A finite element framework for simulating residential burglary in realistic urban geometries

We consider a partial differential equation (PDE) model to predict residential burglary derived from a probabilistic agent-based model through a mean-field limit operation. The PDE model is a nonlinear, coupled system of two equations in two variables (attractiveness of residential sites and density of criminals), similar to the Keller-Segel model for aggregation based on chemotaxis. Unlike previous works, which applied periodic boundary conditions, we enforce boundary conditions that arise naturally from the variational formulation of the PDE problem, i.e., the starting point for the application of a finite element method. These conditions specify the value of the normal derivatives of the system variables at the boundary. For the numerical solution of the PDE problem discretized in time and space, we propose a scheme that decouples the computation of the attractiveness from the computation of the criminal density at each time step, resulting in the solution of two linear algebraic systems per iteration. Through numerous numerical tests, we demonstrate the robustness and computational efficiency of this approach. Leveraging the flexibility allowed by the finite element method, we show results for spatially heterogeneous model parameters and a realistic geometry (city of Chicago). The paper includes a discussion of future perspectives to build multiscale, 'multi-physics' models that can become a tool for the community. The robust and efficient code developed for this paper, which is shared open-source, is intended as the solid base for this broader research program.