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Global threshold dynamics of a stochastic epidemic model incorporating media coverage
- Bin Yang^{1},
- Yongli Cai^{1},
- Kai Wang^{2} and
- Weiming Wang^{1}Email authorView ORCID ID profile
https://doi.org/10.1186/s13662-018-1925-z
© The Author(s) 2018
- Received: 3 September 2018
- Accepted: 6 December 2018
- Published: 12 December 2018
Abstract
In this paper, we investigate the global threshold dynamics of a stochastic SIS epidemic model incorporating media coverage. We give the basic reproduction number \(\mathcal{R}_{0}^{s}\) and establish a global threshold theorem by Feller’s test: if \(\mathcal{R}_{0}^{s}\leq 1\), the disease will die out a.s.; if \(\mathcal{R}_{0}^{s}>1\), the disease will persist a.s. In the case of \(\mathcal{R}_{0}^{s}>1\), we prove the existence, uniqueness, and global asymptotic stability of the invariant density of the Fokker–Planck equations associated with the stochastic model. Via numerical simulations, we find that the average extinction time decreases with the increase of noise intensity σ, and also find that the increasing σ will be beneficial to control the disease spread. Thus, in order to control the spread of the disease, we must increase the intensity of noise σ.
Keywords
- Basic reproduction number
- Feller’s test
- Invariant density
- Fokker–Planck equations
MSC
- 92D30
- 60H10
- 93E15
1 Introduction
It is now widely believed that environmental variations have a critical influence on the spread of the disease [1–4], and stochastic noise plays an indispensable role in the transmission of diseases, especially in a small population. Therefore, it seems more practical to consider stochastic epidemic models [5–20].
- (A1)If one of the following conditionsholds, then the disease will die out with probability one (Theorem 3.1, [22]).$$ R_{0}^{s}< 1 \quad \text{and} \quad \sigma_{2}^{2}< 2 \beta_{1}, \quad \text{or}\quad \sigma_{2}^{2}\geq 2 \beta_{1}, $$
- (A2)
If \(R_{0}^{s}>1\), then the disease will persist with probability one (Theorem 4.1, [22]).
- (B1)Ifthen the disease will die out with probability one (Theorem 3.4, [25]).$$ {R}_{0}^{s}:=\frac{\varLambda \beta_{1}}{\mu (\mu +\gamma)}-\frac{ \sigma^{2}}{2(\mu +\gamma)}< 1, $$
- (B2)Ifhold, then the stochastic process \((S(t),I(t))\) has a unique stationary distribution (Theorem 3.7, [25]).$$ {R}_{0}^{s}>1 \quad \text{and} \quad \sigma^{2}< 2 \mu \min \{1,A\} $$
It is well known that epidemic threshold theorem holds for most deterministic compartmental epidemic models by the basic reproduction number \(R_{0}\) [26]: if \(R_{0}<1\), there is a disease-free equilibrium which is globally asymptotically stable; if \(R_{0}>1\), there exists an endemic equilibrium which is globally asymptotically stable. However, in [A1] for SDE model (1), there is an extra condition \(\sigma_{2}^{2}<2\beta_{1}\), and in [B2] for SDE model (2), there is an extra condition \(\sigma^{2}<2\mu \min \{1,A\}\).
There naturally comes a question: Is there any global threshold theorem for a stochastic epidemic model (e.g., SDE model (1) or (2)) incorporating media coverage?
One goal of this paper is to establish a global threshold theorem for SDE model (4). We will prove that the basic reproduction number \(\mathcal{R}_{0}^{s}\) can be used to govern the stochastic dynamics of SDE model (4).
The other goal is to further study the invariant density of process \(I(t)\). Many long-term asymptotic properties of dynamical systems or random dynamical systems can be described in terms of invariant measure [27] and the density function with respect to Lebesgue measure of the marginals of an invariant measure that can be called an invariant density [28]. If invariant density is \(\mathcal{L}^{1}\) on a set Ω, it satisfies the Fokker–Planck equations (FPE) in the interior of Ω [29]. Hence, we will investigate the FPE associated with (4) and solve the invariant density.
This paper is organized as follows. In Sect. 2, we present the global stochastic threshold theorem. In Sect. 3, in the case of disease persistence, we derive the existence, uniqueness, global stability, and an explicit formula of an invariant density of the Fokker–Planck equation associated with (4). In Sect. 4, we give some numerical examples to show the complicated stochastic dynamics of the model. And in the last section, Sect. 5, we provide a brief discussion and the summary of our main results.
2 Stochastic threshold theorem
In this section, we will focus on the stochastic threshold theorem for model (4). First of all, we state the global existence of the uniqueness and boundedness of the positive solution of model (4).
Theorem 2.1
The proof of Theorem 2.1 is similar to that in [22] or [25]. So we omit it here.
Theorem 2.2
- (i)If \(\mathcal{R}_{0}^{s}\leq 1\), for any given initial value \(I_{0} \in (0, N)\),Namely, the disease will go extinct with probability one.$$ \mathbb{P} \Bigl\{ \lim_{t\rightarrow \infty }I(t)=0 \Bigr\} =1. $$
- (ii)If \(\mathcal{R}_{0}^{s}>1\), for any given initial value \(I_{0} \in (0, N)\),$$ \mathbb{P} \Bigl\{ \sup_{0\leq t< \infty }I(t)=N \Bigr\} =\mathbb{P} \Bigl\{ \inf_{0\leq t< \infty }I(t)=0 \Bigr\} =1. $$
Proof
It follows from [31, Propositions 5.22] and [10, Lemma A.2] that we end the proof. □
3 The properties of invariant density
Theorem 3.1
- (i)the process \(I(t)\) has the ergodic properties, i.e., for any \(\nu_{\sigma }^{s}\)-integrable function G,for all \(I_{0}\in (0,N)\);$$ \mathbb{P}_{I_{0}} \biggl( \lim_{t\rightarrow \infty } \frac{1}{t} \int_{0}^{t}G(I_{\tau })\,d\tau = \int_{0}^{K}G(y)\nu_{\sigma }^{s}(dy) \biggr) =1 $$
- (ii)the invariant density \(p_{\sigma }^{s}\) is globally asymptotically stable in the sense thatand$$ \lim_{t\rightarrow \infty } \int_{0}^{K} \bigl\vert \mathbb{P}(t)g(x)-p_{\sigma }^{s}(x) \bigr\vert \,dx=0,\quad \forall g\in { L_{+}^{1}(0,N)}, $$$$\begin{aligned} L_{+}^{1}\bigl((0,N)\bigr) :=& \biggl\{ w\in L^{1}(\mathcal{R}): \int_{0}^{N}w(x)\,dx=1, w(x)=0 \textit{ for } x\geq N \textit{ or } x\leq 0, \\ &{} \textit{and } w(x)\geq 0 \textit{ for } x\in \mathcal{R} \biggr\} . \end{aligned}$$
- (iii)the unique invariant density \(p_{\sigma }^{s}\) is given bywith$$ p_{\sigma }^{s}(x):=CN^{3-\frac{2\beta_{2}N}{\sigma^{2}(b+N)}} \frac{x ^{c_{0}(\mathcal{R}_{0}^{s}-1)-1}(x+b)^{-\frac{2\beta_{2}N}{\sigma ^{2}(b+N)}}}{(N-x)^{c_{0}(\mathcal{R}_{0}^{s}-1)+3-\frac{2\beta_{2}N}{ \sigma^{2}(b+N)}}}e^{-\frac{c_{0} x}{N-x}} $$(13)and \(c_{0}=\frac{2(\mu +\gamma)}{\sigma^{2}}\).$$ C^{-1}= \int_{0}^{\infty }\bigl(b+(b+N)e^{\xi } \bigr)^{- \frac{2N\beta_{2}}{\sigma^{2}(b+N)}}\bigl(e^{\xi }+1\bigr)^{2} e^{c_{0}( \mathcal{R}_{0}^{s}-1)\xi -c_{0}e^{\xi }}\,d\xi, $$(14)
Proof
4 Numerical results via disease dynamics
From (7), we note that \(\mathcal{R}_{0}^{s}=R_{0}-\frac{ \sigma^{2}}{2(\mu +\gamma)}< R_{0}\), if \(R_{0}<1\), then \(\mathcal{R} _{0}^{s}<1\). We can know that if \(R_{0}<1\), \(I(t)\) goes to extinction for the deterministic model (4) (see [21]); and from Theorem 2.1, \(I(t)\) almost surely tends to zero exponentially with probability one for the stochastic model (4). Therefore we only consider the case of \(R_{0}>1\).
Next we will focus on the role of noise intensity σ on the resulting dynamics for SDE model (4).
4.1 Stochastic disease-free dynamics
First of all, we adopt \(\sigma =0.405\), in this case, \(\mathcal{R} _{0}^{s}=0.97125<1\). From Theorem 2.1(i), we know that the disease \(I(t)\) will go extinct with probability one.
4.2 Stochastic endemic dynamics
From Fig. 4, one can see that the solution to SDE model (4) in the persistent case also suggests for lower σ (e.g., \(\sigma =0.01\) and 0.05), the amplitude of fluctuation is slightly and the oscillations to be more symmetrically distributed (cf. Figs. 4(a) and 4(b)), and the fluctuations are reflected at the stationary distributions. While for higher σ (e.g., \(\sigma =0.1\) and 0.15), the amplitude of fluctuation is remarkable and the distributions of the solutions are skewed (cf. Figs. 4(c) and 4(d)) and the fluctuations are also reflected at the stationary distributions.
5 Concluding remarks
In this paper, we investigate the global threshold dynamics of the SDE SIS epidemic model (4) incorporating media coverage. After defining the basic reproduction number \(\mathcal{R}_{0}^{s}\), we establish a stochastic threshold theorem by using Feller’s test for explosions of solutions to one-dimensional SDE model (4) (see Theorem 2.2). It is worthy to note that in the proof of this main result, we employ Feller’s test [10, 30, 31]. This is very different from the previous results by constructing Lyapunov function to prove the threshold theorem (see, [8, 22, 25]). And in the case of \(\mathcal{R}_{0}^{s}\), i.e., the disease persists with probability one, by studying the FPE associated with SDE model (4), we prove the existence, uniqueness, and global asymptotic stability of the invariant density of the FPE (see Theorem 3.1), which can be useful for us to understand the profile of the distribution of the process \(I(t)\).
Via numerical simulations, in the case of \(\mathcal{R}_{0}^{s}\leq 1\), we find that the average extinction time decreases with the increase of noise intensity σ. And in the case of \(\mathcal{R}_{0}^{s}>1\), we find that the solutions \(I(t)\) of SDE model (4) fluctuate around the endemic equilibrium \(E^{*}=34.4494\) of the deterministic model (6), and finally the distribution of \(I(t)\) seems like a normal stationary distribution. More precisely, for lower σ (e.g., \(\sigma =0.01\) and 0.05), the distribution of \(I(t)\) appears closer to a normal distribution (see Figs. 5(a) and 5(b)); while for higher σ (e.g., \(\sigma =0.1\) and 0.15), the amplitude of fluctuation is remarkable and the distribution of the solutions is positively skewed (cf. Figs. 4(c), 4(d), and 6). Obviously, variance increases with the increase of noise intensity σ, this is the main cause of positive skew distribution (see Figs. 5, 6 and also Figs. 4(c) and 4(d)). In this sense, we can claim that bigger noise σ will be beneficial to make \(I(t)\) stay away from the endemic equilibrium \(E^{*}\) of the deterministic model (6). In other words, increasing σ will be beneficial to control the disease spread. Thus, in order to control the spread of the disease, we must increase the intensity of noise σ.
On the other hand, in Theorem 2.2, we give the global threshold dynamics by Feller’s test for the explosions of solutions to one-dimensional SDE model (4). Unfortunately, this method cannot be used to study a two-dimensional SDE model (e.g., (1) or (2)). And the global threshold dynamics of a high-dimensional SDE model, e.g., (1) or (2), is desirable in the future study.
Declarations
Acknowledgements
The authors would like to thank the anonymous referees for very helpful suggestions and comments which led to the improvement of our original manuscript.
Funding
This research was supported by the National Natural Science Foundation of China (Grant No. 61672013, 11601179 and 11461073), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (16KJB110003), and Huaian Key Laboratory for Infectious Diseases Control and Prevention (HAP201704).
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors 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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