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Infinite Horizon Optimal Control of a SIR Epidemic Under an ICU Constraint
No full textThe 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
Corrigendum to “Optimal control of a SIR epidemic with ICU constraints and target objectives” (Applied Mathematics and Computation (2022) 418, (S0096300321008997), (10.1016/j.amc.2021.126816))
No full textThe authors regret that in the printed version of the above article, some figures (those corresponding to Scenario 2 in Section 6) are missed. We believe that this mistake occurred during the publication process. The missed pictures are the following. [Formula presented] The correct and final version follows. The authors would like to apologise for any inconvenience caused
Optimality of Vaccination for an SIR Epidemic with an ICU Constraint
Full text linkThis paper studies an optimal control problem for a class of SIR epidemic models, in scenarios in which the infected population is constrained to be lower than a critical threshold imposed by the intensive care unit (ICU) capacity. The vaccination effort possibly imposed by the health-care deciders is classically modeled by a control input affecting the epidemic dynamic. After a preliminary viability analysis, the existence of optimal controls is established, and their structure is characterized by using a state-constrained version of Pontryagin’s theorem. The resulting optimal controls necessarily have a bang-bang regime with at most one switch. More precisely, the optimal strategies impose the maximum-allowed vaccination effort in an initial period of time, which can cease only once the ICU constraint can be satisfied without further vaccination. The switching times are characterized in order to identify conditions under which vaccination should be implemented or halted. The uniqueness of the optimal control is also discussed. Numerical examples illustrate our theoretical results and the corresponding optimal strategies. The analysis is eventually extended to the infinite horizon by Γ-convergence arguments
Optimal control of a SIR epidemic with ICU constraints and target objectives
No full textThe aim of this paper is to provide a rigorous mathematical analysis of an optimal control problem with SIR dynamics main feature of our study is the presence of state constraints (related to intensive care units ICU capacity) and strict target objectives (related to the immunity threshold). The first class of results provides a comprehensive description of different zones of interest using viability tools. The second achievement is a thorough mathematical analysis of Pontryagin extremals for the aforementioned problem allowing to obtain an explicit closed-loop feedback optimal control. All our theoretical results are numerically illustrated for a further understanding of the geometrical features and scenarios
Linear models for composite thin-walled beams by Γ–convergence. Part II: closed cross-sections
No full textViability and control of a delayed SIR epidemic with an ICU state constraint
Full text linkThis paper studies viability and control synthesis for a delayed SIR epidemic. The model integrates a constant delay representing an incubation/latency time. The control inputs model nonpharmaceutical interventions, while an intensive care unit (ICU) state-constraint is introduced to reflect the healthcare systema's capacity. The arising delayed control system is analyzed via functional viability tools, providing insights into fulfilling the ICU constraint through feedback control maps. In particular, we consider two scenarios: first, we consider the case of general continuous initial conditions. Then, as a further refinement of our analysis, we assume that the initial conditions satisfy a Lipschitz continuity property, consistent with the considered model. The study compares the (in general, sub-optimal) obtained control policies with the optimal ones for the delay-free case, emphasizing the impact of the delay parameter. The obtained results are supported and illustrated, in a concluding section, by numerical examples
Asymptotic analysis of the Dirichlet fractional Laplacian in domains becoming unbounded
No full textIn this paper we analyze the asymptotic behaviour of the Dirichlet fractional Laplacian (−ΔRn+k)s, with s∈(0,1), on bounded domains in Rn+k that become unbounded in the last k-directions. A dimension reduction phenomenon is observed and described via Γ-convergence
Linear models for composite thin-walled beams by Gamma–convergence. Part I: open cross-sections
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