Application of an Optimal Control Therapeutic Approach for the Memory-Regulated Infection Mechanism of Leprosy through Caputo–Fabrizio Fractional Derivative

Abstract

Leprosy (Hansen’s disease) is an infectious, neglected tropical skin disease caused by the bacterium Mycobacterium leprae (M. leprae). It is crucial to note that the dynamic behavior of any living microorganism such as M. leprae not only depends on the conditions of its current state (e.g., substrate concentration, medium condition, etc.) but also on those of its previous states. In this article, we have developed a three-dimensional mathematical model involving concentrations of healthy Schwann cells, infected Schwann cells, and M. leprae bacteria in order to predict the dynamic changes in the cells during the disease dissemination process; additionally, we investigated the effect of memory on system cell populations, especially on the M. leprae bacterial population, by analyzing the Caputo–Fabrizio fractionalized version of the model. Most importantly, we developed and investigated a fractionalized optimal-control-induced system comprising the combined drug dose therapy of Ofloxacin and Dapsone intended to achieve a more realistic treatment regime for leprosy. The main goal of our research article is to compare this fractional-order system with the corresponding integer-order model and also to distinguish the rich dynamics exhibited by the optimal-control-induced system based on different values of the fractional order ζ∈(0,1). All of the analytical results are validated through proper numerical simulations and are compared with some real clinical data.

Keywords

• leprosy
• mathematical modelling
• optimal control
• memory effect
• Caputo–Fabrizio fractionalized system