Optimal control strategy for the effectiveness of TB treatment taking into account the influence of HIV/AIDS and diabetes

Abstract

The aim of this paper is to present an optimal control problem to reduce the MDR-TB (multidrug-resistant tuberculosis) and XDR-TB (extensively drug-resistant TB) cases, using controls in these compartments and controlling reinfection/reactivation of the bacteria. The model used studies the efficacy of the tuberculosis treatment taking into account the influence of HIV/AIDS and diabetes, and we prove the global stability of the disease-free equilibrium point based on the behavior of the basic reproduction number. Various control strategies are proposed with the combinations of controls. We show the existence of optimal control using Pontryagin’s maximum principle. We solve the optimality system numerically with an algorithm based on forward/backward Runge-Kutta method of the fourth-order. The numerical results indicate that the implementation of the strategy that activates all controls and of type I (starting with the highest controls) is the most cost-effective of the strategies studied. This strategy reduces significantly the number of MDR-TB and XDR-TB cases in all sub-populations, which is an important factor in combating tuberculosis and its resistant strains.