title: Stability analysis and optimal vaccination of an SIR epidemic model.
abstract: Almost all mathematical models of diseases start from the same basic premise: the population can be subdivided into a set of distinct classes dependent upon experience with respect to the relevant disease. Most of these models classify individuals as either a susceptible individual S, infected individual I or recovered individual R. This is called the susceptible-infected-recovered (SIR) model. In this paper, we describe an SIR epidemic model with three components; S, I and R. We describe our study of stability analysis theory to find the equilibria for the model. Next in order to achieve control of the disease, we consider a control problem relative to the SIR model. A percentage of the susceptible populations is vaccinated in this model. We show that an optimal control exists for the control problem and describe numerical simulations using the Runge-Kutta fourth order procedure. Finally, we describe a real example showing the efficiency of this optimal control.
authors.affiliation: Department of Mathematics, Pusan National University, San 30, Geumjeong-Gu, Busan 609-735, South Korea. zaman@pusan.ac.kr
authors.forename: Gul
authors.initials: G
authors.lastname: Zaman
authors.forename: Yong
authors.initials: Y
authors.lastname: Han Kang
authors.forename: Il Hyo
authors.initials: IH
authors.lastname: Jung
mesh_terms: D003198:Computer Simulation; D004198:Disease Susceptibility / Q000453:epidemiology*; D006801:Humans; D007239:Infections / Q000453:epidemiology*; D008954:Models, Biological; D015233:Models, Statistical; D012907:Smoking; D014611:Vaccination