A mathematical model of a criminal-prone society

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2011, 4(1): 193-207. doi: 10.3934/dcdss.2011.4.193

A mathematical model of a criminal-prone society

Juan Carlos Nuño ^1, , Miguel Angel Herrero ^2, and Mario Primicerio ^3,

1.	Departamento de Matemática Aplicada a los Recursos Naturales, E.T.S.I. Montes. Universidad Politécnica de Madrid, 28040 Madrid, Spain
2.	IMI and Departamento de Matemática Aplicada. Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
3.	Dipartimento di Matematica, università degli Studi di Firenze, 55015 Firenze, Italy

Received November 2008 Revised April 2009 Published October 2010

Abstract
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Criminals are common to all societies. To fight against them the community takes different security measures as, for example, to bring about a police. Thus, crime causes a depletion of the common wealth not only by criminal acts but also because the cost of hiring a police force. In this paper, we present a mathematical model of a criminal-prone self-protected society that is divided into socio-economical classes. We study the effect of a non-null crime rate on a free-of-criminals society which is taken as a reference system. As a consequence, we define a criminal-prone society as one whose free-of-criminals steady state is unstable under small perturbations of a certain socio-economical context. Finally, we compare two alternative strategies to control crime: (i) enhancing police efficiency, either by enlarging its size or by updating its technology, against (ii) either reducing criminal appealing or promoting social classes at risk.

Keywords: population dynamics, Criminality, social mobility., nonlinear dynamical system.

Mathematics Subject Classification: Primary: 91D10, 34A34; Secondary: 37N4.

Citation: Juan Carlos Nuño, Miguel Angel Herrero, Mario Primicerio. A mathematical model of a criminal-prone society. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 193-207. doi: 10.3934/dcdss.2011.4.193

References:

[1]	R. A. Araujo and T. B. S. Moreira, A dynamic model of production and traffic of drugs,, Economic Letters, 82 (2004), 371. doi: doi:10.1016/j.econlet.2003.09.015.
[2]	A. A. Berryman, The origins and evolution of predator-prey theory,, Ecology, 73 (1992), 1530. doi: doi:10.2307/1940005.
[3]	M. Campbell and P. Ormerod, Social interactions and the dynamics of crime,, , ().
[4]	E. Durkheim, "Le Crime PhÉnomène Normal. Les Regles de la Méthode Sociologique,", Paris 14 ed. 1960, (1960), 65.
[5]	J. Eck, Police problems: The complexity of problem theory, research and evaluation,, in, 15 (2003).
[6]	M. Felson, "Crime and Nature,", Sage Publications Inc., (2006).
[7]	L. E. Cohen and M. Felson, Social change and crime rate trends: A routine activity approach,, American Sociological Review, 44 (1979), 588. doi: doi:10.2307/2094589.
[8]	C. Lewis (ed.), "Modelling Crime and Offending: Recent Developments in England and Wales,", Occasional paper no. 80, (2003).
[9]	S. Kanazawa and M. C. Still, Why men commit crimes (and why they desist),, Sociological Theory, 18 (2000), 434. doi: doi:10.1111/0735-2751.00110.
[10]	J. C. Nuño, M. A. Herrero and M. Primicerio, A triangle model of criminality,, Physica A, 387 (2008), 2926.
[11]	K. Pease, Science in the service of crime reduction,, in, (2005).
[12]	L. J. Peter and R. Hull, "The Peter Principle. Why Things Always Go Wrong,", William Morrow & Co, (1969).
[13]	A. Quetelet, "Sur L'homme et le Developpement de ses Facultes, ou Essai de Physique Sociale,", Bachelier, (1835).
[14]	L. Real, The kinetics of functional response,, American Naturalist, 111 (1977), 289. doi: doi:10.1086/283161.
[15]	S. Pierazzini, Effect of crimes on a socially structured population,, preprint, ().
[16]	J. E. Strassmann, Rank crime and punishment,, Nature, 432 (2004), 160. doi: doi:10.1038/432160b.
[17]	L. G. Vargo, A note on crime control,, Bull. Math. Biophys., 28 (1966), 375. doi: doi:10.1007/BF02476819.
[18]	H. Zhao, F. Zhilan and C. Castillo-Chavez, The dynamics of poverty and crime,, MTBI-02-08M, (2002), 02.

show all references

References:

[1]	R. A. Araujo and T. B. S. Moreira, A dynamic model of production and traffic of drugs,, Economic Letters, 82 (2004), 371. doi: doi:10.1016/j.econlet.2003.09.015.
[2]	A. A. Berryman, The origins and evolution of predator-prey theory,, Ecology, 73 (1992), 1530. doi: doi:10.2307/1940005.
[3]	M. Campbell and P. Ormerod, Social interactions and the dynamics of crime,, , ().
[4]	E. Durkheim, "Le Crime PhÉnomène Normal. Les Regles de la Méthode Sociologique,", Paris 14 ed. 1960, (1960), 65.
[5]	J. Eck, Police problems: The complexity of problem theory, research and evaluation,, in, 15 (2003).
[6]	M. Felson, "Crime and Nature,", Sage Publications Inc., (2006).
[7]	L. E. Cohen and M. Felson, Social change and crime rate trends: A routine activity approach,, American Sociological Review, 44 (1979), 588. doi: doi:10.2307/2094589.
[8]	C. Lewis (ed.), "Modelling Crime and Offending: Recent Developments in England and Wales,", Occasional paper no. 80, (2003).
[9]	S. Kanazawa and M. C. Still, Why men commit crimes (and why they desist),, Sociological Theory, 18 (2000), 434. doi: doi:10.1111/0735-2751.00110.
[10]	J. C. Nuño, M. A. Herrero and M. Primicerio, A triangle model of criminality,, Physica A, 387 (2008), 2926.
[11]	K. Pease, Science in the service of crime reduction,, in, (2005).
[12]	L. J. Peter and R. Hull, "The Peter Principle. Why Things Always Go Wrong,", William Morrow & Co, (1969).
[13]	A. Quetelet, "Sur L'homme et le Developpement de ses Facultes, ou Essai de Physique Sociale,", Bachelier, (1835).
[14]	L. Real, The kinetics of functional response,, American Naturalist, 111 (1977), 289. doi: doi:10.1086/283161.
[15]	S. Pierazzini, Effect of crimes on a socially structured population,, preprint, ().
[16]	J. E. Strassmann, Rank crime and punishment,, Nature, 432 (2004), 160. doi: doi:10.1038/432160b.
[17]	L. G. Vargo, A note on crime control,, Bull. Math. Biophys., 28 (1966), 375. doi: doi:10.1007/BF02476819.
[18]	H. Zhao, F. Zhilan and C. Castillo-Chavez, The dynamics of poverty and crime,, MTBI-02-08M, (2002), 02.

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