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We consider an epidemiological SIR model and a positive threshold $M$. Usinga parametric expression for the solution curve of the SIR model and the LambertW function, we establish necessary and sufficient conditions on the basicreproduction number $\mathcal{R}_0$ to ensure that the infected population doesnot exceed the threshold $M$. We also propose and analyze different measures toquantify a possible threshold exceedance.

Analysis of infected population threshold exceedance in an SIR epidemiological model