A Two Parameter Closed Form Solution to Kermack and McKendrick’s Integro-Differential Equations

A closed form solution of the full integro-differential equations in the 1927 paper by Kermack and McKendrick (K&M), called herein the KMES, is presented and verified. The solution arises from the use of the time-based weighted averages of K&M’s rate parameters to transform their equations into equivalent, solvable differential equations. Then, by using a network formulation, we find functional forms of the time varying parameters of these new equations, and derive the total case count as the integral of the product of the probability rate of transmission and the number of infectious contacts. Analytical expressions for managing an epidemic, the real-time effective reproduction number, time to peak in new infections, and the final epidemic size, flow directly from the solution. Notably, although the KMES is derived from K&M’s equations, the expressions for time-to-peak and final size are antithetical to currently accepted epidemic concepts The KMES has only two parameters: the basic reproduction number, and the probability rate of disease transmission, both of which can be readily derived from early epidemic data. Using early COVID-19 pandemic data from six different countries to estimate these two parameters, the KMES accurately projects case data from pandemic in the following 30 & 60-day periods with R 2 values >0.93 and 0.78 respectively. In addition, a projection of a typical individual’s contagiousness, derived from the KMES, closely approximates the time course of viral shedding measured in infected persons.