Better methods to predict and prevent the emergence of zoonotic viruses could support future efforts to reduce the risk of epidemics. We propose a network science framework for understanding and predicting human and animal susceptibility to viral infections. Related approaches have so far helped to identify basic biological rules that govern cross-species transmission and structure the global virome. We highlight ways to make modelling both accurate and actionable, and discuss the barriers that prevent researchers from translating viral ecology into public health policies that could prevent future pandemics.