Mobility resilience and recovery dynamics: Parsimonious framework beyond V-shapes

System resilience is increasingly crucial in responding to disruptions from human activities, climate change, earthquakes, pandemics, etc. Existing literature commonly employs V-shaped or variants, such as U-shaped or trapezoid-shaped models, to describe system performance trends, but capturing the non-linear dynamics, identifying steady-state points, and analyzing impacts by interventions have been challenging, especially across multiple resolutions. While the challenges of extending the model to capture multiple waves are acknowledged, this study focuses exclusively on single-wave scenarios. To gain a deeper understanding of system resilience, this paper proposes an Ordinary Differential Equation (ODE): the Double Quadratic Queue (DQQ) model, which is adapted from the fluid-based Polynomial Arrival Queue (PAQ) model proposed by Newell (1982). The DQQ Model offers a parsimonious framework for understanding and estimating the dynamic evolution of disruption and recovery processes. The queue-theoretical ODE is adapted to multiple resolutions from national to route level in complex systems and is capable of identifying steady-state points and quantifying system resilience metrics. To validate the applicability, this paper employs transit mobility data at national, state, and county resolutions from Google COVID-19 Community Mobility Reports and two-way ridership data for RapidRide routes in King County, Seattle. The results reveal that the DQQ model demonstrates a notable improvement over the traditional models, particularly in identifying and approximating stabilization status at the end of the recovery process. Also, this paper conducts regression analysis to examine the correlation between resilience metrics. In addition, to examine the impact of the interventions on the system's recovery capability, statistical analysis is conducted to analyze the impact of the new opening of the H Line to other lines.