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Abstract

In this work, the fractal-fractional Atangana-Baleanu derivative with the Mittag-Leffler kernel is employed to capture the memory and hereditary effects inherent to anthropogenic cutaneous leishmaniasis transmission dynamics. The Banach fixed-point theorem and contraction mapping principle are used to prove the existence and uniqueness of solutions, while Hyers-Ulam stability of the system is analyzed to demonstrate the robustness of solutions with respect to small perturbations. Using a nonlinear least-squares approach, model parameters and fractional order are estimated using epidemiological data from the World Health Organization. The basic reproduction number $R_0 = 0.53$ indicates that the disease is under control after adding new compartments. The global asymptotic stability of the endemic equilibrium is established using a constructed Lyapunov function under suitable conditions.  Furthermore, a two-step Lagrange interpolation-based fractional Adams-Bashforth scheme for the Atangana-Baleanu operator is used to validate the theoretical findings and to highlight the significant influence of the fractional order on the progression of the epidemic. In order to examine the effects of key parameters and assess disease persistence, numerical simulations are presented, including time-series plots, three-dimensional surfaces, and contour plots.

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