Abstract

The concept of fractional calculus was used in this study to develop a mathematical model of tuberculosis
(TB). When compared to integer-order models, fractional order differential equations are a potent tool for
incorporating memory and hereditary properties of dynamical systems. The memory effect plays an
important role in the spread of disease. The presence of memory effects on past events will affect the
spread of disease in the future so that the spread of disease in the future can be controlled. The distance of
memory effect indicates the history of disease spread. Thus, memory effects on the spread of infectious
diseases can be investigated using fractional derivatives. Based on this background, a fractional order
drug-resistant TB model with counseling and case detection was developed using the concept of Caputo
derivatives. In other to describe the model equations, the total population is divided into seven classes:
susceptible individuals, uncounseled and counseled exposed individuals and uncounseled and counseled
infectious individuals, drug-resistant , and recovered individuals. The Generalized Euler Method (GEM) was used to numerically solve the formulated TB model. Graphic results showed that the GEM yielded more accurate results compared to the Runge-
Kutta fourth-order method. Furthermore, numerical simulation demonstrated that the dynamics of TB are
continuously dependent on the order of fractional derivatives.

Key words: Tuberculosis, Caputo derivative, Generalized Euler Method, Fractional Order, Runge-Kutta