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.