Adaptive-memory fractional SIR dynamics via Hermite wavelet spectral methods

The ongoing emergence of infectious diseases has exposed key limitations in classical epidemic models, particularly the standard SIR framework, which fails to account for memory effects that influence real-world disease spread. In this study, we extend the traditional model by incorporating fractional-order dynamics using the Caputo derivative, allowing the inclusion of non-local temporal interactions that better capture disease transmission and recovery behavior. To solve the resulting system efficiently, we apply a high-precision Hermite wavelet-based numerical method, which offers fast convergence and excellent computational accuracy. Our fractional SIR model demonstrates improved realism by reproducing delayed infection peaks and prolonged recovery periods often observed in epidemic data. Furthermore, we introduce a symbolic generalization in which the fractional order $ \tau $ varies with time, reflecting adaptive memory behavior as an epidemic evolves. This addition enhances the model's flexibility and aligns more closely with dynamic public health scenarios. Overall, the proposed framework significantly improves the predictive capabilities of epidemic models and supports more informed intervention planning.