Mathematical home burglary model with stochastic long crime trips and patrolling: Applied to Mexico City

Highlights

• In this article we propose a stochastic extension of the Jones et al. model for the spatial distribution of home burglary.

• The stochastic model includes the fact that a proportion of burglars is willing to make long trips to burgle faraway homes.

• Trips to distant targets are modeled by long-range jumps following a random motion, which is biased by the attractiveness field.

• The stochastic model allows the formation of central hotspots, these kinds of hotspots are seen in burglary data in urban zones.

• Our Stochastic with long-range jumps model are applied to three different scenarios in Mexico City.

Abstract

Mathematical models for predicting the geographical distribution of areas with high rates of home burglary and numerical experiments to evaluate the effectiveness of police patrol routing strategies in reducing crime can support security policymakers in planning more effective patrol routes. We give an overview of the model formulated by Jones et al. (2010) to study the effects of the presence of law enforcement on the formation of home-burglary hotspots. We propose an extension of the model to contemplate that a small proportion of burglars travel far away from their awareness space to burgle houses. We incorporate these trips in the form of long-range stochastic jumps that are biased towards more attractive sites. Simulations suggest that hotspots lose attractiveness as police more influences via deterrence on burglars’ decision making about returning home rather than burglarizing a house. The results indicate that a higher proportion of burglars making long crime journeys might counteract the deterrent effect form the police presence. A case study is carried out to predict at the zone and citywide levels the distribution of areas with the highest home burglary rates in Mexico City. The L-BFGS method is applied to get the values for the parameters in both the Jones et al. model and our model; we use data from home burglaries reported, patrol zones and prison admissions. The minimum of the objective function is slightly lower for our model than for the Jones et al. model. Results obtained through numerical simulations using our model better fit the spatial statistical distribution of home-burglary hotspots for the entire city than for the zones.

Introduction

The Mexico City Observatory on Security and Justice (OCMX) [1], [2] reports that in 2018 Mexico City (CDMX) was going through its fourth home burglary crisis as far back as records on this crime are kept, and mentions that the beginning of 2019 was marked by a very high number of property crimes—home and business burglary, and street robbery—. Concerning home burglary, CDMX ranked 12th nationwide in 2018 and 14th in 2019. In 2018, CDMX’s home burglary rate was $35\%$ higher than the national rate of 63.5 per 100,000 residents. The OCMX notes that during 2019, home burglary ranked fifth most crime committed on an average day. In the opinion of the Director-General of the National Citizen Observatory (ONC) [3], to date there are no national policies that have managed to stop the rise of criminal incidence, nor were able to prove as having positive effects in terms of improving society's quality of life.

The quest for efficient solutions opens the door for researchers to develop models and systems to predict criminals’ movement having the ability to evaluate the effectiveness of strategies in reducing crime incidence rates. Recently, physical and mathematical modeling have provided new insights into specific aspects of urban crime (e.g [4], [5], [6], [7]). Model-driven scenarios may offer decision-making support for security policymakers. Identification of trends and spatial patterns of crime is a concern of several research groups. For this purpose, some studies focus on the spatio-temporal clustering of areas with high rates of crime, known as hotspots. For example, the hotspot maps in Fig. 1 show patterns in the distribution of Basic Geostatistical Areas (AGEBs)—geographical areas occupied by a set of blocks delimited by streets, avenues, walkways or any other feature of easy identification, whose land use is mainly housing, industrial, service or commercial—within CDMX with the highest annual rates of home burglary victimization over the period 2016–2018.

The UCLA burglary model, presented by Short et al. in [8], describes the dynamics that gives rise to the formation of residential burglary hotspots. The model combines the repeat and near-repeat victimization phenomenon and the broken windows effect. Repeat events refer to burglaries occurring several times at the same house [9]. The near-repeat phenomenon suggests that after a home is burgled, neighboring houses may face an increased risk of future burglary victimization [10]. The broken windows effect refers to how social disorganization leads to crime; the theory says that if minor disorders are tolerated in the neighborhood, then community standards break down progressively, producing an environment that encourages criminal behavior wherein more serious crime can prosper [11].

The UCLA model assumes that criminal movement follows a random walk biased towards sites attractive to thieves. The model relies on the premise that burglars tend to commit offenses close to their awareness space, which comprises activity locations, such as home, workplace, and recreation areas; and the pathways that connect them [13]. It also assumes that delinquents are unwilling to travel much farther from their home neighborhood to distant neighborhoods with more desirable targets. Short et al. [8] first explore burglars movement by an agent-based model, where every site corresponds to a target house having a dynamic level of attractiveness. In the continuum limit, the discrete model yields two coupled diffusion equations for the criminal density and the attractiveness.

We find some support for the assumptions of Short et al. in research on the distance to crime. Earlier results [14], [15], [16] conclude that most offenders select crime sites at a distance not too far from their own home. Possible explanations for why offenders’ spatial mobility is limited are based on familiarity and cost-benefit considerations. However, some offenders look for targets farther away from their home area, thus traveling longer distances than others from the same neighborhood to commit a crime. The fact that longer crime trips occur has previously been associated with higher monetary gains [17], but more recently it has been explained by risk and effort-related information. As explained in [18], offenders are disposed to travel long distances, and then to increase travel effort, if detection and arrest risks are low enough to do so.

Chaturapruek et al. [19] extended the UCLA model to include offenders' non-local movement. In the model, burglars occasionally take long-range jumps to high-value targets outside neighborhoods near their awareness space. The offender agents' movement is depicted in the form of biased Lévy flights, which lead to fractional Laplacians in the continuum limit. Instead of allowing criminals to move arbitrarily far, Pan et al. [20] propose to limit jumps range to incorporate limited traveling distances. To exclude arbitrary long jumps, the authors modify the model of Chaturapruek et al. by adding a truncation to the Lévy flights. The continuum limit of the truncated Lévy flights is the Laplace operator.

In the work by Jones et al. [21], the effect of law enforcement agents is incorporated into the UCLA model. The authors examine the discrete agent-based model and the continuum limit thereof. In their model, the very presence of police serves to deter crimes without actually apprehending criminals. The model portrays deterrence by two mechanisms, one of which reduces the offenders’ perceived attractiveness, and the other influences these agents’ decisions directly. Furthermore, Jones et al. devised three crime-reduction patrolling strategies: random walks, cops on the dots, and peripheral interdiction. In the former, the police take random patrol routes to cover places. The second is a target-oriented patrol scheme, involving the deployment of police to high-crime hotspot areas. The latter routing strategy allocates police resources in peripheral areas of hotspots.

The goal of this paper is twofold. First, we extend the Jones et al. continuum model to integrate the occurrence of longer crime trips under the cops-on-the-dots scheme, assuming the contextual factor that police affect criminal agents’ actions. We adopt a stochastic modeling approach for the long crime trips made by a proportion of burglars. The model allows choosing randomly the burglars that will travel as well as their destinations. Second, we attempt to predict hotspots for selected AGEBs in Northwest Central and Southeast Mexico City and the entire city.

The content of the present work is divided in the following way: In Section 2, we introduce the Jones et al. model [21] and carry out numerical implementations for different parameter regimes. We discuss in Section 3 our proposal for adapting the model to capture long trips performed by the burglars. Moreover, to analyze the sensitivity of the parameters, simulations are conducted to illustrate the behavior of the attractiveness under different proportions of burglars that make longer crime trips. In Section 4, a case study is conducted for Mexico City. We construct the initial conditions and perform parameter estimation based on assumptions and statistical data. Finally, in Section 5, we briefly discuss the results and the direction of potential future work.