Noise effects in outbreak statistics: large and small fluctuations

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, 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 proportions of a population infected, given both demographic and parameter noise. Our approach reveals that, unlike other well-known examples of large fluctuations occurring in stochastic systems, the statistics of extreme outbreaks emanate from a full continuum of Hamiltonian paths satisfying unique boundary conditions. Moreover, we find that both the outbreak variance and the probabilities for extreme outbreaks depend sensitively on the source of noise.