Reaction-advection-diffusion systems as tools to explore social dynamics: modeling, analysis, and extensions

Mathematical ecology and biology have been rapidly developing fields. Influenced by their growth and potential effectiveness, other fields have started employing similar mathematical tools. Social sciences are a set of disciplines that complement their natural counterparts in the global effort to understand human nature. An example of how mathematics has been actively utilized in social sciences is crime modeling. We consider one such model, a nonlinear system of partial differential equations, initially developed by a team at UCLA in 2008.  The system has spatially heterogeneous yet persistent steady-state solutions, which are called hotspots in criminology. We propose different real-world suppression strategies and investigate numerically how the suggested approaches effectively eradicate the hotspots. We supplement our discussion with formal theorems on the existence and stability of the bifurcating branches emerging from constant steady-state solutions. Inspired by the model, we also look at the general reaction-advection-diffusion (RAD) systems with a specific application in social sciences. Under certain conditions, we prove the existence of solutions. We conclude by stating the possible extensions and further applications.  This work should be viewed as an example of how mathematics can be helpful as part of a collaborative effort to tackle critical social phenomena, which is part of an effort to quantify human behavior.