  Many pathogens spread primarily via direct contact between infected and
susceptible hosts. Thus, the patterns of contacts or contact network of a
population fundamentally shapes the course of epidemics. While there is a
robust and growing theory for the dynamics of single epidemics in networks, we
know little about the impacts of network structure on long term epidemic or
endemic transmission. For seasonal diseases like influenza, pathogens
repeatedly return to populations with complex and changing patterns of
susceptibility and immunity acquired through prior infection. Here, we develop
two mathematical approaches for modeling consecutive seasonal outbreaks of a
partially-immunizing infection in a population with contact heterogeneity.
Using methods from percolation theory we consider both leaky immunity, where
all previously infected individuals gain partial immunity, and perfect
immunity, where a fraction of previously infected individuals are fully immune.
By restructuring the epidemiologically active portion of their host population,
such diseases limit the potential of future outbreaks. We speculate that these
dynamics can result in evolutionary pressure to increase infectiousness.
