- Research
- Open Access
Stochastic dynamics of an SEIS epidemic model
- Bo Yang^{1}Email author
- Received: 2 May 2016
- Accepted: 26 June 2016
- Published: 31 August 2016
Abstract
In this paper, we investigate the stochastic disease dynamics of an SEIS epidemic model with latent patients and active patients. The two parameters \(R_{0}^{s}\) and \(R_{0}^{*}\) are identified as the disease-free and endemic dynamics of the model. More specifically, we give the almost surely exponential stability of the disease-free equilibrium in terms of \(R_{0}^{s}\), and stochastic endemic dynamics in terms of \(R_{0}^{*}\). The theoretical and numerical results may be useful for studying the dynamics of disease spreading in a randomly fluctuating environment.
Keywords
- epidemic model
- disease-free
- endemic
- exponential stability
1 Introduction
Mathematical models have been an important tool in analyzing the spread and control of infectious diseases since the pioneer work of Kermack and McKendrick [1]. Most of the research literature on these types of models assumes that the disease incubation is negligible so that, once infected, each susceptible individual (in the class S) instantaneously becomes infectious (in the class I) and later recovers (in the class R) with a permanent or temporary acquired immunity [2]. A compartmental model based on these assumptions is customarily called a SIR or SIRS model. Regarding research on the SIR or SIRS models and its generalizations, the reader can refer to [3–5].
- (a)
if \(0<\mathcal{R}_{0}<1\), the disease-free equilibrium \((\Lambda /\mu, 0, 0)\) is globally asymptotically stable, and it is unstable when \(\mathcal{R}_{0}>1\);
- (b)if \(\mathcal{R}_{0}>1\), the endemic equilibrium \((S^{*}, E^{*}, I^{*})\) of model (1) is globally asymptotically stable, where$$ \begin{aligned} &S^{*}=\frac{(\beta+\gamma_{1}+\mu)(\gamma_{2}+\mu)}{\alpha(\beta +\gamma_{1}p+\mu p)}, \\ &E^{*}=\frac{\beta\Lambda\alpha(1-p)^{2}(\gamma_{2}+\mu)}{ \alpha\mu(\beta+\gamma_{1}p+\mu p)(\beta+\mu+\gamma_{1}p+\gamma_{2}(1-p))} \biggl(1-\frac{1}{\mathcal{R}_{0}} \biggr), \\ &I^{*}=\frac{(\beta+\gamma_{1}p+\mu p)E^{*} }{(1-p)(\gamma_{2}+\mu)}. \end{aligned} $$(2)
However, the deterministic approach has some limitations in the mathematical modeling transmission of an infectious disease. Stochastic differential equation (SDE) models play a significant role in various branches of applied sciences including infectious dynamics, as they provide some additional degree of realism compared to their deterministic counterpart [12]. Recently, many authors have introduced parameter perturbation into epidemic models and have studied their dynamics [13–20].
This paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we deduce the conditions which will cause the disease to die out. The condition for the disease to be persistent (i.e., endemic) is given in Section 4. In Section 5, we provide some numerical examples to support our analytic results. In the last section, Section 6, we provide a brief discussion and summary of main results.
2 Preliminaries
Definition 2.1
[21]
Definition 2.2
[22]
The population \(x(t)\) is said to be strongly persistent in the mean if \(\liminf_{t\rightarrow\infty} \frac{1}{t} \int_{0}^{t}x(s)\,ds>0\).
The following lemma is quoted from [16, 23] where it was proved and applied. It plays a similar role in this paper.
Lemma 2.3
To investigate the dynamical behavior of a population model, the first concern is whether the solution of the model is positive and global. Motivated by [14], we can prove the global existence of a solution to model (5). One can obtain the following results.
3 Stochastic disease-free dynamics
Now we present the following theorem, which gives conditions for the almost surely exponential stability of the equilibrium of model (5), which is motivated by [21, 24]. Denote \(\sigma:=\min\{\sigma_{1},\sigma_{2}\}\) and \(X(t):=(E(t),I(t))\). First of all, we give the property of the disease-free (i.e., \(I=0\)) dynamics.
Theorem 3.1
Proof
Remark 3.2
It is noted that \(R_{0}^{s}=\frac{\alpha\beta(1-p) \Lambda}{(\beta +\gamma_{1}+\mu+\sigma^{2}/4) (\mu\gamma_{2}+\mu^{2}+\mu\sigma^{2}/4-\alpha p\Lambda)}< R_{0}\). Therefore, if \(R_{0}<1\), no matter how the noise intensities change, we have the disease-free equilibrium to be almost surely exponentially stable. However, if \(R_{0}>1\), by increasing the values of noise intensities such that \(R_{0}^{s}<1\), the disease-free equilibrium will still be almost surely exponentially stable. That is to say, in this situation, for the deterministic model, there is an endemic equilibrium which is globally stable, but for the stochastic model, there is a stable disease-free equilibrium which means that the disease goes extinct exponentially a.s.
4 Stochastic endemic dynamics
In this section we intend to prove the stochastic endemic dynamics (i.e., persistence of E and I) of model (5) under certain parametric restrictions.
Theorem 4.1
Proof
Remark 4.2
It is noted that \(R_{0}^{*}< R_{0}^{s}< R_{0}\). Therefore, if \(R_{0}^{*}>1\), then \(R_{0}>1\). That is to say, if for stochastic model the disease will be prevalent, for a deterministic model the disease also must be prevalent.
5 Numerical simulations
In this section, we give some numerical simulations to show the effect of noise on the dynamics of model (5) by using the Milstein method mentioned in Higham [26].
- 1.Now we note that these parameters give a value of \({R}_{0} =1.667\) to the basic reproduction number in the deterministic case (i.e., with \(\sigma_{1}=\sigma_{2}=0\)). Consequently the system eventually approaches an endemic equilibrium point \(( 0.0523, 0.18)\) (see Figure 1).
- 2.
- 3.
- 4.
6 Discussions
In this paper, we mainly focus on the SDE version of an SEIS epidemic model with latent patients and active patients. We show that the SDE model has a unique positive global solution and establish some conditions for determining the disease outbreak or extinct. The key parameters are \(R_{0}^{s}\) and \(R_{0}^{*}\), which are all less than the corresponding deterministic version of the basic reproduction number \(R_{0}\).
Theorem 3.1 shows that if \(R_{0}^{S}<1\), the disease will die out (cf. Figure 2). Theorem 4.1 shows that if \(R_{0}^{*}>1\), then the disease will persist (cf. Figure 3). By numerical simulations, we also show that if \(R_{0}^{*}<1<R_{0}^{S}\), the disease will die out (cf. Figure 4). Hence, we can make a conjecture that the behavior of the disease is determined by \(R_{0}^{*}\). It is well known that for deterministic epidemic models, the basic reproduction number \(R_{0}\) determines the prevalence or extinction of the disease. In this paper, we consider the threshold \(R_{0}^{*}\) as the basic reproduction number of model (5). Notice that \(R_{0}^{*}< R_{0}\), and it is possible that \(R_{0}^{*}<1<R_{0}\). This is the case when the deterministic model has an endemic (see Figure 1), while the stochastic model has disease extinction with probability one (see Figure 4). That is to say, in this case, noise can suppress the disease outbreak.
Declarations
Acknowledgements
The author thanks the referees for their important and valuable comments. This work is supported by the Natural Science Foundation of Lanzhou University of Arts and Science.
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
References
- Kermack, WO, McKendrick, AG: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A 115, 700-721 (1927) MATHView ArticleGoogle Scholar
- Li, MY, Graef, JR, Wang, L, Karsai, J: Global dynamics of a SEIR model with varying total population size. Math. Biosci. 160(2), 191-213 (1999) MathSciNetMATHView ArticleGoogle Scholar
- Liu, W, Levin, SA, Iwasa, Y: Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23(2), 187-204 (1986) MathSciNetMATHView ArticleGoogle Scholar
- Durrett, R: Stochastic spatial models. SIAM Rev. 41(4), 677-718 (1999) MathSciNetMATHView ArticleGoogle Scholar
- Hethcote, HW: The mathematics of infectious diseases. SIAM Rev. 42(4), 599-653 (2000) MathSciNetMATHView ArticleGoogle Scholar
- Liu, W, Hethcote, HW, Levin, SA: Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25(4), 359-380 (1987) MathSciNetMATHView ArticleGoogle Scholar
- Smith, HL, Wang, L, Li, MY: Global dynamics of an SEIR epidemic model with vertical transmission. SIAM J. Appl. Math. 62(1), 58-69 (2001) MathSciNetMATHView ArticleGoogle Scholar
- Li, G, Jin, Z: Global stability of an SEI epidemic model. Chaos Solitons Fractals 21(4), 925-931 (2004) MathSciNetMATHView ArticleGoogle Scholar
- Zhang, T, Teng, Z: Global asymptotic stability of a delayed SEIRS epidemic model with saturation incidence. Chaos Solitons Fractals 37(5), 1456-1468 (2004) MathSciNetMATHView ArticleGoogle Scholar
- Connell McCluskey, C: Global stability for a class of mass action systems allowing for latency in tuberculosis. J. Math. Anal. Appl. 338(1), 518-535 (2008) MathSciNetMATHView ArticleGoogle Scholar
- Meng, X, Wu, Z, Zhang, T: The dynamics and therapeutic strategies of a SEIS epidemic model. Int. J. Biomath. 6, 1350029 (2013) MathSciNetMATHView ArticleGoogle Scholar
- Zhao, Y, Jiang, D, Mao, X, Gray, A: The threshold of a stochastic sirs epidemic model in a population with varying size. Discrete Contin. Dyn. Syst., Ser. B 20(4), 1277-1295 (2015) MathSciNetMATHView ArticleGoogle Scholar
- Gray, A, Greenhalgh, D, Hu, L, Mao, X, Pan, J: A stochastic differential equation SIS epidemic model. SIAM J. Appl. Math. 71(3), 876-902 (2011) MathSciNetMATHView ArticleGoogle Scholar
- Lahrouz, A, Omari, L, Kiouach, D: Global analysis of a deterministic and stochastic nonlinear SIRS epidemic model. Nonlinear Anal., Model. Control 16(1), 59-76 (2011) MathSciNetMATHGoogle Scholar
- Yang, Q, Mao, X: Extinction and recurrence of multi-group SEIR epidemic models with stochastic perturbations. Nonlinear Anal., Real World Appl. 14(3), 1434-1456 (2013) MathSciNetMATHView ArticleGoogle Scholar
- Ji, C, Jiang, D: Threshold behaviour of a stochastic SIR model. Appl. Math. Model. 38(21), 5067-5079 (2014) MathSciNetView ArticleGoogle Scholar
- Liu, M, Bai, C, Wang, K: Asymptotic stability of a two-group stochastic SEIR model with infinite delays. Commun. Nonlinear Sci. Numer. Simul. 19(10), 3444-3453 (2014) MathSciNetView ArticleGoogle Scholar
- Cai, Y, Kang, Y, Banerjee, M, Wang, W: A stochastic SIRS epidemic model with infectious force under intervention strategies. J. Differ. Equ. 259(12), 7463-7502 (2015) MathSciNetMATHView ArticleGoogle Scholar
- Li, D, Cui, J, Liu, M, Liu, S: The evolutionary dynamics of stochastic epidemic model with nonlinear incidence rate. Bull. Math. Biol. 77(9), 1705-1743 (2015) MathSciNetMATHView ArticleGoogle Scholar
- Cai, Y, Kang, Y, Banerjee, M, Wang, W: A stochastic epidemic model incorporating media coverage. Commun. Math. Sci. 14(4), 892-910 (2016) MathSciNetMATHView ArticleGoogle Scholar
- Khasminskii, R: Stochastic Stability of Differential Equations, vol. 66. Springer, Berlin (2012) MATHView ArticleGoogle Scholar
- Mandal, PS, Banerjee, M: Stochastic persistence and stationary distribution in a Holling-Tanner type prey-predator model. Physica A 391(4), 1216-1233 (2012) View ArticleGoogle Scholar
- Liu, H, Ma, Z: The threshold of survival for system of two species in a polluted environment. J. Math. Biol. 30(1), 49-61 (1991) MathSciNetMATHView ArticleGoogle Scholar
- Witbooi, PJ: Stability of an SEIR epidemic model with independent stochastic perturbations. Physica A 392(20), 4928-4936 (2013) MathSciNetView ArticleGoogle Scholar
- Mao, X: Stochastic Differential Equations and Their Applications. Horwood, Chichester (1997) MATHGoogle Scholar
- Higham, DJ: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525-546 (2001) MathSciNetMATHView ArticleGoogle Scholar