Theoretical Analysis and Simulation of a Fractional-Order Compartmental Model with Time Delay for the Propagation of Leprosy

This article investigates the propagation of a deadly human disease, namely leprosy. At the outset, the mathematical model is transformed into a fractional-order model by introducing the Caputo differential operator of arbitrary order. A result is established, which ensures the positivity of the fractional-order epidemic model. The stability of the continuous model at different points of equilibria is investigated. The basic reproduction number, R-0, is obtained for the leprosy model. It is observed that the leprosy system is locally asymptotically stable at both steady states when R-0 < 1. On the other hand, the fractional-order system is globally asymptotically stable when R-0 > 1. To find the approximate solutions for the continuous epidemic model, a non-standard numerical scheme is constructed. The main features of the non-standard scheme (such as positivity and boundedness of the numerical method) are also confirmed by applying some benchmark results. Simulations and a feasible test example are presented to discern the properties of the numerical method. Our computational results confirm both the analytical and the numerical properties of the finite-difference scheme.