Extreme outbreak statistics in stochastic populations: large and small fluctuations*

Population dynamics share certain general nonlinear dynamical properties and topology which can be found in human driven disease epidemics, population inversions in lasers, as well as chemical kinetics. In all cases, due to the interactions between individuals, the presence of noise and the underlying nonlinear dynamical topology, there exists the potential for large fluctuations. Here we consider the general problem of calculating the dynamics and likelihood of extensive large transient fluctuations in general stochastic populations for a large class of population models, including outbreaks in the susceptible-infected-recovered (SIR) model and intensity fluctuations in class B lasers. In the limit of large populations, we compute the probability distribution for all extensive outbreaks including those that entail unusually large or small proportions compared to the mean of a population, given both internal 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 optimal paths satisfying unique boundary conditions. Moreover, we find that both the variance and the probabilities for extreme outbreaks depend sensitively on the source of noise.