Brief paperDynamics of a multigroup SIR epidemic model with stochastic perturbation☆
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 , , and be the susceptible, infective, and recovered population at time in the -th group, respectively. If we do not consider the fractions that are immune and vaccinated, and suppose that the death rates of , , and in the -th group are different, then the model is The parameters in the model are summarized in the following list. The values of all the parameters are assumed to be nonnegative, and . 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 for all . Since the dynamics of groups , , have no effects on the transmission dynamics, Guo et al. only analyzed equations involving and of (1.1), i.e., the following system: According to the theory in Guo et al. (2006), system (1.2) always has the disease-free equilibrium . If is irreducible and , then is the unique equilibrium of (1.2) and it is globally stable in , whereas, if is irreducible and , then is unstable and there is an endemic equilibrium of (1.2), which is globally asymptotically stable in int, where , , , and (the spectral radius of ), .
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 , , and by , , and , where , , and are mutual independent standard Brownian motions with , , and , and intensity of white noise , , and , respectively. Then the stochastic system is
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 , 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 , 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 , 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 . Finally, in order to be self-contained, we give an Appendix containing some theory used in the previous sections.
Throughout this paper, let be a complete probability space with a filtration satisfying the usual conditions (i.e. it is right continuous and contains all -null sets). Denote . Consider the -dimensional stochastic differential equation with initial value . denotes -dimensional standard Brownian motion defined on the above probability space. Define the differential operator associated with Eq. (1.4) by . If acts on a function , then , where , . By Itô’s formula, if , then . For (1.4), assume that , for all . So 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
Asymptotic behavior around
It is clear that is the disease-free equilibrium of system (1.1), but not (1.3). For system (1.2), which has been mentioned above, is globally stable if , 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.
References (21)
- et al.
Numerical simulation of stochastic ordinary differential equations in biomathematical modelling
Mathematics and Computers in Simulation
(2004) - et al.
A stochastic model of AIDS and condom use
Journal of Mathematical Analysis and Applications
(2007) - et al.
Global behavior of a multi-group SIS epidemic model with age structure
Journal of Differential Equations
(2005) - et al.
Analysis of a predator–prey model with modified Leslie–Gower and Holling-type II schemes with stochastic perturbation
Journal of Mathematical Analysis and Applications
(2009) - et al.
Qualitative analysis of a stochastic ratio-dependent predator–prey system
Journal of Computational and Applied Mathematics
(2011) - et al.
Sex ratio features of two-group SIR model for asymmetric transmission of heterosexual disease
Mathematical and Computer Modelling
(1996) - et al.
Global-stability problem for coupled systems of differential equations on networks
Journal of Differential Equations
(2010) - et al.
Stability of a stochastic SIR system
Physica A
(2005) - et al.
The influence of external real and white noise on the Lotka–Volterra model
Journal of Biomedical
(1979) - et al.
Ratio-dependent predator–prey model: effect of environmental fluctuation and stability
Nonlinearity
(2005)
Cited by (101)
A Markovian model for the spread of the SARS-CoV-2 virus
2023, AutomaticaA stochastic age-structured HIV/AIDS model based on parameters estimation and its numerical calculation
2021, Mathematics and Computers in SimulationDynamical behaviors of a heroin population model with standard incidence rates between distinct patches
2021, Journal of the Franklin InstituteMathematical perspective of Covid-19 pandemic: Disease extinction criteria in deterministic and stochastic models
2021, Chaos, Solitons and FractalsDynamical behavior of a higher order stochastically perturbed HIV/AIDS model with differential infectivity and amelioration
2020, Chaos, Solitons and Fractals
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.
- 1
Tel.: +86 43185099589; fax: +86 43185098237.