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Controlled Compartmental Models with Time-Varying Population: Normalization, Viability and Comparison

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Abstract

This paper focuses on the characterization of viability zones in compartmental models with varying population size, due both to deaths caused by epidemics and to natural demography. This is achieved with the use of viscosity characterizations of viability and extensively illustrated on several models. An example taking into consideration real data is provided. The paper is completed with a viscosity approach to the optimality of minimal (“greedy”) non-pharmaceutical interventions.

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Data Availability

The data that support the findings of this study are openly available from INSEE (Institut national de la statistique et des études économiques) at https://www.insee.fr/fr/statistiques/. under the references: 2381380, 2383440.

Notes

  1. in a SIR model, the product \(*\) of exposure is infected individuals, in SEIR models \(*\) stands for exposed, etc.

  2. The affinity is then computed proportionally to the concentrations s and i and to the interaction “speed” \(\beta \) tempered with the control u.

  3. Of course, the notion of input (variable) is to be separated from what one might use in automatic sense (in which the controls u are the inputs).

  4. It is then clear for the reader that, in a SI(R) model, in which the input variables are S and I, these restrictions sum up to \({\mathbb {T}}\) being a triangle.

  5. Different authors prefer calling it maximal robust positively invariant zone. Of course, largest and maximal are the same, as are positive(ly) and forward(ly) invariant. We prefer retaining the classical denomination by Aubin.

  6. i.e., search for the smallest lower semicontinuous viscosity supersolution.

  7. According to Cori method and loosely speaking, the basic reproduction number is \(R_0=\beta *c*D\), where \(\beta \) is the risk to get the virus in one contact, c is the number of contacts per time unit (let us say 1!) and D the number of days a person is contagious. In SIR, this corresponds to \(R_0=\frac{\beta }{\gamma }\) since \(\gamma =1/D\) with our notations.

  8. or, equivalently, \(\phi (u)\) for some nonnegative function attaining its strict minimum at \(u_{\max }=1\)

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Funding

Dan Goreac, Juan Li and Junsong Li have been partially supported by the National Key R and D Program of China (No. 2018YFA0703901) and the NSF of P. R. China (Nos. 12031009, 11871037).

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Authors and Affiliations

  1. Laboratoire de Mathématiques Appliquées, Université de Pau, 64012, Pau, France

    Florin Avram

  2. Dipartimento di Scienze Matematiche, Informatiche e Fisiche, via delle Scienze 206, 33100, Udine, Italy

    Lorenzo Freddi

  3. School of Mathematics and Statistics, Shandong University, Weihai, Weihai, 264209, People’s Republic of China

    Dan Goreac, Juan Li & Junsong Li

  4. LAMA, Univ Gustave Eiffel, UPEM, Univ Paris Est Creteil, CNRS, 77447, Marne-la-Valleée, France

    Dan Goreac

  5. Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Qingdao, 266237, People’s Republic of China

    Juan Li

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  1. Florin Avram
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  2. Lorenzo Freddi
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  3. Dan Goreac
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  4. Juan Li
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Corresponding author

Correspondence to Dan Goreac.

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Communicated by Vincenzo Capasso

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Avram, F., Freddi, L., Goreac, D. et al. Controlled Compartmental Models with Time-Varying Population: Normalization, Viability and Comparison. J Optim Theory Appl 198, 1019–1048 (2023). https://doi.org/10.1007/s10957-023-02274-5

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