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The SIR (Susceptible–Infected–Removed) is one of the simplest models for epidemic outbreaks. The present paper derives a novel, simple, analytical asymptotic solution for the I-variable, which is valid on the entire real line. Connections with the Gompertz and Gumbel distributions are also demonstrated. The approach is applied to the ongoing coronavirus disease 2019 (COVID-19) pandemic in four European countries — Belgium, Italy, Sweden, and Bulgaria. The reported raw incidence data from the outbreaks in 2020–2021 have been fitted using constrained least squares. It is demonstrated that the asymptotic solution can be used successfully for parametric estimation either in stand-alone mode or as a preliminary step in the parametric estimation using numerical inversion of the exact parametric solution. [ABSTRACT FROM AUTHOR]