  Mathematical models are instrumental to forecast the spread of pathogens and
to evaluate the effectiveness of non-pharmaceutical measures. A plethora of
optimal strategies has been recently developed to minimize either the infected
peak prevalence (IPP) or the epidemic final size (EFS). While most of the
control strategies optimize a simple cost function along a fixed finite-time
horizon, no consensus has been reached about how to simultaneously handle the
IPP, the EFS, and the avoiding of new cycles of infections rebounding. In this
work, based on the characterization of the dynamical behaviour of SIR-type
models under control actions (including the stability of equilibrium sets, in
terms of the herd immunity), it is studied how to minimize the EFS while
keeping - at any time - the IPP controlled. A procedure is proposed to tailor
non-pharmaceutical interventions by separating transient from stationary
control objectives and the potential benefits of the strategy are illustrated
by a detailed analysis and simulation results related to the COVID-19 pandemic.
