Epidemic management with admissible and robust invariant sets

Abstract

We present a detailed set-based analysis of the well-known SIR and SEIR epidemic models subjected to hard caps on the proportion of infective individuals, and bounds on the allowable intervention strategies, such as social distancing, quarantining and vaccination. We describe the admissible and maximal robust positively invariant (MRPI) sets of these two models via the theory of barriers. We show how the sets may be used in the management of epidemics, for both perfect and imperfect/uncertain models, detailing how intervention strategies may be specified such that the hard infection cap is never breached, regardless of the basic reproduction number. The results are clarified with detailed examples.

Toward more realistic social distancing policies via advanced feedback control

“…g ., [Adolph et al(2021)]) have stimulated a huge number of mathematically oriented investigations among which we select control-theoretic publications: See, e . g ., [Al-Radhawi et al(2022)], [Ames et al(2020)], [Angulo et al(2021)], [Berger(2022)], [Bisiacco & Pillonetto(2021)], [Bisiacco et al(2022)], [Bliman & Duprez(2021)], [Bliman et al(2021)], [Bonnans & Gianatti(2020)], [Borri et al(2021)], [Charpentier et al(2020)], [Dias et al(2022)], [Di Lauro et al(2021a)], [Di Lauro et al(2021b)], [Efimov & Ushirobira(2021)], [Esterhuizen et al(2021)], [Fliess et al(2022)], [Gevertz et al(2021)], [Greene & Sontag(2021)], [Ianni & Rossi(2021)], [Jing et al(2021)], [Köhler et al(2021)], [McQuade et al(2021)], [Morato et al(2020a)], [Morato et al(2020b)], [Morgan et al(2021)], [Morris et al(2021)], [O’Sullivan et al(2020)], [Péni et al(2020)], [Pillonetto et al(2021)], [Sadeghi et al(2021)], [Sontag(2021)], [Stella et al(2022)], [Tsay et al(2020)]. Most diverse viewpoints have been developed.…”

SIR Epidemics With State-Dependent Costs and ICU Constraints: A Hamilton-Jacobi Verification Argument and Dual LP Algorithms

“…Motivated by the recent paper [5], we recall the green maximal zone in which no control is needed to stabilize the infections and the yellow zone in which policies allowing the trajectory to be kept under the level i * do exist. The explicit treatment of the yellow zone (see also [4] for geometrical intuitions) is achieved in [5] by using normal cone (viability) conditions; see also the recent paper [14] for analogous treatments of such sets. We give here some further geometrical insights into the splitting of this yellow zone following the need for strict confinement and some conditional viability properties of such zones (see Proposition 23 in the Appendix).…”

“…We should emphasize that while several authors (e.g. [2], [14]) look into such greedy policies, in connection with the "safety" issues, the conditions on the cost guaranteeing their optimality are not made explicit.…”