A COVID‐19 cluster analysis in an office: Assessing the long‐range aerosol and fomite transmissions with infection control measures

Simulated exposure to severe acute respiratory syndrome coronavirus 2 in the environment was demonstrated based on the actual coronavirus disease 2019 cluster occurrence in an office, with a projected risk considering the likely transmission pathways via aerosols and fomites. A total of 35/85 occupants were infected, with the attack rate in the first stage as 0.30. It was inferred that the aerosol transmission at long‐range produced the cluster at virus concentration in the saliva of the infected cases on the basis of the simulation, more than 108 PFU mL−1. Additionally, all wearing masks effectiveness was estimated to be 61%–81% and 88%–95% reduction in risk for long‐range aerosol transmission in the normal and fit state of the masks, respectively, and a 99.8% or above decline in risk of fomite transmission. The ventilation effectiveness for long‐range aerosol transmission was also calculated to be 12%–29% and 36%–66% reductions with increases from one air change per hour (ACH) to two ACH and six ACH, respectively. Furthermore, the virus concentration reduction in the saliva to 1/3 corresponded to the risk reduction for long‐range aerosol transmission by 60%–64% and 40%–51% with and without masks, respectively.

deduce the transmission cause, estimating prevention control measures' effectiveness for each transmission pathway.The quantitative risk analysis considering multiple transmission pathways has been conducted for influenza (Nicas & Jones, 2009;Nicas & Sun, 2006); MARS (Adhikari et al., 2019); and currently, the SARS-CoV-2 in the healthcare settings (Jones, 2020;Mizukoshi et al., 2021), cruise ship (Azimi et al, 2021), mass gathering events (Murakami et al., 2021(Murakami et al., , 2022;;Yasutaka et al., 2022), and household (Ji et al., 2022).More research work should be conducted to understand the virus-laden droplets and aerosols' behavior in different environmental settings, especially confined spaces (Jayaweera et al., 2020).In this study, the COVID-19 risk was estimated by simulating exposure to SARS-CoV-2 in a similar indoor space in an office environment using the same environmental conditions of a cluster in the office environment as the input values.The infected cases, environmental conditions, and transmission pathways' characteristics were estimated by backcalculating from the actual infection situation.Moreover, transmission decreases through daily adequate preventive behavior (Azuma, Kagi, et al., 2020).Thus, the infection control measures' efficacy was evaluated.The objectives of this study are to verify the quantitative risk from each transmission pathway based on clustered cases, to verify parameters regarding transmissibility such as concentration in the saliva and dose-response function parameter, and to quantify the control measure effects such as masks and ventilation.The key scientific questions were whether long-range aerosol transmission or fomite transmission was dominant quantitatively in COVID-19 cluster cases in an office and how much the COVID-19 risk from each pathway could be reduced by each infection control measure.

Sampling location and data collection
A COVID-19 cluster occurred in an office building in Osaka, Japan.Hence, office environment and occupant information were obtained from interviewing the administrative manager, person in charge of building equipment, and related persons in the office and reports.The office environment information included the office size, ventilation system condition, air conditioning units and shared devices (copy machine and multifunction printers) layout, and partition installation status.The occupant information included the trend of occupants' onset status, seating layout and infected cases distribution, occupants' staying time and mask wearing status in the office, and whether the infected case had lunch with another infected case.

Transmission scenarios
We calculated the risk of COVID-19 onset via long-range aerosol and fomite transmissions given the infection status collected above.As shown in Figure 1 and Figure S1, we divided the office space into Zone 1 of higher attack rate with infected source cases and Zone 2 of lower attack rate without an infected source case.As a scenario, 85 infected and susceptible individuals stayed in the same indoor space (office environment) for 4 days, 8 business hours from 9:00 to 12:00 and from 13:00 to 18:00, respectively, during which SARS-CoV-2 was assumed to be transmitted.Then, we simulated two scenarios; one was assuming that the infection occurred from the long-range aerosol and fomite transmissions (longrange aerosol and fomite transmissions (LF) scenario).In this scenario, the aerosol and droplet transmissions at short-range were not considered because close contact was assumed not to occur due to the seating position distance and the condition of all wearing masks.The other scenario considered the short-range transmission such as the transmission from nearby droplet exposure and aerosol inhalation and the transmission from outside of business hours in addition to the long-range aerosol and fomite transmissions (short-range transmission and LF (SLF) scenario).In this scenario, we assumed that high attack rate in Zone 1 was attributed to the short-range transmission, resulting that the attack rate by long-range aerosol and fomite transmissions in Zone 1 was same as the attack rate in Zone 2. The onset date (first day) was set as the day when the case initially developed COVID-19 symptoms.The number of infected source cases was one on the zeroth-first day, two on the second day, and three on the third day, based on the actual infected cases with transmission possibility from 1 day before the onset, considering secondary or more infection (indicated in Section 3.1.1).Total number of infected source cases was 7.Meanwhile, this number increased to 25 if transmission to others occurred 2 days before the onset; thus, we used this number in the sensitivity analysis.It was assumed that the occupants did not cough or sneeze, and everyone wore masks.

Conditions for infected cases
The virus emission rate in the aerosols generated from the infected case (E a , plaque-forming unit [PFU] h −1 ) was calculated according to the equations by Buonanno, Stabile et al. (2020) and Henriques et al. (2022) as follows: where C v is the virus concentration in the saliva (PFU mL −1 ), and C a is the number concentration of aerosols in the breath (m −3 ).We considered the aerosols diameter as <10 μm based on the qualitative different characteristics with droplets including longer suspension time; penetration of different regions of the airways, especially for the lower respiratory tract; and requirements for different personal protective equipment as stated by Tellier et al. (2019).The aerosols become smaller particles through water evaporation (Johnson et al., 2011).Half of the initial diameter was used as the equilibrium size by Nicas et al. (2005), one third by Netz and Eaton (2020), and 30% by Wang, Alipour et al. (2021) and Henriques et al. (2022).In this study, 30% of the initial diameter was applied based on the ratio of predicted equilibrium diameter to initial diameter of 0.19 and 0.41 for low and high-protein contents, respectively, at 50% relative humidity (Marr et al., 2019;Henriques et al., 2022).
The aerosols concentration emitted during case's breathing and 1 min h −1 talking was used as C a (m −3 ) considering that the infected case talked for 1 min h −1 .People speak about 16,000 words per day in 17 h (Mehl et al., 2007).The average speech rate for conversation of young adult and middle aged ranged in age from 21 to 64 years was 168.4 words per minute (Duchin & Mysak, 1987).Thus, the average talking time in 1 h is calculated as 5.6 min.Considering this value and assuming the infection control were conducted by reducing conversation, we assumed that the talking time in this office was 1 min h −1 .The average talking time in 1 h was used in sensitivity analysis.V a is the volume of an aerosol particle (mL).It was assumed that the aerosol equilibrium diameter was 30% of the initial diameter by desiccation as mentioned above in calculating the volume.Aerosol volume concentration when breathing and voice counting was calculated as 1.1 × 10 −5 and 5.1 × 10 −5 mL m −3 , respectively, using the particle number concentration data with each diameter in breathing and voice counting (Buonanno, Stabile, et al., 2020;Morawska et al., 2009).Thus, C a × V a was calculated as 1.1 × 10 −5 mL m −3 × 58 min/60 min + 5.1 × 10 −5 mL m −3 × 2 min/60 min = 1.2 × 10 −5 mL m −3 (2 min was set considering that voice counting time was half in the experiment (Morawska et al., 2009)).ER is the exhalation rate (0.60 m 3 h −1 ), and M a is the mask removal efficiency for aerosols.M a was set to 0.6 (60%) considering approximately 20%-60% leakage of aerosols with a diameter of 0.3-20 μm for surgical and fabric cloth masks (Onishi et al., 2022).It was noted that the differences from the equation by Buonanno, Stabile et al. (2020) in Equation ( 1) were not containing conversion factor from infectious dose to infectious quantum.
Infectious dose was first calculated, respectively, and then converted to the risk using dose-response relationship in Sections 2.3.6 and 2.4.5 to compare the risk between long-range aerosol and fomite transmissions in this study.Moreover, Equation (1) contained mask removal efficiency (Henriques et al., 2022).

Assumption of two-zone model
As shown in Figure 1 and Figure S1, we divided the office space into Zone 1 of higher attack rate with infected source cases and Zone 2 of lower attack rate without an infected source case.We also assumed that air was changed between the two zones, including indoor airflow.Figure S1 shows the two zones and ventilation system location.The floor area was 514 m 2 , ceiling height was 3 m, and the set value for the air change rate was one air change per hour (ACH) during early stage as described in Section 3.1.3.The total room volume of 1542 m 3 (514 m 2 × 3 m) was divided into 12 groups, A-L (Figure 1), and two (groups A and B) were allocated for the Zone 1 volume (V 1 = 1542 m 3 × 2/12 = 257 m 3 ), and then, 10 (groups C-L) were assigned for Zone 2 volume (V 2 = 1542 m 3 × 10/12 = 1285 m 3 ).There was no ventilation system in Zone 1 and two in Zone 2. It was assumed that air leakage (natural ventilation) rate was 0.14 ACH (95% confidence interval [CI]; <0.01-0.39ACH) according to the office building infiltration data assuming tall buildings and the standard deviation of airtightness data (Emmerich & Persily, 2014).Thus, the air change volumes per hour with air leakages in Zones 1 (F 1 ) and 2 (F 2 ) were set to 36 and 1722 m 3 h −1 , respectively.The air change volume per hour between Zones 1 and 2 (F 12 = F 21 ) was obtained by fitting the ratio of simulated onset risk in Zone 1 against that in Zone 2 to the actual attack rate.

Virus concentration in the air in each zone
A simple well-mixed model was applied to describe the actual infection situation as a typical case and indicate the risk and its reduction as the efficacy of infection control measures clearly.Thus, the virus concentration in the air in each zone (Zone 1: C 1 , Zone 2: C 2 , PFU m −3 ) was expressed by the following equation following the near-field and far-field model (Iizuka et al., 2020;Shinohara et al., 2021;Spencer & Plisko, 2007).It was noted that the concentration in each zone was based on a completely mixed state: where n is the number of infected cases, and R is the virus removal rate other than air change, which is the sum of the particle deposition rate (5.7 × 10 −3 min −1 (Hinds, 1999;Nicas et al., 2005)) and the SARS-CoV-2 inactivation rate in the air (1.1 × 10 −2 min −1 ; van Doremalen et al., 2020) (Buonanno, Stabile, et al., 2020;Yang & Marr, 2011).Moreover, t is the elapsed time (h).The numerical analysis of Equations ( 2) and (3) was performed using the fourth order Runge-Kutta method.Additionally, virus concentration changes in the air over time from 9:00 to 12:00 and 13:00 to 18:00 were also determined.

Exposure dose of the virus via inhalation of aerosols
The average virus concentration in the air for 8 h in 1 day (Zone i: C ave, i , i is 1 or 2) was obtained from the virus concentration in the air (C 1 and C 2 ).The virus exposure dose via aerosol inhalation in each zone (Zone i: D a, i ) is as follows: IR is the inhalation rate (0.60 m 3 h −1 ).M a was used to calculate D a, i , assuming that the inward and outward mask leakages were the same (Bagheri et al., 2021).T is the exposure time (8 h day −1 ).Therefore, the exposure dose for 4 days from zeroth to third day was calculated for each zone.It was assumed that all the viruses in the air of the previous day had been inactivated or eliminated.

Calculation of risk via inhalation of aerosols
The onset risk via aerosol inhalation (R a ) was determined based on the following dose-response relationship: where k L is the dose-response function parameter in exposure on the lower respiratory tract (PFU −1 ).Killingley et al. (2022) conducted human challenge study, resulting that 18 out of 34 participants became infected when inoculated with 10 TCID 50 of wild-type SARS-CoV-2 intranasally.According to this result, the dose-response function parameter was calculated as −ln(1-18/34)/10/(−ln(0.5)) = 0.109 PFU −1 .Thus, we used 0.109 PFU −1 as k L .The dose-response function parameter 0.00246 PFU −1 (95% CI; 0.00128 and 0.00527 PFU −1 ) for SARS-CoV (Watanabe et al., 2010;Mitchell & Weir, 2022) is available as a surrogate of the dose-response function parameter in exposure on the upper respiratory tract (k U , PFU −1 ) of SARS-CoV-2 (Jones, 2020;Mizukoshi et al., 2021;Murakami et al., 2021;Shinohara et al., 2021).Meanwhile, k L could be higher than k U .We thus employed 100 times the value of k U as k L, (Azimi et al., 2021;Mizukoshi et al., 2021;Shinohara et al., 2021), that is, 0.246 PFU −1 (95% CI; 0.128 and 0.527 PFU −1 ) in the sensitivity analysis.The risk was calculated for each day from cumulative exposure.The relationship between k L and quantum emission q was deduced as follows: Dai and Zhao (2020) estimated q to be 14-48 h −1 for the wild-type strain.Corresponding k L is calculated to be 0.13-0.44PFU −1 when C v is 10 7 PFU mL −1 , C a × V a is 1.8 × 10 −5 mL m 3 (5.6 min talking), and ER is 0.60 m 3 h −1 .The employed value of k L 0.109-0.527included this range.Thus, it was considered that the value of 0.109-0.527for k L was reasonable.

2.4
Risk of fomite transmission

Conditions for occupants
We assumed each occupant worked with a personal computer while sitting at their desks during business hours and used a copy machine once an hour.

Conditions for infected cases
The virus load suspended in the air or fallen and deposited on each surface by the infected case's breathing and talking (L f, j , PFU) was calculated as follows: where j indicates state, that is, 1 is Indoor air, 2 is Infected case's desk surfaces, and 3 is Infected case's fingertip.Thus, V a, 1 is the aerosol volume emitted in indoor air during breathing and 1 min h −1 talking (Buonanno, Stabile, et al., 2020;Morawska et al., 2009) and was calculated as 7.3 × 10 −6 mL in 1 h.Additionally, V d, 2 and V d, 3 are the droplet volume with a diameter of >10 μm ejected on touchable surfaces (keyboard keys and a mouse button) and a fingertip from the case's 1 min talking in 1 h.The droplets volume (V d ) with an initial diameter of 32-200 μm and >32 μm ejected from the case were 2.4 × 10 and 3.9 × 10 −3 mL in 1 h, respectively (Chao et al., 2009;Jones, 2020).The emitted droplets deposited or fallen and adhered to the infected case's touchable desk surface and fingertip with the ratio of 2.3 × 10 −3 and 1.0 × 10 −4 , respectively, assuming the droplets distributed over 10,000 cm 2 , and the area of touchable surfaces and a fingertip were 23 and 1 cm 2 , respectively.Thus, V d, 2 and V d, 3 were calculated as V d × 2.3 × 10 −3 and V d × 1.0 × 10 −4 , respectively.M d is droplets' mask removal efficiency.Hence, 80% of the droplets with an initial diameter of 32-200 μm were assumed to be removed (Onishi et al., 2022), and all of the droplets with an initial diameter of >200 μm were assumed to be removed when wearing a mask.
V f, 3 is the transferred facial mucus volume to the infected case's finger.The film thickness was assumed to be 0.5 μm, and the fingertip area was 1 cm 2 .Virus transfer efficiency from fingertips to lips was assumed to be 35% (Nicas & Jones, 2009;Rusin et al., 2002).Touching face mucous membranes frequency by the same fingertip that touched other surfaces was assumed to be 5 h −1 (Kwok et al., 2015;Nicas & Jones, 2009).Touching frequency to nostrils and lips, which were assumed to contain virus in the mucus was two thirds of that (Kwok et al., 2015;Mizukoshi et al., 2021).Thus, V f, 3 was 0.5 μm × 1 cm 2 × 0.35 × 5 h −1 × 2/3 = 5.8 × 10 −5 mL.M f is prevention rate of touching nostrils and lips by a mask and was set to 1.

Modeled pathway of fomite transmission
The fomite transmission pathway was modeled by a Markov chain (Azimi et al., 2021;Jones, 2020;Nicas & Jones, 2009;Nicas & Sun, 2006;Mizukoshi et al., 2021;Murakami et al., 2021).We assumed nine states (1.indoor air; 2. infected case's desk surfaces; 3. infected case's fingertip; 4. copy machine button; 5. susceptible individuals' fingertips; 6. susceptible individual's facial mucous membranes; 7. susceptible individuals' desk surfaces; 8. surface inactivation; and 9. removal from indoor air; Figure S2).The transition rate from the state i to state j (λ ij ) is shown in Table S1.The assumed pathway was that the infected case emitted droplets with an initial diameter of >32 μm by talking (Chao et al., 2009) later adhering to the infected case's touchable desk surfaces (keyboard keys and a mouse button) and a fingertip.Then, virus moved from the infected case's fingertip to the shared device surface (copy machine button) (λ 34 and λ 43 ) and back to the desk surfaces (λ 23 and λ 32 ).The virus in the mucous membranes, such as nostrils and lips, also moved to the infected case's fingertip by touching.The virus was additionally moved when susceptible individuals touched the copy machine button with their fingertips (λ 45 and λ 54 ) and touched their facial mucous membranes with their fingers (λ 56 ).The virus that adhered to the susceptible individuals' fingertips was also moved to the susceptible individuals' desk surfaces (λ 57 and λ 75 ).Aerosols emitted from the infected case's breathing and talking (V a, 1 ) were suspended in the air and deposited on surfaces (λ 12 , λ 13 , λ 14 , λ 15 , and λ 17 ).The virus on the surface was gradually inactivated (λ 28 , λ 38 , λ 48 , λ 58 , and λ 78 ).Indoor air virus was removed by ventilation, deposition, and inactivation (λ 19 ) (Buonanno, Stabile, et al., 2020;Yang & Marr, 2011).Next, the transition probability matrix (p ij ) for a minute time (Δt = 0.0001 min) was obtained from λ ij , and the transition probability (p ij n , n = t/Δt = 6.0 × 10 5 ) after 1 h (t = 60 min) was calculated using the wxMaxima 19.01.2x.

2.4.4
Exposure dose of the virus via fomite The following equation calculated the virus load on state j where D′ f, i (i = 1-7) is the virus load on state i 60 min before (PFU).For L f, j (j = 1-3), the values were calculated using Equations ( 7)-( 9).L f, j (j = 4-7) was 0. D f, j was calculated every hour from 9:00 to 18:00 from the zeroth to third day.It was assumed that between lunch break (12:00-13:00) and nonbusiness hours (18:00-9:00 the next day), the virus was not emitted, and indirect contact transmission did not occur.Thus, L f, j , λ 23 , λ 32 , λ 34 , λ 43 , λ 45 , λ 54 , λ 57 , and λ 75 were 0. The indoor air virus load (D f, 1 ) was assumed to return to 0 each day, and the virus loads attached to the fingers (D f, 3 and D f, 5 ) were assumed to return to 0 after lunch break and each day by hand washing.D f, j was calculated using virus load from each infected case and totaled.

Calculation of risk via fomite
The following formula calculated the risk of onset via fomite (R f ): For k U , 0.109 PFU −1 was used, assuming exposure in the upper respiratory tract via hand contact (Killingley et al., 2022).Additionally, 2.5th percentile value of 0.00128 PFU −1 was used in the sensitivity analysis (Mitchell & Weir, 2022).M f is the reduction rate of exposure by covering the nostrils and lips with a mask set to 0.667 (Kwok et al., 2015;Mizukoshi et al., 2021).

Effect of infection control measures
The COVID-19 risk was calculated when the mask removal efficiency and air change rate were altered in this situation, in evaluating the mask and ventilation effect as infection prevention measures.The aerosols and droplets' mask removal efficiencies were changed to both 0% (no one wore a mask) and 80% and 95%, respectively (a surgical mask in the fit state; Onishi et al., 2022).The air change rate was adjusted to 0.5, 2, and 6 ACH.Additionally, the viral load in the body may be low compared with the unvaccinated case if the infected case had been vaccinated.Thus, we also calculated the risk when the SARS-CoV-2 concentration in the saliva was altered (10 6 -10 10 PFU mL −1 ).

Ethical considerations
This work has received approval for research ethics from the Kindai University Faculty of Medicine Ethics Committee, and a proof/certificate of approval is available upon request (No.R03-099).

Trend in the number of symptomatic cases
The COVID-19 cluster in the work environment occurred in an office in April 2021 in Osaka, Japan.A total of 85 employees were on the same office in this case, where 35 in their   2 shows a trend in the number of new-onset cases.
The infected case with the earliest onset complained of sore throat on Wednesday, April 7, 2021.It was deduced that virus transmission started from April 6, a day before, as SARS-CoV-2 shedding is usually high before and after the COVID-19 onset (He et al., 2020).Thus, a number of days elapsed from the beginning of transmission was displayed, with April 6 as the zeroth day.Moreover, the number of new cases reached its maximum on the 5th-6th days, increasing again on the 10th day.First, transmission could occur during the four weekdays from April 6 (Tuesday) to April 9 (Friday) (zeroth-third days), when almost all occupants were at work.Then, transmission could occur in next weekdays from April 12 (Monday), although many of occupants were absent due to symptom onset.The median incubation period of COVID-19 until February 24, 2020 was estimated to be 5.1 days (95% CI; 4.5-5.8days), and the estimated mean incubation period was 5.5 days (Lauer et al., 2020).The mean and median incubation period were 3.53 and 3.0 days, respectively, for the Alpha variant (Homma et al., 2021).Therefore, symptom onset from first-stage transmission might occur until 5 days after April 9 (April 14, eighth day).Additionally, symptom onset from second-stage transmission might occur after 3 days from April 12 (April 15, ninth day).We grouped the stage of transmission and onset into two stages assuming these infection statuses: before the eight day and after the ninth day.Additionally, one case developed a symptom in 1 day on the first-third days.The infected case who developed a symptom on the second day had a negative test on the fifth day, leading to the transmission possibility to other individuals to be considered low.Moreover, the first case was absent on the third day.In the fourth day, three cases developed a symptom, of which one case developed a symptom at night.Thus, the other two cases had the possibility of transmission to others in the third day the day before the onset.Consequently, results showed one infected source case on the zeroth-first days; two on the second day, including the onset case on the third day; and three on the third day.Total number of infected source cases was 7 and increased to 25 if transmission to others occurred 2 days before the onset.

Spatial distribution of infected cases
Figure 1 shows the seating arrangement in the office and the infected cases' spatial distribution.The infected cases' desks were color-coded until the eight day and from the ninth day because a two-stage transmission was considered.Furthermore, the seats were divided into 12 groups A-L on the basis of their vertical position in the figure, as illustrated in Figure 1.The transmission began with group A based on this classification, and infection was concentrated in groups A and B during the first stage (until the eighth day).The infected cases were distributed in groups C through K in the first stage, apart from groups A and B, including the primary infected cases.Thus, the attack rate for each group was calculated.

Attack rate
Table S2 shows the attack rate for each group.The attack rate of groups A and B in the zone defined as Zone 1 was close (0.64 and 0.63, respectively) regarding the first stage of transmission, and the attack rate of groups C-L in the zone defined as Zone 2 was 0.19 ± 0.12, indicating a difference between the attack rates of Zones 1 and 2. The ratio of the attack rate of Zone 1 to Zone 2 was 3.16, indicating that Zone 1 had about three times higher attack rate than Zone 2.

Office environment and occupants
Figure S1 shows the office floor plan and the ventilation system layout.The floor area was 514 m 2 , the ceiling height was 3 m, and the set value for the air change rate was one ACH during the first stage.Clear panels with a height of 60 cm had been installed on the desks (Figure 1) as an infection control practice.Additionally, all occupants wore masks while inside the room.Considering the lunch status and incubation time from the transmission, one occupant in Zone 1 had a possibility of infection at cafeteria from lunch with an infected case in Zone 1 in the first stage.Thus, we assumed that this occupant was infected outside of business hours in the SLF scenario.

Deduction of infection pathways
The infection pathways of SARS-CoV-2 in the first stage were estimated as follows when considering the transition of new cases, spatial distribution, and the situation of office environment and occupants: First, long-range aerosol transmission can occur as the secondary cases were extensively distributed throughout the office space.The aerosols generated by breathing and talking from the primary case(s) became smaller, suspended in the air, and then inhaled by susceptible individuals in distant seats.Additionally, the spatial SARS-CoV-2 concentration distribution in the air could occur if transmissions of the virus were via aerosols as the attack rate seems to differ depending on the groups classified in Figure 1.Additionally, it is assumed that as all occupants in the office wore masks at the time of cluster occurrence, direct transmission via droplet spray from the infected case to the susceptible individuals was possibly low.Meanwhile, there were shared multifunction devices and a copy machine in the office space (Figure 1), the buttons and panels of which were not wiped or disinfected.Therefore, there was a fomite transmission possibility, that is, the droplets with the larger diameter than that of aerosols emitted from the infected case stuck to the nearby surface or the hand of the infected case, and the virus in the droplets moved to susceptible individuals' faces mucous membranes via the shared devices surfaces.We simulated these assumptions in the LF scenario.
Even if the possibility of nearby exposure and inhalation was low, the resulting infection risk could be high because of high amount of exposure.In addition, the transmission outside of business hours was suggested for one occupant in Zone 1 during lunch with an infected case at cafeteria.In case of the maximum contribution to infection risk from nearby exposure and inhalation and one infected occupant in Zone 1 during lunch, the attack rate by long-range aerosol and fomite transmission in Zone 1 became equal to that in Zone 2. Thus, we also simulated this assumption in the SLF scenario.

3.3
Risk of long-range aerosol transmission

Virus concentration in the air in each zone
The fourth order Runge-Kutta method was used for numerical analysis which showed that the virus concentrations in the air in Zones 1 and 2 changed as shown in Figure S3 (one infected case, all wore masks, virus concentration in the saliva was 10 8.41 PFU mL −1 , and the air change volume per hour between Zones 1 and 2 was 9.45 × 10 2 m 3 h −1 in the LF scenario).Furthermore, the virus content in the air increased in both zones after the infected patient entered the room, reduced during the lunch break (12:00-13:00), and rose again after 13:00.Additionally, a difference in concentration between Zones 1 and 2 was established due to low air change between zones.

Risk of onset and estimated number of cases in each zone
The estimated case number and the ratio of attack rate in Zone 1-2 from the overall risk of both long-range aerosol transmission and fomite transmission matched the actual values in the first stage by exploring the minimum difference values.As a result, the estimated values matched the actual values when the virus concentration in the saliva was 10 8.41 PFU mL −1 (range; 10 6.01 -10 9.52 PFU mL −1 ) and the air change volume between zones was 9.45 × 10 2 m 3 h −1 (range; 9.09 × 10 2 -1.05 × 10 3 m 3 h −1 ) in the LF scenario, and when the virus concentration in the saliva was 10 8.28 PFU mL −1 (range; 10 5.90 -10 9.39 PFU mL −1 ) and the air change volume between zones was 5.56 × 10 4 m 3 h −1 (range; 5.28 × 10 4 -6.12 × 10 4 m 3 h −1 ) in the SLF scenario.Table 1 shows the cumulative onset risk and the estimated cumulative number of onset cases from long-range aerosol transmission in each zone.

Risk of fomite transmission
Table 1 shows the fomite transmission risk when the overall risk matched the actual attack rate for long-range aerosol transmission and fomite transmission.Thus, the maximum risks on the third day in the LF and SLF scenarios are 9.3 × 10 −4 (range; 3.2 × 10 −9 -0.059) and 6.9 × 10 −4 (range; 2.4 × 10 −9 -0.044), respectively, implying that if there were 84 susceptible individuals in the room, less than one case could develop the COVID-19 symptoms.
TA B L E 1 Cumulative onset risk and number of onset cases for transmission via aerosols and fomites.F I G U R E 3 Contribution of the onset risk from each transmission pathway when the virus concentration in the saliva was 10 8.41 PFU mL −1 in the LF scenario (A) and 10 8.28 PFU mL −1 in the SLF scenario (B).

Contribution of the onset risk from each transmission pathway
Figure 3 shows the contribution of the onset risk from each transmission pathway estimated in this study.In the LF sce-nario, the contribution of onset risk in all from the long-range aerosol transmission was highest (99.7%), and the contribution from fomite transmission was 0.3% (Figure 3A).Meanwhile, the contribution of short-range transmission including transmission outside business hours was highest (73.4%) in Zone 1 in the SLF scenario; thus, the long-range aerosol transmission and fomite transmission in all were 60.4% and 0.2%, respectively (Figure 3B).

Inference of the transition of the onset cases
Figure 2 and Table S3 show the estimated number of onset cases for each day.We considered that the infection was caused by long-range aerosol and fomite transmissions inferring the number of cases each day by every zone and exposure days, and the total number daily.Note that the number of cases is distributed by assuming an incubation period of 2-5 days after infection (Homma et al., 2021).An inferred transition of the onset cases with a similar trend to the actual transition was reproduced in the LF scenario as shown in Figure 2.

3.8
Effectiveness of infection control measures

Effectiveness of masks
Tables S4 and S5 show the onset risk and the number of cases after mask removal efficiency and the air change rate were altered for the long-range aerosol transmission and fomite transmission, respectively.Moreover, Figure 5 shows the relationship between the air change rate and the number of onset cases depending on the mask removal efficiency.For example, the number of cases due to long-range aerosol transmission increased to 68 and 63 in the LF and SLF scenarios, respectively, when no one wore a mask at the air change rate of one ACH.Nevertheless, the number of cases decreased to 8 and 5 in the LF and SLF scenarios, respectively, when everyone wore masks with a removal efficiency of 80% for aerosols.Additionally, the number of cases resulting from fomite transmission could increase to 72 (range; <1-84) and 64 (range; <1-82) in the LF and SLF scenarios, respectively, when no one wore a mask.Finally, Table S4 shows the risk reduction rate of wearing masks compared with non-wearing of masks.Results showed that the risk of long-range aerosol transmission reduced by 61%-81% when everyone wore masks with a removal efficiency of 60% for aerosols, whereas by 88%-95% in cases with the removal efficiency of 80%.At the same time, the risk of fomite transmission was reduced by 99.8% or above when everyone wore the masks (Table S5).

Ventilation effectiveness
Figure 5 and Table S4 show that onset cases number resulting from long-range aerosol transmission increased to 29 and 21 when the air change rate was halved (0.5 ACH), decreased to 21 and 12 in case of doubled (two ACH), and decreased to 16 and 6 in case of 6 ACH in the LF and SLF scenarios, respectively, when everyone wore masks with the removal efficiency of 60% for aerosols.The risk reduction rate compared with the air change rate of one ACH was 12%-29% when the air change rate was doubled (two ACH) and 36%-66% in cases of six ACH.The fomite transmission risk was considered not to be affected by the air change rate (the risk reduction rate was below 1%) (Figure 5 and Table S5).

Effectiveness of reducing viral load in the body
Tables S6 and S7 show the onset risk and the number of cases when the cases' virus concentration in the saliva was altered in case of the air change rate of one ACH and a mask removal efficiency of 60% for aerosols and 80% for droplets.The long-range aerosol transmission risk decreased by 60%-64% and 40%-51% and the fomite transmission risk decreased by 67% and 44%-50% with and without masks, respectively, if the virus concentration in the saliva decreased to one third.Figure 6 indicates the relationship between virus concentration in the saliva and the number of cases.Thus, the virus concentration in the saliva should be 10 6.58 PFU mL −1 or less when wearing masks and 10 5.53 PFU mL −1 or less when not wearing masks to prevent the COVID-19 onset.The viral body load should be 1/68 or less when wearing masks and 1/759 or less when not wearing masks to prevent the onsets considering both pathways assuming that the virus concentration in the saliva and the viral body load correlate.

DISCUSSION
The secondary cases spatial distribution in the far zone from the primary infected case(s) was uniform, suggesting transmission independent of the distance in this infection cluster.
It was suggested that long-range aerosol transmission was the dominant cause of the infection cluster in this environment with prevailing adherence to wearing a mask following the simulation of transmission and onset risk in the two scenarios in case of maximum and minimum risk of long-range aerosol and fomite transmissions.On the other hand, all wearing masks could have prevented fomite transmission.
The SARS-CoV-2 wild-type strain concentration in the saliva was assumed to be between 10 2 and 10 4 PFU mL −1 in the range from <1 to >10 7 PFU mL −1 based on reported values (Mizukoshi et al., 2021).Hence, the concentration in the saliva of the primary infected case(s) must be very high (>10 8 PFU mL −1 ) in both LF and SLF scenarios in the situation of matching the actual number of cases, although the past outbreaks did not necessarily require a superspreader with the highest viral load and infectious dose (Buonanno, Morawska, et al., 2020).Two points of view would explain the high virus concentration possibility in the saliva.First, there was an overdispersion of the infected case and ∼10% spread the COVID-19 (Endo, 2020).If this high transmissibility depended on the high virus concentration in the saliva, cluster occurrence from the cases with high virus concentration in the saliva is consistent with the fact.However, >10 8 PFU mL −1 is still high compared with the upper end of the virus concentration in the saliva for the wild-type strain.Therefore, the other reason may be because the virus in this situation was the SARS-CoV-2 variant with a high concentration in the saliva.There is no information on the kinds of variants in this case; however, the Alpha variant may become dominant considering the timing of occurrence (April 2021).For example, Alpha variant positive rate was 78% in Osaka Prefecture in April 12-18 (MHLW, 2021).Furthermore, the early appearance of the Delta variant may be possible.Higher transmissibility (Liu & Rocklöv, 2021;Tanaka et al., 2021) and virus concentration in the saliva (Imai et al., 2021;King et al., 2022) has been reported in these variants than in the wild-type strain.Hence, we inferred that high virus  concentration in this cluster case was possible considering the infected cases dispersion and the high concentration in the saliva of variants.Zhang et al. (2020) identified airborne transmission as the dominant route for the COVID-19 spread based on the trend and mitigation measures in the epicenters.Morawska and Milton (2020) addressed airborne transmission via microdroplets.Moreover, the Centers for Disease Control and Prevention (CDC) (2021) stated that SARS-CoV-2 transmission from inhalation of virus in the air farther than 6 ft from an infectious source can occur.WHO (2021) has suggested that long-range aerosol or long-range airborne transmission is the possible infection pathway.These statements may result from the appearance of new SARS-CoV-2 variants with high transmissibility than the wild-type strain.Edwards et al. (2021) reported that the difference in exhaled aerosols between subjects by three orders of magnitude.In this study, we considered that the generated aerosols differences between cases were included in the difference of diameter change ratio of aerosols by desiccation.The estimated virus concentration in the saliva and resulting risk of fomite transmission became lower than those above estimated values if the volume of aerosols was larger than that considered in this study.Consequently, the cause of long-range aerosol transmission was considered more dominant than the above analysis.
The risk of long-range aerosol transmission was calculated using a dose-response function with the parameter k L in comparing the multiple pathways.Meanwhile, the risk of long-range aerosol transmission is usually calculated by quantum emission q (Azuma, Yanagi, et al., 2020;Bazant & Bush, 2021;Buonanno, Stabile, et al., 2020;Buonanno, Morawska, et al., 2020;Dai & Zhao, 2020;Riley et al., 1978;Shinohara et al., 2022).In the relationship between k L and q, the q is 204 h −1 (range; 23-916 h −1 ) and 151 h −1 (range; 18-679 h −1 ) in the LF and SLF scenarios, respectively, higher than the value 14-48 h −1 estimated for the wild-type strain (Dai & Zhao, 2020).This tendency also suggested the high transmissibility of variants in this period.Azimi et al. (2021) calculated the risk of short-range (i.e., droplets and aerosols within close range) and long-range (i.e., aerosols outside of close-range contact), and fomite transmission on the cruise ship as an example of a risk assessment including long-range transmission.The median (mean) contributions of long-range and fomite transmission modes were estimated to be 41% (35%) and 21% (30%), respectively.Additionally, the contributions before and after quarantine were calculated, which were 42% (34%) and 37% (46%) before quarantine, and 39% (36%) and 0.5% (6%) after quarantine.These indicated that long-range aerosol transmission was more dominant than fomite transmission on the cruise ship.The long-range aerosol transmission and fomite transmission contributions in this study were >99% and 0.3% in the LF scenario, and 60% and 0.2% in the SLF scenario, respectively (Figure 3).This result was close to the ratio after quarantine by Azimi et al. (2021).The rapid systematic review found evidence suggesting the possibility of long distance airborne transmission of SARS-CoV-2 in indoor settings, such as restaurants, workplaces, and venues for choirs (Duval et al., 2022).This result also supported the dominance of long-range aerosol transmission estimated in this study.
The reduction rate of the fomite transmission risk by wearing masks was extremely high (>99.8%).The explanation is that without a mask, many large droplets are emitted, which fall and contaminate the surface, permitting the virus to move into the environment.Conversely, the emission of large droplets could be almost prevented (94% reduction, range; 88%, 99%) and the released virus and its risk of infection decrease when infected cases wear masks (80% reduction).Additionally, masks could prevent the contamination of infected cases' fingers by touching the nostrils and lips containing virus in the mucous membranes.Moreover, susceptible individuals wearing masks prevented touching the fingertip on the nostrils and lips; thus, the risk was reduced to one third by touching only eyes.To clarify the contribution to the risk from each exposure pathway in fomite transmission, virus exposure amount and ratio via aerosols, droplets, and facial mucus with and without masks and the exposure reduction rate compared to the exposure without masks are shown in Table S8.
The attack rate difference between zones was observed, signifying virus concentration difference between zones, estimated due to including the low wind speed.The low wind speed might be partially because of clear desk panels.It was estimated that the onset rate in all was 0.26, and the total onset number was 22 if the concentration distribution in the air did not occur, suggesting that the decrease of total onset number by air agitation was estimated in this environment.This scenario is likely to be the case if the occupants are not ubiquitous in the room.
The air flow pattern affects aerosols according the computational fluid dynamic analysis (Jayaweera et al., 2020;Saw et al., 2021).The attack rate similarity in each group by each zone (Table S2) suggested the well-mixed condition was achieved in each zone, although air flow pattern was not considered in this study.Additionally, this corroborated the validity the model adopted in this study.
The peak onset was delayed by 1 day in Zone 2 in comparing the transition of onset case number between Zones 1 and 2, but the estimated values showed no clear tendency (Table S3).This result is probably because the time for the air to move into Zone 2 from Zone 1 was not considered.
The MHLW, Japan recommended the required volume flow rate (30 m 3 h −1 per person) in the Law for Maintenance of Sanitation in Buildings to resolve poorly ventilated closed spaces (Azuma, Yanagi, et al., 2020;MHLW, 2020).Moreover, the corresponding required volume flow rate was 2550-3330 m 3 h −1 , and the air change rate was 1.7-2.2ACH as the occupants were 85 and the desks were 111 in this environment.The air change rate of one ACH in this office environment was low compared to this value above.The above review also identified the factors that probably contributed to transmission, which were insufficient air replacement, directional air flow, and activities associated with increased aerosols emission, such as singing or speaking loudly (Duval et al., 2022).Insufficient air replacement might be one of the factors in this study.Moreover, it was estimated that the risk from the long-range aerosol transmission was reduced by 16% and 27%, and four and five onset cases decreased in the LF and SLF scenarios, respectively, if the air change rate was two ACH, which is the maximum performance on the ventilation system in this office.This result showed a specific reduction in onset risk; however, the effect is limited.The increasing mask removal efficiency to 80% for aerosols by proper wearing may decrease the long-range aerosol transmission onset risk by 74% and 81% and the onset cases to 7 and 3, less than half in the LF and SLF scenarios, respectively, besides the air change rate increase to two ACH.Contact time reduction time is available in addition to the above infection controls.Bazant and Bush (2021) proposed a guideline of cumulative exposure time to limit COVID-19 indoor airborne transmission.In this study, if the exposure time was limited to zeroth day, onset risk and number from long-range aerosol transmission decreased to 0.055 and 5, respectively, in the LF scenario and 0.031 and 3, respectively, in the SLF scenario (Table 1), and with the air change rate increase to two ACH and increasing mask removal efficiency to 80% for aerosols, the onset risk and number from long-range aerosol transmission decreased to 0.012 and 1, respectively, in the LF scenario and 5.6 × 10 −3 and <1, respectively, in the SLF scenario.
As shown in Figure 6, onset risk is greatly affected by the virus concentration in the saliva.Thus, a significant effect in infection prevention can be projected if the body viral load of an infected case is reduced, and the virus concentration in the saliva is lowered.It was reported that the mean viral load was 2.3 ± 1.7 log 10 copies μL −1 in partially or fully vaccinated participants and 3.8 ± 1.7 log 10 copies μL −1 in unvaccinated participants from December 14, 2020 to April 10, 2021 (Thompson et al., 2021).For the Alpha variant, the viral load was substantially reduced after the vaccination (Levine-Tiefenbrun et al., 2021).In the study, the decrease of 2.8-4.5-fold in viral load in vaccinated individuals was estimated.As mentioned above, one third of the virus concentration in the saliva resulted in the risk of long-range aerosol transmission by 60%-64% and 40%-51% in case of all with and without masks, respectively.The other study reported that, for the Alpha variant, the viral load (indicated inversely by cycle threshold [Ct] values) in individuals after two vaccinations were significantly lower than that in unvaccinated individuals.However, the viral load of individuals after two vaccinations and unvaccinated individuals was similar for the Delta variant (Pouwels et al., 2021).Meanwhile, higher Ct values for vaccinated participants compared with unvaccinated were reported for the Delta variant (Elliott et al., 2021).Additionally, vaccination has been pointed out to cause a rapid decrease in viral load in the upper respiratory tract after infection for the Delta variant (Singanayagam et al., 2022).Thus, it can reduce the virus emission even if infected, resulting in preventing the spread of the disease.
The first limitation of this study is the knowledge of the SARS-CoV-2 variant type of the infected cases in this cluster.There was a possibility of the Alpha variant when considering April 2021.Second, the air change rate was the setting value.Actual value may be less due to various environmental factors, such as short circuits from inlet and outlet and ventilation system performance decrease with time.Additionally, the filtration effect by air conditioning units was not considered.Third, multiple cases were assumed to emit droplets and aerosols with the same high virus concentration in the saliva, which may be overestimated because of the low probability of numerous appearances of the virus cases with a high concentration in the saliva.Fourth, virus transition between infected case's finger and eyes was not considered because SARS-CoV-2 RNA detection potential in tears and conjunctival secretions by PCR essay is fairly low (Güemes-Villahoz et al., 2021).Fifth, the decrease of source case numbers by infection control measures was not considered.Thus, their risk reduction effectiveness was underestimated.Sixth, the transmission in spaces other than office room such as elevators or restrooms was not considered due to the short time in these spaces and recommendation of prevention controls.At last, 10 μm was used to distinguish aerosols and droplets, although the size distinction between aerosols and droplets should be 100 μm with respect to the suspension time to reach a distance of 1-2 m (Wang, Prather et al., 2021).Nevertheless, long-range aerosol transmission with a distance of more than 1-2 m was considered in this study.Furthermore, all occupants wore the mask for which the removal efficiency was assumed to be 80% for droplets with a diameter of <100 μm in this actual case.The size distinction of 10 μm is reasonable, and the impact of aerosols with a diameter of 10-100 μm on the results would be limited.The long-range aerosol and fomite transmissions risks were roughly simulated as the actual cluster in an office environment, suggesting the rising importance of infection control measures for the residual risk of long-range aerosol transmission, especially for SARS-CoV-2 variants that are increasing, in spite of the above limitations.

CONCLUSION
The transmission pathways for this study were hinged on the actual COVID-19 cluster in the office environment, wherein many occupants were in wide-open office space.The key findings were as follows: Long-range transmission from inhalation of aerosols became dominant following the control measures for infection by putting on masks, whereas transmission from indirect contact via fomites was rare.The dominance of long-range aerosol transmission was consistent with the reviewed evidence suggesting the possibility of long distance airborne transmission of SARS-CoV-2 in indoor settings (Azimi et al., 2021;Duval et al., 2022).Additionally, if the infected case(s) with a high concentration in the saliva existed, this transmission may occur.The effectiveness of all wearing masks was a 61%-81% and 88%-95% in the normal and fit state of the masks, respectively, in long-range aerosol transmission and a 99.9% or above reduction in the risk of fomite transmission.Moreover, ventilation was also effective, of which risk reductions for long-range transmission were 12%-29% and 36%-66% with air change rate increases from one ACH to two ACH and six ACH, respectively.The virus concentration reduction in the saliva to one third corresponded to the risk reduction for long-range transmission by 60%-64% and 40%-51% with and without masks, respectively.Thus, infection control measures against long-range aerosol transmission are crucial, especially for SARS-CoV-2 variants, such as increasing mask removal efficiency through proper wearing, ensuring a low virus concentration via ventilation, and possibly, decreasing the virus body load and emission of infected cases, such as early vaccination.

A C K N O W L E D G M E N T S
This study was financially supported by a 2021 All-Kindai University Support Project against COVID-19 (Research project 2) provided by the Kindai University.
Seating layout and infected cases distribution in the office.The onset day is indicated in parentheses.a The day tested positive for asymptomatic case.
days from the start date of transmission Contribution to variance of overall risk and the risk from long-range aerosol transmission and fomite transmission in the LF scenario (A) and SLF scenario (B).
Relationship between air change rate and onset number by the mask removal efficiency in the LF scenario (A) and SLF scenario (B).M a and M d are the mask removal efficiency for aerosols and droplets, respectively.The error bar indicates upper and lower limits for fomite transmission based on the consideration of uncertainty.
Relationship between the virus concentration in the saliva and the onset number with and without masks in the LF scenario (A) and SLF scenario (B).The error bar indicates upper and lower limits for fomite transmission based on the consideration of uncertainty.
The authors declare that there are no conflicts of interest regarding the publication of the present work.O R C I DAtsushi Mizukoshi https://orcid.org/0000-0002-2281-3526Jiro Okumura https://orcid.org/0000-0002-2627-1728Kenichi Azuma https://orcid.org/0000-0002-6382-9807RE F E R E N C E S Trend in the actual number of onset case and estimated number of first-and new-onset cases from long-range aerosol and fomite transmissions in two scenarios.Asymptomatic cases are indicated in the day tested positive.There was one asymptomatic case included in the 8th, 10th, and 13th days, and two asymptomatic cases are included on the 13th day.