We consider the spread of infectious disease through contact networks of Configuration Model type. We assume that the disease spreads through contacts and infected individuals recover into an immune state. We discuss a number of existing mathematical models used to investigate this system, and show relations between the underlying assumptions of the models. In the process we offer simplifications of some of the existing models. The distinctions between the underlying assumptions are subtle, and in many if not most cases this subtlety is irrelevant. Indeed, under appropriate conditions the models are equivalent. We compare the benefits and disadvantages of the different models, and discuss their application to other populations (e.g., clustered networks). Finally we discuss ongoing challenges for network-based epidemic modeling.
Keywords: Configuration Model Networks; Edge-based Compartmental Models; Effective Degree; Epidemic; Network; Pairwise; SIR disease.