Incommensurate fractional epidemic dynamics with compartment-specific memory, treatment response, and physics-informed forecasting

This paper develops an incommensurate fractional-order susceptible-exposed-symptomatic infectious-asymptomatic infectious-treated-recovered (SEIATR) model with compartment-specific Caputo memory. We establish local well-posedness, positivity, boundedness, biological feasibility, disease-free and endemic equilibria, and the basic reproduction number. Local stability is analyzed through the rational-order incommensurate Matignon criterion. A componentwise multi-order L1 scheme is then used for simulation, refinement, and sensitivity analysis. Finally, a physics-informed neural network is employed as a synthetic benchmark for hidden-state reconstruction, parameter and order identification, residual checking, and short-horizon forecasting. The results show that compartment-specific memory mainly affects peak timing, persistence, and treatment load, whereas epidemiological rates govern amplification and threshold behavior.