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Macroscopic Traffic Flow Network Modeling For Wildfire Evacuation: A Game-Theoretic Junction Optimization Approach with Application to Lahaina Fire

Annie Lu, Hong Kiat Tan, Alexander Xue, Alice Koniges, Andrea L. Bertozzi

Abstract

The 2023 Lahaina wildfire killed 102 people on a peninsula served by a single two-lane highway, making exit lane capacity the binding constraint on evacuation time. We model the evacuation as a system of hyperbolic scalar conservation laws on a directed graph with game-theoretic junction conditions that maximize total network flux, an evacuation-calibrated piecewise linear-quadratic flux function, and a loss-driven optimization framework that tunes traffic distribution toward priority corridors. Analytical results on a toy network and numerical simulations of the Lahaina road network reveal a phase transition in exit lane capacity. Additional lanes improve throughput linearly until a computable critical threshold, beyond which no route optimization yields further benefit. For Lahaina, reversing one southbound lane captures nearly all achievable improvement, and a fourth lane can be reserved for emergency vehicles with negligible impact on civilian clearance time. These results provide a rigorous mathematical basis for contraflow recommendations in wildland-urban interface evacuations.

Macroscopic Traffic Flow Network Modeling For Wildfire Evacuation: A Game-Theoretic Junction Optimization Approach with Application to Lahaina Fire

Abstract

The 2023 Lahaina wildfire killed 102 people on a peninsula served by a single two-lane highway, making exit lane capacity the binding constraint on evacuation time. We model the evacuation as a system of hyperbolic scalar conservation laws on a directed graph with game-theoretic junction conditions that maximize total network flux, an evacuation-calibrated piecewise linear-quadratic flux function, and a loss-driven optimization framework that tunes traffic distribution toward priority corridors. Analytical results on a toy network and numerical simulations of the Lahaina road network reveal a phase transition in exit lane capacity. Additional lanes improve throughput linearly until a computable critical threshold, beyond which no route optimization yields further benefit. For Lahaina, reversing one southbound lane captures nearly all achievable improvement, and a fourth lane can be reserved for emergency vehicles with negligible impact on civilian clearance time. These results provide a rigorous mathematical basis for contraflow recommendations in wildland-urban interface evacuations.

Paper Structure

This paper contains 31 sections, 5 theorems, 73 equations, 33 figures, 15 tables.

Table of Contents

  1. Mathematics of Continuum Models for Traffic Flow on a Network
  2. Lighthill-Whitham-Richards (LWR) Model for Traffic Flow on a Road
  3. Traffic Flow on a Network
  4. Definitions and Terminologies
  5. Entropy and Junction Conditions
  6. Numerical Scheme
  7. Modelling Flux Functions for Fire Evacuation
  8. Real World Data and Motivation
  9. Model Description
  10. Scaling for n Lanes
  11. Initial densities
  12. Optimizing Road Networks - An Application of Non-smooth Optimization
  13. Appropriate Choice of Weights
  14. Optimization Algorithm
  15. Numerical Results
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let and hence where $A_j$ is the $j$th row of $A$, and $c_{\text{in}} = \sum_{i=1}^n c_i$ and $c_{\text{out}} = \sum_{j=n+1}^{n+m} c_j$ to represent the total incoming and outgoing capacities. The unique solution to PDE12 is as follows.

Figures (33)

  • Figure 1: Overview of the paper's pipeline, from network modeling and traffic evolution to optimization and simulation of the Lahaina network.
  • Figure 2: An example of a traffic network of interest. Here, the scalar conservation law on each road is solved away from the junction, indicated by the blue directed edges. On the other hand, junctions are labelled with red vertices. Determining the density of cars at the junction would require resolving the junction conditions.
  • Figure 3: An example illustrating how distances along each road (indicated by a number close to each road) and junction (indicated by a number inside each circle representing a junction) are being assigned in a subgraph of the original network containing the exiting road (Hwy30[7]) and junction (h5). In (A), the illustrated network diagram is presented, representing the segment of the real-world network in (B) (reproduced from lahainafirereport).
  • Figure 4: Color legend for LOS classification.
  • Figure 5: A simple toy model. Road 1 is the entrance to the network, and Road 5 is the exit. Roads 2 and 3 represent a path through a residential area with multiple stops, while Road 4 represents a continuous path via a highway.
  • ...and 28 more figures

Theorems & Definitions (9)

  • Theorem 1.1
  • proof : Proof sketch
  • Theorem 1.2
  • Proposition 4.1
  • proof
  • proof
  • Proposition C.1
  • Proposition C.2
  • proof