On the Final Size of Epidemics with Seasonality

We first study an SIR system of differential equations with periodic coefficients describing an epidemic in a seasonal environment. Unlike in a constant environment, the final epidemic size may not be an increasing function of the basic reproduction number R-0 or of the initial fraction of infected people. Moreover, large epidemics can happen even if R-0 < 1. But like in a constant environment, the final epidemic size tends to 0 when R-0 < 1 and the initial fraction of infected people tends to 0. When R-0 > 1, the final epidemic size is bigger than the fraction 1-1/R-0 of the initially nonimmune population. In summary, the basic reproduction number R-0 keeps its classical threshold property but many other properties are no longer true in a seasonal environment. These theoretical results should be kept in mind when analyzing data for emerging vector-borne diseases (West-Nile, dengue, chikungunya) or air-borne diseases (SARS, pandemic influenza); all these diseases being influenced by seasonality.