Exploring the bifurcation and stability analysis of the malaria epidemic model and their environmental impacts; a scheme of piecewise modified ABC fractional derivative

Malaria remains a life-threatening disease and a major global health challenge, particularly in tropical and subtropical regions. In this study, we introduce a novel approach by applying the piecewise modified Atangana-Baleanu-Caputo (mABC) fractional derivative to a malaria transmission model. This operator seamlessly integrates the classical derivative with the modified Atangana-Baleanu operator in the Caputo sense. We divide the interval [0, f2], with f2 is an element of & Ropf;, into two segments: the classical derivative is applied in [0, f1], and the modified differential operator is used in [f1, f2]. This results in the development of the piecewise mABC operator and its corresponding integral. By incorporating this new operator into a malaria model, we explore the crossover behaviors within the system. Our analysis addresses the existence of solutions, the invariant region, the basic reproduction number, bifurcation analysis, sensitivity analysis, and the stability of solutions for the nonlinear piecewise mABC malaria model. To support our theoretical findings, we conduct numerical simulations using a scheme based on Lagrange's interpolation polynomial and compare these results with existing data, providing a deeper understanding of malaria transmission dynamics and the potential implications of the piecewise mABC operator in modeling infectious diseases.