Analytical solutions and parameter estimation of the SIR epidemic model

The SIR (Susceptible-Infected-Removed) model is a simple mathematical model of epidemic outbreaks, yet for decades it evaded the efforts of the community to derive an explicit solution. The present manuscript surveys new analytical results about the SIR model. Recently, it was proven that the explicit solution of the model requires the introduction of a new transcendental special function, related to the Wright's Omega function and Lambert W function. Asymptotic methods and iterative numerical routines suitable for parametric estimation of the SIR model are discussed as well. The methods are exemplified with influenza and COVID-19 datasets.