laboratoire de physique statistique
 
 
laboratoire de physique statistique

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EUROPEAN JOURNAL OF APPLIED MATHEMATICS 


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2010
Modelling the individual and collective dynamics of the propensity to offend - Nadal, J-P and Gordon, M. B. and Iglesias, J. R. and Semeshenko, V.
EUROPEAN JOURNAL OF APPLIED MATHEMATICS 21421-440 (2010)

Abstract : We introduce a general framework for modelling the dynamics of the propensity to offend in a population of (possibly interacting) agents. We consider that each agent has an `honesty index' which parameterizes his probability of abiding by the law. This probability also depends on a composite parameter associated to the attractiveness of the crime outcome and of the crime setting (the context which makes a crime more or less likely to occur, such as the presence or not of a guardian). Within this framework we explore some consequences of the working hypothesis that punishment has a deterrent effect, assuming that, after a criminal act, an agent's honesty index may increase if he is caught and decrease otherwise. We provide both analytical and numerical results. We show that in the space of parameters characterizing the probability of punishment, there are two `phases': one corresponding to a population with a low crime rate and the other to a population with a large crime rate. We speculate on the possible existence of a self-organized state in which, due to the society reaction against crime activities, the population dynamics would be stabilized on the critical line, leading to a wide distribution of propensities to offend in the population. In view of empirical works on the causes of the recent evolution of crime rates in developed countries, we discuss how changes of socio-economic conditions may affect the model parameters, and hence the crime rate in the population. We suggest possible extensions of the model that will allow us to take into account more realistic features.
Self-organised critical hot spots of criminal activity - Berestycki, H. and Nadal, J-P
EUROPEAN JOURNAL OF APPLIED MATHEMATICS 21371-399 (2010)

Abstract : In this paper(1) we introduce a family of models to describe the spatio-temporal dynamics of criminal activity. It is argued here that with a minimal set of mechanisms corresponding to elements that are basic in the study of crime, one can observe the formation of hot spots. By analysing the simplest versions of our model, we exhibit a self-organised critical state of illegal activities that we propose to call a warm spot or a tepid milieu(2) depending on the context. It is characterised by a positive level of illegal or uncivil activity that maintains itself without exploding, in contrast with genuine hot spots where localised high level or peaks are being formed. Within our framework, we further investigate optimal policy issues under the constraint of limited resources in law enforcement and deterrence. We also introduce extensions of our model that take into account repeated victimisation effects, local and long range interactions, and briefly discuss some of the resulting effects such as hysteresis phenomena.