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- Open Access
Asymptotic behavior of a multigroup SIS epidemic model with stochastic perturbation
- Jing Fu^{1},
- Qixing Han^{1, 2}Email author,
- Yuguo Lin^{2, 3} and
- Daqing Jiang^{2, 4}
https://doi.org/10.1186/s13662-015-0406-x
© Fu et al.; licensee Springer. 2015
- Received: 7 November 2014
- Accepted: 6 February 2015
- Published: 8 March 2015
Abstract
In this paper, we introduce stochasticity into a multigroup SIS model. We present the sufficient condition for the exponential extinction of the disease and prove that the noises significantly raise the threshold of a deterministic system. In the case of persistence, we prove that there exists an invariant distribution which is ergodic.
Keywords
- stochastic SIS model
- extinction
- stationary distribution
- ergodicity
MSC
- 60H10
- 93E15
- 34E10
1 Introduction
Mathematical models have become important tools in analyzing the spread and control of infectious diseases. In many epidemiological models it is assumed that the population being considered is uniform and homogeneously mixing. However, when dealing with a heterogeneous population, it is appropriate to divide the whole population into subpopulations, each of which is homogeneous. So far a lot of research has been done on various forms of multigroup models, see, e.g., [1–11].
Recently, Gray et al. [14] discussed a stochastic SIS model (1.2) with one group. They proved that this SIS model has a unique global positive solution and establish conditions for the extinction and persistence of the disease. In the case of persistence they showed the existence of a stationary distribution and derived expressions for its mean and variance.
The aim of this paper is to study the asymptotical behavior of the solutions of system (1.2). The rest of this paper is organized as follows. In Section 2, we show that system (1.2) has a unique nonnegative solution. In Section 3, we present the sufficient condition for the exponential extinction of the disease. Section 4 focuses on the persistence of the disease. We show there is a stationary distribution for system (1.2) and it is ergodic.
Throughout this paper, let $(\mathrm{\Omega},\mathcal{F},{\{{\mathcal{F}}_{t}\}}_{t\ge 0},P)$ be a complete probability space with a filtration ${\{{\mathcal{F}}_{t}\}}_{t\ge 0}$ satisfying the usual conditions (i.e. it is right continuous and ${\mathcal{F}}_{0}$ contains all P-null sets).
2 Existence and uniqueness of positive solution
Theorem 2.1
If \((\beta_{kj})_{n\times n}\) is irreducible, then for any initial value \(Y(0)\in\Gamma\), there is a unique solution \(Y(t)\) of system (1.2) on \(t\geq0\), and the solution will remain in Γ with probability 1.
Proof
Remark 2.1
An \(n\times n\) matrix \((a_{ij})\) is irreducible if for any nonempty subset S of \(\{1,\ldots,n\}\) with a nonempty complement \(S'\), there exist i in S and j in \(S'\) such that \(a_{ij}\neq0\).
Remark 2.2
From the proof of Theorem 2.1, we obtain \(LV\leq M\). Let \(\tilde {V}=V+M\). Then \(L\tilde{V}\leq\tilde{V}\) and it is clear that \(\inf_{Y\in\Gamma\setminus D_{k}}\tilde{V}(Y)\to\infty\) as \(k\to\infty \), where \(D_{k}=(\frac{1}{k},N_{1}-\frac{1}{k})\times(\frac {1}{k},N_{2}-\frac{1}{k})\times\cdots\times(\frac{1}{k},N_{n}-\frac {1}{k})\). Hence, by Remark 2 of Theorem 4.1 of Hasminskii [15], p.86, we find that the solution \(Y(t)\) is a homogeneous Markov process in Γ.
3 Exponential extinction of infectious disease
It is clear that \(P_{0}= (0,0,\ldots,0 )\) is the disease-free equilibrium of system (1.2). For system (1.1), \(P_{0}\) is globally stable if \({\mathcal{R}}_{0}\leq1\). Hence, it is interesting to study the disease-free equilibrium for controlling the infectious disease. In this section, we present sufficient conditions for the disease to extinct exponentially for system (1.2).
Theorem 3.1
Proof
Remark 3.1
4 Ergodicity of system (1.2)
For a deterministic system, we always discuss the global attractivity of the positive equilibrium of the system. However, there is no positive equilibrium for system (1.2). In this section, we show there is a stationary distribution for system (1.2) when the white noise is small, which in turn implies the stability in stochastic sense. To begin with, we present a well-known result, due to Hasminskii [15].
Assumption B
- (B.1)
In the domain U and some neighborhood thereof, the smallest eigenvalue of the diffusion matrix \(A(x)\) is bounded away from zero.
- (B.2)
If \(x\in E_{l}\setminus U\), the mean time τ at which a path issuing from x reaches the set U is finite, and \(\sup_{x\in K}E_{x}\tau<\infty\) for every compact subset \(K\subset E_{l}\).
Lemma 4.1
Theorem 4.1
Proof
5 Conclusions
In this paper, we investigate a multigroup SIS model with the effect of environmental white noise. We obtain the sufficient condition for the extinction of the disease, and we obtain the criteria for the existence of the invariant distribution of system (1.2). Some interesting topics deserve further investigation. For instance, the coefficients of stochastic differential equations can be modeled by fuzzy sets, and this leads to stochastic differential equations with fuzziness [20, 21]. We leave it for future investigation.
Declarations
Acknowledgements
The work was supported by NSFC of China (No. 11371085), the PhD Programs Foundation of Ministry of China (No. 200918) and Nature Science Foundation of Changchun Normal University.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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