A multiscale stochastic criminal behavior model under a hybrid scheme

Chuntian Wang; Yuan Zhang; Chuntian Wang; Yuan Zhang

doi:10.3934/era.2021011

Electronic Research Archive

2021, Volume 29, Issue 4: 2741-2753. doi: 10.3934/era.2021011

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A multiscale stochastic criminal behavior model under a hybrid scheme

Chuntian Wang ^{1
,
,},
Yuan Zhang ^{2
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1.
Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA
2.
School of Mathematical Sciences, Peking University, Beijing 100871, China

Received: 01 June 2020 Revised: 01 January 2021 Published: 22 February 2021
Primary: 60G51; Secondary: 35Q91

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.
- Crime hotspots,
- agent-based crime models,
- piecewise deterministic Markov processes,
- martingale representation,
- independent Poisson clocks
Citation: Chuntian Wang, Yuan Zhang. A multiscale stochastic criminal behavior model under a hybrid scheme[J]. Electronic Research Archive, 2021, 29(4): 2741-2753. doi: 10.3934/era.2021011

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Abstract

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.

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