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Another way of thinking: a review of mathematical models of crime

January 2013

January 2013

Authors:

Balswaroop Bhatt

University of the West Indies, St. Augustine

Donna M. G. Comissiong

University of the West Indies, St. Augustine

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Mathematics TODAY JUNE 2013 131

Another Way of Thinking: A Review of

Mathematical Models of Crime

Joanna Sooknanan, Balswaroop Bhatt FIMA and Donna M.G. Comissiong

∗

Abstract

Mathematical models are useful weapons

in the crime-ﬁghting arsenal. With the

development of cheaper and more pow-

erful computers, mathematical modelling

of systems representing some aspect of

crime or criminal behaviour and the anal-

ysis of the resulting numerical solutions

is becoming more popular. Models may be

and numerical simulation via the computer

must be used to determine their approxi-

mate solutions. These solutions may be in

the form of graphs showing the system’s

behaviour over time as well as its sensitiv-

ity to variations in key model parameters.

Mathematical modelling and numeri-

cal simulation of systems have once again

resulted in a friendship of sorts between

used to guide decision-making, develop policies or to evalu-

ate speciﬁc strategies aimed at reducing crime. This review

provides an introduction to some relatively recent mathematical

models of crime.

1 Introduction

With all the numbers bandied about in the media, one

might be forgiven for thinking that crime rates are up

– and not to any good either. Homicide rates, convic-

tion rates, the increase in robberies with accompanying charts . . .

ﬁgures galore. Statistics has generally gone hand in hand with

crime and all this number crunching has traditionally left a bitter

taste in the public’s mouth. Add mathematics to this mix, and this

bitter taste turns noxious.

Though the popularity of television shows like NUMB3RS

has made mathematics more palatable to the crime-fearing pub-

lic, mathematics is still viewed as a distant, fearsome relative of

criminology. This, despite its long association with the social sci-

ences. The use of mathematics to describe social phenomena has

its roots in the ‘Era of Enlightenment’ in the nineteenth century

with the birth of statistics and probability theory and their use

in the collection and analysis of social statistics. However, this

collaboration was short-lived as the social sciences turned away

from statistics and moved towards a psychological approach to

understanding behaviour [1].

Criminal behaviour and activities have evolved in tandem

with changes in technology. Criminal activities now include

arms, drugs and human traﬃcking, money laundering, cyber-

crime, identity theft and gangs with international links. Crime

has become more sophisticated, organised and transnational.

With the changing nature of crime, traditional approaches to tack-

ling it are fast becoming obsolete and there is a growing need for

a new way of thinking to meet this challenge head on.

2 Mathematical modelling and numerical

simulation: partners in crime

The advent of cheaper, more powerful computers has ushered in

a revolution in mathematics. Mathematical modelling uses math-

ematics to transform real-world systems into abstract models so

as to understand, simulate or make predictions about their be-

haviour. Some of these systems may have no analytical solutions

mathematics and criminology. When modelling criminal be-

haviour and crime, since human behaviour is inherently nonlin-

ear [2], we assume that they may best be described by a nonlinear

system. This is contrary to the models generally used by policy-

makers, who are susceptible to what Ball [1] calls linear think-

ing. This has led to the development of linear models of human

behaviour with their inherent assumptions that cause-and-eﬀect

relationships are identiﬁable and that there is proportionality be-

tween inputs and outputs [1]. These properties make linear sys-

tems particularly useful for prediction and manipulation – hence

their popularity in modelling.

However, when a system contains nonlinear terms, analyti-

cal solutions may be diﬃcult to obtain so that numerical methods

must be used to obtain the approximate solutions. In nonlinear

systems, proportionality does not hold and there is a dispropor-

tional relationship between cause and eﬀect: small changes in key

parameters can trigger large changes in crime rate. Another fea-

ture of nonlinearity is the existence of bifurcation points in key

parameters. The bifurcations give ‘tipping points’ of the system

at which the system may make a sudden transition to a new, very

diﬀerent behaviour.

‘Mathematical modelling and numerical simulation comple-

ment the traditional empirical and experimental approaches to

research’ [3]. Modelling is especially important in criminology

since it helps organise and visualise existing data, identify areas

with missing data and is relatively inexpensive and more practi-

cal than carrying out an actual experiment. Modelling also of-

fers a means of varying conditions so as to conduct social ex-

periments but without the ethics and costs attached to experi-

menting on human beings. The insights provided by the models

may be especially helpful to those in authority who are charged

with the responsibility of designing policies often with a lack of

available data.

3 Crime ‘math’ers

Terminology used to characterise crime and criminal behaviour

seems to lend itself almost naturally to the use of existing math-

ematical models so as to mathematically represent a particular

crime situation. Some of these terms include crime waves, the

spread of crime, crime epidemic, the migration of criminals,

criminals preying upon the population and the formation of crime

hotspots. Two mathematical approaches used in the modelling of

crime and criminal behaviour are described next.

∗

Department of Mathematics and Statistics, The University of the West Indies,

Trinidad and Tobago.

features

Mathematics TODAY JUNE 2013 132

society without crime is possible [22]. While the previous mod-

elling approach was characterised by a ‘top-down’ modelling ap-

proach where the behaviour of the system is described at the start

by a system of equations, the agent-based model is characterised

by a ‘bottom-up’ approach where there is emergent behaviour.

4 Towards another way of thinking

Models may be used to guide decision-making, develop policies

or to evaluate speciﬁc strate-

gies aimed at reducing crime.

In developing models, the

question of model validity

or how well the model rep-

resents the real-world situa-

tion for which it is designed

naturally arises. Model val-

idation techniques include

consultation with experts

about the model design and

its behaviour, parameter

variability-sensitivity analy-

ses of model behaviour and the use of statistical tests and proce-

dures to compare model output for diﬀerent experimental condi-

tions with experimental data.

Experimental data in this research refers to crime data and

statistics. In designing models of crime, there are challenges as-

sociated with the data collection process. Some of these chal-

lenges include case attrition where cases that enter the system get

lost somewhere along the way, lack of data on oﬀenders, lack of

self-report studies, unreported crime due to a lack of trust in the

police and desensitisation to crime which may result in varying

degrees of tolerance to crime. This has led to concerns about

whether crime data should be viewed as representative of the

crime situation in a particular area and may lead to invalid ex-

planations of crime phenomenon and ineﬀectual policies to re-

duce crime [23]. Thus, most of the models reviewed were used

not to predict future trends but for insight into the behaviour

of the system.

In all of the models reviewed, we noted that building a crime

model involved a multidisciplinary approach so as to bridge the

gap between the physical and the social sciences. The ‘ideal type

of the division of labour in quantitative social science would be

one where the sociologist formulates a theory, the mathematician

translates it into a mathematical model, and the statistician pro-

vides the tool for estimating the model’ [24].

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3.1 Modelling via diﬀerential equations

In nature, transport occurs in ﬂuids through the combination of

advection and diﬀusion. Reaction-diﬀusion-advection systems

are used to study the spread of wavelike behaviour in a num-

ber of ﬁelds such as the migration of invasive species, the prop-

agation of genes and the spread of chemical reactions [4]. A

reaction-diﬀusion-advection model has been applied by Stan-

ford researchers Nancy Rodríguez, Henri Berestycki and Lenya

Ryzhik to describe and reduce the spread of crime waves out-

ward from crime hotspots [5]. Reaction-diﬀusion partial diﬀer-

ential equations have also been used to study the formation, dy-

namics, and steady-state properties of crime hotspots and to ex-

plain why these hotspots may either be displaced or eradicated by

police action [6].

Criminal behaviour and violence may also be treated as a

socially infectious disease [7] using concepts borrowed from

epidemiology. Researchers have recognised the propensity for

violent acts to cluster, to spread from one area to another

and to mutate and have suggested applying existing techniques

from mathematical epidemiology to treat the spread of violence

in a population [7].

In one of the earliest papers to acknowledge the social nature

of crime [8], individual crime was treated as a function of expo-

sure to crime-prone peers, where an individual is inﬂuenced in

his choice to commit a criminal act by his perceived probability

of punishment as obtained from his acquaintances. Ormerod,

Mounﬁeld and Smith [9] applied an infectious disease model

consisting of a system of coupled, nonlinear ordinary diﬀerential

equations to violent crime and burglary in the UK. The model di-

vided the population into four groups – three dependent on their

susceptibility to commit crimes and one group representing those

in jail. The model was used to test the eﬀect of crime-ﬁghting

policies on the criminal population.

A similar model was developed for the growth of gangs in

a population [10] by dividing the population into four distinct

groups based on gang status and risk factors with respect to

gang membership. The model examined the impact of vari-

ous crime-ﬁghting strategies by changing parameter values like

imprisonment and recidivism rates and identiﬁed bifurcation

points which resulted in the disappearance of gang members from

the population.

Closely related to infectious disease models are predator-prey

models which also use systems of ordinary diﬀerential equations

and seem to be a natural ﬁt for modelling criminals who ‘prey

upon’ the public. These models have also been used in the inverse

setting to describe the interactions between policemen (preda-

tors) and criminals (prey) and to examine the eﬀects of changes

in policy and law enforcement [11]. Other models include Nuno

et al. [12], who modelled a dynamical system based on routine

activity theory containing a group of motivated oﬀenders Y, suit-

able targets Xand a lack of guardianship. The model consisted

of owners Xwho are the prey, criminals Ywho are the predators

of X, and security guards Zwho are predators of both Xand Y.

Nuno et al. [13] also compared two diﬀerent strategies (upgrading

police forces and increasing social measures) for ﬁghting crime

in a criminal-prone self-protected society divided into ndiﬀer-

ent socio-economic classes. Criminals preying upon the villagers

who banded together in group defence were modelled so that the

criminals switched between areas targeting the less populated ar-

eas [14]. Police eﬀorts to catch criminals were included in the

model by applying constant eﬀort and constant yield harvesting

functions to capture the criminals.

3.2 Agent-based models

Another approach to modelling crime and criminal behaviour in

which numerical simulation plays an important role uses agent-

based models (ABM). These are made up of a collection of au-

tonomous decision-making entities called agents who interact

with each other and their environment, according to a set of spec-

iﬁed behavioural rules [15].

When used to model crime, the agents generally represent

people – criminals, potential victims, police etc. The agents pop-

ulate an artiﬁcial environment that is designed to reﬂect features

such as buildings, a street network, a social network, or barri-

ers to movement etc. The movement and interaction of agents

are deﬁned by either equations or rules [15]. The inherently spa-

tial nature of human movement, interactions and the role of place

in inﬂuencing these interactions are naturally incorporated into

these models.

In crime, agent-based models are popular in investigating the

environmental aspects of criminal behaviour – like the mapping

of crime hotspots and crime displacement [16], street gang rival-

ries [17], street robberies [18] and burglary [19]. Agent-based

models have also been combined with Geographic Information

Systems (GIS) to simulate dynamic spatial systems [20]. Other

applications include the dependence of the frequency of vio-

lence and criminal activity on population size [21] and whether a

Mathematics TODAY JUNE 2013 133

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