doi: 10.3934/era.2021011

A multiscale stochastic criminal behavior model under a hybrid scheme

1. 

Department of Mathematics, The University of Alabama, Tuscaloosa, AL 35487-0350, USA

2. 

School of Mathematical Sciences, Peking University, Beijing 100871, China

* Corresponding author: Chuntian Wang

Received  June 2020 Revised  January 2021 Published  February 2021

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.

Citation: Chuntian Wang, Yuan Zhang. A multiscale stochastic criminal behavior model under a hybrid scheme. Electronic Research Archive, doi: 10.3934/era.2021011
References:
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M. B. ShortM. R. D'OrsognaV. B. PasourG. E. TitaP. J. BrantinghamA. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267.  doi: 10.1142/S0218202508003029.  Google Scholar

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C. Wang, Y. Zhang, A. L. Bertozzi and M. B. Short, A stochastic-statistical residential burglary model with finite size effects, in Active particles, Vol. 2 (eds. N. Bellomo, P. Degond and E. Tadmor), Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, (2019), 245–274.  Google Scholar

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C. WangY. ZhangA. L. Bertozzi and M. B. Short, A stochastic-statistical residential burglary model with independent Poisson clocks, European J. Appl. Math., 32 (2021), 32-58.  doi: 10.1017/S0956792520000029.  Google Scholar

show all references

References:
[1]

A. Alsenafi and A. B. T. Barbaro, A convection–diffusion model for gang territoriality, Phys. A, 510 (2018), 765-786.  doi: 10.1016/j.physa.2018.07.004.  Google Scholar

[2]

R. AzaïsJ.-B. BardetA. GénadotN. Krell and P.-A. Zitt, Piecewise deterministic Markov process - recent results, ESAIM: Proc., 44 (2014), 276-290.  doi: 10.1051/proc/201444017.  Google Scholar

[3]

A. B. T. BarbaroL. Chayes and M. R. D'Orsogna, Territorial developments based on graffiti: A statistical mechanics approach, Physica A: Statistical Mechanics and its Applications, 392 (2013), 252-270.   Google Scholar

[4]

M. H. A. Davis, The representation of martingales of jump processes, SIAM J. Control Optim., 14 (1976), 623-638.  doi: 10.1137/0314041.  Google Scholar

[5]

M. H. A. Davis, Piecewise-deterministic Markov processes: A general class of nondiffusion stochastic models. With discussion, J. Roy. Statist. Soc. Ser. B, 46 (1984), 353-388.  doi: 10.1111/j.2517-6161.1984.tb01308.x.  Google Scholar

[6]

P. Diaconis and D. Freedman, Iterated random functions, SIAM Rev., 41 (1999), 45-76.  doi: 10.1137/S0036144598338446.  Google Scholar

[7]

M. R. D'Orsogna and Perc Matja$\mathop {\rm{z}}\limits^ \circ $, Statistical physics of crime: A review, Physics of Life Rev., 12 (2015), 1-21.  doi: 10.1016/j.plrev.2014.11.001.  Google Scholar

[8]

A. Génadot, Spatio-temporal averaging for a class of hybrid systems and application to conductance-based neuron models, Nonlinear Anal. Hybrid Syst., 22 (2016), 178-190.  doi: 10.1016/j.nahs.2016.03.003.  Google Scholar

[9]

R. A. HegemannL. M. SmithA. B. T. BarbaroA. L. BertozziS. E. Reid and G. E. Tita, Geographical influences of an emerging network of gang rivalries, Physica A: Statistical Mechanics and its Applications, 390 (2011), 3894-3914.  doi: 10.1016/j.physa.2011.05.040.  Google Scholar

[10]

M. Jacobsen, Piecewise deterministic Markov processes, in Point Process Theory and Applications: Marked Point and Piecewise Deterministic Processes, Birkhäuser Boston, (2006), 143–211. doi: 10.1007/0-8176-4463-6_7.  Google Scholar

[11]

J. Jacod and A. V. Skorokhod, Jumping Markov processes, Ann. Inst. H. Poincaré Probab. Statist., 32 (1996), 11-67.   Google Scholar

[12]

N. R. MaglioccaK. McSweeneyS. E. SesnieE. TellmanJ. A. DevineE. A. NielsenZ. Pearson and D. J. Wrathall, Modeling cocaine traffickers and counterdrug interdiction forces as a complex adaptive system, PNAS, 116 (2019), 7784-7792.  doi: 10.1073/pnas.1812459116.  Google Scholar

[13]

C. Z. Marshak, M. P. Rombach, A. L. Bertozzi and M. R. D'Orsogna, Growth and containment of a hierarchical criminal network, Phys. Rev. E, 93 (2016), 022308. doi: 10.1103/PhysRevE.93.022308.  Google Scholar

[14]

D. McMillon, C. P. Simon and J. Morenoff, Modeling the underlying dynamics of the spread of crime, PLOS ONE, e88923, (2014). doi: 10.1371/journal.pone.0088923.  Google Scholar

[15]

G. O. MohlerM. B. ShortP. J. BrantinghamF. P. Schoenberg and G. E. Tita, Self-exciting point process modeling of crime, J. Amer. Statist. Assoc., 106 (2011), 100-108.  doi: 10.1198/jasa.2011.ap09546.  Google Scholar

[16]

G. O. MohlerM. B. ShortS. MalinowskiM. JohnsonG. E. TitaA. L. Bertozzi and P. J. Brantingham, Randomized controlled field trials of predictive policing, J. Amer. Statist. Assoc., 110 (2015), 1399-1411.  doi: 10.1080/01621459.2015.1077710.  Google Scholar

[17]

M. G. Riedler, M. Thieullen and G. Wainrib, Limit theorems for infinite-dimensional piecewise deterministic Markov processes. Applications to stochastic excitable membrane models, Electron. J. Probab., 17 (2012), no. 55, 48 pp. doi: 10.1214/EJP.v17-1946.  Google Scholar

[18]

M. B. ShortM. R. D'OrsognaV. B. PasourG. E. TitaP. J. BrantinghamA. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Math. Models Methods Appl. Sci., 18 (2008), 1249-1267.  doi: 10.1142/S0218202508003029.  Google Scholar

[19]

C. Wang, Y. Zhang, A. L. Bertozzi and M. B. Short, A stochastic-statistical residential burglary model with finite size effects, in Active particles, Vol. 2 (eds. N. Bellomo, P. Degond and E. Tadmor), Model. Simul. Sci. Eng. Technol., Birkhäuser/Springer, Cham, (2019), 245–274.  Google Scholar

[20]

C. WangY. ZhangA. L. Bertozzi and M. B. Short, A stochastic-statistical residential burglary model with independent Poisson clocks, European J. Appl. Math., 32 (2021), 32-58.  doi: 10.1017/S0956792520000029.  Google Scholar

Figure 1.  Plot of the attractiveness $ A ({\textbf{x}}, t) $ for M-IPC Model. For all the cases, the initial conditions (at $ t = 0 $) and parameters are set as $ L = 20 $, $ A0 = 1/30 $, $ A (\textbf{x}, 0) = 0.3 $, and initially a total of $ 640 $ criminal agents are randomly uniformly distributed over $ \mathcal M $. In (a), we set $ c_1 = 7.5 $, $ c_2 = 0.135 $, $ \Gamma = 6 $, $ \theta = 0.04 $, $ \omega = 1.5 $, and $ \eta = 0.0035 $; in (b), $ c_1 = 14.1176 $, $ c_2 = 0.1071 $, $ \Gamma = 6 $, $ \theta = 0.1417 $, $ \omega = 5 $, and $ \eta = 0.0078 $; in (c), $ c_1 = 34 $, $ c_2 = 0.0253 $, $ \Gamma = 8 $, $ \theta = 0.0647 $, $ \omega = 6 $, and $ \eta = 0.00703 $; in (d), $ c_1 = 1.6667 $, $ c_2 = 0.0005 $, $ \Gamma = 0.933 $, $ \theta = 0.0146 $, $ \omega = 0.666 $, and $ \eta = 0.333 $. (a) shows Case 1 Stationary hotspots, (b) shows Case 2.1 Independent hotspots, and (c) shows Case 2.2 Interactive hotspots
Figure 2.  Plot of the attractiveness $ A ({\textbf{x}}, t) $ for M-IPC Model on different random paths, with the same parameters used to create the plots in Fig. 1(c). The same hopspot dynamics is exhibited, which is Case 2.2 Interactive hotspots
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