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Efficient numerical computation of traveler states in explicit mobility-based metapopulation models: Mathematical theory and application to epidemics

Henrik Zunker, René Schmieding, Jan Hasenauer, Martin J. Kühn

Abstract

Metapopulation models are powerful tools for capturing the spatio-temporal spread of infectious diseases. Models that explicitly account for traveler origins and destinations, such as Lagrangian metapopulation models, enable a detailed representation of mobility and traveling subpopulations. However, in densely connected networks, tracking these subpopulations leads to quadratic growth in system size with the number of spatial patches. While specific approaches reducing the effort of traveler state estimation have been proposed, these approaches are either model-specific or heuristic. Here, we introduce a Runge-Kutta (RK) stage-aligned computation of traveler states that leverages the precomputed intermediate stage values of explicit RK methods under the assumption of localized homogeneous mixing. We prove that the resulting numerical solution is identical to that of the standard Lagrangian formulation when solved with the corresponding RK method. For compartments without inflows, we further show that the exact same results can be obtained using a simple algebraic scaling based on the initial traveler share. When embedded in a recently proposed metapopulation framework that combines local dynamics with discrete mobility, the stage-aligned approach eliminates the need for heuristic traveler approximations. In contrast to the standard Lagrangian formulation, the resulting method enables efficient simulations by reducing the global ODE system to linear scaling in the number of patches, while the remaining quadratic interactions are handled through highly efficient algebraic updates. Numerical experiments confirm the theoretical results, demonstrating optimal convergence order. Benchmarks on fully connected networks with up to 1025 patches, 1024 local travel connections, and six age groups achieve speedups of up to 76 and 50 for first- and fourth-order Runge-Kutta methods, respectively.

Efficient numerical computation of traveler states in explicit mobility-based metapopulation models: Mathematical theory and application to epidemics

Abstract

Metapopulation models are powerful tools for capturing the spatio-temporal spread of infectious diseases. Models that explicitly account for traveler origins and destinations, such as Lagrangian metapopulation models, enable a detailed representation of mobility and traveling subpopulations. However, in densely connected networks, tracking these subpopulations leads to quadratic growth in system size with the number of spatial patches. While specific approaches reducing the effort of traveler state estimation have been proposed, these approaches are either model-specific or heuristic. Here, we introduce a Runge-Kutta (RK) stage-aligned computation of traveler states that leverages the precomputed intermediate stage values of explicit RK methods under the assumption of localized homogeneous mixing. We prove that the resulting numerical solution is identical to that of the standard Lagrangian formulation when solved with the corresponding RK method. For compartments without inflows, we further show that the exact same results can be obtained using a simple algebraic scaling based on the initial traveler share. When embedded in a recently proposed metapopulation framework that combines local dynamics with discrete mobility, the stage-aligned approach eliminates the need for heuristic traveler approximations. In contrast to the standard Lagrangian formulation, the resulting method enables efficient simulations by reducing the global ODE system to linear scaling in the number of patches, while the remaining quadratic interactions are handled through highly efficient algebraic updates. Numerical experiments confirm the theoretical results, demonstrating optimal convergence order. Benchmarks on fully connected networks with up to 1025 patches, 1024 local travel connections, and six age groups achieve speedups of up to 76 and 50 for first- and fourth-order Runge-Kutta methods, respectively.
Paper Structure (17 sections, 2 theorems, 34 equations, 4 figures, 2 tables)

This paper contains 17 sections, 2 theorems, 34 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. ODE-based compartmental models
  3. ODE-based Lagrangian metapopulation model
  4. Standard formulation with continuous mobility
  5. Standard formulation with discrete mobility
  6. A graph approach for aggregated dynamics with decoupled traveler-state approximation
  7. An efficient numerical approach for the computation of traveler states
  8. Stage-aligned Runge-Kutta method
  9. Comparison of computational complexity
  10. Numerical results
  11. Description of the considered model
  12. Limitations of traveler computation with auxiliary Euler heuristic
  13. Numerical validation
  14. Assessment of computational efficiency
  15. Conclusion
  16. ...and 2 more sections

Key Result

Theorem 4.1

Let Assumption ass:mixing hold true. Consider $t_a\in\mathbb{R}$ with the next mobility event timing at $t_{a+1}\in\mathbb{R}$ and a compartment $j \in \{1, \dots, N_C\}$ with no inflows. Then, the population share is constant for $t\in[t_a,t_{a+1})$ and it holds that

Figures (4)

  • Figure 1: Visual representation of the generalized piecewise-continuous simulation scheme. During intervals $I_{a}$, the system evolves with all individuals in aggregated dynamics while traveler states are computed on-the-fly with the Runge-Kutta scheme to be used for the aggregated dynamics.
  • Figure 2: Implications for subpopulation updates through the auxiliary Euler step as in kuhn_assessment_2021.(A–B) One-step tests demonstrating the evolution of the exposed subpopulation $E(t)$ under two different initial conditions ($t_{\max}=h = 0.5$ days, $p_{c,S}=p_{c,E}=p_{c,I}=p_{c,R}=0.98$): the solid black curve is the subpopulation reference obtained from the standard Lagrangian model with a fine step size; the blue dashed curve is the single-step auxiliary Euler update; the red dash-dotted curve is the simulated patch total. Yellow shading highlights the gap between the subpopulation and the total, and warning icons indicate Euler overshoot of the patch total. (C) Feasibility map across a uniform traveler fraction $p_c$ (applied equally to S,E,I,R) and step size $h$. Dark cells indicate fewer than $2\%$ violating time points (negativity or overshoot). Yellow cells indicate at least $2\%$ violating time points. (D) Convergence at $p_c=0.95$: The maximum error over time and all compartments decreases approximately linearly with $h$, consistent with first-order behavior (grey dotted slope-1 guideline). Setups are listed in \ref{['tab:setups_params']}.
  • Figure 3: Validation of the traveler state computation scheme with convergence plots for different Runge-Kutta methods.(A) Time series of total population dynamics for compartments $S,E,I,R$ over 100 days (stage-aligned with RK-1 ($h=1$) in solid lines; standard Lagrangian with RK-1 ($h=1$) in dashed lines). (B) Convergence of the maximum relative error for the stage-aligned scheme (RK-1, RK-2, RK-3, and RK-4) using different step sizes $h$. The reference solution is computed with the standard Lagrangian approach using the RK-4 integrator and a step size of $h = 10^{-6}$. Each dotted line shows the theoretical convergence order. (C) Absolute error over time for each compartment $S,E,I,R$ comparing the solution from the stage-aligned approach to the standard Lagrangian solution (both using RK-1 with $h=1$). (D) Absolute error over time for each compartment $S,E,I,R$ comparing the solution from the stage-aligned approach to the standard Lagrangian solution (both using RK-4 with $h=1$). (E) Evolution of the traveler shares $\xi^{(p;q)}_j(t)$ for each compartment over time computed using the stage-aligned solution of order 1 as in (A). (F) Absolute error over time for each compartment $S,E,I,R$ when comparing the hybrid approach (RK-4 for the aggregated system and RK-1 for the computation of traveler states) to the standard Lagrangian integrator (RK-4). Here, the step size is fixed at $h=1.0$.
  • Figure 4: Computational scaling and speedups for traveler updates using a SEIR model (see \ref{['eq:basicSEIR']}) with a step size of $h=0.5$ days for $50$ days.(A–C) Median runtime per simulation (in seconds) against the number of patches for $1$, $3$, and $6$ age groups. Lines show the different strategies (standard Lagrangian with RK-1 and RK-4, stage-aligned with RK-1 and RK-4, auxiliary Euler heuristic, hybrid approach) as explained in the legend. Grey reference lines indicate linear $\mathcal{O}(N_P)$ (dashed) and quadratic $\mathcal{O}(N_P^2)$ (dotted) trends. (D–F) Speedup heatmaps across patches (rows) and age groups (columns). Each cell displays the ratio of median runtimes for the indicated pair: (D) auxiliary Euler heuristic / hybrid approach, (E) standard Lagrangian (RK-4) / stage-aligned (RK-4), (F) Standard Lagrangian (RK-1) / stage-aligned (RK-1). The color scale is logarithmic with a break at 1 (yellow to green $>$ 1: suggested approach faster; red to orange $<$ 1: existing approach faster). Timings are Google Benchmark medians reported as seconds per simulated day. For the parameters and populations used in these simulations see \ref{['tab:setups_params']}.

Theorems & Definitions (6)

  • Theorem 4.1
  • proof
  • Definition 4.2: Runge-Kutta stage-aligned traveler computation
  • Remark 4.3
  • Theorem 4.4
  • proof