Numerical simulations and Hyers-Ulam stability of a novel nonlocal anthropogenic cutaneous leishmaniasis mathematical model
Mathematical Modelling and Numerical Simulation with Applications, vol.6, no.1, pp.26-58, 2026 (Scopus, TRDizin)
- Publication Type: Article / Article
- Volume: 6 Issue: 1
- Publication Date: 2026
- Doi Number: 10.53391/2791-8564.1017
- Journal Name: Mathematical Modelling and Numerical Simulation with Applications
- Journal Indexes: Scopus, Central & Eastern European Academic Source (CEEAS), TR DİZİN (ULAKBİM), Academic Search Ultimate (EBSCO)
- Page Numbers: pp.26-58
- Keywords: existence, fixed point theorems, Hyers-Ulam stability, Mathematical model, uniqueness
- Istanbul Gelisim University Affiliated: Yes
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.