A Hybrid AI-Mathematical approach for epidemic threshold prediction in metapopulation networks: Integrating physics-guided neural networks with spectral graph theory.

Predicting the epidemic threshold [Formula: see text] in contact networks is a central challenge in computational epidemiology. Classical structural approaches based on spectral graph theory-most notably Quenched Mean-Field (QMF) and the recently proposed KSEL (K Spectral Energy of Laplacian) method-deliver fast but approximate predictions. We propose a hybrid AI-mathematical framework that integrates spectral graph features, epidemiological parameters, and epidemiologically motivated soft constraints derived from compartmental theory into a physics-guided neural network (PGNN). Rather than claiming state-of-the-art predictive performance, this work makes three complementary contributions: (i) a rigorous stochastic ground-truth estimation procedure with Monte Carlo uncertainty quantification (median [Formula: see text], n = 775 networks); (ii) a systematic comparative evaluation of seven methods-including tree-based models (Random Forest, Gradient Boosting) trained on the same feature set-revealing the conditions under which deep learning surpasses and falls short of simpler baselines; and (iii) a full ablation study and SHAP interpretability analysis identifying the role of individual spectral features and physical constraints as structured regularisers. Evaluated on 775 synthetic networks spanning Erdős-Rényi, Barabási-Albert, Watts-Strogatz, and regular topologies, Gradient Boosting achieves the best predictive accuracy (R2 = 0.908, RMSE = 0.0731), while the PGNN (R2 = 0.093) offers complementary value through physical consistency and interpretability. These results are established on synthetic benchmarks; application to empirical contact networks (hospital, school, workplace settings) is a natural next step but requires dedicated validation beyond the scope of the present study. Ablation results show that the boundedness constraint is a beneficial regulariser while the stability constraint over-regularises in the low-data regime. Automatic gradient-based calibration of the KSEL coefficient yields topology-dependent optimal values ([Formula: see text]), substantially departing from the universal constant k = 0.3 of prior work.