A SOLUTION TO THE KERMACK AND MCKENDRICK INTEGRO-DIFFERENTIAL EQUATIONS WHICH ACCURATELY PROJECTS COVID-19 CASE DATA USING GOOGLE MOBILITY DATA AS AN INPUT

Abstract

AbstractIn this manuscript, we derive a closed form solution to the full Kermack and McKendrick integro-differential equations (Kermack and McKendrick 1927) which we call the KMES. We demonstrate the veracity of the KMES using the Google Residential Mobility Measure to accurately project case data from the Covid 19 pandemic and we derive many useful, but previously unknown, analytical expressions for characterizing and managing an epidemic. These include expressions for the viral load, the final size, the effective reproduction number, and the time to the peak in infections. The KMES can also be cast in the form of a step function system response to the input of new infections; and that response is the time series of total infections.Since the publication of Kermack and McKendrick’s seminal paper (1927), thousands of authors have utilized the Susceptible, Infected, and Recovered (SIR) approximations; expressions putatively derived from the integro-differential equations to model epidemic dynamics. Implicit in the use of the SIR approximation are the beliefs that there is no closed form solution to the integro-differential equations, and that the approximation is a special case which adequately reproduces the dynamics of the integro-differential equations mapped onto the physical world. However, the KMES demonstrates that the SIR approximations are not adequate representations of the integro-differential equations, and we therefore suggest that the KMES obsoletes the need for the SIR approximations by providing not only a new mathematical perspective, but a new understanding of epidemic dynamics.