September  2015, 10(3): 421-441. doi: 10.3934/nhm.2015.10.421

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

1. 

Department of Mathematics, Faculty Sciences, King Abdulaziz University, Jeddah, Saudi Arabia

2. 

Department of Mathematical Sciences, Politecnico of Torino, Corso Duca degli Abruzzi 24, 10129, Torino, Italy

3. 

Centro de Investigación y Estudios de Matemática (CONICET), Medina Allende s/n, 5000 Córdoba, Argentina

4. 

Departamento de Matemática Aplicada, Facultad de Ciencias, Universidad de Granada, 18071 Granada

Received  October 2014 Revised  January 2015 Published  July 2015

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.
Citation: Nicola Bellomo, Francesca Colasuonno, Damián Knopoff, Juan Soler. From a systems theory of sociology to modeling the onset and evolution of criminality. Networks & Heterogeneous Media, 2015, 10 (3) : 421-441. doi: 10.3934/nhm.2015.10.421
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. doi: 10.3934/krm.2008.1.249. Google Scholar

[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. doi: 10.1016/j.aml.2011.09.043. Google Scholar

[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, (1997). Google Scholar

[4]

K. D. Baily, Sociology and the New System Theory - Towards a Theoretical Synthesis,, Suny Press, (1994). Google Scholar

[5]

P. Ball, Why Society is a Complex Matter: Meeting Twenty-first Century Challenges with a New Kind of Science,, Springer-Verlag, (2012). doi: 10.1007/978-3-642-29000-8. Google Scholar

[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. doi: 10.1073/pnas.0711437105. Google Scholar

[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. doi: 10.1142/S0218202507002431. Google Scholar

[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. doi: 10.3934/krm.2013.6.459. Google Scholar

[9]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512300049. Google Scholar

[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). doi: 10.1142/S0218202511400069. Google Scholar

[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. doi: 10.1142/S021820251350053X. Google Scholar

[12]

N. Bellomo and M. Pulvirenti, Eds., Modeling in Applied Sciences - A Kinetic Theory Approach,, Birkhäuser, (2000). doi: 10.1007/978-1-4612-0513-5. Google Scholar

[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. doi: 10.1142/S0218202512500650. Google Scholar

[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). doi: 10.1371/journal.pone.0085531. Google Scholar

[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. doi: 10.1137/130922744. Google Scholar

[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. doi: 10.1073/pnas.0610172104. Google Scholar

[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. doi: 10.1142/S0218202513500656. Google Scholar

[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, 594 (2013), 155. doi: 10.1090/conm/594/11789. Google Scholar

[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. Google Scholar

[20]

E. De Angelis, On the mathematical theory of post-Darwinian mutations, selection, and evolution,, Math. Models Methods Appl. Sci., 24 (2014), 2723. doi: 10.1142/S0218202514500353. Google Scholar

[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. doi: 10.1142/S0218202509003838. Google Scholar

[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. doi: 10.1142/S0218202514500237. Google Scholar

[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). doi: 10.1371/journal.pone.0061458. Google Scholar

[24]

M. D'Orsogna and M. Perc, Statistical physics of crime: A review,, Phys. Life Rev., 12 (2014), 1. Google Scholar

[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. doi: 10.1098/rspa.2009.0239. Google Scholar

[26]

P. Fajnzlber, D. Lederman and N. Loayza, Inequality and violent crime,, J. Law Econ., 45 (2002), 1. doi: 10.1086/338347. Google Scholar

[27]

M. Felson, What every mathematician should know about modelling crime,, Eur. J. Appl. Math., 21 (2010), 275. doi: 10.1017/S0956792510000070. Google Scholar

[28]

S. Harrendorf, M. Heiskanen and S. Malby, International Statistics on Crime and Justice,, European Institute for Crime Prevention and Control, (2010). Google Scholar

[29]

D. Helbing, Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes,, 2nd edition, (2010). doi: 10.1007/978-3-642-11546-2. Google Scholar

[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. doi: 10.1177/073401689301800203. Google Scholar

[31]

E. Jager and L. Segel, On the distribution of dominance in populations of social organisms,, SIAM J. Appl. Math., 52 (1992), 1442. doi: 10.1137/0152083. Google Scholar

[32]

A. P. Kirman and N. J. Vriend, Learning to be loyal. A study of the Marseille fish market,, in Interaction and Market Structure, (2000), 33. doi: 10.1007/978-3-642-57005-6_3. Google Scholar

[33]

D. Knopoff, On the modeling of migration phenomena on small networks,, Math. Models Methods Appl. Sci., 23 (2013), 541. doi: 10.1142/S0218202512500558. Google Scholar

[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. doi: 10.1142/S0218202513400137. Google Scholar

[35]

R. M. May, Uses and abuses of mathematics in biology,, Science, 303 (2004), 790. doi: 10.1126/science.1094442. Google Scholar

[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. doi: 10.1007/s10955-012-0678-4. Google Scholar

[37]

G. Mohler and M. Short, Geographic profiling form kinetic models of criminal behavior,, SIAM J. Appl. Math., 72 (2012), 163. doi: 10.1137/100794080. Google Scholar

[38]

M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life,, Harvard University Press, (2006). Google Scholar

[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. doi: 10.3934/dcdss.2011.4.193. Google Scholar

[40]

H. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[41]

P. Ormerod, Crime: Economic incentives and social networks,, IEA Hobart Paper, 151 (2005), 1. doi: 10.2139/ssrn.879716. Google Scholar

[42]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods,, Oxford University Press, (2013). Google Scholar

[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. doi: 10.3934/dcdss.2014.7.807. Google Scholar

[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). doi: 10.1103/PhysRevE.82.066114. Google Scholar

[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. doi: 10.1142/S0218202508003029. Google Scholar

[46]

H. A. Simon, Models of Bounded Rationality: Empirically Grounded Economic Reason,, Volume 3, (1997). Google Scholar

[47]

P. E. Tetlock, Thinking the unthinkable: Sacred values and taboo cognitions,, Trends Cogn. Sci., 7 (2003), 320. doi: 10.1016/S1364-6613(03)00135-9. Google Scholar

show all references

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. doi: 10.3934/krm.2008.1.249. Google Scholar

[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. doi: 10.1016/j.aml.2011.09.043. Google Scholar

[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, (1997). Google Scholar

[4]

K. D. Baily, Sociology and the New System Theory - Towards a Theoretical Synthesis,, Suny Press, (1994). Google Scholar

[5]

P. Ball, Why Society is a Complex Matter: Meeting Twenty-first Century Challenges with a New Kind of Science,, Springer-Verlag, (2012). doi: 10.1007/978-3-642-29000-8. Google Scholar

[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. doi: 10.1073/pnas.0711437105. Google Scholar

[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. doi: 10.1142/S0218202507002431. Google Scholar

[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. doi: 10.3934/krm.2013.6.459. Google Scholar

[9]

N. Bellomo, B. Piccoli and A. Tosin, Modeling crowd dynamics from a complex system viewpoint,, Math. Models Methods Appl. Sci., 22 (2012). doi: 10.1142/S0218202512300049. Google Scholar

[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). doi: 10.1142/S0218202511400069. Google Scholar

[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. doi: 10.1142/S021820251350053X. Google Scholar

[12]

N. Bellomo and M. Pulvirenti, Eds., Modeling in Applied Sciences - A Kinetic Theory Approach,, Birkhäuser, (2000). doi: 10.1007/978-1-4612-0513-5. Google Scholar

[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. doi: 10.1142/S0218202512500650. Google Scholar

[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). doi: 10.1371/journal.pone.0085531. Google Scholar

[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. doi: 10.1137/130922744. Google Scholar

[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. doi: 10.1073/pnas.0610172104. Google Scholar

[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. doi: 10.1142/S0218202513500656. Google Scholar

[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, 594 (2013), 155. doi: 10.1090/conm/594/11789. Google Scholar

[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. Google Scholar

[20]

E. De Angelis, On the mathematical theory of post-Darwinian mutations, selection, and evolution,, Math. Models Methods Appl. Sci., 24 (2014), 2723. doi: 10.1142/S0218202514500353. Google Scholar

[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. doi: 10.1142/S0218202509003838. Google Scholar

[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. doi: 10.1142/S0218202514500237. Google Scholar

[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). doi: 10.1371/journal.pone.0061458. Google Scholar

[24]

M. D'Orsogna and M. Perc, Statistical physics of crime: A review,, Phys. Life Rev., 12 (2014), 1. Google Scholar

[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. doi: 10.1098/rspa.2009.0239. Google Scholar

[26]

P. Fajnzlber, D. Lederman and N. Loayza, Inequality and violent crime,, J. Law Econ., 45 (2002), 1. doi: 10.1086/338347. Google Scholar

[27]

M. Felson, What every mathematician should know about modelling crime,, Eur. J. Appl. Math., 21 (2010), 275. doi: 10.1017/S0956792510000070. Google Scholar

[28]

S. Harrendorf, M. Heiskanen and S. Malby, International Statistics on Crime and Justice,, European Institute for Crime Prevention and Control, (2010). Google Scholar

[29]

D. Helbing, Quantitative Sociodynamics. Stochastic Methods and Models of Social Interaction Processes,, 2nd edition, (2010). doi: 10.1007/978-3-642-11546-2. Google Scholar

[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. doi: 10.1177/073401689301800203. Google Scholar

[31]

E. Jager and L. Segel, On the distribution of dominance in populations of social organisms,, SIAM J. Appl. Math., 52 (1992), 1442. doi: 10.1137/0152083. Google Scholar

[32]

A. P. Kirman and N. J. Vriend, Learning to be loyal. A study of the Marseille fish market,, in Interaction and Market Structure, (2000), 33. doi: 10.1007/978-3-642-57005-6_3. Google Scholar

[33]

D. Knopoff, On the modeling of migration phenomena on small networks,, Math. Models Methods Appl. Sci., 23 (2013), 541. doi: 10.1142/S0218202512500558. Google Scholar

[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. doi: 10.1142/S0218202513400137. Google Scholar

[35]

R. M. May, Uses and abuses of mathematics in biology,, Science, 303 (2004), 790. doi: 10.1126/science.1094442. Google Scholar

[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. doi: 10.1007/s10955-012-0678-4. Google Scholar

[37]

G. Mohler and M. Short, Geographic profiling form kinetic models of criminal behavior,, SIAM J. Appl. Math., 72 (2012), 163. doi: 10.1137/100794080. Google Scholar

[38]

M. A. Nowak, Evolutionary Dynamics. Exploring the Equations of Life,, Harvard University Press, (2006). Google Scholar

[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. doi: 10.3934/dcdss.2011.4.193. Google Scholar

[40]

H. Othmer, S. R. Dunbar and W. Alt, Models of dispersal in biological systems,, J. Math. Biol., 26 (1988), 263. doi: 10.1007/BF00277392. Google Scholar

[41]

P. Ormerod, Crime: Economic incentives and social networks,, IEA Hobart Paper, 151 (2005), 1. doi: 10.2139/ssrn.879716. Google Scholar

[42]

L. Pareschi and G. Toscani, Interacting Multiagent Systems: Kinetic Equations and Monte Carlo Methods,, Oxford University Press, (2013). Google Scholar

[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. doi: 10.3934/dcdss.2014.7.807. Google Scholar

[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). doi: 10.1103/PhysRevE.82.066114. Google Scholar

[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. doi: 10.1142/S0218202508003029. Google Scholar

[46]

H. A. Simon, Models of Bounded Rationality: Empirically Grounded Economic Reason,, Volume 3, (1997). Google Scholar

[47]

P. E. Tetlock, Thinking the unthinkable: Sacred values and taboo cognitions,, Trends Cogn. Sci., 7 (2003), 320. doi: 10.1016/S1364-6613(03)00135-9. Google Scholar

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