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Contagious processes, such as spread of infectious diseases, socialbehaviors, or computer viruses, affect biological, social, and technologicalsystems. Epidemic models for large populations and finite populations onnetworks have been used to understand and control both transient andsteady-state behaviors. Typically it is assumed that after recovery from aninfection, every agent will either return to its original susceptible state oracquire full immunity to reinfection. We study the network SIRI(Susceptible-Infected-Recovered-Infected) model, an epidemic model for thespread of contagious processes on a network of heterogeneous agents that canadapt their susceptibility to reinfection. The model generalizes existingmodels to accommodate realistic conditions in which agents acquire partial orcompromised immunity after first exposure to an infection. We prove necessaryand sufficient conditions on model parameters and network structure thatdistinguish four dynamic regimes: infection-free, epidemic, endemic, andbistable. For the bistable regime, which is not accounted for in traditionalmodels, we show how there can be a rapid resurgent epidemic after what lookslike convergence to an infection-free population. We use the model and itspredictive capability to show how control strategies can be designed tomitigate problematic contagious behaviors.

Adaptive Susceptibility and Heterogeneity in Contagion Models on Networks