Stochastic SIRS epidemic model
- Standard incidence
- Extinction
- Periodic solution
- Stationary distribution and ergodicity
MSC
- 60H10
- 65C30
- 91B70
1 Introduction
This paper is arranged as follows. In Sect. 2, we give some basic knowledge which are used in this paper. In Sect. 3, for the system (2), the criteria for extinction of the disease are obtained and then we show that the system has at least one nontrivial positive T-periodic solution under some conditions. In Sect. 4, for the system (3), we proposed a sufficient condition of the disease extinction. Meanwhile, the existence of a unique ergodic stationary distribution is proved. Finally, we conclude the main result briefly and make some numerical simulations in Sect. 5.
2 Preliminaries
Lemma 2.1
([37])
Lemma 2.2
The proof is similar to Lemma 3.1 in Liu et al. [28], here we omit it.
Lemma 2.3
([37])
- (i)
\(q_{ij}>0\) for each \(i\neq j\);
- (ii)for any \(k\in\mathbb{S}\), \(D(x,k)=(d_{ij}(x,k))_{n\times n}\) is symmetric and obeywith some constant \(\kappa_{0}\in(0,1]\) for any \(x\in\mathbb{R}^{n}\);$$ \kappa_{0}|\xi|^{2}\leq\bigl\langle D(x,k)\xi,\xi\bigr\rangle \leq\kappa_{0}^{-1}|\xi |^{2} \quad \textit{for all }\xi\in\mathbb{R}^{n}, $$
- (iii)there exists a nonempty open set \(\mathcal{D}\) with compact closure, and for any \(k\in\mathbb{S}\), there is a nonnegative function \(V(\cdot,k):\mathcal{D}^{C}\rightarrow\mathbb{R}\) such that$$\mathcal{L}V(x,k)\leq-1\quad \textit{for any }(x,k)\in\mathcal{D}^{C} \times\mathbb{S}. $$
3 Extinction of the disease and the periodic solution for system (2)
For the non-autonomous stochastic system (2), we investigate the extinction criteria of the disease, firstly. Define \(R_{1}(t)=\beta(t)-(\gamma(t)+d(t)+\alpha(t)+\frac{\sigma_{2}^{2}(t)}{2})\) and \(\langle f\rangle_{T}=\frac{1}{T}\int_{0}^{T}f(s)\,ds\), where f is an integral function on \([0,+\infty)\). Then we have the following conclusion.
Theorem 3.1
The disease \(I(t)\) will go to extinction exponentially almost surely when \(\langle R_{1}(t)\rangle_{T}<0\).
Proof
Next, we consider the existence of nontrivial positive T-periodic solution of system (2). To simplify, we denote \(g^{u}=\sup_{t\in[0,+\infty)}g(t)\), \(g^{l}=\inf_{t\in[0,+\infty)}g(t)\), where g is a bounded function on \([0,+\infty)\).
Theorem 3.2
If \(\langle R_{2}(t)\rangle_{T}>0\), then system (2) admits at least one positive T-periodic solution.
Proof
To prove Theorem 3.2, we should construct a \(C^{2}\)-function \(V(t,x)\) which is T-periodic in t and a closed set \(U\in\mathbb {R}_{+}^{3}\) satisfy the conditions in Lemma 2.1.
4 Extinction of the disease and the ergodic stationary distribution for system (3)
For the system with regime switching, we will explore the extinction of the disease and the existence of an ergodic stationary distribution. Let \((S(t),I(t),R(t),r(t))\) be the solution of system (3) with initial value \((S(0),I(0),R(0),r(0))\in\mathbb {R}_{+}^{3}\times\mathbb{S}\).
Theorem 4.1
If \(R_{1}^{*}<1\), then \(\lim_{t\rightarrow\infty}I(t)=0\) a.s.
Proof
Next, we shall establish sufficient conditions for the existence of an ergodic stationary distribution of system (3).
Theorem 4.2
If \(R_{2}^{*}>1\), then system (3) has a unique ergodic stationary distribution.
Proof
5 Conclusions and numerical simulations
In this paper, we proposed a stochastic non-autonomous SIRS epidemic model with periodic coefficients (model (2)), and a stochastic epidemic model perturbed by telegraph noise (model (3)). Then the dynamic behaviors of the two models are studied.
- (1)
If \(\langle R_{1}(t)\rangle_{T}<0\), then the disease will go to extinction almost surely.
- (2)
If \(\langle R_{2}(t)\rangle_{T}>0\), then system (2) has at least one positive T-periodic solution.
- (1)
If \(R_{1}^{*}<1\), the disease \(I(t)\) will go to extinction exponentially with probability 1.
- (2)
If \(R_{2}^{*}>1\), then the solution of system (3) has a unique ergodic stationary distribution.
To verify the correctness of the theoretical analysis, we will give some examples with computer simulations.
Example 1
Sample paths of \((S(t),I(t),R(t))\) with initial conditions \((S(0),I(0),R(0))=(3,2.5,2)\)
Sample paths of \((S(t),I(t),R(t))\) with initial conditions \((S(0),I(0),R(0))=(0.3,0.3,0.3)\)
Example 2
Sample paths of \((S(t),I(t),R(t))\) with initial conditions \((S(0),I(0),R(0))=(1.5,2.5,2.5)\)
Computer simulation of a single path of Markov chain \(r(t)\) and its corresponding solutions \((S(t),I(t),R(t))\) for the system (3). The last three pictures are the histograms of the path
Declarations
Acknowledgements
The authors thank the editors and the anonymous referees for their careful reading and valuable comments.
Availability of data and materials
Data sharing not applicable to this article as all data sets are hypothetical during the current study.
Funding
This work is supported by SDUST Research Fund (2014TDJH102).
Authors’ contributions
All authors worked together to produce the results and 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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