Bi-dimensional crime model based on anomalous diffusion with law enforcement effect

Francisco Javier Martínez-Farías,Arturo Hernández-Hernández,Eduardo Rangel-Cortes,Anahí Alvarado-Sánchez

doi:10.53391/mmnsa.2022.01.003

Abstract

Several models based on discrete and continuous fields have been proposed to comprehend residential criminal dynamics. This study introduces a two-dimensional model to describe residential burglaries diffusion, employing Lèvy flights dynamics. A continuous model is presented, introducing bidimensional fractional operator diffusion and its differences with the 1-dimensional case. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even fractional derivative order is modified. We also provide qualitative evidence that steady-state approximation in one dimension by series expansion is insufficient to capture similar original system behavior. At least for the case where series coefficients have a linear relationship with derivative order. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even if fractional derivative order is modified. Two dynamic regimes emerge in maximum and total attractiveness magnitude as a result of fractional derivative changes, these regimes can be understood as considerations about different urban environments. Finally, we add a Law enforcement component, embodying the "Cops on dots" strategy; in the Laplacian diffusion dynamic, global attractiveness levels are significantly reduced by Cops on dots policy but lose efficacy in Lèvy flight-based diffusion regimen. The four-step Preditor-Corrector method is used for numerical integration, and the fractional operator is approximated, getting the advantage of the spectral methods to approximate spatial derivatives in two dimensions.

Highlights

The present work is motivated by the impact that insecurity produces for an urban area; there are different types of crimes, and each one must be studied to later design prevention policies
We have made a numerical study of the fractional model for the bi-dimensional case of home thieves dynamics, incorporating the police effect and using the Cops on dots strategy
It is observed that applying a Predictor-Corrector 4 schedule to a random initial attractiveness distribution, the number and hotspot magnitude are related to the derivative order s of the fractional operator

Summary

Introduction

The present work is motivated by the impact that insecurity produces for an urban area; there are different types of crimes, and each one must be studied to later design prevention policies. The present work shows a two-dimensional extension of the Chaturapruek continuous model, incorporating law enforcement with Cops on dots strategy. These models have been published at the agent level (discrete) for the two-dimensional case, using Lévy flights, for example Brantingham et al [1], but this work proposes a deduction for the continuous two-dimensional case based on the one already made by Chaturapruek for one dimension. The main aim is to show numerically that hotspot solutions in attractiveness bi-dimensional fields are preserved by varying derivative order (in not truncated Lévy flights) using a spectral approximation to bi-dimensional fractional derivative operator In the final of this manuscript, a nomenclature section is presented

Continuum fractional model formulation

Approximate analysis for stationary solutions in one dimension

Numerical analysis of fractional order in 2-dimensions

Law enforcement of fractional bi-dimensional model

Conclusions