Asymptotic behavior of global positive solution to a stochastic SIR model incorporating media coverage
- Miaochan Zhao^{1} and
- Huitao Zhao^{1}Email author
https://doi.org/10.1186/s13662-016-0884-5
© Zhao and Zhao 2016
Received: 28 January 2016
Accepted: 29 May 2016
Published: 8 June 2016
Abstract
We study the basic dynamical features of a stochastic SIR epidemic model incorporating media coverage. Firstly, we discuss the positivity and boundedness of solutions of the model within deterministic environment and then investigate the asymptotical stability and global stability of equilibria of deterministic model. Secondly, we show that the stochastic model has a unique global positive solution and that this solution oscillates around the equilibria of the deterministic model under certain conditions. Finally, we give some numerical simulations to illustrate our analytical results.
Keywords
1 Introduction
Mathematical models plays an important role in the study of epidemiology, which provides understanding of the underlying mechanisms that influence the spread of disease, and in the process, it suggests control strategies. Various epidemic models have been proposed and explored extensively, and great progress has been achieved in the studies of disease control and prevention (see, e.g., [1–4]).
In the following, unless otherwise specified, we assume that \((\Omega, \mathcal{F}, \{\mathcal{F}_{t} \}_{t\geqslant0}, P)\) is a complete probability space with filtration \(\{\mathcal{F}_{t} \}_{t\geqslant0}\) satisfying the usual conditions (i.e., it is increasing and right continuous, and \(\mathcal{F}_{0}\) contains all P-null sets). Let \(B_{i}(t)\), \(i=1,2,3\), be Brownian motions defined on this probability space. Also, let \(\mathbb{R}^{3}_{+}=\{\mathbf{x}\in\mathbb{R}^{3}, x_{i}>0 \mbox{ for all } 1\leqslant i \leqslant3 \}\) and \(\mathbf {x}(t)=(S(t), I(t), R(t))^{T}\).
The rest of the paper is organized as follows. In Section 2, we first show the positivity and boundedness of the deterministic model (1.3); the existence and stability of equilibria of model (1.3) is also investigated in this section. In Section 3, we first study the existence of the global positive solution of the stochastic model (1.4), and then, we investigate the asymptotic behavior around the equilibria of model (1.3). In Section 4, we give some numerical simulations to support the theoretical prediction. In Section 5, a brief discussion is given.
2 Deterministic model
2.1 Positivity and boundedness
In this subsection, we study the positivity and boundedness of solutions of system (1.3) with initial condition (2.1).
Theorem 2.1
Solutions of system (1.3) with initial condition (2.1) are positive for all \(t \geqslant0\).
Proof
Proof
2.2 Equilibria and their existence
Note that \(\beta_{1} \geqslant\beta_{2}>0\). From (2.12) we can easily prove that \(G(I)\) is decreasing and \(H(I)\) is increasing. Hence, from (2.10)-(2.12) we can verify that if \(R_{0}>1\), then the two curves \(G(I)\) and \(H(I)\) have only one positive intersection in \([0,\frac{\Lambda}{\mu}]\), which gives only one endemic equilibrium. However, if \(R_{0}<1\), then it follows that the two curves \(G(I)\) and \(H(I)\) have no intersection in \([0,+\infty)\), which implies that there is no endemic equilibria.
From the discussion above we obtain the following:
Theorem 2.3
If \(R_{0}<1\), then system (1.3) has no endemic equilibria. If \(R_{0}>1\), then system (1.3) has only one endemic equilibrium.
2.3 Stability of equilibria
In this subsection, by analyzing the corresponding characteristic equations we discuss the local stability of a disease-free equilibrium and endemic equilibrium of system (1.3), respectively.
Theorem 2.4
Next, we discuss the global stability of \(E_{0}\). It is easy to prove the following lemma.
Lemma 2.1
The plane \(S+I+R= \frac{\Lambda}{\mu}\) is an invariant manifold of system (1.3), which is globally attractive in \(\mathbb{R}^{3}_{+}\).
Theorem 2.5
If \(R_{0}<1\), then the disease-free equilibrium \(E_{0}\) is globally asymptotically stable.
Proof
Let \((S(t),I(t),R(t))\) be any positive solution of system (1.3) with initial condition (2.1).
In the following, we suppose that \(R_{0}>1\) and \(E^{*}\) is an endemic equilibrium satisfying Eqs. (2.4)-(2.6).
Theorem 2.6
If \(R_{0}>1\), then the endemic equilibrium \(E^{*}\) is globally stable in \(\mathbb{R}^{3}_{+}\).
Proof
When \(R_{0}>1\), by Theorem 2.1, \(E_{0}\) is a hyperbolic unstable saddle point and repels solutions in its neighborhood. Due to the hyperbolicity of \(E_{0}\), it is not part of any cycle chain in \(\mathbb {R}^{3}_{+}\). Thus, every bounded forward orbit of (1.3) in \(\mathbb{R}^{3}_{+}\) converges to the unique endemic equilibrium \(E^{*}\). Therefore, \(E^{*}\) is globally asymptotically stable. The proof is complete. □
3 Stochastic model
In this section, we first show that the solution of system (1.4) is global and nonnegative. As we know, in order for a stochastic differential equation to have a unique global (i.e., without explosion in finite time) solution for any given initial value, the coefficients of the equation are generally required to satisfy the linear growth condition and local Lipschitz condition [18]. However, the coefficients of Eq. (1.4) do not satisfy the linear growth condition, though they are locally Lipschitz continuous, so the solution of Eq. (1.4) may explode in finite time [18]. Using the Lyapunov analysis method (mentioned in [18]), it is easy to show that the solution of Eq. (1.4) is positive and global.
Theorem 3.1
For any given initial value \((S(0),I(0),R(0))\in\mathbb {R}^{3}_{+}\), there is a unique positive solution \((S(t),I(t),R(t))\) of model (1.4) on \(t\geqslant0\), and the solution remains in \(\mathbb{R}^{3}_{+}\) with probability 1, namely \((S(t),I(t),R(t)) \in \mathbb{R}^{3}_{+}\) for all \(t\geqslant0\) almost surely.
From the discussion of Section 2, for the deterministic model (1.3), there is a disease-free equilibrium \(E_{0}=(\frac{\Lambda }{d},0,0)\), which is globally stable if \(R_{0}=\frac{ \beta_{1}\Lambda }{\mu(\mu+\gamma)}<1\). However, for the stochastic model (1.4), \(E_{0}=(\frac{\Lambda}{d},0,0)\) is no longer a disease-free equilibrium. In this subsection, we investigate the asymptotic behavior around \(E_{0}\).
Theorem 3.2
Proof
Remark 3.1
From Theorem 3.2 we can conclude that if \(R_{0}<1\) and condition (3.1) holds, then the solution of Eq. (1.4) will fluctuate around the disease-free equilibrium of Eq. (1.3).
3.1 Asymptotic behavior around the endemic equilibrium of the deterministic model
In this subsection, we assume that \(R_{0} >1\). Then model (1.3) has a single endemic equilibrium \(E^{*}\), but for model (1.4), \(E^{*}\) is not an endemic equilibrium. Similarly, we also expect to find out whether or not the solution goes around \(E^{*}\). The following result gives a positive answer.
Theorem 3.3
Proof
4 Numerical simulations
Example 4.1
In this case, we set \(\Lambda=15\), \(\beta_{1}=0.0008\), \(\beta_{2}=0.0006\), \(m=30\), \(\mu=0.05\), \(\gamma=0.2 \), where ‘year’ is used as the unit of time [20].
From Eq. (2.3) we compute \(R_{0}=0.96<1\). From the discussion of Section 2 we know that system (1.3) has only one disease-free equilibrium \(E_{0}(300,0,0)\), which is globally asymptotically stable.
Example 4.2
In this case, we set \(\Lambda=15\), \(\beta_{1}=0.002\), \(\beta_{2}=0.0018\), \(m=30\), \(\mu=0.05\), \(\gamma=0.2 \), where ‘year’ is used as the unit of time [20].
From Eq. (2.3) we compute \(R_{0}=2.4>1\). From the discussion of Section 2 we know that system (1.3) has an unstable disease-free equilibrium \(E_{0}(300,0,0)\) and a globally asymptotically stable endemic equilibrium \(E^{*}=(197.1728, 20.5654, 82.2617)\).
5 Discussion
In this paper, we proposed a stochastic SIR epidemic model incorporating media coverage. We first investigated the positivity and boundedness of the solution of model (1.3). We showed that the solution of model (1.3) with the initial condition (2.1) is positive and bounded. Our results also show that, when \(R_{0}<1\), model (1.3) has only one disease-free equilibrium and, when \(R_{0}>1\), model (1.3) has a disease-free equilibrium and an endemic equilibrium. Then, we studied the stability of the disease-free and endemic equilibria. Our results show that the disease-free equilibrium is globally stable when the basic reproduction number \(R_{0}<1\) and is unstable when \(R_{0}>1\). This result shows that media coverage cannot change the basic feature of the SIR epidemic model (1.1) [21]. That is to say, disease eventually disappears when the basic reproduction number \(R_{0}<1\); however, the epidemic eventually becomes an endemic disease when \(R_{0}>1\).
Section 3 deals with the stochastic differential equations; by using suitable Lyapunov functions we show that the solution of the stochastic model is positive and global, and this solution oscillates around the equilibria of the deterministic model under certain conditions. That is to say, if the effects of environmental stochastic perturbations are smaller enough than the natural death rate, then the solution of the stochastic model (1.4) oscillates around the disease-free equilibrium when \(R_{0}<1\); however, the solution of the stochastic model (1.4) oscillates around the endemic equilibrium when \(R_{0}>1\). Moreover, the numerical results also suggest that the fluctuations reduce as the noise level decreases.
Declarations
Acknowledgements
We are grateful to thank the editors and anonymous referees for their careful reading and constructive suggestions which lead to truly significant improvement of the manuscript. This research is supported by the National Natural Science Foundation of China (Nos. 11061016, 11461036), the Science and Technology Department of Henan Province (No. 152300410230), and the Doctoral Research Foundation of Zhoukou Normal University (No. ZKNU2014126).
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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