Near-exact explicit asymptotic solution of the SIR model well above the epidemic threshold

Kozyreff, G.

A simple and explicit expression of the solution of the SIR epidemiological model of Kermack and McKendrick is constructed in the asymptotic limit of large basic reproduction numbers $ro$. The proposed formula yields good qualitative agreement already when $rogeq3$ and rapidly becomes quantitatively accurate as larger values of $ro$ are assumed. The derivation is based on the method of matched asymptotic expansions, which exploits the fact that the exponential growing phase and the eventual recession of the outbreak occur on distinct time scales. From the newly derived solution, an analytical estimate of the time separating the first inflexion point of the epidemic curve from the peak of infections is given.