  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.
