Risk-evolution forecasting of epidemic transmission with physics-informed neural networks for a modified susceptible–infected–quarantined–recovered–susceptible model

Epidemic transmission involves nonlinear mechanisms, heterogeneous contacts, and multiple delays, which challenges conventional approaches to balance mechanistic fidelity and predictive accuracy. An integrated framework is developed by coupling a modified Susceptible–Infected–Quarantined–Recovered–Susceptible compartmental model with physics-informed neural networks. The model introduces an adaptive nonlinear incidence function and dual delay mechanisms, including incubation delay and decision-to-quarantine delay. The basic reproduction number is derived analytically via the next-generation matrix method, with stability proofs for both disease-free and endemic equilibria. A physics-informed neural-network-based inverse-problem solver is constructed to infer latent states and time-varying parameters by minimizing a composite objective that penalizes both data mismatch and equation residuals, while uncertainty is assessed through repeated training under perturbed observations. Numerical experiments examine how alternative delay values reshape epidemic outcomes, including final size, peak magnitude, and outbreak duration, thereby clarifying the distinct transmission pathways regulated by each delay. Validation with early-phase empirical data from the Wuhan coronavirus disease 2019 outbreak indicates that the framework yields accurate reconstruction of time-varying parameters and reliable short-term trend forecasting, supporting risk-informed decision making and safety evaluation in epidemic scenarios.