This special issue of the European journal of applied mathematics features research articles that involve the application of mathematical methodologies to the modelling of a broad range of problems related to crime and security. Some specific topics in this issue include recent developments in mathematical models of residential burglary, a dynamical model for the spatial spread of riots initiated by some triggering event, the analysis and development of game-theoretic models of crime and conflict, the study of statistically based models of insurgent activity and terrorism using real-world data sets, models for the optimal strategy of police deployment under realistic constraints, and a model of cyber crime as related to the study of spiking behaviour in social network cyberspace communications. Overall, the mathematical and computational methodologies employed in these studies are as diverse as the specific applications themselves and the scales (spatial or otherwise) to which they are applied. These methodologies range from statistical and stochastic methods based on maximum likelihood methods, Bayesian equilibria, regression analysis, self-excited Hawkes point processes, agent-based random walk models on networks, to more traditional applied mathematical methods such as dynamical systems and stability theory, the theory of Nash equilibria, rigorous methods in partial differential equations and travelling wave theory, and asymptotic methods that exploit disparate space and time scales.

We examine the game theoretic properties of a model of crime first introduced by Short et al. (2010 Phys. Rev. E82, 066114) as the SBD Adversarial Game. We identify the rationalizable strategies and one-shot equilibria under multiple equilibrium refinements. We further show that SBD's main result about the effectiveness of defecting-punishers (“Informants”) in driving the system to evolve to the cooperative equilibrium under an imitation dynamic generalizes to a best response dynamic, though only under certain parameter regimes. The nature of this strategy's role, however, differs significantly between the two dynamics: in the SBD imitation dynamic, Informants are sufficient but not necessary to achieve the cooperative equilibrium, while under the best response dynamic, Informants are necessary but not sufficient for convergence to cooperation. Since a policy of simply converting citizens to Informants will not guarantee success under best response dynamics, we identify alternative strategies that may help the system reach cooperation in this case, e.g., the use of moderate but not too severe punishments on criminals.

Recently, many security-related problems have gained increasing attention from a quantitative perspective. In this paper, we propose a game-theoretical approach to model the interaction between police forces and delinquents in public places. In the well-known Stackelberg game, a leader is faced with only one follower. However, in our application, the police are simultaneously faced with many offenders, who may be organized or act independently of each other. This application motivates the development of two games: a classical leader-follower interaction between police and organized criminals on the one hand and a novel approach between the leader and selfishly acting offenders on the other. It is of special interest that the effect of crime displacement under police surveillance be anticipated by the proposed models. Results using data from a simulated environment emphasise how these models can provide decision support for policing outperforming traditional strategies.

We discuss the emergence of common mathematical patterns governing the timing and severity of insurgent and terrorist attacks, across geographic scales and including cyberspace. We present mathematical models that provide a generative explanation of these patterns. Despite wide variations in the underlying settings and circumstances, the ubiquity of these patterns suggests there is a common way in which groups of humans fight each other. Our empirical findings follow from the analysis of myriad state-of-the-art datasets with resolution at the level of individual attacks, while our mathematical modelling involves numerical and analytical solutions of fission–fusion dynamics together with progress curve analysis.

In this paper, a unique dataset of improvised explosive device attacks during “The Troubles” in Northern Ireland (NI) is analysed via a Hawkes process model. It is found that this past dependent model is a good fit to improvised explosive device attacks yielding key insights about the nature of terrorism in NI. We also present a novel approach to quantitatively investigate some of the sociological theory surrounding the Provisional Irish Republican Army which challenges previously held assumptions concerning changes seen in the organisation. Finally, we extend our use of the Hawkes process model by considering a multidimensional version which permits both self and mutual-excitations. This allows us to test how the Provisional Irish Republican Army responded to past improvised explosive device attacks on different geographical scales from which we find evidence for the autonomy of the organisation over the six counties of NI and Belfast. By incorporating a second dataset concerning British Security Force (BSF) interventions, the multidimensional model allows us to test counter-terrorism (CT) operations in NI where we find subsequent increases in violence.

Using mathematical methods to understand and model crime is a recent idea that has drawn considerable attention from researchers during the last five years. From the plethora of models that have been proposed, perhaps the most successful one has been a diffusion-type differential equations model that describes how the number of criminals evolves in a specific area. We propose a more detailed form of this model that allows for two distinct criminal types associated with major and minor crime. Additionally, we examine a stochastic variant of the model that represents more realistically the ‘generation’ of new criminals. Numerical solutions from both models are presented and compared with actual crime data for the Greater Manchester area. Agreement between simulations and actual data is satisfactory. A preliminary statistical analysis of the data also supports the model's potential to describe crime.

Crime reduction and prevention strategies are vital for policymakers and law enforcement to face inevitable increases in urban crime rates as a side effect of the projected growth of urban population by the year 2030. Studies conclude that crime does not occur uniformly across urban landscapes but concentrates in certain areas. This phenomenon has drawn attention to spatial crime analysis, primarily focusing on crime hotspots, areas with disproportionally higher crime density. In this paper, we present CrimeTracer1, a personalized random walk-based approach to spatial crime analysis and crime location prediction outside of hotspots. We propose a probabilistic model of spatial behaviour of known offenders within their activity spaces. Crime Pattern Theory concludes that offenders, rather than venture into unknown territory, frequently select targets in or near places they are most familiar with as part of their activity space. Our experiments on a large real-world crime dataset show that CrimeTracer outperforms all other methods used for location recommendation we evaluate here.

In this paper, we explore some of the various issues that may occur in attempting to fit a dynamical systems (either agent- or continuum-based) model of urban crime to data on just the attack times and locations. We show how one may carry out a regression analysis for the model described by Short et al. (2008, Math. Mod. Meth. Appl. Sci.) by using simulated attack data from the agent-based model. It is discussed how one can incorporate the attack data into the partial differential equations for the expected attractiveness to burgle and the criminal density to predict crime rates between attacks. Using this predicted crime rate, we derive a likelihood function that one can maximise in order to fit parameters and/or initial conditions for the model. We focus on carrying out data assimilation for two different parameter regions, namely in the case where stationary and non-stationary crime hotspots form. It is found that the likelihood function is ‘flat’ for large ranges of parameters, and that this has major implications for crime forecasting. Hence, we look at how one might carry out a goodness-of-fit and forecasting analysis for crime rates given the range of parameter fits. We show how one can use the Kolmogorov–Smirnov statistic to assess the goodness-of-fit. The dynamical systems analysis of the partial differential equations proves invaluable to understanding how the crime rate forecasts depend on the parameters and their sensitivity. Finally, we outline several interesting directions for future research in this area where we believe that the combination of dynamical systems modelling, analysis, and data assimilation can prove effective in developing policing strategies for urban crime.

Crime prevention is a major goal of law-enforcement agencies. Often, these agencies have limited resources and officers available for patrolling and responding to calls. However, patrolling and police visibility can influence individuals to not perform criminal acts. Therefore, it is necessary for the police to optimize their patrolling strategies to deter the most crime. Previous studies have created agent-based models to simulate criminal and police agents interacting in a city, indicating a “cops on the dots” strategy as a viable method to mitigate large amounts of crime. Unfortunately, police departments cannot allocate all of the patrolling officers to seek out these hotspots, particularly since they are not immediately known. In large cities, it is often necessary to keep a few officers in different areas of the city, frequently divided up into beats. Officers need to respond to calls, possibly not of a criminal nature. Therefore, we modify models for policing to account for these factors. Through testing the policing strategies for various hotspot types and number of police agents, we found that the methods that performed the best varied greatly according to these factors.

Electronic communications, as well as other categories of interactions within social networks, exhibit bursts of activity localised in time. We adopt a self-exciting Hawkes process model for this behaviour. First we investigate parameter estimation of such processes and find that, in the parameter regime we encounter, the choice of triggering function is not as important as getting the correct parameters once a choice is made. Then we present a relaxed maximum likelihood method for filling in missing data in records of communications in social networks. Our optimisation algorithm adapts a recent curvilinear search method to handle inequality constraints and a non-vanishing derivative. Finally we demonstrate the method using a data set composed of email records from a social network based at the United States Military Academy. The method performs differently on this data and data from simulations, but the performance degrades only slightly as more information is removed. The ability to fill in large blocks of missing social network data has implications for security, surveillance, and privacy.

In both historical and modern conflicts, space plays a critical role in how interactions occur over time. Despite its importance, the spatial distribution of adversaries has often been neglected in mathematical models of conflict. In this paper, we propose an entropy-maximising spatial interaction method for disaggregating the impact of space, employing a general notion of ‘threat’ between two adversaries. This approach addresses a number of limitations that are associated with partial differential equation approaches to spatial disaggregation. We use this method to spatially disaggregate the Richardson model of conflict escalation, and then explore the resulting model with both analytical and numerical treatments. A bifurcation is identified that dramatically influences the resulting spatial distribution of conflict and is shown to persist under a range of model specifications. Implications of this finding for real-world conflicts are discussed.

This paper is concerned with modelling the dynamics of social outbursts of activity, such as protests or riots. In this sequel to our work in Berestycki et al. (Networks and Heterogeneous Media, vol. 10, no. 3, 1–34), written in collaboration with J-P. Nadal, we model the effect of restriction of information and explore its impact on the existence of upheaval waves. The system involves the coupling of an explicit variable representing the intensity of rioting activity and an underlying (implicit) field of social tension. We prove the existence of global solutions to the Cauchy problem in ${\mathbb R}^d$ as well as the existence of traveling wave solutions in certain parameter regimes. We furthermore explore the effects of heterogeneities in the environment with the help of numerical simulations, which lead to pulsating waves in certain cases. We analyse the effects of periodic domains as well as the barrier problem with the help of numerical simulations. The barrier problem refers to the potential blockage of a wavefront due to a spatial heterogeneity in the system which leads to an area of low excitability (referred to as the barrier). We conclude with a variety of open problems.

The existence, stability, and dynamics of localized patterns of criminal activity are studied for the reaction–diffusion model of urban crime introduced by Short et al. (Math. Models. Meth. Appl. Sci.18(Suppl.), (2008), 1249–1267). In the singularly perturbed limit of small diffusivity ratio, this model admits hotspot patterns, where criminal activity of high amplitude is localized within certain narrow spatial regions. By using a combination of asymptotic analysis and numerical path-following methods, hotspot equilibria are constructed on a finite 1-D domain and their bifurcation properties analysed as the diffusivity of criminals is varied. It is shown, both analytically and numerically, that new hotspots of criminal activity can be nucleated in low-crime regions with inconspicuous crime activity gradient when the spatial extent of these regions exceeds a critical threshold. These nucleations are referred to as “peak insertion” events, and for the steady-state problem, they occur near a saddle-node bifurcation point characterizing hotspot equilibria. For the time-dependent problem, a differential algebraic (DAE) system characterizing the slow dynamics of a collection of hotspots is derived, and the results compared favourably with full numerical simulations of the PDE system. The asymptotic theory to construct hotspot equilibria, and to derive the differential algebraic system for quasi-steady patterns, is based on the resolution of a triple-deck structure near the core of each hotspot and the identification of so-called switchback terms.