Spatio-temporal averaging for a class of hybrid systems and application to conductance-based neuron models

Genadot, Alexandre

doi:10.1016/j.nahs.2016.03.003

Nonlinear Analysis: Hybrid Systems

2016

DOI: 10.1016/j.nahs.2016.03.003

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Spatio-temporal averaging for a class of hybrid systems and application to conductance-based neuron models

Alexandre Genadot

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Cited by 2 publications

References 24 publications

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A multiscale stochastic criminal behavior model and the convergence to a piecewise-deterministic-Markov-process limit

Cai

Wang

Zhang

2022

Math. Models Methods Appl. Sci.

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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.

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A multiscale stochastic criminal behavior model and the convergence to a piecewise-deterministic-Markov-process limit

Cai

Wang

Zhang

2022

Math. Models Methods Appl. Sci.

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A multiscale stochastic criminal behavior model under a hybrid scheme

Wang

Zhang

2021

era

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<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>

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Spatio-temporal averaging for a class of hybrid systems and application to conductance-based neuron models

Cited by 2 publications

References 24 publications

A multiscale stochastic criminal behavior model and the convergence to a piecewise-deterministic-Markov-process limit

A multiscale stochastic criminal behavior model and the convergence to a piecewise-deterministic-Markov-process limit

A multiscale stochastic criminal behavior model under a hybrid scheme

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