Figure 4

Periodicity of epidemics driven by fluctuations in transmissibility. The figure plots the function σ(τ) = 2^2ax^a(1 - x)^a, with p = 5 and a = 2, where τ measures time in host generations. The value x = pτ - floor(pτ), where floor(pτ) is the largest integer less than or equal to pτ. This definition causes x to increase linearly from zero to one and then be reset to zero, with p cycles in each time unit. We use this function for σ, which is a modified form of the beta probability distribution function, to provide a convenient form of cycling for numerical analysis. We did not use sinusoidal functions for seasonality, because those functions drop too sharply at the end of seasons and rise too rapidly at the start of seasons, rather than having more realistically shaped long, flat interseaonal periods as described by the functions that we used.