Qualitative features of a nonlinear, nonlocal, agent-based PDE model with applications to homelessness

Abstract

In this paper, we develop a continuum model for the movement of agents on a lattice, taking into account location desirability, local and far-range migration, and localized entry and exit rates. Specifically, our motivation is to qualitatively describe the homeless population in Los Angeles. The model takes the form of a fully nonlinear, nonlocal, nondegenerate parabolic partial differential equation. We derive the model and prove useful properties of smooth solutions, including uniqueness and L2-stability under certain hypotheses. We also illustrate numerical solutions to the model and find that a simple model can be qualitatively similar in behavior to observed homeless encampments.