1 00:00:10,780 --> 00:00:12,950 PROFESSOR: So now we've included all the key physics 2 00:00:12,950 --> 00:00:14,900 in our model-- 3 00:00:14,900 --> 00:00:19,640 sedimentation, deactivation of virus, filtration of the air, 4 00:00:19,640 --> 00:00:22,340 ventilation flows. 5 00:00:22,340 --> 00:00:25,940 And now, we have to include one last piece of information, 6 00:00:25,940 --> 00:00:27,610 which is that all of those quantities, 7 00:00:27,610 --> 00:00:29,270 or many of them that we've discussed, 8 00:00:29,270 --> 00:00:31,820 have a dependence on the size of the droplets. 9 00:00:31,820 --> 00:00:34,130 And the size of the droplets and number of the droplets 10 00:00:34,130 --> 00:00:37,620 is a strong function of the type of respiration 11 00:00:37,620 --> 00:00:40,690 which is being performed by the infected person who 12 00:00:40,690 --> 00:00:43,130 is spewing out or emitting these droplets, 13 00:00:43,130 --> 00:00:46,040 and also the respiration of the susceptible person who's 14 00:00:46,040 --> 00:00:48,650 breathing those droplets in. 15 00:00:48,650 --> 00:00:53,150 So here is a sketch of what droplet distributions look 16 00:00:53,150 --> 00:00:55,910 like that have been measured for different kinds 17 00:00:55,910 --> 00:00:58,950 of expiratory activities. 18 00:00:58,950 --> 00:01:02,420 So this is shown kind of on a log scale, 19 00:01:02,420 --> 00:01:04,370 at least what I'm attempting to sketch. 20 00:01:04,370 --> 00:01:07,430 So the dotted line here, roughly speaking, 21 00:01:07,430 --> 00:01:09,470 is separating the aerosol droplets, 22 00:01:09,470 --> 00:01:12,710 which are less than several microns in radius 23 00:01:12,710 --> 00:01:16,530 and the large droplets, which are bigger than that. 24 00:01:16,530 --> 00:01:20,240 And what you can see here is that as you change 25 00:01:20,240 --> 00:01:22,580 from resting breathing, which is let's 26 00:01:22,580 --> 00:01:25,460 say through the nose or even resting breathing 27 00:01:25,460 --> 00:01:28,010 through the mouth, this distribution starts to go up. 28 00:01:28,010 --> 00:01:30,440 When you start speaking, there's a significant increase 29 00:01:30,440 --> 00:01:31,789 in the emissions. 30 00:01:31,789 --> 00:01:35,450 Presumably because there is a fragmentation and breakup 31 00:01:35,450 --> 00:01:38,970 of mucus which is surrounding the vocal cords in the pharynx. 32 00:01:38,970 --> 00:01:43,160 Which leads to significant aerosol emissions, which do not 33 00:01:43,160 --> 00:01:45,200 evaporate and do not follow the Wells curve, 34 00:01:45,200 --> 00:01:46,940 but actually survive. 35 00:01:46,940 --> 00:01:48,950 And then that becomes much more extreme 36 00:01:48,950 --> 00:01:53,330 as we get towards singing and any prolonged vocalizations. 37 00:01:53,330 --> 00:01:57,610 In fact, also, this size here depends 38 00:01:57,610 --> 00:01:59,509 on the volume of speaking. 39 00:01:59,509 --> 00:02:04,790 So this is loud speaking, and somewhere below here 40 00:02:04,790 --> 00:02:05,570 is whispering. 41 00:02:05,570 --> 00:02:07,370 So basically, the amount that you release 42 00:02:07,370 --> 00:02:11,690 is a strong function of the volume of speech. 43 00:02:11,690 --> 00:02:13,790 And whether you're vocalizing or not 44 00:02:13,790 --> 00:02:16,460 makes a big difference, where vocalization leads to many more 45 00:02:16,460 --> 00:02:18,250 droplets being formed that are aerosols 46 00:02:18,250 --> 00:02:20,150 that remain in the air. 47 00:02:20,150 --> 00:02:23,660 And singing is actually probably almost as bad as you can get, 48 00:02:23,660 --> 00:02:26,100 in terms of spewing out droplets. 49 00:02:26,100 --> 00:02:28,400 What's interesting is that all these distributions have 50 00:02:28,400 --> 00:02:30,740 a peak or the most probable value 51 00:02:30,740 --> 00:02:32,360 is actually less than a micron. 52 00:02:32,360 --> 00:02:35,480 So it's around maybe half a micron 53 00:02:35,480 --> 00:02:38,750 in diameter, which is even a quarter micron in radius. 54 00:02:38,750 --> 00:02:39,420 But it can vary. 55 00:02:39,420 --> 00:02:42,500 So it's somewhere in the range of, 56 00:02:42,500 --> 00:02:45,680 let's just say 0.3 to one micron is somewhere 57 00:02:45,680 --> 00:02:47,900 where you'll find the peak of these distributions. 58 00:02:47,900 --> 00:02:51,650 And then they drop off, but in such a way that they're not-- 59 00:02:51,650 --> 00:02:53,960 they're dropping off fairly quickly in number, 60 00:02:53,960 --> 00:02:57,860 but since the larger droplets carry more volume, like R 61 00:02:57,860 --> 00:03:01,220 cubed, it turns out they're not decaying quite so quickly 62 00:03:01,220 --> 00:03:03,020 as a function of volume. 63 00:03:03,020 --> 00:03:04,790 But still, they are decaying and the peak 64 00:03:04,790 --> 00:03:06,550 is in the aerosol range. 65 00:03:06,550 --> 00:03:08,930 So clearly, there is a strong size dependence there which 66 00:03:08,930 --> 00:03:10,320 you must take into account. 67 00:03:10,320 --> 00:03:11,400 But that's not it. 68 00:03:11,400 --> 00:03:13,730 We've already talked about other quantities which 69 00:03:13,730 --> 00:03:15,350 also strong size dependence. 70 00:03:15,350 --> 00:03:17,690 So we've talked about infectivity. 71 00:03:17,690 --> 00:03:19,370 If we think about a water droplet, 72 00:03:19,370 --> 00:03:22,250 I was calculating the time for the virion 73 00:03:22,250 --> 00:03:24,710 to get out of the water droplet by diffusion. 74 00:03:24,710 --> 00:03:28,910 And we saw that even up to 10 micron-type sizes, the virion 75 00:03:28,910 --> 00:03:30,350 can still escape. 76 00:03:30,350 --> 00:03:32,300 On the other hand, if it's in mucus, 77 00:03:32,300 --> 00:03:36,560 as the droplets that are emitted from-- especially vocalization 78 00:03:36,560 --> 00:03:39,110 patterns-- then you'll find that, as we've already 79 00:03:39,110 --> 00:03:42,170 calculated, that cutoff where you start to not 80 00:03:42,170 --> 00:03:43,790 have time for the virus to get out 81 00:03:43,790 --> 00:03:45,750 might be on the order of microns. 82 00:03:45,750 --> 00:03:47,210 And so we're certainly going to see 83 00:03:47,210 --> 00:03:48,680 that the aerosols are infectious, 84 00:03:48,680 --> 00:03:50,750 and that has been shown experimentally. 85 00:03:50,750 --> 00:03:54,350 But we may not find that they're as infectious in the larger 86 00:03:54,350 --> 00:03:55,130 drop form. 87 00:03:55,130 --> 00:03:56,510 So that's something we could also 88 00:03:56,510 --> 00:03:58,350 take into account in the model. 89 00:03:58,350 --> 00:04:01,190 And then of course masks and air filters 90 00:04:01,190 --> 00:04:03,380 have a strong dependence on size, 91 00:04:03,380 --> 00:04:05,510 and that's included in their ratings. 92 00:04:05,510 --> 00:04:08,480 So for example, when we go from an N95 mask 93 00:04:08,480 --> 00:04:10,820 to various forms of cloth face covering, 94 00:04:10,820 --> 00:04:14,420 there's a lot of experimental data showing the transmission 95 00:04:14,420 --> 00:04:20,709 of those different droplet sizes through different materials. 96 00:04:20,709 --> 00:04:23,960 And in addition to the material, there's also the fit factor. 97 00:04:23,960 --> 00:04:25,700 So you can have a really great material. 98 00:04:25,700 --> 00:04:29,790 If you measure it, it's 99% removing the aerosol droplets. 99 00:04:29,790 --> 00:04:31,450 But then you put on your face and you've 100 00:04:31,450 --> 00:04:33,200 got a big gap over your nose or you let 101 00:04:33,200 --> 00:04:36,050 it sink down below your nose. 102 00:04:36,050 --> 00:04:38,190 And suddenly, you're letting a lot out as well. 103 00:04:38,190 --> 00:04:40,430 So the fit of the mask is also very important, 104 00:04:40,430 --> 00:04:42,710 and somehow ends up being included in this quantity. 105 00:04:42,710 --> 00:04:44,620 But clearly, there's a strong size dependence. 106 00:04:44,620 --> 00:04:47,630 Although, I would point out, both these quantities 107 00:04:47,630 --> 00:04:50,300 are not varying too much over the aerosol range. 108 00:04:50,300 --> 00:04:52,250 So if we're focusing on aerosols, 109 00:04:52,250 --> 00:04:55,100 then the variation isn't as much as you might think, actually. 110 00:04:55,100 --> 00:04:57,290 So what happens with the filters is 111 00:04:57,290 --> 00:04:59,030 when you get to bigger than 10 microns, 112 00:04:59,030 --> 00:05:00,890 like 10 millimeter scale, then there's 113 00:05:00,890 --> 00:05:02,930 a physical blocking of those droplets 114 00:05:02,930 --> 00:05:04,580 and they get condensed and collected 115 00:05:04,580 --> 00:05:07,160 on the fibers of the material. 116 00:05:07,160 --> 00:05:09,770 And so there's almost 100% of filtration 117 00:05:09,770 --> 00:05:11,390 at the scale of millimeters. 118 00:05:11,390 --> 00:05:14,750 So when you have a large cough or sneeze, 119 00:05:14,750 --> 00:05:17,780 if you are wearing a mask-- even a fairly poor mask-- a lot 120 00:05:17,780 --> 00:05:18,620 of it gets filtered. 121 00:05:18,620 --> 00:05:21,740 Yes, some drops do get through and that's been observed, 122 00:05:21,740 --> 00:05:24,800 but it's very significantly filtered at the high sizes. 123 00:05:24,800 --> 00:05:27,110 On the other hand, when you get down to the aerosols, 124 00:05:27,110 --> 00:05:30,760 then the masks may not be filtering so strongly. 125 00:05:30,760 --> 00:05:32,260 And the same thing with the filters. 126 00:05:32,260 --> 00:05:35,510 So the HEPA filter is intended to remove 127 00:05:35,510 --> 00:05:38,840 the aerosol droplets, even those which are below a micron. 128 00:05:38,840 --> 00:05:41,659 But the various MERV ratings of filters 129 00:05:41,659 --> 00:05:45,450 are not really going after the aerosols as much, but still 130 00:05:45,450 --> 00:05:46,580 have a significant removal. 131 00:05:46,580 --> 00:05:49,220 Maybe in the order of tens of a percent, 132 00:05:49,220 --> 00:05:52,290 or maybe up to even 90%, depending on the rating. 133 00:05:52,290 --> 00:05:54,960 So the point is, all these quantities are size dependent. 134 00:05:54,960 --> 00:05:59,570 So we have to go back and revisit our model including 135 00:05:59,570 --> 00:06:00,660 that size dependence. 136 00:06:00,660 --> 00:06:03,290 OK, so it just makes things a little bit more complicated, 137 00:06:03,290 --> 00:06:04,680 but not too much. 138 00:06:04,680 --> 00:06:08,680 So let's go back yet again to our mass balance, 139 00:06:08,680 --> 00:06:12,600 but now we're going to do it in a radius resolved fashion. 140 00:06:18,270 --> 00:06:21,050 And so for this, I'm in need a C of R and t, 141 00:06:21,050 --> 00:06:23,750 and I'm going to use the same notation, C, even though I've 142 00:06:23,750 --> 00:06:26,240 just changed the meaning here by including the R. 143 00:06:26,240 --> 00:06:29,540 And this is going to be a radius resolved concentration 144 00:06:29,540 --> 00:06:30,230 of virions. 145 00:06:40,960 --> 00:06:42,790 So that means it's a number of virions 146 00:06:42,790 --> 00:06:46,659 per air volume per radius, OK? 147 00:06:46,659 --> 00:06:49,750 And so, now when we write the mass balance, instead of 148 00:06:49,750 --> 00:06:52,850 a dC dt with an order derivative, 149 00:06:52,850 --> 00:06:54,310 it's actually a partial derivative. 150 00:06:54,310 --> 00:06:56,830 Because C is now a function of R and t. 151 00:06:56,830 --> 00:07:00,850 So I have a partial dC dt, and then 152 00:07:00,850 --> 00:07:03,970 if I divide through by volume on the other side, 153 00:07:03,970 --> 00:07:06,580 I have a P which now depends on R. 154 00:07:06,580 --> 00:07:08,470 So the production rate of droplets 155 00:07:08,470 --> 00:07:10,810 is certainly very strongly size-dependent, 156 00:07:10,810 --> 00:07:13,970 and then all these other factors may come in as well. 157 00:07:13,970 --> 00:07:16,310 Divided by volume. 158 00:07:16,310 --> 00:07:21,590 And then, we have a lambda C, which now depends 159 00:07:21,590 --> 00:07:25,830 on R times concentration. 160 00:07:25,830 --> 00:07:30,590 So this is now what happens to our mass balance. 161 00:07:30,590 --> 00:07:34,100 And this lambda C of R-- 162 00:07:34,100 --> 00:07:35,640 what does that look like? 163 00:07:35,640 --> 00:07:44,110 Well, it's lambda a times one plus-- 164 00:07:44,110 --> 00:07:45,940 and then we have the sedimentation effect. 165 00:07:45,940 --> 00:07:51,720 And I'm going to write this as R over RC squared. 166 00:07:51,720 --> 00:07:53,670 Because we know the sedimentation of velocity 167 00:07:53,670 --> 00:07:55,350 scales like R squared. 168 00:07:55,350 --> 00:07:57,370 I'll come back to this in just a moment. 169 00:07:57,370 --> 00:07:59,130 This is sedimentation. 170 00:07:59,130 --> 00:08:01,030 That's the tricky part. 171 00:08:01,030 --> 00:08:03,770 It is strongly size-dependent. 172 00:08:03,770 --> 00:08:10,470 Plus lambda v and plus Pf lambda f, which of course, 173 00:08:10,470 --> 00:08:14,240 depends on R as well. 174 00:08:14,240 --> 00:08:16,100 Lambda v in principle could depend on R, 175 00:08:16,100 --> 00:08:17,890 although we haven't really considered that. 176 00:08:17,890 --> 00:08:23,410 We've been thinking of more just spontaneous or maybe chemical 177 00:08:23,410 --> 00:08:24,020 disinfectants. 178 00:08:24,020 --> 00:08:25,250 In the case of chemical disinfectants, 179 00:08:25,250 --> 00:08:27,650 maybe that's size-dependent too because whether the virion is 180 00:08:27,650 --> 00:08:30,020 attacked by a chemical depends on how big the droplet is. 181 00:08:30,020 --> 00:08:32,510 So in principle, this could also be size-dependent. 182 00:08:32,510 --> 00:08:34,309 So basically, everything is size-dependent. 183 00:08:34,309 --> 00:08:38,570 Except for lambda a as the airflow, air changed rate. 184 00:08:38,570 --> 00:08:41,630 And that, of course, does not depend on size. 185 00:08:41,630 --> 00:08:44,169 Now, what have I done here in this RC? 186 00:08:44,169 --> 00:08:49,820 So I've written here that if I write lambda s of R 187 00:08:49,820 --> 00:08:52,280 is the sedimentation rate, that's 188 00:08:52,280 --> 00:08:56,990 vs of R divided by the height. 189 00:08:56,990 --> 00:08:58,700 So basically, that's what we talked about 190 00:08:58,700 --> 00:09:04,000 before as the rate at which particles are sedimenting 191 00:09:04,000 --> 00:09:05,990 onto the horizontal surfaces. 192 00:09:05,990 --> 00:09:09,400 Where, remember H is v over a. 193 00:09:09,400 --> 00:09:11,050 And then what I've done here-- notice, 194 00:09:11,050 --> 00:09:13,390 so what was supposed to be here is lambda s of R, 195 00:09:13,390 --> 00:09:14,940 but then I factored out lambda a. 196 00:09:14,940 --> 00:09:21,220 So what this term is here is lambdas s of R over lambda a. 197 00:09:21,220 --> 00:09:25,720 That is by definition-- 198 00:09:25,720 --> 00:09:30,090 so that's the same as vs of R over va, 199 00:09:30,090 --> 00:09:33,010 were va is the air velocity. 200 00:09:33,010 --> 00:09:35,970 So that's Q, the air flow rate, the outdoor air flow 201 00:09:35,970 --> 00:09:37,630 rate divided by a. 202 00:09:37,630 --> 00:09:40,350 And then we're writing that as R over RC 203 00:09:40,350 --> 00:09:43,900 squared because we know that vs scales like R squared. 204 00:09:43,900 --> 00:09:48,060 So when I factor this out, I can define a quantity RC. 205 00:09:48,060 --> 00:09:52,530 And what does that RC turn out to be? 206 00:09:52,530 --> 00:09:56,470 That is-- basically, you can actually just 207 00:09:56,470 --> 00:09:58,310 see by working it through this form formula. 208 00:09:58,310 --> 00:10:03,510 Actually, I'll just leave it here as a definition basically. 209 00:10:03,510 --> 00:10:06,420 Now, then we can also now say what 210 00:10:06,420 --> 00:10:09,080 happens to these kinds of droplet distributions. 211 00:10:09,080 --> 00:10:10,710 So this is kind of the important thing, 212 00:10:10,710 --> 00:10:16,130 is that lambda C is something which 213 00:10:16,130 --> 00:10:18,920 has some R dependence from filtration, as we're sketching 214 00:10:18,920 --> 00:10:20,930 here, but it has also very strong R 215 00:10:20,930 --> 00:10:22,250 dependence from sedimentation. 216 00:10:22,250 --> 00:10:25,010 So we don't show it here, but the sedimentation velocity kind 217 00:10:25,010 --> 00:10:26,640 of increases like R squared. 218 00:10:26,640 --> 00:10:28,470 So that's a pretty big dependence. 219 00:10:28,470 --> 00:10:29,930 And so what that means is that if I 220 00:10:29,930 --> 00:10:32,420 take some of these initial droplet distributions 221 00:10:32,420 --> 00:10:34,040 that are coming from breathing-- so 222 00:10:34,040 --> 00:10:36,800 imagine this is kind of the drop of distribution 223 00:10:36,800 --> 00:10:39,170 soon after the aerosols leave the mouth, 224 00:10:39,170 --> 00:10:41,690 after the initial evaporation has taken place, 225 00:10:41,690 --> 00:10:44,090 but the Wells curve has been disrupted by the fact 226 00:10:44,090 --> 00:10:49,200 that there's mucus with lots of solute and charged molecules. 227 00:10:49,200 --> 00:10:52,550 And so the droplets have reached an equilibrium distribution 228 00:10:52,550 --> 00:10:55,510 that looks something like is shown here. 229 00:10:55,510 --> 00:10:58,180 If I then ask myself what happens at a later time, 230 00:10:58,180 --> 00:10:59,660 how does that build up? 231 00:10:59,660 --> 00:11:02,830 So let's say one person in the room who is-- basically 232 00:11:02,830 --> 00:11:04,930 one of these curves is describing 233 00:11:04,930 --> 00:11:06,540 the distribution of droplets that they 234 00:11:06,540 --> 00:11:09,700 are emitting from breathing. 235 00:11:09,700 --> 00:11:12,360 So let's ask ourselves what happens to the concentration 236 00:11:12,360 --> 00:11:14,230 profile in those cases. 237 00:11:14,230 --> 00:11:20,890 So there is an RC here and R. And basically 238 00:11:20,890 --> 00:11:23,300 as you can see, when R is less than RC, 239 00:11:23,300 --> 00:11:25,930 then the sedimentation term is small. 240 00:11:25,930 --> 00:11:27,520 And essentially it's just lambda a. 241 00:11:27,520 --> 00:11:31,600 So that means that the removal of the infectious particles 242 00:11:31,600 --> 00:11:35,710 is dominated by ventilation, which is not size dependent. 243 00:11:35,710 --> 00:11:37,720 So when you're less than RC, which 244 00:11:37,720 --> 00:11:40,720 is another way of defining the aerosol range, 245 00:11:40,720 --> 00:11:45,010 that is where you don't have a strong size dependence. 246 00:11:45,010 --> 00:11:49,000 But when you're in the large drop category, of course 247 00:11:49,000 --> 00:11:51,340 those drops are sedimenting. 248 00:11:51,340 --> 00:11:53,110 And in fact, this term can be very large. 249 00:11:53,110 --> 00:11:56,620 If we go out to 10 microns or even up to a millimeter, 250 00:11:56,620 --> 00:11:58,540 the sedimentation rate is incredibly fast. 251 00:11:58,540 --> 00:12:00,670 And those droplets are very quickly removed 252 00:12:00,670 --> 00:12:02,470 and do not end up swirling around the room, 253 00:12:02,470 --> 00:12:04,330 as we've been describing. 254 00:12:04,330 --> 00:12:07,170 So let's imagine we take-- let's say it's somebody speaking. 255 00:12:07,170 --> 00:12:11,030 And let's say that, just for illustration purposes, 256 00:12:11,030 --> 00:12:13,880 let's say it initially looks like this. 257 00:12:18,600 --> 00:12:20,630 So this is kind of the initial profile that's 258 00:12:20,630 --> 00:12:22,820 in the room in early times. 259 00:12:22,820 --> 00:12:26,920 Now at first, until we get to a time inverse of lambda C, 260 00:12:26,920 --> 00:12:28,760 that's the concentration or relaxation time. 261 00:12:28,760 --> 00:12:31,010 Over that time scale, the concentration 262 00:12:31,010 --> 00:12:32,690 is going to be building. 263 00:12:32,690 --> 00:12:35,720 But it's building the fastest in the aerosol range 264 00:12:35,720 --> 00:12:38,310 because those guys are not being removed. 265 00:12:38,310 --> 00:12:40,260 These guys, though, are being removed. 266 00:12:40,260 --> 00:12:42,780 So instead of increasing, it doesn't 267 00:12:42,780 --> 00:12:45,420 increase that much because they're also being removed. 268 00:12:45,420 --> 00:12:49,340 So what happens at a later time is that this goes up, 269 00:12:49,340 --> 00:12:52,070 but not so much over there. 270 00:12:52,070 --> 00:12:55,970 And it gets more and more peaked in the aerosol range 271 00:12:55,970 --> 00:13:04,510 because here you have basically fast removal of large drops. 272 00:13:04,510 --> 00:13:10,990 But here, you have slow buildup of aerosols. 273 00:13:10,990 --> 00:13:14,350 So if you wait for a timescale-- the time it takes here, 274 00:13:14,350 --> 00:13:18,290 time is basically lambda C inverse. 275 00:13:18,290 --> 00:13:20,170 So we call that tC or tau C, that's 276 00:13:20,170 --> 00:13:22,120 the timescale for buildup. 277 00:13:22,120 --> 00:13:23,740 That's the inverse of this. 278 00:13:23,740 --> 00:13:27,500 That's how long it takes to essentially reach steady state. 279 00:13:27,500 --> 00:13:32,930 And where you really get the biggest increase is 280 00:13:32,930 --> 00:13:34,460 among the aerosol, because over here 281 00:13:34,460 --> 00:13:35,750 you're basically losing them. 282 00:13:35,750 --> 00:13:37,670 Now, there are filtration effects too, 283 00:13:37,670 --> 00:13:39,050 and there's also the infectivity. 284 00:13:39,050 --> 00:13:41,000 So there are other competing size dependencies 285 00:13:41,000 --> 00:13:43,120 that might actually emphasize the large scale here, 286 00:13:43,120 --> 00:13:45,300 so I want to make sure we're clear about that. 287 00:13:45,300 --> 00:13:47,210 But these size dependencies are bounded. 288 00:13:47,210 --> 00:13:49,520 For example, the mass filtration is sort of 289 00:13:49,520 --> 00:13:51,020 bounded by one, right? 290 00:13:51,020 --> 00:13:53,110 But R squared just keeps going. 291 00:13:53,110 --> 00:13:54,950 So if you go to larger and larger particles, 292 00:13:54,950 --> 00:13:57,090 they sediment faster and faster and faster. 293 00:13:57,090 --> 00:13:57,770 OK. 294 00:13:57,770 --> 00:13:58,880 And so it really is true-- 295 00:13:58,880 --> 00:14:00,650 I didn't draw this very well, but if you 296 00:14:00,650 --> 00:14:02,880 get to like large sizes on the order of, 297 00:14:02,880 --> 00:14:05,750 let's say millimeters, those droplets really 298 00:14:05,750 --> 00:14:07,370 never build up in the air because they 299 00:14:07,370 --> 00:14:09,830 sediment almost immediately. 300 00:14:09,830 --> 00:14:12,770 When you cough or sneeze, the largest droplets fall 301 00:14:12,770 --> 00:14:14,210 and it happens very quickly. 302 00:14:14,210 --> 00:14:16,970 It might happen in a few seconds or a minute. 303 00:14:16,970 --> 00:14:18,650 And we're talking about timescales here 304 00:14:18,650 --> 00:14:20,270 that are on the order of the air change 305 00:14:20,270 --> 00:14:22,410 rate, which might be on the order of hours 306 00:14:22,410 --> 00:14:24,360 or tens of minutes. 307 00:14:24,360 --> 00:14:28,100 So basically, this size-dependent relaxation 308 00:14:28,100 --> 00:14:30,950 tends to kind of sharpen distribution right 309 00:14:30,950 --> 00:14:32,500 into the aerosol range. 310 00:14:32,500 --> 00:14:33,920 Which is, again, why we're talking 311 00:14:33,920 --> 00:14:36,740 about indoor airborne transmission being dominated 312 00:14:36,740 --> 00:14:38,330 by these aerosols that are swirling 313 00:14:38,330 --> 00:14:40,010 around the well-mixed room. 314 00:14:40,010 --> 00:14:42,320 And that's very different from transmission 315 00:14:42,320 --> 00:14:44,600 through coughs or sneezes of large droplets, which 316 00:14:44,600 --> 00:14:48,170 are only very briefly present and then they sediment out. 317 00:14:48,170 --> 00:14:50,600 So if you're not standing in the way of that cough, 318 00:14:50,600 --> 00:14:53,390 or if the cough is blocked by a mask or a shield, 319 00:14:53,390 --> 00:14:55,850 then you really don't have to worry as much about that form 320 00:14:55,850 --> 00:14:56,960 of transmission. 321 00:14:56,960 --> 00:14:58,970 But as shown here, you do have to worry 322 00:14:58,970 --> 00:15:01,400 about the aerosols which are building up in the air 323 00:15:01,400 --> 00:15:04,460 and are very strongly related to all these different factors 324 00:15:04,460 --> 00:15:07,820 that we've been talking about. 325 00:15:07,820 --> 00:15:09,620 And in fact, maybe I should also sketch-- 326 00:15:09,620 --> 00:15:11,240 just to help understand this picture, 327 00:15:11,240 --> 00:15:16,040 maybe I should draw lambda C as a function of R. 328 00:15:16,040 --> 00:15:20,650 So where this RC is-- again, I was kind of saying it in words, 329 00:15:20,650 --> 00:15:24,520 but it's dividing a situation when you're less than RC, 330 00:15:24,520 --> 00:15:26,920 where basically these quantities aren't varying 331 00:15:26,920 --> 00:15:30,160 too much, to the situation above where 332 00:15:30,160 --> 00:15:31,520 it's growing like R squared. 333 00:15:31,520 --> 00:15:34,180 So basically a much faster relaxation, again, 334 00:15:34,180 --> 00:15:38,140 separating aerosols from large drops. 335 00:15:38,140 --> 00:15:40,900 So if we substitute the Stokes velocity 336 00:15:40,900 --> 00:15:44,230 that we've been talking about earlier into this formula here, 337 00:15:44,230 --> 00:15:46,780 we can actually derive what RC is. 338 00:15:46,780 --> 00:15:50,470 And it comes out to be the square root 339 00:15:50,470 --> 00:15:56,710 because it is defined with RC squared of nine halves, that's 340 00:15:56,710 --> 00:15:58,420 from the Stokes formula. 341 00:15:58,420 --> 00:16:02,740 We have lambda a, the air change rate; 342 00:16:02,740 --> 00:16:05,410 the effective height of the room or ceiling height; 343 00:16:05,410 --> 00:16:07,750 the viscosity of air; and the denominator 344 00:16:07,750 --> 00:16:14,980 is two times the density of the liquid and gravity. 345 00:16:14,980 --> 00:16:17,230 And this number it turns out is also 346 00:16:17,230 --> 00:16:19,240 on the order of a few microns. 347 00:16:19,240 --> 00:16:21,970 So it does depend on what lambda a is, 348 00:16:21,970 --> 00:16:26,590 but lambda a on the order of tens of minutes to hours, this 349 00:16:26,590 --> 00:16:27,650 is a kind of range. 350 00:16:27,650 --> 00:16:28,480 It can be smaller. 351 00:16:28,480 --> 00:16:30,640 It can be bigger. 352 00:16:30,640 --> 00:16:31,660 In fact, it could be-- 353 00:16:31,660 --> 00:16:35,970 maybe it could even be like 0.5 microns up to 5 microns. 354 00:16:35,970 --> 00:16:39,480 0.5 to 5 microns is probably more accurate. 355 00:16:39,480 --> 00:16:42,510 But basically, it's sitting there right about-- 356 00:16:42,510 --> 00:16:45,180 as I was trying to sketch here-- in the barrier between aerosols 357 00:16:45,180 --> 00:16:46,800 and non aerosols. 358 00:16:46,800 --> 00:16:49,470 And also below RC is actually where 359 00:16:49,470 --> 00:16:55,440 the peak of the distributions is from typical expiratory 360 00:16:55,440 --> 00:16:56,560 activities. 361 00:16:56,560 --> 00:16:59,130 And so the point is that those are not being affected 362 00:16:59,130 --> 00:17:01,340 very much by sedimentation. 363 00:17:01,340 --> 00:17:02,970 Now the last thing I'll mention is that 364 00:17:02,970 --> 00:17:06,329 as soon as we have these kind of size-dependent properties, what 365 00:17:06,329 --> 00:17:08,550 happens to our calculation of transmission? 366 00:17:08,550 --> 00:17:10,710 I'll just put the mathematical formula on the board 367 00:17:10,710 --> 00:17:13,140 here without really dwelling on it. 368 00:17:13,140 --> 00:17:15,930 But the way you can plot what I've 369 00:17:15,930 --> 00:17:20,650 sketched down here for an actual distribution shown here 370 00:17:20,650 --> 00:17:23,990 would be to substitute it into the formulas we had before. 371 00:17:23,990 --> 00:17:29,190 So for example, what is the time-dependent transmission 372 00:17:29,190 --> 00:17:30,180 rate? 373 00:17:30,180 --> 00:17:33,860 Well remember, that's Qb times the integral 374 00:17:33,860 --> 00:17:37,960 over all the sizes now of all the size-dependent qualities. 375 00:17:37,960 --> 00:17:46,820 So it's PM of R, C of R and t, and C I of R dR, where 376 00:17:46,820 --> 00:17:48,720 CI is the infectivity. 377 00:17:48,720 --> 00:17:52,170 So basically, as you solve this [INAUDIBLE] equation here, 378 00:17:52,170 --> 00:17:55,470 there's sort of change of the concentration field, 379 00:17:55,470 --> 00:17:56,910 as I've sketched. 380 00:17:56,910 --> 00:17:59,430 You have to then integrate these curves 381 00:17:59,430 --> 00:18:00,780 against these other factors-- 382 00:18:00,780 --> 00:18:03,780 the mass factor and the infectivity factor-- 383 00:18:03,780 --> 00:18:05,670 in order to figure out the transmission 384 00:18:05,670 --> 00:18:10,270 rate at that moment through the well-mixed air. 385 00:18:10,270 --> 00:18:14,170 And if we solve this equation here, 386 00:18:14,170 --> 00:18:17,020 we can actually substitute back in and get our formula 387 00:18:17,020 --> 00:18:29,140 for beta, which is that beta of t is Qb squared over v. 388 00:18:29,140 --> 00:18:32,950 So here I'm substituting P of R. So remember before, 389 00:18:32,950 --> 00:18:37,510 P was basically nd times C times the infectivity. 390 00:18:37,510 --> 00:18:41,080 So that was basically the production rate of virions 391 00:18:41,080 --> 00:18:42,550 and we've already seen that before. 392 00:18:42,550 --> 00:18:45,220 And so when I kind of substitute the solution that we've already 393 00:18:45,220 --> 00:18:47,500 derived, but just keeping track of the fact 394 00:18:47,500 --> 00:18:49,420 that these quantities are all radius-dependent 395 00:18:49,420 --> 00:18:51,480 and appear under the integral, we 396 00:18:51,480 --> 00:18:53,800 are left with the integral from zero 397 00:18:53,800 --> 00:19:01,360 to infinity of Pn of R squared Cv CI of R. Cv 398 00:19:01,360 --> 00:19:04,870 is the concentration of virions per liquid volume or mucous 399 00:19:04,870 --> 00:19:06,460 volume in the droplet. 400 00:19:06,460 --> 00:19:09,850 CI of R is the infectivity. 401 00:19:09,850 --> 00:19:15,940 And let's see, then we have also an nd of R vd of R, 402 00:19:15,940 --> 00:19:18,880 where vd of R is the droplet size. 403 00:19:18,880 --> 00:19:24,240 All divided by lambda C of R, which is all this stuff. 404 00:19:24,240 --> 00:19:28,410 And this integrand is multiplied by the exponential relaxation, 405 00:19:28,410 --> 00:19:34,200 one minus e to the lambda C of Rt dR. 406 00:19:34,200 --> 00:19:38,010 So this is your general formula for 407 00:19:38,010 --> 00:19:41,520 the time-dependent transmission rate in a room where 408 00:19:41,520 --> 00:19:44,610 you can select an actual distribution that corresponds 409 00:19:44,610 --> 00:19:47,880 to the infected persons breathing activity 410 00:19:47,880 --> 00:19:49,920 and tells you the kinds of aerosol droplets 411 00:19:49,920 --> 00:19:52,210 that they are emitting, and then you basically 412 00:19:52,210 --> 00:19:53,760 have all these other parameters to do 413 00:19:53,760 --> 00:20:00,720 with filtration and airflow and ventilation, viral deactivation 414 00:20:00,720 --> 00:20:01,690 and sedimentation. 415 00:20:01,690 --> 00:20:03,360 And you finally have to do this integral 416 00:20:03,360 --> 00:20:06,050 and you end up with the transmission rate.