A Neural Network Approach to Fractional-Order Reaction–Diffusion Epidemic Model Based on SEIAR Dynamics

This study presents the SEIAR epidemic model within the Caputo time-fractional derivative of order α and a diffusion operator in one dimension. The modified formulation also contains two measures of effort; u1, which concerns the adoption of infection prevention measures, such as taking health precautions in gatherings and working to decrease social connections, and u2, which concerns the ability of susceptible individuals to be vaccinated. A rigorous theoretical analysis of the fractional order model (FOM) is conducted, including the existence, positivity, and boundedness of projected solutions. The system behavior is examined by evaluating equilibrium points (EPs); the disease-free equilibrium (DFE) is locally asymptotically stable (LAS) and globally asymptotically stable (GAS) if the effective reproduction number (ERN) Re 1. The forward sensitivity index formula is used to describe the sensitivity analysis of the ERN parameters. The dynamics of the susceptible, symptomatic, and asymptomatic compartments are obtained through several numerical simulations. Responses to the proposed scenarios for u1 and u2 are depicted in 3D and 2D plots. Furthermore, the “memory effect†of the Caputo fractional operator demonstrates faster convergence to the steady state compared to the classical case α=1. The neural network (NN) is employed as a data-driven surrogate solver to increase the efficiency of computation and provide an independent numerical output. The Levenberg –Marquardt algorithm was used to train the NN. Finally, the best validation performance (BVP), regression, and error histogram (EH) are presented to confirm the accuracy and efficiency of the suggested NN results.