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 $t\to \infty$ 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.

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• Bifurcations
• Complex systems
• Discontinuous phase transition
• Epidemic
• Network phase transitions
• Physics & society
• Spreading