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Abstract
It has been known that epidemic outbreaks in the SIR model on networks are described
by phase transitions. Despite the similarity with percolation transitions, whether
an epidemic outbreak occurs or not cannot be predicted with probability one in the
thermodynamic limit. We elucidate its mechanism by deriving a simple Langevin equation
that captures an essential aspect of the phenomenon. We also calculate the probability
of epidemic outbreaks near the transition point.