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Hybrid epidemic simulation framework coupling equation-based and individual-based models

Jaeyoung Kwak, Michael H. Lees, Chin Chun Ooi, Wentong Cai

arXiv:2609.35162v1physics.soc-phcs.CYcs.MAq-bio.PE

TL;DR

Mass gatherings create localized transmission, yet event-specific contact patterns and downstream citywide spread remain insufficiently characterized in prior models. This study links empirical event contacts to Madrid's metropolitan SEIR model and commuter network; simulations show larger, faster outbreaks and earlier, more synchronized district invasion, with event seeding explaining only part of the association.

  • Problem

    How event-specific contact heterogeneity translates into metropolitan epidemic invasion remains insufficiently characterized, despite mass gatherings' potential to increase respiratory transmission opportunities.

    Source: Introduction · pp. 2–3

  • Method

    The framework couples empirically derived individual contact networks during gatherings to stochastic metropolitan SEIR spread through an event-generated infection seed and commuter mobility.

    Source: Summary and Discussion · p. 20

  • Results

    Mass gatherings amplified epidemic magnitude and progression and advanced and synchronized district invasion, with effects varying by event characteristics and background transmissibility.

    Source: Summary and Discussion · pp. 20, 22

  • Takeaways & Limitations

    Reducing transmission during gatherings may reduce infections entering urban transmission and potentially delay subsequent epidemic invasion.

    Source: Summary and Discussion · p. 21

    The simulations omit ventilation, airflow, and surface persistence, hold mobility and epidemiological parameters constant, and exclude behavioral adaptation.

    Source: Summary and Discussion · p. 22

Abstract

from arXiv · show

Mass gathering events like concerts, sports matches, and festivals bring many people into close contact within a short period, creating localized bursts of infection that can shape epidemic outcomes across an entire city. To evaluate how these transient transmission events translate into broader urban impacts, we developed a simulation model linking event-scale contact dynamics with citywide commuting networks. Using Madrid, Spain, as a case study, we compared several types of gatherings and examined how their effects changed under different levels of disease transmissibility. We found that mass gatherings consistently amplified outbreak magnitude, accelerated progression, advanced district-level arrival times, and synchronized spatial spread. Remarkably, while the initial seed size generated at the event accounted for much of this acceleration, post-event transmission conditions provided complementary predictive signal regarding invasion timing. These findings demonstrate that mitigating transmission during mass gatherings can yield downstream public health benefits by delaying broader spatial spread. More generally, this multiscale framework offers a tool to evaluate how temporary, localized contact shifts produce longer-lasting consequences for urban populations.

Introduction

Mass gatherings create close, heterogeneous contacts that can seed infections which spread through urban mobility. This study addresses the mismatch between event-level contact detail and citywide epidemic modeling by coupling the two scales and examining downstream invasion in Madrid.

Introduction

  • Crowd density, prolonged contact, and intensified social interaction can increase opportunities for respiratory transmission, making the wider effects of temporary gatherings a public-health concern.The paper notes that mass-gathering restrictions were among the principal non-pharmaceutical interventions during the COVID-19 pandemic.Source: Introduction · p. 2
  • Metapopulation models efficiently represent urban transmission but may not resolve event-specific contacts, while citywide individual-based models carry substantially greater computational and data requirements.The hybrid strategy selectively assigns fine resolution to heterogeneous event transmission and tractable population-level modeling to subsequent spread.Source: Introduction · pp. 2–3
  • Prior MGE models often represented detailed event contacts and subsequent metropolitan mobility separately, leaving the link from event-generated infections to urban invasion insufficiently characterized.The study identifies quantifying how variation in the event-generated seed relates to invasion timing and spatial progression as a specific gap.Source: Introduction · p. 3
  • The framework transfers infections generated on empirical event contact networks into a stochastic metropolitan SEIR model for spread through the urban population and commuter network.This allocates individual-level resolution to the gathering and population-level modeling to subsequent citywide spread.Source: Introduction · p. 4
  • The study uses Madrid population distributions, commuter flows, and empirical contact networks to compare gathering scenarios across epidemic seeding, magnitude, progression, and spatial spread.Survival analysis examines district invasion timing and its relationship to the event-generated seed and subsequent transmission conditions, with robustness assessed across transmission settings.Source: Introduction · p. 4

Results

The results examine how mass gatherings relate to the magnitude, timing, and spatial progression of epidemic invasion across Madrid, and how initial seeding and subsequent transmission conditions inform invasion timing. Earlier district-level invasion was associated with gatherings across contrasting seed locations and persisted after adjustment for initial seed size; these findings are statistical rather than causal.

Spatiotemporal Dynamics of Epidemics

Epidemic arrival time

  • All MGE scenarios advanced district-level epidemic arrival, with the largest reductions under AMS dance and arrival patterns closest to baseline under Leipzig 3.Leipzig 3 also produced smaller arrival-time shifts across most districts, consistent with effective contact mitigation reducing subsequent spread.Source: Epidemic arrival time · pp. 9–11, Figure 5 · p. 9, Figure 7 · p. 11
  • MGEs synchronized invasion across districts and shortened median arrival intervals; AMS dance showed the strongest synchronization, while some seed–destination pairs had advances exceeding 30 days.Stronger Leipzig crowd-control measures reduced synchronization; arrival-time acceleration varied with both seed location and destination district.Source: Epidemic arrival time · pp. 10–11

Statistical analysis of epidemic arrival timing

  • The event-associated invasion hazard was 2.90 in M0 and 2.66 after seed-size adjustment, while log-transformed seed size had a hazard ratio of 1.45 per unit increase.Pre-invasion infectious pressure added information beyond event status and seed size (M2 event and seed hazard ratios ≈2.67 and ≈1.47); the event association persisted after adjustment.Source: Statistical analysis of epidemic… · pp. 14–16
  • M2 with a 14-day pressure window achieved the best model fit (ΔAIC = 0) and discrimination (C-index = 0.634), compared with M0 (ΔAIC = 9,611; C-index = 0.575).M1 improved on M0 (ΔAIC = 1,435; C-index = 0.618); adding pre-invasion pressure further improved performance, with the 14-day window best among those tested.Source: Statistical analysis of epidemic… · p. 15, Figure 11 · p. 15
  • At βevent = 0.1, adjusted hazard ratios rose from 0.98 at R0 = 1.5 to 1.99 at R0 = 3.0, while adjustment weakened βevent’s association.The pattern is consistent with event transmission partly influencing invasion through infections generated at the gathering, while R0 shapes subsequent propagation.Source: Statistical analysis of epidemic… · pp. 17, 19
  • After seed-size adjustment, gathering-event hazard ratios remained above 1 across District 0 (2.66), District 7 (1.74), and District 18 (1.95).The positive association held across seed locations with contrasting spatial and network characteristics, although its magnitude varied.Source: Statistical analysis of epidemic… · pp. 17–18, Figure 13 · p. 18
  • Across Districts 0, 7, and 18, adjustment increased C-index from 0.575 to 0.618, 0.553 to 0.627, and 0.559 to 0.622, respectively.AIC decreased by 8,176, 18,514, and 15,258, respectively, indicating better AIC-based fit for the adjusted model at each location.Source: Statistical analysis of epidemic… · p. 18
  • Model M2, particularly with a 14-day averaging window, improved fit and discrimination over M1, suggesting recent transmission history adds information about invasion timing.Pre-invasion pressure reflects the recent transmission environment, not a distinct causal pathway; the findings describe statistical relationships in the simulated system.Source: Statistical analysis of epidemic… · pp. 19–20

Summary and Discussion

The Madrid case study links event-scale individual-based simulations with metropolitan epidemic spread. Mass gatherings amplified and accelerated outbreaks, shifted and synchronized district-level invasion, and their association with earlier arrival reflected both event-generated seeding and subsequent transmission conditions.

Summary and Discussion

  • MGEs amplified transmission and accelerated progression versus the no-event baseline, with the strongest relative effects at intermediate R0.The relative effect was smaller under low or high background transmissibility.Source: Summary and Discussion · p. 20
  • MGEs advanced district-level arrivals and synchronized spatial spread, with effects varying by event scenario, seed location, and destination district.AMS dance produced the strongest acceleration, while Leipzig 3 departed less from baseline.Source: Summary and Discussion · pp. 20–21
  • HR 2.90 fell to 2.66 after adjustment for initial seed size, while seed size remained strongly associated with earlier invasion.The attenuation indicates that event-generated seeding explains part, but not all, of the observed association.Source: Summary and Discussion · p. 21
  • Pre-invasion infectious pressure improved statistical performance, particularly with a 14-day averaging window, and added arrival-timing information beyond event status and seed size.The Cox models are statistical evidence consistent with the simulated mechanisms, not independent causal identification.Source: Summary and Discussion · p. 21
  • Reducing transmission during gatherings may reduce infections entering the wider urban system and potentially delay invasion; the framework can evaluate strategies across event and epidemiological conditions.Suggested measures include crowd-density management, ventilation, screening, and vaccination.Source: Summary and Discussion · pp. 21–22
  • The simulations omit repeated or clustered gatherings, behavioral adaptation, and several sources of individual heterogeneity, limiting the modeled scenarios.The study also assumes constant mobility and epidemiological parameters during each simulation.Source: Summary and Discussion · p. 22

Materials and Methods

The methods couple district-level stochastic SEIR dynamics with individual-based event transmission, using Madrid mobility data and empirical crowd-contact networks. District invasion timing is analyzed with sequential Cox models to assess event-generated seeding and recent transmission pressure across varied community and event transmission conditions.

Overview

  • The hybrid workflow couples a district-level metapopulation SEIR model of prevalence and commuter mixing with a proximity-based individual event model, updating the citywide state after each time step.On non-event days, the population follows stochastic metapopulation SEIR dynamics similar to Ref..Source: Overview · p. 22

Dataset

  • The event-scale contact networks come from pedestrian trajectory data, while the city-scale model uses Madrid’s 21 districts, resident populations, and normalized daily commuter flows.Scenarios include Amsterdam dance and football events and Leipzig concert experiments with baseline, moderate, and high contact mitigation.Source: Dataset · pp. 23–24

Stochastic metapopulation SEIR model

  • The stochastic metapopulation model tracks home–destination subpopulations in discrete steps of ∆t = 0.5 days, with compartment transitions sampled from binomial distributions.For daily analyses, transmission pressure averages the away- and home-phase forces of infection.Source: Stochastic metapopulation SEIR model · pp. 24, 26

Individual-based model for event transmission

  • Event contacts are represented as weighted network edges for attendee pairs within 1.5 m, with edge weight τab measuring cumulative contact duration.Infection probability depends on cumulative exposure time through a Poisson process, followed by Bernoulli transitions from susceptible to exposed; newly exposed attendees rejoin the metapopulation.Source: Individual-based model for event… · p. 27

Hybrid coupling mechanism

  • Event-day transmission is modeled by splitting attendees from the citywide population, simulating event contacts alongside background SEIR spread, then reintegrating residents into their home districts and mobility flows.Imported infectious attendees seed event transmission but are removed afterward; infections among resident attendees continue through community spread.Source: Overview · p. 23, Hybrid coupling mechanism · p. 28

Statistical analysis of district-level epidemic invasion timing

  • District invasion is defined as the first time a district has at least one infectious individual; Kaplan–Meier and log-rank methods compare arrival-time distributions across scenarios.Districts not invaded by the simulation horizon are right-censored; Cox models use replicate-clustered robust standard errors to account for dependence among districts sharing an epidemic realization.Source: Statistical analysis of district-level… · pp. 28–29
  • Nested Cox models assess the MGE association with invasion timing, then account for event-generated seed size and recent transmission pressure rather than selecting a single predictive model.Transmission pressure is averaged over the 3-, 7-, and 14-day windows preceding district arrival.Source: Statistical analysis of district-level… · pp. 29–30
  • Model performance was assessed with AIC and Harrell’s C-index, with the C-index treated as a within-sample discrimination measure rather than out-of-sample validation.Hazard ratios used 95% confidence intervals from a cluster-robust sandwich variance estimator accounting for observations from the same simulation replicate.Source: Statistical analysis of district-level… · pp. 30–31
  • Model M1 was the principal adjusted model for mechanistic interpretation of event-generated seeding, while M2 was a secondary exploratory extension assessing additional information from recent transmission pressure.Because pre-invasion pressure was defined retrospectively relative to observed arrival, M2 was not interpreted as evidence of a causal effect.Source: Statistical analysis of district-level… · pp. 30–31
  • Sensitivity analysis varied R0 from 1.0 to 3.0 and βevent from 0.1 to 0.5, producing 25 conditions compared categorically against R0 = 1.0 and βevent = 0.1.Parallel Cox models P0–P2 assessed transmission-condition associations before and after adjustment for event-generated seed and recent transmission pressure; P0 and P1 hazard ratios were summarized with heatmaps.Source: Statistical analysis of district-level… · p. 31

Supporting information

The supporting information identifies Madrid’s district-level population and mobility data, then characterizes how seed location shapes district-level epidemic arrival times in baseline and mass-gathering scenarios.

Seed-location-dependent epidemic arrival times

  • The analysis uses each of Madrid’s 21 districts as a seed and averages destination arrival times over 1,000 simulation runs.Arrival time is defined for each seed–destination pair and run before averaging.Source: Seed-location-dependent epidemic… · pp. 33–34

Population and mobility data

Seed-location-dependent epidemic arrival times

  • In the baseline scenario, arrival time at the same destination can vary substantially with the epidemic seed district.The 21 × 21 matrix excludes diagonal cells because the seed district is the initial condition, not a subsequent arrival.Source: Seed-location-dependent epidemic… · p. 34, Figure 18 · p. 35
  • The mass-gathering scenario substantially and unevenly changes mean arrival times across seed–destination pairs, with some pairs showing earlier arrivals.The effect depends on both the initial epidemic location and the destination district.Source: Figure 19 · p. 36, Seed-location-dependent epidemic… · p. 37
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