A KdV-SIR equation and its analytical solutions: An application for COVID-19 data analysis

Abstract

To describe the time evolution of infected persons associated with an epidemic wave, we recently derived the KdV-SIR equation that is mathematically identical to the Kortewegde Vries (KdV) equation in the traveling wave coordinate and that represents the classical SIR model under a weakly nonlinear assumption. This study further discusses the feasibility of applying the KdV-SIR equation and its analytical solutions together with COVID-19 data in order to estimate a peak time for a maximum number of infected persons. To propose a prediction method and to verify its performance, three types of data were generated based on COVID-19 raw data, using the following procedures: (1) a curve fitting package, (2) the empirical mode decomposition (EMD) method, and (3) the 28-day running mean method. Using the produced data and our derived formulas for ensemble forecasts, we determined various estimates for growth rates, providing outcomes for possible peak times. Compared to other methods, our method mainly relies on one parameter, σo (i.e., a time independent growth rate), which represents the collective impact of a transmission rate (β) and a recovery rate (ν). Utilizing an energy equation that describes the relationship between the time dependent and independent growth rates, our method offers a straightforward alternative for estimating peak times in ensemble predictions.