This special issue is one of the very first dedicated to crime modelling in a journal of applied mathematics. It emphasizes one of the new areas at the Social Science frontier, where modelling and mathematical tools are put to use with a view to shed light on phenomena previously thought to be outside of their reach. Pioneering research is increasingly being carried out in many different areas in the life sciences or social sciences, often under the heading of the study of complex systems. When addressing issues regarding society, individuals or the collective behaviours of humans, several questions naturally arise about the modelling enterprise. What is the nature and role of modelling in social sciences? What is one to expect from these new approaches? The case of economics, which has relied on mathematics for a very long time now, can serve as a paradigm for what is happening in other social sciences.

This paper by a criminologist explains why it makes more sense to model criminal acts than to model criminals, how many preconceptions about crime can mislead modellers and offers some simple crime modelling ideas. Many opportunities for simulation now exist, and new opportunities for real-data modelling are emerging. The author suggests mathematical models of crime, including offender foraging for crime targets, as a rich area for future research.

We are interested in the possible contributions of mathematical modelling of crime. We refer to numerous and quite recent papers that analyse and discuss empirical data in an attempt to discover stylised trends worthy of being understood through simple models. We summarise part of this literature and try to understand the reasons of important discrepancies in their conclusions. Then we present some recent modelling attempts that may help to understand the large variance in the statistical-based conclusions.

In this paper, we argue that quantitative empirical research to explain and predict criminal and related behaviour can benefit greatly from explicit theories of action linking individual and contextual factors in the causation of crime. Such theories foster a systematic selection of causal variables for data collection and hypothesis testing instead of a more indiscriminate accumulation of ‘risk factor’ correlates. Moreover, action theory encourages statistical modelling of crime causation beyond the most common linear regression. This paper illustrates both points by estimating two empirical models – a conventional logistic model and a Rasch model – on scenario response data concerning youth violence. The findings of this study show that the extent to which young people indicate a violent response to a provocation is dependent on their (law relevant) morality and ability to exercise self-control as well as the deterrent qualities (monitoring) of the setting.

Crime is disproportionally concentrated in few areas. Though long established, there remains uncertainty about the reasons for variation in the concentration of similar crime (repeats) or different crime (multiples). Wholly neglected have been composite crimes when more than one crime types coincide as parts of a single event. The research reported here disentangles area crime concentration into repeats, multiple and composite crimes. The results are based on estimated bivariate zero-inflated Poisson regression models with covariance structure which explicitly account for crime rarity and crime concentration. The implications of the results for criminological theorizing and as a possible basis for more equitable police funding are discussed.

Decades of research demonstrate that crime is concentrated at a range of spatial scales. Such findings have clear implications for crime forecasting and police resource allocation models. More recent work has also shown that crime clusters in space and time with a regularity that might improve methods of crime prediction. In this paper I review some of the available evidence and provide illustrations of the types of analysis – spatial and spatio-temporal – conducted hitherto. With a few exceptions, the application of formal Mathematics in the study of space–time patterns of crime has been rather limited, and so a central aim of the paper is to stimulate interest in this area of research.

In this paper1 we introduce a family of models to describe the spatio-temporal dynamics of criminal activity. It is argued here that with a minimal set of mechanisms corresponding to elements that are basic in the study of crime, one can observe the formation of hot spots. By analysing the simplest versions of our model, we exhibit a self-organised critical state of illegal activities that we propose to call a warm spot or a tepid milieu2 depending on the context. It is characterised by a positive level of illegal or uncivil activity that maintains itself without exploding, in contrast with genuine hot spots where localised high level or peaks are being formed. Within our framework, we further investigate optimal policy issues under the constraint of limited resources in law enforcement and deterrence. We also introduce extensions of our model that take into account repeated victimisation effects, local and long range interactions, and briefly discuss some of the resulting effects such as hysteresis phenomena.

We review the Short model of urban residential burglary derived from taking the continuum limit of two difference equations – one of which models the attractiveness of individual houses to burglary, and the other of which models burglar movement. This leads to a system of non-linear partial differential equations. We propose a change to the Short model and also add deterrence caused by the presence of uniformed officers to the model. We solve the resulting system of non-linear partial differential equations numerically and present results both with and without deterrence.

We introduce a general framework for modelling the dynamics of the propensity to offend in a population of (possibly interacting) agents. We consider that each agent has an ‘honesty index’ which parameterizes his probability of abiding by the law. This probability also depends on a composite parameter associated to the attractiveness of the crime outcome and of the crime setting (the context which makes a crime more or less likely to occur, such as the presence or not of a guardian). Within this framework we explore some consequences of the working hypothesis that punishment has a deterrent effect, assuming that, after a criminal act, an agent's honesty index may increase if he is caught and decrease otherwise. We provide both analytical and numerical results. We show that in the space of parameters characterizing the probability of punishment, there are two ‘phases’: one corresponding to a population with a low crime rate and the other to a population with a large crime rate. We speculate on the possible existence of a self-organized state in which, due to the society reaction against crime activities, the population dynamics would be stabilized on the critical line, leading to a wide distribution of propensities to offend in the population. In view of empirical works on the causes of the recent evolution of crime rates in developed countries, we discuss how changes of socio-economic conditions may affect the model parameters, and hence the crime rate in the population. We suggest possible extensions of the model that will allow us to take into account more realistic features.

The inspection game has been widely used as a model for crime deterrence. In this article, we shall present a broad review of the literature and an original result. The original result concerns the interaction between a single, long-run inspector and a population of short-run, boundedly rational would-be criminals. This approach gives rise to an optimal control problem that can be solved explicitly. We show that a forward-looking inspector obtains the same payoff he would obtain if he could commit to a probability of inspection. The consequences of this result for the economics of crime deterrence are also explored.

Human societies are formed by different socio-economical classes which are characterized by their contribution to, and their share of, the common wealth available. Cheaters, defined as those individuals that do not contribute to the common wealth but benefit from it, have always existed, and are likely to be present in all societies in the foreseeable future. Their existence brings about serious problems since they act as sinks for the community wealth and deplete resources which are always limited and often scarce.

To fight cheaters, a society can invest additional resources to pursue one or several aims. For instance, an improvement in social solidarity (e.g. by fostering education) may be sought. Alternatively, deterrence (e.g. by increasing police budget) may be enhanced. Then the following questions naturally arise: (i) how much to spend and (ii) how to allocate the expenditure between both strategies above. This paper addresses this general issue in a simplified setting, which however we believe of some interest. More precisely, we consider a society constituted by two productive classes and an unproductive one, the cheaters, and proposes a dynamical system that describes their evolution in time. We find it convenient to formulate our model as a three-dimensional ordinary differential equation (ODE) system whose variables are the cheater population, the total wealth and one of the productive social classes. The stationary values of the cheater population and the total wealth are studied in terms of the two parameters: φ (how much to invest) and s (how to distribute such expenditure). We show that it is not possible to simultaneously minimize the cheater population and maximize the total wealth with respect to φ and s. We then discuss the possibility of defining a compromise function to find suitable values of φ and s that optimize the response to cheating. In our opinion, this qualitative approach may be of some help to plan and implement social strategies against cheating.