  We consider an epidemiological SIR model and a positive threshold $M$. Using
a parametric expression for the solution curve of the SIR model and the Lambert
W function, we establish necessary and sufficient conditions on the basic
reproduction number $\mathcal{R}_0$ to ensure that the infected population does
not exceed the threshold $M$. We also propose and analyze different measures to
quantify a possible threshold exceedance.
