Explicit Formulae for the Peak Time of an Epidemic from the SIR Model

Reducing the peak time of an epidemic disease in order for slowing down the eventual dynamics and getting prepared for the unavoidable epidemic wave is utmost significant to fight against the risks of a contagious epidemic disease. To serve to this purpose, the well-documented infection model of SIR is examined in the current research to propose an analytical approach for providing an explicit formula associated with a straightforward computation of peak time of outbreak. Initially, the time scale from the relevant autonomous SIR epidemic model is formulated analytically via an integral based on the fractions of susceptible and infected compartments. Afterwards, through a series expansion of the logarithmic term of the resultant integrand, the peak time is shown to rely upon the fraction of susceptible, the infectious ratio as well as the initial fractions of ill and susceptible individuals. The approximate expression is shown to rigorously capable of capturing the time threshold of illness for an epidemic from the semi-time SIR epidemiology. Otherwise, it is also successful to predict the peak time from a past history of a disease when all-time epidemic model is adopted. Accuracy of the derived expressions are initially confirmed by direct comparisons with recently reported approximate formulas in the literature. Several other epidemic disease samples including the COVID-19 often studied in the recent literature are eventually attacked with favourable performance of the presented formulae for assessing the peak time occurrence of an epidemic. A quick evaluation of the peak time of a disease certainly enables the governments to take early effective epidemic precautions.