Analysis of Infected Population Threshold Exceedance in an SIR Epidemiological Model

Abstract

Since the beginning of the COVID-19 outbreak, much attention has been given to the idea of flattening the curve of cases to reduce the harmful effects of an overloaded medical system. In this context, it is relevant to determine conditions to ensure that the health care threshold capacity will not be exceeded. If such a situation is unavoidable, it would be useful to effectively quantify the potential negative impact produced. In this paper, 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 properties of the Lambert W function, we establish necessary and sufficient conditions on the basic reproduction number R₀ to ensure that the infected population I does not exceed M. We also introduce and numerically analyze, five different quantities to measure the impact caused by a possible threshold exceedance.