Stability analysis and optimal control of an SIR epidemic model with vaccination

This paper focuses on the study of a nonlinear mathematical SIR epidemic model with a vaccination program. We have discussed the existence and the stability of both the disease free and endemic equilibrium. Vaccine induced reproduction number is determined and the impact of vaccination in reducing the vaccine induced reproduction number is discussed. Then to achieve control of the disease, a control problem is formulated and it is shown that an optimal control exists for our model. The optimality system is derived and solved numerically using the Runge-Kutta fourth order procedure.