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 ℛ₀ or of the initial fraction of infected people. Moreover, large epidemics can happen even if ℛ₀<1. But like in a constant environment, the final epidemic size tends to 0 when ℛ₀<1 and the initial fraction of infected people tends to 0. When ℛ₀>1, the final epidemic size is bigger than the fraction 1−1/ℛ₀ of the initially nonimmune population. In summary, the basic reproduction number ℛ₀ 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.