Stable Neural-Augmented SEIR Models for Reliable Epidemic Forecasting: Theory and Applications

Introduction: Neural-augmented compartmental epidemic models extend classical SEIR dynamics with flexible neural corrections that capture unmodeled heterogeneities such as behavioural adaptation, seasonal forcing, and reporting artifacts while retaining mechanistic interpretability. This hybridization raises foundational questions about well-posedness, equilibrium structure, stability of disease-free and endemic states, robustness of reproduction-number thresholds, and parameter identifiability. Methods: A neural-augmented SEIR framework is developed under mild continuity, Lipschitz, and boundedness assumptions on the neural terms. The analysis combines classical ordinary differential equation theory with Lyapunov stability methods, Jacobian-based linearization and perturbation bounds, and structural identifiability techniques formulated in terms of effective, neural-modified parameters and reproduction numbers. Results: Global existence and uniqueness of solutions, positivity and forward invariance of the epidemiological state space, and persistence of both disease-free and endemic equilibria under neural-modified effective parameters are established. Sharp Lyapunov- and Jacobian-based conditions are derived that guarantee robust local, global, and fixed-time stability in the presence of neural perturbations and delineate when neural augmentation preserves or distorts classical reproduction-number thresholds. Under suitable observability assumptions, mechanistic parameters are shown to be structurally identifiable, while neural adaptations are identifiable at the functional level but not at the level of individual network weights. Numerical SEIR case studies illustrate that small neural corrections preserve classical thresholds, moderate perturbations yield robustly stable dynamics, and large corrections can induce oscillatory outbreaks. Conclusion: The results provide a rigorous mathematical foundation for neural-augmented epidemic models, demonstrating that stability and interpretability can be preserved when neural components are appropriately constrained and offering explicit regularization criteria for safe use of hybrid mechanistic–neural models in epidemic forecasting and inference.