On Optimal Control Analysis of Pneumonia

Abstract

The optimal control of pneumonia disease is very important in the management of the disease within a given population. Pneumonia is one of the leading causes of death worldwide, especially among children below 5 years old, the elderly above 65 years old and people with a weaker immune system. In this research work, the basic mathematical properties of a deterministic SEIR model for pneumonia disease were first presented. These properties include the invariant region, disease free equilibrium, basic reproduction number and the disease endemic equilibrium. The optimal control problem, which is the main focus of this work, was ushered in and thoroughly dealt with using the Pontryagin's Maximum Principle. The control measures include the prevention effort of the susceptible class, the vaccination intervention strategy and the treatment intervention strategy. The Hamiltonian of the control problem was defined and used together with the adjoint equations to obtain the optimality system and the optimal values of the controls. Numerical simulations were carried out using various combinations of these control measures (prevention, vaccination and treatment). The results were presented and compared to determine the best strategy that should be taken in order to eliminate the disease from a given population within a desired period of time. It is observed that the combination with all three controls gives the best intervention strategy for the elimination of the disease.