Fundamental Bound on Epidemic Overshoot in the SIR Model

We derive an exact upper bound on the epidemic overshoot for the Kermack-McKendrick SIR model. This maximal overshoot value of 0.2984... occurs at $R_0^*$ = 2.151... . In considering the utility of the notion of overshoot, a rudimentary analysis of data from the first wave of the COVID-19 pandemic in Manaus, Brazil highlights the public health hazard posed by overshoot for epidemics with $R_0$ near 2. Using the general analysis framework presented within, we then consider more complex SIR models that incorporate vaccination.