Modeling the propagation of riots, collective behaviors and epidemics

Henri Berestycki; Samuel Nordmann; Luca Rossi; Henri Berestycki; Samuel Nordmann; Luca Rossi

doi:10.3934/mine.2022003

Mathematics in Engineering

2022, Volume 4, Issue 1: 1-53. doi: 10.3934/mine.2022003

Previous Article Next Article

Research article Special Issues

Modeling the propagation of riots, collective behaviors and epidemics

1.
Centre d'analyse et de mathématique sociales, EHESS - CNRS, 54, boulevard Raspail, Paris, France
2.
Senior Visiting Fellow, Institute for Advanced Study, Hong Kong University of Science and Technology, Hong Kong
3.
Department of Applied Mathematics, Tel Aviv University, Tel Aviv, Israel

Received: 11 March 2020 Accepted: 01 August 2020 Published: 17 March 2021

This paper is concerned with a family of Reaction-Diffusion systems that we introduced in ^[15], and that generalizes the SIR type models from epidemiology. Such systems are now also used to describe collective behaviors. In this paper, we propose a modeling approach for these apparently diverse phenomena through the example of the dynamics of social unrest. The model involves two quantities: the level of social unrest, or more generally activity, $ u $, and a field of social tension $ v $, which play asymmetric roles. We think of $ u $ as the actually observed or explicit quantity while $ v $ is an ambient, sometimes implicit, field of susceptibility that modulates the dynamics of $ u $. In this article, we explore this class of model and prove several theoretical results based on the framework developed in ^[15], of which the present work is a companion paper. We particularly emphasize here two subclasses of systems: tension inhibiting and tension enhancing. These are characterized by respectively a negative or a positive feedback of the unrest on social tension. We establish several properties for these classes and also study some extensions. In particular, we describe the behavior of the system following an initial surge of activity. We show that the model can give rise to many diverse qualitative dynamics. We also provide a variety of numerical simulations to illustrate our results and to reveal further properties and open questions.
- epidemiology models,
- SIR model,
- threshold phenomenon,
- social systems,
- reaction-diffusion systems,
- contagion,
- traveling waves,
- speed of propagation
Citation: Henri Berestycki, Samuel Nordmann, Luca Rossi. Modeling the propagation of riots, collective behaviors and epidemics[J]. Mathematics in Engineering, 2022, 4(1): 1-53. doi: 10.3934/mine.2022003

Related Papers:

Abstract

This paper is concerned with a family of Reaction-Diffusion systems that we introduced in ^[15], and that generalizes the SIR type models from epidemiology. Such systems are now also used to describe collective behaviors. In this paper, we propose a modeling approach for these apparently diverse phenomena through the example of the dynamics of social unrest. The model involves two quantities: the level of social unrest, or more generally activity, $ u $, and a field of social tension $ v $, which play asymmetric roles. We think of $ u $ as the actually observed or explicit quantity while $ v $ is an ambient, sometimes implicit, field of susceptibility that modulates the dynamics of $ u $. In this article, we explore this class of model and prove several theoretical results based on the framework developed in ^[15], of which the present work is a companion paper. We particularly emphasize here two subclasses of systems: tension inhibiting and tension enhancing. These are characterized by respectively a negative or a positive feedback of the unrest on social tension. We establish several properties for these classes and also study some extensions. In particular, we describe the behavior of the system following an initial surge of activity. We show that the model can give rise to many diverse qualitative dynamics. We also provide a variety of numerical simulations to illustrate our results and to reveal further properties and open questions.

References

[1]	K. Afassinou, Analysis of the impact of education rate on the rumor spreading mechanism, Physica A, 414 (2014), 43–52. doi: 10.1016/j.physa.2014.07.041
[2]	P. G. Altbach, M. Klemencic, Student activism remains a potent force worldwide, International Higher Education, 76 (2014), 2–3.
[3]	R. M. Anderson, The population dynamics of infectious diseases: theory and applications, Boston, MA: Springer, 1982.
[4]	H. Arendt, Crises of the republic : lying in politics; civil disobedience; on violence; thoughts on politics and revolution, Harcourt Brace Jovanovich, 1972.
[5]	D. Aronson, H. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. Math., 30 (1978): 33–76.
[6]	M. Bages, P. Martinez, Existence of pulsating waves in a monostable reaction-diffusion system in solid combustion, Discrete Cont. Dyn. B, 14 (2010): 817–869.
[7]	N. T. J. Bailey, The mathematical theory of infectious diseases and its applications, 2 Eds., London: Griffin, 1975.
[8]	F. M. Bass, A new product growth for model consumer durables, Manage. Sci., 15 (1969), 215–227.
[9]	P. Baudains, A. Braithwaite, S. D. Johnson, Spatial patterns in the 2011 London riots, Policing, 7 (2013), 21–31. doi: 10.1093/police/pas049
[10]	H. Berestycki, L. Caffarelli, L. Nirenberg, Further qualitative properties for elliptic equations in unbounded domains, Ann. Scuola Norm. Sci., 25 (1997), 69–94.
[11]	H. Berestycki, F. Hamel, H. Matano, Bistable travelling waves around an obstacle, Commun. Pure Appl. Math., 62 (2009), 729–788. doi: 10.1002/cpa.20275
[12]	H. Berestycki, B. Larrouturou, Quelques aspects mathématiques de la propagation des flammes prémélangées, London: Pitman, 1991.
[13]	H. Berestycki, J. P. Nadal, N. Rodríguez, A model of riot dynamics: shocks, diffusion, and thresholds, Netw. Heterog. Media, 10 (2015), 1–34. doi: 10.3934/nhm.2015.10.1
[14]	H. Berestycki, B. Nicolaenko, B. Scheurer, Traveling wave solutions to combustion models and their singular limits, SIAM J. Math. Anal., 16 (1985), 1207–1242. doi: 10.1137/0516088
[15]	H. Berestycki, S. Nordmann, L. Rossi, Activity/Susceptibility systems, 2020, preprint.
[16]	H. Berestycki, N. Rodriguez, Analysis of a heterogeneous model for riot dynamics: the effect of censorship of information, Eur. J. Appl. Math., 27 (2016), 554–582. doi: 10.1017/S0956792515000339
[17]	H. Berestycki, N. Rodríguez, L. Ryzhik, Traveling wave solutions in a reaction-diffusion model for criminal activity, Multiscale Model. Simul., 11 (2013), 1097–1126. doi: 10.1137/12089884X
[18]	H. Berestycki, L. Rossi, N. Rodríguez, Periodic cycles of social outbursts of activity, J. Differ. Equations, 264 (2018), 163–196. doi: 10.1016/j.jde.2017.09.005
[19]	S. Bhattacharya, K. Gaurav, S. Ghosh, Viral marketing on social networks: an epidemiological perspective, Physica A, 525 (2019), 478–490. doi: 10.1016/j.physa.2019.03.008
[20]	L. Bonnasse-Gahot, H. Berestycki, M. A. Depuiset, M. B. Gordon, S. Roché, N. Rodriguez, et al. Epidemiological modelling of the 2005 French riots: a spreading wave and the role of contagion, Sci. Rep., 8 (2018), 107. doi: 10.1038/s41598-017-18093-4
[21]	J. P. Bouchaud, Crises and collective socio-economic phenomena: simple models and challenges, J. Stat. Phys., 151 (2013), 567–606. doi: 10.1007/s10955-012-0687-3
[22]	D. Braha, Global civil unrest: contagion, self-organization, and prediction, PLoS ONE, 7 (2012), 1–9.
[23]	S. L. Burbeck, W. J. Raine, M. J. Stark, The dynamics of riot growth: an epidemiological approach, J. Math. Sociol., 6 (1978), 1–22. doi: 10.1080/0022250X.1978.9989878
[24]	B. Cao, S. H. Han, Z. Jin, Modeling of knowledge transmission by considering the level of forgetfulness in complex networks, Physica A, 451 (2016), 277–287. doi: 10.1016/j.physa.2015.12.137
[25]	P. Caroca, C. Cartes, T. P. Davies, J. Olivari, S. Rica, K. Vogt, The anatomy of the 2019 Chilean social unrest, Chaos, 30 (2020), 073129. doi: 10.1063/5.0006307
[26]	D. J. Daley, D. G. Kendall, Epidemics and rumors, Nature, 204, (1964), 1118.
[27]	T. P. Davies, H. M. Fry, A. G. Wilson, S. R. Bishop, A mathematical model of the London riots and their policing, Sci. Rep., 3, (2013), 1303.
[28]	K. Dietz, Epidemics and rumours: a survey, Journal of the Royal Statistical Society. Series A (General), 130 (1967), 505–528. doi: 10.2307/2982521
[29]	R. Ducasse, L. Rossi, Blocking and invasion for reaction-diffusion equations in periodic media, Calc. Var., 57, (2018), 142.
[30]	M. Edmonds, How Riots Work, 2011. Available from: https://people.howstuffworks.com/riot.htm.
[31]	J. M. Epstein, Modeling civil violence: an agent-based computational approach, PNAS, 99 (Suppl 3) (2002), 7243–7250.
[32]	G. Fibich, Bass-SIR model for diffusion of new products in social networks, Phys. Rev. E, 94 (2016), 32305.
[33]	G. Fibich, Diffusion of new products with recovering consumers, SIAM J. Appl. Math., 77 (2017), 1230–1247. doi: 10.1137/17M1112546
[34]	M. Fonoberova, V. A. Fonoberov, I. Mezic, J. Mezic, P. J. Brantingham, A. Societies, et al. Nonlinear dynamics of crime and violence in urban settings an agent-based model of civil violence, JASSS, 15 (2012), 1–15.
[35]	N. Gaumont, M. Panahi, D. Chavalarias, Reconstruction of the socio-semantic dynamics of political activist Twitter networks – Method and application to the 2017 French presidential election, PLoS ONE, 13 (2018), e0201879. doi: 10.1371/journal.pone.0195800
[36]	S. Gavrilets, Collective action problem in heterogeneous groups, Phil. Trans. R. Soc. B, 370 (2015), 20150016. doi: 10.1098/rstb.2015.0016
[37]	W. Goffman, V. A. Vaun, A. Newill, Generalization of epidemic theory an application to the transmission of ideas, Nature, 204 (1964), 225–228. doi: 10.1038/204225a0
[38]	J. N. C. Gonçalves, H. S. Rodrigues, M. T. T. Monteiro, A contribution of dynamical systems theory and epidemiological modeling to a viral marketing campaign, In: Intelligent systems design and applications, Cham: Springer, 2017,974–983.
[39]	M. Granovetter, Threshold models of collective behavior, Am. J. Soc., 83 (1978), 1420–1443. doi: 10.1086/226707
[40]	N. Gurley, D. K. Johnson, Viral economics: an epidemiological model of knowledge diffusion in economics, Oxford Economic Papers, 69 (2017), 320–331. doi: 10.1093/oep/gpw044
[41]	H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599–653. doi: 10.1137/S0036144500371907
[42]	H. W. Hethcote, Three basic epidemiological models, In: Applied mathematical ecology, Berlin, Heidelberg: Springer, 1989,119–144.
[43]	F. C. Hoppensteadt, Mathematical methods of population biology, Cambridge: Cambridge University Press, 1982.
[44]	A. Huo, N. Song, Dynamical interplay between the dissemination of scientific knowledge and rumor spreading in emergency, Physica A, 461 (2016), 73–84. doi: 10.1016/j.physa.2016.05.028
[45]	R. A. Jeffs, J. Hayward, P. A. Roach, J. Wyburn, Activist model of political party growth, Physica A, 442 (2016), 359–372. doi: 10.1016/j.physa.2015.09.002
[46]	D. I. Kaiser, M. A. Bettencourt, A. Cintro, C. Castillo-Chavez, The power of a good idea: Quantitative modeling of the spread of ideas from epidemiological models, Physica A, 364 (2006), 513–536. doi: 10.1016/j.physa.2005.08.083
[47]	K. Kawachi, Deterministic models for rumor transmission, Nonlinear Anal. Real, 9 (2008), 1989–2028. doi: 10.1016/j.nonrwa.2007.06.004
[48]	W. Kermack, A. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. A, 115 (1927), 700–721. doi: 10.1098/rspa.1927.0118
[49]	I. Z. Kiss, M. Broom, P. G. Craze, I. Rafols, Can epidemic models describe the diffusion of topics across disciplines?, J. Informetr., 4 (2010), 74–82. doi: 10.1016/j.joi.2009.08.002
[50]	J. Lang, H. De Sterck, The Arab spring: A simple compartmental model for the dynamics of a revolution, Math. Soc. Sci., 69 (2014), 12–21. doi: 10.1016/j.mathsocsci.2014.01.004
[51]	M. Lewis, S. V. Petrovskii, J. Potts, The mathematics behind biological invasions, Springer International Publishing, 2016.
[52]	M. Lynch, The Arab uprising : the unfinished revolutions of the new Middle East, PublicAffairs, 2013.
[53]	H. MacGregor, M. Leach, A. Wilkinson, M. Parker, COVID-19 – a social phenomenon requiring diverse expertise, 2020. Available from: https://www.ids.ac.uk/opinions/covid-19-a-social-phenomenon-requiring-diverse-expertise/.
[54]	V. Mendez, S. Fedotov, H. Werner, Reaction-transport systems, Springer-Verlag Berlin Heidelberg, 2010.
[55]	J. Miller, Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes, Infect. Dis. Model., 2 (2017), 35–55.
[56]	D. Mistry, Q. Zhang, N. Perra, A. Baronchelli, Committed activists and the reshaping of status-quo social consensus, Phys. Rev. E Stat. Nonlin. Soft Matter Phys., 92 (2015), 042805. doi: 10.1103/PhysRevE.92.042805
[57]	D. Moritz Marutschke, H. Ogawa, Clustering scientific publication trends in cultural context using epidemiological model parameters, Procedia Technology, 18 (2014), 90–95. doi: 10.1016/j.protcy.2014.11.018
[58]	A. Morozov, S. Petrovskii, S. Gavrilets, The Yellow Vests Movement - a case of long transient dynamics?, 2019, 10.31235/osf.io/tpyux.
[59]	E. N. Nepomuceno, D. F. Resende, M. J. Lacerda, A survey of the individual-based model applied in biomedical and epidemiology research, J. Biomed. Res. Rev., 1 (2018), 11–24.
[60]	M. Perrie, The Russian Peasant Movement of 1905–1907: Its social composition and revolutionary significance, Past & Present, 57 (1972), 123–155.
[61]	S. Petrovskii, W. Alharbi, A. Alhomairi, A. Morozov, Modelling population dynamics of social protests in time and space : the reaction-diffusion approach, Mathematics, 8 (2020), 78. doi: 10.3390/math8010078
[62]	R. M. Raafat, N. Chater, C. Frith, Herding in humans, Trends Cogn. Sci., 13 (2009), 420–428. doi: 10.1016/j.tics.2009.08.002
[63]	H. S. Rodrigues, M. Fonseca, Viral marketing as epidemiological model, In: Proceedings of the 15th International Conference on Computational and Mathematical Methods in Science and Engineering, CMMSE, Cadiz, July, 2015.
[64]	S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, In: Mathematics for life science and medicine, Springer, 2007, 97–122.
[65]	F. J. Santonja, A. C. Tarazona, R. J. Villanueva, A mathematical model of the pressure of an extreme ideology on a society, Comput. Math. Appl., 56 (2008), 836–846. doi: 10.1016/j.camwa.2008.01.001
[66]	A. Schussman, S. A. Soule, Process and protest: accounting for individual protest participation, Social Forces, 84 (2005), 1083–1108. doi: 10.1353/sof.2006.0034
[67]	C. I. Siettos, L. Russo, Mathematical modeling of infectious disease dynamics, Virulence, 4 (2013), 295–306. doi: 10.4161/viru.24041
[68]	J. Skaza, B. Blais, Modeling the infectiousness of Twitter hashtags, Physica A, 465 (2017), 289–296. doi: 10.1016/j.physa.2016.08.038
[69]	L. M. Smith, A. L. Bertozzi, P. J. Brantingham, G. E. Tita, M. Valasik, Adaptation of an ecological territorial model to street gang spatial patterns in los angeles, Discrete Cont. Dyn. A, 32 (2012), 3223–3244. doi: 10.3934/dcds.2012.32.3223
[70]	D. A. Snow, R. Vliegenthart, C. Corrigall-Brown, Framing the French riots: a comparative study of frame variation, Social Forces, 86 (2007), 385–415. doi: 10.1093/sf/86.2.385
[71]	M. J. A. Stark, W. J. Raine, S. L. Burbeck, K. K. Davison, Some empirical patterns in a riot process, Am. Sociol. Rev., 39 (1974), 865–876. doi: 10.2307/2094159
[72]	E. Vynnycky, R. G. White, An introduction to infectious disease modelling, Oxford: Oxford University Press, 2010.
[73]	L. Wang, B. C. Wood, An epidemiological approach to model the viral propagation of memes, Appl. Math. Model., 35 (2011), 5442–5447. doi: 10.1016/j.apm.2011.04.035
[74]	Wikipedia, Mouvement des Gilets jaunes. Available from: https://fr.wikipedia.org/wiki/Mouvement_des_Gilets_jaunes.
[75]	J. Woo, H. Chen, Epidemic model for information diffusion in web forums: experiments in marketing exchange and political dialog, SpringerPlus, 5 (2016), 66. doi: 10.1186/s40064-016-1675-x
[76]	C. Yang, N. Rodriguez, A numerical perspective on traveling wave solutions in a system for rioting activity, Appl. Math. Comput., 364 (2020), 124646.
[77]	P. A. Yurevich, M. A. Olegovich, S. V. Mikhailovich, P. Y. Vasilievich, Modeling conflict in a social system using diffusion equations, Simulation, 94 (2018), 1053–1061. doi: 10.1177/0037549718761573
[78]	L. Zhao, J. Wang, Y. Chen, Q. Wang, J. Cheng, H. Cui, SIHR rumor spreading model in social networks, Physica A, 391 (2012), 2444–2453. doi: 10.1016/j.physa.2011.12.008
[79]	L. Zhao, W. Xie, H. O. Gao, X. Qiu, X. Wang, S. Zhang, A rumor spreading model with variable forgetting rate, Physica A, 392 (2013), 6146–6154. doi: 10.1016/j.physa.2013.07.080

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)