An Age-Structured Population Approach for the Mathematical Modeling of Uruail Burglaries

We propose a nonlinear model for the dynamics of urban burglaries which takes into account the deterring effect of the police. The model focuses on the timing of criminal activity rather than on the spatial spreading of burglaries and it is inspired in the age-dependent population dynamics. The structuring variables are the time elapsed between two consecutive offenses committed by a burglar or suffered by a house. The main ingredients of the model are the propensity of burglars to commit a crime and the rate at which houses are being burgled. These rates are taken as general as possible to allow different scenarios, including the widely used repeat victimization pattern. The dissuasive effect of the active police deployment is introduced by means of a memory term that depends on the number of the last committed burglaries. The asymptotic behavior of the model and the existence of a globally stable equilibrium are determined thanks to a suitable change of variables that involves a continuous rescaling of the time variable. This new approach provides some interesting analytic results on the equilibrium and the expected times between consecutive offenses. Numerical simulations are shown to illustrate these results.