Stability analysis and optimal vaccination of an SIR epidemic model

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 (SIRSIR) model. In this paper, we describe an SIRSIR 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 SIRSIR 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.