The COVID-19 pandemic has demonstrated how disruptive emergent disease outbreaks can be and how useful epidemic models are for quantifying risks of local outbreaks. Here we develop an analytical approach to calculate the dynamics and likelihood of outbreaks within the canonical Susceptible-Exposed-Infected-Recovered and more general models, including COVID-19 models, with fixed population sizes. We compute the distribution of outbreak sizes including extreme events, and show that each outbreak entails a unique, depletion or boost in the pool of susceptibles and an increase or decrease in the effective recovery rate compared to the mean-field dynamics -- due to finite-size noise. Unlike extreme events occurring in long-lived metastable stochastic systems, the underlying outbreak distribution depends on a full continuum of optimal paths, each connecting two unique non-trivial fixed-points, and thus represents a novel class of extreme dynamics.
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