As with most crime, the highest rates of burglary occur in urban communities since large metropolitan areas generally boast more concentrated wealth. Big cities also allow burglars to maintain anonymity and evade authority while offering ample opportunities for discreet disposal of stolen property. Burglars observe their target cities with the careful attention of urban planners, taking note of public spaces, roadways, building architecture, behavior patterns, and tenant schedules. Although law enforcement is making concerted efforts to address and prevent burglary, frequent offenses in major metropolises continue to unsettle city-dwellers.
Existing mathematical models typically examine burglaries in residential, suburban environments, where similarly-structured houses with predictable lattice alignments are hotspots for repeated criminal activity. Some are agent-based, others utilize differential equations, and still others account for the effect of police presence. These models suggest that residential burglars prefer revisiting previously-burgled houses--or those with similar architecture--because they are already familiar with layout, security features, and availability of goods. Thus, if a home or its neighboring residence is robbed, repeat or near-repeat victimization heightens that home's attractiveness. While this phenomenon--upon which most models are based--occurs throughout the world, it is much more common in suburban districts. Flexible models that include alternate patterns of victimization are especially desirable when considering urban burglary.
Their work is inspired by age-dependent population models, which study a population's evolution in time based on the physiological ages of its individuals. "Our model puts the emphasis on when--rather than where--the burglaries will take place and on the type of victimized houses, represented by their 'age,'" Avinyó said. A burglar's age is the amount of time since his most recent offense, while a house's age is the amount of time since it was last burglarized. The likelihood of robbery acts as a function of a burglar's age, and a house's susceptibility is a function of that house's age. When a burglar commits a crime, the ages of both the house and the burglar reset to zero. These details add a level of heterogeneity to the populations of houses and burglars.
Because behavioral hypotheses related to repeat and near-repeat victimization limit a model's customizability, Saldaña et al. consider general functions for the recurrence rate (tendency of burglars to commit a crime) and victimization rate (rate at which houses are robbed). "Contrary to previous models where repeat and near-repeat victimization theories are widely considered, our model is compatible with different scenarios," Ripoll said. "Our age-structured model is a conceptually different approach in comparison with agent-based models where burglars are just seen as "particles" entering and leaving the system randomly. Less a priori assumptions are needed."
After establishing these fundamentals, Saldaña et al. modify their initial predator-prey system to account for active police response to criminal activity. "The organization of police resources and the allocation of police units is one of the main concerns of police departments," Ripoll said. "Nowadays many police departments tend to allocate police resources to help citizens feel secure. They ask police patrols to move randomly around different areas so that people develop a (sometimes false) sense of security. But this has long been observed to be quite an inefficient way to employ police resources."
Ultimately, Saldaña et al.'s nonlinear model of urban burglary offers more flexibility than traditional models based on spatio-temporal descriptions of criminal activity. "Our model is simple enough to provide some explicit formulae for relationships between different aspects of the dynamics," Pellicer said. "These can be contrasted with real data--for instance, the mean time between two consecutive burglaries of the same house and the mean time between two consecutive offenses committed by the same burglar--under different police strategies." When studying burglaries in a particular city, researchers must adjust their model's parameters and functions to correlate both qualitatively and quantitatively with real data.
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Source article: Saldaña, J., Aguareles, M., Avinyó, A., Pellicer, M., & Ripoll, J. (2018). An Age-Structured Population Approach for the Mathematical Modeling of Urban Burglaries. SIAM J. Appl. Dynam. Syst. To be published.
SIAM Journal on Applied Dynamical Systems