Optimal Vaccination of an Endemic Model with Variable Infectivity and Infinite Delay

In this work, we consider a nonlinear SEIR (susceptible, exposed, infectious, and removed) endemic model, which describes the dynamics of the interaction between susceptible and infected individuals in a population. The model represents the disease evolution through a system of nonlinear differential equations with variable infectivity which determines that the infectivity of an infected individual may not be constant during the time after infection. To control the spread of infection and to find a vaccination schedule for an endemic situation, we use optimal control strategies which reduce the susceptible, exposed, and infected individuals and increase the total number of recovered individuals. In order to do this, we introduce the optimal control problem with a suitable control function using an objective functional. We first show the existence of an optimal control for the control problem and then derive the optimality system. Finally the numerical simulations of the model is identified to fit realistic measurement which shows the effectiveness of the model.