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Biosystems. 2008 Sep;93(3):240-9. doi: 10.1016/j.biosystems.2008.05.004. Epub 2008 May 23.

Stability analysis and optimal vaccination of an SIR epidemic model.

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1
Department of Mathematics, Pusan National University, San 30, Geumjeong-Gu, Busan 609-735, South Korea. zaman@pusan.ac.kr

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.

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