Mathematics of Crime

There is an extensive applied mathematics literature developed for problems in the biological and physical sciences. Our understanding of social science problems from a mathematical standpoint is less developed, but also presents some very interesting problems. This lecture uses crime as a case study for using applied mathematical techniques in a social science application and covers a variety of mathematical methods that are applicable to such problems. We will review recent work on agent based models, differential equations, variational methods for inverse problems and statistical point process models. From an application standpoint we will look at problems in residential burglaries and gang crimes. Examples will consider both "bottom up" and "top down" approaches to understanding the mathematics of crime, and how the two approaches could converge to a unifying theory. Predicting Criminal Incidents Using Geographic, Demographic, and Twitter-derived Information Donald E. Brown, University of Virginia Abstract: Predictive policing seeks to anticipate the times and locations of crimes to better allocate law enforcement resources to combat these crimes. The key to predictive policing is modeling that combines available data to forecast or estimate the areas most threatened by crimes at different times. We have developed models that integrate geographic, demographic, and social media information from a specific area of interest to produce the needed predictions. In this presentation, I describe our approach to this predictive modeling, which combines spatial-temporal generalized additive models (STGAM) with a new approach to text mining. We use the STGAM to predict the probability of criminal activity at a given location and time within the area of interest. Our new approach to text mining combines Latent Dirichlet Allocation (LDA) with Latent Semantic Indexing (LSI) to identify and use key topics in social media relevant to criminal activity. We use social media since these data provide a rich, event-based context for criminal incidents. I present our application of this approach to actual criminal incidents in Charlottesville, Virginia. Our results indicate that this combined modeling approach outperforms models that only use geographic and demographic data. Predictive policing seeks to anticipate the times and locations of crimes to better allocate law enforcement resources to combat these crimes. The key to predictive policing is modeling that combines available data to forecast or estimate the areas most threatened by crimes at different times. We have developed models that integrate geographic, demographic, and social media information from a specific area of interest to produce the needed predictions. In this presentation, I describe our approach to this predictive modeling, which combines spatial-temporal generalized additive models (STGAM) with a new approach to text mining. We use the STGAM to predict the probability of criminal activity at a given location and time within the area of interest. Our new approach to text mining combines Latent Dirichlet Allocation (LDA) with Latent Semantic Indexing (LSI) to identify and use key topics in social media relevant to criminal activity. We use social media since these data provide a rich, event-based context for criminal incidents. I present our application of this approach to actual criminal incidents in Charlottesville, Virginia. Our results indicate that this combined modeling approach outperforms models that only use geographic and demographic data. Space Dependence in Fighting Cheaters L. Meacci, Juan C. Nuno and M. Primicerio, Universita degli Studi di Firenze and Universidad Politecnica de Madrid Abstract: In this paper we study the dynamics of a population where the individuals can either be contributors (tax payers) or no contributors (tax evaders or cheaters). We introduce a 2D cellular automaton on which the probability of transition from one of the above states to the other is the sum of the local effect and of the global field effect. The model also includes the policy that allocates a fraction of the budget to fight tax evasion. This scheme allowed us to simulate the cases in which inhomogeneous strategies in contrasting tax evasion is applied in a region and the case in which cooperative policies are adopted by neighbor societies. In this paper we study the dynamics of a population where the individuals can either be contributors (tax payers) or no contributors (tax evaders or cheaters). We introduce a 2D cellular automaton on which the probability of transition from one of the above states to the other is the sum of the local effect and of the global field effect. The model also includes the policy that allocates a fraction of the budget to fight tax evasion. This scheme allowed us to simulate the cases in which inhomogeneous strategies in contrasting tax evasion is applied in a region and the case in which cooperative policies are adopted by neighbor societies. Point Process Methods for Crime Hotspots George Mohler, Santa Clara University Abstract: This talk focuses on the application of point process methods to crime and security data. We will discuss semiand nonparametric models, as well as their estimation using Expectation-Maximization algorithms. We conclude the talk with some results from a randomized controlled trial in Los Angeles where police patrols are determined each day using a semi-parametric self-exciting point process. This talk focuses on the application of point process methods to crime and security data. We will discuss semiand nonparametric models, as well as their estimation using Expectation-Maximization algorithms. We conclude the talk with some results from a randomized controlled trial in Los Angeles where police patrols are determined each day using a semi-parametric self-exciting point process. Development of a Social-Behavior Modeling and Simulation Framework for Assessing Strategies in Response to Crisis AH1N1 in Chile Jaime H. Ortega, J. Amaya, F. Padilla, M. Escobar, Universidad de Chile Abstract: The occurrence of catastrophic phenomena such as pandemic, natural disasters or social crisis, detrimentally affects a nation's population and its environment. If the authorities responsible for managing national infrastructure, such as public health, transportation, educational and communications networks, defense forces and law enforcement agencies, among others, are not sufficiently prepared to face these kinds of events, their responses may not be timely sufficient nor appropriate to mitigate their effects. Hence, it is vitally important to have a system capable for simulating the social and economical consequences of natural and unnatural disasters. Such a system can provide criteria for strategic decision-making and helps to mitigate the impact on health and safety of the population and the national economy. The occurrence of catastrophic phenomena such as pandemic, natural disasters or social crisis, detrimentally affects a nation's population and its environment. If the authorities responsible for managing national infrastructure, such as public health, transportation, educational and communications networks, defense forces and law enforcement agencies, among others, are not sufficiently prepared to face these kinds of events, their responses may not be timely sufficient nor appropriate to mitigate their effects. Hence, it is vitally important to have a system capable for simulating the social and economical consequences of natural and unnatural disasters. Such a system can provide criteria for strategic decision-making and helps to mitigate the impact on health and safety of the population and the national economy. In this work we present some suitable quantitative tools, based on mathematical modeling, computational algorithm and social analysis to analyze the social behavior under an emergency. We consider particularly the occurrence of a known pandemic (e.g. H1N1) in Chile in during 2009. We take into account the capacities of public and private organizations to face the disaster, their possible actions and the people's reactions to the emergency. This system can simulate scenarios considering different actions as a vaccines program, transport regulations, among others, allowing us to measure the social behavior and population response. We note that this analysis can be extended to another kind of emergencies with a suitable source of data, for instance, natural disasters, social crisis among others. Quasilinear systems and residential burglary Raul Manasevich, CEAMOS & U Chile Abstract: In this talk we will present some results for systems of equations modeling residential burglary. In this talk we will present some results for systems of equations modeling residential burglary. For the parabolic system model proposed by Andrea Bertozzi et-al, we study the equilibrium case. By using bifurcation theory we show that this system does support pattern formation. We also give some results concerning stability of the bifurcating patterns. These results correspond to a joint work with Chris Cosner and Steve Cantrel from the University of Miami. The model has been recently modified by Pitcher giving rise to a new parabolic system of equations. We show some results for this system that contain a condition for existence of global solutions. This work corresponds to a collaboration with Philippe Souplet and Quoc Hung Phan from Paris 13. How criminal defectors may lead the way to a peaceful society Dr.Maria-Rita R D'Orsogna, California State University at Northridge Abstract: Traditional models of human cooperation are usually cast in the form of a prisoner's dilemma, where although cooperation may be beneficial, players may choose to "defect" and pursue selfish goals. In this talk we consider an adversarial evolutionary game developed for criminal activity where players not only choose whether or not to cooperate for the common good but also whether or not to actively harm others by committing crimes. The introduction of this new choice gives rise to four possible strategies among which the so called "informant", a player who cooperates while still committing crimes. We find two possible equilibration regimes, a defection-dominated and an ideal, cooperation-dominated one and show that the number of informants is crucial in determining which of these two regimes is achieved. Since large numbers of informants lead to the ideal cooperative society we also study their Traditional models of human cooperation are usually cast in the form of a prisoner's dilemma, where although cooperation may be beneficial, players may choose to "defect" and pursue selfish goals. In this talk we consider an adversarial evolutionary game developed for criminal activity where players not only choose whether or not to cooperate for the common good but also whether or not to actively harm others by committing crimes. The introduction of this new choice gives rise to four possible strategies among which the so called "informant", a player who cooperates while still committing crimes. We find two possible equilibration regimes, a defection-dominated and an ideal, cooperation-dominated one and show that the number of informants is crucial in determining which of these two regimes is achieved. Since large numbers of informants lead to the ideal cooperative society we also study their active recruitment from the overall society, by considering differential recruitment costs and benefits, via an optimal control problem where finite resources are included. We discuss our results in the context of extreme adversarial societies, such as those marred by wars, insurgencies and organized crime. The Stability of Steady-State Hot-Spot Patterns for Reaction-Diffusion Models of Urban Crime