A Hilfer Fractional Vaccination Epidemic Model: Stability Analysis and a Physics-Informed Neural Network Validated on COVID-19 Data

Abstract

A Hilfer generalized fractional epidemic model with vaccination and asymptomatic transmission is proposed and analysed. The Hilfer operator interpolates between the Riemann-Liouville and Caputo derivatives through a type parameter. Therefore, the model captures hereditary memory and nonlocal temporal effects through both the fractional order and the type. An equivalent Volterra integral formulation is derived and used to prove the existence and uniqueness of solutions by fixed-point theory, together with positivity, boundedness, and a positively invariant region. The disease-free and endemic equilibria are obtained. The basic reproduction number is computed by the next-generation matrix method and local stability is established. Then, a fractional physics-informed neural network is developed on the Volterra residual and verified against a predictor-corrector scheme. The same network is used in inverse mode to calibrate the model against reported COVID-19 incidence for Italy during the autumn 2021 vaccination period, achieving a coefficient of determination of 0.99. Sensitivity analysis and a quantitative comparison with existing fractional approaches round out the study, which offers an accurate and flexible framework for memory-dependent epidemic systems.