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.