Spatial modeling of crime dynamics: Patch and reaction–diffusion compartmental systems

Abstract

We study the dynamics of abstract models for crime evolution. The population is divided into three compartments, taking into account the participation in crime and incarceration. Individuals transit between the three segments, assuming that having more contact with criminally active people increases one's risk of learning and acquiring the same traits; essentially, crime is regarded as a social epidemic. In the literature, there are several models of this type, based on spatial homogeneity and ordinary differential equations. However, these ideas have not been extended to account for spatial variability. Here, we achieve this target with discrete and continuous forms of space: patch and reaction–diffusion compartmental systems, respectively. We build the models and focus on the effect of the basic reproduction number on the long-term dynamics of crime (persistence or disappearance). Several theoretical results are presented, which are supported by numerical simulations.