Fast-variable reduction to complex contagion unifies epidemic models

Abstract

Across biological and social complex systems, models of epidemics or contagions incorporate complexity through network correlations, adaptive behavior, higher-order interactions, or co-circulating strains. Tracking these mechanisms lead to high-dimensional mathematical systems that resist analytical treatment. We show that a broad class of models admits a systematic reduction via fast-variable elimination. The central insight is that complexity can be captured by fast variables equilibrating much faster than the epidemic, leading to ``emergent complex contagion'' where the force of infection depends on prevalence. The epidemic transition follows a universal equation and is continuous or discontinuous whether the force of infection decays or intensifies with prevalence. Our results reveal diverse epidemic models share a common skeleton such that mechanisms that share a characteristic form can be macroscopically indistinguishable.