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

Dynamic characterization of control SIR-type systems and optimal single-interval control