Predator-prey oscillations can shift when diseases become endemic.

In epidemiology, knowing when a disease is endemic is important. This is usually done by finding the basic reproductive number, R(0), using equilibrium-based calculations. However, oscillatory dynamics are common in nature. Here, we model a disease with density dependent transmission in an oscillating predator-prey system. The condition for disease persistence in predator-prey cycles is based on the time-average density of the host and not the equilibrium density. Consequently, the time-averaged basic reproductive number R(0)¯ is what determines whether a disease is endemic, and not on the equilibrium-based basic reproductive number R(0)(*). These findings undermine any R(0) analysis based solely on steady states when predator-prey oscillations exist for density dependent diseases.