Crime modeling with truncated Lévy flights for residential burglary models

Statistical agent-based models for crime have shown that repeat victimization can lead to predictable crime hotspots (see e.g. M. B. Short, M. R. D’Orsogna, 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.), then a recent study in one-space dimension (S. Chaturapruek, J. Breslau, D. Yazdi, T. Kolokolnikov and S. G. McCalla, Crime modeling with Lévy flights, SIAM J. Appl. Math. 73 (2013) 1703–1720.) shows that the hotspot dynamics changes when movement patterns of the criminals involve long-tailed Lévy distributions for the jump length as opposed to classical random walks. In reality, criminals move in confined areas with a maximum jump length. In this paper, we develop a mean-field continuum model with truncated Lévy flights (TLFs) for residential burglary in one-space dimension. The continuum model yields local Laplace diffusion, rather than fractional diffusion. We present an asymptotic theory to derive the continuum equations and show excellent agreement between the continuum model and the agent-based simulations. This suggests that local diffusion models are universal for continuum limits of this problem, the important quantity being the diffusion coefficient. Law enforcement agents are also incorporated into the model, and the relative effectiveness of their deployment strategies are compared quantitatively.