Minimizing the epidemic final size while containing the infected peak prevalence in SIR systems

Juan Sereno; Alejandro Anderson; Antonio Ferramosca; Esteban A Hernandez-Vargas; Alejandro Hernán González

doi:10.1016/j.automatica.2022.110496

Minimizing the epidemic final size while containing the infected peak prevalence in SIR systems

Automatica (Oxf). 2022 Oct:144:110496. doi: 10.1016/j.automatica.2022.110496. Epub 2022 Jul 30.

Authors

Juan Sereno¹, Alejandro Anderson¹, Antonio Ferramosca², Esteban A Hernandez-Vargas^{3

4}, Alejandro Hernán González¹

Affiliations

¹ Institute of Technological Development for the Chemical Industry (INTEC), CONICET-Universidad Nacional del Litoral (UNL), Guemes 3450, Santa Fe, 3000, Argentina.
² Department of Management, Information and Production Engineering, University of Bergamo, Via Marconi 5, Dalmine (BG), 24044, Italy.
³ Instituto de Matemáticas, UNAM, Boulevard Juriquilla 3001, Querétaro, 76230, Mexico.
⁴ Frankfurt Institute for Advanced Studies, Ruth-Moufang-Str. 1, 60438, Frankfurt am Main, 76230, Germany.

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 ( $I P P$ ) or the epidemic final size ( $E F S$ ). 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 $I P P$ and the $E F S$ , 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 $E F S$ while keeping the $I P P$ 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.

Keywords: Epidemic final size; Herd immunity; Infected peak prevalence; Optimal control; SIR model.