Abstract
Mathematical models are critical to understand the spread of pathogens in a population and evaluate the effectiveness of non-pharmaceutical interventions (NPIs). A plethora of optimal strategies has been recently developed to minimize either the infected peak prevalence ( IPPIPP<math><mrow><mi>I</mi> <mi>P</mi> <mi>P</mi></mrow> </math> ) or the epidemic final size ( EFSEFS<math><mrow><mi>E</mi> <mi>F</mi> <mi>S</mi></mrow> </math> ). While most of them optimize a simple cost function along a fixed finite-time horizon, no consensus has been reached about how to simultaneously handle the IPPIPP<math><mrow><mi>I</mi> <mi>P</mi> <mi>P</mi></mrow> </math> and the EFSEFS<math><mrow><mi>E</mi> <mi>F</mi> <mi>S</mi></mrow> </math> , while minimizing the intervention's side effects. In this work, based on a new characterization of the dynamical behaviour of SIR-type models under control actions (including the stability of equilibrium sets in terms of herd immunity), we study how to minimize the EFSEFS<math><mrow><mi>E</mi> <mi>F</mi> <mi>S</mi></mrow> </math> while keeping the IPPIPP<math><mrow><mi>I</mi> <mi>P</mi> <mi>P</mi></mrow> </math> controlled at any time. A procedure is proposed to tailor NPIs by separating transient from stationary control objectives: the potential benefits of the strategy are illustrated by a detailed analysis and simulation results related to the COVID-19 pandemic.