  A well-known characteristic of pandemics such as COVID-19 is the high level
of transmission heterogeneity in the infection spread: not all infected
individuals spread the disease at the same rate and some individuals
(superspreaders) are responsible for most of the infections. To quantify this
phenomenon requires the analysis of the effect of the variance and higher
moments of the infection distribution. Working in the framework of stochastic
branching processes, we derive an approximate analytical formula for the
probability of an outbreak in the high variance regime of the infection
distribution, verify it numerically and analyze its regime of validity in
various examples. We show that it is possible for an outbreak not to occur in
the high variance regime even when the basic reproduction number $R_0$ is
larger than one and discuss the implications of our results for COVID-19 and
other pandemics.
