has made mathematics more palatable to the crime-fearing pub- ... With the changing nature of crime, traditional approaches to tack- ling it are fast becoming ...
features Another Way of Thinking: A Review of Mathematical Models of Crime Joanna Sooknanan, Balswaroop Bhatt FIMA and Donna M.G. Comissiong∗ Abstract and numerical simulation via the computer must be used to determine their approxiMathematical models are useful weapons mate solutions. These solutions may be in in the crime-fighting arsenal. With the the form of graphs showing the system’s development of cheaper and more powbehaviour over time as well as its sensitiverful computers, mathematical modelling ity to variations in key model parameters. of systems representing some aspect of crime or criminal behaviour and the analMathematical modelling and numeriysis of the resulting numerical solutions cal simulation of systems have once again © Mila Gligoric | Dreamstime.com is becoming more popular. Models may be resulted in a friendship of sorts between used to guide decision-making, develop policies or to evalu- mathematics and criminology. When modelling criminal beate specific strategies aimed at reducing crime. This review haviour and crime, since human behaviour is inherently nonlinprovides an introduction to some relatively recent mathematical ear [2], we assume that they may best be described by a nonlinear system. This is contrary to the models generally used by policymodels of crime. makers, who are susceptible to what Ball [1] calls linear thinking. This has led to the development of linear models of human 1 Introduction behaviour with their inherent assumptions that cause-and-effect ith all the numbers bandied about in the media, one relationships are identifiable and that there is proportionality bemight be forgiven for thinking that crime rates are up tween inputs and outputs [1]. These properties make linear sys– and not to any good either. Homicide rates, convic- tems particularly useful for prediction and manipulation – hence tion rates, the increase in robberies with accompanying charts . . . their popularity in modelling. figures galore. Statistics has generally gone hand in hand with However, when a system contains nonlinear terms, analyticrime and all this number crunching has traditionally left a bitter cal solutions may be difficult to obtain so that numerical methods taste in the public’s mouth. Add mathematics to this mix, and this must be used to obtain the approximate solutions. In nonlinear bitter taste turns noxious. systems, proportionality does not hold and there is a disproporThough the popularity of television shows like NUMB3RS tional relationship between cause and effect: small changes in key has made mathematics more palatable to the crime-fearing pub- parameters can trigger large changes in crime rate. Another fealic, mathematics is still viewed as a distant, fearsome relative of ture of nonlinearity is the existence of bifurcation points in key criminology. This, despite its long association with the social sci- parameters. The bifurcations give ‘tipping points’ of the system ences. The use of mathematics to describe social phenomena has at which the system may make a sudden transition to a new, very its roots in the ‘Era of Enlightenment’ in the nineteenth century different behaviour. ‘Mathematical modelling and numerical simulation complewith the birth of statistics and probability theory and their use in the collection and analysis of social statistics. However, this ment the traditional empirical and experimental approaches to collaboration was short-lived as the social sciences turned away research’ [3]. Modelling is especially important in criminology from statistics and moved towards a psychological approach to since it helps organise and visualise existing data, identify areas with missing data and is relatively inexpensive and more practiunderstanding behaviour [1]. Criminal behaviour and activities have evolved in tandem cal than carrying out an actual experiment. Modelling also ofwith changes in technology. Criminal activities now include fers a means of varying conditions so as to conduct social exarms, drugs and human trafficking, money laundering, cyber- periments but without the ethics and costs attached to expericrime, identity theft and gangs with international links. Crime menting on human beings. The insights provided by the models has become more sophisticated, organised and transnational. may be especially helpful to those in authority who are charged With the changing nature of crime, traditional approaches to tack- with the responsibility of designing policies often with a lack of ling it are fast becoming obsolete and there is a growing need for available data. a new way of thinking to meet this challenge head on.
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3 Crime ‘math’ers
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 mathematics to transform real-world systems into abstract models so as to understand, simulate or make predictions about their behaviour. Some of these systems may have no analytical solutions
Terminology used to characterise crime and criminal behaviour seems to lend itself almost naturally to the use of existing mathematical 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.
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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 number of fields such as the migration of invasive species, the propagation of genes and the spread of chemical reactions [4]. A reaction-diffusion-advection model has been applied by Stanford researchers Nancy Rodríguez, Henri Berestycki and Lenya Ryzhik to describe and reduce the spread of crime waves outward from crime hotspots [5]. Reaction-diffusion partial differential equations have also been used to study the formation, dynamics, and steady-state properties of crime hotspots and to explain 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].
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In one of the earliest papers to acknowledge the social nature of crime [8], individual crime was treated as a function of exposure 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
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equations to violent crime and burglary in the UK. The model divided 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 various 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 (predators) 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 , suitable targets X and a lack of guardianship. The model consisted of owners X who are the prey, criminals Y who are the predators of X, and security guards Z who are predators of both X and 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 n different 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 areas [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 agentbased models (ABM). These are made up of a collection of autonomous decision-making entities called agents who interact with each other and their environment, according to a set of specified behavioural rules [15]. When used to model crime, the agents generally represent people – criminals, potential victims, police etc. The agents populate an artificial environment that is designed to reflect features such as buildings, a street network, a social network, or barriers to movement etc. The movement and interaction of agents are defined by either equations or rules [15]. The inherently spatial 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 rivalries [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 violence and criminal activity on population size [21] and whether a
society without crime is possible [22]. While the previous modelling approach was characterised by a ‘top-down’ modelling approach 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 strategies aimed at reducing crime. In developing models, the question of model validity or how well the model represents the real-world situation for which it is designed naturally arises. Model validation techniques include consultation with experts about the model design and © Bram Janssens its behaviour, parameter Dreamstime.com variability-sensitivity analyses of model behaviour and the use of statistical tests and procedures to compare model output for different experimental conditions with experimental data. Experimental data in this research refers to crime data and statistics. In designing models of crime, there are challenges associated with the data collection process. Some of these challenges 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 explanations of crime phenomenon and ineffectual policies to reduce 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 provides the tool for estimating the model’ [24]. References 1 Ball, P. (2003) The physical modelling of human social sciences, Complexus, vol. 1, pp. 190–206. 2 Brown, C. (1995) Serpents in the Sand: Essays on the Nonlinear Nature of Politics and Human Destiny, University of Michigan Press, Michigan. 3 Castiglione, F. (2006) Agent based modeling, Scholarpedia, vol. 1, no. 10, p. 1562. 4 Stober-Stanford, D. (2013) To Stop Crime Wave, Change Attitudes, Psychology One News. 5 Berestycki, H. and Rodriguez, N. and Ryzhik, L. Traveling Wave Solutions in a Reaction-Diffusion Model for Criminal Activity, submitted for publication 2013.
6 Short, M.B., Brantingham, P.J., Bertozzi, A.L. and Tita, G.E. (2010) Dissipation and displacement of hotspots in reaction-diffusion models of crime, Proceedings of the National Academy of Sciences, vol. 107, no. 9, pp. 3961–3965. 7 Patel, D.M., Simon, M.A. and Taylor, R.M. (2012) Contagion of violence: Workshop summary, in Institute of Medicine and National Research Council, National Academies Press, Washington, DC. 8 Sah, R.K. (1991) Social osmosis and patterns of crime, Journal of Political Economy, vol. 99, no. 6, pp. 1272–1295. 9 Ormerod, P., Mounfield, C. and Smith, L. (2001) Nonlinear modelling of burglary and violent crime in the UK, in Modelling Crime and Offending: Recent Developments in England and Wales, vol. 80, Home Office of the Research, Development and Statistics Directorate, London. 10 Sooknanan, J., Bhatt, B.S., Comissiong, D.M.G. (2013) Catching a gang: A mathematical model of the spread of gangs in a population, Int. Journal of Pure and Applied Math, vol. 83, no. 1, pp. 25–44. 11 Vargo, L. (1966) A note on crime control, Bulletin of Mathematical Biology, vol. 83, no. 3, pp. 375–378. 12 Nuno, J.C., Herrero, M.A. and Primicerio, M. (2008) A triangle model of criminality, Physica A: Statistical Mechanics and its Applications, vol. 387, no. 12, pp. 2926–2936. 13 Nuno, J.C., Herrero, M.A. and Primicerio, M. (2011) A mathematical model of criminal-prone society, Discrete Continuous Dynamical Systems Series S, vol. 4, no. 1, pp. 193–207. 14 Sooknanan, J., Bhatt, B.S. and Comissiong, D.M.G. (2012) Criminals treated as predators to be harvested: A two prey one predator model with group defense, prey migration and switching, Journal of Mathematics Research, vol. 4, pp. 92–106. 15 Bonabeau, E. (2002) Agent-based modeling: Methods and techniques for simulating human systems, Proceedings of the National Academy of Sciences of the USA, vol. 99, pp. 7280–7287. 16 Bosse, T., Gerritsen, C., Hoogendoorn, M., Jaffry, S.W. and Treur, J. (2011) Agent-based vs. population-based simulation of displacement of crime: A comparative study, Web Intelligence and Agent Systems, vol. 9, no. 2, 147–160. 17 Hegemann, R.A., Smith, L.M., Barbaro, A.B., Bertozzi, A.L., Reid, S.E. and Tita, G.E. (2011) Geographical influences of an emerging network of gang rivalries, Physica A: Statistical Mechanics and its Applications, vol. 390, pp. 3894–3914. 18 Groff, E. (2007) Simulation for theory testing and experimentation: An example using routine activity theory and street robbery, Journal of Qualitative Criminology, vol. 23, no. 2, pp. 75–103. 19 Malleson, N., Heppenstall, A. and See L. (2010) Crime reduction through simulation: An agent-based model of burglary, Computers, Environment and Urban Systems, vol. 34, no. 3, pp. 236–250. 20 Liu, L. and Eck, J. (2008) Artificial Crime Analysis Systems: Using Computer Simulations and Geographic Information Systems, Idea Group Inc. 21 Fonoberova, M., Fonoberov, V.A., Mezic, I., Mezic, J. and Brantingham, P.J. (2012) Nonlinear dynamics of crime and violence in urban settings. Journal of Artificial Societies and Social Simulation, vol. 15, no. 1, p. 2. 22 Winoto, P. (2003) A simulation of the market for offenses in multiagent systems: Is zero crime rate attainable? in Proceedings of the 3rd International Conference on Multi-Agent-Based Simulation II MABS’02, Springer-Verlag, Berlin, Heidelberg, pp. 181–193. 23 Eck, J. and Liu, L. (2008) Contrasting simulated and empirical experiments in crime prevention, Journal of Experimental Criminology, vol. 4, no. 3, pp. 195–213. 24 Backman, O. and Edling, C. (1999) Mathematics matters: On the absence of mathematical models in quantitative sociology, Acta Sociologica, vol. 42, pp. 69–78. Mathematics TODAY
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