  Although modeling studies are focused on the control of SIR-based systems
describing epidemic data sets (particularly the COVID-19), few of them present
a formal dynamic characterization in terms of equilibrium sets and stability.
Such concepts can be crucial to understand not only how the virus spreads in a
population, but also how to tailor government interventions such as social
distancing, isolation measures, etc. The objective of this work is to provide a
full dynamic characterization of SIR-type systems under single-interval control
actions and, based on it, to find the control action that produces the smallest
number of infected individuals at the end of the epidemic that avoids second
wave outbreaks. %Because of its simplicity, the latter result is intended to be
just a reference/baseline for more complex control strategies related to
general nonpharmaceutical measure (\textit{i.e}., those accounting for the
health system capacity, the number of deaths, etc.). Simulations illustrate the
benefits of the aforementioned results in terms of the herd immunity threshold.
