A Solution to the Kermack and McKendrick Integro-Differential Equations

In 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 independent data from the Covid 19 pandemic and derive many previously unknown and useful 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 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 more complex integro-differential equations, that the approximation adequately reproduces the dynamics of the integro-differential equations, and that herd immunity always exists. However, the KMES demonstrates that the SIR models are not adequate representations of the integro-differential equations, and herd immunity is not guaranteed. We suggest that the KMES obsoletes the need for the SIR approximations; and provides a new level of understanding of epidemic dynamics.