Elsevier

Automatica

Volume 48, Issue 1, January 2012, Pages 121-131
Automatica

Brief paper
Dynamics of a multigroup SIR epidemic model with stochastic perturbation

https://doi.org/10.1016/j.automatica.2011.09.044Get rights and content

Abstract

In this paper, we introduce stochasticity into a multigroup SIR (susceptible, infective, and recovered) model. The stochasticity in the model is introduced by parameter perturbation, which is a standard technique in stochastic population modeling. In the deterministic models, the basic reproduction number R0 is a threshold which completely determines the persistence or extinction of the disease. We carry out a detailed analysis on the asymptotic behavior of the stochastic model, also regarding of the value of R0. If R01, the solution of the model is oscillating around a steady state, which is the disease-free equilibrium of the corresponding deterministic model, whereas, if R0>1, there is a stationary distribution, which means that the disease will prevail.

Introduction

Epidemiology is the study of the spread of diseases with the objective of tracing factors that are responsible for or contribute to their occurrence. Significant progress has been made in the theory and application by mathematical research. Considering different contact patterns, distinct numbers of sexual partners, or different geography, amongst other factors, it is more proper to divide individual hosts into groups in modeling epidemic diseases. Therefore, multigroup models have been  proposed in the literature to describe the transmission dynamics of infectious diseases in heterogeneous host populations, and much research has been done on various forms of multigroup models; see, for example, Beretta and Capasso (1986), Feng, Huang, and Castillo-Chavez (2005), Huang, Cooke, and Castillo-Chavez (1992), Koide and Seno (1996). The global stability of the unique endemic equilibrium, which is one of main mathematical challenges in analyzing multigroup models, has been proved through the Lyapunov functional technique.

In Guo, Li, and Shuai (2006), Guo et al. characterized a multigroup SIR (susceptible, infective, and recovered) model. Let Sk(t), Ik(t), and Rk(t) be the susceptible, infective, and recovered population at time t in the k-th group, respectively. If we do not consider the fractions that are immune and vaccinated, and suppose that the death rates of Sk, Ik, and Rk in the k-th group are different, then the model is {Sk̇=Λkj=1nβkjSkIjdkSk,Ik̇=j=1nβkjSkIj(ϵk+γk)Ik,Rk̇=γkIkδkRk,k=1,2,,n. The parameters in the model are summarized in the following list. The values of all the parameters are assumed to be nonnegative, and dk,Λk>0. Considering that the death rates of the compartments which are infected and recovered are usually no less than that of the susceptible, we assume that dkmin{ϵk,δk} for all k. Since the dynamics of groups Rk, k=1,2,,n, have no effects on the transmission dynamics, Guo et al. only analyzed 2n equations involving Sk and Ik of (1.1), i.e., the following system: {Sk̇=Λkj=1nβkjSkIjdkSk,Ik̇=j=1nβkjSkIj(ϵk+γk)Ik,k=1,2,,n. According to the theory in Guo et al. (2006), system (1.2) always has the disease-free equilibrium P0=(S10,0,S20,0,,Sn0,0). If B=(βkj) is irreducible and R01, then P0 is the unique equilibrium of (1.2) and it is globally stable in Γ, whereas, if B=(βkj) is irreducible and R0>1, then P0 is unstable and there is an endemic equilibrium P=(S1,I1,S2,I2,,Sn,In) of (1.2), which is globally asymptotically stable in intΓ, where Sk0=Λkdk, k=1,2,,n, Γ={(S1,I1,,Sn,In)R+2n:SkΛkdk,Sk+IkΛkdk,k=1,2,,n}, and R0=ρ(M0) (the spectral radius of M0), M0=M(S10,S20,,Sn0)=(βkjSk0ϵk+γk)n×n.

However, in the real world, epidemic systems are inevitably affected by environmental noise. Hence the deterministic approach has some limitations in mathematically modeling the transmission of an infectious disease, and it is quite difficult to predict the future dynamics of the system accurately. This happens due to the fact that deterministic models do not incorporate the effect of a fluctuating environment. Stochastic differential equation 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 (Arnold et al., 1979, Bandyopadhyay and Chattopadhyay, 2005, Carletti et al., 2004). In reality, the parameters involved with the modeling approach of ecological systems are not absolute constants, and they always fluctuate around some average values due to continuous fluctuation in the environment. As a result, the parameters in the model exhibit continuous oscillation around some average values but do not attain fixed values with the advancement of time. Therefore, many authors have studied epidemic dynamics with parameter perturbations, such as Dalal et al., 2007, Ji et al., 2009, Ji et al., 2011, Tornatore et al., 2005.

Considering environmental noise, in this paper, we introduce randomness into the model by replacing the parameters dk, ϵk, and δk by dkdk+αkḂ1k(t), ϵkϵk+βkḂ2k(t), and δkδk+σkḂ3k(t), where B1k(t), B2k(t), and B3k(t) are mutual independent standard Brownian motions with B1k(0)=0, B2k(0)=0, and B3k(0)=0, and intensity of white noise αk20, βk20, and σk20, respectively. Then the stochastic system is {dSk(t)=[Λkj=1nβkjSk(t)Ij(t)dkSk(t)]dtαkSk(t)dB1k(t),dIk(t)=[j=1nβkjSk(t)Ij(t)(ϵk+γk)Ik(t)]dtβkIk(t)dB2k(t),dRk(t)=[γkIk(t)δkRk(t)]dtσkRk(t)dB3k(t),k=1,2,,n.

In accordance with Guo et al. (2006), we investigate the asymptotic behavior of system (1.3). When studying epidemic systems, we are interested in two problems: one is when the disease will die out, and the other is when the disease will prevail and persist in a population. For a deterministic system, the problems are usually solved by showing the stability of the two equilibria under different conditions. But, there is neither a disease-free equilibrium nor an endemic equilibrium of system (1.3). How can we measure whether the disease will die out or prevail? Since there is no equilibrium of system (1.3), it is impossible to expect the solution to tend to a steady state. Therefore, we try to explore some weak stability. Since system (1.3) is a disturbed form of system (1.1), it is reasonable to consider that it is stable if the solution is oscillating around the equilibrium of system (1.1) for a long time. In detail, if R01, we conclude that the distance between the solution of (1.3) and the disease-free equilibrium of (1.1) is no more than a value which is proportional to the white noise. If R0>1, we show that there is a stationary distribution. One of the main features in our proof is using an appropriate Lyapunov function, and this is more delicate in multisystems.

The rest of this paper is organized as follows. Section 2 shows there is a unique positive solution of (1.3). In Section 3, if R01, we show that the solution is oscillating around the disease-free equilibrium of (1.1), and that the intensity of fluctuation is proportional to the white noise. Section 4 focuses on the persistence of the disease. By choosing appropriate Lyapunov functions, we show that there is a stationary distribution for (1.3) and that it is ergodic if R0>1. Finally, in order to be self-contained, we give an Appendix containing some theory used in the previous sections.

Throughout this paper, let (Ω,,{t}t0,P) be a complete probability space with a filtration {t}t0 satisfying the usual conditions (i.e. it is right continuous and 0 contains all P-null sets). Denote R+3n={xR3n:xk>0,1k3n},R̄+3n={xR3n:xk0,1k3n}. Consider the d-dimensional stochastic differential equation dx(t)=f(x(t),t)dt+g(x(t),t)dB(t)on tt0 with initial value x(t0)=x0Rd. B(t) denotes d-dimensional standard Brownian motion defined on the above probability space. Define the differential operator L associated with Eq. (1.4) by L=t+k=1dfk(x,t)xk+12k,j=1d[gT(x,t)g(x,t)]kj2xkxj. If L acts on a function VC2,1(Rd×R̄+;R̄+), then LV(x,t)=Vt(x,t)+Vx(x,t)f(x,t)+12trace[gT(x,t)Vxx(x,t)g(x,t)], where Vt=Vt,Vx=(Vx1,,Vxd), Vxx=(2Vxkxj)d×d. By Itô’s formula, if x(t)Rd, then dV(x(t),t)=LV(x(t),t)dt+Vx(x(t),t)g(x(t),t)dB(t). For (1.4), assume that f(0,t)=0, g(0,t)=0 for all tt0. So x(t)0 is a solution of (1.4), called the trivial solution or equilibrium position.

Section snippets

Existence and uniqueness of the nonnegative solution

To investigate the dynamical behavior of a population model, the first concern is whether the solution is positive and has global existence. Hence, in this section, we mainly use the Lyapunov analysis method (as mentioned in Dalal et al., 2007) to show that the solution of system (1.3) is positive and global. For the multigroup model, the main difficulty is to choose suitable parameters in defining the Lyapunov function, which is found by graph theory. From now on, we denote the solution (S1(t),

Asymptotic behavior around P̃0

It is clear that P̃0=(Λ1d1,0,0,Λ2d2,0,0,,Λndn,0,0) is the disease-free equilibrium of system (1.1), but not (1.3). For system (1.2), which has been mentioned above, P0 is globally stable if R01, which means that the disease will die out after some period of time. Hence, it is interesting to study the disease-free equilibrium for controlling infectious disease. Since there is no disease-free equilibrium of system (1.3), in this section, we show that the solution is oscillating in a small

Ergodicity of system (1.3)

When studying epidemic dynamical systems, we are interested in when the disease will prevail and persist in a population. For a deterministic model, this is usually solved by showing that the endemic equilibrium is a global attractor or is globally asymptotically stable. But, for system (1.3), there is no endemic equilibrium. In this section, we show that there is a stationary distribution based on the theory of Hasminskii (1980) (see the Appendix), which reveals that the disease will prevail.

Chunyan Ji received her M.S. degree in Applied Mathematics from Northeast Normal University, Changchun, China, in 2006. From 2007, she has been working towards her Ph.D. degree Northeast Normal University, and she will graduate July 2011. Her research interests include the theory and application of stochastic differential equations and stochastic ecosystems, as well as estimation and control problems.

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Chunyan Ji received her M.S. degree in Applied Mathematics from Northeast Normal University, Changchun, China, in 2006. From 2007, she has been working towards her Ph.D. degree Northeast Normal University, and she will graduate July 2011. Her research interests include the theory and application of stochastic differential equations and stochastic ecosystems, as well as estimation and control problems.

Daqing Jiang received his M.S. degree in Applied Mathematics from Jilin University, Changchun, China, in 1991. Since 1991, he has worked in the School of Mathematics and Statistics, Northeast Normal University, where he is currently a Full Professor. He received his Ph.D. degree in Probability and Statistics from Northeast Normal University, Changchun, China, in 2003. His current research interests include the theory and application of stochastic differential equations, stochastic ecosystems, and problems of estimation and hypothesis testing of parameters in the ecosystem.

Qingshan Yang received his Ph.D. degree from Wuhan University, Wuhan, China in 2008. Since 2008, he has worked in the department of Mathematics and statistics of Northeast Normal University. His research interests include stochastic processes and stochastic applications.

Ningzhong Shi studied in the Science Department of Kyushu University from 1982 to 1989, and he received his Ph.D. degree in 1989. Form 1989 to 1997, he worked in Northeast Normal University, China. From 1997 to 1998, he worked as a professor in Yiqiao University, Japan. Since 1998, he has been the president of Northeast Normal University. His research interests include multivariate Analysis, umbrella order restrictions, and convex analysis.

This work was supported by the Ministry of Education of China (No. 109051), the Ph.D. Programs Foundation of Ministry of China (No. 200918) and NSFC of China (No. 10971021), and the Graduate Innovative Research Project of NENU (No. 09SSXT117). This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor George Yin under the direction of Editor Ian R. Petersen.

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