  Most models of epidemic spread, including many designed specifically for
COVID-19, implicitly assume mass-action contact patterns and undirected contact
networks, meaning that the individuals most likely to spread the disease are
also the most at risk to receive it from others. Here, we review results from
the theory of random directed graphs which show that many important quantities,
including the reproduction number and the epidemic size, depend sensitively on
the joint distribution of in- and out-degrees ("risk" and "spread"), including
their heterogeneity and the correlation between them. By considering joint
distributions of various kinds, we elucidate why some types of heterogeneity
cause a deviation from the standard Kermack-McKendrick analysis of SIR models,
i.e., so-called mass-action models where contacts are homogeneous and random,
and some do not. We also show that some structured SIR models informed by
realistic complex contact patterns among types of individuals (age or activity)
are simply mixtures of Poisson processes and tend not to deviate significantly
from the simplest mass-action model. Finally, we point out some possible policy
implications of this directed structure, both for contact tracing strategy and
for interventions designed to prevent superspreading events. In particular,
directed graphs have a forward and backward version of the classic "friendship
paradox" -- forward edges tend to lead to individuals with high risk, while
backward edges lead to individuals with high spread -- such that a combination
of both forward and backward contact tracing is necessary to find
superspreading events and prevent future cascades of infection.
