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 − t 2 /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.

Перевод пока недоступен