# The effect of local ventilation on airborne viral transmission in indoor
spaces

Alexander Pretty1<EMAIL_ADDRESS>Ian M. Griffiths2  Zechariah Lau1,2
Katerina Kaouri1  1School of Mathematics, Cardiff University, Cardiff, CF24
4AG, UK 2Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

###### Abstract

We incorporate local ventilation effects into a spatially dependent
generalisation of the Wells–Riley model of airborne viral transmission.
Aerosol production and removal through ventilation (global and local),
biological deactivation, and gravitational settling as well as transport
around a recirculating air-conditioning flow and turbulent mixing are modelled
using an advection–diffusion–reaction equation. The local ventilation effects
are compared with the equivalent global ventilation and we find that the
streamlines of the airflow provide insight into when the global ventilation
model is a good approximation. When the agreement between ventilation models
is poor, we find that the global ventilation model generally overestimates the
infection risk.

###### keywords:

##  1 Introduction

The importance of ventilation in reducing the indoor transmission of
infectious diseases was first highlighted by Nightingale (1860). When an
infectious person breathes, talks, coughs, or sneezes, disease-carrying
particles are emitted. In poorly ventilated spaces, small particles known as
_aerosols_ can remain airborne for several hours, transmitting the disease to
susceptible people when inhaled. During the COVID-19 pandemic, a key shift in
understanding and mitigating transmission of the SARS-CoV-2 virus was the
recognition of airborne transmission (Morawska & Milton, 2020), most likely
responsible for superspreader outbreaks in a restaurant (Ho, 2021), courtroom
(Vernez et al., 2021), choir practice (Miller, 2021), and meat processing
plant (Günther, 2020).

There are two main approaches to modelling airborne transmission: Wells–Riley
models and Computational Fluid Dynamics (CFD). Wells–Riley models (Riley et
al., 1978) assume a well-mixed-room (WMR), meaning aerosols are
instantaneously transported throughout the room. Due to its high computational
speed, this approach can be readily applied at the start of an epidemic. This
was the case for COVID-19 (Buonanno et al., 2020; Dai & Zhao, 2020; Lelieveld,
2020). However, the WMR assumption is not always appropriate and cannot
provide any information on the spatial variation of the concentration.

CFD models simulate the (usually turbulent) airflow in a room. The
computational demand is high, so many CFD studies at the start of the COVID-19
pandemic focused on relatively short time frames (less than 5 minutes)
(Shafaghi et al., 2020; Vuorinen, 2020) whereas airborne transmission
typically occurs over hours. Some CFD models have simulated aerosol evolution
for up to an hour (Shao et al., 2021), but the high computational times means
that CFD can not easily inform up-to-date decisions in a quickly developing
epidemic.

Lau et al. (2022) model the spatiotemporal evolution of aerosols in a room
using an advection–diffusion–reaction (ADR) equation under the assumption of a
recirculating airflow. Unlike Wells–Riley models, the ADR model accounts for
the spatial variation of concentration and infection risk. Moreover, the
simplified 2D airflow allows fast simulations so the ADR model can be quickly
deployed in a fast-changing epidemic. In Lau et al. (2022), aerosol removal by
ventilation is modelled as a global sink, as in the Wells–Riley type models.
This assumption produces good agreement with real-life scenarios ventilated by
inbuilt air-conditioning (AC) units (Lau et al., 2022).

For rooms with poor or non-existent AC, air purifiers can increase overall
aerosol removal. The effectiveness of air purifiers depends strongly on their
location (Burgmann & Janoske, 2021; Narayanan & Yang, 2021), an effect that
cannot be captured by the WMR assumption or the global sink of Lau et al.
(2022). Moreover, many CFD studies of air purifiers report prohibitively long
computational times; for example Dbouk et al. (2021) report 7 days to simulate
a 2.5 minute event in a domestic setting. While air purifiers are not
technically classified as ventilation, since they do not provide fresh air
from outdoors, the American Society of Heating, Refrigeration and Air-
Conditioning (ASHRAE) recently incorporated air cleaning devices in their
measure of _equivalent clean airflow_ when risk of disease transmission is
high (e.g. during an epidemic) (ASHRAE, 2023).

In this paper, we introduce a spatially local ventilation model to the
methodology of Lau et al. (2022). While all ventilation systems (including
doors, windows and AC units) have local effects, we are motivated by common
air purifier designs and introduce a cylindrical device that draws air in
through the top and expels clean air from the bottom. This device will be an
addition to the existing inbuilt AC system.

The paper is organised as follows. The modelling framework, incorporating a
local ventilation system, is presented in §2. In §3, we compare the average
aerosol concentration in the room predicted by the local and global
ventilation models. In §4 we consider the infection risk to individuals at
specific locations and compare the ventilation models. Conclusions and
suggestions for future work are provided in §5.

##  2 Modelling framework

###  2.1 Advection–diffusion–reaction (ADR) equation

Consider a 3-dimensional (3D) room with dimensions $L_{x}$, $L_{y}$, $L_{z}$,
as depicted in figure 1(a). Following Lau et al. (2022), we assume a
recirculating flow produced by a single AC vent along the top corner and
introduce the arclength coordinate $\xi$, which follows the recirculating
loop. The distance between the recirculation layers is $L_{z}/2$ (van Hooff et
al., 2013) and the total arclength is $2L_{x}$. A cylindrical local
ventilation system with radius $r$ is introduced, which extends over both
recirculating layers: Air is drawn into an inlet in the upper layer and
expelled from an outlet in the lower layer. We will refer to this device as a
_purifier_.

Figure 1(b) shows the computational domain $(\xi,y$). The left/right halves
correspond to the upper/lower layers and the purifier inlet and outlet appear
as circles removed from the domain, with boundaries

$\displaystyle\partial_{\mathrm{in}}=\\{(\xi,y):|(\xi,y)-(x_{p},y_{p})|=r\\},\quad\partial_{\mathrm{out}}=\\{(\xi,y):|(\xi,y)-(2L_{x}-x_{p},y_{p})|=r\\},$
(1a,b)

where $(x_{p},y_{p}$) denotes the location of the purifier in the
$(x,y)$-plane, and we will assume throughout this work that the purifier is in
the centre of the room. Although the purifier design is motivated by real-life
devices (Dbouk et al., 2021), in this quasi-3D model the device extends the
entire height of the room, which is significantly taller than real-life
purifiers. However, this simplification allows for a direct comparison with
the model of Lau et al. (2022) while still offering useful and practical
insights.

Figure 1: A 3D room with dimensions $L_{x}$, $L_{y}$, $L_{z}$ is shown in (a).
A cylindrical local ventilation system crosses the two recirculating layers
and the arclength coordinate $\xi$ follows the recirculating loop. The
computational domain $(\xi,y)$ is shown in (b).

Consider a single infectious individual standing at
$\boldsymbol{x}_{0}=(x_{0},y_{0})$ and talking continuously. Talking produces
around 10 times as many aerosols as breathing (Asadi et al., 2019): a
reasonable worst-case scenario for an asymptomatic person. We assume, as in
Lau et al. (2022), that the concentration of aerosols, $\mathcal{C}(\xi,y,t)$,
is governed by the ADR equation

$\frac{\partial\mathcal{C}}{\partial
t}+\nabla\cdot(\boldsymbol{v}\mathcal{C})-\nabla\cdot(K\nabla\mathcal{C})=R\delta(\xi-
x_{0})\delta(y-y_{0})-(\lambda+\beta+\sigma)\mathcal{C},$ (2)

where $\boldsymbol{v}=(u,v)$ is a vector field describing the airflow around
the recirculating loop; $K$ is the eddy diffusion coefficient; $R$ is the
(constant) aerosol production rate; and $\lambda$, $\beta$, $\sigma$ are
global removal rates due to ventilation, biological deactivation, and
gravitational settling, respectively. Parameter values are given in table 1;
more details are provided in Lau et al. (2022).

The recirculating loop is central to the modelling framework so the AC must be
switched on. Following Lau et al. (2022), we model this inbuilt ventilation
with the global removal term $\lambda$ and set an air-exchange rate of 0.72
air changes per hour (ACH): the ‘poor ventilation’ scenario of Lau et al.
(2022) (from classroom data, Guo et al., 2008). This reflects a broken or
poorly maintained AC system that moves air around but is ineffective at
removing aerosols.

Parameter | Symbol | Value | Source
---|---|---|---
Room length | $L_{x}$ | 8 m | Lau et al. (2022)
Room width | $L_{y}$ | 8 m | Lau et al. (2022)
Room height | $L_{z}$ | 3 m | Lau et al. (2022)
Room volume | $V$ | 192 m3 | $L_{x}\times L_{y}\times L_{z}$
AC airflow speed | $u_{0}$ | 0.15 ms-1 | (ASHRAE, 2020)
Aerosol emission rate (talking) | $R$ | 5 aerosols/s | (Lau et al., 2022)
Virus deactivation rate | $\beta$ |  $1.7\times 10^{-4}$ s-1 | (van Doremalen, 2020)
Gravitational settling rate | $\sigma$ |  $1.1\times 10^{-4}$ s-1 | (De Oliveira et al., 2021)
Air-exchange rate | $\lambda$ | 0.72 ACH: $2\times 10^{-4}$ s-1 | (Guo et al., 2008)
|  | 1.4 ACH: $4.0\times 10^{-4}$ s-1 | see table 2
|  | 6 ACH: $1.7\times 10^{-3}$ s-1 | see table 2
Eddy diffusion coefficient | $K$ | 0.72 ACH: $5.3\times 10^{-3}$ m2s-1 |  (Foat et al., 2020), (6)
|  | 1.4 ACH: $1.0\times 10^{-2}$ m2s-1 |  (Foat et al., 2020), (6)
|  | 6 ACH: $4.5\times 10^{-2}$ m2s-1 |  (Foat et al., 2020), (6)
Breathing rate | $\rho$ |  $1.3\times 10^{-4}$ m3s-1 | (Hallett et al., 2020)
Infectivity constant | $I$ | 0.0069 | (Lau et al., 2022)
Location of purifier centre | $(x_{p},y_{p})$ | (4,4) m | Room centre: $(L_{x}/2,L_{y}/2)$
Radius of purifier | $r$ | 0.1 m |
Table 1: Parameters and their values.

Aerosol production begins at $t=0$, so we set the initial condition
$\mathcal{C}(\xi,y,0)=0$. Periodic conditions across the left and right
boundaries ($\xi=0,2L_{x}$) complete the recirculating loop,

$\displaystyle\mathcal{C}(0,y,t)=\mathcal{C}(2L_{x},y,t),\quad\frac{\partial\mathcal{C}}{\partial\xi}(0,y,t)=\frac{\partial\mathcal{C}}{\partial\xi}(2L_{x},y,t),$
(3a,b)

and there is no flux through the walls at $y=0,L_{y}$,

$\frac{\partial\mathcal{C}}{\partial
y}(\xi,0,t)=\frac{\partial\mathcal{C}}{\partial y}(\xi,L_{y},t)=0.$ (4)

At the purifier inlet, aerosols are carried out of the domain by advection (no
diffusive flux), and no aerosols enter the domain at the outlet (no total
flux),

$\displaystyle\boldsymbol{\hat{n}}\cdot(K\nabla\mathcal{C})=0\;\;\mathrm{on}\;\;\partial_{\mathrm{in}},\quad\boldsymbol{\hat{n}}\cdot(\boldsymbol{v}\mathcal{C}-K\nabla\mathcal{C})=0\;\;\mathrm{on}\;\;\partial_{\mathrm{out}},$
(5a,b)

where $\boldsymbol{\hat{n}}$ denotes the unit vector normal to each boundary,
directed out of the domain.

It is unclear from the methodology of Foat et al. (2020) how to determine an
eddy diffusion coefficient $K$ for this scenario since the AC vent and
purifier outlet have different surface areas. We assume that $K$ is related to
the total air-exchange rate as follows,

$K=(\lambda+\lambda_{p})\sqrt[3]{\frac{V^{2}}{2}},$ (6)

where $\lambda_{p}$ is the air-exchange rate of the purifier (discussed
below), so that the value of $K$ is the same for equivalent global and local
ventilation levels.

Following Lau et al. (2022), we assume that the majority of aerosols remain
within the recirculating loop and are well-mixed over this height. Hence, the
concentration in aerosols/m3 is given by

$C(x,y,t)=\frac{\mathcal{C}(\xi=x,y,t)+\mathcal{C}(\xi=2L_{x}-x,y,t)}{L_{z}/2}.$
(7)

The risk of infection to a susceptible person at any $(x,y)$ is then
calculated using

$P(x,y,t)=1-\mathrm{exp}\left[-I\int_{0}^{t}\rho
C(x,y,\tau)\,\mathrm{d}\tau\right],$ (8)

where $I$ is the infectivity constant of the virus and $\rho$ is the breathing
rate (see table 1).

###  2.2 Airflow simulations

In Lau et al. (2022), aerosols are advected around the recirculating loop at
constant speed $u_{0}$ (table 1). Here, we assume that the recirculating loop
remains coherent in the presence of a purifier, leading to a modified
$\boldsymbol{v}=(u,v)$ such that air enters and leaves the purifier with a
specified constant speed $v_{p}$,

$\displaystyle\boldsymbol{v}\cdot\boldsymbol{\hat{n}}=v_{p}\;\;\mathrm{on}\;\;\partial_{\mathrm{in}},\quad\boldsymbol{v}\cdot\boldsymbol{\hat{n}}=-v_{p}\;\;\mathrm{on}\;\;\partial_{\mathrm{out}}.$
(9a,b)

We also require that $\boldsymbol{v}$ satisfies the periodic boundary
condition

$\boldsymbol{v}(0,y)=\boldsymbol{v}(2L_{x},y).$ (10)

A vector field that satisfies (9) and (10) is determined using the Shear
Stress Transport (SST) turbulent flow solver (Menter, 1994) in COMSOL (a
laminar flow solver is not suitable since $\Rey>$ 10 000). The resulting 2D
flow in the $(\xi,y)$-plane does not account for the inherently 3D structure
of turbulent flow. However, the spreading of aerosols by small-scale turbulent
eddies is accounted for by the eddy diffusion coefficient $K$.

Imposing no-slip and no-penetration conditions at the walls,

$\boldsymbol{v}(\xi,0)=\boldsymbol{v}(\xi,L_{y})=(0,0),$ (11)

we run the SST solver until a steady-state is reached. This steady velocity,
$\boldsymbol{v}$, is then used in the ADR equation (2). The results are
compared against those of Lau et al. (2022) for: (i) no purifier, (ii) a
switched off purifier ($v_{p}=0$). There is good agreement provided

$\max_{y}u(\xi=0,y)=u_{0},$ (12)

which is imposed by setting a suitable pressure gradient over the periodic
boundaries. Several turbulent models were compared and all resulted in a
similar concentration distribution $\mathcal{C}$.

Two purifier settings are considered based on the clean air delivery rate
(CADR) of purifiers used in experimental and computational studies. We define
a weak purifier with a CADR of 140 m3h-1, representative of devices for small
spaces such as domestic rooms (Dbouk et al., 2021) and individual offices; and
a strong purifier with a CADR of 1000 m3h-1, representative of devices for
larger spaces such as classrooms (Kähler et al., 2020) and open-plan offices.

Let $Q$ denote the flow-rate through the device in m3s-1. The CADR (stated in
m3h-1 by convention) is given by $\eta Q$ where $\eta$ is the filter efficacy.
We assume that 100% of the aerosols that enter the purifier are trapped by the
filter, so $\eta=1$ and the CADR and flow-rate $Q$ are thus equivalent. For a
cylindrical purifier (circumference $2\upi r$) with an inlet half the height
of the room ($L_{z}/2$),

$Q=\upi rL_{z}v_{p}.$ (13)

We hence compute the velocity $v_{p}=Q/(\upi rL_{z})$ for a given $Q$ (see
table 2).

Parameter | Symbol | Weak purifier | Strong purifier | Source
---|---|---|---|---
Flow-rate (CADR) | $Q$ | 0.039 m3s-1 (140 m3h-1) | 0.28 m3s-1 (1000 m3h-1) | *
Air velocity into purifier | $v_{p}$ | 0.04 ms-1 | 0.3 ms-1 | (13)
Purifier air-exchange rate | $\lambda_{p}$ |  $2.0\times 10^{-4}$ s-1 |  $1.5\times 10^{-3}$ s-1 | $Q/V$
Total air-exchange rate | $\lambda_{\mathrm{tot}}$ |  $4.0\times 10^{-4}$ s-1 |  $1.7\times 10^{-3}$ s-1 | $\lambda+\lambda_{p}$
Equivalent global ACH |  | 1.4 ACH | 6 ACH | 3600$\lambda_{\mathrm{tot}}$
Table 2: Parameters for the two purifier settings with $\lambda=2\times
10^{-4}$ s-1 (0.72 ACH).
* Weak purifier: Dbouk et al. (2021), Strong purifier: Kähler et al. (2020).

Each purifier is compared with an equivalent increase in the global removal
term $\lambda$. The air-exchange rate associated with each purifier is given
by $\lambda_{p}=Q/V$, which we add to the global removal rate of the AC unit
(0.72 ACH) to determine a total air-exchange rate,
$\lambda_{\mathrm{tot}}=\lambda+\lambda_{p}$ (see table 2). The weak purifier
doubles the total air-exchange to 1.4 ACH, less than half the recommended
ventilation for classrooms (3 ACH for 30 occupants: Lau et al., 2022; ASHRAE,
2022). The total air-exchange rate for the strong purifier is 6 ACH, exceeding
this recommendation and also sufficient to meet the guidelines for times of
heightened infection risk provided the number of occupants is halved (ASHRAE,
2023). Hereon in, we will refer to the global ventilation model by the ACH and
local ventilation model by the purifier strength.

The streamlines of the airflow $\boldsymbol{v}$ are shown in figure 2 for both
purifiers. The flow is broadly unidirectional for the weak purifier (figure
2a). Regions in which the streamlines are directed into or out of the purifier
are shaded and have a greater area for the strong purifier (figure 2b). In
both cases, these regions meet at the periodic boundary.

Figure 2: The airflow streamlines in the $(\xi,y)$-plane are shown for (a) the
weak purifier ($v_{p}=0.04$) and (b) the strong purifier ($v_{p}=0.3$). The
shaded regions indicate streamlines that pass through the purifier inlet
(left) and the purifier outlet (right).

###  2.3 Computational speed

For each airflow simulation (no purifier, weak purifier, strong purifier) to
reach a steady state, computation takes approximately 10 minutes. For the ADR
equation (2), an event of 4 hours takes around 5 minutes to run, including
calculation of the infection risk (8). At these computational speeds, advice
and guidance can be quickly updated with new information during a fast-
changing epidemic. Simulations were performed on a Lenovo IdeaPad Flex 5
laptop, with a 1.3 GHz 4-core Intel Core i7-1065G7 processor and 8 GB of RAM.

##  3 Average aerosol concentration

We define the average aerosol concentration in the room as

$\bar{C}(t)=\frac{1}{V}\iint_{\Omega}\mathcal{C}(\xi,y,t)\,\mathrm{d}\xi\,\mathrm{d}y,$
(14)

where $\Omega$ denotes the computational domain depicted in figure 1(b).
Taking appropriate integrals of (2) and applying the divergence theorem gives

$\frac{\partial\bar{C}}{\partial
t}=\frac{R}{V}-(\lambda+\beta+\sigma)\bar{C}-\frac{1}{V}\oint_{\partial_{\mathrm{in}}}v_{p}\mathcal{C}\,\mathrm{d}l.$
(15)

When $v_{p}\neq 0$, the boundary integral (describing aerosol removal by the
purifier) depends on the values of $\mathcal{C}$ on the purifier inlet
boundary $\partial_{\mathrm{in}}$ (1). Hence, $\bar{C}$ depends on the
location of the infectious source $\boldsymbol{x}_{0}$ and the vector field
$\boldsymbol{v}$. When $v_{p}=0$ there is no local ventilation (as in Lau et
al., 2022), the boundary integral vanishes, and (15) is equivalent to the
Wells–Riley model used in Miller (2021). In this case the solution to (15) is
given by

$\bar{C}(t)=C^{*}\left[1-\mathrm{e}^{-(\lambda+\beta+\sigma)t}\right],\;\;\mathrm{where}\;\;C^{*}=\frac{R}{(\lambda+\beta+\sigma)V}.$
(16)

Hence, $\bar{C}$ does not depend on $\boldsymbol{x}_{0}$ or $\boldsymbol{v}$
for the global ventilation cases, and $\bar{C}\rightarrow C^{*}$ as
$t\rightarrow\infty$.

We express the location of the infectious source relative to the purifier
location as

$\boldsymbol{x}_{0}=(x_{0},y_{0})=(x_{p}-d\,\mathrm{cos}\,\theta,y_{p}+d\,\mathrm{sin}\,\theta),$
(17)

where $d$ is the distance from the purifier and $\theta$ is the angle (in
degrees) from the line $y=L_{y}/2$. The problem is symmetric in this line so
we consider only $\theta\in[0,180]$. Figure 3(a) shows all choices of
$(d,\theta)$ considered here.

Figure 3: The infectious source locations, $\boldsymbol{x}_{0}$ (17), are
depicted by filled circles in (a) and the regions where all streamlines are
directed into the purifier inlet (figure 2) are shaded for each purifier. For
these $\boldsymbol{x}_{0}$, the average aerosol concentration, $\bar{C}$ (14),
after 4 hours is shown for the weak ($+$) and the strong ($\times$) purifiers
in (b). The global ventilation cases are also shown in (b), depicted as
horizontal lines (labelled with the ACH).

To compare the long-time behaviour of the different ventilation models,
$\bar{C}$ is depicted in figure 3(b) after an event of 4 hours, ensuring
$\mathcal{C}$ reaches a steady-state in every case. For the global ventilation
models, $\bar{C}$ is depicted by horizontal lines, which agree with $C^{*}$
(16). For most $(d,\theta)$ choices, there is good agreement between each
purifier and the equivalent global ventilation. For the weak purifier, the
only significant deviation is when $\theta=0$ (for all values of $d$). For the
strong purifier, the discrepancy is significant for $d=1$ (for all values of
$\theta$) and for small $\theta$ when $d=2,3$. Where the results differ the
most, global ventilation predicts a greater $\bar{C}$ than the equivalent
local ventilation. When $d=2,3$, the global ventilation model predicts a
larger $\bar{C}$ than the purifiers for some $\theta$, but the discrepancy is
relatively small.

Figure 3(b) shows that $\bar{C}$ increases with distance from the purifier
$d$, but there is also dependence on $\theta$ that is related to the
streamlines of $\boldsymbol{v}$. The regions where the streamlines enter each
purifier are shaded in figure 3(a), and $\bar{C}$ is notably lower when
$\boldsymbol{x}_{0}$ is within (or close to) these regions. For values of
$\boldsymbol{x}_{0}$ outside these regions, there is less variation in
$\bar{C}$ and closer agreement with the equivalent global ventilation.

##  4 Infection risk to susceptible people nearby

Our spatially varying ADR model allows us to determine the infection risk to
susceptible people at specific locations (8). We express the location of a
susceptible person $\boldsymbol{x}_{s}$ as

$\boldsymbol{x}_{s}=(x_{s},y_{s})=(x_{p}-d_{s}\,\mathrm{cos}(\theta+\phi),y_{p}+d_{s}\,\mathrm{sin}(\theta+\phi)),$
(18)

where $\phi$ is the angle between $\boldsymbol{x}_{0}$ and
$\boldsymbol{x}_{s}$, and $d_{s}$ is the distance from the purifier. We
consider a susceptible person directly opposite the infectious person,
$\phi=180$, and a susceptible person left of the infectious person, $\phi=90$
(depicted in figure 4). We restrict interest to $d=1$, the case with the
greatest discrepancy between the local and global ventilation models in figure
3. To reflect a scenario in which individuals are in close proximity, we also
set $d_{s}=1$. This is reflective of classrooms, restaurants, galleries and
other social events, with 1 hour being a representative event duration.
However, in reality, these purifiers are potentially too large and noisy for
use in such a setting.

Figure 4: The infection risk, $P$ (8), after 1 hour to the susceptible person
(a) opposite ($\phi=180$) and (b) left ($\phi=90$) of the infectious person
(18). Global ventilation cases are labelled with the ACH (open symbols) and
the equivalent local ventilation (purifier) is depicted by filled symbols of
the same shape. Schematics in the lower axes depict the scenario for each
$\theta$.

Figure 4 shows the infection risk to susceptible people opposite and left of
the infectious person after 1 hour, with the scenario for each $\theta$
depicted on the lower axes. Figure 4(a) is symmetric around $\theta=180$
because the problem is symmetric in the line $y=L_{y}/2$. Moreover, the
infection risk to a susceptible person right of the infectious person
($\phi=-90$) can be deduce from figure 4(b) by reflection in this line
($\theta\rightarrow 360-\theta$).

After 1 hour, the infection risk is below 40% in all cases and the infection
risk is lower with the purifiers than with the equivalent global ventilation.
The weak purifier has had only a marginal effect when compared against the
baseline example of 0.72 ACH, and shows close agreement with the corresponding
global ventilation of 1.4 ACH. The infection risk is lower for the strong
purifier than for the equivalent global ventilation of 6 ACH for all values of
$\theta$. The greatest discrepancy is when $\theta=0$, with the strong
purifier predicting half the infection risk of the 6 ACH global ventilation
(figure 4a,b).

The streamlines of $\boldsymbol{v}$ again provide insight into these results.
For the global ventilation models and the weak purifier, peaks occur when
$y_{0}=y_{s}$ due to the broadly unidirectional flow around the recirculating
loop, with lower peaks when the aerosols travel further (e.g. $\theta=180$ in
figure 4a). For the strong purifier, the largest infection risk ($\theta=90$
in figure 4b) corresponds to the susceptible person being directly downstream
from the infectious person according to the significantly modified streamlines
in this case (figure 2b).

Under the WMR assumption, the infection risk is calculated based on the
average aerosol concentration $\bar{C}$ (16). This approach predicts the
following infection risks: 8.8% for 0.72 ACH, 7.5% for 1.4 ACH, and 3.6% for 6
ACH. This is a significant underestimate compared to the corresponding values
in figure 4 since the close proximity to the infectious source results in a
concentration significantly greater than the room average at all times.

##  5 Summary and conclusions

The recent ADR model for airborne virus transmission (Lau et al., 2022) was
modified to incorporate a local ventilation system motivated by air purifiers.
A weak purifier (CADR = 140 m3h-1) and a strong purifier (CADR = 1000 m3h-1)
were compared against equivalent increases in the global ventilation (1.4 ACH
and 6 ACH, respectively).

For each purifier, the average aerosol concentration after reaching a steady-
state was compared against the equivalent global ventilation, with good
agreement in most cases (figure 3). The infection risk to susceptible people
near to the purifier after 1 hour was also considered: The weak purifier
showed close agreement with the global ventilation model, whereas the strong
purifier predicted a lower infection risk than the equivalent global
ventilation (figure 4). The largest discrepancies between the local and global
ventilation models were observed when the infectious person was located inside
or near to the regions where the airflow streamlines are directed into the
purifier inlet (figure 2).

When modelling airborne transmission, Wells–Riley models (e.g. Miller, 2021)
offer great computational speed but are highly simplistic, whereas CFD models
(particularly those that track individual particles, e.g. Dbouk et al., 2021)
provide significant detail at great computational expense. The ADR model of
Lau et al. (2022) offers a compromise, with greater detail than Wells–Riley
models (spatial variation) at low computational cost. By adding further
complexity to the problem, the present model is able to explore the effects of
local ventilation over hours, the time-frame over which airborne transmission
occurs, while retaining relatively small computational times.

In future, this computational speed could facilitate a more thorough
investigation of the problem, exploring factors such as the purifier location,
size, and strength. The model could be further developed by incorporating a
spatially varying eddy diffusion coefficient or an unsteady airflow. Building
on this work, the local ventilation effects of air-conditioning, windows,
doors, and other purifier designs could be explored in a similar manner.

[Acknowledgements]The authors wish to acknowledge the contributions of Dr.
Aaron English, Dr. Raquel González Fariña and Dr. Attila Kovács. We are
grateful to Siân Grant for generating Figure 1(a).

[Funding]A.P. and K.K. gratefully acknowledge funding from a Sêr Cymru
‘Tackling COVID-19’ grant, awarded by the Welsh Government. I.M.G. is grateful
to the Royal Society for funding through a University Research Fellowship.

[Declaration of interests]The authors report no conflict of interest.

## References

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