  Compartmental models of epidemics are widely used to forecast the effects of
communicable diseases such as COVID-19 and to guide policy. Although it has
long been known that such processes take place on social networks, the
assumption of random mixing is usually made, which ignores network structure.
However, super-spreading events have been found to be power-law distributed,
suggesting that the underlying networks may be scale free or at least highly
heterogeneous. The random-mixing assumption would then produce an
overestimation of the herd-immunity threshold for given $R_0$; and a (more
significant) overestimation of $R_0$ itself. These two errors compound each
other, and can lead to forecasts greatly overestimating the number of
infections. Moreover, if networks are heterogeneous and change in time,
multiple waves of infection can occur, which are not predicted by random
mixing. A simple SIR model simulated on both Erd\H{o}s-R\'enyi and scale-free
networks shows that details of the network structure can be more important than
the intrinsic transmissibility of a disease. It is therefore crucial to
incorporate network information into standard models of epidemics.
