Abstract
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 R0. The proposed formula yields good qualitative agreement already when R0 ⩾ 3 and rapidly becomes quantitatively accurate as larger values of R0 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 can dramatically alter the time scales of the outbreak.
Competing Interest Statement
The authors have declared no competing interest.
Funding Statement
G.K. is a Research Associate with the Belgian Fonds de la Recherche Scientifique (FNRS).
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Footnotes
The work has been extended to include the SEIR model in addition to the SIR model. It contains a new section and new figures specifically dedicated to the SEIR model.
Data Availability
No data used