Dynamical analysis of heterogeneous epidemic models with discrete and continuous contact variation

Discrete representations of contact heterogeneity have been widely studied in epidemic modeling, but their limitations in capturing contact-level variation motivate continuous formulations. While recent extensions to continuous heterogeneity yield compartmental systems with strongly nonlinear terms in closed populations, key dynamical properties of such models remain unresolved. We develop a network-inspired framework that constructs continuous epidemic models through discrete-like deconvolution and assortative mixing approximations, employing moment generating function to reformulate them into tractable, edge-analogous compartmental systems for complete dynamical characterization. This approach enables us to derive analytical solutions for compartmental models with strongly nonlinear terms under various contact heterogeneity distributions, explicitly obtaining the basic reproduction number, equilibria along with their stability, as well as the final size and duration. Comprehensive numerical validation confirms consistency between theoretical predictions and simulated epidemic progression across all models. The framework reveals how contact heterogeneity and approximation methods influence epidemic progression, offering new insights and methodological foundations for modeling heterogeneous disease transmission.