# American Institute of Mathematical Sciences

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

## A mathematical model of a criminal-prone society

Received  November 2008 Revised  April 2009 Published  October 2010

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.
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
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##### References:
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