- Research
- Open Access
Dynamical analysis of a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis
- Anqi Miao^{1},
- Xinyang Wang^{1},
- Tongqian Zhang^{1, 2}Email authorView ORCID ID profile,
- Wei Wang^{3} and
- BG Sampath Aruna Pradeep^{4}
https://doi.org/10.1186/s13662-017-1289-9
© The Author(s) 2017
- Received: 16 April 2017
- Accepted: 19 July 2017
- Published: 4 August 2017
Abstract
In this paper, a stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis is proposed and analysed. We explain the effects of stochastic disturbance on disease transmission. To this end, firstly, we investigated the dynamic properties of the system neglecting stochastic disturbance and obtained the threshold and the conditions for the extinction and the permanence of two kinds of epidemic diseases by considering the stability of the equilibria of the deterministic system. Secondly, we paid prime attention on the threshold dynamics of the stochastic system and established the conditions for the extinction and the permanence of two kinds of epidemic diseases. We found that there exists a significant difference between the threshold of the deterministic system and that of the stochastic system. Moreover, it has been established that the persistent of infectious disease analysed by use of deterministic system becomes extinct under the same conditions due to the stochastic disturbance. This implies that a stochastic disturbance has significant impact on the spread of infectious diseases and the larger stochastic disturbance leads to control the epidemic diseases. In order to illustrate the dynamic difference between the deterministic system and the stochastic system, there have been given a series of numerical simulations by using different noise disturbance coefficients.
Keywords
- stochastic SIS epidemic model
- double epidemic hypothesis
- Beddington-DeAngelis incidence rate
- extinction
- permanence
MSC
- 60H10
- 65C30
- 91B70
1 Introduction
It is well known that random noise factors play an important role in the transmission of infectious diseases. Therefore, many scholars [36–43] have studied the impact of the stochastic epidemic system, various stochastic perturbation approaches have been introduced into epidemic models and have obtained excellent results. For example, the authors of [44–49] have considered a stochastic epidemic model with a Markov transform. A class of epidemic model which shows the effect of the random white noise has been studied by the researchers in the articles [50–64]. Further, in [65–70], the authors studied a class of stochastic epidemic models, in which the stochastic white noise is assumed to be proportional to S, I and R. It can be seen following the literature that the authors of the articles [71, 72] analysed a stochastic epidemic model with two different kinds of perturbation. A stochastic epidemic model with Lévy jumps has been proposed and studied by the researchers [73–75]; the authors investigated stochastic perturbation around the positive equilibria of deterministic models (see, for example, [40, 41, 76–78]). Although there were limited numbers of publications in the recent literature considering time delay and stochastic behavior, the authors in [77] paid attention on the stochastic epidemic model with time delay.
2 Preliminaries and lemmas
In this section, we will give some notations, definitions and some lemmas which will be used for analysing our main results.
Throughout this paper, let \((\Omega,\mathcal{F}, \{\mathcal{F}\}_{t\geq 0}, \mathcal{P})\) be a complete probability space with a filtration \(\{ \mathcal{F}_{t}\}_{t\geq0}\) satisfying the usual conditions (i.e. it is increasing and right continuous while \(\mathcal{F}_{0}\) contains all \(\mathcal{P}\)-null sets \(R_{+}^{3}=\{ x_{i}>0,i=1,2,3\}\)). The function \(B(t)\) denotes a scalar Brownian motion defined on the complete probability space Ω. For an integrable function f on \([0,+\infty)\), define \(\langle f(t)\rangle =\frac{1}{t}\int_{0}^{t}f(\theta)\, \mathrm{d}\theta\).
Definition 2.1
- (i)
The diseases \(I_{1}(t)\) and \(I_{2}(t)\) are said to be extinctive if \(\lim_{t\rightarrow+\infty}I_{1}(t)=0\) and \(\lim_{t\rightarrow+\infty}I_{2}(t)=0\).
- (ii)
The diseases \(I_{1}(t)\) and \(I_{2}(t)\) are said to be permanent in mean if there exist two positive constants \(\lambda_{1}\) and \(\lambda_{2}\) such that \(\liminf_{t\rightarrow+\infty}\langle I_{1}(t)\rangle\geq\lambda_{1}\) and \(\liminf_{t\rightarrow+\infty }\langle I_{2}(t)\rangle\geq\lambda_{2}\).
Lemma 2.1
For any initial value \((S(0), I_{1}(0), I_{2}(0))\in R^{3}_{+}\), there exists a unique solution \((S(t), I_{1}(t), I_{2}(t))\) to system (4) on \(t\geq0\), and the solution will remain in \(R^{3}_{+}\) with probability 1, namely, \((S(t), I_{1}(t), I_{2}(t))\in R^{3}_{+}\) for all \(t\geq0\) almost surely.
Proof
If this statement is false, then there is a pair of constants \(T>0\) and \(\delta\in(0,1)\) such that \(P\{\tau_{0}\leq T\}>\delta\). Hence, there is a positive constant \(\varepsilon_{1}\leq\varepsilon_{0}\) such that \(P\{\tau _{\varepsilon}\leq T\}\) for any positive \(\varepsilon\leq\varepsilon_{1}\).
Lemma 2.2
Denote \(\Gamma=\{(S(t), I_{1}(t), I_{2}(t))\in R_{+}^{3}: S(t), I_{1}(t), I_{2}(t)\leq\frac{A}{d}, t\geq0\}\), then Γ is an invariant set on system (3) or (4).
Proof
By Lemma 2.2 and the strong law of large numbers for martingales [36], we can obtain the following lemma.
Lemma 2.3
3 Dynamics of deterministic system (3)
Theorem 3.1
- (i)
if \(\mathcal{R}_{1}<1\) and \(\mathcal{R}_{2}<1\), then both diseases go extinct and system (3) has a unique stable ‘diseases-extinction’ equilibrium \(E_{0}\);
- (ii)
if \(\mathcal{R}_{1}>1\) and \(\mathcal{R}_{2}<1\), then the disease \(I_{2}\) goes extinct and system (3) has a unique stable equilibrium \(E_{1}\);
- (iii)
if \(\mathcal{R}_{1}<1\) and \(\mathcal{R}_{2}>1\), then the disease \(I_{1}\) goes extinct and system (3) has a unique stable equilibrium \(E_{2}\); and
- (iv)
when condition H holds, and if \(\mathcal{R}_{1}>1\) and \(\mathcal{R}_{2}>1\), then \(E^{*}\) is a unique stable equilibrium, which implies both diseases of system (3) are permanent.
Proof
Similarly, we can show that if \(\mathcal{R}_{1}<1\) and \(\mathcal {R}_{2}>1\), then the equilibrium \(E_{2}\) of system (3) is stable.
Now, let us prove that the positive equilibrium \(E^{*}\) is stable as \(\mathcal{R}_{1}>1\) and \(\mathcal{R}_{2}>1\).
4 Dynamics of stochastic system (4)
4.1 Extinction
In this section, we are going to explore the conditions which lead to the extinction of two infectious diseases mentioned in the system (4) under a white noise stochastic disturbance.
Theorem 4.1
Proof
Remark 4.1
Theorem 4.1 shows that when \(\sigma_{i}>\frac {\beta_{i}}{\sqrt{2(d+\alpha_{i}+r_{i})}}\), \(i=1,2\), two infectious diseases of system (4) die out almost surely, that is to say, large white noise stochastic disturbance can lead to the two epidemics to be extinct. Therefore, we always assume that the white noise stochastic disturbance is not too large in the rest of this paper.
Theorem 4.2
Proof
Remark 4.2
Theorem 4.1 and Theorem 4.2 show that two diseases will die out if the white noise disturbance is sufficiently larger or \(\mathcal{R}_{i}^{*}<1\) and the white noise disturbance is not large. Note that the expressions for \(\mathcal{R}_{i}^{*}\) for \(i=1,2\) which are the threshold values of system (4) are strictly different compared with the thresholds \(\mathcal{R}_{i}\) of system (3), This implies that the conditions which are needed to have \(I_{i}(t)\) for \(i=1,2\) gone in extinction in deterministic system (3) are stronger than in the corresponding stochastic system (4).
4.2 Permanence in mean
Theorem 4.3
- (i)If \(\mathcal{R}_{1}^{*}>1\), \(\mathcal{R}_{2}^{*}<1\) and \(\sigma_{2}\leq \sqrt{\frac{\beta_{2}(A a_{2}+d)}{A}}\), then the disease \(I_{2}\) goes extinct and the disease \(I_{1}\) is permanent in mean, moreover, \(I_{1}\) satisfies$$ \liminf_{t\rightarrow+\infty}\bigl\langle I_{1}(t)\bigr\rangle \geq \frac {(Aa_{1}+d)(d+\alpha_{1}+r_{1})}{\beta_{1}(d+\alpha_{1})+b_{1}d(d+\alpha _{1}+r_{1})}\bigl(\mathcal{R}_{1}^{*}-1\bigr). $$
- (ii)If \(\mathcal{R}_{2}^{*}>1\), \(\mathcal{R}_{1}^{*}<1\) and \(\sigma_{1}\leq \sqrt{\frac{\beta_{1}(A a_{1}+d)}{A}}\), then the disease \(I_{1}\) goes extinct and the disease \(I_{2}\) is permanent in mean, moreover, \(I_{2}\) satisfies$$ \liminf_{t\rightarrow+\infty}\bigl\langle I_{2}(t)\bigr\rangle \geq \frac {(Aa_{2}+d)(d+\alpha_{2}+r_{2})}{\beta_{2}(d+\alpha_{2})+b_{2}d(d+\alpha _{2}+r_{2})}\bigl(\mathcal{R}_{2}^{*}-1\bigr). $$
- (iii)If \(\mathcal{R}_{1}^{*}>1\) and \(\mathcal{R}_{2}^{*}>1\), then two infectious diseases \(I_{1}\) and \(I_{2}\) are permanent in mean, moreover, \(I_{1}\) and \(I_{2}\) satisfywhere$$ \liminf_{t\rightarrow+\infty}\bigl\langle I_{1}(t)+I_{2}(t) \bigr\rangle \geq\frac {1}{\Delta_{\max}}\sum_{i=1}^{2}a_{i}(d+ \alpha_{i}+r_{i}) \bigl(\mathcal {R}_{i}^{*}-1 \bigr), $$$$\Delta_{\max}=\sum_{i=1}^{2} \biggl[\frac{\beta_{1}+\beta_{2}}{d}(d+\alpha _{i})+b_{i}(d+ \alpha_{i}+r_{i}) \biggr]. $$
Proof
By Lemma 2.3, we get \(\lim_{t\rightarrow+\infty}\frac{M(t)}{t}=0\). According to Lemma 2.2, one can see that \(I_{1}(t)\leq\frac {A}{d}\). Thus, one has \(\lim_{t\rightarrow+\infty}\frac{I_{1}(t)}{t}=0\), \(\lim_{t\rightarrow +\infty}\frac{\ln I_{1}(t)}{t}=0\) as \(I_{1}(t)\geq1\) and \(\lim_{t\rightarrow+\infty}\Theta(t)=0\).
Remark 4.3
Theorem 4.3 shows that both diseases will prevail if the white noise disturbances are small enough such that \(\mathcal{R}_{i}^{*}>1\), conversely, if the white noise disturbances are large enough, then both diseases will become extinct. This implies that the stochastic disturbance may cause epidemic diseases to die out.
5 Conclusion and simulations
This paper proposed two \(SIS\) epidemic models with Beddington-DeAngelis incidence rate and double epidemic hypothesis from the point of view of deterministic and stochastic aspect. The threshold dynamics of both two systems were investigated and the conditions for extinction and permanence of both epidemic diseases were obtained. From Theorems 4.1 and 4.2, it can be seen that there is a significant difference between the thresholds of the stochastic system and the deterministic system, from which it can be concluded that the conditions for two epidemic diseases to go to extinction in the stochastic system are weaker than those of the deterministic system.
To illustrate the dynamic difference between the deterministic system and the stochastic system, we next carry out some numerical simulations of these cases with respect to different noise disturbance intensity using the Euler Maruyama (EM) method [36, 79].
Declarations
Acknowledgements
This work is supported by Shandong Provincial Natural Science Foundation (No. ZR2015AQ001), National Natural Science Foundation of China (No. 11371230), Project for Higher Educational Science and Technology Program of Shandong Province (No. J13LI05), Research Funds for Joint Innovative Center for Safe and Effective Mining Technology and Equipment of Coal Resources by Shandong Province and SDUST Research Fund (2014TDJH102).
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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