Fractal–Fractional Modeling of SEIR Epidemic Dynamics Using the Atangana–Baleanu Derivative: Existence, Ulam–Hyers Stability, and Numerical Simulations

This paper introduces a fractal–fractional SEIR epidemic model based on the Atangana–Baleanu derivative in the Caputo sense augmented by fractal scaling. The fractional order α∈(0,1] captures memory effects while the fractal dimension β∈(0,1] accounts for irregular contact networks. We prove global existence, uniqueness, positivity, and boundedness of solutions via fixed-point arguments and establish global Ulam–Hyers stability. An adapted second-order Adams–Bashforth–Moulton predictor-corrector scheme with explicit weights is derived and verified. Numerical simulations across representative (α,β) pairs reveal that decreasing either parameter delays epidemic peaks, reduces peak intensity (with β exerting a stronger damping effect), prolongs tails, and induces irregular oscillations—features absent from classical or pure-fractional SEIR models. These results provide a rigorous and reproducible framework for forecasting emerging infections in heterogeneous populations and carry direct implications for targeted public health interventions.