Temporal persistence thresholds for discontinuous higher-order contagion
Abstract
Higher-order contagion models show that group interactions can create tipping points: small outbreaks die out, while larger ones persist. In temporal systems, group interactions do not simply exist or not exist; they appear, disappear, and sometimes persist over time. This paper investigates when that persistence is strong enough to matter qualitatively. A minimal SIS contagion model with ordinary pairwise transmission and an additional triadic reinforcement mechanism is studied. The key assumption is that higher-order transmission requires a candidate triangle to be active in two consecutive observation windows. This gives a persistence-dependent reinforcement term derived from an explicit microscopic rule. Under a homogeneous mean-field closure, closed-form expressions are obtained for the bistability boundary, the critical persistence threshold, and the unstable seed-size threshold. The result is an analytical criterion showing when repeated small-group interactions are persistent enough to generate tipping-point epidemic behavior in the model. Finite-population simulations reproduce the persistence-controlled transition qualitatively, while explicit sparse-network simulations reveal a systematic reduction in realized triadic exposure compared to the homogeneous closure.