From a systems theory of sociology to modeling the onset and evolution of criminality

Nicola  Bellomo; Francesca  Colasuonno; Damián  Knopoff; Juan Soler

doi:10.3934/nhm.2015.10.421

Article Contents

2015, Volume 10, Issue 3: 421-441. Doi: 10.3934/nhm.2015.10.421

This issue Previous Article Special issue on modeling and control in social dynamics Next Article A model of riots dynamics: Shocks, diffusion and thresholds

From a systems theory of sociology to modeling the onset and evolution of criminality

1.
Department of Mathematics, Faculty Sciences, King Abdulaziz University, Jeddah
2.
Department of Mathematical Sciences, Politecnico of Torino, Corso Duca degli Abruzzi 24, 10129, Torino
3.
Centro de Investigación y Estudios de Matemática (CONICET), Medina Allende s/n, 5000 Córdoba
4.
Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada

Received: October 2014

Revised: January 2015

Published: September 2015

Abstract / Introduction Related Papers Cited by

Abstract

This paper proposes a systems theory approach to the modeling of onset and evolution of criminality in a territory. This approach aims at capturing the complexity features of social systems. Complexity is related to the fact that individuals have the ability to develop specific heterogeneously distributed strategies, which depend also on those expressed by the other individuals. The modeling is developed by methods of generalized kinetic theory where interactions and decisional processes are modeled by theoretical tools of stochastic game theory.

Keywords:

Mathematics Subject Classification: 35Q91, 91C99, 91D10.

Citation:

Related Papers

Cited by

References

[1]	G. Ajmone Marsan, N. Bellomo and M. Egidi, Towards a mathematical theory of complex socio-economical systems by functional subsystems representation, Kinet. Relat. Models, 1 (2008), 249-278.doi: 10.3934/krm.2008.1.249.
[2]	L. Arlotti, E. De Angelis, L. Fermo, M. Lachowicz and N. Bellomo, On a class of integro-differential equations modeling complex systems with nonlinear interactions, Appl. Math. Lett., 25 (2012), 490-495.doi: 10.1016/j.aml.2011.09.043.
[3]	W. B. Arthur, S. N. Durlauf and D. A. Lane, Eds., The Economy as an Evolving Complex System II, Studies in the Sciences of Complexity, XXVII, Addison-Wesley, 1997.
[4]	K. D. Baily, Sociology and the New System Theory - Towards a Theoretical Synthesis, Suny Press, 1994.
[5]	P. Ball, Why Society is a Complex Matter: Meeting Twenty-first Century Challenges with a New Kind of Science, Springer-Verlag, Heidelberg, 2012.doi: 10.1007/978-3-642-29000-8.
[6]	M. Ballerini, N. Cabibbo, R. Candelier, A. Cavagna, E. Cisbani, I. Giardina, V. Lecomte, A. Orlandi, G. Parisi, A. Procaccini, M. Viale and V. Zdravkovic, Interaction ruling animal collective behavior depends on topological rather than metric distance: Evidence from a field study, Proc. Natl. Acad. Sci. USA, 105 (2008), 1232-1237.doi: 10.1073/pnas.0711437105.
[7]	N. Bellomo, A. Bellouquid, J. Nieto and J. Soler, Multicellular biological growing systems: Hyperbolic limits towards macroscopic description, Math. Models Methods Appl. Sci., 17 (2007), 1675-1692.doi: 10.1142/S0218202507002431.
[8]	N. Bellomo, M. A. Herrero and A. Tosin, On the dynamics of social conflicts: Looking for the Black Swan, Kinet. Relat. Mod., 6 (2013), 459-479.doi: 10.3934/krm.2013.6.459.
[9]	N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint, Math. Models Methods Appl. Sci., 22 (2012), paper No.1230004.doi: 10.1142/S0218202512300049.
[10]	N. Bellomo and J. Soler, On the mathematical theory of the dynamics of swarms viewed as complex systems, Math. Models Methods Appl. Sci., 22 (2012), paper No.1140006.doi: 10.1142/S0218202511400069.
[11]	N. Bellomo, D. Knopoff and J. Soler, On the difficult interplay between life, "complexity'', and mathematical sciences, Math. Models Methods Appl. Sci., 23 (2013), 1861-1913.doi: 10.1142/S021820251350053X.
[12]	N. Bellomo and M. Pulvirenti, Eds., Modeling in Applied Sciences - A Kinetic Theory Approach, Birkhäuser, Boston, 2000.doi: 10.1007/978-1-4612-0513-5.
[13]	A. Bellouquid, E. De Angelis and D. Knopoff, From the modeling of the immune hallmarks of cancer to a black swan in biology, Math. Models Methods Appl. Sci., 23 (2013), 949-978.doi: 10.1142/S0218202512500650.
[14]	B. Berenji, T. Chou and M. D'Orsogna, Recidivism and rehabilitation of criminal offenders: A carrot and stick evolutionary games, PLOS ONE, 9 (2014), 885531.doi: 10.1371/journal.pone.0085531.
[15]	H. Berestycki, J. Wei and M. Winter, Existence of symmetric and asymmetric spikes of a crime hotspot model, SAM J. Math. Anal., 46 (2014), 691-719.doi: 10.1137/130922744.
[16]	L. M. A. Bettencourt, J. Lobo, D. Helbing, C. Kohnert and G. B. West, Growth, innovation, scaling, and the pace of life in cities, Proc. Natl. Acad. Sci. USA, 104 (2007), 7301-7306.doi: 10.1073/pnas.0610172104.
[17]	J. J. Bissell, C. C. S. Caiado, M. Goldstein and B. Straughan, Compartmental modelling of social dynamics with generalized peer incidence, Math. Models Methods Appl. Sci., 24 (2014), 719-750.doi: 10.1142/S0218202513500656.
[18]	F. Colasuonno and M. C. Salvatori, Existence and uniqueness of solutions to a Cauchy problem modeling the dynamics of socio-political conflicts, in Recent Trends in Nonlinear Partial Differential Equations I: Evolution Problems (eds. J. B. Serrin, E. L. Mitidieri and V. D. Radulescu), Series Cont. Math. AMS, Providence, USA, Contemporary Mathematics, 594 (2013), 155-165.doi: 10.1090/conm/594/11789.
[19]	T. Davies, H. Fry, A. Wilson and S. Bishop, A Mathematical Model of the London Riots and Their Policing, Scientific Report, 2013.doi: 10.1038/srep01303.
[20]	E. De Angelis, On the mathematical theory of post-Darwinian mutations, selection, and evolution, Math. Models Methods Appl. Sci., 24 (2014), 2723-2742.doi: 10.1142/S0218202514500353.
[21]	S. De Lillo, M. Delitala and M. C. Salvatori, Modelling epidemics and virus mutations by methods of the mathematical kinetic theory for active particles, Math. Models Methods Appl. Sci., 19 (2009), 1405-1425.doi: 10.1142/S0218202509003838.
[22]	M. Dolfin and M. Lachowicz, Modeling altruism and selfishness in welfare dynamics: The role of nonlinear interactions, Math. Models Methods Appl. Sci., 24 (2014), 2361-2381.doi: 10.1142/S0218202514500237.
[23]	M. D'Orsogna, R. Kendall, M. McBride and M. B. Short, Criminal defectors lead to the emergence of cooperation in an experimental,adversarial game, PLOS ONE, 8 (2013), e61458.doi: 10.1371/journal.pone.0061458.
[24]	M. D'Orsogna and M. Perc, Statistical physics of crime: A review, Phys. Life Rev., 12 (2014), 1-21.
[25]	B. Düring, P. Markowich, J.-F. Pietschmann and M.-T. Wolfram, Boltzmann and Fokker-Planck equations modelling opinion formation in the presence of strong leaders, P. R. Soc. London, 465 (2009), 3687-3708.doi: 10.1098/rspa.2009.0239.
[26]	P. Fajnzlber, D. Lederman and N. Loayza, Inequality and violent crime, J. Law Econ., 45 (2002), 1-39.doi: 10.1086/338347.
[27]	M. Felson, What every mathematician should know about modelling crime, Eur. J. Appl. Math., 21 (2010), 275-281.doi: 10.1017/S0956792510000070.
[28]	S. Harrendorf, M. Heiskanen and S. Malby, International Statistics on Crime and Justice, European Institute for Crime Prevention and Control, affiliated with the United Nations (HEUNI), 2010.
[29]	D. Helbing, Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes, 2nd edition, Springer, Berlin Heidelberg, 2010.doi: 10.1007/978-3-642-11546-2.
[30]	C. C. Hsieh and M. D. Pugh, Poverty, income inequality, and violent crime: A meta-analysis of recent aggregate data studies, Crim. Just. Rev., 18 (1993), 182-202.doi: 10.1177/073401689301800203.
[31]	E. Jager and L. Segel, On the distribution of dominance in populations of social organisms, SIAM J. Appl. Math., 52 (1992), 1442-1468.doi: 10.1137/0152083.
[32]	A. P. Kirman and N. J. Vriend, Learning to be loyal. A study of the Marseille fish market, in Interaction and Market Structure, Lecture Notes in Economics and Mathematical Systems, 484, Springer-Verlag, Heidelberg, 2000, 33-56.doi: 10.1007/978-3-642-57005-6_3.
[33]	D. Knopoff, On the modeling of migration phenomena on small networks, Math. Models Methods Appl. Sci., 23 (2013), 541-563.doi: 10.1142/S0218202512500558.
[34]	D. Knopoff, On a mathematical theory of complex systems on networks with application to opinion formation, Math. Models Methods Appl. Sci., 24 (2014), 405-426.doi: 10.1142/S0218202513400137.
[35]	R. M. May, Uses and abuses of mathematics in biology, Science, 303 (2004), 790-793.doi: 10.1126/science.1094442.
[36]	S. McCalla, M. Short and P. J. Brantingham, The effects of sacred value networks within and evolutionary, adversarial game, J. Stat. Phys., 151 (2013), 673-688.doi: 10.1007/s10955-012-0678-4.
[37]	G. Mohler and M. Short, Geographic profiling form kinetic models of criminal behavior, SIAM J. Appl. Math., 72 (2012), 163-180.doi: 10.1137/100794080.
[38]	M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life, Harvard University Press, 2006.
[39]	J. C. Nuño, M. A. Herrero and M. Primicerio, A mathematical model of a criminal-prone society, Discr. Cont. Dyn. Syst. S, 4 (2011), 193-207.doi: 10.3934/dcdss.2011.4.193.
[40]	H. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems, J. Math. Biol., 26 (1988), 263-298.doi: 10.1007/BF00277392.
[41]	P. Ormerod, Crime: Economic incentives and social networks, IEA Hobart Paper, 151 (2005), 1-54.doi: 10.2139/ssrn.879716.
[42]	L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods, Oxford University Press, USA, 2013.
[43]	P. Pucci and M. C. Salvatori, On an initial value problem modeling evolution and selection in living systems, Disc. Cont. Dyn. Syst. S, 7 (2014), 807-821.doi: 10.3934/dcdss.2014.7.807.
[44]	M. B. Short, P. J. Brantingham and M. R. D'Orsogna, Cooperation and punishment in an adversarial game: How defectors pave the way to a peaceful society, Phys. Rev. E, 82 (2010), 066114, 7pp.doi: 10.1103/PhysRevE.82.066114.
[45]	M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267.doi: 10.1142/S0218202508003029.
[46]	H. A. Simon, Models of Bounded Rationality: Empirically Grounded Economic Reason, Volume 3, MIT Press, Cambridge, MA, 1997.
[47]	P. E. Tetlock, Thinking the unthinkable: Sacred values and taboo cognitions, Trends Cogn. Sci., 7 (2003), 320-324.doi: 10.1016/S1364-6613(03)00135-9.