A Hybrid Integro-Differential Equation And Network Based Model Of Epidemics

A closed form solution of the full Kermack and McKendrick integro-differential equations (Kermack and McKendrick 1927), called the KMES, is presented and verified. The solution is derived by combining network concepts with the integro-differential equations. This solution has two parameters: one describing disease transmissibility and a second characterizing population interactions. The verified solution leads directly to useful, previously unknown, analytical expressions which characterize an epidemic. These include novel expressions for the effective reproduction number, time to peak in new infections, and the final size. Using COVID -19 data from six countries, the transmissibility parameter is estimated and subsequently used to estimate the normalized contagiousness of an individual, a close approximation to viral shedding measured in infected persons. The population interaction parameter is estimated using the Google Residential Mobility Measure. With these parameter estimations, the KMES accurately projects case data from the COVID-19 pandemic in six countries over a 60-day period with R2 values above 0.85. As to performance over longer periods, the KMES projects the Covid-19 total case data from the United States 21 days in advance over an 18-month period with a Mean Absolute Percentage Error of 4.1%. The KMES also accurately identifies the beginnings and peaks of outbreaks within multi month periods in case data from 4 countries.