Mathematical Modeling and Control of the Cell Dynamics in Leprosy

Abstract

Leprosy (Hansen’s disease) is an infectious, neglected tropical disease caused by the bacterium or scientifically Mycobacterium Leprae (M. Leprae). Every year approximately 214,000 new cases of leprosy are reported globally. Leprosy affects the skin or more precisely epidermal layer but M. Leprae specifically targets the Schwann cells of myelinated axons. In this article, we have proposed a three dimensional mathematical model involving the concentrations of healthy Schwann cells, infected Schwann cells and Bacteria (M. Leprae) to predict the dynamical changes of the cells during this disease progression. We have also studied the control mathematical model by introducing two bounded control parameters into our model. For detail analysis of this control system, by considering a minimization problem for the concentration of infected Schwann cells and Bacteria, we have applied the Pontryagin maximum principle. As a result of this analysis, the properties of the optimal controls and their possible types have been established. All analytical outcomes have been verified by numerical simulations.