Numerical investigation of the effect of macro control measures on epidemics transport via a coupled PDE crowd flow - epidemics spreading dynamics model

A.I. Delis; N. Bekiaris-Liberis

REVIEW 4 major objections 5 minor 38 references

Numerical investigation of the effect of macro control measures on epidemics transport via a coupled PDE crowd flow - epidemics spreading dynamics model

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper builds a coupled PDE model of crowd flow, airborne contagion, and ventilation, and uses it to rank macro control measures by how many people become exposed during an evacuation.

desk verdict Worth a careful look for the ventilation-direction reversal, but Eq. (19) has a units and boundary-condition problem that undermines the quantitative exposure comparisons until fixed. read the letter →

arxiv 2411.16223 v1 pith:UD6MG4NM submitted 2024-11-25 q-bio.PE cs.NAmath.NA

A.I. Delis , N. Bekiaris-Liberis This is my paper · ORCID

classification q-bio.PEcs.NAmath.NA

keywords crowdflowepidemicspreadingmacroscopicmodelsnumericalsolutionventilationcontrolSEISVmodelpotentialinfectioncoefficient

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether a handful of macroscopic levers—ventilation rate and direction, pedestrians' maximum speed, average spacing, and masked or vaccinated fractions—can be ranked by their effect on disease exposure in confined evacuating crowds. To answer it, the authors couple three PDE components: a second-order crowd-flow model with Eikonal route choice, a Susceptible-Exposed-Infected-Susceptible-Vaccinated contagion model advected by the crowd, and a drift-reaction-diffusion infection coefficient driven by a potential-flow ventilation field. In simulated rooms of one and two exits and a larger corridor, the model predicts that total exposed individuals decrease with higher ventilation rate, airflow against walking direction, higher maximum speed, larger pressure coefficient (that is, larger average interpersonal distance), and inclusion of masked or vaccinated individuals. The same tests also reveal limits: when exits are wide or multiple, spacing matters less, and with narrow exits very high walking speeds can worsen clogging and exposure. If these trends hold, the model gives building operators and safety planners a quantitative way to compare interventions before deploying them.

What carries the argument

The carrying object is the infection-coefficient field $\beta(x,t)$, a transportable quantity that converts infected presence into new exposures: the source term for exposed density is $\beta_I\rho_S$. The field is advected by the ventilation velocity $U_G$, diffuses with effective turbulent viscosity $\sigma$, decays by aerosol settling $\nu\beta$, and is replenished proportionally to infected density $\rho_I/\rho$. Computing $U_G$ as the gradient of a Laplace potential $\Psi$ with Neumann duct boundary conditions is what lets ventilation rate and direction enter the model without resolving indoor turbulence. The crowd-flow component supplies both the advection velocity for the epidemic densities and the Eikonal-based desired direction toward exits, so the spatial density profile—not just a uniform mixing assumption—controls where exposure happens.

What would settle it

Replace the potential-flow air field with a measured or turbulence-resolving airflow in the same geometries and compare predicted exposed totals; if the direction of airflow that minimizes exposure changes, the ventilation conclusions are artifacts of the ideal airflow assumption.

Watch

Extended reading notes

Core claim

This paper's central claim is that epidemic transport in a closed, evacuating space can be described by a single coupled system in which the infection field is not imposed by hand but emerges from crowd motion, ventilation, and infected density. The force of infection is $\beta_I = i_0\beta(x,t)$, where $\beta$ solves a drift-reaction-diffusion equation $\beta_t + \nabla\cdot(\beta U_G) = \nabla\cdot(\sigma\nabla\beta) - \nu\beta + \rho_I/\rho$, and $U_G = \nabla\Psi$ is a steady potential-flow velocity field obtained from Laplace's equation with duct-like boundary conditions. Advecting the SEISV densities with the crowd velocity from a second-order macroscopic flow model closes the loop. In the numerical tests, the predicted exposed total falls when ventilation rate rises, when air moves against pedestrian flow, when maximum speed rises, when pressure coefficient $C_0$ rises (larger average distances), and when masked or vaccinated pedestrians are present; the direction effect reverses in large crowds near an exit, where a clean-air pocket of the size seen in smaller rooms no longer reaches most of the crowd.

Load-bearing premise

The ventilation air is treated as a steady, smooth, incompressible potential flow; real indoor airflow is turbulent and altered by the crowd itself, so the exposure reductions attributed to ventilation rate and direction rest on this idealization.

Editorial extensions

If this is right

If the model is right, ventilation rate can be treated as a tunable exposure-reduction knob, but its direction should be chosen per geometry; airflow against walking direction helps small single-exit rooms, while a pattern that sweeps clean air across a large waiting area helps big crowds.
Faster pedestrian movement reduces exposure mainly by shortening evacuation time; the benefit saturates or reverses when a narrow exit turns higher speed into stronger clogging.
Enforcing larger average interpersonal distance (higher $C_0$) lowers exposure and evacuation time in single-exit congestion, yet contributes little when multiple or wide exits already keep densities low.
A fraction of masked or vaccinated pedestrians reduces the exposed total additively with the other levers, bringing the lowest observed exposure in the tests to about 4% of the crowd.
Real-time spatial density and infection-field profiles, rather than room-averaged totals, are the natural signals for deciding when to change ventilation or issue spacing and speed recommendations.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the potential-flow airflow is the linchpin of the ventilation conclusions; replacing it with a turbulence-resolving airflow solver in the same geometries would directly test whether the direction rankings survive real indoor air motion.
Because the model outputs full spatiotemporal density and infection fields, it invites a closed-loop control formulation—ventilation rate, direction, and speed or spacing recommendations updated from measured crowd state—which the paper mentions as a possibility but does not implement.
The simulations fix recovery, latency, and vaccination rates at zero; extending the SEISV equations with nonzero recovery, latency, and vaccination rates would determine whether the same control rankings hold for longer exposures or partially immune populations.
A tracer-gas experiment in a ventilated room with a controlled walking crowd could probe the predicted clean-air pocket at the exit and the reversal of direction preference in large crowds.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on arXiv:2411.16223. It's a numerical study of a coupled PDE model: a second-order pedestrian flow model, an SEISV epidemic model, and a drift-diffusion equation for the infection coefficient, with ventilation entering as a potential flow. The paper systematically tests ventilation rate and direction, maximum pedestrian speed, a pressure coefficient that controls average spacing, and masked/vaccinated individuals. The most interesting result is that pushing airflow against pedestrian motion helps in small crowds but can hurt in large crowds because the clean-air pocket near the exit is too small. That non-obvious interaction is the paper's real contribution.

The good: this is a sensible assembly of known components, and the parameter exploration is broader than earlier work by Salam, Bock, and colleagues. The numerics are standard (Roe with MUSCL, Rusanov for transport equations, fast sweeping for the Eikonal), and the qualitative behavior is plausible. No parameters were fitted to generate the exposure numbers; they come from the stated equations.

The soft spot is load-bearing. Equation (19), which evolves β, is not a well-posed IBVP: the paper gives only the initial condition β(x,0)=0 and no boundary conditions on the ducts or walls. Since ventilation enters exactly through the inflow and outflow boundaries, the choice of BC for β will change the clean-air pockets that drive the exposure comparisons. The stress-test note also checks out: βI = i0 β must have units of 1/time to make sense in the SEISV equations, so β has units of 1/time; then β_t has units of 1/time^2 while the source term ρI/ρ is dimensionless. One term in (19) is dimensionally inconsistent no matter how you read it. This means the quantitative exposure percentages, and possibly some rankings of ventilation direction, are not uniquely determined until the equation is repaired. That's a bigger deal than the potential-flow idealization, which the authors acknowledge as an imitation.

Two smaller issues: no grid-convergence or sensitivity analysis, and no code or data released. Given the paper is purely numerical, both should be addressed.

Net: the model is worth publishing after revision. The direction-reversal finding is interesting enough that a serious referee should look at it. Ask for a corrected Eq. (19), explicit boundary conditions, and a convergence study. I'd take the qualitative conclusions with a grain of salt until then, but I wouldn't desk-reject it.

Referee Report

4 major / 5 minor

Summary. This paper presents a coupled PDE model for crowd flow and epidemic spreading in confined spaces. The crowd component is a second-order Payne-Whitham-style model with an Eikonal equation for the desired walking direction; the epidemic component is a macroscopic SEISV model; infection transmission is represented through a drift-diffusion equation for the infection coefficient, with air flow obtained from a potential-flow solution of Laplace's equation. The authors implement this model with finite volume schemes (Roe for the crowd model and Rusanov for the advection equations, with MUSCL reconstruction and a fast sweeping method for the Eikonal equation) and run numerical experiments for one-exit, two-exit, and corridor geometries, varying ventilation rate and direction, maximum pedestrian speed, pressure coefficient, exit width, and the fraction of masked/vaccinated individuals. The main reported findings are that total exposure decreases with higher ventilation rate, with air flow against pedestrian motion, with higher maximum speed, with higher pressure coefficient, and with masked/vaccinated individuals, with a noted reversal for large crowds near an exit.

Significance. If taken as a deliberately idealized modeling study, the paper is useful: it constructs a complete coupled PDE system, states all parameter values, and performs a broad sweep of control scenarios without fitting parameters to reproduce the exposure percentages. The numerical infrastructure is standard and the parameter choices are mostly traceable to the cited literature. The main value is qualitative hypothesis generation for macro-control measures in evacuating crowds. However, the quantitative exposure percentages and the rankings of ventilation scenarios are not yet fully grounded, because the infection-coefficient equation has an apparent dimensional inconsistency and missing boundary conditions, and because no grid-convergence or sensitivity study is reported. The ventilation conclusions also rest entirely on an inviscid potential-flow idealization of indoor air movement.

major comments (4)

[Section 2.2, Eq. (19)] Equation (19) is not dimensionally consistent as written. Since beta_I = i0 beta in Eqs. (15)-(17) is an infection rate with units 1/time, beta itself has units 1/time when i0 is dimensionless. Then beta_t, the advection term, and the diffusion term each carry units 1/time^2, while the source term rho_I/rho is dimensionless. If beta is instead interpreted as dimensionless, the differential terms carry 1/time and the source term remains dimensionless. Either way, one term in Eq. (19) has mismatched units. This directly affects the quantitative exposure percentages in Section 4 and the interpretation of i0. Please introduce an explicit rate constant in the source term, state the units of i0, sigma, nu, and beta, and rerun or justify the simulations with a consistent formulation.
[Section 3, boundary conditions paragraph] No boundary conditions are given for the infection-coefficient equation (19). The ghost-cell conditions listed in Section 3 apply to the crowd flow model only, and Section 2.2 supplies only beta(x,0)=0. Boundary conditions matter most at the inflow and outflow ducts, where the clean-air pockets and ventilation-direction effects in Figs. 7-12 and 22 are controlled by how beta is transported into and out of the domain. A Dirichlet condition beta=0 at supply ducts versus homogeneous Neumann conditions at walls and exhausts can change both the magnitude of the exposed fraction and the ranking of airflow directions. Please specify the boundary conditions used in the simulations and test the sensitivity of the reported trends to that choice.
[Section 4] No grid-convergence or sensitivity study is reported. All one-exit and two-exit runs use Delta x = 0.05 m and Delta t = 2e-3 s, while the corridor runs use Delta x = 0.1 m and Delta t = 5e-3 s, with no refinement study. Many of the reported exposure differences are only a few percentage points, for example the reductions of up to about 6% in Section 4.1.2, so numerical dissipation and dispersion could be of the same order as the effects being claimed. Please add at least one mesh-refinement test for each geometry and report the resulting exposure percentages to confirm the conclusions are not grid-dependent.
[Section 2.2, Eqs. (20)-(21)] The ventilation velocity field U_G is obtained as a potential flow of an inviscid, irrotational, incompressible fluid. This ignores turbulence, crowd-air interactions, and duct geometry, and the spatial structure of U_G is exactly what determines the 'pocket of clean air' near the exit that drives the ventilation-direction conclusions in Figs. 7-8 and Sections 4.1-4.3. The paper should state this limitation more prominently and, for at least one scenario, compare the potential-flow velocity field with a more realistic field, such as a RANS solution or measured data, to show that the qualitative trends are robust.

minor comments (5)

[Section 2.2, Eq. (20)] Equation (20) contains a typo: 'where Psi(x is' should be 'where Psi(x) is'.
[Section 2.2, after Eq. (18)] The sentence defining the initial condition states rho0 = rhoS0 + rhoE0 + rhoS0 + rhoV0; the third term should presumably be rhoI0.
[Section 3, CFL condition] In the CFL condition (29), 'therm' should be 'term'.
[Appendix] The limiter is named 'van Albaba' but should be 'van Albada'.
[Figure 1 caption] The caption says 'Equilibrium speed-density relations for flow (left) and velocity (right)', but both panels appear to be speed-density relations; please clarify what distinguishes the two panels.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the reported simulation outcomes are direct outputs of the stated coupled PDE model, with no fitted-input predictions, self-citation chains, or imported uniqueness claims.

full rationale

The paper does not fit parameters to reproduce the reported exposure percentages; all results are computed from the stated PDEs with declared parameter values (e.g., i0 = 0.04, ν = 0.5, σ = 1.2e-3), and no data or target outcomes are used to tune the model. The ventilation, speed, spacing, and vaccination findings are consequences of the model equations and of the chosen simulation scenarios, not of parameters reverse-engineered from the results. The model equation for β, Eq. (19), is introduced following external references [27] and [26], which are not the present authors' own works, so no load-bearing self-citation is present. No uniqueness theorem is invoked, and no ansatz is smuggled in through a self-citation chain. The potential lack of boundary conditions and the apparent unit inconsistency in Eq. (19) are validity or correctness concerns that could undermine confidence in the quantitative predictions, but they do not make the predictions equivalent to the model inputs by construction; an ill-posed equation is a different failure mode from circularity. Overall, the derivation chain is self-contained as a numerical modeling study, and no circular step can be exhibited from the paper's own equations.

Assumptions & free parameters 9 free parameters · 6 assumptions · 0 invented entities

The central results depend on many hand-picked parameters and strong modeling assumptions, notably the potential-flow airflow and the unvalidated infection-coefficient PDE. No entity beyond the model variables is introduced.

free parameters (9)

alpha in speed-density relation = 7.5
Chosen constant in V(rho)=umax exp(-alpha (rho/rhomax)^2); sets critical density and affects congestion and evacuation timing.
C0 (anticipation factor) = 0.5, 0.8, 1.2
Pressure-like coefficient in crowd momentum equations; interpreted as average interpersonal distance; varied to test spacing controls.
tau (relaxation time) = 0.6 s
Drives velocity relaxation to equilibrium speed in Eq. (2); set by hand for all runs.
i0 (infectivity scaling) = 0.04
Multiplies the beta field to obtain infection rate betaI; value taken from prior work, not independently fitted.
nu (aerosol settling rate) = 0.5
Decay rate in Eq. (19); authors assume 50% of aerosol settles per second; chosen, not measured.
sigma (effective turbulent viscosity) = 1.2e-3
Diffusion coefficient for beta, taken from literature; affects spreading of infection coefficient.
umax (maximum pedestrian speed) = 1.4 and 2 m/s
Scenario parameter tested as a speed control measure.
uin (ventilation speed) = 5 and 10 m/s
Boundary normal derivative for potential flow; scenario parameter for ventilation rate.
initial infected fraction = 0.25 and 0.1 of initial density
Sets the number of infected pedestrians at t=0; scenario parameter.

assumptions (6)

domain assumption Pedestrian flow is described by a second-order macroscopic Payne-Whitham-type model with isotropic pressure and the specified speed-density relation.
Invoked in Section 2.1, Eqs. (1)-(5). This is a modeling choice, not derived from first principles.
domain assumption Pedestrians choose their desired direction from the negative gradient of the Eikonal solution, i.e., they have global knowledge of the density field and seek the shortest travel time.
Section 2.1, Eqs. (6)-(8), following prior literature. Strong assumption about pedestrian information and route choice.
domain assumption The ventilation-induced air flow is inviscid, incompressible, and irrotational, so it is fully described by a potential flow from Laplace's equation.
Section 2.2, Eqs. (20)-(21). This is the weakest physical simplification; real indoor airflow is turbulent and affected by crowd and obstacles.
domain assumption The infection coefficient beta evolves according to the drift-reaction-diffusion equation with source rho_I/rho and constant parameters, independent of the crowd dynamics except through the source.
Section 2.2, Eq. (19), adopted from [27]. No validation that this PDE represents aerosol transport or infection pressure.
domain assumption During the time horizon, recovery, progression from exposed to infected, and vaccination are negligible (kappa=theta=xi=0).
Section 2.2, after Eq. (18). Thus the model tracks cumulative exposure, not actual infections.
standard math The numerical schemes (Roe with entropy fix, Rusanov, MUSCL reconstruction, fast sweeping method) converge to the exact solution of the PDE system for the chosen grids and time steps.
Section 3. The methods are established in the literature, but no convergence study is provided in this paper.

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Cite this review

Pith. "Pith review of Numerical investigation of the effect of macro control measures on epidemics transport via a coupled PDE crowd flow - epidemics spreading dynamics model." pith.science (2026). https://pith.science/paper/UD6MG4NM

@misc{pith2026241116223,
  author       = {Pith},
  title        = {Pith review of: Numerical investigation of the effect of macro control measures on epidemics transport via a coupled PDE crowd flow - epidemics spreading dynamics model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UD6MG4NM}},
  note         = {Machine review of arXiv:2411.16223}
}

abstract

This work aims to provide an approach to the macroscopic modeling and simulation of pedestrian flow, coupled with contagion spreading, towards numerical investigation of the effect of certain, macro-control measures on epidemics transport dynamics. To model the dynamics of the pedestrians, a second-order macroscopic model, coupled with an Eikonal equation, is used. This model is coupled with a macroscopic Susceptible-Exposed-Infected-Susceptible-Vaccinated (SEISV) contagion model, where the force-of-infection $\beta$ coefficient is modeled via a drift-diffusion equation, which is affected by the air-flow dynamics due to the ventilation. The air-flow dynamics are obtained assuming a potential flow that can imitate the existence of ventilation in the computational domain. Numerical approximations are considered for the coupled model along with numerical tests and results. In particular, we investigate the effect of employment of different, epidemics transport control measures, which may be implemented through real-time manipulation of i) ventilation rate and direction, ii) maximum speed of pedestrians, and iii) average distances between pedestrians, and through iv) incorporation in the crowd of masked or vaccinated individuals. Such simulations of disease spreading in a moving crowd can potentially provide valuable information about the risks of infection in relevant situations and support the design of systematic intervention/control measures.

Figures

Figures reproduced from arXiv: 2411.16223 by the authors.

**Figure 2.** Time evolution of the total number of pedestrians [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗

**Figure 3.** Total density profiles (top), infection coefficient profile ( [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗

Figures from the paper (21 more)

**Figure 4.** Figure 4: Total density profiles (top), infection coefficient profile ( [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]

**Figure 5.** Figure 5: Time evolution of the percentage of exposed pedestrians [PITH_FULL_IMAGE:figures/full_fig_p015_5.png]

**Figure 6.** Figure 6: Velocity field UG vectors (in blue) and the corresponding streamlines for a ventilated room along the pedestrians’ moving direction (left) and against (right) for uin = 10 m/s. (clogging) near the exit, especially for smaller values of C0, this “pocket” of clean air ne…

**Figure 7.** Figure 7: Total density profiles (top), infection coefficient profile ( [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]

**Figure 8.** Figure 8: Total density profiles (top), infection coefficient profile ( [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]

**Figure 9.** Figure 9: Percentage of exposed pedestrians in a ventilated room fo [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

**Figure 10.** Figure 10: Percentage of exposed pedestrians in a ventilated room [PITH_FULL_IMAGE:figures/full_fig_p018_10.png]

**Figure 11.** Figure 11: Percentage of exposed pedestrians in a ventilated room [PITH_FULL_IMAGE:figures/full_fig_p019_11.png]

**Figure 12.** Figure 12: Percentage of exposed pedestrians in a ventilated room [PITH_FULL_IMAGE:figures/full_fig_p019_12.png]

**Figure 13.** Figure 13: Two exits: Time evolution of the total number of pedestr [PITH_FULL_IMAGE:figures/full_fig_p020_13.png]

**Figure 14.** Figure 14: Two exits: Total density profiles (top), infection coeffic [PITH_FULL_IMAGE:figures/full_fig_p021_14.png]

**Figure 15.** Figure 15: Two exits: Total density profiles (top), infection coeffic [PITH_FULL_IMAGE:figures/full_fig_p022_15.png]

**Figure 16.** Figure 16: Two exits: Total density profiles (top), infection coeffic [PITH_FULL_IMAGE:figures/full_fig_p023_16.png]

**Figure 17.** Figure 17: Two exits: Percentage of exposed pedestrians in a vent [PITH_FULL_IMAGE:figures/full_fig_p024_17.png]

**Figure 18.** Figure 18: Corridor: Time evolution of the total mass [PITH_FULL_IMAGE:figures/full_fig_p025_18.png]

**Figure 19.** Figure 19: Corridor with a 4 m exit: Total density profiles (top), infection coefficient profile (middle), and exposed pedestrians’ profile at different time instances with umax = 1.4 m/s, C0 = 0.5, and with no ventilation. and an equation for computing the infection coefficient…

**Figure 20.** Figure 20: Corridor with a 4 m exit: Total density profiles (top), infection coefficient profile (middle), and exposed pedestrians’ profile at different time instances with umax = 1.4 m/s, C0 = 1.2, and with no ventilation. evacuation, as ventilation along the pedestrians flow m…

**Figure 21.** Figure 21: Corridor with a 4 m exit: Velocity field UG vectors (in blue) and the corresponding streamlines (in red), with uin = 10 m/s for pattern I (top) and pattern II (bottom). [2] Juan Pablo Agnelli, Bruno Buffa, Dami´an Knopoff, and Germ´an Torres. A spatial kinetic model o…

**Figure 22.** Figure 22: Corridor with a 4 m exit: Percentage of exposed pedestrians with umax = 1.4 m/s for three different values of C0. [7] S. Buchm¨uller and U. Weidmann. Parameters of pedestrians, pedestrian traffic and walking facilities. IVT Schriftenreihe. Institut f¨ur Verkehrsplanu…

**Figure 23.** Figure 23: Corridor with a 4 m exit: Total density profiles (top), infection coefficient profile (middle), and exposed pedestrians’ profile at different time instances with umax = 1.4 m/s, C0 = 0.5, and air-flow pattern I. [12] F. S. Hanseler and S. Hoogendoorn. Optimal crowd ma…

**Figure 24.** Figure 24: Corridor with a 4 m exit: Total density profiles (top), infection coefficient profile (middle), and exposed pedestrians’ profile at different time instances with umax = 1.4 m/s, C0 = 0.5, and air-flow pattern II. [18] D. Kim and A. Quaini. Coupling kinetic theory appr…

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