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Journal of Convex Analysis 31 (2024), No. 2, 525--562
Copyright Heldermann Verlag 2024

Infinite Horizon Optimal Control of a SIR Epidemic Under an ICU Constraint

Lorenzo Freddi
Dip. di Scienze Matematiche, Informatiche e Fisiche, Universitá di Udine, Udine, Italy

Dan Goreac
(1) School of Mathematics & Statistics, Shandong University, Weihai, P.R.China
(2) Université Gustave Eiffel, LAMA (UMR 8050), UPEM, UPEC, CNRS, Marne-la-Vallée, France

The aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem of a SIR epidemic on an infinite horizon. A state constraint related to intensive care units (ICU) capacity is imposed and the objective functional linearly depends on the state and the control. After preliminary asymptotic and viability analyses, a Γ-convergence argument is developed to reduce the problem to a finite horizon allowing to use a state constrained version of Pontryagin's theorem to characterize the structure of the optimal controls. Illustrating examples and numerical simulations are given according to the available data on Covid-19 epidemic in Italy.

Keywords: Optimal control, SIR, Gamma-convergence, Pontryagin principle, state constraints, viability, epidemics, infinite horizon.

MSC: 49J45, 49K15, 49K21, 92D30.

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