Bi-dimensional crime model based on anomalous diffusion with law enforcement effect

Year 2022, , 26 - 40, 31.03.2022

Francisco Javier Martínez-farías Anahi Alvarado-sánchez Eduardo Rangel-cortes Arturo Hernández-hernández

https://doi.org/10.53391/mmnsa.2022.01.003

Cited By: 7

Abstract

Several models based on discrete and continuous fields have been proposed to comprehend residential criminal dynamics. This study introduces a two-dimensional model to describe residential burglaries diffusion, employing Lèvy flights dynamics. A continuous model is presented, introducing bidimensional fractional operator diffusion and its differences with the 1-dimensional case. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even fractional derivative order is modified. We also provide qualitative evidence that steady-state approximation in one dimension by series expansion is insufficient to capture similar original system behavior. At least for the case where series coefficients have a linear relationship with derivative order. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even if fractional derivative order is modified. Two dynamic regimes emerge in maximum and total attractiveness magnitude as a result of fractional derivative changes, these regimes can be understood as considerations about different urban environments. Finally, we add a Law enforcement component, embodying the "Cops on dots" strategy; in the Laplacian diffusion dynamic, global attractiveness levels are significantly reduced by Cops on dots policy but lose efficacy in Lèvy flight-based diffusion regimen. The four-step Preditor-Corrector method is used for numerical integration, and the fractional operator is approximated, getting the advantage of the spectral methods to approximate spatial derivatives in two dimensions.

Keywords

Residential burglary, Lévy flights, fractional operator, anomalous diffusion, hotspots, law enforcement

References

• Brantingham, P. & Tita, G. Offender Mobility and crime pattern formation from first principles. (In L. Liu, & J. Eck (Ed.), Artificial Crime Analysis Systems: Using Computer Simulations,2008).
• Short, M., D’Orsogna, M., Pasour, V., Tita, G., Brantingham, P., Bertozzi, A. & Chayes, L. A Statistical Model of Criminal Behavior. Mathematical Models And Methods In Applied Sciences, 18, 1249-1267, (2008).
• Short, M.B., Bertozzi, A.L., & Brantingham, P.J. Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression. SIAM Journal on Applied Dynamical Systems, 9(2), 462-483, (2010).
• Jones, P.A., Brantingham, P.J., & Chayes, L.R. Statistical models of criminal behavior: the effects of law enforcement actions. Mathematical Models and Methods in Applied Sciences, 20(supp01), 1397-1423, (2010).
• Berestycki, H. & Nadal, J.P. Self-organised critical hot spots of criminal activity. European Journal Of Applied Mathematics, 21(4-5), 371-399, (2010).
• Chaturapruek, S., Breslau, J., Yazdi, D., Kolokolnikov, T. & McCalla, S.G. Crime modeling with Lévy flights. SIAM Journal On Applied Mathematics, 73(4), 1703-1720, (2013).
• Kolokolnikov, T., Ward, M. & Wei, J. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete And Continuous Dynamical Systems - Series B, 19, 1373, (2014).
• Zipkin, J.R., Short, M.B. & Bertozzi, A.L. Cops on the dots in a mathematical model of urban crime and police response. Discrete And Continuous Dynamical Systems - Series B, 19(5), 1479, (2014).
• Camacho, A., Lee, H.R.L & Smith, L.M. Modelling policing strategies for departments with limited resources. European Journal Of Applied Mathematics, 27(3), 479-501, (2016).
• Gu, Y., Wang, Q., & Yi, G. Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect. European Journal Of Applied Mathematics, 28(1), 141- 78, (2017).
• Pan, C., Li, B., Wang, C., Zhang, Y., Geldner, N., Wang, L. & Bertozzi, A.L. Crime modeling with truncated Lévy flights for residential burglary models. Mathematical Models And Methods In Applied Sciences, 28(09), 1857-1880, (2018).
• Wang, Q., Wang, D. & Feng, Y. Global well-posedness and uniform boundedness of urban crime models: One-dimensional case. Journal Of Differential Equations, 269(7), 6216-6235, (2020).
• Rodríguez, N. & Winkler, M. Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation. Mathematical Models And Methods In Applied Sciences, 30(11), 2105-2137, (2020).
• Kang, K., Kolokolnikov, T. & Ward, M. The stability and dynamics of a spike in the 1D Keller–Segel model. IMA Journal Of Applied Mathematics, 72(2), 140-162, (2007).
• Mei, L. & Wei, J. The existence and stability of spike solutions for a chemotax is system modeling crime pattern formation. Mathematical Models And Methods In Applied Sciences, 30(09), 1727-1764, (2020).
• Kondo, S. & Miura, T. Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation. Science, 329(5999), 1616-1620, (2010).
• Keller, E. & Segel, L. Traveling bands of chemotactic bacteria: A theoretical analysis. Journal Of Theoretical Biology, 30(2), 235-248, (1971).
• Biler, P. & Wu, G. Two-dimensional chemotaxis models with fractional diffusion. Mathematical Methods In The Applied Sciences, 32(1), 112-126, (2009).
• Calvó-Armengol, A. & Zenou, Y. Social Networks and Crime Decisions: The Role of Social Structure in Facilitating Delinquent Behavior. International Economic Review, 45(3), 939-958, (2004).
• Nec, Y. Spike-Type Solutions to One Dimensional Gierer–Meinhardt Model with Lévy Flights. Studies In Applied Mathematics, 129(3), 272-299, (2012).
• Cruz-García, S., Martínez-Farías, F., Santillán-Hernández, A. & Rangel, E. Mathematical home burglary model with stochastic long crime trips and patrolling: Applied to Mexico City. Applied Mathematics And Computation, 396, 125865, (2021).
• Levajkovic, T., Mena, H. & Zarfl, M. Lévy processes, subordinators and crime modeling. Novi Sad Journal Of Mathematics, 46(2), 65-86, (2016).
• Cohen, L. & Felson, M. Social Change and Crime Rate Trends: A Routine Activity Approach. American Sociological Review, 44, 588-608, (1979).
• Wilson, J.Q. & Kelling, G. L. Broken windows. The Atlantic, 249, 29-38, (1982).
• Kilbas, A.A., Srivastava, H.M. & Trujillo, J. Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier, North-Holland Mathematics Studies, (2006).
• Kopriva, D. Implementing Spectral Methods for Partial Differential Equations. Springer, (2009).
• Osgood, B.G. Lectures on the Fourier Transform and Its Applications. SIAM, (2019).
• Trefethen, L.N. Spectral methods in MATLAB, volume 10 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 24, (2000).
• Butcher, J. Numerical Methods for Ordinary Differential Equations. John Wiley, 2015.
• Ghrist, M.L., Fornberg, B. & Reeger, J.A. Stability ordinates of Adams predictor-corrector methods. BIT Numerical Mathematics, 55(3), 733-750, (2015).
• Lindfield, G.R. & Penny, J.E. Numerical methods: using MATLAB. Elsevier, 2012.
• Tadjeran, C. & Meerschaert, M.M. A second-order accurate numerical method for the two-dimensional fractional diffusion equation. Journal Of Computational Physics, 220(2), 813-823, (2007).
• Lapidus, L. & Seinfeld, J. 4 Predictor-Corrector Methods. Numerical Solution Of Ordinary Differential Equations, 74, 152-241, (1971).
• Rodriguez, N., & Bertozzi, A. Local existence and uniqueness of solutions to a PDE model for criminal behavior. Mathematical Models and Methods in Applied Sciences, 20(supp01), 1425-1457, (2010).
• Cantrell, R.S., Cosner, C. & Manásevich, R. Global Bifurcation of Solutions for Crime Modeling Equations. SIAM Journal On Mathematical Analysis, 44(3), 1340-1358, (2012).
• Tse, W.H. & Ward, M.J. Hotspot formation and dynamics for a continuum model of urban crime. European Journal Of Applied Mathematics, 27(3), 583-624, (2016).
• Rodríguez, N., Wang, Q. & Zhang, L. Understanding the Effects of On- and Off-Hotspot Policing: Evidence of Hotspot, Oscillating, and Chaotic Activities. SIAM Journal On Applied Dynamical Systems, 20(4), 1882-1916, (2021).