Phase transition in the susceptible-infected model on hypernetworks

We derive the master equations for the Susceptible-Infected (SI) model on general hypernetworks with N-body interactions. We solve these equations exactly for infinite d-regular hypernetworks, and obtain an explicit solution for the expected infection level as a function of time. The solution shows that the epidemic spreads out to the entire population as t→∞ if and only if the initial infection level exceeds a positive threshold value. This phase transition is a high-order interaction effect, which is absent with pairwise interactions.