Mathematical model with fractional order derivatives for Tuberculosis taking into account its relationship with HIV/AIDS and Diabetes

In this paper, we present a mathematical model for the study of resistance to tuberculosis treatment using fractional derivatives in the Caputo sense. This model takes into account the relationship between Tuberculosis, HIV/AIDS, and diabetes and differentiates resistance cases into MDR-TB (multidrug-resistant tuberculosis) and XDR-TB (extensively drug-resistant tuberculosis). We present the basic results associated with the model and study the behavior of the disease-free equilibrium points in the different sub-populations, TB-Only, TB-HIV/AIDS, and TB-Diabetes. We performed computational simulations for different fractional orders (α-values) using an Adams-Bashforth-Moulton type predictor-corrector PECE method. Among the results obtained, we have that the MDR-TB cases in all sub-populations decrease at the beginning of the study for the different α-values. In XDR-TB cases in the TB-Only sub-population, there is a decrease in the number of cases. XDR-TB cases in the TB-HIV/AIDS sub-population have differentiated behavior depending on α. This knowledge helps to design an effective control strategy. The XDR-TB cases in diabetics increased throughout the study period and outperformed all resistant compartments for the different α-values. We recommend special attention to the control of this compartment due to this growth.