  Uncertainty can be classified as either aleatoric (intrinsic randomness) or
epistemic (imperfect knowledge of parameters). The majority of frameworks
assessing infectious disease risk consider only epistemic uncertainty. We only
ever observe a single epidemic, and therefore cannot empirically determine
aleatoric uncertainty. Here, we characterise both epistemic and aleatoric
uncertainty using a time-varying general branching process. Our framework
explicitly decomposes aleatoric variance into mechanistic components,
quantifying the contribution to uncertainty produced by each factor in the
epidemic process, and how these contributions vary over time. The aleatoric
variance of an outbreak is itself a renewal equation where past variance
affects future variance. We find that, superspreading is not necessary for
substantial uncertainty, and profound variation in outbreak size can occur even
without overdispersion in the offspring distribution (i.e. the distribution of
the number of secondary infections an infected person produces). Aleatoric
forecasting uncertainty grows dynamically and rapidly, and so forecasting using
only epistemic uncertainty is a significant underestimate. Therefore, failure
to account for aleatoric uncertainty will ensure that policymakers are misled
about the substantially higher true extent of potential risk. We demonstrate
our method, and the extent to which potential risk is underestimated, using two
historical examples.
