Asymptotic solutions of the SIR and SEIR models well above the epidemic threshold

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 ${\mathsf R_0}$. The proposed formula yields good qualitative agreement already when ${\mathsf R_0}\geq 3$ and rapidly becomes quantitatively accurate as larger values of ${\mathsf R_0}$ 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. Finally, we use the same method on the SEIR model and find that the inclusion of the ‘exposed’ population in the model can dramatically alter the time scales of the outbreak.