Controlling Propagation of epidemics via mean-field games

The coronavirus disease 2019 (COVID-19) pandemic is changing and impacting lives on a global scale. In this paper, we introduce a mean-field game model in controlling the propagation of epidemics on a spatial domain. The control variable, the spatial velocity, is first introduced for the classical disease models, such as the SIR model. For this proposed model, we provide fast numerical algorithms based on proximal primal-dual methods. Numerical experiments demonstrate that the proposed model illustrates how to separate infected patients in a spatial domain effectively.

[1]  E Weinan,et al.  A mean-field optimal control formulation of deep learning , 2018, Research in the Mathematical Sciences.

[2]  P. Lions,et al.  Mean field games , 2007 .

[3]  Yves Achdou,et al.  Mean Field Games for Modeling Crowd Motion , 2018, Computational Methods in Applied Sciences.

[4]  Bilal Ilyas,et al.  TRAVELING WAVES FOR A SIMPLE DIFFUSIVE EPIDEMIC MODEL , 1995 .

[5]  M. Burger,et al.  Mean field games with nonlinear mobilities in pedestrian dynamics , 2013, 1304.5201.

[6]  István Faragó,et al.  Electronic Journal of Qualitative Theory of Differential Equations Qualitatively Adequate Numerical Modelling of Spatial Sirs-type Disease Propagation , 2022 .

[7]  Peter E. Caines,et al.  Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle , 2006, Commun. Inf. Syst..

[8]  W. O. Kermack,et al.  A contribution to the mathematical theory of epidemics , 1927 .

[9]  Diogo A. Gomes,et al.  Mean Field Games Models—A Brief Survey , 2013, Dynamic Games and Applications.

[10]  Odo Diekmann,et al.  Run for your life; a note on the asymptotic speed of propagation of an epidemic : (preprint) , 1979 .

[11]  Levon Nurbekyan,et al.  Computational methods for nonlocal mean field games with applications , 2020, 2004.12210.

[12]  Jeehyun Lee,et al.  Optimal control problem of an SIR reaction-diffusion model with inequality constraints , 2020, Math. Comput. Simul..

[13]  Antonin Chambolle,et al.  A First-Order Primal-Dual Algorithm for Convex Problems with Applications to Imaging , 2011, Journal of Mathematical Imaging and Vision.

[14]  P. Lions,et al.  Jeux à champ moyen. I – Le cas stationnaire , 2006 .

[15]  S. Sethi,et al.  Optimal Control of Some Simple Deterministic Epidemic Models , 1978 .

[16]  Wuchen Li,et al.  Solving Large-Scale Optimization Problems with a Convergence Rate Independent of Grid Size , 2018, SIAM J. Numer. Anal..

[17]  Pierre-Louis Lions,et al.  Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis , 2013, 1305.6323.

[18]  Jianhong Wu,et al.  Travelling waves of a diffusive Kermack–McKendrick epidemic model with non-local delayed transmission , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  Jean-David Benamou,et al.  Augmented Lagrangian Methods for Transport Optimization, Mean Field Games and Degenerate Elliptic Equations , 2015, J. Optim. Theory Appl..

[20]  Shigui Ruan,et al.  Spatial-Temporal Dynamics in Nonlocal Epidemiological Models , 2007 .

[21]  P. Lions,et al.  Jeux à champ moyen. II – Horizon fini et contrôle optimal , 2006 .

[22]  Settapat Chinviriyasit,et al.  Numerical modelling of an SIR epidemic model with diffusion , 2010, Appl. Math. Comput..

[23]  Levon Nurbekyan,et al.  A machine learning framework for solving high-dimensional mean field game and mean field control problems , 2020, Proceedings of the National Academy of Sciences.

[24]  O. Bjørnstad,et al.  Travelling waves and spatial hierarchies in measles epidemics , 2001, Nature.

[25]  H. Thieme.  A model for the spatial spread of an epidemic , 1977, Journal of mathematical biology.

[26]  Dante Kalise,et al.  Proximal Methods for Stationary Mean Field Games with Local Couplings , 2016, SIAM J. Control. Optim..

[27]  A. Settati,et al.  Dynamics and optimal control of a non-linear epidemic model with relapse and cure , 2018 .

[28]  Henri Berestycki,et al.  Propagation of Epidemics Along Lines with Fast Diffusion , 2020, Bulletin of Mathematical Biology.

[29]  Anthony F. Bartholomay.  Mathematics and Computer Science in Biology and Medicine , 1966 .

[30]  Thomas Caraco,et al.  Stage‐Structured Infection Transmission and a Spatial Epidemic: A Model for Lyme Disease , 2002, The American Naturalist.

[31]  Antonin Chambolle,et al.  On the ergodic convergence rates of a first-order primal–dual algorithm , 2016, Math. Program..

[32]  R. Rosenthal,et al.  Anonymous sequential games , 1988 .

[33]  HighWire Press.  Proceedings of the Royal Society of London. Series A, Containing papers of a mathematical and physical character , 1934 .

[34]  Wirawan Chinviriyasit,et al.  Numerical Modelling of Influenza Model with Diffusion , 2014 .

[35]  Zhongjun Ma,et al.  Dynamic stability of an SIQS epidemic network and its optimal control , 2019, Commun. Nonlinear Sci. Numer. Simul..

[36]  Anders Källén,et al.  Thresholds and travelling waves in an epidemic model for rabies , 1984 .

[37]  Levon Nurbekyan,et al.  APAC-Net: Alternating the Population and Agent Control via Two Neural Networks to Solve High-Dimensional Stochastic Mean Field Games , 2020, ArXiv.