  Contagious processes, such as spread of infectious diseases, social
behaviors, or computer viruses, affect biological, social, and technological
systems. Epidemic models for large populations and finite populations on
networks have been used to understand and control both transient and
steady-state behaviors. Typically it is assumed that after recovery from an
infection, every agent will either return to its original susceptible state or
acquire full immunity to reinfection. We study the network SIRI
(Susceptible-Infected-Recovered-Infected) model, an epidemic model for the
spread of contagious processes on a network of heterogeneous agents that can
adapt their susceptibility to reinfection. The model generalizes existing
models to accommodate realistic conditions in which agents acquire partial or
compromised immunity after first exposure to an infection. We prove necessary
and sufficient conditions on model parameters and network structure that
distinguish four dynamic regimes: infection-free, epidemic, endemic, and
bistable. For the bistable regime, which is not accounted for in traditional
models, we show how there can be a rapid resurgent epidemic after what looks
like convergence to an infection-free population. We use the model and its
predictive capability to show how control strategies can be designed to
mitigate problematic contagious behaviors.
