Dynamical Analysis and SDEPINN-Based Modeling of a Fractional-Order Stochastic SIS Epidemic Model

This paper proposes a novel susceptible–infected–susceptible (SIS) epidemic model incorporating a fractional-order term and white-noise perturbations. Several analytical results concerning its dynamical behavior are established. Firstly, we prove the existence and uniqueness of model solutions using the Carathéodory approximation. Secondly, the extinction of disease is rigorously proven based on the properties of the quadratic function. Meanwhile, the Ulam–Hyers stability of the model is derived by using stochastic Gronwall-type inequalities and the stochastic analysis techniques. Furthermore, the solution of the proposed fractional-order stochastic model can be approximated by that of the corresponding averaged model under suitable averaging conditions. Then, numerical simulations are conducted to illustrate the effects of key parameters on disease extinction, stability, and long-term dynamical behavior. To improve its adaptability to real-world epidemic dynamics, this paper embeds dynamical constraints into neural networks and constructs a stochastic physics-informed identification framework for seasonal infectious diseases. Empirical results based on monthly influenza data from Xinjiang show that the proposed framework can capture epidemic trends, seasonal peaks, and fitting uncertainty. These results provide a theoretically grounded and practically applicable approach for infectious disease modeling under memory effects and stochastic perturbations.