Analysis of Piecewise Terminal Fractional System: Theory and Application to TB Treatment Model with Drug Resistance Development

Researchers have devised numerous methods to model intricate behaviors in phenomena that unfold in multiple stages. This work focuses on a specific category of piecewise hybrid terminal systems characterized by delay. To account for hereditary memory effects, which are absent in standard integer-order systems, our framework partitions the time interval into two distinct phases. The initial phase employs the classical derivative, while the subsequent phase utilizes the Atangana–Baleanu–Caputo (ABC) fractional derivative. We establish conditions that guarantee both the existence and uniqueness of solutions through the application of suitable fixed-point arguments. Furthermore, Hyers–Ulam (H-U) stability is investigated to ascertain the robustness and reliability of the derived solutions. To exemplify these theoretical findings, we present a fractional-order tuberculosis treatment model that incorporates the development of drug resistance, alongside a general numerical example. Numerical simulations reveal that changes in the fractional order influence the dynamic behavior of the disease.