A fractional–order model with prevention and isolation optimal control measures to reduce the transmission of Tuberculosis

In this research paper, we develop a fractional mathematical model consisting of a system of four fractional differential equations (FDEs) utilizing the Caputo operator. This model aims to capture the key epidemiological characteristics of Tuberculosis infection and its transmission dynamics. To assess the well-posedness of the model, we examine the existence, positivity, uniqueness, and boundedness of solutions. We also identify the disease-free and endemic equilibrium points (EPs) and analyze their local stability, alongside conducting a bifurcation analysis. To determine whether the infection is spreading within the population, we calculate the basic reproduction number ((Formula presented.)) and perform a sensitivity analysis to identify the primary epidemiological parameters that influence the proposed model. Furthermore, we implement a fractional optimal control problem (FOCP) for the proposed model utilizing the maximum principle of Pontryagin (PMP). This includes control variables that represent prevention measures against Tuberculosis transmission, such as isolation and protective strategies. We establish the necessary optimality conditions (NOCs) for this FOCP. Numerical simulations are conducted and presented graphically to illustrate the effects of various optimal control strategies on the transmission dynamics of Tuberculosis. The results indicate that all proposed control measures contribute to limiting the spread of the disease to some degree, with the most effective approach being the combination of all control efforts.