In this paper, we have developed a compartment of epidemic model with vaccination. We
have divided the total population into five classes, namely susceptible, exposed, infective,
infective in treatment and recovered class. We have discussed about basic properties of the
system and found the basic reproduction number (R0) of the system. The stability analysis of
the model shows that the system is locally as well as globally asymptotically stable at
disease-free equilibrium E0 when R0< 1. When R0> 1 endemic equilibrium E1 exists and …

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 …