Optimal strategy of vaccination & treatment in an SIR epidemic model

Abstract

In this work, we propose a susceptible–infected–recovered (SIR) epidemic model which describes the interaction between susceptible and infected individuals in a community and analyze the SIR epidemic model through the optimal control theory and mathematical analysis. In addition, we present some possible strategies to prevent the spread of some infection causing epidemic in the society. In order to do this, we introduce an optimal control problem with an objective functional, where two control functions, vaccination and treatment have been used as control measures for susceptible and infected individuals. We show the existence of an optimal control pair for the optimal control problem and derive the optimality condition. Finally we consider a smoking epidemic model to illustrate our theoretical results with some numerical simulations, which use real data collected in April and May 2004 from 300 male students at three vocational technical high schools in Korean metropolitan areas. Our analysis suggests that two control strategies are more effective than only one control strategy in controlling the increase of male student smokers in Korean metropolitan areas.