  Motivated by recent epidemic outbreaks, including those of COVID-19, we solve
the canonical problem of calculating the dynamics and likelihood of extensive
outbreaks in a population within a large class of stochastic epidemic models
with demographic noise, including the Susceptible-Infected-Recovered (SIR)
model and its general extensions. In the limit of large populations, we compute
the probability distribution for all extensive outbreaks, including those that
entail unusually large or small (extreme) proportions of the population
infected. Our approach reveals that, unlike other well-known examples of rare
events occurring in discrete-state stochastic systems, the statistics of
extreme outbreaks emanate from a full continuum of Hamiltonian paths, each
satisfying unique boundary conditions with a conserved probability flux.
