A highly accurate peak time formula of epidemic outbreak from the SIR model


Chinese Journal of Physics, vol.84, pp.39-50, 2023 (SCI-Expanded) identifier

  • Publication Type: Article / Article
  • Volume: 84
  • Publication Date: 2023
  • Doi Number: 10.1016/j.cjph.2023.05.009
  • Journal Name: Chinese Journal of Physics
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, INSPEC, zbMATH
  • Page Numbers: pp.39-50
  • Keywords: Analytical formula, COVID-19, Pade approximant, Peak time of an infection, SIR model
  • Hacettepe University Affiliated: Yes


Forecasting the epidemic peak time right from the origination of a disease is vital to take over dynamical behaviour of its spread over time. The decision of isolation, social distance and lock down strategic progresses does all rely on an accurate prediction of the peak time so that reduction of the time of peak or of the infected size of population will be made possible. Therefore, recent efforts concentrated on deriving elaborative and analytically accessible expressions representing the peak time of the infected compartment from the classical SIR epidemic mathematical model. In this research paper, two closed-form formulae are introduced to yield a straightforward computation of peak time of an infectious disease with no restrictions on the SIR quantities. In addition to this, the calculations can be implemented on a usual calculator, without requiring the use of advanced mathematical functions, having provided the initial fractions of infected and susceptible populations as well as the recovery to infectious ratio. A comparison including the COVID-19 data is fulfilled with the very recent formulas available in the open literature. With the proposed new scalings, evaluation of the peak time is reduced only to two parameter space and the accuracy of the present formulas in reduced form is ultimately confirmed yielding an error of order of magnitude 10−4 valid for the complete regime of the set of SIR model parameters. Even in the case of an endemic, the past peak time of the illness can also be captured accurately by the given formulae. Two simple approximations in terms of usual geometric series are also provided. These can be safely used with a pocket calculator without sophisticated laboratory equipments.