  The basic reproductive number -- $R_0$ -- is one of the most common and most
commonly misapplied numbers in public health. Although often used to compare
outbreaks and forecast pandemic risk, this single number belies the complexity
that two different pathogens can exhibit, even when they have the same $R_0$.
Here, we show how to predict outbreak size using estimates of the distribution
of secondary infections, leveraging both its average $R_0$ and the underlying
heterogeneity. To do so, we reformulate and extend a classic result from random
network theory that relies on contact tracing data to simultaneously determine
the first moment ($R_0$) and the higher moments (representing the
heterogeneity) in the distribution of secondary infections. Further, we show
the different ways in which this framework can be implemented in the
data-scarce reality of emerging pathogens. Lastly, we demonstrate that without
data on the heterogeneity in secondary infections for emerging infectious
diseases like COVID-19, the uncertainty in outbreak size ranges dramatically.
Taken together, our work highlights the critical need for contact tracing
during emerging infectious disease outbreaks and the need to look beyond $R_0$
when predicting epidemic size.
