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. The KMES can be cast in the form of a step function response to the input of new infections; and that response is the time series of the total infections. 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 diagnosing and managing an epidemic. These include new expressions for the viral load, the final size, the effective reproduction number, and the time to the peak in 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 which are 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, as we explicate in this manuscript, the KMES demonstrates that the SIR models are not adequate representations of the integro-differential equations, and herd immunity is not guaranteed. Our conclusion is that the KMES obsoletes the need for the SIR approximations; and provides a new level of understanding of epidemic dynamics.
Susceptible–infectious–recovered (SIR) models are widely used for estimating the dynamics of epidemics and project that social distancing “flattens the curve”, i.e., reduces but delays the peak in daily infections, causing a longer epidemic. Based on these projections, individuals and governments have advocated lifting containment measures such as social distancing to shift the peak forward and limit societal and economic disruption. Paradoxically, the COVID-19 pandemic data exhibits phenomenology opposite to the SIR models’ projections. Here, we present a new model that replicates the observed phenomenology and quantitates pandemic dynamics with simple and actionable analytical tools for policy makers. Specifically, it offers a prescription of achievable and economically palatable measures for ending an epidemic.One Sentence SummaryThe SIR epidemic models are wrong; a new model offers achievable and economically viable measures for ending an epidemic.
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