Abstract

I review several special cases of epidemics spreading through random networks which reduce to simple solutions based on ordinary differential equations. This reveals a link between traditional mass action models of epidemics in which contacts are instantaneous and uncorrelated, and networks which have dynamically rearranging ties. I then present several applications to HIV and Influenza. A simple modification to our equations allows us to model sero-sorting of HIV positive individuals, which is the tendency to rearrange relationships to those with matching infection status. Other extensions to the model are explored, including diffusion in networks with clustering (transitivity), and heterogeneous susceptibility and infectiousness. We find that ODE approximations using probability generating functions are more precise and computationally tractable than corresponding systems based on pair approximation.