Determinants of periodicity in seasonally driven epidemics.

Seasonality strongly affects the transmission and spatio-temporal dynamics of many infectious diseases, and is often an important cause for their recurrence. However, there are many open questions regarding the intricate relationship between seasonality and the complex dynamics of infectious diseases it gives rise to. For example, in the analysis of long-term time-series of childhood diseases, it is not clear why there are transitions from regimes with regular annual dynamics, to regimes in which epidemics occur every two or more years, and vice-versa. The classical seasonally-forced SIR epidemic model gives insights into these phenomena but due to its intrinsic nonlinearity and complex dynamics, the model is rarely amenable to detailed mathematical analysis. Making sensible approximations we analytically study the threshold (bifurcation) point of the forced SIR model where there is a switch from annual to biennial epidemics. We derive, for the first time, a simple equation that predicts the relationship between key epidemiological parameters near the bifurcation point. The relationship makes clear that, for realistic values of the parameters, the transition from biennial to annual dynamics will occur if either the birth-rate (μ) or basic reproductive ratio (R(0)) is increased sufficiently, or if the strength of seasonality (δ) is reduced sufficiently. These effects are confirmed in simulations studies and are also in accord with empirical observations. For example, the relationship may explain the correspondence between documented transitions in measles epidemics dynamics and concomitant changes in demographic and environmental factors.