The global spread of coronavirus disease 2019 (COVID-19) is overwhelming many health-care systems. As a result, epidemiological models are being used to inform policy on how to effectively deal with this pandemic. The majority of existing models assume random diffusion but do not take into account differences in the amount of interactions between individuals, i.e. the underlying human interaction network, whose structure is known to be scale-free. Here, we demonstrate how this network of interactions can be used to predict the spread of the virus and to inform policy on the most successful mitigation and suppression strategies. Using stochastic simulations in a scale-free network, we show that the epidemic can propagate for a long time at a low level before the number of infected individuals suddenly increases markedly, and that this increase occurs shortly after the first hub is infected. We further demonstrate that mitigation strategies that target hubs are far more effective than strategies that randomly decrease the number of connections between individuals. Although applicable to infectious disease modelling in general, our results emphasize how network science can improve the predictive power of current COVID-19 epidemiological models.