  We present a generic epidemic model with stochastic parameters, in which the
dynamics self-organize to a critical state with suppressed exponential growth.
More precisely, the dynamics evolve into a quasi-steady-state, where the
effective reproduction rate fluctuates close to the critical value one, as
observed for different epidemics. The main assumptions underlying the model are
that the rate at which each individual becomes infected changes stochastically
in time with a heavy-tailed steady state. The critical regime is characterized
by an extremely long duration of the epidemic. Its stability is analyzed both
numerically and analytically.
