Analysis of a Fractional Nonlinear SIR Model with Atangana-Baleanu Derivatives

Abstract

We present a fractional nonlinear SIR epidemic model based on the Atangana--Baleanu derivative in the Caputo sense. By incorporating memory and non-local effects, the model offers a more realistic description of disease transmission than classical integer-order formulations. Existence, uniqueness, and Hyers--Ulam stability are established using fixed point theory and generalized Grönwall inequalities, while equilibrium analysis highlights the role of the basic reproduction number. A stable Adams-Bashforth-Moulton predictor--corrector scheme is developed, and numerical experiments confirm accuracy, convergence, and the impact of fractional dynamics on epidemic peaks and persistence. These results underscore the value of fractional operators in epidemiology and point toward integration with artificial intelligence for predictive health modeling.