A stochastic SIR epidemic model with density dependent birth rate
- Ling Zhu^{1, 2} and
- Hongxiao Hu^{3}Email author
https://doi.org/10.1186/s13662-015-0669-2
© Zhu and Hu 2015
Received: 9 July 2015
Accepted: 30 September 2015
Published: 23 October 2015
Abstract
In this paper, we introduce stochasticity into a model of SIR with density dependent birth rate. We show that the model possesses non-negative solutions as desired in any population dynamics. We also carry out the globally asymptotical stability of the equilibrium through the stochastic Lyapunov functional method if \(R_{0}\le1\). Furthermore, when \(R_{0}>1\), we give the asymptotic behavior of the stochastic system around the endemic equilibrium of the deterministic model and show that the solution will oscillate around the endemic equilibrium. We consider that the disease will prevail when the white noise is small and the death rate due to disease is limited.
Keywords
1 Introduction
However, population systems are often subject to environmental noise (see [12, 13]), which is ignored by the deterministic models. Hence stochastic model has come to play an important role in infectious dynamics. Nowadays, the investigations of the epidemic models perturbed by the white noise have been engaging in a lot of attentions of many authors, and Beddington and May [14] assumed the environmental noise of the systems is proportional to the variables. Carletti [15] investigated stochastic perturbation around the positive equilibrium. Mao et al. [16–19] assumed the parameters in models are suffering stochastic perturbation.
Stochastic SIR models have been investigated in recent work. Tornatore et al. [17] proposed a stochastic SIR model with or without distributed time delay, they gave a sufficient condition for the asymptotic stability of the disease-free equilibrium. They only showed that the introduction of noise modifies the threshold of system for an epidemic to occur by numerical simulations. Lin and Jiang [18] considered a stochastic SIR model with perturbed disease transmission coefficient. They presented sufficient conditions for the disease to get extinct exponentially. In the case of persistence, they analyzed the long-time behavior of densities of the distributions of the solution and proved that the densities of the solution can converge in L^{1} to an invariant density under appropriate conditions. Also they found the support of the invariant density. Specially, when the intensity of white noise is relatively small, they gave a new threshold for an epidemic to occur. Ji et al. [19] discussed a two-group SIR model with the transmission parameter subject to white noise, while Yu et al. [20] investigated a two-group SIR model with stochastic perturbation around the positive equilibrium. But until now, few scholars have considered a stochastic SIR model with logistic growth.
The remaining parts of this paper are as follows. In the next section we show the existence and uniqueness of a global positive solution of model (3). In Section 3, we analyze the stochastically asymptotic stability in the large of the disease-free equilibrium. In Section 4, we study the dynamics of system (3) around the endemic equilibrium of the deterministic model. Finally, in Section 5, numerical simulations are carried out.
2 Non-negative solutions
When we study a dynamical behavior, a global solution is important for the system. In this section, we show that the solution of (3) is global and nonnegative. As we know, for a stochastic differential equation to have a unique global (i.e., no explosion in a finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and the local Lipschitz condition is a sufficient condition (see [21, 22]). Although the coefficients of system (3) satisfy local Lipschitz condition, they do not satisfy the linear growth condition, so the solution of system (3) may explode at a finite time. In this section, we will use the Lyapunov analysis method mentioned in [16] to show that the solution of system (3) is positive and global.
Theorem 1
There is a unique solution \((I(t),R(t),N(t))\) of system (3) on \(t\ge0\) for any initial value \((I(0),R(0),N(0))\in\mathbb{R}_{+}^{3}\), and the solution will remain in \(\mathbb{R}_{+}^{3}\) with probability 1, namely \((I(t),R(t),N(t))\in\mathbb{R}_{+}^{3}\) for all \(t\ge0\) almost surely.
Proof
Remark 1
Theorem 1 has shown that for any initial value \((I(0),R(0),N(0))\in \mathbb{R}^{3}_{+}\), system (3) has a unique global solution \((I(t),R(t),N(t))\in\mathbb{R}^{3}_{+}\) a.s., and by (5) if \(N(0)\le K\), then \(N(t)\le K\), so the region \(\varOmega ^{*}=\{(I,R,N)\in\mathbb{R}_{+}^{3}, I+R\le N\le K\}\) is a positively invariant set with respect to (3).
From now on, we always assume that \((I(0),R(0),N(0))\in \varOmega ^{*}\).
3 Stochastically asymptotical stability in the large of the disease-free equilibrium
Obviously, \(E_{0}=(0,0,K)\) which is called the disease-free equilibrium, is a solution of system (3). We divide this section into globally asymptotically stability at this point mainly by a stochastic Lyapunov function. First, we give a lemma (see [21]).
Lemma 1
If there exists a positive-definite decrescent radially unbounded function \(V(x,t)\in C^{2,1}(\mathbb{R}^{d}\times[t_{0},\infty);\mathbb{\bar{R}}_{+})\) such that \(LV(x,t)\) is negative-definite, then the trivial solution of (7) is stochastically asymptotically stable in the large.
Theorem 2
Assume \(R_{0}=\frac{\beta K}{\delta}\le1\), then the solution \((0,0,K)\) of system (3) is stochastically asymptotically stable in the large.
Proof
Theorem 2 means that when \(R_{0}\le1\) the disease will die out after some period of time. This the phenomenon of interest to us studying an epidemic dynamical system. Another phenomenon we are interested in is when the disease will prevail and persist in a population, which will be discussed in the next section.
4 Asymptotic behavior around the endemic equilibrium of the deterministic model
Different from the deterministic system, there is no endemic equilibrium in system (3), so we cannot see whether the disease prevails through the endemic equilibrium. But (3) is a perturbation system of (2) which has an endemic equilibrium \(E^{*}\). We tend to study the behavior around \(E^{*}\) to reflect whether the disease will prevail.
Before proving the main theorem we put forward a lemma (see [21]).
Lemma 2
Theorem 3
Proof
Remark 2
Theorem 3 shows that if \(R_{0}>1\), \(\alpha<\frac{r}{2+(\frac{1}{1+\frac {\gamma}{\mu}})}\), the solution of system (3) goes around \(E^{*}\) for a long time while the intensity of the white noise is weak. Therefore according to [11], \(E^{*}\) is globally asymptotically stable when \(R_{0}>1\), \(\alpha\le\min\{2\mu,\frac{1}{2}r\}\). In this sense, as long as α is small properly, we consider the disease to prevail.
5 Numerical simulations
Declarations
Acknowledgements
The authors would like to extend their sincere gratitude to their supervisor, Jifa Jiang, for his instructive advice and useful suggestions on their thesis. The authors would like to express their gratitude to all those who have helped them during the writing of this paper. This work was supported by National Natural Science Foundation of China under Grant 11371252, National Natural Science Foundation of China under Grant 11401382, Research and Innovation Project of Shanghai Education Committee under Grant 14zz120, the Program of Shanghai Normal University under Grant tDZL121 and the Talent Program of Anhui Agriculture University under Grant wd2015-10.
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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