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 Erdős-Rényi 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.