Contact rate epidemic control of COVID-19: an equilibrium view

Romuald Elie¹ ; Emma Hubert¹ ; Gabriel Turinici²

¹ LAMA – UMR 8050, Université Gustave Eiffel, Champs-sur-Marne, France
² CEREMADE – UMR 7534, Université Paris Dauphine, PSL Research University, France

Mathematical modelling of natural phenomena, Tome 15 (2020), article no. 35.

Résumé

We consider the control of the COVID-19 pandemic through a standard SIR compartmental model. This control is induced by the aggregation of individuals’ decisions to limit their social interactions: when the epidemic is ongoing, an individual can diminish his/her contact rate in order to avoid getting infected, but this effort comes at a social cost. If each individual lowers his/her contact rate, the epidemic vanishes faster, but the effort cost may be high. A Mean Field Nash equilibrium at the population level is formed, resulting in a lower effective transmission rate of the virus. We prove theoretically that equilibrium exists and compute it numerically. However, this equilibrium selects a sub-optimal solution in comparison to the societal optimum (a centralized decision respected fully by all individuals), meaning that the cost of anarchy is strictly positive. We provide numerical examples and a sensitivity analysis, as well as an extension to a SEIR compartmental model to account for the relatively long latent phase of the COVID-19 disease. In all the scenario considered, the divergence between the individual and societal strategies happens both before the peak of the epidemic, due to individuals’ fears, and after, when a significant propagation is still underway.

Détail

DOI : 10.1051/mmnp/2020022

Bibliographie

[1] A. Abakuks An optimal isolation policy for an epidemic J. Appl. Prob 1973 247 262

[2] A. Abakuks Optimal immunisation policies for epidemics Adv. Appl. Prob 1974 494 511

[3] Y. Achdou, M. Laurière Mean field type control with congestion Appl. Math. Optim 2016 393 418

[4] L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the space of probability measures, 2nd edn. Birkhäuser, Basel (2008).

[5] S. Anand, K. Hanson Disability-adjusted life years: a critical review J. Health Eco 1997 685 702

[6] R.M. Anderson and R.M. May, Infectious Diseases of Humans Dynamics and Control. Oxford University Press, Oxford (1992).

[7] R.M. Anderson, T.D. Hollingsworth and D.J. Nokes, Mathematical models of transmission and control, Vol. 2. Oxford University Press, Oxford (2009).

[8] N. Bacaer Un modéle mathématique des débuts de l’épidémie de coronavirus en France MMNP 2020 29

[9] M. Bardi and I. Capuzzo-Dolcetta, Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations. Systems Control: Foundations Applications. Birkhäuser Boston Inc., Boston, MA (1997).

[10] C.T. Bauch, D.J.D. Earn Vaccination and the theory of games Proc. Natl. Acad. Sci. USA 2004 13391 13394

[11] H. Behncke Optimal control of deterministic epidemics Optim. Control Appl. Methods 2000 269 285

[12] A. Bressan, F. Rampazzo Impulsive control systems with commutative vector fields J. Optim. Theory Appl. 1991 67 83

[13] B. Buonomo, A. D’Onofrio, D. Lacitignola. Global stability of an SIR epidemic model with information dependent vaccination Math. Biosci 2008 9 16

[14] V. Capasso, Mathematical Structures of Epidemic Systems. Lecture Notes in Biomathematics. Springer-Verlag, Berlin (1993).

[15] V. Capasso, G. Serio A generalization of the Kermack-McKendrick deterministic epidemic model Math. Biosci 1978 43 61

[16] R. Carmona, J.P. Fouque, L.H. Sun Mean field games and systemic risk Commun. Math. Sci 2015 911 933

[17] R. Carmona, F. Delarue, et al. Probabilistic Theory of Mean Field Games with Applications I-II. Springer, Berlin (2018).

[18] G. Dal Maso, F. Rampazzo On systems of ordinary differential equations with measures as controls Differ. Integral Equ. 1991 739 765

[19] A. Danchin, T.W.P. Ng and G. Turinici, A new transmission route for the propagation of the SARS-CoV-2 coronavirus. Preprint medRxiv: 20022939v1 (2020).

[20] R. Djidjou-Demasse, Y. Michalakis, M. Choisy, M.T. Sofonea and S. Alizon, Optimal COVID-19 epidemic control until vaccine deployment. Preprint medRxiv: 20049189v3 (2020).

[21] J. Dolbeault and G. Turinici, Heterogeneous social interactions and the COVID-19 lockdown outcome in a multi-group SEIR model. Preprint arXiv:2005.00049 (2020).

[22] A. D’Onofrio, P. Manfredi, E. Salinelli Vaccinating behaviour, information, and the dynamics of SIR vaccine preventable diseases Theor. Popul. Biol 2007 301 317

[23] A. D’Onofrio, P. Manfredi, E. Salinelli. Fatal SIR diseases and rational exemption to vaccination Math. Med. Biol 2008 337 357

[24] R. Elie, E. Hubert, T. Mastrolia and D. Possamaï, Mean-field moral hazard for optimal energy demand response management. Preprint arXiv:1902.10405 (2019).

[25] R. Élie, T. Ichiba and M. Laurière, Large banking systems with default and recovery: A mean field game model. Preprint arXiv:2001.10206 (2020).

[26] E.P. Fenichel, C. Castillo-Chavez, M.G. Ceddia, G. Chowell, P.A.G. Parra, G.J. Hickling, G. Holloway, R. Horan, B. Morin, C. Perrings, M. Springbornh, L. Velazqueze, C. Villalobos Adaptive human behavior in epidemiological models Proc. Natl. Acad. Sci 2011 6306 6311

[27] S. Ghader, J. Zhao, M. Lee, W. Zhou, G. Zhao, L. Zhang, Observed mobility behavior data reveal social distancing inertia. Preprint arXiv:2004.14748 (2020).

[28] O. Guéant, J.-M. Lasry and P.-L. Lions, Mean field games and applications, in Paris-Princeton lectures on mathematical finance 2010. Springer, Berlin (2011) 205–266.

[29] E. Hansen, T. Day Optimal control of epidemics with limited resources J. Math. Biol 2011 423 451

[30] M. Huang, R. Malhamé, P. Caines Large population stochastic dynamic games: Closed–loop McKean–Vlasov systems and the Nash certainty equivalence principle Commun. Inf. Syst 2006 221 252

[31] M. Huang, P. Caines, R. Malhamé An invariance principle in large population stochastic dynamic games J. Syst. Sci. Complex 2007 162 172

[32] M. Huang, P. Caines, R. Malhamé Large–population cost–coupled LQG problems with nonuniform agents: Individual–mass behavior and decentralized ε–Nash equilibria IEEE Trans. Auto. Control 2007 1560 1571

[33] M. Huang, P. Caines and R. Malhamé, The Nash certainty equivalence principle and McKean–Vlasov systems: An invariance principle and entry adaptation, in 46th IEEE Conference on Decision and Control (2007) 121–126.

[34] E. Hubert, G. Turinici Nash-MFG equilibrium in a SIR model with time dependent newborn vaccination Ric. Mat 2018 227 246

[35] L. Laguzet, G. Turinici Global optimal vaccination in the SIR model: properties of the value function and application to cost-effectiveness analysis Math. Biosci 2015 180 197

[36] L. Laguzet, G. Turinici Individual Vaccination as Nash Equilibrium in a SIR Model with Application to the 2009-2010 Influenza A (H1N1) Epidemic in France Bull. Math. Biol. 2015 1955 1984

[37] L. Laguzet, G. Turinici and G. Yahiaoui, Equilibrium in an individual-societal SIR vaccination model in presence of discounting and finite vaccination capacity, in New Trends in Differential Equations, Control Theory and Optimization: Proceedings of the 8th Congress of Romanian Mathematicians (2016) 201–214.

[38] J.-M. Lasry, P.-L. Lions Jeux à champ moyen. I–Le cas stationnaire C R Math. 2006 619 625

[39] J.-M. Lasry, P.-L. Lions Jeux à champ moyen. II–Horizon fini et contrôle optimal C R Math. 2006 679 684

[40] J.-M. Lasry, P.-L. Lions Mean field games Jpn. J. Math 2007 229 260

[41] S. Lenhart and J.T. Workman, Optimal control applied to biological models. CRC Press, Boca Raton (2007).

[42] Q. Li, X. Guan, P. Wu, X. Wang, L. Zhou, Y. Tong, R. Ren, K.S.M. Leung, E.H.Y. Lau, J.Y. Wong, X. Xing, N. Xiang, Y. Wu, C. Li, Q. Chen, D. Li, T. Liu, J. Zhao, M. Liu, W. Tu, C. Chen, L. Jin, R. Yang, Q. Wang, S. Zhou, R. Wang, H. Liu, Y. Luo, Y. Liu, G. Shao, H. Li, Z. Tao, Y. Yang, Z. Deng, B. Liu, Z. Ma, Y. Zhang, G. Shi, T.T.Y. Lam, J.T. Wu, G.F. Gao, B.J. Cowling, B. Yang, G.M. Leung, Z. Feng Early Transmission Dynamics in Wuhan, China, of Novel Coronavirus-Infected Pneumonia New Engl. J. Med 2020 1199 1207

[43] R. Morton, K.H. Wickwire On the optimal control of a deterministic epidemic Adv. Appl. Prob 1974 622 635

[44] T. Ng, G. Turinici, A. Danchin A double epidemic model for the SARS propagation BMC Infect. Dis 2003 19

[45] A. Perasso An introduction to the basic reproduction number in mathematical epidemiology ESAIM: Proc. Surv 2018 123 138

[46] A.B. Piunovskiy, D. Clancy An explicit optimal intervention policy for a deterministic epidemic model Optim. Control Appl. Methods 2008 413 428

[47] A.S. Poznyak, Topics of functional analysis, in Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques, edited by A.S. Poznyak. Elsevier, Oxford (2008) 451–498.

[48] A. Rizzo, M. Frasca, M. Porfiri Effect of individual behavior on epidemic spreading in activity-driven networks Phys. Rev. E 2014 042801

[49] F.D. Sahneh, F.N. Chowdhury, C.M. Scoglio On the existence of a threshold for preventive behavioral responses to suppress epidemic spreading Sci. Rep 2012 632

[50] F. Salvarani, G. Turinici Optimal individual strategies for influenza vaccines with imperfect efficacy and durability of protection Math. Biosci. Eng 2018 629

[51] F. Sassi Calculating QALYs, comparing QALY and DALY calculations Health Policy Plan 2006 402 408

[52] S.P. Sethi, P.W. Staats Optimal control of some simple deterministic epidemic models Oper. Res. Soc. J 1978 129 136

[53] G. Silva, R. Vinter Necessary conditions for optimal impulsive control problems SIAM J. Control Optim. 1997 1829 1846

[54] G. Turinici Metric gradient flows with state dependent functionals: The Nash-MFG equilibrium flows and their numerical schemes Nonlinear Anal 2017 163 181

[55] G. Turinici and A. Danchin, The SARS Case Study: An Alarm Clock? in Encyclopedia of Infectious Diseases. John Wiley Sons, Ltd, New Jersey (2006) 151–162.

[56] V. Volpert, M. Banerjee, A. D’Onofrio, T. Lipniacki, S. Petrovskii, V.C. Tran Coronavirus: scientific insights and societal aspects MMNP 2020 E2

[57] Z. Wang, C.T. Bauch, S. Bhattacharyya, A. D’Onofrio, P. Manfredi, M. Perc, N. Perra, M. Salathé, D. Zhao Statistical physics of vaccination Phys. Rep 2016 1 113

[58] K. Wickwire Optimal isolation policies for deterministic and stochastic epidemics Math. Biosci 1975 325 346

[59] R. Zeckhauser, D. Shepard Where now for saving lives Law Contemp. Probl 1976 5

[60] E. Zeidler, Applied Functional Analysis: Applications to Mathematical Physics. Applied Mathematical Sciences. Springer, New York (2012).