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 lambda is close to 1. It is shown that with a 'critical scaling' lambda approximate to 1 + a/n(1/3), m approximate to bn(1/3), where a is the initial number of susceptibles and ra is the initial number of infected, then K/n(2/3) has a limit distribution when n --> infinity. 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.