The Behavior of an SVIR Epidemic Model with Stochastic Perturbation

Abstract and Applied Analysis, Jun 2014

We discuss a stochastic SIR epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction number . We deduce the globally asymptotic stability of the disease-free equilibrium when and the perturbation is small, which means that the disease will die out. When , we derive that the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model in time average. The key to our analysis is choosing appropriate Lyapunov functions.

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The Behavior of an SVIR Epidemic Model with Stochastic Perturbation

The Behavior of an SVIR Epidemic Model with Stochastic Perturbation Yanan Zhao1,2,3 and Daqing Jiang1,3 1School of Mathematics and Statistics, Northeast Normal University, Changchun, Jilin 130024, China 2School of Science, Changchun University, Changchun, Jilin 130022, China 3College of Science, China University of Petroleum (East China), Qingdao 266580, China Received 18 October 2013; Revised 4 May 2014; Accepted 6 May 2014; Published 24 June 2014 Academic Editor: Cristina Pignotti Copyright © 2014 Yanan Zhao and Daqing Jiang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We discuss a stochastic SIR epidemic model with vaccination. We investigate the asymptotic behavior according to the perturbation and the reproduction number . We deduce the globally asymptotic stability of the disease-free equilibrium when and the perturbation is small, which means that the disease will die out. When , we derive that the disease will prevail, which is measured through the difference between the solution and the endemic equilibrium of the deterministic model in time average. The key to our analysis is choosing appropriate Lyapunov functions. 1. Introduction Epidemiology is the study of the spread of diseases with the objective to trace factors that are responsible for or contribute to their occurrence. Mathematical models are used extensively in the study of epidemiological phenomena. Most models for the transmission of infectious diseases descend from the classical SIR model of Kermack and McKendrick established in 1927; see [1]. In recent years, many researchers have discussed the SIR model allowing vaccination, that is, the SVIR model. In the epidemiology, vaccines are extremely important and widely used in the modern day world and have been proved to be the most effective and cost-efficient method of preventing infectious diseases such as measles, polio, diphtheria, tetanus, pertussis, and tuberculosis. Routine vaccination is now provided in all developing countries against all these diseases. Li and Ma [2] discuss an SIS model with vaccination. The results in [2] show that the system always has the disease-free equilibrium . If the basic reproduction number , then is the unique equilibrium and it is globally stable. If , then is unstable and there is an endemic equilibrium which is globally asymptotically stable under a sufficient condition. In addition, much research has been done on SVIR models; see [3–6]. In light of these results, complete determination of the global dynamics of these models is essential for their application and further development. In fact, epidemic models are inevitably affected by environmental white noise which is an important component in realism, because it can provide an additional degree of realism in comparison to their deterministic counterparts. Recent advances in stochastic differential equations enable us to introduce stochasticity into the model of biological phenomena, whether it is a random noise in the system of differential equations or environmental fluctuations in parameters. Modeling population dynamics in random environments is a way of studying the fluctuations of population size that has been affected by the stochasticity of external factors. Recently several authors have studied stochastic biological systems; see [7–9]. In addition, some stochastic epidemic models have been studied by many authors; see [10–15]. Parameter perturbations on the transmission rate are considered in [10, 12, 14, 16]. Tornatore et al. [12] study the stability of disease-free equilibrium of a stochastic SIR model with or without distributed time delay, and the same discussion is extended to a SIRS model by Lu [10]. To our knowledge, due to the complexity of stochastic SVIR models, this is the first time that a stability analysis for such system with white noise stochastic perturbations around the transmission rate is performed. In this paper, we will discuss the stochastic SVIR model as follows: SVIR models are formulated by dividing the population size into four distinct groups, , , , and , where is the numbers of a population susceptible to the disease, is the number of infective members, is the number of vaccinated members, and is the number of who have been removed from the possibility of infection through full immunity. The parameters in the model are summarized in the following list::a constant input of new members into the population per unit time;:transmission coefficient between compartments and ;:natural death rate of compartments;:the proportional coefficient of vaccinated individuals for the susceptible;:recovery rate of infectious individuals;:the rate of losing their immunity for vaccinated individuals;:disease-caused death rate of infectious individuals. All parameter values are ass (...truncated)


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Yanan Zhao, Daqing Jiang. The Behavior of an SVIR Epidemic Model with Stochastic Perturbation, Abstract and Applied Analysis, 2014, 2014, DOI: 10.1155/2014/742730