S. Yusef Shafi

S. Yusef Shafi

We present a condition that guarantees spatially uniformity in the solution trajectories of a diffusively-coupled compartmental ODE model, where each compartment represents a spatial domain of components interconnected through diffusion terms with like components in different compartments. Each set of like components has its own weighted undirected graph describing the topology of the interconnection between compartments. The condition makes use of the Jacobian matrix to describe the dynamics of each compartment as well as the Laplacian eigenvalues of each of the graphs. We discuss linear matrix inequalities that can be used to verify the condition guaranteeing spatial uniformity, and apply the result to a coupled oscillator network. Next we turn to reaction-diffusion PDEs with Neumann boundary conditions, and derive an analogous condition guaranteeing spatial uniformity of solutions. The paper contributes a relaxed condition to check spatial uniformity that allows individual components to have their own specific diffusion terms and interconnection structures.

Yu Chen, S. Yusef Shafi, Yi-fan Chen

Traffic evacuation plays a critical role in saving lives in devastating disasters such as hurricanes, wildfires, floods, earthquakes, etc. An ability to evaluate evacuation plans in advance for these rare events, including identifying traffic flow bottlenecks, improving traffic management policies, and understanding the robustness of the traffic management policy are critical for emergency management. Given the rareness of such events and the corresponding lack of real data, traffic simulation provides a flexible and versatile approach for such scenarios, and furthermore allows dynamic interaction with the simulated evacuation. In this paper, we build a traffic simulation pipeline to explore the above problems, covering many aspects of evacuation, including map creation, demand generation, vehicle behavior, bottleneck identification, traffic management policy improvement, and results analysis. We apply the pipeline to two case studies in California. The first is Paradise, which was destroyed by a large wildfire in 2018 and experienced catastrophic traffic jams during the evacuation. The second is Mill Valley, which has high risk of wildfire and potential traffic issues since the city is situated in a narrow valley.