The final size of a nearly critical epidemic, and the first passage time of a Wiener process to a parabolic barrier

The distribution of the final size, K, in a general SIR epidemic model is considered in a situation when the critical parameter λ is close to 1. It is shown that with a ‘critical scaling’ λ ≈ 1 + a/n^1/3, m ≈ bn^1/3, where n is the initial number of susceptibles and m is the initial number of infected, then K/n^2/3 has a limit distribution when n → ∞. It can be described as that of T, the first passage time of a Wiener process to a parabolic barrier b + at − t²/2. The proof is based on a diffusion approximation. Moreover, it is shown that the distribution of T can be expressed analytically in terms of Airy functions using the spectral representation connected with Airy's differential equation.