An Optimal Control of Vaccination Applied to Whooping Cough Model

Abstract

Whooping cough is a vaccine preventable public health problem caused by Bordetella Pertussis and Bordetella Parapertussis. It is a highly contagious respiratory disease. In this paper firstly we have incorporated a vaccination rate into whooping cough model. Simulations show that as we go on increasing the vaccination rate, the period of infection decreases. Secondly we have applied an optimal control on a whooping cough model. Our objective is to minimize the infected population. For this we have developed an optimal control strategy. We have constructed an objective functional comprising of Lagrangian and a Hamiltonian. The objective functional along with adjoint equations, transversality conditions, optimality conditions and model equations constitute an optimal control strategy. We have simulated this optimal control problem using the Forward-Backward Sweep Method by incorporating the RK-4 Method. Simulations show that the susceptible and infected populations are reduced under an optimal control where as the recovered individuals are increased with the application of control. At the end we have mentioned some prominent challenges the whole world must address on urgent basis to control this disease.