key: cord-0910395-ecflw1yc
authors: Li, Xiangdong; Lester, Daniel; Rosengarten, Gary; Aboltins, Craig; Patel, Milan; Cole, Ivan
title: A spatiotemporally resolved infection risk model for airborne transmission of COVID-19 variants in indoor spaces
date: 2021-12-23
journal: Sci Total Environ
DOI: 10.1016/j.scitotenv.2021.152592
sha: a2db4011ee762168c57feb27cad49b193c4388ce
doc_id: 910395
cord_uid: ecflw1yc

The classic Wells-Riley model is widely used for estimation of the transmission risk of airborne pathogens in indoor spaces. However, the predictive capability of this zero-dimensional model is limited as it does not resolve the highly heterogeneous spatiotemporal distribution of airborne pathogens, and the infection risk is poorly quantified for many pathogens. In this study we address these shortcomings by developing a novel spatiotemporally resolved Wells-Riley model for prediction of the transmission risk of different COVID-19 variants in indoor environments. This modelling framework properly accounts for airborne infection risk by incorporating the latest clinical data regarding viral shedding by COVID-19 patients and SARS-CoV-2 infecting human cells. The spatiotemporal distribution of airborne pathogens is determined via computational fluid dynamics (CFD) simulations of airflow and aerosol transport, leading to an integrated model of infection risk associated with the exposure to SARS-CoV-2, which can produce quantitative 3D infection risk map for a specific SARS-CoV-2 variant in a given indoor space. Application of this model to airborne COVID-19 transmission within a hospital ward demonstrates the impact of different virus variants and respiratory PPE upon transmission risk. With the emergence of highly contagious SARS-CoV-2 variants such as the Delta and Omicron strains, respiratory PPE alone may not provide effective protection. These findings suggest a combination of optimal ventilation and respiratory PPE must be developed to effectively control the transmission of COVID-19 in healthcare settings and indoor spaces in general. This generalised risk estimation framework has the flexibility to incorporate further clinical data as such becomes available, and can be readily applied to consider a wide range of factors that impact transmission risk, including location and movement of infectious persons, virus variant and stage of infection, level of PPE and vaccination of infectious and susceptible individuals, impacts of coughing, sneezing, talking and breathing, and natural and mechanised ventilation and filtration.

Since the outbreak of the 2019 coronavirus disease pandemic, indoor spaces have been the major venue for the disease to spread from person to person (Kenarkoohi et al., 2020; Noorimotlagh et al., 2021; Sodiq et al., 2021) . COVID-19 is a respiratory disease caused by the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Spread of COVID-19 is affected by many factors including physical and socio-demographic factors 4 Due to the multidisciplinary nature of aerosol transport, there currently is some inconsistency regarding aerosol-related terminology across different disciplines (Tang et al., 2021) . To avoid any ambiguity, this study uses the following definitions: (i) Droplets are defined as hydrated respiratory droplets that contain water and may be airborne or free falling. (ii) Droplet nuclei or particles are defined as dehydrated respiratory droplets that are airborne and water-free solid particles. (iii) Aerosols are defined as hydrated droplets or dehydrated droplet nuclei that are airborne.

Droplets emitted from human respiratory activities are distributed over a wide size range, typically from 100nm to 1000μm (Wölfel et al., 2020) , as shown in Figure 1 (hollow symbols and dashed curves). Given that a SARS-CoV-2 virus particle is ~100nm in diameter ( Bar-On et al., 2020) , even submicron droplets can carry a few virions (a virion is a complete virus particle that consists of an RNA core and a protein coat and is the extracellular infectious form of SARS-CoV-2 ) and thus spread COVID-19. Although large droplets (>100μm) contain more virions and so have a higher probability to trigger an infection if inhaled by a susceptible person, they can quickly deposit on to the ground or other surfaces (Yan et al., 2019b) , and so do not play a major role in airborne transmission. Conversely, although small particles (<5μm) contain fewer virions, these can remain airborne for prolonged periods and thus cause "long-distance transmission" (Kwon et al., 2020) . Sehrawat and Rouse (Sehrawat and Rouse, 2021) argued that the most dangerous particle size is around 0.4μm since these particles can access the lower lungs and alveoli, leading to increased infection risk and potentially damaging and lethal lesions. Sitting in between these "small" and "large" size regimes are medium-sized particles (5-100 μm) whose movement is jointly controlled by gravitational and inertial forces, and forces imposed by 5 the background airflows (e.g., drag and lift forces, turbulent dispersion, etc) (Yan et al., 2019b) .

As expelled sputum droplets contain 95-99% water, they can dehydrate in the air and ultimately become desiccated droplet nuclei (or particles) which are 20-34% of the original hydrated size (Stadnytskyi et al., 2020) , as illustrated by the solid symbols and curves in Figure 1 (data from (Chao et al., 2009) and (Li et al., 2018) ). Droplet evaporation in aerosol plumes is a highly complex process that is strongly affected by the ambient conditions, especially the local humidity and temperature surrounding the droplets and mixing of the vapour plume generated by the evaporating droplets (Villermaux et al., 2017) . Thus the rate of evaporation of droplets can vary over several orders of magnitude in different indoor environments and depends strongly upon the fine-scale turbulent structure of the exhalation event (Chong et al., 2021; Villermaux et al., 2017) .

These processes mean that the evaporation and transport of droplets is highly variable, and particles which may otherwise be free-falling in one indoor space can become airborne for much longer periods in another. The challenge of airborne disease transmission modelling is to develop a framework that can account for these complex dynamics but is flexible enough to be readily applied to a wide range of practical indoor scenarios.

The Wells-Riley model (Riley et al., 1978) is widely used to estimate the transmission risk of airborne diseases, and has recently been extensively used (Li et al., 2021b; Liu et al., 2021; Zhang and Lin, 2021) for the evaluation of COVID-19 transmission. This model is based on the concept of a "quantum of infection" (Wells, 1955) under the assumption that the air in a given space is well-mixed and contains certain "quanta" of infectious pathogens (e.g., bacterium or virus). The probability of a susceptible person being infected accumulatively increases as they inhale more quanta over a period of exposure time. Although the Wells-Riley model is flexible and simple to 6 use, this model is compromised as the quanta of infectious pathogens is often ill-defined for many viruses, and the assumption of well-mixed air conditions is often far from representative as sources of aerosolised pathogens are localised to infectious persons. Hence the classic Wells-Riley model does not resolve the spatiotemporal inhomogeneity of the airborne pathogen distribution and thus infection risk, and the loosely defined infectious quanta cannot be clearly quantified from clinical data. Sze and Chao (Sze To and Chao, 2010) suggested that in order to obtain effective risk assessments using the Wells-Riley model, the characteristics of a specific pathogen and the process of pathogen dispersion in a given indoor environment must be carefully addressed.

This study aims to address these shortcomings by developing a spatiotemporally resolved model of airborne COVID-19 transmission risk that addresses the above requirements. Clinical data of viral shedding by COVID-19 patients, and biomedical data regarding SARS-CoV-2 infecting human cells are formulated and incorporated into the basic Wells-Riley framework in Section 2, forming a novel, fit-for-purpose quantitative prediction model for airborne COVID-19 transmission risk. Computational fluid dynamics (CFD) simulations are used to predict the transport of droplet nuclei, providing accurate spatiotemporal distribution of virus-laden aerosol particles. These aerosol distributions serve as an input into the extended Wells-Riley model, forming an integrated model capable of predicting the spatially resolved infection risk associated 7 of scenarios.

The most significant contribution of this study is the development of a theoretical framework in which engineering designs and biomedical data can be incorporated to quantify epidemiological risks. To the best of our knowledge, the presented model is the first of its type and can be easily adapted for other airborne diseases. However, due to the lack of clinical and virological data regarding COVID-19 and SARS-CoV-2, some assumptions are not firmly validated by experimental evidence, such as the linear relationship between the HID 50 and TCID 50 indices, and ignoring natural virus decay in the air. The model accuracy will increase when the relevant data become available.

According to the original model proposed by Wells (Wells, 1955) and Riley (Riley et al., 1978) , the probability of infection via airborne particles is given by a Poisson process, where the infection probability P is dependent upon the total "infectious quanta" of pathogen inhaled n quantum , which is given by the product of the number concentration of quanta in the air c quantum , the susceptible person's pulmonary ventilation volumetric flow rate p, and the exposure time t e as 

This equation assumes that the infectious quanta are removed from the room air only by the ventilation airflow. In fact, however, many factors such as the accelerated decay of pathogens in air, ultraviolet deactivation, deposition of pathogen-laden particles, air purification, as well as chemical and thermal conditions will affect the pathogen viability and quantum concentration (Fisk et al., 2005) . Over the past decades, a number of sink terms have been added to Eq. 2 to account for these factors (Sze To and Chao, 2010 

indoor airspace. This assumption is rarely satisfied due to the highly heterogeneous nature of indoor airflows and localised nature of infection sources such as individual patients. Hence the distribution of quantum concentration and infection risk typically exhibit strong spatial and temporal variations, which must be resolved to accurately quantify infection risk. With the increasing popularity of computational fluid dynamics (CFD) in modelling indoor air flow and contaminant transport in recent decades, the Wells-Riley model has been extended to 3D using CFD-generated aerosol concentration fields. Yan et al (Yan et al., 2017) developed a 3D predictive model to analyse airborne transmission risks in airliner cabins based on the Wells-Riley framework coupled with Lagrangian particle simulations. Srivastava et al (Srivastava et al., 2021) recently developed an analogous 3D model for COVID-19 infection evaluation in office buildings using Eulerian CFD simulations. In these studies the average concentration of infectious quanta c quantum in (1) is replaced by the spatiotemporal concentration c quantum (x,t), greatly improving the spatial resolution of the Wells-Riley model.

Another major drawback of the Wells-Riley model is that the infectious quantum has not been properly quantified. A properly quantified infectious quantum is important because different diseases have different transmissibility, therefore require different pathogen dose to initiate an infection (Sze To and Chao, 2010) . Moreover, the dose of exposure also affects whether the outcome is subclinical, tissue damaging or even lethal following the infection (Sehrawat and Rouse, 2021) . However, there currently is a severe lack of data regarding what represents an infectious quantum for a given disease or pathogen type. Although this problem has been pointed out by Sze and Chao (Sze To and Chao, 2010) over a decade ago, there has been little progress in this field to date. Existing studies typically use hypothetical infectious quantum values, which J o u r n a l P r e -p r o o f severely limits the accuracy of risk predictions. This situation calls for accurate virological and clinical data to facilitate accurate quantification of the Wells-Riley model. This paper will present a unified framework in which the infectious quantum is quantified using clinical data of viral shedding by COVID-19 patients and virological data of SARS-CoV-2 infecting human cells. The local quantum concentration c quantum (Eq. 1) in the air is characterised using temporospatial CFD simulations of aerosol transport considering a range of influencing factors including the HVAC scheme and wearing of PPE. This contributes to a fit-for-purpose predictive model for temporospatial infection risk of different SARS-CoV-2 variants in indoor spaces, and thus overcomes the limitations identified above.

To properly quantify the infection risk of SARS-CoV-2 in the Wells-Riley model, it is necessary to link the number of infectious quanta n quantum to clinical data. Under the assumption that the number n quantum of infectious quanta inhaled scales linearly with the number n virion of virions inhaled, then these variables may be related as virions (Iwami et al., 2012) , while for influenza A virus the unit is 1,000 to 6,000 virions with a mean value of 4,000 (Parker et al., 2015; Yan et al., 2019a) . TCID 50 data for SARS-CoV-2 has only recently been reported, with variation over a very wide range (between 10 3 and 10 5 virions) depending on the virus variant (Liotti et al., 2021; Pollock et al., 2021; Sender et al., 2021) . It is expected that as further studies are conducted that these data shall become more robust, but similar 

where the coefficient δ represents the ratio of HID 50 to TCID 50 (δ=HID 50 /TCID 50 ). For influenza, the HID 50 unit is reported to be 1 to 126 TCID 50 units with the median value of δ =5 (Nikitin et al., 2014) and δ = 0.5 to 3 for those not immunised. For SARS-CoV-2, δ is unknown but we currently assume it to be the same as that of the influenza virus (δ =5) in lieu of more detailed information.

In the intial stage of the COVID-19 pandemic, Wölfel et al ( 2020) continuously monitored the J o u r n a l P r e -p r o o f SARS-CoV-2 viral load in the sputum of COVID-19 patients and found that viral shedding was very high during the first week of symptoms, followed by a slowly descending viral load in the following 3 weeks. Over a 28-day period, the mean viral load in the patients' sputum was 7.0×10 6 ribonucleic acid (RNA) copies/mL, with a maximum of 2.35×10 9 copies/mL. The number of virions has been found (Sender et al., 2021) to be roughly equal to the number of RNA copies.

Assuming each genome is associated with a virion, Stadnytskyi et al ( 2020) calculated that the probability that a 50-μm hydrated droplet contains at least one infectious virion is ~37%. For a 10-μm and 3.5-μm hydrated droplet, the probability drops to 0.37% and 0.01%, respectively. This suggests a linear relationship between the virion count in a respiratory droplet and the volume of the droplet, which can be expressed as

where n p,virion is the number of virions contained in a droplet with an initial hydrated diameter of d p,0 and c RNA is the viral load of an infected person's respiratory fluid (RNA copies per unit volume).

As a droplet evaporates and ultimately becomes a desiccated particle, it shrinks to a diameter of d p although the number of virions contained in it does not change. Therefore, for air with a local particle volume fraction of c(x,t), the local number concentration of virions is

and so the local number concentration of infectious quanta is

Substitution into Eq 1 yields a spatiotemporal Wells-Riley model for the local infection risk of J o u r n a l P r e -p r o o f SARS-CoV-2 based upon the local particle concentration.

where the coefficient η S (0< η S <1) is the filtration efficiency of respiratory PPE worn by susceptible persons. Conversely, respiratory PPE worn by infected person reduces the load of expelled aerosols and so is accounted for during the modelling of particle release and transport using CFD, and ultimately impacts c(x, t). CFD can accurately simulate physical aerosol transport and thus directly account for the quantum sink terms in Eq. 3 associated with particle deposition and filtration in a physically consistent manner. Similarly, if significant, impacts such as particle decay and UV degradation can incorporated into the CFD simulations via appropriate models.

However, changes in the viability of the suspended virions with the aerosols is not considered in this study due to the very strong viability of SARS-CoV-2 in the environment. Van Doremalen et al (van Doremalen et al., 2020) studied the viability of the virus in aerosols, and found that the virus remained highly viable throughout the duration of their experiment (3 hours), with a reduction in the infectious titre from 10 3.5 to 10 2.7 TCID 50 per litre of air, equivalent to a half-life of 1.1 to 1.2 hours, compared to the typical particle residence time of 5-15 minutes in a well-ventilated room with an air exchange rate of 12 air changes per hour (ACH) (Tang et al., 2021) . Therefore, the rate of degradation is regarded as very small and may be negligible under normal mandated air exchange rates. In addition, due to the lack of clinical and virological data regarding COVID-19 and SARS-CoV-2, some assumptions are not firmly validated by experimental evidence, such as the linear relationship between the HID 50 and TCID 50 indices, and ignoring natural virus decay in the air. The model accuracy will increase when the relevant data become available.

It is important also to note that the modified Wells-Riley model (11) assumes that the infection risk only depends upon the total viral load inhaled (given by the total volume of inhaled aerosols) and does not depend upon the particle size distribution of inhaled aerosols. Several studies have

shown (Inthavong et al., 2013) that the size of inhaled aerosols governs the deposition of such aerosols in lungs and upper airway, with smaller particles penetrating the lower airway and alveoli, causing an elevated risk of infection. A particle size-dependent transmission coefficient ξ is required to properly incorporate these effects into the Wells-Riley model, but such clinical data is not yet available and so this issue is beyond the scope of this study.

Equation 11 between aerosols), and so resolves spatiotemporal evolution of the aerosol particle size distribution. The Lagrangian method can also account for various interphase transport phenomena such as interfacial forces and droplet evaporation. It is widely used in the studies of indoor particulate contaminants (Yan et al., 2017) including the transmission of COVID-19 (Li et al., 2021b; Liu et al., 2021) . However, additional post-processing procedures (such as smoothing kernels) must be implemented to convert the discrete particle trajectories to a continuous particle concentration field (Yan et al., 2017; Zhang and Chen, 2007 ) before the risk model can be applied, and large particle numbers must be advected to provide sufficient resolution to avoid conversion errors (Evrard et al., 2021) .

In contrast, the Eulerian models treat the particulate phase as a continuous pseudo-fluid that interpenetrates the air, thus their transport can be modelled using a set of coupled conservation equations, with the particle concentration directly calculated. Depending on how the conservation equations are solved for the particulate phase, the Eulerian models have different variants, such as the two-fluid model, algebraic slip model (ASM), multiple size group (MUSIG), and many others.

The two-fluid model solves two sets of conservation equations, one for each phase, with interphase transport terms being included in the conservation equations to account for the interphase transport processes. The algebraic slip model treats the air-particle mixture as a single-phase fluid, thus only solves one set of momentum conservation equations for the mixture and a further scalar conservation equation for the particulate phase volume fraction. Momentum transfer between phases is encoded via an algebraic "slip velocity" within the particulate phase volume fraction conservation equation. In comparison, the two-fluid model can produce more physically robust predictions but is computationally more expensive than the algebraic slip model.

The two-fluid and algebraic slip models are somewhat limited in that they can only model a single representative particle size (encoded in the momentum transfer terms), and so cannot resolve evolution of the particle size distribution. MUlti-SIze-Group (MUSIG) models (Li et al., 2010; Yuan et al., 2016) overcome this restriction by using a series of momentum equations for various particle size "bins" (with different momentum exchange terms for each) to resolve the evolving particle size distribution, but the resultant large equation system is computationally expensive to solve. The governing equations and related closure equations of the above models have been extensively elaborated in the literature (Li et al., 2015; Mikko et al., 1996) and will not be repeated here. In principle, the spatiotemporal Wells-Riley model (11) can be used with any appropriate CFD method for a given application. However, to provide some guidance as to method selection, we evaluate the Lagrangian (LM), two-fluid (TFM) and algebraic slip (ASM) models in terms of accuracy and computational cost to simulate the transport of micron-sized particles in a ventilated chamber against experimental data measured via phase Doppler anemometry (PDA) (Chen et al., 2006) . Details regarding the model setup and numerical procedure can be found in (Li et al., 2015) , and only data comparison are shown here for conciseness. Figure 2 compares the particle concentration distribution predicted from CFD simulations against the experimental measurements. It shows that all the three two-phase flow models achieve good agreement with an average predictive error smaller than 5%. The computational time for each model on a desktop computer with a 4-core CPU (3.3GHz base speed) and 16GB RAM is shown

in Table 1 . It shows that the algebraic slip model (ASM) only needs 180 minutes to obtain the particle concentration field, whereas the two fluid model (TFM) requires 300 minutes due to J o u r n a l P r e -p r o o f solution of the two momentum phase equations. The Lagrangian model (LM) needs a considerably longer compute time (540 mins) due to the need to solve the particle advection equations, and the subsequent post-processing to convert the discrete particle locations into a continuous particle concentration distribution.

Despite the clear advantages in computational overhead, it is still worth reiterating that the algebraic slip model is relatively simplified, while the two-fluid and Lagrangian models provide more physically robust approaches to particle transport in the air by accounting for phenomena such as lift, Magnus and Basset forces. In addition, these models also allow for dynamic models of droplet evaporation. Furthermore, the algebraic slip model and two-fluid model should only be used when the aerosol plume can be validly represented by a single particle size, i.e. when evolution of the particle size distribution is negligible. Thus, it is important to select appropriate multiphase flow models and associated closures for the physical scenario at hand and the simulation fidelity demanded by the application.

As a demonstration, the spatiotemporal Wells-Riley model (11) 

Shown in Figure 3 is the hospital ward of consideration, which includes 4 patient rooms, a nursing To reduce the computational overhead of the model, two separate computational domains were created, one for Patient Room 3 and the other for the common ward area, rather than computing the whole ward section, as shown in Figure 4 . It is assumed that the aerosol concentrations in the leaked air from each room are equal as they are all single-bed rooms containing one infected person. This allows us to only simulate one patient room, and hence reduce the CPU time. The patient room model will be used to analyse the transport characteristics of aerosol particles so that the particle size can be properly characterised and their transport characteristics can be actually predicted. The room model will also generate important data to properly specify boundary conditions of the work area model.

The computational domains shown in Figure 4 were discretised using an unstructured tetrahedral mesh with prism inflation layers applied at all solid surfaces for improved modelling of near-wall flows, heat transfer and aerosol deposition. Mesh independence is achieved at 2.2 million and 10.8 million mesh elements for the patient room and work area models, respectively, as further J o u r n a l P r e -p r o o f increasing the mesh count to 2.8 and 12.2 million elements respectively results in <0.5% change in the predicted air velocity at randomly selected locations.

The ventilation and heat load parameters of the ward, as well as the respiration and aerosol emission data of the occupants (e.g., healthcare worker and patients) are summarised in Table 2 .

The ventilation parameters are selected based on the ASHRAE Standard 55-2004 55- (ANSI, 2004 .

The total human metabolic heat is ~90 W/person, of which around 40% (36W) is dissipated through convection in a typical indoor environment (de Dear et al., 1997) . This study only considers the convective component as other components (e.g., radiative, sweating, etc) are not expected to remarkably affect the airflow field and aerosol transport. The average human pulmonary ventilation rate is 6 L/min (Carroll, 2007) , which corresponds to a total mass flux of exhaled aerosol particles by a patient is 2.0×10 -10 kg/s, as calculated in Section 3.2.

As discussed in Section 1, the particle size strongly affects the transport characteristics of aerosol particles, therefore proper characterisation of the particle size is critical to effective modelling.

Previous studies (Li et al., 2018) have demonstrated that isolated droplets as large as 100μm only need 5 seconds to become completely desiccated in typical indoor conditions (25 o C, 50% relative humidity), however this evaporation time can be extended significantly in the presence of high humidity or for dense sprays where the surrounding vapor field significantly retards evaporation (Chong et al., 2021; Villermaux et al., 2017) . Under the above fast-evaporation conditions, this evaporation time reduces to 0.1s for droplets with initial size of 10μm. As a result, over 80% of the respiratory droplets will be desiccated and thus become airborne droplet nuclei (<10μm) within 1.0s and more than 95% of the droplets will shrink to be less than 30μm in few seconds. In J o u r n a l P r e -p r o o f comparison, the particle residence time in a well-ventilated hospital room with an air exchange rate of 12 air changes per hour (ACH) is 5-15 minutes (Tang et al., 2021) , rending the droplet dehydration time negligible. Therefore, this study neglects the process of droplet dehydration, and assumes that droplets become desiccated particles immediately after being emitted. As the density and load of non-volatile compounds in human sputum respectively is 1400 kg/m 3 and 1.8% (Nicas et al., 2005) , the diameter d p of a dehydrated droplet nucleus is estimated to be 26.2% of its original diameter d p,0 . As a result, more than 95% of the particles will be smaller than 30μm according to Figure 1 .

To understand the transport characteristics of different-sized particles, CFD simulations using the Lagrangian model are first performed with the ejected particle size distribution given in Figure 1 for human speaking. The trajectories of particles with different sizes are shown in Figure 5 . The results show that as the particles disperse, the tendency to deposit increases with particle size, such that 1.6-μm particles follow the air flow and remain airborne with minimal deposition ( Figure   5 (a)), but 11.8-μm particles are found to deposit on the floor and room walls ( Figure 5(b) ).

Deposition becomes dominant for 22.9-μm and 29.5-μm particles ( Figure 5 (c) and (d)), where most of these sized particles settle in a small area around the patient mouth, and some of the 22.9-μm particles settle on the floor. As a result, almost all particles larger than 20μm settle in the room and cannot pass through the door gaps and enter the work area. These results demonstrate it is typically safe to ignore particles larger than 20μm in terms of airborne disease transmission risk, and so these particles shall not be considered further in this study.

According to Figure 1 , although particles larger than 20μm comprise a large mass fraction (>80%) of the exhaled aerosol, they only comprise a small number faction (<5%). The number and mass J o u r n a l P r e -p r o o f fractions distribution of particles smaller than 20μm are shown in Figure 6 .

The algebraic slip model is selected to predict the aerosol transport due to its good accuracy and relatively low computational cost. However, due to the inherent limit of the model, a representative particle size rather than the particle size distribution as shown in Figure 6 must be used. Therefore, the size and mass distributions shown in Figure 6 are then used to calculate a representative volume-weighted mean particle diameter (d p,mean ). If the particle size range is binned into i size groups and each group contains j i particles, the volume-weighted mean particle diameter d p,mean then satisfies

resulting in d p,mean = 6.62μm. The volume-weighted mean particle diameter d p,mean is incorporated in the algebraic model to predict the transport of particles in the air. The predicted particle distribution and deposition pattern are compared with those generated from the Lagrangian model using the particle size distribution shown in Figure 6 . As shown in Figure 7 , both models predict a highly heterogeneous particle concentration field in the room, with a local high concentration region directly above the patient's head. The particles quickly disperse as the distance from the patient increases. The ASM with the representative particle mean diameter d p,mean predicts a very similar spatial aerosol distribution to that of the Lagrangian model. Similarly, the particle deposition patterns predicted from both models are also very similar, resulting in a net particle deposition rate at the floor of 5.1×10 -12 and 6.2×10 -12 kg/s, for the algebraic slip model and Lagrangian model respectively. Although these comparisons do not extensively test evolution of the particle size distribution, these results indicate the algebraic slip model is sufficiently accurate for demonstration of the spatiotemporal Wells-Riley model.

Journal Pre-proof 22 For SARS-CoV-2, the TCID 50 unit is reported to range from 10 3 and 10 5 virions with an average value of 10 4 virions (Liotti et al., 2021; Pollock et al., 2021; Sender et al., 2021) . For the original SARS-CoV-2 variants such as Alpha and Beta, the TCID 50 unit is around 10,000 (Sender et al., 2021) . The Delta variant possesses mutated spike proteins that have a stronger ability to bind to human cells . The study by Scudellari (Scudellari, 2021) found that in the Alpha variant of SARS-CoV-2, around 50% of spike proteins are primed to infect a human cell. This percentage rises to greater than 75% in the Delta variant, making it need fewer virions to start an infection (i.e., a smaller TCID 50 unit) compared to non-Delta variants. As a result, TCID 50 unit of the Delta variant is around 4,000 (Liotti et al., 2021) .

Infection risk in the patient room is quantified in terms of the 3D distribution of infection J o u r n a l P r e -p r o o f probability given by (11) using an assumed total exposure time of 1 hour (t e = 3600s) and a viral load of C RNA = 2.35×10 9 copies/mL (the peak value of the original variants measured by Wölfel et al (Wölfel et al., 2020) ). As shown in Figure 8 , the computations reveal that the infection risk has a strongly heterogeneous distribution similar to that of the particle concentration field (Figure   7(b) ). As expected, the region above the infected person represents the highest infection risk. The infection risk is relatively low when the TCID 50 unit is large (1×10 5 virions) even the infected person does not wear a mask and the susceptible person only wears a surgical mask with a filtration efficiency of 0.715. The area-averaged mean infection probability in the horizontal plane at the typical nose height of the susceptible person (H = 1.65m) is as low as 0.028. As the TCID 50 unit decreases and the viruses become more contagious, the infection risk quickly increases. When the TCID 50 unit drops to 1×10 3 , the mean infection probability at the nose height becomes more than 16 times higher than that of TCID 50 = 1×10 5 .

Apart from the TCID 50 unit, the viral load c RNA is another critical parameter determining the transmissibility. The viral load reported in the literature are distributed in a very wide range (10 2 to 10 11 RNA copies/mL (Miller et al., 2021) ) and can vary with different virus variants. Li et al (Li et al., 2021a) reported that the viral load produced by the Delta variant is over 1000 times higher than the original 19A/19B strain, and this variation is one of the main causes of the Delta variant's high transmissibility. To investigate the effects of viral load on the infection risk, computations are performed over the range of 6×10 5 to 1×10 11 copies/mL. The lower limit of viral load (6×10 5 copies/mL) represents the minimum viral load to trigger an infection in vitro (Li et al., 2021a) and

the upper limit (1×10 11 copies/mL) is the maximum value reported in the literature (Miller et al., 2021) . Figure 9 shows the infection risk pattern in the room under different viral loads. The

24 infection probability is very low at low viral loads (Figure 9(a) and (b) ), but gradually increases as the viral load becomes larger and begin to display heterogeneous distribution patterns ( Figure   9 (c)). At very high viral load (1×10 11 RNA copies/mL), a significant part of the room becomes a "red zone" (Figure 9 (d)) with a mean infection probability of 0.767 for a one-hour exposure.

To comprehensively analyse the relative significance of the TCID 50 unit and viral load, parametric studies are performed. The computations are completed with an exposure time of 1.0 hour (t e = 3600s) and the susceptible person wears a surgical mask (η S = 0.715 (Clapp et al., 2021) ). The predicted mean infection risk at the nose height is shown in Figure 10 , which indicates that the infection risk is very low at small viral loads (c RNA < 10 7 copies/mL), but increases nonlinearly as the viral load becomes larger. This results in an apparent "threshold viral load", beyond which the infection risk rapidly increases, and this threshold viral load increases with TCID 50 value.

The parametric studies show either a large viral load or small TCID 50 unit can lead to a high infection risk. Highly contagious variants of SARS-CoV-2 such as Delta are typically characterised by high viral loads and small TCID 50 values, rending them much more contagious than the original variants.

Using a TCID 50 unit of 4,000 virions for the Delta variant (Liotti et al., 2021) and 10,000 virions for the original variants (Sender et al., 2021) , the corresponding infection risks of the variants are estimated and plotted in Figure 10 . With the same aerosol concentration, the infection risk of the Delta variant is significantly higher than the original variants. Moreover, given that the viral load produced by the Delta strain is 3 orders of magnitude larger than the original SARS-CoV-2

variants (Li et al., 2021a) , the mean and maximum viral load produced by the Delta variant is estimated to be 8.82×10 9 and 2.96×10 12 copies/mL respectively based on the study of Wölfel et al J o u r n a l P r e -p r o o f (Wölfel et al., 2020) . Consequently, the average infection probability of the Delta variant is estimated to be more than 200 times greater than the original variants, as illustrated by the red dots in Figure 10 . If we consider the maximum viral load, the transmissibility of the Delta variant is even more striking. Newly emerged variants such as the C.1.2 (Scheepers et al., 2021) and the B.1.1529 strains are reported to have highly mutated spike proteins, which could lead to dramatically increased infection risk.

If the susceptible person wears a surgical mask (η s = 0.715), this can provide good protection from the original SARS-CoV-2 variants, but the protection is limited in the case of the Delta variant, which means that more effective protection strategies need to be developed urgently.

The use of respiratory personal protective equipment (PPE) by both the infected and susceptible persons is also investigated. Computations were completed with the infected person wearing either no mask (η I = 0) or a mask with a filtration efficiency of 0.9 (η I = 0.9). This means the net particle injection rate from each infected person into the air reduces from 2.0×10 -10 kg/s to 2.0×10 -11 kg/s. Surgical masks and NIOSH-approved N95 respirators are selected for the susceptible person, with respective filtration efficiencies of η s = 0.715 and η s = 0.98 (Clapp et al., 2021) . Other computational parameters are selected for the Delta variant (c RNA = 8.82×10 9 copies/mL, TCID 50 =4,000), an exposure time of t e = 3600s is used, and the predicted infection risk distribution is shown in Figure 11 .

As expected, the results indicate that wearing of facial masks by the infected and/or susceptible persons markedly reduces the infection risk. As shown in Figure 11 (a), when the infected person is not wearing a mask and the susceptible person only wears a surgical mask, the room has a local J o u r n a l P r e -p r o o f infection risk approaching 1.0 over a one-hour exposure. The mean infection probability at the nose height is 0.415. However, if the susceptible person wears a N95 respirator, the risk drops significantly to 0.118 (Figure 11(b) ). It is also important for the infected person to wear a facial mask in order to reduce aerosol emission and hence source of infection. When 90% of the emitted particles are captured by the mask, the mean infection probability drops from 0.415 (Figure 11 (a)) to 0.152 even if the susceptible person only wears an ordinary surgical mask (Figure 11(c) ). The mean risk further drops to 0.017 if the susceptible person wears a properly fitted N95 respirator ( Figure 11(d) ). The results clearly demonstrate the efficacy of respiratory PPE for both patients and healthcare workers in healthcare environments.

Parametric studies are also conducted to analyse the effectiveness of respiratory PPE over a broad parametric range, as shown in Figure 12 . Figure 12 (Figure 12(d) ). As the viral load exceeds 10 9 copies/mL, the efficacy of surgical masks is very limited (Figure 12(b) and (e)). If the viral load further increases, N95 respirators will lose their high level of protection too, as shown in Figure 12 (c) and (f).

The computations clearly demonstrate the benefits for both the infected and susceptible person to properly wear respiratory PPE. However, respiratory PPE is effective only when the viral load is low and TCID 50 is large. At high viral loads and small TCID 50 units, respiratory PPE can only provide very limited protection. In fact, PPE locates at the bottom of the NIOSH's hierarchy of hazard controls ( (NIOSH), 2015) , meaning it is the least effective method for hazard control. As highly contagious variants such as the Delta (Scudellari, 2021) and Lambda (Kimura et al., 2021) variants are increasingly causing concerns all over the world, more effective protection strategies must be developed.

In this subsection we determine the infection risk in common ward area arising from infectious patients in the ward rooms. To achieve this, the predicted air and aerosol mass flow rates at the door gap of the patient room are extracted and applied as boundary conditions for the CFD model of the common ward area. CFD computations are completed with the exposure time ranging from 10 minutes to 8 hours, representing the shortest and longest possible time that a staff member may stay in the work area. Other computational parameters include transmissivity for the Delta variant of TCID 50 =4,000 and c RNA = 8.82×10 9 copies/mL (Li et al., 2021a) . All the susceptible persons are J o u r n a l P r e -p r o o f wearing a N95 respirator (η S = 0.98).

The air flow field in the horizontal plane at the nose height (H=1.65m) and 3D particle concentration field in the work area are shown in Figure 13 . The computations predict an evenly distributed air flow field with an average air speed of 0.11 m/s. Local high-speed regions can be seen near the door gaps and ventilation diffusers (Figure 13(a) ). It also can be seen that the aerosol that escapes from the patient rooms is carried by the ventilation air flow, creating a heterogeneous particle concentration field. Relatively high-concentration regions are observed in the corridor close to Patient Rooms 1 and 2 ( Figure 13 (b)) where two room doors are close to each other, and fresh air is not directly supplied to. Compared to that in the patient rooms (Figure 7) , the aerosol concentration in the work area is around 2 orders of magnitude lower. The spatial distribution of airborne infection risk over an 8-hour duration in the common ward area is shown in Figure 13 (c).

Similar to the aerosol concentration field, the spatial infection risk distribution is highly heterogeneous throughout the common ward area.

The mean infection risk in the horizontal plane at the breathing height (H = 1.65m) in the common ward area is calculated using different values of exposure time t e and viral load c RNA , as shown in The computations show that although the infection risk in the work area is significantly lower than that in the patient rooms, it is still high for an 8-hour exposure if the virus has a small TCID 50 unit or can produce a high viral load, as is the case for the Delta variant (Li et al., 2021a) . In these circumstances, conventional respiratory PPE such as surgical masks and N95 respirators are not able to provide effective protection. Although Figures 12 and 14 clearly demonstrate that reducing aerosol emission and/or inhalation via respiratory PPE is an effective approach to lowered infection risk, further improvement of the filtration efficiency of masks mat not be possible. From Eq.7, the ratio (δ) of HID 50 to TCID 50 is correlated to the level of immunisation. An elevated vaccination level increasesδ and hence reduces infection probability. This is important for the development of efficient strategies to fight the COVID-19 pandemic (Priesemann et al., 2021) . In addition, engineering controls such as optimal ventilation design can significantly reduce expsure to airborne pathogens in indoor environments. However, many contemporary buildings including healthcare facilities adopt a mixing ventilation scheme, which enhances the mixing of air and so promotes spread of COVID-19 in the indoor environment. This issue has been identified by many investigators (Lepore et al., 2021; Li et al., 2021b; Sodiq et al., 2021) and in public media. Meikov (Melikov, 2020) suggests that to fight COVID-19, we need a paradigm shift in ventilation design.

This issue of ventilation optimisation shall be topic of future research, and the spatiotemporal Wells-Riley model developed herein is well suited to the development of such mitigation strategies. (2) This study highlights the significance of wearing respiratory PPE by both the infected and susceptible persons. Respiratory PPE can provide very good protection when the viral load is low and the variant is not very contagious. However, with an increasing viral load J o u r n a l P r e -p r o o f and/or smaller TCID 50 unit, the efficacy of respiratory PPE quickly decreases. For highly contagious variants such as the Delta variant, respiratory PPE can only provide very limited protection in healthcare settings. As highly contagious variants of SARS-CoV-2 are increasingly causing concerns all over the world, a paradigm shift in ventilation and improved level of vaccination is needed to reduce the risk in high-risk environments.

☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:

J o u r n a l P r e -p r o o f Comparison of (a), (b) particles in the airand (c), (d) particle deposition pattern for the Lagrangian and algebraic slip models respectively. Note that (a) shows the particle traveling time and (b) shows the particle concentration field. (c) and (d) show the particle deposition rate. The figure shows a properly selected representative particle size is able to achieve a satisfactory prediction. (a) c RNA = 6×10 5 copies/mL (mean risk at the nose height < 0.001) (b) c RNA = 1×10 7 copies/mL (mean risk at the nose height = 0.008) (c) c RNA = 1×10 9 copies/mL (mean risk at the nose height = 0.088) (d) c RNA = 1×10 11 copies/mL (mean risk at the nose height = 0.767) J o u r n a l P r e -p r o o f 

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The authors gratefully acknowledge funding from RMIT University through the Restart Initiative and Enabling Capability Platforms and the Victorian Government through its Victorian Higher Education State Investment Fund Pool A.