Social Distancing as a Population Game in Networked Social Environments

Abstract

While social living is considered to be an indispensable part of human life in today's ever-connected world, social distancing has recently received much public attention on its importance since the outbreak of the coronavirus pandemic. In fact, social distancing has long been practiced in nature among solitary species, and been taken by human as an effective way of stopping or slowing down the spread of infectious diseases. Here we consider a social distancing problem for how a population, when in a world with a network of social sites, decides to visit or stay at some sites while avoiding or closing down some others so that the social contacts across the network can be minimized. We model this problem as a population game, where every individual tries to find some network sites to visit or stay so that he/she can minimize all his/her social contacts. In the end, an optimal strategy can be found for everyone, when the game reaches an equilibrium. We show that a large class of equilibrium strategies can be obtained by selecting a set of social sites that forms a so-called maximal r-regular subnetwork. The latter includes many well studied network types, which are easy to identify or construct, and can be completely disconnected (with=0 ) for the most strict isolation, or allow certain degrees of connectivities (with r>0) for more flexible distancing. We derive the equilibrium conditions of these strategies, and analyze their rigidity and flexibility on different types of r-regular subnetworks. We also extend our model to weighted networks, when different contact values are assigned to different network sites.

Keywords

Social distancing / epidemic and pandemic prevention / population games / social networks / regular networks / optimal distancing strategies / rigidity / flexibility, and fragility of distancing strategies

References

[1] Wang M., Zhou W., and Wu Z., Equilibrium distributions of populations of biological species on networks of social sites, Journal of Biological Dynamics, 13: 74-98, 2019.

[2] Chakradhar S., To fight coronavirus spread, the U.S. may extend 'social distancing' measures, but it comes at a cost, STAT News, February 17, 2020.

[3] Kissler S., Tedijanto C., Goldstein E., et al., Projecting the transmission dynamics of SARS-CoV-2 through the post-pandemic period, Science, online April 14, 2020, doi:10.1126/science.abb5793.

[4] Long N., From social distancing to social containment: reimagining sociality for the coronavirus pandemic, Medicine Anthropology Theory, ISSN 2405-691x (submitted), 2020.

[5] Mann C., Pandemics leave us forever altered - What history can tell us about the long-term effects of the coronavirus, The Atlantic: IDEAS, June 2020.

[6] Miller G., Social distancing prevents infections, but it can have unintended consequences, Science News, March 16, 2020.

[7] Thunstrome L., Newbold S., Finnoff D., et al., The benefits and costs of using social distancing to flatten the curve for COVID-19, Journal of Benefit Cost Analysis, online March 27, 2020, doi:10.2139/ssrn. 3561934.

[8] Krause, J. and Ruxton, G., Living in Groups, Oxford University Press, 2002.

[9] Alcock J., Animal Behavior:An Evolutionary Approach, Sinauer Associates Inc., 2009.

[10] Combs S., These wild animals also practice social distancing to avoid getting sick, National Geographic News, March 24, 2020.

[11] Hatchett R., Mecher C., and Lipsitch M., Public health interventions and epidemic intensity during the 1918 influenza pandemic, PNAS, 104: 7582-7587, 2007.

[12] Roth D., Henry B., Social distancing as a pandemic influenza prevention measure, National Collaborating Center for Infectious Diseases, Winnipeg, MB, Canada, July, 2011.

[13] Fenichel E., Economic considerations of social distancing and behavioral based policies during an epidemic, Journal of Health Economics, 32: 440-451, 2013.

[14] Ahmed F., Zviedrite N., and Uzicanin A., Effectiveness of workplace social distancing measures in reducing influenza transmission: A systematic review, BMC Public Heath, 8: 518-530, 2018.

[15] Faherty L., Schwartz H., Ahmed F., Zheteyeva Y., Uzicanin A., and Uscher-Pines L., School and preparedness officials' perspectives on social distancing practices to reduce influenza transmission during a pandemic: Considerations to guide future work, Preventive Medicine Reports, 14: 100871, 2019.

[16] Wilder-Smith A. and Freedman D, Isolation, quarantine, social distancing and community containment: Pivotal role for old-style public heath measures in the novel coronavirus (2019nCoV) outbreak, Journal of Travel Medicine, 2020: 1-4.

[17] Eubank S., Guclu H., Anil Kumar V., Marathe M., Srinivasan A., Toroczkai Z., and Wang N., Modeling disease outbreaks in realistic urban social networks, Nature, 429: 180-184, 2004.

[18] Del Valle S., Hethcote H., Hyman J., and Castillo-Chavez C., Effects of behavioral changes in a smallpox attack model, Mathematical Biosciences, 195: 238-251, 2005.

[19] Glass R., Glass L., Beyeler W., and Min H., Targeted social distancing design for pandemic influenza, Emerging Infectious Diseases, 12: 1671-1681, 2006.

[20] Meyers L., Contact network epidemiology: Bond percolation applied to infectious disease prediction and control, Bulletin of American Mathematics Society, 44: 63-86, 2007.

[21] Caley P., Philp D., and McCracken K., Quantifying social distancing arising from pandemic influenza, Journal of Royal Society, Interface, 5: 631-639, 2008.

[22] Kelso J., Milne G., and Kelly H., Simulation suggests that rapid activation of social distancing can arrest epidemic development due to a novel strain of influenza, BMC Public Health, 9: 1, 2009.

[23] Funk S., Salathe M., and Jansen V., Modeling the influence of human behavior on the spread of infectious diseases: A review, Journal of Royal Society, Interface, 7: 1247-1256, 2010.

[24] Reluga T., Game theory of social distancing in response to an epidemic, PLoS Computational Biology, 6: c1000793, 2010.

[25] Chen F., Jiang M., Rabidoux S., and Robinson S., Public avoidance and epidemics: Insights from an economic model, Journal of Theoretical Biology, 278: 107-119, 2011.

[26] Valdez L., Buono C., Macri P., and Braunstein L., Intermittent social distancing strategies for epidemic control, Physics Review E, 85: 036108, 2012.

[27] Bhattacharyya S. and Reluga T., Game dynamic model of social distancing while cost of infection varies with epidemic burden, IMA Journal of Applied Mathematics, 84: 23-43, 2019.

[28] Eubank S., Eckstrand I., Lewis B., Venkatramanan S., Marathe M., and Barrett C., Commentary on Ferguson, et al., "Impact on non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand", Bulletin of Mathematics Biology, DOI: https://doi.org/10.1007/s11538-020-00726-x.2020.

[29] Ferguson N., Laydon D., Nadjati-Gilani N., et al., Impact of non-pharmaceutical interventions (NPIs) to reduce COVID-19 mortality and healthcare demand, Imperial College COVID 19 Response Team, London, March 16, 2020.

[30] Hellewell J., Abbott S., Gimma A., et al., Feasibility of controlling COVID-19 outbreaks by isolation of cases and contacts, The LANCET Global Health, 8: E488-E496, 2020.

[31] Kiesha P., Liu Y., Russell T., et al., The effect of controlling strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: A modeling study, LANCET PUBLIC Health, DOI: https://doi.org/10.1016/S2468-2667(20)30073-6.2020.

[32] McCombs A. and Kadelka C., A model-based evaluation of the efficacy of COVID-19 social distancing, testing and hospital triage policies, medRxiv: doi: https://doi.org/10.1101/2020.04.20.20073213.2020.

[33] Sanche S., Lin Y., Xu C., et al., The Novel coronavirus, 2019-nCoV, is highly contagious and more infectious than initially estimated, arXir.org: 2002, 03268, 2020.

[34] Bondy A. and Murty U., Graph Theory, Springer, 2008.

[35] Newman M., Networks, Oxford University Press, 2018.

[36] Reluga T. and Galvani A., A general approach for population games with applications to vaccination, Mathematical Biosciences, 230: 67-78, 2011.

[37] Reluga T., Equilibria of an epidemic game with piecewise linear social distancing cost, Bulletin of Mathematical Biology, 75: 1961-1984, 2013.

[38] Barrett C., Eubank S., Anil Kumar V., and Marathe M., Understanding large-scale social and infrastructure Networks: A simulation-based approach, SIAM News, 37, May, 2004.

[39] Eubank S., Network based models of infectious disease spread, Japanese Journal of Infectious Diseases, 58: S9-S13, 2005.

[40] Eubank S., Anil Kumar V., Marathe M., Srinivasan A., and Wang N., Structure of social networks and their impact on epidemics, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 70: 181-213, 2006.

[41] Barrett C., Bisset K., Eubank S., Feng X., Marathe M., EpiSimdemics: An efficient algorithm for simulating the spread of infectious disease over large realistic social networks, SC08: Proceedings of the 2008ACM/IEEE Conference on Supercomputing, 1-12, 2008.

[42] Meyers L., Newman M., and Pourbohloul B., Predicting epidemics on direct contact networks, Journal of Theoretical Biology, 240: 400-418, 2006.

[43] Volz E. and Meyers L., Susceptible-infected-recovered epidemics in dynamic contact networks, Proceedings of the Royal Society B, 274: 2925-2933, 2007.

[44] Volz E. and Meyers L., Epidemic thresholds in dynamic contact networks, Journal of the Royal Society, Interface, 6: 233-241, 2009.

[45] Bansal S., Read J., Pourbohloul B., Meyers L., The dynamic nature of contact networks in infectious disease epidemiology, Journal of Biological Dynamics, 4: 478-489, 2010.

[46] Weibull J. W., Evolutionary Game Theory, The MIT Press, 1995.

[47] Hofbauer J. and Sigmund K., Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.

[48] Fletcher R., Practical Optimization, Wiley, 2000.

[49] Nocedal J. and Wright S., Numerical Optimization, Springer, 2006.

[50] Brauer F. and Castillo-Chavez C., Mathematical Models in Population Biology and Epidemiology, 2nd Edition, Springer 2011.

[51] Lewis T., Network Science: Theory and Applications, Wiley, 2009.

[52] Cormen T., Leiserson C., Rivest R., and Stein C., Introduction to Algorithms, Third Edition, The MIT Press, 2009.

[53] Erickson J., Algorithms, Independent Publisher, 2019.

[54] Garey M. and Johnson D., Computers and Intractability: A Guide to the Theory of NPCompleteness, Freeman, 1979.

[55] Stockmeyer L. and Vazirani V., NP-completeness of some generalizations of the maximum matching problem, Information Processing Letters, 15: 14-19, 1982.

[56] Cameron K., Induced matchings, Discrete Applied Mathematics, 24: 97-102, 1989.

[57] Cardoso D., Kaminski M., and Lozin V., Maximum k-regular induced subgraphs, Rutcor Research Report (RRR), 3, 2006.

[58] Gupta S., Raman V., and Saurabh S, Fast exponential algorithms for maximum r-regular induced subgraph problems, in S. Arun-Kumar and N. Garg (Eds): FSTTCS 2006: Lecture Notes in Computer Science, 4337: 139-151, 2006.

[59] Bomze I., Evolution towards the maximum clique, Journal of Global Optimization, 10: 143-164, 1997.

[60] Vickers G. and Cannings C., On the number of stable equilibria in a one-locus, multi-allelic system, Journal of Theoretical Biology, 131: 273-277, 1988.