A multiscale stochastic criminal behavior model under a hybrid scheme

Abstract

<p style='text-indent:20px;'>Crime in urban environment is a major social problem nowadays. As such, many efforts have been made to develop mathematical models for this type of crime. The pioneering work [M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, Math. Models Methods Appl. Sci., 18, (2008), pp. 1249-1267] establishes an agent-based human-environment interaction model of criminal behavior for residential burglary, where aggregate pattern formation of "hotspots" is quantitatively studied for the first time. Potential offenders are assumed to interact with environment according to well-known criminology and sociology notions. However long-term simulations for the coupled dynamics are computationally costly due to all components evolving on slow time scales. In this paper, we introduce a new-generation criminal behavior model with separated spatio-temporal scales for the agent actions and the environment parameter reactions. The computational cost is reduced significantly, while the essential stochastic features of the pioneering model are preserved. Moreover, the separation of scales brings the model into the theoretical framework of piecewise deterministic Markov processes (PDMP). A martingale approach is applicable which will be useful to analyze both stochastic and statistical features of the model in subsequent studies.</p>