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A MEAN FIELD GAME ANALYSIS OF SIR DYNAMICS WITH VACCINATION

Published online by Cambridge University Press:  13 November 2020

Josu Doncel Open the ORCID record for Josu Doncel [Opens in a new window] ,
Nicolas Gast Open the ORCID record for Nicolas Gast [Opens in a new window]  and
Bruno Gaujal
Josu Doncel
Affiliation:
University of the Basque Country, Leioa48940, Spain E-mail: josu.doncel@ehu.eus
Nicolas Gast
Affiliation:
INRIA and Université Grenoble Alpes, CNRS, LIG, F-38000Grenoble, France E-mails: nicolas.gast@inria.fr; bruno.gaujal@inria.fr
Bruno Gaujal
Affiliation:
INRIA and Université Grenoble Alpes, CNRS, LIG, F-38000Grenoble, France E-mails: nicolas.gast@inria.fr; bruno.gaujal@inria.fr
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Abstract

We analyze a mean field game model of SIR dynamics (Susceptible, Infected, and Recovered) where players choose when to vaccinate. We show that this game admits a unique mean field equilibrium (MFE) that consists in vaccinating at a maximal rate until a given time and then not vaccinating. The vaccination strategy that minimizes the total cost has the same structure as the MFE. We prove that the vaccination period of the MFE is always smaller than the one minimizing the total cost. This implies that, to encourage optimal vaccination behavior, vaccination should always be subsidized. Finally, we provide numerical experiments to study the convergence of the equilibrium when the system is composed by a finite number of agents ($N$) to the MFE. These experiments show that the convergence rate of the cost is $1/N$ and the convergence of the switching curve is monotone.

Keywords

applied probabilityprobabilistic networksstochastic modeling
Type
Research Article
Information
Probability in the Engineering and Informational Sciences , Volume 36 , Issue 2 , April 2022 , pp. 482 - 499
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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