Bistability and Resurgent Epidemics in Reinfection Models

Spreading processes that propagate through local interactions have been studied in multiple fields (e.g., epidemiology, complex networks, social sciences) using the susceptible-infected-recovered (SIR) and susceptible-infected-susceptible (SIS) frameworks. SIR assumes individuals acquire full immunity to the infection after recovery, while SIS assumes individuals acquire no immunity after recovery. However, in many spreading processes individuals may acquire only partial immunity to the infection or may become more susceptible to reinfection after recovery. We study a model for reinfection called Susceptible-Infected-Recovered-Infected (SIRI). The SIRI model generalizes the SIS and SIR models and allows for study of systems in which the susceptibility of agents changes irreversibly after first exposure to the infection. We show that when the rate of reinfection is higher than the rate of primary infection, the SIRI model exhibits bistability with a small difference in the initial fraction of infected individuals determining whether the infection dies out or spreads through the population. We find this critical value and show that when the infection does not die out there is a resurgent epidemic in which the number of infected individuals decays initially and remains at a low level for an arbitrarily long period of time before rapidly increasing toward an endemic equilibrium in which the fraction of infected individuals is non-zero.