key: cord-0852327-no3cy092 authors: Jia, Wei; Wei, Jianjian; Cheng, Pan; Wang, Qun; Li, Yuguo title: Exposure and respiratory infection risk via the short-range airborne route date: 2022-05-10 journal: Build Environ DOI: 10.1016/j.buildenv.2022.109166 sha: bbddebc081a978f3ac4b631f650b6af3dc818219 doc_id: 852327 cord_uid: no3cy092 Leading health authorities have suggested short-range airborne transmission as a major route of severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2). However, there is no simple method to assess the short-range airborne infection risk or identify its governing parameters. We proposed a short-range airborne infection risk assessment model based on the continuum model and two-stage jet model. The effects of ventilation, physical distance and activity intensity on the short-range airborne exposure were studied systematically. The results suggested that increasing physical distance and ventilation reduced short-range airborne exposure and infection risk. However, a diminishing return phenomenon was observed when the ventilation rate or physical distance was beyond a certain threshold. When the infectious quantum concentration was less than 1 quantum/L at the mouth, our newly defined threshold distance and threshold ventilation rate were independent of quantum concentration. We estimated threshold distances of 0.59, 1.1, 1.7 and 2.6 m for sedentary/passive, light, moderate and intense activities, respectively. At these distances, the threshold ventilation was estimated to be 8, 20, 43, and 83 L/s per person, respectively. The findings show that both physical distancing and adequate ventilation are essential for minimising infection risk, especially in high-intensity activity or densely populated spaces. Keywords: Short-range airborne transmission, Wells-Riley model, ventilation rate, physical 52 distance, interrupted jet, COVID-19 53 54 1. Introduction 55 56 As the coronavirus disease 2019 (COVID-19) pandemic continues, major health authorities, 57 including the World Health Organization (WHO) and the US Centers for Disease Control and 58 Prevention, have at least partially recognised the importance of airborne transmission of 59 severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) since October 2020 (WHO, 60 2020) and of short-range airborne transmission since late April 2021 (WHO, 2021). 61 Recognising the short-range airborne route is important for determining appropriate 62 interventions, as wearing N95 masks has become necessary for healthcare workers working 63 in the presence of COVID-19 patients in healthcare settings and improved building 64 ventilation has become an important infection control measure in the community. Here, we 65 define airborne transmission as exposure to exhaled fine aerosols or droplet nuclei with 66 diameters less than 5 m that contain infectious viruses and eventually lead to infection. 67 Short-range airborne transmission is defined as direct inhalation exposure of a susceptible 68 person, through the mouth or nose, to expired virus-containing droplets or aerosols smaller 69 than 50 m in the expired jet of an infected person. Large droplet transmission has been 70 shown to be insignificant or less important compared to short-range airborne transmission 71 . The concept of airborne transmission traditionally implies long-range 72 airborne transmission. Identifying the presence of a short-range airborne transmission route 73 by health authorities is a major milestone in determining appropriate interventions for 74 respiratory infections. 75 76 The Wells-Riley equation has been widely used to describe the infection risk via the long-77 range airborne route in indoor spaces. Riley et al. (1978) presented the first model, assuming 78 fully mixed room air with a uniform distribution of infectious pathogens. At a steady state, 79 the probability of infection is 80 81 where n is the number of infectors, is the infectious quantum generation rate by one 84 infector (quanta/min), p is the average pulmonary ventilation rate or inhalation rate of a 85 susceptible individual (m 3 /min), ∆ is the exposure duration (min), and is the room 86 ventilation flow rate or strictly speaking, the effective ventilation rate including filtration 87 (m 3 /min). 88 89 The definition of an infectious quantum arises from the need to resolve the challenge of not 90 knowing the number of infectious virus particles. A quantum is defined as the number of 91 virus particles needed to produce a probability of 63.2% (or exactly, 1-1/e) of infecting a 92 susceptible individual (Wells, 1955) . Once an outbreak is detected, the infectious quantum 93 generation rate may be determined using Equation (1) or its variants if the ventilation rate at 94 the time of infection can be determined. A transient version of the equation was derived by 95 Rudnick and Milton (2003) and others. For cases of non-uniform distribution, Qian et al. 96 (2009) developed an approach using computational fluid dynamics and the infectious 97 quantum concept. 98 99 There are at least two challenges in analysing short-range infection risk. First, the infectious 100 quantum generation rates are expected to differ for short-and long-range airborne 101 transmission. Compared with the long-range airborne transmission, the airborne aerosols 102 involved in the short-range airborne route can be larger within the expired jet of any 103 individual, and more viable virus particles may exist in these aerosols expired by an infected 104 individual (Xie et al., 2007; Gralton et al., 2011) . People are constantly in close contact with 105 others in their daily life . The exposure duration and physical distance 106 during close contact are difficult to determine. The conventional approach for determining the 107 quantum generation rate for long-range transmission using Equation (1) cannot be directly 108 applied to short-range transmission. 109 110 Second, the dilution factors in the expired jet zone and in the rest of the room are governed 111 by different mechanisms. The key mechanism for controlling long-range airborne 112 transmission is the dilution by ventilation, filtration and settling, as described by the term 113 in the Wells-Riley equation (1) . The dilution factor for a jet at distance from jet origin 114 may be defined as = 0 , i.e., the ratio of the jet flow rate at distance to the jet flow rate 115 at the jet origin. Alternatively, = 0 , which is the ratio of the average concentrations of the 116 infectious quanta at the origin and at distance , can be used to calculate the dilution factor. 117 In general, infectious droplets evaporate, with larger droplets settling, and infectious viruses 118 may be deactivated while traveling in the expired jet. Hence, 0 may not equal 0 . Dilution in 119 the expired jet differs based on whether the surrounding air is clean or polluted. combined a simple steady jet into the short-range airborne plume transmission. Although 140 experimental data exist for the starting jet and puff theories , to the best 141 of our knowledge, there are no data on the dilution factor of such a two-stage jet, which is 142 essential for estimating the short-range inhalation exposure of expired aerosols. 143 144 In this study, a dilution factor of the two-stage expired jet was derived for the first time. We 145 integrated the continuum model described by We focus on inhalation transmission here. Particles with a diameter of smaller than 50 µm 156 follow the expired jet streamline . Hence the spatial variation of virus-157 containing airborne particle concentration can be characterized by the two-stage jet dilution 158 model. We assumed that the mouth opening was a nozzle with a diameter (m) (Figure 1 ). 159 The expired airflow rate at the mouth of an infected individual for the duration of exhalation 160 is 0 (m 3 /s or L/s), where the exhalation rate 0 is the exhaled/inhaled flow rate during a 161 period (e.g., a minute or a day) as defined by the US Environmental Protection Agency (EPA) 162 (2011), rather than during the exhalation period. For a breathing cycle with a period , the 163 exhalation duration is , the inhalation duration is − , and the coefficient is defined as 164 = . When exhalation and inhalation durations are equal (Figure 2 ), = 2. In the 165 calculations here, a coefficient of = 2 was used. 166 167 The jet flow rate increases to (m 3 /s or L/s) at a streamwise penetration distance due to 168 entrainment of the surrounding air. We aimed to estimate the jet dilution factor = 0 at 169 any distance . The dilution factor in this study is the average dilution factor of the cross-170 section at distance . Knowledge of the dilution factor allowed us to estimate the short-range 171 exposure of a susceptible individual by direct inhalation of the expired airflows of an infected 172 individual. Dilution occurs due to continuous turbulent entrainment as an expired jet 173 develops. 174 175 For a steady jet originating from a round opening, the dilution factor is 176 where is the diameter of the mouth opening. Equation (2) is valid at distances of ≥ 6.2 180 (Lee and Chu, 2012) . For a typical mouth diameter of 20 mm, ≥ 0.12 m. Buoyancy, 181 stratification, and cross flows are known to affect the jet dilution factor (Lee and Chu, 2012). 182 183 An interrupted jet is driven by an intermittent momentum source. Buoyancy forces can also 184 exist but were ignored here for simplicity. A sudden instantaneous release of the momentum 185 produces a puff. In the case of an expired flow, instead of instantaneous release, a release 186 with a finite duration occurs, followed by a short duration of inspiration, after which, the 187 cycle repeats. The inspiration flow has a potential flow pattern, drawing air from all 188 directions. Recall that the steady-jet dilution ratio is obtained using the basic mass and momentum 195 conservation principle, with an additional assumption that the radial penetration distance 196 grows linearly with the streamwise penetration distance (Lee and Chu, 2012) . A similar 197 approach was used here to derive the dilution formula for the interrupted jet. 198 199 We first approximated a realistic inhalation/exhalation profile to a square cycle (Figure 2b ). 200 The specification of uniform velocity at the mouth allowed us to use existing jet data 201 . Jets with an irregular velocity profile (e.g., Figure 2a an ideal 4-s breathing cycle in this idealised model. 209 210 We assumed calm surrounding air and used the experimental data of Sangras et al. (2002) and 211 Diez et al. (2003) , including both starting jets and puffs. The exhalation phase (t = 0-2 s in 212 Figure 2 ) is the jet-like stage, which is dominated by conservation of the momentum flux. 213 After the exhalation phase (t > 2 s), the exhaled flow enters a puff-like stage, which is 214 dominated by conservation of the momentum force. 215 216 At the jet-like stage, the streamwise penetration distance is proportional to the square root 217 of the lapsed time , and the radial penetration distance is proportional to the streamwise 218 penetration distance . Note that the radial penetration distance here was not the Gaussian 219 width described by Lee and Chu (2012, page 37), but the visible radii, i.e., the top-hat width. 220 Including the correction of the jet virtual origin, we have 221 where is the streamwise penetration distance of the expired jet geometric centre from the 226 mouth; is the radial penetration distance at distance ; 0 is the distance from the virtual 227 origin of the jet-like stage to the mouth ( 0 = 2 1 ); and 0 is the corresponding virtual time 228 ). The two empirical parameters 1 and 1 are the radial penetration 229 coefficient and the streamwise penetration coefficient in the jet-like stage, respectively, and The detailed derivation process is shown in Supplementary Information C2. 256 257 The transition point * from the jet-like stage to the puff-like stage was determined using 258 Equation (3) at transition time * (e.g., * = 2 s in Figure 2 ). 259 260 where * is the streamwise penetration distance at the transition point. The dilution factor * 263 at transition point * was estimated using Equation (5) as * = 2 1 ( * + 0 ) . At the puff-like stage, the streamwise penetration distance is proportional to the fourth root 266 of the lapsed time , and the radial penetration distance is proportional to the streamwise 267 penetration distance . A puff cloud may be best described as an ellipsoid with a radius and a height where = 286 9 4 (Scorer, 1957) . The cloud volume is estimated as In the puff-like stage, the dilution factor at distance is calculated as the product of the 289 dilution factor * at transition point * and the volume ratio of the puff cloud at distance to 290 the puff cloud at distance * , i.e., = ( * 0 ) ( * ). The dilution factor in the jet-like stage is, thus, similar to the dilution factor in a steady jet, 295 but the dilution factor in the puff-like stage is different from that in a steady jet. The dilution 296 J o u r n a l P r e -p r o o f factor during the puff-like stage depends on the released volume of air during the exhalation 297 period of one breathing cycle. 298 299 If the surrounding air is clean, the dilution factor described above can be used directly to 301 evaluate short-range inhalation exposure. In an enclosed environment, aerosols in the 302 surrounding air may be simultaneously entrained into the expired airflows. 303 304 Such an entrainment effect is considered in the continuum model derived by . 305 This continuum model may be improved using the newly derived dilution factor described 306 above for a two-stage jet. We divided a room space into two zones: the jet zone (e.g., ≤ 2 307 m for certain respiratory activities) and the room zone (i.e., the rest of the room). The volume 308 of the jet zone (typically 0.15 m 3 for a 2-m expired jet cone) is much smaller than the volume 309 of the room zone (> 30 m 3 ). 310 311 312 313 Figure 3 . A simple model of the continuum from short-range to long-range inhalation routes. 314 (a) A simple jet model assuming the expired jet is steady (a1) or interrupted (a2); (b) The jet 315 zone with a variable distance x in (a), and the room zone. The jet model may be modelled 316 using the ideal steady jet model of (a1) and the two-stage jet model of (a2). 317 318 Consider exhaled virus-containing aerosols with an average concentration of 0 at the jet 319 origin (the mouth), an average concentration of at distance , and an average 320 concentration of in the room zone. When the infectious quantum is used, the concentration 321 is measured in quanta/m 3 or quanta/L, the room ventilation rate is (L/s), and the ambient 322 (outdoor) concentration is zero. 323 324 The steady-state and time-averaged macroscopic mass balance equations for the exhaled 325 virus-containing aerosol concentrations in the room zone and the jet zone, respectively, 326 become 327 328 331 332 J o u r n a l P r e -p r o o f where = + + + , = is the equivalent ventilation rate due to particle 333 settling, is the deposition rate (h -1 ) (Supplementary Information B), V is the room air 334 volume (m 3 ), is the equivalent ventilation rate due to filtration, is the equivalent 335 ventilation rate due to virus deactivation, and is the fraction of infectious aerosols in 336 suspended aerosols in the expired jet that remain suspended in the room zone. Note that the 337 exhaled flow rate 0 during a period (e.g., a minute or a day) as defined by the US EPA 338 (2011) was used here, not 0 . Correspondingly, the jet flow rate 0 at distance should 339 also be understood as the corresponding jet flow rate when the exhaled flow rate is 0 at the 340 mouth. Hence, the dilution factor here is calculated as = 0 0 . Note that the effects of 341 particle deposition and virus deactivation were not considered in the jet zone in Equation 342 (15), but were considered in the room zone in Equation (14). 343 344 The size range of suspended aerosols is less in the room zone than in the jet zone due to air 345 speed differences between the two zones ( (14) and (15), we obtained 354 355 In the room zone, when is infinite, the average concentration at distance becomes 367 368 This reflects the nature of the continuity of exposure to respiratory droplets from a close 371 range to a long range. 372 373 We considered the worst situation in which breathing patterns of the infector and the 376 susceptible individual involved in a conversation are synchronic (Figure 4 ). In such a 377 J o u r n a l P r e -p r o o f situation, the susceptible individual is in close proximity, and his/her inhalation occurs at the 378 same time that the exhaled puff arrives. 379 380 381 382 Figure 4 . The assumed worst condition in which a susceptible person inhales exactly when 383 the exhaled air of the infected person arrives at his/her mouth. 384 385 We can directly integrate the newly obtained dilution factors with the quantum concentration 386 equations. 387 388 The short-range Wells-Riley equation for one infector in a room is: 389 390 In outdoor situations or when ventilation is infinite, 393 394 The long-range equation for one infector in the room zone is: 397 398 We assumed that all individuals in a room can be presented by an average person, i.e., the 401 exhalation rate of the infected individual equals the inhalation rate of the susceptible 402 individual ( 0 = ). The short-and long-range equations can be further simplified, such that 403 Equation (19) becomes 404 405 The long-range equation then becomes 408 409 The difference between the long-and short-range infection risk can be seen in Equations (22) 412 and (23). When the infection risk is low, we can simplify Equation (22) concentration of 0 , which is now diluted by a factor of . The direct inhalation exposure 416 1 0 ∆ at a close range is approximately proportional to the inhalation/exhalation flow rate 417 , although the dilution factor is somewhat affected by (see later discussion of Figure 8 ). 418 The indirect inhalation (1 − 1 ) 0 2 ∆ at a close range is approximately proportional to 419 2 . The significance of this re-entrained 'long-range' term is determined by the effective 420 ventilation rate . The 'long-range' exposure, calculated using Equation (23), is 421 approximately proportional to 2 , as the dilution factor is somewhat affected by . 422 423 In theory, we can use the long-range Wells-Riley equation (21) to determine the long-range 424 quantum generation rate 0 0 from long-range exposure data and use the short-range Wells-425 Riley equation (19) to determine the short-range quantum generation rate 0 0 from short-426 range exposure data if an outbreak existed with data for both short-and long-range airborne 427 transmission. From another perspective, if the short-range infection risk is known for a certain distance, 439 activity intensity, close-range exposure duration, and ventilation rate, the short-range 440 infectious quantum concentration can also be determined. However, such data do not exist. 441 The infectious quantum concentration of initial expired air was estimated as 0.1 quanta/L 442 using the closest dataset in Chu et al. (2020) (see Figure S4 in Supplementary Each ideal breathing cycle (Figure 4) Figure 7 shows the idealised puff trains of the expired air flows, using the data presented in 514 Figure 5 . As each train reaches a distance (a stop) downstream, the volume of air increases, 515 and the concentration of the discharge at the origin is diluted. Only two complete breathing 516 cycles (duration of 8 s) are shown in Figure 7 . The expired air volume reduces as it travels for 517 more than 2 s ( Figure 7b ). As shown in Figure 7i , the two sequential air volumes are closer at 518 8 s after the first discharge. When these two air volumes are sufficiently close, the two 519 vortices may merge. This is identical to the observed jet-like flow in the numerical simulation As expected, both the steady jet and the two-stage jet models predicted that the dilution factor 535 always increases with an increase in the streamwise penetration distance. The dilution factor 536 of a steady jet increased linearly with the streamwise penetration distance, with a slope that 537 was independent of activity intensity. The steady-jet-estimated dilution factor was less than 538 the dilution factor predicted by the two-stage jet model (Figure 8 ). This is an important 539 observation, as it suggested that the steady-state model used by Li inhalation/exhalation rates listed in Table 1 were used. Note that for the steady jet model, the 568 dilution factor is 1 when the distance is less than 0.12 m. The jet or puff flow in Maghami et al. chamber. The former was accompanied by a weak co-flow (a 0.5% of jet velocity) while the 571 latter had no co-flow. 572 573 To further verify the developed dilution model, we predicted the exposure index / and 574 compared it with the measured data from the literature (Figure 9 ). As our room zone model 575 was based on a fully mixing assumption, we chose only those data with mixing ventilation or 576 with manikin heads in the mixing zone when displacement ventilation was used. In these 577 experiments, one can assume = 1. A reasonable agreement was observed, suggesting that 578 the developed dilution factor formula and its integration with the continuum model 579 reasonably predicted short-range exposure. and activity intensities 596 597 We compared the estimated normalised concentrations of the expired aerosols at any 598 streamwise penetration distance and various ventilation rates using the new two-stage jet 599 model and the steady-state jet model for standard activity (exhalation rate, 0.1 L/s) in Figure 600 10 Two major observations can be made. First, the normalised concentration of the expired 611 aerosols predicted by the steady jet model (thin lines in Figure 10 ) is higher than the 612 corresponding concentration predicted by the two-stage jet model for all conditions (thick 613 lines in Figure 10 ). This can be explained by an underestimate of the dilution factor by the 614 steady jet model. In contrast to the steady jet model, the normalised concentration profile 615 predicted by the two-stage jet model was not smooth, with a transition point at approximately 616 0.3 m for standard activity with an exhalation rate of 0.1 L/s. A ventilation rate less than 5 L/s 617 per person gave a significantly higher concentration than outdoor conditions (with an infinite 618 ventilation rate as shown by the red dashed curves), whereas a ventilation rate higher than 5 619 L/s per person did not. To be safe, we suggest a threshold ventilation rate of 10 L/s per person 620 for standard activity with an exhalation/inhalation rate of 0.1 L/s so that the indoor infection 621 risk at a short range is similar to the risk outdoors where the ventilation rate is infinite. This 622 suggestion was also made in our previous study . 623 624 The short-range normalised concentration profiles when the exhalation rate changes from 0.1 625 L/s are shown in Figure S3 . The short-range normalised concentrations are determined by the 626 exhalation rate, ventilation rate, and dilution factor. Unlike the steady jet model, the dilution 627 factor estimated by the two-stage jet model depends on the exhalation rate of the infector. A 628 higher activity intensity results in a lower dilution factor in the puff-region (Figure 8) . 629 Simultaneously, a higher activity intensity results in a greater rebreathed fraction ( = 0 ) if 630 the ventilation rates are kept constant. Thus, a higher short-range normalised concentration, 631 ( 0 (1 − 1 ) + 1 ), is predicted for heavier activities than for standard activities if the 632 ventilation rates are kept constant ( Figure S3 ). 633 634 635 The short-range airborne infection risk was estimated using a quantum concentration of 0.1 638 quantum/L and an exposure duration of 42 s as an example. Figure 11 shows the predicted 639 profiles of the short-range airborne infection risk for standard activity (inhalation rate, 0.1 640 L/s). The data clearly show that a ventilation rate less than 5 L/s per person introduces a 641 significantly higher infection risk than outdoor conditions. This conclusion is consistent with 642 the exposure estimates presented in Figure 10 . 643 644 We examined whether it was possible to estimate a threshold distance and threshold 645 ventilation rate when the quantum concentration at the mouth was unknown. Answering this 646 question is complicated by two factors, namely, the distance and ventilation rate, which 647 jointly affect the short-range airborne infection risk. When we evaluated the partial derivative 648 as a function of distance, the curves for different ventilation rates collapsed into almost 649 one curve. We attempted to identify a relatively flat region by determining when has a 650 small value. Outdoor infection risk is known to be low at a typical close-contact distance, 651 which is approximately 0.7 m. At = 0.7 m, we found that = −0.2%, which may be 652 located in a relatively flat region of the curve (Figure 11a ). This corresponds to an infection 653 risk of 0.82% for a ventilation rate of 10 L/s per person. At = 0.7 m and = 10 L/s per 654 person, we also found that = −0.02%, which may also be located in a relatively flat 655 region of the curve (Figure 11b ). Thus, we determined a threshold distance of 0.7 m and a 656 threshold ventilation rate of 10 L/s per person for standard activity (inhalation rate, 0.1 L/s). 657 Further, two parameters, namely, = −0.2% with an infinite ventilation rate and = 658 −0.02% at the corresponding threshold distance, were regarded as the benchmarks for 659 determining the threshold distance and threshold ventilation rates for other activities. We further estimated the short-range infection risk for different activity intensities using the 676 infectious quantum concentration of 0.1 quanta/L and exposure time of 42 s as an example 677 ( Figure S6 ). In all settings, we determined the threshold distance for = −0.2% with an 678 infinite ventilation rate. The threshold distance was then found to be 0.59 m for 679 sedentary/passive activity, 1.1 m for light activity, 1.7 m for moderate activity, and 2.6 m for 680 intense activity. Once a threshold distance was found, the corresponding threshold ventilation 681 rate was determined using = −0.02% at the corresponding threshold distance. The 682 threshold ventilation rate was 8 L/s per person for sedentary activity, 20 L/s per person for 683 light activity, 43 L/s per person for moderate activity, and 83 L/s per person for intense 684 activity ( Figure 12) . It was clear that the threshold ventilation rate that we identified was 685 proportional to the inhalation rate. The partial derivative approach predicted a threshold 686 ventilation rate that was identical to the ventilation rate that gives a constant rebreathed 687 fraction of 0.01 (Table 1 ). 688 689 Note that the partial derivative method we used to determine the threshold distance and 690 threshold ventilation rate did not lead to a constant infection risk under threshold conditions. 691 The risk of infection under threshold conditions was 0.79% for sedentary/passive activity, 692 0.82% for standard activity (inhalation rate, 0.1 L/s), 1.14% for light activity, 1.74% for 693 moderate activity, and 2.61% for intense activity. The explanation for this finding is that the 694 obtained threshold ventilation rates for all activity intensities led to a constant rebreathed 695 fraction, , of 0.01, i.e., the concentration of the expired aerosols was the same after dilution. 696 When the infection risk is low, Equation (20) can be simplified by Taylor series expansion to 697 ≈ 1 0 ∆ + (1 − 1 ) 0 2 ∆ = ( + 1− ) 0 ∆ . From this simplified formula, the 698 short-range airborne infection risk is approximately proportional to the inhalation volume 699 ∆ if ventilation rates for different activity intensities are kept at threshold ventilation rates 700 (i.e., a constant rebreathed fraction, ). However, higher-intensity activity had a larger 701 dilution factor at the corresponding threshold distance, i.e., 54-fold at 0.59 m for 702 sedentary/passive activity, 69-fold at 0.7 m for standard activity, 115-fold at 1.1 m for light 703 activity, 211-fold at 1.7 m for moderate activity, and 383-fold at 2.6 m for intense activity. 704 This led to a decrease in short-range exposure for higher-intensity activity at the threshold 705 distance and ventilation rate compared to the proportional relationship with the inhalation 706 volume ∆ . In contrast, the long-range exposure was proportional to the inhalation volume 707 ∆ at the threshold ventilation rate. 708 709 710 711 Figure 12 . Estimated short-range infection risk for four activity intensities and the 712 corresponding physical distance threshold, with ventilation rates ranging from 0.1 to 500 L/s 713 per person, and a partial derivative of infection risk against the ventilation rate using the 714 equations presented in Supplementary Information C3. 715 716 717 4. Discussion 718 719 Our study is the first to present a dilution factor formula for a two-stage jet, and our estimated 722 dilution factor was in reasonable agreement with measured or simulated data from the 723 literature ( Figure 8 ). The dilution factor formula was further confirmed by comparing the 724 predicted short-range exposure with measured data from the literature (Figure 9 ). To the best 725 of our knowledge, such a formula has not previously been reported, but it is required to assess 726 the short-range exposure and infection risk of respiratory infections. The traditional steady jet 727 model does not consider the effect of respiratory activity on the dilution 728 factor. In the steady jet model, the dilution factor is not a function of activity intensity, but is 729 constant, and the predicted dilution factors are 8, 16, and 32 at distances of 0.5, 1, and 2 m, 730 respectively. The two-stage jet model predicted a dilution factor that depends on activity 731 intensity (exhalation rate, Figure 8 ). For light activity, the predicted dilution factors were 12, 732 94, and 748 at distances of 0.5, 1, and 2 m, respectively ( Figure 5 ). The dilution factors for 733 intense activity were smaller than those for light activity, but still significantly higher than 734 those predicted by the steady jet model. For the rest scenarios, the dilution factor predicted in 735 the puff-like stage at 2 m was up to 79 times higher with the two-stage jet model than with 736 the steady jet model. 737 738 Using an existing continuum model of short-and long-range airborne transmission of SARS-739 CoV The new model predictions suggest that the short-range infection risk with intense activity is 760 significantly high when the ventilation rate and physical distance requirements cannot be 761 satisfied. This is expected, as high infection rates have been observed in gyms, fitness 762 centres, and dance floors during the on-going COVID-19 pandemic. Our new model 763 predicted that outdoor infection may also occur with intense activity at close range, such as 764 during sports activities or heavy labour. The data suggested that for light activities, the 765 outdoor infection risk due to short-range inhalation was low; however, during intense activity, 766 short-range inhalation transmission is possible. This has implications for intervention 767 measures at outdoor events. Various approaches have been used to determine the threshold physical distance for infection 775 control . Different countries have also adopted different physical distances 776 from 1 to 2 m (Shulman, 2020) . Our data support the use of a threshold distance of 1 m for 777 light activities in the presence of sufficient ventilation, but a higher threshold distance for 778 higher-intensity activities. A much higher ventilation rate is also needed to reduce the short-779 range airborne infection risk associated with high-intensity activities. 780 781 Different close-contact distances between people are chosen for various social purposes. The 782 average distance between people is approximately 0.7 m. One commonly cited, but informal, 783 explanation for keeping a minimum distance is to avoid body odour or expired gas odour 784 from other people. The provision of sufficient ventilation is also known to minimise body 785 odour. If this is correct, then the commonly used minimum ventilation standard of 10 L/s per 786 person is sufficient. At a standard activity intensity, with an exhalation flow rate of 0.1 L/s, a 787 ventilation rate of 10 L/s per person results in a 100-fold dilution for the room average odour 788 concentration. Thus, a dilution of 100-fold may lead to sufficient dilution of expired air to 789 avoid odour problems. In the steady jet model, the normalised concentration was estimated to 790 be 0.03, or a dilution of only 33-fold, at a physical distance of 2 m in outdoor conditions. If a 791 100-fold dilution is required to avoid body odour, and body odour can be avoided at a 792 standard distance of 0.7 m, then the estimate by the steady jet model is markedly out of scale. 793 This is expected, as a realistic expired jet is not a steady jet. load and droplet release rate may also be used to estimate the quantum generation rate 871 (Buonanno et al., 2020) . This method has the potential to be developed for estimating both 872 short-and long-range quantum generation rates. However, additional data, such as the 873 survival of the virus in the short-range expired jet, are needed. This is a difficult problem to 874 solve, as the residence time of the aerosols in the expired jet is less than 1 or only a few 875 seconds. Virus survival characteristics within such a short duration are difficult to obtain. 876 There are no data for virus survival in aerosols within the first few seconds of their release. In 877 addition, the long-range quantum generation rate should be much less than the short-range 878 quantum generation rate, given that the size range of suspended aerosols within a close range 879 may be larger than the size range of suspended aerosols in the rest of the room (Xie et al., In this study, we made a crude estimate of the short-range infectious quantum generation rate 883 using data reported by Chu et al. (2020) . However, complexities exist when using these data. 884 The observed infection at close range reported by Chu et al. (2020) may have been due to the 885 lack of adherence to the threshold distance, such that at least some infections may have 886 occurred at a close distance. In such situations, our estimated short-range quantum generation 887 rate may have been over-estimated. It is also possible that the average close-contact exposure 888 period may be longer or shorter than our assumed values. A secondary infection may also be 889 the result of both short-and long-range exposure. A Monte Carlo method may be used in 890 future studies to perform such estimates, but more reliable data are needed. However, no 891 reliable short-range quantum generation rate exists. Thus, a reliable approach to estimate 892 short-and long-range quantum generation rates is needed. 893 894 4.4 Limitations of the study 895 896 There are several major limitations of this study. The short-range Wells-Riley model assumes 897 that the emission rate of bioeffluents or the number of exhaled droplets is proportional to the 898 exhalation rates for different activity intensities, and that the infectious quantum 899 concentration at the mouth is constant. For subjects infected with the human rhinovirus, the 900 exhaled particle concentration (< 10 µm) is likely to be proportional to minute ventilation at a 901 close range (Fabian et al., 2010) ; however, it remains unknown how the number and size 902 distribution of droplets exhaled by healthy subjects or subjects infected with a respiratory 903 virus vary when physical activity intensity and respiratory activity change. It is also unknown 904 how the virus concentration is distributed in different sizes of particles, even though Milton et 905 al. (2013) found that, for influenza virus, fine particles may contain a much higher number of 906 viral copies than coarse particles. Hence, data are needed on how the number and size 907 distribution of exhaled droplets change with activity intensity/mode, and how the infectious 908 quantum concentration at the mouth changes with exhaled droplet size. The dilution analysis 909 of short-range airborne infection risk in Section 3.4 was based on ventilation rate per infector 910 only, without considering effective dilution due to aerosols settling, filtration and virus 911 deactivation as suggested in Equation (14). Our model (Equation 22) is appliable to the 912 situation with multiple infectors, and in such situations, the overall room dilution air flow rate 913 should be used as in Equation (14). 914 915 Several physical and biological assumptions were made in this study. Our model did not 916 consider particle settling or virus deactivation at a close range. A more accurate model needs 917 to consider dispersion of individual droplets in the expired jet. Evaporation or dehydration is 918 known to occur at a close range. Virus deactivation over a short range may need to be 919 considered. The jet trajectory changes due to buoyancy, but being ignored here, but the 920 dilution factor formula may still be applied along the trajectory. Our model simplified the 921 breathing profile as a square cycle, which may lead to an underestimation of the streamwise 922 penetration distance of the jet-like stage and an overestimation of the dilution factor at any 923 distance in the puff-like stage. Hence, the required physical distance and ventilation rate may 924 be underestimated. A calm air surrounding was assumed to avoid considering ambient 925 turbulent dispersion; however, in reality, room air is not usually calm. Once the jet is 926 destroyed by the surrounding air flows, the short-range transmission merges into long-range 927 transmission. The change in streamlines when approaching the exposed individual affect the 928 inhalation exposure (Chen et al., 2020). 929 930 Speech can 983 produce jet-like transport relevant to asymptomatic spreading of virus Clustering and superspreading potential of SARS-CoV-2 infections 987 in Hong Kong Measurement of breathing rate and volume in routinely performed 989 daily activities Ventilation for Acceptable Indoor Air Quality. 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(2021). 3000 workers tested as Covid outbreak disrupts Hong Kong's 3 rd 1053 runway mega project Concentration field measurements within 1057 isolated turbulent puffs The role of particle 1059 size in aerosolised pathogen transmission: a review Community transmission of SARS-CoV-2 at three fitness facilities-Hawaii Transmission of SARS-CoV-2 1064 in the karaoke room: an outbreak of COVID-19 in Estimation of risk due to low doses of microorganisms: a comparison 1067 of alternative methodologies Cluster of coronavirus disease associated 1069 with fitness dance classes Turbulent Jets and Plumes: A Lagrangian 1075 Approach Poor ventilation worsens short-range airborne 1077 transmission of respiratory infection Probable airborne transmission of SARS-CoV-2 in a poorly ventilated 1080 restaurant Revisiting physical 1082 distancing threshold in indoor environment using infection-risk-based modeling Hong Kong's construction sites could be Covid-19 transmission hotspots, 1087 experts say, but blanket work stoppages are unlikely to help. South China Morning Post 1093 and Noakes, C. (2021). 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World Health Organization (WHO). (2021). Coronavirus disease (COVID-19): How is it 1140 transmitted? How far droplets can 1143 move in indoor environments--revisiting the Wells evaporation-falling curve Close 1146 contact behavior in indoor environment and transmission of respiratory infection. Indoor 1147 Air Evidence for lack of transmission by close contact and surface touch in a restaurant 1150 outbreak of COVID-19 Using the newly developed dilution factor formula, we estimated a dilution factor that 933 depends on the exhalation flow rate. In the jet-like stage, the newly estimated dilution factor 934 was similar to the dilution factor predicted by the steady jet model. However, the dilution 935 factor markedly increased in the puff-like stage for all activity intensities, with light activity 936 corresponding to a higher dilution than intense activity for a simple reason, i.e., more intense 937 activity has a later transition point from jet-like to puff-like stages. The higher estimated 938 dilution factor indicates a more rapid decrease of the normalised concentration of virus-939 containing particles within expired flows of the corresponding short-range exposure. 940 941The newly developed dilution factor using the two-stage expired jet model has enabled the 942 development of a simple short-range infection risk model. The infectious quantum generation 943 rate differs between short-range and long-range airborne infection. The short-range quantum 944 generation rate remains to be determined. Following this uncertainty, we propose to use a 945 partial derivative approach to estimate the threshold distance and threshold ventilation rate. 946The partial derivative approach has been shown to be valid when the infectious quantum 947 concentration is less than 1 quantum/L at the mouth, or when the equivalent quantum 948 generation rate is less than 360 quanta/h at an exhaled flow rate of 0.