Attack rates of seasonal epidemics

We study the attack rate, that is the total fraction of the population infected each year, for a disease with seasonally varying transmission rate. The attack rate is shown to be governed by both the reproductive number, reflecting the transmissibility of the disease, and the birth rate, which provides a source of new susceptibles. For the case of epidemics which have an annual period (like the seasonality), we prove inequalities which show that the attack rate is close to that of the non-seasonal model, so that it is nearly independent of the strength of the forcing, despite the fact that the shape of the epidemic curve depends strongly on the degree of seasonality of the forcing. Numerical simulations show that this holds to an even stronger extent than is implied by our rigorous results. When the system has subharmonic or chaotic solutions, we show that similar results hold when the attack rate is replaced by the average attack rate over several years. Consequences of these findings for analyzing the effect of vaccination in seasonally-forced models are noted.