1 00:00:10,500 --> 00:00:13,030 PROFESSOR: And so now let's consider the spreading 2 00:00:13,030 --> 00:00:16,329 of infections, disease indoors. 3 00:00:16,329 --> 00:00:19,960 So the McKendrick-Kermack models and various compartment models 4 00:00:19,960 --> 00:00:23,710 that we just described start from the assumption essentially 5 00:00:23,710 --> 00:00:25,660 of a well-mixed population, where there's 6 00:00:25,660 --> 00:00:27,580 no spatial dependence, no network 7 00:00:27,580 --> 00:00:30,460 dependence of connectivity of people with each other 8 00:00:30,460 --> 00:00:32,890 but rather sort of everyone is interacting with everyone 9 00:00:32,890 --> 00:00:35,050 in some sort of average sense. 10 00:00:35,050 --> 00:00:36,940 So that philosophy has also been brought 11 00:00:36,940 --> 00:00:39,820 to bear on spreading events indoors 12 00:00:39,820 --> 00:00:42,720 where a little bit stronger assumption is made 13 00:00:42,720 --> 00:00:45,940 is that the transmission is occurring through the air 14 00:00:45,940 --> 00:00:47,360 and the air is well-mixed. 15 00:00:47,360 --> 00:00:49,600 So not only are the people well-mixed in the sense 16 00:00:49,600 --> 00:00:52,180 that they can interact with each other regardless 17 00:00:52,180 --> 00:00:55,150 of their distance apart or anywhere placed in the room, 18 00:00:55,150 --> 00:00:56,860 but the reason it's a valid approximation 19 00:00:56,860 --> 00:00:58,540 is because the air itself is well-mixed. 20 00:00:58,540 --> 00:00:59,920 And so we've already been talking 21 00:00:59,920 --> 00:01:01,570 about the air, the well-mixed air, 22 00:01:01,570 --> 00:01:03,940 and we've calculated transmission rate. 23 00:01:03,940 --> 00:01:07,990 Now let's start to connect that to disease-spreading. 24 00:01:07,990 --> 00:01:13,710 So one concept there would be to consider 25 00:01:13,710 --> 00:01:17,950 that the initial number of infected people, i0, 26 00:01:17,950 --> 00:01:26,410 enters a room or an indoor space with n persons, 27 00:01:26,410 --> 00:01:34,229 including the infectors, for a time, tau. 28 00:01:34,229 --> 00:01:36,360 So we now have introduced a finite timescale, which 29 00:01:36,360 --> 00:01:37,610 is the time spent in the room. 30 00:01:37,610 --> 00:01:38,410 So you have a room. 31 00:01:38,410 --> 00:01:41,160 Some people are in it, and we're imagining an infected person 32 00:01:41,160 --> 00:01:43,320 comes in, and they spend a certain amount of time, 33 00:01:43,320 --> 00:01:44,340 and then they leave. 34 00:01:44,340 --> 00:01:47,420 And during that time we'd like to ask ourselves what happened 35 00:01:47,420 --> 00:01:48,509 with all the other people. 36 00:01:48,509 --> 00:01:50,180 How many people were infected? 37 00:01:50,180 --> 00:01:51,150 OK? 38 00:01:51,150 --> 00:01:57,150 So this tau kind of acts like the inverse of the removal 39 00:01:57,150 --> 00:02:04,530 rate, but we also will assume that gamma tau is 40 00:02:04,530 --> 00:02:05,460 much less than 1. 41 00:02:05,460 --> 00:02:09,240 So if we think of gamma as the recovery rate or the death rate 42 00:02:09,240 --> 00:02:12,330 from the disease, so that's the removal quantity 43 00:02:12,330 --> 00:02:14,910 at the population level, then we're 44 00:02:14,910 --> 00:02:17,510 assuming the time spent in the room is small compared to that. 45 00:02:17,510 --> 00:02:19,350 So in other words, while you're in the room, 46 00:02:19,350 --> 00:02:21,660 nobody is going to be recovering or dying. 47 00:02:21,660 --> 00:02:23,910 We're just simply going to be transmitting the disease 48 00:02:23,910 --> 00:02:25,110 for a certain amount of time. 49 00:02:25,110 --> 00:02:28,079 So the r is gone. 50 00:02:28,079 --> 00:02:40,420 On the other hand, we can also consider an exposed compartment 51 00:02:40,420 --> 00:02:42,579 of the population, the group in the room. 52 00:02:45,520 --> 00:02:48,100 So the exposed group is one which has essentially 53 00:02:48,100 --> 00:02:50,560 been exposed to the pathogen, so let's say 54 00:02:50,560 --> 00:02:54,910 they've inhaled a critical quantity of the virus, 55 00:02:54,910 --> 00:02:57,520 but they have not yet become themselves infectious. 56 00:02:57,520 --> 00:02:59,020 So they are already exposed. 57 00:02:59,020 --> 00:03:01,370 They are going to come down with the disease. 58 00:03:01,370 --> 00:03:05,290 So they are going to be eventually symptomatic 59 00:03:05,290 --> 00:03:09,160 and potentially infectious, but they may not be infectious yet. 60 00:03:09,160 --> 00:03:22,610 So the exposed group basically has received the disease. 61 00:03:22,610 --> 00:03:28,360 So it's been transmitted to them but is not yet 62 00:03:28,360 --> 00:03:29,900 themselves infectious. 63 00:03:32,650 --> 00:03:36,610 So that's another sort of group that we want to think about. 64 00:03:36,610 --> 00:03:41,140 So another way to say it, they're not yet contagious. 65 00:03:41,140 --> 00:03:43,690 So they're going to get sick, but they're not contagious yet. 66 00:03:43,690 --> 00:03:47,470 So I should put here contagious is another word you 67 00:03:47,470 --> 00:03:50,020 could think about using. 68 00:03:50,020 --> 00:03:53,130 So we've gotten rid of r, the recovered compartment, 69 00:03:53,130 --> 00:03:54,610 but we've added e. 70 00:03:54,610 --> 00:04:00,670 So instead of the SEIR model, we have the SEI model. 71 00:04:00,670 --> 00:04:03,880 Now, where we can write down the same kinds 72 00:04:03,880 --> 00:04:05,020 of equations as before. 73 00:04:05,020 --> 00:04:10,770 ds dt is minus beta, and I'll write beta of t, 74 00:04:10,770 --> 00:04:12,580 si, because another complexity here 75 00:04:12,580 --> 00:04:15,220 is we know is that as soon as the infected person comes in 76 00:04:15,220 --> 00:04:17,779 if it's airborne transmission, it takes time 77 00:04:17,779 --> 00:04:19,660 through the fluid mechanics that we were just 78 00:04:19,660 --> 00:04:22,450 describing for the concentration to build up. 79 00:04:22,450 --> 00:04:24,280 So the transmission rate is not constant. 80 00:04:24,280 --> 00:04:25,300 We've actually already calculated 81 00:04:25,300 --> 00:04:27,850 that transmission rate, but it now enters this equation. 82 00:04:30,620 --> 00:04:37,490 So this is now the susceptible people are getting exposed. 83 00:04:37,490 --> 00:04:40,540 And so when that sort of reaction happens, if you will, 84 00:04:40,540 --> 00:04:46,090 the exposed population then grows, 85 00:04:46,090 --> 00:04:48,460 and then there's a certain rate with which 86 00:04:48,460 --> 00:04:51,580 exposed people can become themselves infectious 87 00:04:51,580 --> 00:04:53,170 to others. 88 00:04:53,170 --> 00:04:54,920 And that rate we'll call alpha. 89 00:04:54,920 --> 00:04:58,060 So we can remove exposed people at a rate alpha, 90 00:04:58,060 --> 00:05:02,800 and when they're removed they become infected people. 91 00:05:02,800 --> 00:05:06,220 And so then the number of infections grows. 92 00:05:06,220 --> 00:05:09,490 So this is a very closely-related model 93 00:05:09,490 --> 00:05:11,740 that we could solve. 94 00:05:11,740 --> 00:05:16,810 And in order to couple this to airborne disease-spreading, 95 00:05:16,810 --> 00:05:20,560 we need to go back to our model of the airborne pathogen. 96 00:05:20,560 --> 00:05:24,540 So, for example, the virions per air volume which we called c, 97 00:05:24,540 --> 00:05:26,860 and, in fact, I'll even write this as a dc dt 98 00:05:26,860 --> 00:05:30,820 with a partial derivative, which is p, which depends 99 00:05:30,820 --> 00:05:34,720 on the size, r, divided by-- 100 00:05:34,720 --> 00:05:38,740 actually this should be v-- 101 00:05:38,740 --> 00:05:42,460 minus lambda c of rc, so that was 102 00:05:42,460 --> 00:05:46,030 the model we just talked about, which we have to solve. 103 00:05:46,030 --> 00:05:52,510 And then the beta of t is the breathing rate 104 00:05:52,510 --> 00:06:00,220 times the integral over all these sizes of c of r and t 105 00:06:00,220 --> 00:06:01,840 and then ci, and of course there could 106 00:06:01,840 --> 00:06:09,280 be masks, which might also be size dependent, dr. 107 00:06:09,280 --> 00:06:12,940 So we tie it together this way. 108 00:06:12,940 --> 00:06:15,640 This kind of model, which is typically 109 00:06:15,640 --> 00:06:17,830 run without accounting for the size dependents, 110 00:06:17,830 --> 00:06:20,930 but that has been done by some authors as well, 111 00:06:20,930 --> 00:06:23,650 generally falls under the category of Wells-Riley models. 112 00:06:28,480 --> 00:06:31,430 And I mentioned here that Wells actually 113 00:06:31,430 --> 00:06:34,550 was a real pioneer here, already starting in the 1930s. 114 00:06:34,550 --> 00:06:38,520 Did careful studies of disease transmission, 115 00:06:38,520 --> 00:06:41,240 including for flu and other viral diseases, 116 00:06:41,240 --> 00:06:44,150 and really was one of the initial proponents 117 00:06:44,150 --> 00:06:47,780 of the idea of infectious air, that there are particles 118 00:06:47,780 --> 00:06:50,120 suspended in the air which can become infectious. 119 00:06:50,120 --> 00:06:51,800 He also was the one who pioneered 120 00:06:51,800 --> 00:06:57,440 the Wells curve, which explored evaporation versus settling. 121 00:06:57,440 --> 00:07:02,170 And in 1955, he introduced this kind of a model 122 00:07:02,170 --> 00:07:04,760 where he was taking into account the balance of the production 123 00:07:04,760 --> 00:07:07,310 rate, p, and then the ventilation rate in order 124 00:07:07,310 --> 00:07:12,110 to describe the buildup of the infectious droplets in the air. 125 00:07:12,110 --> 00:07:17,270 And then the transmission is connected in this way here. 126 00:07:17,270 --> 00:07:20,330 And I should say that in Wells-Riley modeling, 127 00:07:20,330 --> 00:07:27,710 usually e is ignored, and so we normally just 128 00:07:27,710 --> 00:07:32,480 go straight from s to i. 129 00:07:32,480 --> 00:07:36,190 So I'll say e is neglected, and what that really 130 00:07:36,190 --> 00:07:41,470 means is that the incubation rate here, this alpha, 131 00:07:41,470 --> 00:07:44,930 is the incubation rate. 132 00:07:44,930 --> 00:07:49,310 So it's the rate at which an exposed person becomes 133 00:07:49,310 --> 00:07:52,190 infectious or contagious. 134 00:07:52,190 --> 00:07:56,700 That alpha t is much less than 1. 135 00:07:56,700 --> 00:07:59,040 So it's essentially the Wells-Riley model, 136 00:07:59,040 --> 00:08:01,440 it's the slow incubation limit. 137 00:08:04,200 --> 00:08:06,550 But we've written down something more general here, 138 00:08:06,550 --> 00:08:10,680 which does also allow for the possibility of exposed 139 00:08:10,680 --> 00:08:12,340 people that have not yet-- 140 00:08:12,340 --> 00:08:14,340 that could become infectious, so we'll come back 141 00:08:14,340 --> 00:08:16,360 to that in just a moment. 142 00:08:16,360 --> 00:08:18,870 So let's kind of summarize the results 143 00:08:18,870 --> 00:08:21,490 of the Wells-Riley approach. 144 00:08:21,490 --> 00:08:30,780 So we'll consider the slow incubation limit, which 145 00:08:30,780 --> 00:08:35,340 is alpha tau much less than 1. 146 00:08:35,340 --> 00:08:39,750 And so in that case, if there's slow incubation, 147 00:08:39,750 --> 00:08:43,820 during the time of the infected person being in the room, 148 00:08:43,820 --> 00:08:48,650 there is no time for new people to become infectious. 149 00:08:48,650 --> 00:08:51,290 So it's really just like a fixed source. 150 00:08:51,290 --> 00:08:53,460 So it's essentially saying that i 151 00:08:53,460 --> 00:08:57,490 is equal to i0, which is a constant. 152 00:08:57,490 --> 00:08:59,790 So even though the disease is being transmitted, 153 00:08:59,790 --> 00:09:02,050 there is no kind of amplification effect 154 00:09:02,050 --> 00:09:03,790 because the people who have been exposed 155 00:09:03,790 --> 00:09:07,600 are not able to themselves infect others 156 00:09:07,600 --> 00:09:10,570 because the incubation is too slow. 157 00:09:10,570 --> 00:09:13,870 And this model here, where you sort 158 00:09:13,870 --> 00:09:20,880 of couple these equations with the si equations basically, 159 00:09:20,880 --> 00:09:33,160 that was done by Gammaitoni and Nucci in 1997. 160 00:09:33,160 --> 00:09:34,990 So you can see starting from Wells, 161 00:09:34,990 --> 00:09:37,450 we start to move towards this picture of actually coupling 162 00:09:37,450 --> 00:09:41,170 the dynamics of the pathogen to the transmission. 163 00:09:41,170 --> 00:09:44,170 And there have been a number of models since then 164 00:09:44,170 --> 00:09:45,880 which have even taken into account some 165 00:09:45,880 --> 00:09:47,470 of these aspects of the different drop 166 00:09:47,470 --> 00:09:49,540 sizes and other effects that we've 167 00:09:49,540 --> 00:09:54,130 been discussing that were not initially considered by Wells. 168 00:09:54,130 --> 00:09:56,930 So that's this. 169 00:09:56,930 --> 00:10:00,200 So the first thing to do is to deal with beta of t. 170 00:10:00,200 --> 00:10:02,150 So how are we going to do that? 171 00:10:02,150 --> 00:10:04,770 So if I put beta t on the other side 172 00:10:04,770 --> 00:10:06,190 here, if I kind of divide it down, 173 00:10:06,190 --> 00:10:07,780 you can see that I really would like 174 00:10:07,780 --> 00:10:13,840 to define a new differential of a variable t hat, which 175 00:10:13,840 --> 00:10:17,980 is beta of t times dt. 176 00:10:17,980 --> 00:10:21,370 That would be my definition of a new variable. 177 00:10:21,370 --> 00:10:23,080 And so when i integrate this, it tells me 178 00:10:23,080 --> 00:10:25,180 that I should really switch to a new time 179 00:10:25,180 --> 00:10:28,390 variable, which is not just t, but it's 180 00:10:28,390 --> 00:10:31,900 the integral of beta dt. 181 00:10:31,900 --> 00:10:33,480 So it's a time-like variable. 182 00:10:33,480 --> 00:10:36,310 And once I do that I'm essentially sweeping this guy 183 00:10:36,310 --> 00:10:38,690 into the derivative there. 184 00:10:38,690 --> 00:10:41,700 And if I also assume that i is roughly equal to i0, 185 00:10:41,700 --> 00:10:43,910 then what I have now is a much simpler equation here. 186 00:10:43,910 --> 00:10:46,030 I go from this non-linear equation 187 00:10:46,030 --> 00:10:48,790 with time-dependent coefficients to something 188 00:10:48,790 --> 00:10:51,920 which is now linear. 189 00:10:51,920 --> 00:11:01,120 And we have ds dt hat is approximately equal to beta-- 190 00:11:05,480 --> 00:11:16,770 sorry-- which is minus i0s. 191 00:11:21,610 --> 00:11:23,540 So, again, our i is roughly constant. 192 00:11:23,540 --> 00:11:25,940 Our beta got swept into the time. 193 00:11:25,940 --> 00:11:29,650 So this is just kind of a simple exponential relaxation. 194 00:11:29,650 --> 00:11:34,060 And the initial value is that s at time equals 0 is n minus i0. 195 00:11:34,060 --> 00:11:35,800 So there are n people. 196 00:11:35,800 --> 00:11:38,920 i0 of them are the initial infected ones. 197 00:11:38,920 --> 00:11:41,290 And so this a pretty easy equation to solve. 198 00:11:41,290 --> 00:11:43,630 It's just an exponential relaxation, which we've already 199 00:11:43,630 --> 00:11:45,440 seen a few times before. 200 00:11:45,440 --> 00:11:50,080 And so the solution is that s is n 201 00:11:50,080 --> 00:11:53,710 minus i0, the initial value of s, 202 00:11:53,710 --> 00:11:58,030 times e to the minus i0t hat. 203 00:11:58,030 --> 00:12:03,560 So it's i0, integral 0 to t of beta dt. 204 00:12:07,070 --> 00:12:08,660 And then we can also write-- 205 00:12:08,660 --> 00:12:11,090 so that's the s, and the e is just 206 00:12:11,090 --> 00:12:13,720 if we have fast incubation-- 207 00:12:13,720 --> 00:12:15,770 or sorry, slow incubation, this term is gone, 208 00:12:15,770 --> 00:12:18,710 and we basically just have e as sort of just-- 209 00:12:18,710 --> 00:12:21,500 s and e have to add up to n basically. 210 00:12:21,500 --> 00:12:29,720 And so we're left with e of t is n minus i and then 1 minus, 211 00:12:29,720 --> 00:12:33,020 and let me actually write this quantity up here 212 00:12:33,020 --> 00:12:45,340 as little q of t, where little q of t is i0 times 213 00:12:45,340 --> 00:12:48,010 the time integral of beta dt. 214 00:12:56,610 --> 00:12:59,020 And what is this q here? 215 00:12:59,020 --> 00:13:04,630 This q can be thought of as the number of infection quanta 216 00:13:04,630 --> 00:13:07,570 per time that are released by the infectors 217 00:13:07,570 --> 00:13:09,970 because their i's are infectors. 218 00:13:09,970 --> 00:13:13,490 Each one of them is infecting another individual person 219 00:13:13,490 --> 00:13:15,920 at a rate beta, and that beta is time-dependent, 220 00:13:15,920 --> 00:13:18,260 so you have to integrate in time. 221 00:13:18,260 --> 00:13:20,980 And so this is telling me the total number 222 00:13:20,980 --> 00:13:26,860 of people that would be infected by those infected people 223 00:13:26,860 --> 00:13:29,720 if you weren't running into all of the limits that 224 00:13:29,720 --> 00:13:30,890 are described by this model. 225 00:13:30,890 --> 00:13:33,670 So what is happening is that if someone gets exposed, 226 00:13:33,670 --> 00:13:35,680 they can't be exposed again, so there's 227 00:13:35,680 --> 00:13:37,270 some numbers there that are changing, 228 00:13:37,270 --> 00:13:41,220 so you can't keep passing an infection quantum to somebody 229 00:13:41,220 --> 00:13:42,930 and have them get infected over and over. 230 00:13:42,930 --> 00:13:44,500 So that doesn't happen. 231 00:13:44,500 --> 00:13:47,500 But what this is measuring is somehow the number of times 232 00:13:47,500 --> 00:13:49,420 that somehow there's been an attempted 233 00:13:49,420 --> 00:13:52,000 infection or an expected infection 234 00:13:52,000 --> 00:13:54,640 if a person were susceptible. 235 00:13:54,640 --> 00:14:02,010 So this q of t here is the infection quanta 236 00:14:02,010 --> 00:14:12,260 as defined by Wells transmitted in time t. 237 00:14:12,260 --> 00:14:14,750 So up to time t, there are i0 infectors-- 238 00:14:14,750 --> 00:14:16,610 that number's not changing-- and the way 239 00:14:16,610 --> 00:14:18,860 you can think about this kind of airborne transmission 240 00:14:18,860 --> 00:14:22,520 is they're essentially spewing out infection quanta. 241 00:14:22,520 --> 00:14:26,030 Each one of them if it hits a potentially susceptible person 242 00:14:26,030 --> 00:14:28,820 will infect them, but the number of susceptible people 243 00:14:28,820 --> 00:14:29,580 is changing. 244 00:14:29,580 --> 00:14:31,340 So it's not like you keep getting more infections. 245 00:14:31,340 --> 00:14:32,450 Eventually you run out of people, 246 00:14:32,450 --> 00:14:33,860 and you can't keep infecting them, 247 00:14:33,860 --> 00:14:35,810 but that's one way to think about this. 248 00:14:35,810 --> 00:14:39,240 And we've already calculated this concept 249 00:14:39,240 --> 00:14:41,870 of infection quanta in the context of the breathing. 250 00:14:41,870 --> 00:14:44,360 We've defined a quantity, which is 251 00:14:44,360 --> 00:14:48,080 the number of infection quanta per volume of exhaled breath. 252 00:14:48,080 --> 00:14:49,880 We've also talked about the quanta emission 253 00:14:49,880 --> 00:14:52,830 rates for people and connected that back to the droplets. 254 00:14:52,830 --> 00:14:55,100 And so here's how you see how that quantity enters 255 00:14:55,100 --> 00:14:59,490 into the disease-spreading models. 256 00:14:59,490 --> 00:15:03,060 And in fact, we can sketch what happens with s and i here. 257 00:15:03,060 --> 00:15:06,680 So basically if we start with s, for example, starts 258 00:15:06,680 --> 00:15:12,690 at the value n minus i0 and as you go in time it decays-- 259 00:15:12,690 --> 00:15:15,440 well, the decay rate is basically 260 00:15:15,440 --> 00:15:17,910 set by sort of the average beta. 261 00:15:17,910 --> 00:15:19,910 So this is kind of like some kind of maybe 262 00:15:19,910 --> 00:15:21,720 if you're getting close to steady state, 263 00:15:21,720 --> 00:15:24,860 then beta inverse is sort of what this timescale is. 264 00:15:24,860 --> 00:15:28,100 But what's actually happening is you cut it off 265 00:15:28,100 --> 00:15:30,110 at a certain time, tau. 266 00:15:30,110 --> 00:15:32,060 So that's the time that you're in the room. 267 00:15:32,060 --> 00:15:34,130 In that time, the number of susceptible people 268 00:15:34,130 --> 00:15:41,860 has gone down, and the number of exposed people goes up. 269 00:15:41,860 --> 00:15:43,780 But then because the incubation rate is slow, 270 00:15:43,780 --> 00:15:46,640 the exposed people never become infectious in this model. 271 00:15:46,640 --> 00:15:51,130 So the number of infectors is fixed. 272 00:15:51,130 --> 00:15:55,160 And so since we have slow incubation 273 00:15:55,160 --> 00:15:56,960 and since the number of infectors is fixed, 274 00:15:56,960 --> 00:15:58,500 another way to look at this is notice 275 00:15:58,500 --> 00:16:00,530 this non-linear equation, which had s times i, 276 00:16:00,530 --> 00:16:02,210 it became i0 times s. 277 00:16:02,210 --> 00:16:04,640 So it became a linear response. 278 00:16:04,640 --> 00:16:06,860 And so another way to think about this limit 279 00:16:06,860 --> 00:16:11,340 is this is the limit of linear response. 280 00:16:15,190 --> 00:16:18,570 So the Wells-Riley models are basically 281 00:16:18,570 --> 00:16:21,540 linear response models, which are typically 282 00:16:21,540 --> 00:16:25,160 not taking into account a growth in the number of infectors, 283 00:16:25,160 --> 00:16:27,000 which would lead to kind of an amplification 284 00:16:27,000 --> 00:16:29,010 of disease-spreading in a room. 285 00:16:29,010 --> 00:16:32,130 And that's often justified because the time someone 286 00:16:32,130 --> 00:16:35,460 spends in a room is often times a lot less than the incubation 287 00:16:35,460 --> 00:16:37,560 time, which could be on the order of days. 288 00:16:37,560 --> 00:16:40,290 People might only spend a few hours in the room. 289 00:16:40,290 --> 00:16:43,170 On the other hand, there are situations such as classrooms, 290 00:16:43,170 --> 00:16:48,360 long-term care facilities and homes, prisons, workplaces, 291 00:16:48,360 --> 00:16:50,460 where people are exposed to each other for days 292 00:16:50,460 --> 00:16:52,020 or weeks or months actually. 293 00:16:52,020 --> 00:16:55,300 They may go home in between, but there's a constant exposure. 294 00:16:55,300 --> 00:16:59,370 And so you may need to worry about these other sorts 295 00:16:59,370 --> 00:17:03,030 of dynamics, even in an indoor setting actually. 296 00:17:03,030 --> 00:17:05,410 So we'll come back to that. 297 00:17:05,410 --> 00:17:07,500 Now another important concept I'd like to get here 298 00:17:07,500 --> 00:17:10,400 is what is the early rate of infection. 299 00:17:10,400 --> 00:17:15,210 So this quantity here is q. 300 00:17:15,210 --> 00:17:16,290 I wrote n minus 1 here. 301 00:17:16,290 --> 00:17:17,040 That was the case. 302 00:17:17,040 --> 00:17:18,490 Actually more generally this should 303 00:17:18,490 --> 00:17:20,190 be n minus i0 because often times we're 304 00:17:20,190 --> 00:17:22,740 thinking of one infector, so I kind of jumped to that, 305 00:17:22,740 --> 00:17:25,339 but more generally it'd be n minus i0, 306 00:17:25,339 --> 00:17:28,050 And this quantity here is also s0. 307 00:17:28,050 --> 00:17:30,660 That's the initial number of susceptible people. 308 00:17:30,660 --> 00:17:33,060 So basically you take the initial number of susceptible, 309 00:17:33,060 --> 00:17:35,400 and they slowly become exposed based 310 00:17:35,400 --> 00:17:39,670 on how many quanta they've consumed. 311 00:17:39,670 --> 00:17:44,400 Now, the definition of quanta of infection 312 00:17:44,400 --> 00:17:46,110 often is tied to this equation. 313 00:17:46,110 --> 00:17:53,920 This essentially is the Wells equation, derived in 1955. 314 00:17:53,920 --> 00:17:56,240 And Wells actually defined quanta from this. 315 00:17:56,240 --> 00:18:00,520 He said that one quantum infection corresponds 316 00:18:00,520 --> 00:18:05,590 to a probability of transmission of 63% 317 00:18:05,590 --> 00:18:10,960 because if you put q equals minus 1, 1 minus e 318 00:18:10,960 --> 00:18:13,930 to the minus 1 is 63%. 319 00:18:13,930 --> 00:18:16,270 So basically he said a quantum-- 320 00:18:16,270 --> 00:18:19,480 so if someone asks you what is a quantum infection-- 321 00:18:19,480 --> 00:18:26,760 a quantum is basically 63% chance of infection. 322 00:18:26,760 --> 00:18:28,180 But the problem with that thinking 323 00:18:28,180 --> 00:18:32,620 though is that in some other situation where there is maybe 324 00:18:32,620 --> 00:18:34,960 incubation going on, I might get more infected people. 325 00:18:34,960 --> 00:18:37,090 So I can't just count the infected people in a room 326 00:18:37,090 --> 00:18:41,560 and assume that I'm getting a measure of infection quanta. 327 00:18:41,560 --> 00:18:44,680 Infection quanta actually are defined by beta. 328 00:18:44,680 --> 00:18:46,300 That's an important thing. 329 00:18:46,300 --> 00:18:47,530 So really what is beta? 330 00:18:47,530 --> 00:18:51,460 It's actually the rate of infection quantum transfer 331 00:18:51,460 --> 00:18:53,850 from infectious to susceptible. 332 00:18:53,850 --> 00:18:54,670 Sorry. 333 00:18:54,670 --> 00:18:56,080 It's si basically. 334 00:18:56,080 --> 00:18:58,120 Oh actually did I-- 335 00:18:58,120 --> 00:18:59,230 no, that's correct, yeah. 336 00:18:59,230 --> 00:19:01,700 So it's the rate of basically becoming exposed. 337 00:19:01,700 --> 00:19:02,200 Excuse me. 338 00:19:02,200 --> 00:19:03,700 So it's from susceptible to exposed. 339 00:19:03,700 --> 00:19:05,890 So that rate basically defines beta, 340 00:19:05,890 --> 00:19:09,790 and that is giving you the rate of infection quanta transfer. 341 00:19:09,790 --> 00:19:12,460 How many people actually get infected or exposed 342 00:19:12,460 --> 00:19:14,140 involves solving some set of equations 343 00:19:14,140 --> 00:19:15,730 like this, which might not be the same 344 00:19:15,730 --> 00:19:17,170 as the Wells-Riley model. 345 00:19:17,170 --> 00:19:18,640 So that's an important thing to keep in mind, 346 00:19:18,640 --> 00:19:19,800 and we'll come back to that. 347 00:19:19,800 --> 00:19:26,930 So this is basically Wells' definition of a quantum 348 00:19:26,930 --> 00:19:29,270 is that basically there's a 63% chance that it's 349 00:19:29,270 --> 00:19:30,770 going to infect somebody, but that's 350 00:19:30,770 --> 00:19:33,110 completely tied to this linear response exponential 351 00:19:33,110 --> 00:19:33,840 relaxation. 352 00:19:33,840 --> 00:19:38,000 It's not really, I don't think, is the appropriate definition. 353 00:19:38,000 --> 00:19:41,240 Now, another thing we can ask is, what about at early times. 354 00:19:47,530 --> 00:19:50,380 And that would be the case where the expected number of quanta 355 00:19:50,380 --> 00:19:52,810 transferred is less than 1. 356 00:19:52,810 --> 00:19:56,380 So at early times you haven't really seen a lot of infection 357 00:19:56,380 --> 00:19:57,710 take place. 358 00:19:57,710 --> 00:19:59,230 This will be very important for us 359 00:19:59,230 --> 00:20:01,700 because we're going to come to safety guidelines, 360 00:20:01,700 --> 00:20:04,000 and in my opinion the right way to think about a safety 361 00:20:04,000 --> 00:20:07,630 guideline is you don't want one person to cause 362 00:20:07,630 --> 00:20:08,980 one or more infections. 363 00:20:08,980 --> 00:20:13,120 You'd like the expectation to be that less than one person will 364 00:20:13,120 --> 00:20:14,200 get infected. 365 00:20:14,200 --> 00:20:16,570 So this, what I'm calling early times here, 366 00:20:16,570 --> 00:20:18,790 in the epidemiological model is actually 367 00:20:18,790 --> 00:20:20,950 very relevant for, like, safety guidelines. 368 00:20:20,950 --> 00:20:22,450 You don't want to deal with the case 369 00:20:22,450 --> 00:20:23,740 where there's rampant infection. 370 00:20:23,740 --> 00:20:25,780 You just want to say if an infected person comes 371 00:20:25,780 --> 00:20:28,140 in the room, are they going to infect anybody else. 372 00:20:28,140 --> 00:20:29,890 And you want that to be a low probability. 373 00:20:29,890 --> 00:20:32,080 So this is a very relevant limit. 374 00:20:32,080 --> 00:20:35,170 Now, if I take that limit here, I can expand this exponential, 375 00:20:35,170 --> 00:20:38,920 and I find that e of t at early times 376 00:20:38,920 --> 00:20:42,720 behaves like s0 times q of t if s0 377 00:20:42,720 --> 00:20:44,530 is the number of susceptibles. 378 00:20:44,530 --> 00:20:47,680 And if I plug in what we have here, 379 00:20:47,680 --> 00:20:54,620 that's n minus i0 times i0 times the integral in time of beta. 380 00:20:57,310 --> 00:21:00,250 And so this is the expected number 381 00:21:00,250 --> 00:21:04,630 of people that will be infected by i0 infected people. 382 00:21:04,630 --> 00:21:07,970 So the ratio of these two things is very important. 383 00:21:07,970 --> 00:21:12,530 So if I look at e in time tau. 384 00:21:12,530 --> 00:21:14,740 I should say we do this integral-- typically 385 00:21:14,740 --> 00:21:17,560 we want to go up to dd which is the time that the person spends 386 00:21:17,560 --> 00:21:21,460 in the room, and we divide by the initial number, 387 00:21:21,460 --> 00:21:23,380 then that's basically telling us sort 388 00:21:23,380 --> 00:21:26,280 of what's the reproductive number of the room. 389 00:21:26,280 --> 00:21:28,600 i0 people come in, and the question 390 00:21:28,600 --> 00:21:31,810 is do more than i0 people come out that are infected. 391 00:21:31,810 --> 00:21:33,160 So have you infected others? 392 00:21:33,160 --> 00:21:35,600 So that's the ratio that you really care about. 393 00:21:35,600 --> 00:21:40,460 And so I would call this the indoor reproductive number. 394 00:21:40,460 --> 00:21:46,570 And in this case, if we pick i0 equal to 1, 395 00:21:46,570 --> 00:21:55,060 then this is n minus 1, 0 to tau beta dt. 396 00:21:55,060 --> 00:21:57,270 So that's going to be an important thing, which we'll 397 00:21:57,270 --> 00:22:00,630 come back to later, which is the definition of kind of what 398 00:22:00,630 --> 00:22:03,030 makes a room actually safe. 399 00:22:03,030 --> 00:22:06,390 Well, what we'd like to do is have this quantity be 400 00:22:06,390 --> 00:22:09,420 much less than 1 because I want to say if one infector comes 401 00:22:09,420 --> 00:22:11,520 in the room and everybody is currently 402 00:22:11,520 --> 00:22:14,280 susceptible and healthy, I want to make sure 403 00:22:14,280 --> 00:22:17,580 that less than one person actually gets infected 404 00:22:17,580 --> 00:22:19,110 And there'll be some tolerance maybe 405 00:22:19,110 --> 00:22:21,690 on how low I'd like that to be, but that's 406 00:22:21,690 --> 00:22:23,040 a very important concept. 407 00:22:23,040 --> 00:22:26,400 And here you see it comes out of the slow incubation limit. 408 00:22:26,400 --> 00:22:28,320 What I'd like to show you next is 409 00:22:28,320 --> 00:22:31,710 that the very same indoor reproductive number actually 410 00:22:31,710 --> 00:22:35,070 occurs for any model including the opposite limit 411 00:22:35,070 --> 00:22:37,820 of fast incubation.