A fractional-order tuberculosis model with efficient and cost-effective optimal control interventions

Abstract

This study is concerned with applying fractional calculus in modelling tuberculosis (TB) transmission dynamics. A new six-dimensional fractional-order mathematical model in the sense of the Caputo derivative operator is formulated to capture memory effects in the spread process of tuberculosis. Some key features such as vaccination, forces of TB infection, and exogenous re-infection by two actively infectious classes are considered. Banach fixed point theory is employed to prove the existence and uniqueness of a solution for the fractional-order model. Equilibrium points and effective reproduction number of the model are determined. Analysis reveals that the model exhibits backward bifurcation where stable TB-free and TB-present equilibrium points co-exist when the effective reproduction number is below unity. Some sensitive parameters of the model are identified to help design intervention measures against the disease. Hence, four time-dependent controls, namely advocacy effort against infection and re-infection, vaccination control, prophylaxis against latent infection, and treatment control, are considered to form a fractional-order optimal control model. The controls are characterized using Pontryagin’s maximum principle. Efficiency and cost-effectiveness assessments are conducted to quantify the most efficient and cost-effective intervention plan to minimize TB spread in the population. Behaviours of the fractional-order TB model at varying values of the order of the Caputo derivative are extensively explored to show memory effects in the transmission dynamics of the disease.