Also at ]Department of Aerospace Engineering, University of Michigan

# On the Utility of a Well-Mixed Model for Predicting Disease Transmission on
an Urban Bus

Zhihang Zhang  Department of Naval Architecture and Marine Engineering,
University of Michigan  Jesse Capecelatro  [  Department of Mechanical
Engineering, University of Michigan  Kevin Maki<EMAIL_ADDRESS>Department
of Naval Architecture and Marine Engineering, University of Michigan

###### Abstract

The transport of virus-laden aerosols from a host to a susceptible person is
governed by complex turbulent airflow, and physics related to breathing,
coughing and sneezing, mechanical and passive ventilation, thermal buoyancy
effects, surface deposition, masks, and air filtration. In this paper, we
study the infection risk via airborne transmission on an urban bus using
unsteady Reynolds-Averaged Navier–Stokes equations and a passive-scalar model
of the virus-laden aerosol concentration. Results from these simulations are
directly compared to the widely-used well mixed model, and show significant
differences in the concentration field and number of inhaled particles.
Specifically, in the limit of low mechanical ventilation rate, the well-mixed
model will over-predict concentration far from the infected passenger, and
substantially underpredict concentration near the infected passenger. The
results reported herein also apply to other enclosed spaces.

††preprint: AIP/123-QED

##  I Introduction

Accurately predicting the transmission of airborne diseases is critical for
assessing the associated risk to individuals and to inform decision making
with regard to societal controls. For example, it is now recognized that
airborne transmission is the dominant transmission mode of SARS-CoV-2 (Zhang
_et al._ , 2020), which is responsible for the COVID-19 pandemic. The majority
of so-called ‘super-spreading’ events during the COVID-19 pandemic has taken
place in indoor settings with poor ventilation (see, e.g., Setti  _et al._ ,
2020; Shen  _et al._ , 2020; Morawska  _et al._ , 2020; Miller  _et al._ ,
2020). The precise infectious dose an individual would be exposed to in a
given setting depends on the turbulent nature of the airflow that carries the
virus from one person to another and the various factors that contribute to
the underlying fluid dynamics.

Due to its simplicity, the well-mixed model (Riley, Murphy, and Riley, 1978;
Miller  _et al._ , 2020) remains the predominant tool to predict transmission
for risk assessment. It is based primarily on the assumption that the
concentration in an enclosed space is spatially uniform such that an ordinary-
differential equation can be written as a balance of the generation rate due
to infected persons, and the destruction due to causes such as natural decay
of infectivity, removal due to ventilation, filtration, or deposition, etc.
Miller  _et al._ (2020); Bazant and Bush (2021) use the well-mixed model to
analyze transmission for indoor spaces, and include data from super spreader
events that have occurred throughout the COVID-19 pandemic. However, indoor
spaces are rarely well mixed, and developing simple models for predicting
exposure of virus-laden aerosols under realistic settings remains challenging.

Computational fluid dynamics (CFD) is increasingly being used to study the
detailed flow physics of airborne diseases. Talaat  _et al._ (2021) use steady
Reynolds-average Navier–Stokes (RANS) simulations to predict the flow field in
the cabin of a Boeing 737 airplane to assess mitigation and safety of air
travel. Several recent papers presented detailed analyses of the flow physics
of a cough using direct numerical simulation (Fabregat  _et al._ , 2021;
Monroe  _et al._ , 2021; Liu  _et al._ , 2021). Dbouk and Drikakis (2021,
2020) use Eulerian–Lagrangian particle tracking to study the physics of
coughing, effectiveness of face masks, and the transmission risk in an
elevator. While detailed studies using RANS or large-eddy simulation (LES) are
able to predict the complex flow and pathways of virus-laden aerosols, it is
not practical to simulate all indoor spaces and airflow or occupancy
arrangements.

Lau  _et al._ (2021) refined the well-mixed mode by adding advection and
diffusion terms to the equation governing the concentration. The challenge
with this approach is how to effectively model advection, since it is
inherently nonlinear in an Eulerian setting, and depends on many parameters,
including specific room geometry, people and their movement within the
enclosed space, and effects of mechanical and natural ventilation. Guo  _et
al._ (2021) combined the well-mixed model and the Wells–Riley equation with
CFD simulations to assess the need to control spread in hospital spaces. They
found that the well-mixed model, which does not consider spatial variability,
fails to accurately predict risk, and does not provide the spatial details
that are needed to space people within a given room or enclosed zone.

This paper uses detailed simulations that model turbulence and the geometry of
the enclosed space to predict the transport of virus-laden aerosols throughout
the passenger cabin of an urban bus. The mechanical ventilation system on the
bus is used to create two different flow scenarios, one in which forced
convection and turbulence dominate the aerosol transport, and another in which
the ventilation system plays a much smaller role such that diffusion and
natural convection play a more important role. The detailed numerical results
are compared with estimates from a well-mixed model, and it is shown that
while this model is instructive for general guidance, it ignores the spatial
dependence of aerosols and this limits its accuracy.

##  II Modeling of Aerosol Transport

In this paper, we consider the smallest aerosols only, those with diameter
less than $10$ $\upmu$m. Under this assumption, aerosols are represented with
the continuum field $C=C(\mathbf{x},t)$, which is transported passively with
the unsteady RANS (URANS) velocity field Zhang  _et al._ (2021). Details on
both the well-mixed model and CFD simulations are summarized herein.

#### Well-Mixed Model

As described above, a common approach to modeling the aerosol in a confined
space employs the well-mixed assumption in which the concentration field does
not vary in space. It is not clear what is required for aerosols to be
perfectly mixed, yet this assumption is widely used since the resulting model
is so simple. The well mixed concentration field $\tilde{C}=\tilde{C}(t)$
satisfies the model equation

$\frac{{\rm d}\tilde{C}}{{\rm d}t}=\frac{\lambda}{V}-\gamma\tilde{C},$ (1)

where $V$ is the volume of the enclosed space, $\gamma$ is the loss-rate
coefficient that accounts for the decay, deposition, loss due to ventilation,
filtration, etc. Here, the tilde denotes a spatially averaged quantity. The
source of aerosols is represented by the emission rate $\lambda$. The solution
for the well-mixed concentration is simply

$\tilde{C}=\frac{\lambda}{V\gamma}\left(1-e^{-\gamma t}\right).$ (2)

The equilibrium value at large $t$ of the concentration is
$\tilde{C}_{\infty}=\lambda/V\gamma$, where $\gamma^{-1}$ is the time scale
for the spatially uniform concentration to reach the equilibrium value. The
relationship between the spatial average and local concentration is

$\tilde{C}=\frac{1}{V}\int_{V}C(\mathbf{x},t)\,{\rm d}V,$ (3)

which is the link between the CFD simulations and the well-mixed model
prediction.

#### Emission of respiratory droplets

An important quantity in any airborne transmission and risk study is the rate
at which the host produces virus-laden aerosols. Vuorinen  _et al._ (2020);
Miller  _et al._ (2020); Mittal, Meneveau, and Wu (2020); Bazant and Bush
(2021) all present a range of virus shedding rate from an infected individual.
The emission rate is defined as the number of virus-laden aerosols per time as
$\lambda$ s-1. It depends on the stage of infection, masking, and breathing
rate, and this quantity is difficult to accurately assess. For modeling in
CFD, we use a continuous breathing model due to the fact that the breathing
period is of order seconds and the time horizon of the analysis is minutes. In
this work, the shedding rate is assumed to be $\lambda=50$ s-1, and breathing
rate $\dot{V}_{b}=0.1$ l-s-1. The breathing rate depends strongly on the
activity of the person, and can range from 6-100 l-min-1 (Mittal, Meneveau,
and Wu, 2020).

#### Susceptible Person Disease Contraction

The contraction of the disease by a susceptible person via the airborne route
requires inhalation of the virus-laden aerosols. The minimum number of
aerosols that are required for one to become ill is not clearly understood.
The range is thought to be between 100 and 1000 (Vuorinen  _et al._ , 2020),
and methods like the Wells–Riley (Riley, Murphy, and Riley, 1978) (which uses
a well-mixed assumption), or dose-response functions (Bazant and Bush, 2021)
describe the probability of infection. In this work, the minimum infective
dose (MID) of $N_{{\rm b,crit}}\approx 50$ is used based on the analysis of 20
COVID-19 spreading events (Kolinski and Schneider, 2020). Also, the
Wells–Riley approach is used when comparing the CFD results with the well-
mixed model. The Wells-Riley assumes that the concentration is uniform
throughout space in the enclosed volume, and the value of the concentration is
the long-time equilibrium value from the well-mixed model. The probability of
inhaling an infectious dose is calculated as (Riley, Murphy, and Riley, 1978,
Eqn. 1):

$P=1-e^{rt},$ (4)

where $r=\tilde{C}_{\infty}\dot{V_{b}}$. The present CFD results will give
direct insight into the validity of the underlying assumptions used in the
Wells–Riley equation.

The number of inhaled aerosols, $N_{b}$, is related to the concentration field
as

$N_{b}(\mathbf{x},t)=\int C(\mathbf{x},t)\dot{V_{b}}{\rm d}t.$ (5)

If the well-mixed model is used, the concentration and number of inhaled
particles is no longer a function of space, and the integral can be solved
directly to determine the number of aerosols inhaled as

$N_{b}(t)=\frac{\dot{V_{b}}\lambda}{V\gamma}\left[t+\frac{1}{\gamma}\left(e^{-\gamma
t}-1\right)\right].$ (6)

This equation shows that for long time $t\gamma\gg 1$ the number of inhaled
aerosols grows linearly. On the other hand, for the initial time, that is
$t\gamma\ll 1$, the solution is proportional to $t^{2}$, _i.e._

$N_{b}(t)=\frac{\dot{V}_{b}\lambda}{2V}t^{2}\mbox{ for }t\gamma\ll 1.$ (7)

The quadratic nature of the initial period is acutely problematic and exposure
for time up to $t\gamma\approx 1$ should be considered to be quadratic
according to the well-mixed assumption.

Equations (2) and (6) constitute the description of virus transport in indoor
spaces that is used throughout the literature and policy making throughout the
world. In this paper, we compare the well-mixed formulation with the more
accurate CFD prediction of the time and spatially varying concentration field
in order to determine whether the well-mixed model is appropriate for
accurately predicting risk of infection.

#### URANS-based Numerical Method

In this work, the turbulent airflow inside an urban bus is determined using
the URANS equations. The effects of turbulence are represented with the
$k-\epsilon$ turbulence model. Buoyancy effects due to temperature differences
are considered through a Buossinesq approximation. Aerosols are modeled with a
continuum approximation. The number of inhaled aerosols is computed for all
locations within the confined space through numerical solutions of Eq. (5).
The method allows for details of the geometry of the enclosed space and for
the aerodynamic effects of the ventilation system. The customized solver is
based on the OpenFOAM opensource CFD platform. The concentration field $C$ is
governed by the convection-diffusion equation:

$\frac{\partial{C}}{\partial{t}}+\nabla\cdot(\mathbf{u}C)-\nabla\cdot\left(D_{\rm
eff}\nabla{C}\right)=0,$ (8)

where $D_{\rm eff}=\nu_{t}/{\rm Sc}_{t}+\nu/{\rm Sc}$, and ${\rm Sc}_{t}={\rm
Sc}=1$ are the turbulent and laminar Schmidt numbers, respectively. Additional
details can be found in the paper Zhang  _et al._ (2021). The custom solver is
available on Github.111https://github.com/zhihangz/covid-transmission

##  III Description of Bus and Numerical Setup

The University of Michigan operates 54 urban busses to transport students,
faculty, staff, and visitors across campus. Each bus can hold up to 70
passengers (35 seated and the rest standing). Aerosol transport inside the bus
is governed by the heating ventilation air conditions (HVAC) system, thermal
effects due to HVAC and passengers, the opening of windows and doors, and
passenger movement. The well-mixed model accounts for HVAC and windows and
doors.

The bus nominal dimensions are $12.1\times 2.58\times 2.95$ m and the interior
of the bus is approximately 2,000 ft3 (56.6 m3). The HVAC system consists of a
single fan in the rear of the bus that gathers air from the interior through a
single vent on the rear bulkhead, and distributes the conditioned air through
a system of supply vents that are oriented downward along the header rails on
both sides of the bus. See Figure 1(a) for a view inside the bus with the HVAC
return vent and supply vents. The HVAC system can supply conditioned air at a
maximal rate of 2,500 ft3-min-1 (70.8 m3-min-1), and the conditioned air
includes 20% that is added from outside. There are 42 supply vents, each with
dimension of $9\times 1$ in ($0.229\times 0.0254$ m), and the return vent is
$4\times 1.5$ ft ($1.22\times 0.457$ m).

(a) Bus interior with HVAC return and supply, and passenger arrangement.

(b) Top view of the bus interior.

Figure 1: Bus interior used in the CFD simulations.

The interior geometry of the bus is digitized with a laser scanner and used to
generate a mesh that includes all of the interior geometry including seating,
steps, and hand rails. Manikins are placed in the seats, and three standing
passengers are located near the front, in the middle, and in the far rear of
the bus. The standing passengers are used as hosts, and it is assumed that
their emission rate is $\lambda=50$ s-1. The continuous breathing model is
used, and the exhalation rate is $\dot{V}_{b}=0.1$ l-s-1.

A mesh refinement study was performed and the converged results on a mesh of
5.87 million unstructured cells are used in the present analysis. The largest
cells in the domain are cubes of edge length 125 mm, and the smallest cells on
the surfaces such as the mouths of the passengers and the HVAC supply vents
are 4 and 2 mm, respectively.

The turbulence intensity and length scale for the mouth of the passengers is
10% and 7.5 mm. The intensity and length scale for the HVAC supply vents is
2.5% and 5 mm. The temperature of the conditioned air is 20∘C, and the oral
temperature of 37∘C is applied to the exhaled breath.

The boundary conditions on the HVAC supply vents enforce a downward velocity
set by the fan flow-rate, and the aerosol concentration assumes an even
redistribution of that which exits the passenger cabin through the HVAC return
vents, with 20% replaced by fresh uncontaminated air.

Four simulations are performed corresponding to two settings of the HVAC
system, and two locations of the infected passenger. The bus HVAC system when
set at the maximal rate generates a strong current from front to rear in the
bus, whereas when at ten percent of the maximum it is much less dominant. The
single HVAC return vent at the rear of the bus drives a net rearward flow in
the passenger cabin, and to analyze the role of the HVAC system the passengers
are organized into two groups corresponding to either in front or behind the
infected host. Table 1 summarizes the four simulations.

For each HVAC rate, a precursor simulation of length three minutes is run to
establish equilibrium airflow conditions in the bus before the infected
passenger begins to exhale virus-laden aerosols.

Table 1: Simulation Parameters  Run | Host Position | HVAC Rate
---|---|---
1 | Forward | 100%
2 | Middle | 100%
3 | Forward | 10%
4 | Middle | 10%

##  IV Results

###  IV.1 Flow Field

The transport of the virus from host to susceptible via the airborne route
depends on the turbulent fluid velocity field that is driven by breathing,
ventilation, and thermal effects. Two HVAC settings are investigated, and Fig.
2 shows the instantaneous velocity vector fields, with color contours of the
mean velocity in the $x$ direction, where $x$ points from the front to the
rear of the bus. Upward and downward currents are observed for both HVAC rates
that indicate the airflow should effectively mix the concentration in the
passenger cabin. In addition, for both cases the rearward velocity increases
as one moves closer to the HVAC return at the back of the bus, but this effect
is much stronger for the higher HVAC rate. This is an important effect on how
the HVAC system influences the transmission of the disease and the number of
infected passengers. Another interesting phenomenon is the appearance of local
circulation in front of the host for the lower HVAC rate while rearward
currents present in the case with the higher HVAC rate. This causes the
aerosols to be trapped in the front of the cabin in Run 3, which can be
observed in the contours of inhaled aerosols.

(a) Run 1: HVAC at maximum rate.

(b) Run 3: HVAC at 10% of maximum rate.

Figure 2: Instantaneous velocity vectors of mean $x$ velocity on center plane
of the bus at $t$ = 15 min.

###  IV.2 Aerosol Concentration

The aerosol concentration at different probe locations is plotted as a
function of time for each of the four simulations in Fig. 3. There are a total
of 42 probes representing 42 susceptible people, including 35 seated
passenger, the driver, and 6 standing passengers through the centerline of the
bus. The probes are all placed near the mouths. The solution to the well-mixed
model Eq. (2) is plotted in the thick blue line. The loss-rate coefficient for
the maximal HVAC setting is $\gamma=4.17\times 10^{-3}$ s-1 ($\gamma^{-1}=240$
s) and the equilibrium concentration is $\tilde{C}_{\infty}=212$ m-3 and for
the low HVAC setting they are $\gamma=4.17\times 10^{-4}$ s-1
($\gamma^{-1}=2398$ s), $\tilde{C}_{\infty}=2119$ m-3. It is interesting to
note that the general trend of the concentration follows that predicted by the
well-mixed model for the highest HVAC rate, whereas the disagreement is
significant for the low HVAC rate. For Run 1, the concentration predicted by
the CFD model exhibits large variability due to turbulence. Also, there is a
time-lag in the concentration field that corresponds to the time it takes for
the first exhaled aerosols to reach the particular passenger. In the well-
mixed model this occurrs instantaneously. The time shift depends on the
distance from the host to susceptible passenger, and can be as large as $60$
s. This means that for a susceptible passenger that is far away from an host,
it is only a matter of a minute until contaminated air is inhaled, regardless
of the distance between the two passengers. Also note that the maximum
concentration at any instant in time can be significantly greater than that
predicted by the well-mixed model. Finally, the concentration approaches
equilibrium in the 15 min exposure time of the longest bus ride
($t\gamma\approx 3.75$).

Similar observations can be made for Run 2 where the host is located in the
middle of the bus.

The results for Run 3 with the low HVAC rate and the host forward show that
the well-mixed model and the CFD predictions differ significantly (Figs.
3e-f). For almost all passengers, the well-mixed model overpredicts the
concentration, except for the two passengers in front of the host. For these
two passengers, the concentration is significantly higher due to the low
rearward flux from the HVAC system causing the concentration to accumulate in
the front of the bus. In the 15 min exposure, the concentration is still
steadily increasing, and much larger than the equilibrium value for the high
HVAC Runs 1 and 2.

Run 4 is similar to 3, but the infected passenger is located in the middle of
the bus. Figs. 3g-h show the time histories of this run, and the agreement
between the well-mixed model and the CFD is close. Similar to runs 1 and 2,
the instantaneous concentration can be much greater than that predicted by the
well-mixed model. Also, the time lag for the low HVAC rate is increased to
almost 100 s due to the reduced rearward flow induced by the HVAC fan.

(a) Run 1: in front of the host

(b) Run 1: behind the host

(c) Run 2: in front of the host

(d) Run 2: behind the host

(e) Run 3: in front of the host

(f) Run 3: behind the host

(g) Run 4: in front of the host

(h) Run 4: behind the host

Figure 3: Time history of aerosol concentration at different passenger
locations (black lines), compared with the well-mixed model (blue lines).

The well-mixed model assumes that the concentration of the virus is
instantaneously averaged in space. The CFD computations yield the time-
accurate evolution of the spatially evolving concentration field. To analyze
the way the virus spreads throughout the bus, the spatial mean and standard
deviation are calculated according to

$\displaystyle\tilde{C}_{\rm CFD}(t)$ $\displaystyle=$
$\displaystyle\frac{1}{V}\int C(\mathbf{x},t)d\mathbf{x}$ (9)
$\displaystyle{\sigma}(t)$ $\displaystyle=$
$\displaystyle\left[\frac{1}{V}\int\left(C(\mathbf{x},t)-\tilde{C}_{\rm
CFD}\right)^{2}d\mathbf{x}\right]^{1/2}.$ (10)

The mean and standard deviation of the concentration are plotted as a function
of time in Fig. 4, compared with the well-mixed model. Although the well-mixed
model gives close average values, the large values of standard deviation
indicate that the spatial distribution exhibits strong non-uniformity. This is
more clearly shown by the Probability Density Function (PDF) of concentration
in Fig. 5, which samples over the entire domain. The well-mixed model assumes
uniform distribution hence has only a single value at each time, while the
distribution from the RANS simulations is spread out over several orders of
magnitude. For each run, more than one peak can be observed in the PDF. This
may indicate different processes that influence the concentration field, such
as diffusion and convection. The peaks shown by the simulations are to the
left of the value from the well-mixed model, which means the well-mixed model
overpredicts the concentration over the majority of space whereas there exits
pockets of high concentration that are represented by the long tails of the
PDF.

(a) Run 1

(b) Run 2

(c) Run 3

(d) Run 4

Figure 4: Time history of average and standard deviation of concentration.

(a) Run 1

(b) Run 2

(c) Run 3

(d) Run 4

Figure 5: PDF of concentration at different time instants.

###  IV.3 Inhaled Aerosols

The number of inhaled aerosols is determined through numerical solution of Eq.
(5). Figure 6 shows the time history of $N_{b}$ for each passenger, together
with the solution of Eq. (6) that assumes the aerosols are well-mixed. The CFD
results are colored according to the relative position in the bus.

Figure 6 shows the number of inhaled aerosols increases faster than linearly
for the entire 15 min exposure window. Some of the passengers in the front of
the bus are exposed to higher concentrations, and are the first to exceed the
MID threshold in as soon as 450 s. For the passengers to the rear of the bus,
the concentration predicted by the CFD is less than that of the well-mixed
model, and they do not exceed the MID in the 15 min ride.

Run 2 shows similar behavior as Run 1. The high HVAC rate dominates the
airflow, the turbulence mixes the aerosol well, and the agreement between the
well-mixed model and the CFD is strong.

For Run 3, the HVAC rate is reduced, which localizes the aerosol field. Most
passengers inhale fewer aerosols, with the exception of two passengers near
the host. The two that exceed the MID do so in the first 300 s on the bus,
whereas the most of the others inhale no more than 15 aerosols on the entire
ride.

Figure 6 shows the result for Run 4. A hallmark of the low HVAC rate is the
large fluctuation in the velocity and concentration fields, and hence the
number of inhaled aerosols. Here, many of the passengers inhale more than the
MID, the first of which after only 200 s on the bus.

The exposure to the virus in the bus can be visualized by the contour of the
quantity $N_{b}$. This is shown in Fig. 7 for the plane of 1.7 m above the
lower deck at $t=15$ min. The contours are spaced logarithmically, and the
contour of the MID is shown in white.

Figure 6: Time history of number of inhaled aerosols at different passenger
locations, compared with the well-mixed model (blue lines). Top to bottom are
Runs 1–4.

(a) Run 1

(b) Run 2

(c) Run 3

(d) Run 4

Figure 7:  Contours of inhaled aerosols at $t=15$ min. The white contour lines
represent the critical number of inhaled aerosols $N_{\rm b,crit}=50$.

###  IV.4 Number of Infected Passengers

Guidance on how to safely operate busses can be summarized by calculating the
number of susceptible passengers that become infected under different
conditions. In this section, this quantity is determined by counting the
number of passengers that inhale more than the MID. Comparison is also made
with the Wells–Riley (Riley, Murphy, and Riley, 1978) prediction of number of
infected passengers (Eqn. 4). This equation uses the well-mixed assumption to
predict the equilibrium concentration. Also, we introduce a simple mask model
based on Mittal, Meneveau, and Wu (2020) to assess their effectiveness while
being worn on the bus.

Figure 8 shows the number of passengers that inhale more than MID for each
run. Note that the Wells–Riley prediction for 100% HVAC rate corresponds to
Runs 1 and 2, and the 10% HVAC rate curve corresponds to Runs 3 and 4. There
are a total of 42 susceptible passengers. This figure highlights the
complexity of the transmission process. Starting from Run 1, there is a
rapidly increasing number of infections, reaching the value of 19 by the end
of the ride. If the infected person is in the middle of the bus (Run 2), the
number of infected decreases by 50%, whereas the well-mixed assumption and
Wells–Riley model do not distinguish the effect of infected passenger
location. For this specific HVAC design, conditioned air is supplied uniformly
along the roof of the bus, but the return is only in the rear. This sets up a
net rearward convection velocity field, which excludes passengers in the front
of the bus from inhaling infectious particles (other than those that are mixed
with fresh air and recirculated to the passenger compartment).

Run 3 shows that only 2 passengers become ill, yet 27 contract the disease in
Run 4. For both these runs the HVAC rate is minimal, and thus less forced
mixing by the mechanical ventilation system. For both of these runs, the plume
of high-concentration remains close to the infected passenger. When they are
in the front of the bus, few other passengers are within range, whereas when
they are in the middle of the bus, the high concentration field encompasses
more than half the passengers.

(a) no mask

(b) everyone wears a handmade mask.

Figure 8: Number of infected passengers as a function of time, compared with
Wells–Riley model.

#### Simple mask model

The previous results are all based on the assumption that the host emits
aerosols at a high rate. A key strategy to combat the spread of the disease is
to wear a face covering or mask, which has the two-fold effect of reducing the
number of aerosols emitted from the host, and reducing the number that are
inhaled by the susceptible passenger. The physics of masks are complex for
many reasons and are the subject of intense scientific study. In order to
qualitatively assess the influence of masks, the results are reanalyzed using
a simple model based on (Dbouk and Drikakis, 2020; Mittal, Meneveau, and Wu,
2020). The mask model assumes that a fraction of both the emitted and inhaled
aerosols are blocked by the mask. Since mask effectiveness depends on many
factors, including the type of mask and nature of how it is worn, two values
of mask effectiveness are investigated. To represent a well-fitted surgical
mask it is assumed that 90% of the aerosols are blocked, and to more
conservatively estimate the effectiveness of masks a lower effectiveness of
30% is also investigated.

When the 90% effective mask is worn by all passengers, there are no
transmissions predicted by either the CFD or the well-mixed model.

Figure 8(b) shows the number of infected passengers as a function of time when
each person wears a handmade mask with effectiveness of 30%. The CFD
simulations show that for masks are effective, the reduction in transmissions
is significant when compared to Fig. 8. Also, the difference between the
Wells–Riley model and the CFD is significant, where the model shows all
passengers will become infected at $t\approx 880$ s for the lower HCAC rate.

##  V Summary

In this paper, the transient nature of aerosol dispersion within an urban bus
is studied with CFD and a well-mixed model. Two different mechanical
ventilation settings are analyzed corresponding to strong and weak mechanical
ventilation. The spatial mean and standard deviation of the concentration are
computed as a function of time, and for all cases the standard deviation is
significantly larger than the mean, at least for the initial time.

For the high HVAC setting, the concentration throughout the cabin follows the
general trend that is predicted by the well-mixed model, although the model
either over or underpredicts the CFD data depending on location of the host.
The number of infected aerosols is also calculated, and the trends of the CFD
and well-mixed model are similar to each other, but again depending on the
location of the host and susceptible passenger the well-mixed model either
over or underpredicts number of inhaled aerosols. This means that while the
well-mixed model should be used with caution, and its accuracy depends on many
factors including relative position of the infectious individual and the
airflow in the enclosed space.

Finally, a minimum infective dose is used to count the number of infected
passengers as a function of time and compared to the Wells–Riley model. The
differences between the detailed simulations and the well-mixed model are
significant. The number of infected passengers grows in time as the aerosols
travel throughout the passenger compartment, whereas the Wells–Riley model
assumes uniform concentration throughout and significantly overpredicts the
number of transmissions.

At low HVAC rate, the results are differ even more. In the absence of
mechanical ventilation, mixing is done via buoyancy and diffusion. This leads
to more localized concentration of the aerosol around the host, and
consequently the well-mixed model is a worse approximation of the
concentration in the bus. Interesting, when the infected person is in the
front, the CFD predicts the fewest number of transmissions among the four
cases analyzed herein. At the same low HVAC setting but with the infected
person in the middle, the CFD predicts the largest number of transmissions. As
the mechanical ventilation rate becomes lower, the well-mixed model becomes a
worse indicator of the concentration in the bus for the passengers, this is
true when considering them individually, and as a whole.

Finally, a mask model is implemented assess their role on mitigation. When a
mask with 90% effectiveness is worn by all passengers, there are no
transmissions on the bus for all cases considered in the paper. When a 30%
effective mask is worn by all, the number of transmission is significantly
reduced compared to when no masks are worn, although there are significant
differences between the CFD and the Wells–Riley equation.

In summary, the air flow inside the cabin of an urban bus is complex and plays
a critical role on the transmissibility of airborne diseases. In some
circumstances, especially at higher HVAC rates, a simple well-mixed model does
a reasonable job predicting the trends in mean concentration. However, it is
unable to capture the variation about the mean, which the CFD results
demonstrated to be significant. Incorporating spatial information into a
simple model is the focus of future work.

###### Acknowledgements.

We wish to acknowledge the support of the University of Michigan College of
Engineering and the Advanced Research Computing Technology Services for
providing the computational resources to perform the simulations.

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