Affiliation:
1. School of Mathematical Sciences, Peking University, Beijing, P. R. China
2. Department of Mathematics, The University of Alabama, Tuscaloosa, USA
Abstract
Ever since the pioneering human–environment interaction model of criminal behavior [M. B. Short, M. R. DOrsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci. 18 (2008) 1249–1267] was published, many mathematical agent-based residential burglary models have been proposed. In order to reach an improved balance among model accuracy, analysis simplicity and real-world data fitting tractability, we introduce in this paper a multi-scale hybrid interacting-particle-system model of criminal behavior in a discrete setting. We assume that agents’ actions are governed by independent Poisson clocks, while the environment variable evolves on a separate finer discrete spatial-temporal scale. Furthermore, as we refine the second scale to its scaling limit, the hybrid system converges to a piecewise deterministic Markov process (PDMP). Through a martingale approach and infinitesimal generator analysis, we provide a formal derivation for the convergence. Computer simulations of coupled hybrid and PDMP systems both exhibit spatio-temporal aggregates of crime and show excellent agreement between the two, which supports our theoretical derivation of the scaling limit. The methodology and results we develop here indicate a way to establish connections for the proposed crime models.
Publisher
World Scientific Pub Co Pte Ltd
Subject
Applied Mathematics,Modeling and Simulation