1 00:00:00,060 --> 00:00:02,500 The following content is provided under a Creative 2 00:00:02,500 --> 00:00:04,019 Commons license. 3 00:00:04,019 --> 00:00:06,360 Your support will help MIT OpenCourseWare 4 00:00:06,360 --> 00:00:10,730 continue to offer high-quality educational resources for free. 5 00:00:10,730 --> 00:00:13,330 To make a donation or view additional materials 6 00:00:13,330 --> 00:00:17,212 from hundreds of MIT courses, visit MIT OpenCourseWare 7 00:00:17,212 --> 00:00:17,837 at ocw.mit.edu. 8 00:00:20,975 --> 00:00:22,350 PROFESSOR: So today, what we want 9 00:00:22,350 --> 00:00:25,402 to do is we want to talk about two related topics. 10 00:00:25,402 --> 00:00:26,860 The first is going to this question 11 00:00:26,860 --> 00:00:28,670 of the evolution of virulence and how 12 00:00:28,670 --> 00:00:31,830 to model host-parasite interactions more broadly. 13 00:00:31,830 --> 00:00:34,580 That's kind of modeled on chapter 11 of Martin's book. 14 00:00:34,580 --> 00:00:38,040 We'll focus on the first half of it for the discussions today. 15 00:00:38,040 --> 00:00:41,320 This is in the context of when a given host can only 16 00:00:41,320 --> 00:00:46,320 have one strain of the parasite or virus or whatnot 17 00:00:46,320 --> 00:00:49,890 inside that body. 18 00:00:49,890 --> 00:00:51,630 The model presented in Martin's book 19 00:00:51,630 --> 00:00:56,590 is very similar to classic models in epidemiology, which 20 00:00:56,590 --> 00:00:59,750 are the so-called SIR type models, 21 00:00:59,750 --> 00:01:02,130 where you divide up the host population 22 00:01:02,130 --> 00:01:05,200 into whether the they are sensitive-- i.e., 23 00:01:05,200 --> 00:01:09,360 non-infected-- infected, or resistant. 24 00:01:09,360 --> 00:01:11,120 And then, we'll draw the parallels 25 00:01:11,120 --> 00:01:15,030 of how we get from the model that you read about in Martin's 26 00:01:15,030 --> 00:01:17,990 book to the classic SIR models. 27 00:01:17,990 --> 00:01:21,450 But in both of these cases, the fundamental parameter 28 00:01:21,450 --> 00:01:24,690 that drives these things is this R0 parameter. 29 00:01:24,690 --> 00:01:29,989 It tells us about the expected number of new cases 30 00:01:29,989 --> 00:01:32,280 that will result when you introduce one infected member 31 00:01:32,280 --> 00:01:33,664 into the population. 32 00:01:33,664 --> 00:01:37,460 AUDIENCE: Sorry, will you be taking about super-infections? 33 00:01:37,460 --> 00:01:39,280 PROFESSOR: Only a little bit but, I 34 00:01:39,280 --> 00:01:40,920 would say, depending on time. 35 00:01:40,920 --> 00:01:43,530 But if you're interested in the super-infection discussion 36 00:01:43,530 --> 00:01:46,156 more, we can talk about it after class, maybe. 37 00:01:46,156 --> 00:01:46,655 All right. 38 00:01:49,609 --> 00:01:51,150 And so, for the second half of class, 39 00:01:51,150 --> 00:01:52,733 what we're going to do though is we're 40 00:01:52,733 --> 00:01:56,320 going to talk about the possible evolutionary benefits of sex. 41 00:01:56,320 --> 00:01:59,700 And in particular, we'll talk about this hypothesis, which 42 00:01:59,700 --> 00:02:02,110 is one of the reigning hypotheses for why it might 43 00:02:02,110 --> 00:02:05,640 be that sex is as widespread as it is, which is the Red Queen 44 00:02:05,640 --> 00:02:10,970 hypothesis, from Lewis Carroll's novel. 45 00:02:10,970 --> 00:02:13,730 And we're going to discuss this paper that you guys read about, 46 00:02:13,730 --> 00:02:15,188 "Running with the Red Queen," which 47 00:02:15,188 --> 00:02:18,290 I think has a nice discussion of this debate 48 00:02:18,290 --> 00:02:20,130 and, then, some nice experiments looking 49 00:02:20,130 --> 00:02:23,880 at experimental coevolution between the C. elegans worm 50 00:02:23,880 --> 00:02:27,420 and it's infecting parasite, which is a serratia bacterium. 51 00:02:30,190 --> 00:02:31,880 Any questions before we get going? 52 00:02:37,770 --> 00:02:42,390 OK, what I want to do is start by discussing 53 00:02:42,390 --> 00:02:45,630 this model in Martin's book. 54 00:02:45,630 --> 00:02:49,860 But also, there's a little bit of this philosophical question. 55 00:02:49,860 --> 00:02:53,600 Any time, that you are modeling, you always 56 00:02:53,600 --> 00:02:55,869 had to make decisions about which of the details 57 00:02:55,869 --> 00:02:57,910 you want to try to model and which of the details 58 00:02:57,910 --> 00:02:59,290 you do not want to model. 59 00:02:59,290 --> 00:03:01,750 And depending upon the situation, 60 00:03:01,750 --> 00:03:03,890 it may be that some assumptions are more or less 61 00:03:03,890 --> 00:03:05,510 appropriate than others. 62 00:03:05,510 --> 00:03:11,920 Now, in the model that Martin wrote down-- well, 63 00:03:11,920 --> 00:03:15,150 we'll try to figure out what the assumptions are here. 64 00:03:15,150 --> 00:03:18,140 So we have what you might think of 65 00:03:18,140 --> 00:03:20,840 as some sensitive individuals. 66 00:03:20,840 --> 00:03:24,240 Plus, the infected individuals are going 67 00:03:24,240 --> 00:03:26,590 to interact at some rate, beta. 68 00:03:26,590 --> 00:03:30,230 So this is how the sensitive become infected. 69 00:03:30,230 --> 00:03:33,300 And it results in, now, two infected individuals. 70 00:03:36,020 --> 00:03:37,920 Now of course, each of these individuals 71 00:03:37,920 --> 00:03:41,810 will, say, have some lifespan or die at some rate. 72 00:03:41,810 --> 00:03:47,200 All right, so the sensitive or uninfected individuals 73 00:03:47,200 --> 00:03:48,840 die at rate u. 74 00:03:48,840 --> 00:03:53,290 Whereas, infected individuals, others-- an increase 75 00:03:53,290 --> 00:03:56,658 in the death-rate, described by some virulence, v. OK? 76 00:04:01,817 --> 00:04:03,230 OK. 77 00:04:03,230 --> 00:04:05,136 Now, the model as written-- what's 78 00:04:05,136 --> 00:04:06,760 going to be the fate of the population? 79 00:04:11,870 --> 00:04:13,774 AUDIENCE: Everyone all dies. 80 00:04:13,774 --> 00:04:14,730 PROFESSOR: Yeah. 81 00:04:14,730 --> 00:04:16,579 Everyone's going to die, right? 82 00:04:16,579 --> 00:04:18,279 And that's even true in the-- it's not 83 00:04:18,279 --> 00:04:22,910 even that the population's dying as a result of the infection. 84 00:04:22,910 --> 00:04:26,216 Because even in the absence of any infected individuals, 85 00:04:26,216 --> 00:04:27,340 you just have people dying. 86 00:04:27,340 --> 00:04:31,270 So you need to have some way of keeping the population going, 87 00:04:31,270 --> 00:04:33,080 so you can study it perhaps. 88 00:04:33,080 --> 00:04:36,830 What we're going to assume is that sensitive individuals, 89 00:04:36,830 --> 00:04:39,770 or uninfected individuals, will enter the population just 90 00:04:39,770 --> 00:04:40,650 at some rate, k. 91 00:04:44,330 --> 00:04:50,120 Now, in terms of the philosophical question, 92 00:04:50,120 --> 00:04:52,680 in the beginning of the chapter, Martin 93 00:04:52,680 --> 00:04:55,820 talks a bit about this question of microparasites 94 00:04:55,820 --> 00:04:58,460 versus macroparasites. 95 00:04:58,460 --> 00:05:02,425 And I can somebody remind us, what's the distinction? 96 00:05:02,425 --> 00:05:06,695 and For what kind of parasite might this 97 00:05:06,695 --> 00:05:07,790 be intended to model? 98 00:05:12,560 --> 00:05:13,080 Yes? 99 00:05:13,080 --> 00:05:20,460 AUDIENCE: Well, what I got from it is that microparasites 100 00:05:20,460 --> 00:05:27,348 are on the order of single-cellular organisms, 101 00:05:27,348 --> 00:05:34,256 generally things that have much shorter reproductive steps, 102 00:05:34,256 --> 00:05:35,679 I guess. 103 00:05:35,679 --> 00:05:37,220 They reproduce a lot more frequently. 104 00:05:37,220 --> 00:05:38,768 PROFESSOR: That's right. 105 00:05:38,768 --> 00:05:40,184 AUDIENCE: Whereas, macro-parasites 106 00:05:40,184 --> 00:05:43,642 are like the [INAUDIBLE] or the tapeworm or something, 107 00:05:43,642 --> 00:05:48,088 which would not necessarily reproduce 108 00:05:48,088 --> 00:05:49,570 a lot inside the host. 109 00:05:49,570 --> 00:05:50,710 PROFESSOR: Mm-hm. 110 00:05:50,710 --> 00:05:51,210 Right. 111 00:05:51,210 --> 00:05:53,690 And I would say, that just given this distinction 112 00:05:53,690 --> 00:05:56,420 between the microparasites, that might be viruses and bacteria, 113 00:05:56,420 --> 00:05:57,836 as compared to the macroparasites, 114 00:05:57,836 --> 00:06:00,720 that are things like tapeworms and so forth, it's 115 00:06:00,720 --> 00:06:03,950 not obvious from that that you would have two 116 00:06:03,950 --> 00:06:05,520 different modeling frameworks. 117 00:06:05,520 --> 00:06:11,050 But what is the argument that is made in Martin's book? 118 00:06:11,050 --> 00:06:13,030 Or can you think up an argument for why 119 00:06:13,030 --> 00:06:15,270 it is that it might be this kind of model you would 120 00:06:15,270 --> 00:06:16,561 want to use for microparasites? 121 00:06:19,145 --> 00:06:19,726 Yes. 122 00:06:19,726 --> 00:06:21,101 AUDIENCE: Because we don't really 123 00:06:21,101 --> 00:06:22,568 care about-- he mentioned something 124 00:06:22,568 --> 00:06:24,035 that the microparasites reproduce 125 00:06:24,035 --> 00:06:26,480 in large numbers in infected individuals. 126 00:06:26,480 --> 00:06:29,169 So we don't have to keep track of the internal state 127 00:06:29,169 --> 00:06:31,380 of someone that's infected. 128 00:06:31,380 --> 00:06:32,510 PROFESSOR: That's right. 129 00:06:32,510 --> 00:06:35,440 And some of it's, maybe, even a historical thing. 130 00:06:35,440 --> 00:06:40,590 There might be huge numbers of viruses-- a flu virus or so-- 131 00:06:40,590 --> 00:06:42,440 in an infected individual. 132 00:06:42,440 --> 00:06:44,430 And in some ways, maybe, the number 133 00:06:44,430 --> 00:06:48,409 of viruses that is in that host is not the most relevant thing. 134 00:06:48,409 --> 00:06:49,950 And it's certainly would be much more 135 00:06:49,950 --> 00:06:51,730 complicated to try to keep track of that. 136 00:06:51,730 --> 00:06:57,635 And so, if you can get meaningful predictions-- rather 137 00:06:57,635 --> 00:07:00,260 than keeping track of the number of viruses, say, in each host, 138 00:07:00,260 --> 00:07:03,570 instead you just put the host into different classes-- 139 00:07:03,570 --> 00:07:06,199 sensitive and infected, for example. 140 00:07:06,199 --> 00:07:07,990 Later, we'll talk about what happens if you 141 00:07:07,990 --> 00:07:10,150 have a resistant type of class. 142 00:07:10,150 --> 00:07:13,730 But the idea there is that there's, maybe, even also 143 00:07:13,730 --> 00:07:15,550 some separation of time scales. 144 00:07:15,550 --> 00:07:16,830 Because you get infected. 145 00:07:16,830 --> 00:07:22,376 And kind of quickly, you're just sick and may be infective. 146 00:07:22,376 --> 00:07:23,750 But at some rate, you get better. 147 00:07:23,750 --> 00:07:26,120 And it's not that you'll necessarily 148 00:07:26,120 --> 00:07:28,890 gain very much by keeping track of the precise number 149 00:07:28,890 --> 00:07:30,450 of viruses in the host. 150 00:07:30,450 --> 00:07:33,050 Of course, this is ultimately an experimental observational 151 00:07:33,050 --> 00:07:34,840 question of whether this sort of model 152 00:07:34,840 --> 00:07:36,552 provides you inside that you're going 153 00:07:36,552 --> 00:07:38,510 to need to make sense of these diseases, right? 154 00:07:38,510 --> 00:07:40,995 AUDIENCE: And then it also seems like your method 155 00:07:40,995 --> 00:07:43,977 of transmission of macroparasites 156 00:07:43,977 --> 00:07:45,480 can be very different. 157 00:07:45,480 --> 00:07:46,480 PROFESSOR: That's right. 158 00:07:46,480 --> 00:07:48,510 So the mechanism of transmission depends 159 00:07:48,510 --> 00:07:52,310 very much on the disease that you're studying. 160 00:07:52,310 --> 00:07:55,200 And the macroparasites, in many cases, 161 00:07:55,200 --> 00:07:58,260 they're transmitted not from direct interactions 162 00:07:58,260 --> 00:08:02,450 between the hosts, but through the environment or something 163 00:08:02,450 --> 00:08:02,950 else. 164 00:08:07,120 --> 00:08:09,140 It's also, perhaps, just worth pointing out 165 00:08:09,140 --> 00:08:14,260 that parasites are just a ubiquitous aspect of life. 166 00:08:14,260 --> 00:08:19,290 So you can name an organism, and you can pretty much 167 00:08:19,290 --> 00:08:21,560 be guaranteed that there's going to be some notion 168 00:08:21,560 --> 00:08:23,650 of a parasite on that organism. 169 00:08:23,650 --> 00:08:26,860 And there can be multiple layers of this. 170 00:08:26,860 --> 00:08:30,590 So we certainly have many parasites. 171 00:08:30,590 --> 00:08:34,630 We're infected by many viruses and bacteria and other things. 172 00:08:34,630 --> 00:08:37,830 But bacteria-- we think of them as being very small-- 173 00:08:37,830 --> 00:08:42,289 they're also preyed upon by these phage, which 174 00:08:42,289 --> 00:08:46,842 is a parasite that targets specifically bacteria. 175 00:08:46,842 --> 00:08:48,800 So it's not just that it's an incidental thing. 176 00:08:48,800 --> 00:08:52,570 But these are really viruses that have evolved specifically 177 00:08:52,570 --> 00:08:55,920 to divide in bacteria. 178 00:08:55,920 --> 00:08:58,560 And we didn't really talk about this very much. 179 00:08:58,560 --> 00:09:02,890 But one of the classic models for cooperation and cheating 180 00:09:02,890 --> 00:09:05,690 is based on what you could think about as 181 00:09:05,690 --> 00:09:10,120 some sort of parasitic sub-population within phage. 182 00:09:10,120 --> 00:09:12,110 So this is a classic paper by Lin Chow where 183 00:09:12,110 --> 00:09:14,480 he showed that if you evolved phage and bacteria 184 00:09:14,480 --> 00:09:18,290 in a condition where many phage infect a given bacteria, then, 185 00:09:18,290 --> 00:09:22,394 you can evolve what you could think of as cheater 186 00:09:22,394 --> 00:09:23,560 strategies or cheater phage. 187 00:09:23,560 --> 00:09:26,270 Because these are phage that maybe 188 00:09:26,270 --> 00:09:28,680 can't reproduce on their own, but have shorter genomes 189 00:09:28,680 --> 00:09:32,350 and can out-replicate the normal phage. 190 00:09:32,350 --> 00:09:35,550 So if both of these end up in a single bacterial cell, 191 00:09:35,550 --> 00:09:38,560 then these cheater phage can spread 192 00:09:38,560 --> 00:09:40,810 by taking advantage of, say, the replication machinery 193 00:09:40,810 --> 00:09:42,060 from the rest of the phage. 194 00:09:42,060 --> 00:09:44,740 So in some ways, you might call that some sort of DNA 195 00:09:44,740 --> 00:09:46,172 parasite or so. 196 00:09:46,172 --> 00:09:48,630 So there's really parasites in many, many different levels. 197 00:09:53,965 --> 00:09:55,280 AUDIENCE: [INAUDIBLE]. 198 00:09:55,280 --> 00:09:55,905 PROFESSOR: Yes? 199 00:09:55,905 --> 00:09:58,330 AUDIENCE: [INAUDIBLE] ultimately think about is, k, 200 00:09:58,330 --> 00:10:01,260 is this really the number-- 201 00:10:01,260 --> 00:10:04,050 PROFESSOR: Yeah, you know, I would 202 00:10:04,050 --> 00:10:08,470 say that the k is, in some ways, not a very satisfying feature 203 00:10:08,470 --> 00:10:09,120 of this model. 204 00:10:09,120 --> 00:10:12,670 Because it makes it feel that the model is very special, 205 00:10:12,670 --> 00:10:13,170 right? 206 00:10:13,170 --> 00:10:15,265 AUDIENCE: But I mean, in a natural population 207 00:10:15,265 --> 00:10:19,740 and rate at which people are born, is that the same change? 208 00:10:19,740 --> 00:10:21,460 Or-- 209 00:10:21,460 --> 00:10:23,060 PROFESSOR: Yes. 210 00:10:23,060 --> 00:10:25,320 That was the example I was going to give. 211 00:10:25,320 --> 00:10:31,170 It's not clear-- of course, it requires somebody 212 00:10:31,170 --> 00:10:33,370 to give birth to kids, right? 213 00:10:33,370 --> 00:10:36,705 So in that sense, modeling this as a constant number per unit 214 00:10:36,705 --> 00:10:39,660 of time-- which is what we're doing-- this rate k, 215 00:10:39,660 --> 00:10:40,730 is a little bit funny. 216 00:10:40,730 --> 00:10:42,480 Because then, what you'd really want to do 217 00:10:42,480 --> 00:10:44,970 is say, oh, maybe it's these guys that 218 00:10:44,970 --> 00:10:47,130 give birth at some rate or so, if you really 219 00:10:47,130 --> 00:10:48,420 wanted to be accurate. 220 00:10:48,420 --> 00:10:50,470 So I'd say that this is, in some ways, just 221 00:10:50,470 --> 00:10:53,142 a mathematical simplification so that we can get 222 00:10:53,142 --> 00:10:54,350 at the heart of the dynamics. 223 00:10:54,350 --> 00:10:57,020 And what we'll see is that, in these SIR models, 224 00:10:57,020 --> 00:11:00,180 you don't invoke anything like this. 225 00:11:00,180 --> 00:11:03,510 But rather, what you do is you assume that, at some rate, 226 00:11:03,510 --> 00:11:07,430 infected individuals don't just die. 227 00:11:07,430 --> 00:11:09,500 But they become resistant. 228 00:11:09,500 --> 00:11:12,560 And then, maybe later, they become sensitive again. 229 00:11:12,560 --> 00:11:14,956 So you need some way of being sure that it's not 230 00:11:14,956 --> 00:11:19,080 the case that everybody just always dies. 231 00:11:19,080 --> 00:11:21,700 So in some ways, this is more mathematical convenience. 232 00:11:21,700 --> 00:11:23,589 And the basic conclusions end up being 233 00:11:23,589 --> 00:11:25,130 very robust to these sorts of things. 234 00:11:32,420 --> 00:11:34,250 All right, so in these models, it's 235 00:11:34,250 --> 00:11:37,180 always good to be clear about how 236 00:11:37,180 --> 00:11:40,280 we go from this framework to something that is more 237 00:11:40,280 --> 00:11:42,870 of a differential equation. 238 00:11:42,870 --> 00:11:44,500 And what we can do is we can think 239 00:11:44,500 --> 00:11:49,920 about these uninfected individuals, i.e. 240 00:11:49,920 --> 00:11:51,605 the S-es as compared to the infected. 241 00:11:55,060 --> 00:11:57,335 And we're going to have these guys be x. 242 00:12:01,680 --> 00:12:05,040 And this S and I. 243 00:12:05,040 --> 00:12:08,670 So the way that the x will be changing 244 00:12:08,670 --> 00:12:11,135 is that we're assuming that there's always 245 00:12:11,135 --> 00:12:12,510 some influx of individuals, which 246 00:12:12,510 --> 00:12:15,035 could be birth or migration or something else, 247 00:12:15,035 --> 00:12:17,537 that are just always entering. 248 00:12:17,537 --> 00:12:19,120 But then, there's going to be two ways 249 00:12:19,120 --> 00:12:21,670 that x is going to decrease. 250 00:12:21,670 --> 00:12:25,780 One is that there's just a death rate that is resulting 251 00:12:25,780 --> 00:12:27,627 in the absence of infection. 252 00:12:27,627 --> 00:12:29,085 But then, also, there's going to be 253 00:12:29,085 --> 00:12:30,110 some rate of infection, which is going 254 00:12:30,110 --> 00:12:31,470 to be proportional to beta. 255 00:12:31,470 --> 00:12:33,960 So this is the simplest way that you 256 00:12:33,960 --> 00:12:37,890 can imagine capturing this element 257 00:12:37,890 --> 00:12:42,870 that the infected individuals can transmit the infection 258 00:12:42,870 --> 00:12:45,090 to the sensitive individuals. 259 00:12:45,090 --> 00:12:50,290 So we're modeling them as a well-mixed population, 260 00:12:50,290 --> 00:12:51,780 just like in chemical reactions. 261 00:12:51,780 --> 00:12:54,500 And somehow, the rate of infection 262 00:12:54,500 --> 00:12:58,080 is proportional to the frequency that they hit each other. 263 00:12:58,080 --> 00:13:02,810 Certainly, the simplest kind of model you can imagine. 264 00:13:02,810 --> 00:13:06,250 Whereas, the infected individuals-- well, 265 00:13:06,250 --> 00:13:09,530 we're going to have an increased rate of death. 266 00:13:09,530 --> 00:13:12,180 So this is a simplified way to write it. 267 00:13:17,120 --> 00:13:20,230 So this is really that there's a minus, u plus v times y. 268 00:13:20,230 --> 00:13:23,130 So this is just the death rate. 269 00:13:23,130 --> 00:13:27,530 But then, any individual that leaves the sensitive class-- 270 00:13:27,530 --> 00:13:31,460 this minus beta xy-- enters the infected class. 271 00:13:36,120 --> 00:13:37,860 This makes a lot of sense, I think. 272 00:13:37,860 --> 00:13:42,310 Now, the question is, can we make sense of what's going on? 273 00:13:42,310 --> 00:13:49,215 Now, you saw in your reading what this R0 parameter was. 274 00:13:49,215 --> 00:13:50,960 And you should always remember that it's 275 00:13:50,960 --> 00:13:54,970 defined as this thing of, if you introduce 276 00:13:54,970 --> 00:13:57,460 one infected individual into a population 277 00:13:57,460 --> 00:13:59,540 of sensitive individuals, what is 278 00:13:59,540 --> 00:14:03,430 the mean number of new infections that you get? 279 00:14:03,430 --> 00:14:07,730 And it makes sense that the key thing is whether that R0 280 00:14:07,730 --> 00:14:09,026 is greater or less than 1. 281 00:14:09,026 --> 00:14:10,400 Because if it's greater than one, 282 00:14:10,400 --> 00:14:13,204 that leads to this exponential explosion 283 00:14:13,204 --> 00:14:14,370 of the infected individuals. 284 00:14:14,370 --> 00:14:17,500 It doesn't mean that everyone's going to die, necessarily. 285 00:14:17,500 --> 00:14:18,580 We'll get into that. 286 00:14:18,580 --> 00:14:20,760 But if R0 is less than 1, then you 287 00:14:20,760 --> 00:14:23,710 expect that infection to die out. 288 00:14:23,710 --> 00:14:27,380 So why is it that if R0 is greater than 1, 289 00:14:27,380 --> 00:14:29,130 and you introduce one infected individual, 290 00:14:29,130 --> 00:14:33,149 it doesn't necessarily lead to a wipe-out 291 00:14:33,149 --> 00:14:34,190 of the entire population? 292 00:14:42,860 --> 00:14:44,822 Or maybe it does. 293 00:14:44,822 --> 00:14:46,030 This model is a little funny. 294 00:14:46,030 --> 00:14:47,988 Because you always have an individual entering. 295 00:14:51,581 --> 00:14:52,080 But-- 296 00:14:52,080 --> 00:14:54,415 AUDIENCE: If the [INAUDIBLE] is really virulent, 297 00:14:54,415 --> 00:14:56,872 then only infected individuals die before-- 298 00:14:56,872 --> 00:14:57,580 PROFESSOR: Right. 299 00:14:57,580 --> 00:14:59,984 So if it's very virulent, then the infected individuals 300 00:14:59,984 --> 00:15:00,650 may die quickly. 301 00:15:00,650 --> 00:15:01,930 And this gets into this question of there 302 00:15:01,930 --> 00:15:03,804 may be some trade-offs in terms of virulence. 303 00:15:03,804 --> 00:15:07,190 And we'll talk more about, then, the evolution of virulence. 304 00:15:07,190 --> 00:15:12,820 But there's a wide variety of classes of models, 305 00:15:12,820 --> 00:15:16,020 not just the ones where you have k entering. 306 00:15:16,020 --> 00:15:20,461 But the question somehow is-- just because R0 307 00:15:20,461 --> 00:15:21,960 is greater than 1, that doesn't mean 308 00:15:21,960 --> 00:15:25,667 that the entire population will necessarily become infected. 309 00:15:25,667 --> 00:15:27,750 Because we have this idea of an exponential growth 310 00:15:27,750 --> 00:15:32,740 of the infected population if R0 is greater than 1. 311 00:15:32,740 --> 00:15:35,330 So why is it that it's not necessarily 312 00:15:35,330 --> 00:15:40,780 going to happen that the entire population is-- well, 313 00:15:40,780 --> 00:15:43,411 this question's a little bit ill-posed in this model. 314 00:15:43,411 --> 00:15:47,259 AUDIENCE: Is it because the number of sensitive individuals 315 00:15:47,259 --> 00:15:49,320 becomes very low at some point and-- 316 00:15:49,320 --> 00:15:50,320 PROFESSOR: That's right. 317 00:15:50,320 --> 00:15:52,010 And I think this is the basic intuition. 318 00:15:52,010 --> 00:15:56,440 As more and more members of the population become infected, 319 00:15:56,440 --> 00:15:58,610 then that could, in principle, reduce 320 00:15:58,610 --> 00:16:04,760 to the possibilities for new individuals to be susceptible. 321 00:16:09,770 --> 00:16:15,310 And I'm using susceptible and sensitive interchangeably. 322 00:16:15,310 --> 00:16:18,990 And so eventually, this exponential growth 323 00:16:18,990 --> 00:16:20,951 of the population can be limited in some way. 324 00:16:20,951 --> 00:16:22,700 We'll maybe look at this a little bit more 325 00:16:22,700 --> 00:16:23,960 in this SIR model. 326 00:16:23,960 --> 00:16:25,636 Because it's more clear. 327 00:16:25,636 --> 00:16:29,890 AUDIENCE: So we basically collect a rate. 328 00:16:29,890 --> 00:16:31,404 R0 is a rate, right? 329 00:16:34,258 --> 00:16:34,758 [INAUDIBLE] 330 00:16:34,758 --> 00:16:36,240 PROFESSOR: No. 331 00:16:36,240 --> 00:16:38,440 It's a number. 332 00:16:38,440 --> 00:16:40,894 It's the expected number of new infections 333 00:16:40,894 --> 00:16:43,185 that you get when you introduce one infected individual 334 00:16:43,185 --> 00:16:45,681 into the population. 335 00:16:45,681 --> 00:16:46,180 And we're-- 336 00:16:46,180 --> 00:16:47,801 AUDIENCE: It's like, period. 337 00:16:47,801 --> 00:16:51,730 So there's not, like, per-unit time or-- 338 00:16:51,730 --> 00:16:55,030 PROFESSOR: It's a number, period. 339 00:16:55,030 --> 00:16:57,940 AUDIENCE: OK, so if you have R equal to 1, 340 00:16:57,940 --> 00:16:59,386 you expect that when you introduce 341 00:16:59,386 --> 00:17:01,073 an infected individual into a population 342 00:17:01,073 --> 00:17:04,051 of sensitive individuals, you will get one other infected 343 00:17:04,051 --> 00:17:04,550 individual. 344 00:17:04,550 --> 00:17:04,710 PROFESSOR: That's right. 345 00:17:04,710 --> 00:17:07,240 So that's kind of the neutrally-stable situation. 346 00:17:07,240 --> 00:17:09,516 If R0 is 1, then you add one infected individual 347 00:17:09,516 --> 00:17:12,099 and you expect to get one other one, so you have a random walk 348 00:17:12,099 --> 00:17:14,970 and-- well, in general, it will randomly 349 00:17:14,970 --> 00:17:16,221 go extinct, eventually. 350 00:17:16,221 --> 00:17:16,720 Yes? 351 00:17:16,720 --> 00:17:19,672 AUDIENCE: What is the lifetime of the infected-- 352 00:17:19,672 --> 00:17:21,530 PROFESSOR: So it depends. 353 00:17:21,530 --> 00:17:23,560 And so, that's what we're going to do right now 354 00:17:23,560 --> 00:17:27,339 is see if we can reconstruct what this R0 is equal to 355 00:17:27,339 --> 00:17:28,860 in this model. 356 00:17:28,860 --> 00:17:30,720 AUDIENCE: Another quick question-- 357 00:17:30,720 --> 00:17:34,060 what's the distribution of the-- so-- 358 00:17:34,060 --> 00:17:34,690 PROFESSOR: Yes. 359 00:17:34,690 --> 00:17:35,340 AUDIENCE: --a deterministic role-- 360 00:17:35,340 --> 00:17:36,360 PROFESSOR: Yes, exactly. 361 00:17:36,360 --> 00:17:38,317 So this is very interesting. 362 00:17:38,317 --> 00:17:40,400 The question is, what is going to the distribution 363 00:17:40,400 --> 00:17:44,160 of the number of infected individuals? 364 00:17:44,160 --> 00:17:45,920 And in this model, we're assuming 365 00:17:45,920 --> 00:17:49,690 that every infected individuals is the same. 366 00:17:49,690 --> 00:17:51,590 So we should be able to-- I wish that we 367 00:17:51,590 --> 00:17:54,580 just, on the wall somewhere, had our five different standard 368 00:17:54,580 --> 00:17:56,860 probability distributions, so that we could always 369 00:17:56,860 --> 00:17:58,980 go back to them. 370 00:17:58,980 --> 00:18:01,180 So the question is, in this model, 371 00:18:01,180 --> 00:18:04,486 if you introduce one infected individual in the population, 372 00:18:04,486 --> 00:18:06,360 how many new infected individuals do you get? 373 00:18:06,360 --> 00:18:08,800 R0 tells you about the mean. 374 00:18:08,800 --> 00:18:11,270 But will you always get the mean? 375 00:18:11,270 --> 00:18:13,270 No. 376 00:18:13,270 --> 00:18:15,160 So let's all think about this for 10 seconds. 377 00:18:15,160 --> 00:18:17,500 And we will verbally yell out what 378 00:18:17,500 --> 00:18:19,250 we think the distribution will be 379 00:18:19,250 --> 00:18:21,610 of the number of new infections from a single infected 380 00:18:21,610 --> 00:18:22,110 individual. 381 00:18:24,755 --> 00:18:25,255 OK? 382 00:18:31,960 --> 00:18:32,870 Verbally, ready! 383 00:18:32,870 --> 00:18:34,146 Five, (WHISPERING) four-- 384 00:18:36,730 --> 00:18:37,480 AUDIENCE: Poisson. 385 00:18:37,480 --> 00:18:39,722 AUDIENCE: Piosson. 386 00:18:39,722 --> 00:18:41,384 AUDIENCE: Exponential 387 00:18:41,384 --> 00:18:43,550 PROFESSOR: All right, everybody thinks it's Poisson. 388 00:18:43,550 --> 00:18:44,549 Why would it be Poisson? 389 00:18:47,670 --> 00:18:50,539 AUDIENCE: Because you have a rate-- 390 00:18:50,539 --> 00:18:51,622 AUDIENCE: It's not a rate. 391 00:18:51,622 --> 00:18:53,598 AUDIENCE: It's not? 392 00:18:53,598 --> 00:18:55,862 AUDIENCE: The initial population is much larger-- 393 00:18:55,862 --> 00:18:56,570 PROFESSOR: Right. 394 00:18:56,570 --> 00:18:58,510 OK, so the idea is that we imagine 395 00:18:58,510 --> 00:19:00,490 we're some infected individual. 396 00:19:00,490 --> 00:19:04,770 And there's some rate that we are infecting others. 397 00:19:04,770 --> 00:19:09,110 Now, if I ask you the question, how many individuals will I 398 00:19:09,110 --> 00:19:13,990 infect in the next 10 days? 399 00:19:13,990 --> 00:19:17,030 Or we could do 21 days if you guys like that. 400 00:19:17,030 --> 00:19:19,760 So all right, if I ask, how many do I 401 00:19:19,760 --> 00:19:23,010 infect in the next 10 days? 402 00:19:23,010 --> 00:19:24,680 Now, I'm assuming that I stay alive. 403 00:19:24,680 --> 00:19:26,349 But let's say, assuming I stay alive, 404 00:19:26,349 --> 00:19:28,140 how many do I infect over the next 10 days? 405 00:19:28,140 --> 00:19:33,750 That's going to be distributed as-- that is Poisson. 406 00:19:33,750 --> 00:19:36,280 But the question you're asking is a different one. 407 00:19:36,280 --> 00:19:39,440 You're asking, what is going to be 408 00:19:39,440 --> 00:19:41,690 the distribution of the total number of new infections 409 00:19:41,690 --> 00:19:43,710 that I cause? 410 00:19:43,710 --> 00:19:48,130 And this is precisely the same situation 411 00:19:48,130 --> 00:19:50,930 that we've analyzed lots and lots of times. 412 00:19:50,930 --> 00:19:52,155 What does it look like? 413 00:19:52,155 --> 00:19:53,580 AUDIENCE: Geometric distribution. 414 00:19:53,580 --> 00:19:54,205 PROFESSOR: Hmm? 415 00:19:54,205 --> 00:19:55,480 AUDIENCE: Geometric. 416 00:19:55,480 --> 00:19:56,750 PROFESSOR: OK, yes. 417 00:19:56,750 --> 00:19:57,910 It's going to be a geometric distribution. 418 00:19:57,910 --> 00:19:58,470 But why? 419 00:19:58,470 --> 00:20:01,890 AUDIENCE: Well, because you can have an infection and then 420 00:20:01,890 --> 00:20:02,640 another infection. 421 00:20:02,640 --> 00:20:04,050 And so then, it's, like, multiples of ten. 422 00:20:04,050 --> 00:20:05,050 PROFESSOR: That's right. 423 00:20:05,050 --> 00:20:08,530 So we have an infected individual. 424 00:20:08,530 --> 00:20:10,120 There's two things that can happen. 425 00:20:12,930 --> 00:20:14,600 He's going to die at some rate. 426 00:20:14,600 --> 00:20:21,560 And there's this other rate, which 427 00:20:21,560 --> 00:20:25,367 is going to go as beta times x or so, 428 00:20:25,367 --> 00:20:27,200 telling us about the rate of new infections. 429 00:20:27,200 --> 00:20:28,908 And we want to know, how many times do we 430 00:20:28,908 --> 00:20:34,310 go around this loop before we degrade, or die, or something, 431 00:20:34,310 --> 00:20:35,849 disappear from the population? 432 00:20:35,849 --> 00:20:37,140 Does this look at all familiar? 433 00:20:37,140 --> 00:20:38,170 AUDIENCE: Mm-hmm. 434 00:20:38,170 --> 00:20:38,460 PROFESSOR: All right. 435 00:20:38,460 --> 00:20:40,145 Were you guys the same students that 436 00:20:40,145 --> 00:20:41,345 were here for the first half of the class? 437 00:20:41,345 --> 00:20:42,240 AUDIENCE: Ha. 438 00:20:42,240 --> 00:20:43,038 PROFESSOR: Yeah? 439 00:20:43,038 --> 00:20:43,538 No. 440 00:20:43,538 --> 00:20:47,031 AUDIENCE: And so, R0 is like the [INAUDIBLE]? 441 00:20:47,031 --> 00:20:52,021 It's like the number of-- 442 00:20:52,021 --> 00:20:54,017 AUDIENCE: It's the mean number for bursts. 443 00:20:54,017 --> 00:20:56,530 AUDIENCE: You haven't evaluated when the population-- 444 00:20:56,530 --> 00:21:00,342 PROFESSOR: So R0 is the mean size of protein bursts, 445 00:21:00,342 --> 00:21:02,050 in the context of this other model, which 446 00:21:02,050 --> 00:21:03,258 was-- what was the situation? 447 00:21:05,471 --> 00:21:07,720 And this one's a really good one for you guys to know. 448 00:21:07,720 --> 00:21:08,719 It's going to be useful. 449 00:21:13,230 --> 00:21:17,970 So we saw this exact model in the context of gene expression. 450 00:21:17,970 --> 00:21:20,381 And what was the situation that we-- 451 00:21:20,381 --> 00:21:21,880 AUDIENCE: Production of [INAUDIBLE]. 452 00:21:21,880 --> 00:21:22,877 PROFESSOR: Production of-- 453 00:21:22,877 --> 00:21:24,160 AUDIENCE: --proteins from a single mRNA. 454 00:21:24,160 --> 00:21:26,610 PROFESSOR: Yeah, the production of proteins from a single mRNA. 455 00:21:26,610 --> 00:21:26,840 Right? 456 00:21:26,840 --> 00:21:29,130 Because remember, we had this thing where we had the mRNA. 457 00:21:29,130 --> 00:21:30,850 And we said, oh, well the mRNA is going 458 00:21:30,850 --> 00:21:32,450 to be degraded at some rate. 459 00:21:32,450 --> 00:21:35,732 But also, it's going to be translated at some rate. 460 00:21:35,732 --> 00:21:37,190 So the distribution number of times 461 00:21:37,190 --> 00:21:39,149 that it's translated before it degrades 462 00:21:39,149 --> 00:21:40,190 is going to be geometric. 463 00:21:40,190 --> 00:21:42,500 Because we go around this loop some number of times. 464 00:21:42,500 --> 00:21:46,050 All right, so this is the same thing. 465 00:21:46,050 --> 00:21:52,150 And the paper that I put as supplementary reading, 466 00:21:52,150 --> 00:21:54,870 by Jamie Lloyd-Smith? 467 00:21:54,870 --> 00:21:57,200 I always get his name and Jamie Lloyd Wright-- right? 468 00:21:57,200 --> 00:21:58,075 Something-- mixed up. 469 00:21:58,075 --> 00:21:59,241 But yeah, Jamie Lloyd-Smith. 470 00:21:59,241 --> 00:21:59,770 Smith. 471 00:21:59,770 --> 00:22:05,050 So he was studying the dynamics of infections 472 00:22:05,050 --> 00:22:07,110 when you have this thing where there's 473 00:22:07,110 --> 00:22:11,100 intrinsic variation in, say, the infectivity of an individual. 474 00:22:11,100 --> 00:22:13,380 Because here, you get this geometric distribution, 475 00:22:13,380 --> 00:22:17,010 even though all the individuals are, in principal, identical. 476 00:22:17,010 --> 00:22:20,140 Now, the question is, if there's some distributions of, say, 477 00:22:20,140 --> 00:22:23,650 infectivities, then you'll get an even broader 478 00:22:23,650 --> 00:22:26,585 distribution of resulting number of new infections. 479 00:22:29,470 --> 00:22:31,670 So there's this classic thing of Typhoid Mary. 480 00:22:34,390 --> 00:22:38,346 She's was a nurse-- OK, now I don't remember the story. 481 00:22:38,346 --> 00:22:38,846 Yeah? 482 00:22:38,846 --> 00:22:39,840 AUDIENCE: She was a cook. 483 00:22:39,840 --> 00:22:40,589 PROFESSOR: A cook. 484 00:22:40,589 --> 00:22:41,400 Oh, a cook, nurse-- 485 00:22:41,400 --> 00:22:43,733 AUDIENCE: [INAUDIBLE] so she cooked for a lot of people. 486 00:22:43,733 --> 00:22:46,120 PROFESSOR: OK, so she was somehow resistant to typhoid. 487 00:22:46,120 --> 00:22:48,852 But then, she was cooking for other people and, so then, 488 00:22:48,852 --> 00:22:50,060 caused a bunch of infections. 489 00:22:50,060 --> 00:22:50,560 Is that--? 490 00:22:50,560 --> 00:22:51,366 OK. 491 00:22:51,366 --> 00:22:52,740 Yeah, so this would be an example 492 00:22:52,740 --> 00:22:54,810 of a very infective individual that's 493 00:22:54,810 --> 00:22:58,590 beyond the assumptions in this model. 494 00:22:58,590 --> 00:23:00,800 And as you can imagine, if you have variations 495 00:23:00,800 --> 00:23:04,490 in this infectivity, then what it does, for a given R0-- 496 00:23:04,490 --> 00:23:07,080 so if you fix R0, then you have a broader distribution 497 00:23:07,080 --> 00:23:07,720 of infectivity. 498 00:23:07,720 --> 00:23:11,270 What it means is that a larger fraction of the infections 499 00:23:11,270 --> 00:23:12,310 will go extinct. 500 00:23:12,310 --> 00:23:16,826 But those that get going will be explosive. 501 00:23:16,826 --> 00:23:18,700 If you're curious about these sorts of ideas, 502 00:23:18,700 --> 00:23:21,380 you should look at this optional reading paper 503 00:23:21,380 --> 00:23:24,697 that I put out there. 504 00:23:24,697 --> 00:23:26,280 All right, so I just want to be clear. 505 00:23:26,280 --> 00:23:33,221 This is geometric number of new infections, distribution 506 00:23:33,221 --> 00:23:33,970 of new infections. 507 00:23:39,550 --> 00:23:40,050 Yes? 508 00:23:40,050 --> 00:23:43,410 AUDIENCE: So is this just one cycle? 509 00:23:43,410 --> 00:23:45,350 Like, one-- 510 00:23:45,350 --> 00:23:49,320 PROFESSOR: So we're talking about the number of infections 511 00:23:49,320 --> 00:23:51,915 that result when you just add one infected individual 512 00:23:51,915 --> 00:23:52,685 to the population. 513 00:23:52,685 --> 00:23:55,222 AUDIENCE: OK, so then, you don't think about, afterwards, 514 00:23:55,222 --> 00:23:56,972 what happens to those infected individuals 515 00:23:56,972 --> 00:23:58,064 and if they infect-- 516 00:23:58,064 --> 00:24:00,230 PROFESSOR: Well, we are not yet thinking about them. 517 00:24:00,230 --> 00:24:03,460 Although, in this case, those are also geometrically 518 00:24:03,460 --> 00:24:04,260 distributed. 519 00:24:04,260 --> 00:24:06,690 But what you expect is that the mean of those things 520 00:24:06,690 --> 00:24:07,620 will change. 521 00:24:07,620 --> 00:24:10,970 Because the number of susceptible or whatnot 522 00:24:10,970 --> 00:24:14,300 individuals-- that's going to change. 523 00:24:14,300 --> 00:24:17,200 So the mean number is going to change. 524 00:24:17,200 --> 00:24:19,381 But the distributions will still be geometric. 525 00:24:19,381 --> 00:24:21,786 AUDIENCE: But in total, that won't be geometric anymore. 526 00:24:21,786 --> 00:24:23,619 Because if we're looking at the total number 527 00:24:23,619 --> 00:24:27,077 of infected individuals after learning about the geometric-- 528 00:24:27,077 --> 00:24:27,660 PROFESSOR: No. 529 00:24:27,660 --> 00:24:31,100 So just because each of these sub-steps is geometric 530 00:24:31,100 --> 00:24:34,910 does not mean that you end up with a geometric distribution. 531 00:24:34,910 --> 00:24:40,977 Indeed, let's say that i put in 20 infected individuals 532 00:24:40,977 --> 00:24:41,810 into the population. 533 00:24:41,810 --> 00:24:43,990 And I ask, what's going to be the distribution 534 00:24:43,990 --> 00:24:46,750 of the number of infections caused immediately 535 00:24:46,750 --> 00:24:47,770 from those 20? 536 00:24:47,770 --> 00:24:49,030 That's going to be what? 537 00:24:49,030 --> 00:24:51,010 [INTERPOSING VOICES] 538 00:24:51,010 --> 00:24:54,020 Yeah, and for 20, it's going to be basically Gaussian. 539 00:24:54,020 --> 00:24:58,414 OK, well if I said 100, well definitely Gaussian. 540 00:24:58,414 --> 00:25:00,330 It's a gamma distribution that looks very much 541 00:25:00,330 --> 00:25:04,062 look a Gaussian, in that case. 542 00:25:04,062 --> 00:25:05,837 All right? 543 00:25:05,837 --> 00:25:09,240 All right. 544 00:25:09,240 --> 00:25:11,610 So we've been talking about the definition of this R0. 545 00:25:11,610 --> 00:25:15,495 But of course, we should figure out what it is. 546 00:25:15,495 --> 00:25:18,900 We want R0 is equal to-- and what 547 00:25:18,900 --> 00:25:26,262 I'm going to tell you is that there's a 1 over u plus v. 548 00:25:26,262 --> 00:25:27,720 But then, there's some other terms. 549 00:25:27,720 --> 00:25:30,330 And I've unfortunately lost my notes. 550 00:25:30,330 --> 00:25:33,479 So you guys are going to have to help me figure this out. 551 00:25:33,479 --> 00:25:35,020 And what you're going to do is you're 552 00:25:35,020 --> 00:25:36,603 going to take advantage of your cards. 553 00:25:36,603 --> 00:25:38,390 And again, put things in the numerators 554 00:25:38,390 --> 00:25:39,884 and the denominators, corresponding 555 00:25:39,884 --> 00:25:41,800 to how I'm supposed to fill out this equation. 556 00:25:44,039 --> 00:25:46,580 You can start thinking about it while I give you the options. 557 00:26:01,800 --> 00:26:02,820 OK. 558 00:26:02,820 --> 00:26:05,809 So I guess you could recapitulate this 559 00:26:05,809 --> 00:26:07,600 by just putting B and C, although maybe you 560 00:26:07,600 --> 00:26:09,772 need it more than once. 561 00:26:09,772 --> 00:26:10,480 Do what you will. 562 00:26:10,480 --> 00:26:13,404 Do you understand the question? 563 00:26:13,404 --> 00:26:14,820 There's going to be something else 564 00:26:14,820 --> 00:26:15,790 I'm going to put right here. 565 00:26:15,790 --> 00:26:16,756 And I want to know-- there are going 566 00:26:16,756 --> 00:26:18,290 to be somethings in the numerator, somethings 567 00:26:18,290 --> 00:26:19,081 in the denominator. 568 00:26:22,120 --> 00:26:24,380 I'm going to give you 30 seconds to think about it. 569 00:26:24,380 --> 00:26:26,714 Because it's important to be able to reason 570 00:26:26,714 --> 00:26:27,630 your way through this. 571 00:27:13,786 --> 00:27:16,136 All right, do need more time? 572 00:27:16,136 --> 00:27:18,080 Yep, OK. 573 00:27:18,080 --> 00:27:19,680 I'll give you another 15 seconds. 574 00:27:46,021 --> 00:27:48,420 All right, let's go ahead and vote. 575 00:27:48,420 --> 00:27:49,370 Ready? 576 00:27:49,370 --> 00:27:54,040 Three, two, one. 577 00:27:54,040 --> 00:27:55,320 All right. 578 00:27:55,320 --> 00:27:55,820 I like it! 579 00:27:55,820 --> 00:27:58,064 We're really looking quite nice. 580 00:27:58,064 --> 00:27:59,480 So there's a claim that it's going 581 00:27:59,480 --> 00:28:08,336 to be AD over B. We should be writing a beta k over u. 582 00:28:08,336 --> 00:28:10,770 All right. 583 00:28:10,770 --> 00:28:12,880 Can somebody explain how they got there? 584 00:28:21,710 --> 00:28:22,957 Yes, please. 585 00:28:22,957 --> 00:28:29,747 AUDIENCE: Well, it's going to be-- these two-- 586 00:28:29,747 --> 00:28:30,580 PROFESSOR: OK, yeah. 587 00:28:30,580 --> 00:28:31,640 This helped, right? 588 00:28:31,640 --> 00:28:34,390 OK, good, perfect. 589 00:28:34,390 --> 00:28:38,392 So this thing is what? 590 00:28:38,392 --> 00:28:41,242 What is this term here? 591 00:28:41,242 --> 00:28:43,034 AUDIENCE: That's the death rate. 592 00:28:43,034 --> 00:28:43,830 PROFESSOR: Right. 593 00:28:43,830 --> 00:28:48,095 Which means that the one over it is the expected lifetime 594 00:28:48,095 --> 00:28:49,970 of an infected individual. 595 00:28:49,970 --> 00:28:53,360 So the definition of R0 is, you put an infected individual 596 00:28:53,360 --> 00:28:57,027 into a population of susceptibles. 597 00:28:57,027 --> 00:28:58,610 Now, we want to know, OK, well there's 598 00:28:58,610 --> 00:29:01,860 a expected lifetime of this infected individual, which 599 00:29:01,860 --> 00:29:02,915 is given by this. 600 00:29:02,915 --> 00:29:04,540 And then, we have to think about, well, 601 00:29:04,540 --> 00:29:07,081 what's the rate that we're going to be infecting individuals? 602 00:29:07,081 --> 00:29:10,850 And that's going to be beta times x. 603 00:29:10,850 --> 00:29:15,590 But what we want to know is, is x before we add any infection? 604 00:29:15,590 --> 00:29:19,251 And without any infection, then we just have a rate of entry 605 00:29:19,251 --> 00:29:20,250 and, then, a death rate. 606 00:29:20,250 --> 00:29:22,067 So it's just k over u. 607 00:29:22,067 --> 00:29:22,567 OK? 608 00:29:25,850 --> 00:29:28,960 Now, the key thing in all of these epidemiological models 609 00:29:28,960 --> 00:29:33,190 is whether this R0 is greater or less than 1. 610 00:29:33,190 --> 00:29:36,340 And that's going to tell us whether the disease becomes 611 00:29:36,340 --> 00:29:40,920 endemic or not, whether, at steady state, 612 00:29:40,920 --> 00:29:46,140 we have a population of infected individuals. 613 00:29:46,140 --> 00:29:52,060 So R0, greater than one, means it's in an endemic population. 614 00:30:05,530 --> 00:30:09,884 Now, in this model, we can then ask-- 615 00:30:09,884 --> 00:30:12,060 [LAUGHTER] 616 00:30:13,551 --> 00:30:16,036 AUDIENCE: That really is really funny. 617 00:30:16,036 --> 00:30:17,530 AUDIENCE: That is the hard part. 618 00:30:17,530 --> 00:30:18,405 PROFESSOR: All right. 619 00:30:18,405 --> 00:30:22,300 So do you like A, donuts, B, cupcakes, or C, carrots? 620 00:30:22,300 --> 00:30:24,074 [INTERPOSING VOICES] 621 00:30:24,074 --> 00:30:24,990 Yeah, it's quite nice. 622 00:30:24,990 --> 00:30:28,130 Does anybody have any notion of why somebody might 623 00:30:28,130 --> 00:30:30,570 have-- so Sam likes cupcakes. 624 00:30:30,570 --> 00:30:32,000 That's good to know. 625 00:30:32,000 --> 00:30:36,130 Yeah, they're so pretty that I actually feel bad erasing it. 626 00:30:36,130 --> 00:30:37,661 OK, well, we'll leave it up there -- 627 00:30:37,661 --> 00:30:38,160 [LAUGHTER] 628 00:30:38,160 --> 00:30:41,090 --for a little bit longer. 629 00:30:41,090 --> 00:30:43,240 Everybody can just smile, because they know 630 00:30:43,240 --> 00:30:45,348 that the drawing is back there. 631 00:30:45,348 --> 00:30:46,782 [INTERPOSING VOICES] 632 00:30:47,740 --> 00:30:49,530 All right. 633 00:30:49,530 --> 00:30:54,880 So in this model, the fact that R0 634 00:30:54,880 --> 00:30:56,520 is this parameter that tells us about 635 00:30:56,520 --> 00:30:58,894 whether there's going to be an epidemic and then, indeed, 636 00:30:58,894 --> 00:31:01,640 whether later the disease will be endemic, 637 00:31:01,640 --> 00:31:04,240 does that already tell us that R0 is 638 00:31:04,240 --> 00:31:05,785 what's maximized by selection? 639 00:31:08,980 --> 00:31:10,231 No. 640 00:31:10,231 --> 00:31:10,730 Right? 641 00:31:10,730 --> 00:31:14,020 What was the key thing that Martin does in the chapter 642 00:31:14,020 --> 00:31:16,260 in order to try to understand something about what 643 00:31:16,260 --> 00:31:17,520 strain is selected for? 644 00:31:36,430 --> 00:31:37,140 Yes? 645 00:31:37,140 --> 00:31:42,475 AUDIENCE: Are you thinking of when he introduces [INAUDIBLE]? 646 00:31:42,475 --> 00:31:45,020 PROFESSOR: Right. 647 00:31:45,020 --> 00:31:48,160 Yes, although, the part of the chapter your thinking about is, 648 00:31:48,160 --> 00:31:49,582 I think, the second half. 649 00:31:49,582 --> 00:31:51,040 It's talking about super-infection, 650 00:31:51,040 --> 00:31:53,331 where there's all these different types, and craziness, 651 00:31:53,331 --> 00:31:54,110 and so forth. 652 00:31:54,110 --> 00:31:56,800 But the initial insight about what 653 00:31:56,800 --> 00:31:58,580 is going to be selected for comes 654 00:31:58,580 --> 00:32:00,688 from a simpler model than that. 655 00:32:04,680 --> 00:32:07,340 So you don't have to think about all those many parasites 656 00:32:07,340 --> 00:32:09,910 and those triangles and all the craziness. 657 00:32:09,910 --> 00:32:12,228 Instead, there was a simpler model that-- yeah? 658 00:32:12,228 --> 00:32:14,618 AUDIENCE: If you have multiple parasites, 659 00:32:14,618 --> 00:32:18,560 then study states only that one parasite exists. 660 00:32:18,560 --> 00:32:19,560 PROFESSOR: That's right. 661 00:32:19,560 --> 00:32:22,240 And so, what he does is he just writes down 662 00:32:22,240 --> 00:32:28,180 this model of where he now allows two different parasites 663 00:32:28,180 --> 00:32:29,860 to be spreading the population. 664 00:32:29,860 --> 00:32:32,380 And at the beginning-- well, we'll write it down. 665 00:32:32,380 --> 00:32:33,890 And we want to be clear. 666 00:32:33,890 --> 00:32:36,080 There's a very important assumption that he makes. 667 00:32:36,080 --> 00:32:42,400 What he's going to find is that selection maximizes R0, 668 00:32:42,400 --> 00:32:43,988 the basic reproductive ratio. 669 00:32:47,749 --> 00:32:49,165 And it's going to be in this model 670 00:32:49,165 --> 00:32:50,560 that I'm writing down now. 671 00:32:50,560 --> 00:32:52,530 So there's this x dot. 672 00:32:52,530 --> 00:32:53,950 And things look very similar. 673 00:33:17,806 --> 00:33:18,350 All right. 674 00:33:18,350 --> 00:33:20,349 Now, the question is, what is the key assumption 675 00:33:20,349 --> 00:33:22,580 that we're making in writing down these equations? 676 00:33:36,606 --> 00:33:37,105 Thought? 677 00:33:40,869 --> 00:33:42,910 AUDIENCE: One cannot have more than one type of-- 678 00:33:42,910 --> 00:33:43,909 PROFESSOR: That's right. 679 00:33:43,909 --> 00:33:47,710 A host cannot be infected by more than one type of strain. 680 00:33:47,710 --> 00:33:49,830 So that's very, very important. 681 00:33:49,830 --> 00:33:52,514 So there's no super-infection, as they say. 682 00:33:59,554 --> 00:34:00,970 And depending on the disease, this 683 00:34:00,970 --> 00:34:04,170 could either be a better or worse assumption. 684 00:34:04,170 --> 00:34:07,330 But then, what is it that is defining these two 685 00:34:07,330 --> 00:34:08,150 strains then? 686 00:34:08,150 --> 00:34:09,483 In what ways are they different? 687 00:34:13,919 --> 00:34:15,889 AUDIENCE: In virulence. 688 00:34:15,889 --> 00:34:19,489 PROFESSOR: Their virulence is different. 689 00:34:19,489 --> 00:34:21,430 And what is virulence again? 690 00:34:21,430 --> 00:34:23,514 AUDIENCE: How likely it is that it could kill you. 691 00:34:23,514 --> 00:34:24,513 PROFESSOR: That's right. 692 00:34:24,513 --> 00:34:26,159 It's the additional mortality that 693 00:34:26,159 --> 00:34:29,717 is caused by being affected by that strain. 694 00:34:29,717 --> 00:34:31,550 Is that the only way that they're different? 695 00:34:31,550 --> 00:34:32,092 AUDIENCE: No. 696 00:34:32,092 --> 00:34:32,675 PROFESSOR: No. 697 00:34:32,675 --> 00:34:33,683 What else is different? 698 00:34:33,683 --> 00:34:34,116 AUDIENCE: The infectivity. 699 00:34:34,116 --> 00:34:35,600 PROFESSOR: The infectivity, right. 700 00:34:35,600 --> 00:34:38,850 So the betas are also different. 701 00:34:38,850 --> 00:34:41,190 And it's imported to note that in this model-- 702 00:34:41,190 --> 00:34:42,750 it's a very simple model-- but we're 703 00:34:42,750 --> 00:34:46,360 allowing the strains to be different in these two ways. 704 00:34:46,360 --> 00:34:49,090 And I think that it's very intuitive 705 00:34:49,090 --> 00:34:53,080 to just say that, oh, well, you want to have a larger beta, 706 00:34:53,080 --> 00:34:55,000 all other things equal. 707 00:34:55,000 --> 00:34:57,650 Because you would like to spread. 708 00:34:57,650 --> 00:35:00,480 But I'd say, maybe, it's not as obvious what 709 00:35:00,480 --> 00:35:02,160 happens in terms of virulence. 710 00:35:02,160 --> 00:35:03,730 And then, of course, if you think 711 00:35:03,730 --> 00:35:07,200 about these two parameters-- in any biological context 712 00:35:07,200 --> 00:35:08,510 they may be coupled. 713 00:35:08,510 --> 00:35:13,176 In which case, things are more subtle. 714 00:35:13,176 --> 00:35:18,016 AUDIENCE: So how reasonable is something of something 715 00:35:18,016 --> 00:35:20,440 of [INAUDIBLE] on the strain? 716 00:35:20,440 --> 00:35:22,972 PROFESSOR: Yeah, I think that this depends on the disease. 717 00:35:22,972 --> 00:35:23,888 AUDIENCE: [INAUDIBLE]. 718 00:35:23,888 --> 00:35:27,372 Is it that you're trading virulence and new response 719 00:35:27,372 --> 00:35:27,872 and-- 720 00:35:27,872 --> 00:35:31,740 PROFESSOR: Right, so that's somehow the argument. 721 00:35:31,740 --> 00:35:38,510 And I'd say that, depending on whether you're 722 00:35:38,510 --> 00:35:42,380 thinking about the host at the level of an organism or a cell, 723 00:35:42,380 --> 00:35:45,920 then this would correspond to very different worlds. 724 00:35:45,920 --> 00:35:49,680 So certainly, in the context of viral infections 725 00:35:49,680 --> 00:35:51,870 and individual cells, there are various mechanisms 726 00:35:51,870 --> 00:35:55,020 where, if one strain gets in, then other strains 727 00:35:55,020 --> 00:35:57,960 have trouble getting in. 728 00:35:57,960 --> 00:36:00,920 Or if they do get in, they can't do anything. 729 00:36:00,920 --> 00:36:03,446 So I think, this really depends on the biology 730 00:36:03,446 --> 00:36:05,821 of the situation, whether this is a reasonable assumption 731 00:36:05,821 --> 00:36:06,321 or not. 732 00:36:20,242 --> 00:36:22,200 So I'm not going to go through all of the math. 733 00:36:22,200 --> 00:36:26,150 Because it ends up being a little bit involved. 734 00:36:26,150 --> 00:36:31,610 But the condition for the mutual invasibility of the two strains 735 00:36:31,610 --> 00:36:33,300 is a little bit subtle. 736 00:36:33,300 --> 00:36:37,130 So I do want to talk about that a little bit. 737 00:36:37,130 --> 00:36:39,930 And so, in general, in this model 738 00:36:39,930 --> 00:36:53,900 there's some equilibrium, x star and y star. 739 00:36:53,900 --> 00:36:57,677 And indeed, there will generally be damped oscillations 740 00:36:57,677 --> 00:36:58,510 to this equilibrium. 741 00:37:03,470 --> 00:37:05,480 So this is in the model where there's just 742 00:37:05,480 --> 00:37:09,460 a single strain that's described by some beta 743 00:37:09,460 --> 00:37:12,740 and some v. Of course, you can also 744 00:37:12,740 --> 00:37:18,640 think about the equilibria, E1 and E2, that would result when 745 00:37:18,640 --> 00:37:24,050 you have-- so E1 is what would happen at the equilibrium 746 00:37:24,050 --> 00:37:30,550 when you have x star evaluated for the particular parameters 747 00:37:30,550 --> 00:37:32,850 of strain one, i.e. 748 00:37:32,850 --> 00:37:36,810 evaluated for beta1 and for v1. 749 00:37:36,810 --> 00:37:39,120 x star-- and that's also y star. 750 00:37:39,120 --> 00:37:41,120 Whereas, you would have a different equilibrium, 751 00:37:41,120 --> 00:37:46,030 E2-- so this is a x star and a y star-- 752 00:37:46,030 --> 00:37:49,110 evaluated at beta2 and v2. 753 00:37:49,110 --> 00:37:50,890 Do you understand what I'm saying? 754 00:37:50,890 --> 00:37:52,390 So these are the equilibria that you 755 00:37:52,390 --> 00:37:54,645 would have if this was the only strain that 756 00:37:54,645 --> 00:37:57,350 was present in the population or if this was the only strain 757 00:37:57,350 --> 00:37:59,620 in the population. 758 00:37:59,620 --> 00:38:01,880 Now, it's not obvious that those-- the equilibria-- 759 00:38:01,880 --> 00:38:04,010 are the result when you have both strains 760 00:38:04,010 --> 00:38:05,670 present in the population. 761 00:38:05,670 --> 00:38:07,560 But that's what we want to try to figure out. 762 00:38:10,510 --> 00:38:12,895 But certainly, if you only added strain one 763 00:38:12,895 --> 00:38:15,520 and you didn't add strain 2, you would come to equilibrium, E1. 764 00:38:20,300 --> 00:38:25,745 So the question is, if we start out at equilibrium E1, 765 00:38:25,745 --> 00:38:27,370 how is it we can determine what happens 766 00:38:27,370 --> 00:38:32,040 if we now add an infected individual, 767 00:38:32,040 --> 00:38:34,510 but infected by strain two? 768 00:38:43,730 --> 00:38:46,895 So what we want to know is-- strain 2 can invade. 769 00:38:52,799 --> 00:38:54,340 And really, what we're thinking about 770 00:38:54,340 --> 00:38:58,190 is a situation-- these are the equilibria if it's 771 00:38:58,190 --> 00:39:03,155 the case that R1 is greater than 1 and R2 is greater than 1. 772 00:39:03,155 --> 00:39:05,100 Because in some ways, it's clear what happens. 773 00:39:05,100 --> 00:39:09,430 If both of the R1 and R2 are less than 1, then what happens? 774 00:39:12,995 --> 00:39:14,120 AUDIENCE: It would die out. 775 00:39:14,120 --> 00:39:15,160 PROFESSOR: They both die out. 776 00:39:15,160 --> 00:39:17,368 If one of them is greater than 1, one is less than 1. 777 00:39:17,368 --> 00:39:20,660 Then, right, the strain that's below 1 778 00:39:20,660 --> 00:39:23,230 goes-- so the only interesting or the only the only 779 00:39:23,230 --> 00:39:25,460 non-obvious question, somehow, is what happens 780 00:39:25,460 --> 00:39:28,800 if they're both larger than 1? 781 00:39:28,800 --> 00:39:30,465 What that means is that, for example, 782 00:39:30,465 --> 00:39:32,590 if you had a population of susceptible individuals, 783 00:39:32,590 --> 00:39:36,030 and you had one infected by strain one, 784 00:39:36,030 --> 00:39:38,040 one infected by strain two, then, 785 00:39:38,040 --> 00:39:40,420 in the deterministic differential equation limit, 786 00:39:40,420 --> 00:39:43,470 they would both spread. 787 00:39:43,470 --> 00:39:45,939 So they're both exponentially growing. 788 00:39:45,939 --> 00:39:47,480 Does that already tell us that you're 789 00:39:47,480 --> 00:39:51,380 going to have coexistence of the two strains? 790 00:39:51,380 --> 00:39:52,116 No. 791 00:39:52,116 --> 00:39:54,240 It means that they're both exponentially spreading. 792 00:39:54,240 --> 00:39:56,800 But who knows what's going to happen later? 793 00:39:56,800 --> 00:39:59,250 And indeed, what you can show is that only one of the two 794 00:39:59,250 --> 00:40:00,250 strains is going to win. 795 00:40:00,250 --> 00:40:03,280 And it's the strain with the higher reproductive ration, 796 00:40:03,280 --> 00:40:05,810 the higher R0, or in this case, here. 797 00:40:08,291 --> 00:40:10,457 So how is it that we can determine if strain two can 798 00:40:10,457 --> 00:40:13,510 invade equilibrium one? 799 00:40:13,510 --> 00:40:15,795 It's if-and-only-if something. 800 00:40:15,795 --> 00:40:18,170 Does anybody remember what this condition ended up being? 801 00:40:30,988 --> 00:40:34,470 AUDIENCE: y2 dot at E1 has to be positive? 802 00:40:34,470 --> 00:40:39,010 PROFESSOR: All right, y2 dot at E1 has to be positive. 803 00:40:39,010 --> 00:40:39,540 Yes. 804 00:40:39,540 --> 00:40:41,370 And this ends up being equivalent, I think, 805 00:40:41,370 --> 00:40:43,190 to what he writes. 806 00:40:43,190 --> 00:40:44,390 So let's write. 807 00:40:44,390 --> 00:40:45,770 I'll write and tell you. 808 00:40:45,770 --> 00:40:49,540 So the way that he wrote it, is that it's y2 809 00:40:49,540 --> 00:40:52,761 dot, with respect to y2. 810 00:40:52,761 --> 00:40:55,260 But I think that this is really equivalent to what you said. 811 00:41:00,310 --> 00:41:02,850 Because you said, OK, y2 dot has to be something, right? 812 00:41:02,850 --> 00:41:04,632 But then, y2 dot evaluated what? 813 00:41:04,632 --> 00:41:06,090 And you would say, evaluated at E1. 814 00:41:06,090 --> 00:41:11,244 But then, at E1-- what is y2 dot evaluated at E1? 815 00:41:11,244 --> 00:41:12,170 AUDIENCE: 0. 816 00:41:12,170 --> 00:41:13,890 PROFESSOR: 0, right? 817 00:41:13,890 --> 00:41:17,280 Because at E1, there is 0 y2. 818 00:41:17,280 --> 00:41:19,865 So then, of course, y2 dot is 0. 819 00:41:19,865 --> 00:41:23,340 So it's almost what you want, but not quite. 820 00:41:23,340 --> 00:41:25,640 So this condition, which looks very weird, 821 00:41:25,640 --> 00:41:30,210 is really saying that, if we are at E1-- so there's 822 00:41:30,210 --> 00:41:34,770 no infected type-two infections-- what we want to do 823 00:41:34,770 --> 00:41:36,840 is add a little bit of y2. 824 00:41:36,840 --> 00:41:39,360 And then, we want to ask, what is y2 dot? 825 00:41:39,360 --> 00:41:41,000 And this derivative, evaluated at E1, 826 00:41:41,000 --> 00:41:44,080 somehow allows you to do that. 827 00:41:44,080 --> 00:41:46,220 And do we want this to be greater than 0? 828 00:41:46,220 --> 00:41:48,060 Or less than 0? 829 00:41:48,060 --> 00:41:49,661 We're going to do verbal answer. 830 00:41:49,661 --> 00:41:50,160 Ready? 831 00:41:50,160 --> 00:41:52,100 Three, two, one. 832 00:41:52,100 --> 00:41:52,850 AUDIENCE: Greater. 833 00:41:52,850 --> 00:41:53,750 PROFESSOR: Greater, right. 834 00:41:53,750 --> 00:41:55,400 Because if it's saying you add a y2, 835 00:41:55,400 --> 00:41:56,710 you want y2 to start growing. 836 00:41:59,610 --> 00:42:01,450 So you want this thing to be greater than 0. 837 00:42:01,450 --> 00:42:03,870 And this looks really crazy. 838 00:42:03,870 --> 00:42:05,370 But it's actually pretty easy to do. 839 00:42:05,370 --> 00:42:07,286 Because you take the derivative of this thing, 840 00:42:07,286 --> 00:42:08,240 with respect to y2. 841 00:42:08,240 --> 00:42:11,070 And you just get the thing in parentheses, right? 842 00:42:11,070 --> 00:42:12,700 But you evaluated it at E1. 843 00:42:12,700 --> 00:42:15,400 So it's beta2. 844 00:42:15,400 --> 00:42:23,954 And this is x star at E1, minus u, minus v2. 845 00:42:23,954 --> 00:42:25,620 And this thing has to be greater than 0. 846 00:42:28,910 --> 00:42:34,340 But this is x at E1. 847 00:42:34,340 --> 00:42:38,710 That's u plus v1 divided by beta1. 848 00:42:38,710 --> 00:42:47,650 So what we have is beta 2, then, u plus v1 divided by beta1. 849 00:42:47,650 --> 00:42:50,220 And this thing has to be greater than-- we'll move this over 850 00:42:50,220 --> 00:42:53,830 to the other side-- u plus v2. 851 00:42:53,830 --> 00:42:57,680 OK, so this is not so horrible, right? 852 00:42:57,680 --> 00:43:00,610 There's a u plus v1, beta1, right? 853 00:43:00,610 --> 00:43:04,839 I just want to make sure I write-- (WHISPERING) u plus v1. 854 00:43:04,839 --> 00:43:07,130 (NORMAL VOICE) OK, we have to these in the denominators 855 00:43:07,130 --> 00:43:07,671 now, somehow. 856 00:43:10,730 --> 00:43:13,550 So we'll put divide by both of these things. 857 00:43:13,550 --> 00:43:18,250 So we have a 1 divided by a u plus v2. 858 00:43:18,250 --> 00:43:21,350 1 divided by u plus v1. 859 00:43:21,350 --> 00:43:23,820 So we put these things in the denominator. 860 00:43:23,820 --> 00:43:26,550 And now, the beta1 is going to come up here. 861 00:43:26,550 --> 00:43:30,550 So we have a beta1 and a beta2. 862 00:43:30,550 --> 00:43:33,420 And there still is a greater-than sign. 863 00:43:33,420 --> 00:43:36,250 Did I screw anything up yet? 864 00:43:36,250 --> 00:43:36,750 Maybe? 865 00:43:36,750 --> 00:43:37,920 No? 866 00:43:37,920 --> 00:43:38,420 All right. 867 00:43:41,240 --> 00:43:43,020 Now, we can do something wonderful, 868 00:43:43,020 --> 00:43:45,100 which is we can just multiply by k divided by u. 869 00:43:51,918 --> 00:43:53,490 And so what does this say? 870 00:43:56,822 --> 00:43:58,730 AUDIENCE: R2 is greater than R1. 871 00:43:58,730 --> 00:44:00,790 PROFESSOR: R2 is greater than R1, right? 872 00:44:03,359 --> 00:44:04,900 And what were we trying to calculate? 873 00:44:07,671 --> 00:44:09,170 How did we get started on this math? 874 00:44:12,044 --> 00:44:13,960 AUDIENCE: What can [INAUDIBLE] the strains. 875 00:44:13,960 --> 00:44:15,418 PROFESSOR: Right, we wanted to ask, 876 00:44:15,418 --> 00:44:19,000 strain two can invade the equilibrium one, 877 00:44:19,000 --> 00:44:31,457 if and only if R2 is greater than-- now, 878 00:44:31,457 --> 00:44:32,540 that's interesting, right? 879 00:44:32,540 --> 00:44:34,164 So that's saying that, if you start out 880 00:44:34,164 --> 00:44:37,290 with the endemic strain one and you add the strain 881 00:44:37,290 --> 00:44:39,590 two in there, it's going to be able to invade 882 00:44:39,590 --> 00:44:42,000 if it's R2 is greater than R1. 883 00:44:42,000 --> 00:44:43,720 That's not obvious. 884 00:44:43,720 --> 00:44:45,600 Because R2 and R1, that was telling us 885 00:44:45,600 --> 00:44:47,317 about a different situation. 886 00:44:47,317 --> 00:44:48,900 That was telling us about what happens 887 00:44:48,900 --> 00:44:52,520 when you add those infected individuals into a population 888 00:44:52,520 --> 00:44:54,260 of susceptibles. 889 00:44:54,260 --> 00:44:57,590 But this also ends up telling us about what 890 00:44:57,590 --> 00:45:00,860 happens when the strains are competing against each other. 891 00:45:00,860 --> 00:45:05,302 So this is so simple that it feels trivial. 892 00:45:05,302 --> 00:45:07,260 I'm sure that if you understood it well enough, 893 00:45:07,260 --> 00:45:08,093 it would be trivial. 894 00:45:08,093 --> 00:45:11,540 But you'd have to think about it more than I have. 895 00:45:11,540 --> 00:45:13,110 So I think this is surprising. 896 00:45:13,110 --> 00:45:15,900 Now, you can also ask the same question, 897 00:45:15,900 --> 00:45:26,010 which is about whether strain one can invade strain two. 898 00:45:26,010 --> 00:45:27,570 And that is kind of the same thing. 899 00:45:27,570 --> 00:45:32,840 So it all comes down to the orderings of R2 and R1. 900 00:45:32,840 --> 00:45:38,390 And what you find is that, if this condition is satisfied, 901 00:45:38,390 --> 00:45:42,140 strain two will drive strain one out of the population. 902 00:45:47,996 --> 00:45:50,904 AUDIENCE: What if they strains are the same? 903 00:45:50,904 --> 00:45:52,320 PROFESSOR: So if they're the same, 904 00:45:52,320 --> 00:45:55,050 then you can get coexistence. 905 00:45:55,050 --> 00:45:58,920 But as Martin says, it's non-generic, 906 00:45:58,920 --> 00:46:02,380 in a sense that it's kind of a coincidence, 907 00:46:02,380 --> 00:46:04,810 or it's of measure 0, the situations 908 00:46:04,810 --> 00:46:10,960 in which the two will have the same R parameters. 909 00:46:10,960 --> 00:46:12,674 But of course, you could imagine-- 910 00:46:12,674 --> 00:46:14,840 just like we talked about all this neutral evolution 911 00:46:14,840 --> 00:46:16,770 business-- things are nearly neutral 912 00:46:16,770 --> 00:46:19,730 and so forth, da-da-da-- you can kind of invoke similar ideas. 913 00:46:19,730 --> 00:46:22,310 Because you can imagine that, as these two become closer 914 00:46:22,310 --> 00:46:23,810 and closer to each other, it's going 915 00:46:23,810 --> 00:46:27,165 to take longer and longer for the more-fit strain 916 00:46:27,165 --> 00:46:28,880 to out-compete the less-fit strain. 917 00:46:31,820 --> 00:46:36,310 Now, even though this is such a simple condition 918 00:46:36,310 --> 00:46:41,560 and the R parameters have such a simple physical origin 919 00:46:41,560 --> 00:46:45,260 or mathematical origin, the actual expression for the Rs 920 00:46:45,260 --> 00:46:46,880 is kind of complicated. 921 00:46:46,880 --> 00:46:48,829 But again, it's a combination of all 922 00:46:48,829 --> 00:46:50,620 of these different parameters, whether it's 923 00:46:50,620 --> 00:46:52,860 the beta1, B1, or beta2, v2. 924 00:47:06,190 --> 00:47:08,205 Which board do you think is least useful? 925 00:47:13,530 --> 00:47:16,079 Well, I don't want this one anymore. 926 00:47:16,079 --> 00:47:17,870 But in particular, we want to say something 927 00:47:17,870 --> 00:47:21,510 about the evolution of virulence in this model 928 00:47:21,510 --> 00:47:24,577 and what the expectation is. 929 00:47:24,577 --> 00:47:26,160 So what we're going to do is I'm going 930 00:47:26,160 --> 00:47:29,180 to give you some different situations, in terms 931 00:47:29,180 --> 00:47:34,220 of how the infectivity depends upon the virulence. 932 00:47:34,220 --> 00:47:36,390 And then, you guys will get to tell me 933 00:47:36,390 --> 00:47:38,185 how the virulence will evolve. 934 00:47:55,520 --> 00:48:24,720 Now, this is virulence, v. All right, so the first situation 935 00:48:24,720 --> 00:48:30,230 is if beta as a function of the virulence-- so 936 00:48:30,230 --> 00:48:35,250 the infectivity as a function of virulence is, we'll say, 937 00:48:35,250 --> 00:48:37,110 some beta0. 938 00:48:37,110 --> 00:48:40,570 So first, the question is, if the infectivity does not 939 00:48:40,570 --> 00:48:47,940 depend upon the virulence, where will virulence go to? 940 00:48:51,750 --> 00:48:55,460 You have equation, various places that'll help you. 941 00:49:07,270 --> 00:49:09,170 Do you need more time? 942 00:49:09,170 --> 00:49:09,670 No? 943 00:49:09,670 --> 00:49:14,820 All right, ready, three, two, one. 944 00:49:14,820 --> 00:49:16,860 OK, we've got many A's. 945 00:49:16,860 --> 00:49:17,460 That's great. 946 00:49:21,690 --> 00:49:25,550 So if it's the case that the parasite, 947 00:49:25,550 --> 00:49:28,360 regardless of how rapidly it's killing the individual, 948 00:49:28,360 --> 00:49:31,220 has the same rate of getting to other individuals, 949 00:49:31,220 --> 00:49:37,220 then we want to make the R0 to get as large as possible. 950 00:49:37,220 --> 00:49:40,134 So then, we make v go as small as possible. 951 00:49:40,134 --> 00:49:42,050 That's the mathematical thing you can look at. 952 00:49:42,050 --> 00:49:43,350 And why does this make sense? 953 00:49:51,093 --> 00:49:52,592 AUDIENCE: You're killing people off, 954 00:49:52,592 --> 00:49:54,089 and you're infecting most people. 955 00:49:54,089 --> 00:49:55,090 But-- 956 00:49:55,090 --> 00:49:55,940 PROFESSOR: Yeah. 957 00:49:55,940 --> 00:50:00,480 So from the standpoint of the parasite, 958 00:50:00,480 --> 00:50:03,190 in this situation where the infectivity doesn't depend 959 00:50:03,190 --> 00:50:05,080 on the virulence, well in that case, 960 00:50:05,080 --> 00:50:06,570 you don't want to kill your host. 961 00:50:06,570 --> 00:50:09,530 Because the longer that the host lives, 962 00:50:09,530 --> 00:50:11,380 the more other individuals you can infect. 963 00:50:14,060 --> 00:50:19,310 And this is the basic statement underlying the statement 964 00:50:19,310 --> 00:50:22,960 you often hear that a well-adapted parasite does not 965 00:50:22,960 --> 00:50:25,236 harm its host. 966 00:50:25,236 --> 00:50:26,860 And I think this is one of those things 967 00:50:26,860 --> 00:50:29,620 that you can write down a simple model 968 00:50:29,620 --> 00:50:32,390 and convince yourself it should be true. 969 00:50:32,390 --> 00:50:34,360 You can maybe find a few case studies 970 00:50:34,360 --> 00:50:36,180 where it seems to be true. 971 00:50:36,180 --> 00:50:40,179 But then, you always have to be careful about how strongly you 972 00:50:40,179 --> 00:50:41,720 should believe such a statement based 973 00:50:41,720 --> 00:50:43,110 on those kinds of evidence. 974 00:50:43,110 --> 00:50:46,500 Because it's a very simple model. 975 00:50:46,500 --> 00:50:49,770 It's making an assumption that is very likely not true 976 00:50:49,770 --> 00:50:53,130 and, actually, many assumptions that are very likely not true. 977 00:50:53,130 --> 00:50:54,710 And there are counterexamples. 978 00:50:54,710 --> 00:50:57,590 So Martin talks about malaria, which 979 00:50:57,590 --> 00:51:00,110 humans have had for millions of years, 980 00:51:00,110 --> 00:51:05,450 and still causes a lot of problems. 981 00:51:05,450 --> 00:51:06,950 And so you might imagine, what would 982 00:51:06,950 --> 00:51:11,510 happen if the infectivity is a function of the virulence? 983 00:51:11,510 --> 00:51:13,960 Maybe it's just proportional to the virulence. 984 00:51:13,960 --> 00:51:15,850 And this kind of would make sense. 985 00:51:15,850 --> 00:51:19,590 Because you say, oh well, let's imagine that the number 986 00:51:19,590 --> 00:51:25,560 of viruses in the host-- that somehow, the virulence 987 00:51:25,560 --> 00:51:27,070 is proportional to that number. 988 00:51:27,070 --> 00:51:30,182 And also, the infectivity is proportional to that number. 989 00:51:30,182 --> 00:51:31,640 In that case, infectivity will just 990 00:51:31,640 --> 00:51:38,330 scale linearly with v. It's kind of another reasonable world. 991 00:51:38,330 --> 00:51:42,540 So in this situation, where does the virulence evolve to? 992 00:51:42,540 --> 00:51:44,700 We'll think about this for 10 seconds. 993 00:52:01,100 --> 00:52:01,660 Ready? 994 00:52:01,660 --> 00:52:03,480 Three, two, one. 995 00:52:11,130 --> 00:52:13,780 All right, so now we actually have votes that 996 00:52:13,780 --> 00:52:16,010 are pretty distributed around. 997 00:52:19,310 --> 00:52:21,880 OK, I'm just going to write down the expression. 998 00:52:21,880 --> 00:52:23,720 Just because I think we can do it quickly. 999 00:52:23,720 --> 00:52:27,399 So now, the R0 is going to be given by-- we're 1000 00:52:27,399 --> 00:52:28,690 going to be thinking about a 1. 1001 00:52:28,690 --> 00:52:31,120 There's a u plus v down here. 1002 00:52:31,120 --> 00:52:33,730 But beta-- we're going to put beta here. 1003 00:52:33,730 --> 00:52:39,173 This is an a times v, and then a k over u. 1004 00:52:39,173 --> 00:52:41,140 Right? 1005 00:52:41,140 --> 00:52:46,161 So if we plot this as a function of v, what does it look like? 1006 00:52:46,161 --> 00:52:47,862 AUDIENCE: It monotonically increases. 1007 00:52:47,862 --> 00:52:49,320 PROFESSOR: Monotonically increases. 1008 00:52:49,320 --> 00:52:52,420 This is a Michaelis-Menten type form 1009 00:52:52,420 --> 00:52:58,680 where the R0 is a function of v. It starts out 1010 00:52:58,680 --> 00:52:59,885 linear and, then, saturates. 1011 00:53:04,170 --> 00:53:05,770 Oops, this is not v. This is R0. 1012 00:53:08,540 --> 00:53:14,706 If we want to maximize R0, then v goes to infinity, right? 1013 00:53:20,060 --> 00:53:22,430 There was one other model that Martin talked about, 1014 00:53:22,430 --> 00:53:25,745 which was the situation if the infectivity itself. 1015 00:53:25,745 --> 00:53:28,550 It had a kind of Michealis-Menten type form. 1016 00:53:28,550 --> 00:53:32,390 And we still have an a there. 1017 00:53:32,390 --> 00:53:38,630 So it's some a,v c plus v. So this would be the situation 1018 00:53:38,630 --> 00:53:41,360 where, initially, the infectivity increases with 1019 00:53:41,360 --> 00:53:41,870 virulence. 1020 00:53:41,870 --> 00:53:46,110 But beyond some virulence, you no longer 1021 00:53:46,110 --> 00:53:47,484 get an increase in infectivity. 1022 00:53:47,484 --> 00:53:49,650 So for example, if it's the case that every time you 1023 00:53:49,650 --> 00:53:51,884 sneeze on somebody you're going to infect them, 1024 00:53:51,884 --> 00:53:54,300 then it doesn't matter if you double the number of viruses 1025 00:53:54,300 --> 00:53:54,966 you have. 1026 00:53:54,966 --> 00:53:56,590 You've still saturated the infectivity. 1027 00:54:00,700 --> 00:54:07,400 Now, I'll just tell you that, in this case, 1028 00:54:07,400 --> 00:54:11,540 there ends up being some intermediate infectivity 1029 00:54:11,540 --> 00:54:14,850 that you evolve to, which comes here. 1030 00:54:14,850 --> 00:54:16,750 And indeed, this makes sense. 1031 00:54:16,750 --> 00:54:31,590 As c goes to infinity, so if c is very large, 1032 00:54:31,590 --> 00:54:34,217 then it just looks like this. 1033 00:54:34,217 --> 00:54:35,550 It's divided by something large. 1034 00:54:35,550 --> 00:54:38,530 But in terms of the scaling with v, it's just linear in v. 1035 00:54:38,530 --> 00:54:42,670 And that means that the evolved virulence goes very large. 1036 00:54:46,575 --> 00:54:49,850 Does that make sense? 1037 00:54:49,850 --> 00:54:50,350 No? 1038 00:54:50,350 --> 00:54:51,320 OK. 1039 00:54:51,320 --> 00:54:54,560 So I guess what I'm saying is that, if you're 1040 00:54:54,560 --> 00:55:00,120 in the region where c is somehow very large, then 1041 00:55:00,120 --> 00:55:02,080 you end up with the infectivity just being 1042 00:55:02,080 --> 00:55:04,700 proportional to the virulence. 1043 00:55:04,700 --> 00:55:07,580 Because the virulence, the v here in the denominator, 1044 00:55:07,580 --> 00:55:09,510 doesn't really contribute very much. 1045 00:55:09,510 --> 00:55:12,076 And that means that it's going to be the same kind 1046 00:55:12,076 --> 00:55:13,200 of functional form as this. 1047 00:55:13,200 --> 00:55:14,670 Except if it's a divided by c. 1048 00:55:14,670 --> 00:55:16,950 But it's still proportional to the virulence. 1049 00:55:16,950 --> 00:55:21,772 And that leads to a larger and larger evolved virulence. 1050 00:55:21,772 --> 00:55:24,240 OK? 1051 00:55:24,240 --> 00:55:24,890 Yep? 1052 00:55:24,890 --> 00:55:25,390 Yes? 1053 00:55:25,390 --> 00:55:29,471 AUDIENCE: Is it possible to say something 1054 00:55:29,471 --> 00:55:32,894 in actual infection, what the data is? 1055 00:55:32,894 --> 00:55:33,872 Is it virulunce? 1056 00:55:33,872 --> 00:55:34,880 Or can you [INAUDIBLE]-- 1057 00:55:34,880 --> 00:55:35,546 PROFESSOR: Yeah. 1058 00:55:35,546 --> 00:55:37,296 AUDIENCE: --other than just these verbal-- 1059 00:55:37,296 --> 00:55:38,600 PROFESSOR: Yeah, right. 1060 00:55:38,600 --> 00:55:43,160 So I think, from real data, I think people do argue about it. 1061 00:55:43,160 --> 00:55:52,180 But I think it seems to be there is some sense that it 1062 00:55:52,180 --> 00:55:54,020 does plateau. 1063 00:55:54,020 --> 00:55:58,460 But it's an increasing function of v, but sub-linear. 1064 00:55:58,460 --> 00:56:00,729 And the question is how strong of a statement 1065 00:56:00,729 --> 00:56:01,520 you can make there. 1066 00:56:01,520 --> 00:56:03,769 Because also, in many cases, there is super-infection. 1067 00:56:03,769 --> 00:56:07,460 And super-infection tends to lead to even higher virulence 1068 00:56:07,460 --> 00:56:10,510 than what you would expect from this. 1069 00:56:10,510 --> 00:56:14,340 Because you're competing against viruses in the same host. 1070 00:56:14,340 --> 00:56:19,570 So you have to out-compete the other parasite in the host, 1071 00:56:19,570 --> 00:56:21,340 as well as go on. 1072 00:56:21,340 --> 00:56:23,260 And you don't pay the full cost associated 1073 00:56:23,260 --> 00:56:25,210 with keeping the host alive. 1074 00:56:25,210 --> 00:56:26,710 So when you have super-infection, 1075 00:56:26,710 --> 00:56:29,981 there's this notion that the better strategy, 1076 00:56:29,981 --> 00:56:31,480 from the standpoint of the parasite, 1077 00:56:31,480 --> 00:56:34,336 can be to just evolve very high virulence. 1078 00:56:34,336 --> 00:56:35,710 So you're going to kill the host. 1079 00:56:35,710 --> 00:56:37,340 But you can get out quickly. 1080 00:56:37,340 --> 00:56:41,170 So then, you can imagine that the less-virulent strain 1081 00:56:41,170 --> 00:56:44,380 was kind of stranded in that host that died. 1082 00:56:44,380 --> 00:56:46,770 So from that standpoint, you can think about, 1083 00:56:46,770 --> 00:56:48,650 in some cases, for a parasite, evolving 1084 00:56:48,650 --> 00:56:52,620 low-virulence is somehow like some cooperative behavior. 1085 00:56:52,620 --> 00:56:56,810 Because you're keeping this host alive. 1086 00:56:56,810 --> 00:56:59,010 You're using the resources in a wise way, 1087 00:56:59,010 --> 00:57:01,240 so that you can infect other individuals. 1088 00:57:01,240 --> 00:57:03,356 But then, that kind of strategy is 1089 00:57:03,356 --> 00:57:05,230 susceptible to these cheating strategies that 1090 00:57:05,230 --> 00:57:07,340 just have high virulence and kill the host 1091 00:57:07,340 --> 00:57:09,645 and, then, get on to a new host. 1092 00:57:09,645 --> 00:57:11,730 Yeah, so I think there are enough of these issues 1093 00:57:11,730 --> 00:57:16,370 that it's hard to take any of this too seriously. 1094 00:57:16,370 --> 00:57:18,590 I would say that, perhaps, where this field 1095 00:57:18,590 --> 00:57:22,375 has had the biggest impact is in the context of vaccinations. 1096 00:57:30,000 --> 00:57:34,840 Because you can really measure R0 for many different diseases. 1097 00:57:34,840 --> 00:57:36,430 And I think that's somehow easier 1098 00:57:36,430 --> 00:57:37,846 to measure than many other things. 1099 00:57:37,846 --> 00:57:42,700 Because you can try to do tracing of infections. 1100 00:57:42,700 --> 00:57:45,620 So if you think about somebody that gets infected 1101 00:57:45,620 --> 00:57:48,040 and moves to a new city or lands in a new city, 1102 00:57:48,040 --> 00:57:50,840 you can try to figure out who they infected. 1103 00:57:50,840 --> 00:57:55,250 So then, you can go and measure these R0 parameters. 1104 00:57:55,250 --> 00:57:58,410 And of course, the diseases that we worry about 1105 00:57:58,410 --> 00:58:01,810 tend to have R0s larger than 1. 1106 00:58:01,810 --> 00:58:04,630 And how large they are tells you about how difficult 1107 00:58:04,630 --> 00:58:09,470 the vaccination will be in order to be successful 1108 00:58:09,470 --> 00:58:12,800 and to remove the disease from the population. 1109 00:58:12,800 --> 00:58:15,520 So if you're curious, after class, you can come up. 1110 00:58:15,520 --> 00:58:18,550 And there's a nice table that I have here 1111 00:58:18,550 --> 00:58:21,550 for smallpox, measles, whooping cough, German measles, 1112 00:58:21,550 --> 00:58:23,881 chickenpox, diphtheria, scarlet fever, mumps, 1113 00:58:23,881 --> 00:58:24,630 and poliomielitis. 1114 00:58:27,240 --> 00:58:34,456 Estimates of R0s-- and they kind of range 5 to 15, 1115 00:58:34,456 --> 00:58:36,680 to give you a sense. 1116 00:58:36,680 --> 00:58:40,820 And so, if you want to remove the disease 1117 00:58:40,820 --> 00:58:43,300 from the population, what is it that changes? 1118 00:58:43,300 --> 00:58:47,230 --in terms of vaccination, in order to remove the disease. 1119 00:58:55,260 --> 00:58:55,760 Yes? 1120 00:58:55,760 --> 00:58:58,586 AUDIENCE: In R0, then, the [INAUDIBLE] for u, 1121 00:58:58,586 --> 00:59:02,230 as the population available [INAUDIBLE]. 1122 00:59:02,230 --> 00:59:03,230 PROFESSOR: That's right. 1123 00:59:03,230 --> 00:59:07,189 So by vaccinating you're removing 1124 00:59:07,189 --> 00:59:08,980 susceptible individuals from the population 1125 00:59:08,980 --> 00:59:11,420 and making them resistant somehow. 1126 00:59:11,420 --> 00:59:15,670 And the R0 parameter tells you about what fraction 1127 00:59:15,670 --> 00:59:17,670 of the population you have to vaccinate in order 1128 00:59:17,670 --> 00:59:20,830 to remove the parasite from the population. 1129 00:59:20,830 --> 00:59:23,230 And so, basically, you have to vaccinate 1130 00:59:23,230 --> 00:59:29,000 a fraction, a percentage, p that's greater than 1 minus 1 1131 00:59:29,000 --> 00:59:30,630 over R0. 1132 00:59:30,630 --> 00:59:33,850 So as R0 gets very large, it means 1133 00:59:33,850 --> 00:59:37,770 that you have to vaccinate, essentially, everybody. 1134 00:59:37,770 --> 00:59:39,787 So if you have R0 of, say, five-- 1135 00:59:39,787 --> 00:59:41,620 which is typical of many of these diseases-- 1136 00:59:41,620 --> 00:59:44,170 it's saying you have to vaccinate 1137 00:59:44,170 --> 00:59:46,260 80% of the population. 1138 00:59:46,260 --> 00:59:48,084 You can never get to 100%. 1139 00:59:48,084 --> 00:59:49,500 And that's why it's very difficult 1140 00:59:49,500 --> 00:59:53,860 to get rid of these diseases with large R0s. 1141 00:59:53,860 --> 00:59:58,730 And incidentally, they note that smallpox has an R0 of 3 to 5. 1142 01:00:01,330 --> 01:00:03,960 And this always depends on the environment. 1143 01:00:03,960 --> 01:00:06,420 But in the case where they measured, 1144 01:00:06,420 --> 01:00:10,380 smallpox R0 is 3 to 5. 1145 01:00:10,380 --> 01:00:12,027 And this is sort of small, as compared 1146 01:00:12,027 --> 01:00:13,110 to many of these diseases. 1147 01:00:13,110 --> 01:00:17,660 And this is telling us that smallpox is easier 1148 01:00:17,660 --> 01:00:19,360 to get rid of via vaccinations than many 1149 01:00:19,360 --> 01:00:20,360 of these other diseases. 1150 01:00:20,360 --> 01:00:24,940 And indeed, the vaccination procedures 1151 01:00:24,940 --> 01:00:28,882 have been more successful in smallpox than the others. 1152 01:00:28,882 --> 01:00:31,740 AUDIENCE: Do you know what is 3, 4? 1153 01:00:31,740 --> 01:00:32,750 PROFESSOR: I don't. 1154 01:00:32,750 --> 01:00:35,070 But maybe in the next 20 minutes, 1155 01:00:35,070 --> 01:00:38,580 somebody can Google this. 1156 01:00:38,580 --> 01:00:41,450 We can estimate this right now, though. 1157 01:00:41,450 --> 01:00:42,840 We've had, what? 1158 01:00:42,840 --> 01:00:45,710 --five Ebola patients come to the United States. 1159 01:00:45,710 --> 01:00:49,470 And we've gotten two or three infections. 1160 01:00:49,470 --> 01:00:51,815 So I'll say, 3/5. 1161 01:00:51,815 --> 01:00:53,342 [LAUGHTER] 1162 01:00:53,342 --> 01:00:55,300 It obviously depends on the environment, right? 1163 01:00:55,300 --> 01:00:57,040 AUDIENCE: Yeah. 1164 01:00:57,040 --> 01:00:57,665 PROFESSOR: Yes. 1165 01:00:57,665 --> 01:00:59,623 AUDIENCE: I mean, I guess that was my question. 1166 01:00:59,623 --> 01:01:01,449 You're not going to take someone with Ebola 1167 01:01:01,449 --> 01:01:05,097 and throw them in New York City and just measure how 1168 01:01:05,097 --> 01:01:06,235 many people they infect. 1169 01:01:06,235 --> 01:01:06,690 PROFESSOR: That's right. 1170 01:01:06,690 --> 01:01:07,145 AUDIENCE: So-- 1171 01:01:07,145 --> 01:01:08,060 AUDIENCE: That would be [INAUDIBLE]. 1172 01:01:08,060 --> 01:01:09,005 AUDIENCE: Like-- 1173 01:01:09,005 --> 01:01:09,921 AUDIENCE: [INAUDIBLE]. 1174 01:01:09,921 --> 01:01:12,327 PROFESSOR: Yeah, but the thing is, you 1175 01:01:12,327 --> 01:01:14,160 don't have to do anything that's so immoral. 1176 01:01:14,160 --> 01:01:15,630 Because what you're interested in 1177 01:01:15,630 --> 01:01:18,309 is the R0 for individual in an actual environment 1178 01:01:18,309 --> 01:01:19,850 that they're actually going to be in. 1179 01:01:19,850 --> 01:01:24,210 So this is a situation where the doctor comes back from Africa 1180 01:01:24,210 --> 01:01:27,170 after working with Doctors Without Borders. 1181 01:01:27,170 --> 01:01:29,704 In this day and age, he knows that he has to watch out 1182 01:01:29,704 --> 01:01:30,620 for a fever, da-da-da. 1183 01:01:30,620 --> 01:01:32,910 And then, if he gets a fever, he calls in. 1184 01:01:32,910 --> 01:01:34,440 And he's brought to the hospital. 1185 01:01:34,440 --> 01:01:36,240 And that's the world that we're interested in of what 1186 01:01:36,240 --> 01:01:36,950 the R0 is. 1187 01:01:36,950 --> 01:01:40,050 It's not the world in which there's fevers everywhere 1188 01:01:40,050 --> 01:01:43,470 and nobody knows. 1189 01:01:43,470 --> 01:01:45,280 So the R0 in the United States is 1190 01:01:45,280 --> 01:01:50,108 going to be much lower than the R0 somewhere else. 1191 01:01:50,108 --> 01:01:51,056 Yeah? 1192 01:01:51,056 --> 01:01:53,599 AUDIENCE: Is there enough of R0 for a disease that you 1193 01:01:53,599 --> 01:01:54,848 don't transmit between people? 1194 01:01:54,848 --> 01:01:56,952 You get malaria of the plague-- 1195 01:01:56,952 --> 01:01:57,910 PROFESSOR: Yeah, right. 1196 01:01:57,910 --> 01:02:03,024 So I think that you could try to generate a similar kind of R0 1197 01:02:03,024 --> 01:02:03,940 for those to diseases. 1198 01:02:03,940 --> 01:02:05,990 Although, it's going to be very muddled, 1199 01:02:05,990 --> 01:02:08,540 in the case of where you have all the vectors and so forth. 1200 01:02:08,540 --> 01:02:13,830 Because it's not even clear-- yeah, 1201 01:02:13,830 --> 01:02:16,360 I'm hesitant to say too much. 1202 01:02:16,360 --> 01:02:18,736 Because I don't know anything. 1203 01:02:18,736 --> 01:02:20,155 AUDIENCE: According to Wikipedia, 1204 01:02:20,155 --> 01:02:23,315 it's like 1.5 to 2.5. 1205 01:02:23,315 --> 01:02:24,190 PROFESSOR: For Ebola? 1206 01:02:24,190 --> 01:02:24,550 AUDIENCE: Yeah. 1207 01:02:24,550 --> 01:02:25,216 PROFESSOR: Here? 1208 01:02:25,216 --> 01:02:26,522 Or in-- 1209 01:02:26,522 --> 01:02:31,076 AUDIENCE: It says the 2014 West Africa aggregate. 1210 01:02:31,076 --> 01:02:33,876 PROFESSOR: Ah, so this is in West Africa then. 1211 01:02:33,876 --> 01:02:34,750 AUDIENCE: Apparently. 1212 01:02:34,750 --> 01:02:35,458 PROFESSOR: Right. 1213 01:02:35,458 --> 01:02:38,460 Yeah And it has obviously spread exponentially, 1214 01:02:38,460 --> 01:02:40,779 which means it had to have been larger than 1. 1215 01:02:40,779 --> 01:02:43,320 And I think it's important to remember that, just because you 1216 01:02:43,320 --> 01:02:47,130 have a chart with a bunch of R0s, this is not set in stone. 1217 01:02:47,130 --> 01:02:50,640 Public policy, hygiene, and everything changes this. 1218 01:02:50,640 --> 01:02:52,850 And we'd like to drive it down. 1219 01:02:52,850 --> 01:02:54,258 Was there a question in the back? 1220 01:02:54,258 --> 01:02:56,620 AUDIENCE: I just was going to say about two. 1221 01:02:56,620 --> 01:02:57,995 PROFESSOR: Same thing, about two. 1222 01:02:57,995 --> 01:03:02,170 OK, so that means that we could actually, in principle, 1223 01:03:02,170 --> 01:03:03,950 vaccinate against Ebola. 1224 01:03:03,950 --> 01:03:07,580 We'd only have to get over 50% of the population vaccinated 1225 01:03:07,580 --> 01:03:08,230 in West Africa. 1226 01:03:08,230 --> 01:03:11,940 And then, we can maybe make it so it cannot spread and become 1227 01:03:11,940 --> 01:03:12,750 an epidemic. 1228 01:03:12,750 --> 01:03:17,434 Of course, we need to have a vaccine first. 1229 01:03:17,434 --> 01:03:19,046 AUDIENCE: [INAUDIBLE]. 1230 01:03:19,046 --> 01:03:20,670 PROFESSOR: So I think I'm going to skip 1231 01:03:20,670 --> 01:03:22,350 the discussion of SIR models. 1232 01:03:22,350 --> 01:03:24,670 Because you are going to play with some of them 1233 01:03:24,670 --> 01:03:27,140 in the context of your problem set. 1234 01:03:27,140 --> 01:03:30,180 And if you just Google SIR, you can find it. 1235 01:03:30,180 --> 01:03:31,712 And a very similar kind of intuition 1236 01:03:31,712 --> 01:03:32,920 that you get from this model. 1237 01:03:35,895 --> 01:03:38,270 Because I do want to spend the last 15 minutes, at least, 1238 01:03:38,270 --> 01:03:39,728 talking about the evolution of sex. 1239 01:03:39,728 --> 01:03:41,850 Because it is an interesting topic. 1240 01:03:41,850 --> 01:03:47,126 And I think the paper is a nice discussion of it. 1241 01:03:54,920 --> 01:04:00,540 So can somebody say why it is that this is a puzzle at all? 1242 01:04:11,408 --> 01:04:13,813 Yeah? 1243 01:04:13,813 --> 01:04:15,854 AUDIENCE: In almost all the situations we imagine 1244 01:04:15,854 --> 01:04:19,806 and things that can be [INAUDIBLE] introduce much 1245 01:04:19,806 --> 01:04:21,501 faster, even violating the [INAUDIBLE]-- 1246 01:04:21,501 --> 01:04:22,500 PROFESSOR: That's right. 1247 01:04:22,500 --> 01:04:25,634 AUDIENCE: --80% [INAUDIBLE] right? 1248 01:04:25,634 --> 01:04:29,130 PROFESSOR: So sex is costly, and in particular, if you have 1249 01:04:29,130 --> 01:04:33,130 this obligate bi-parental sex. 1250 01:04:33,130 --> 01:04:37,795 In particular, there's the so-called twofold cost 1251 01:04:37,795 --> 01:04:38,295 of males. 1252 01:04:41,670 --> 01:04:44,200 Because you can imagine comparing these two 1253 01:04:44,200 --> 01:04:47,660 populations, one of which has both males and females. 1254 01:04:47,660 --> 01:04:51,984 And one of them is just, maybe, reproducing asexually, 1255 01:04:51,984 --> 01:04:53,900 or parthenogenetically, or hermaphroditically, 1256 01:04:53,900 --> 01:04:56,310 or what not. 1257 01:04:56,310 --> 01:04:59,810 And so, if you have a male and a female, 1258 01:04:59,810 --> 01:05:01,570 then on average, if they have two kids, 1259 01:05:01,570 --> 01:05:03,999 you end up with another male and a female. 1260 01:05:03,999 --> 01:05:06,040 And of course, this could be many different males 1261 01:05:06,040 --> 01:05:06,539 and females. 1262 01:05:06,539 --> 01:05:08,990 So you don't have to have any sibling anything. 1263 01:05:08,990 --> 01:05:15,125 But if, every generation, each female 1264 01:05:15,125 --> 01:05:18,610 is giving birth to two progeny, then you 1265 01:05:18,610 --> 01:05:20,610 end up with a constant population size. 1266 01:05:20,610 --> 01:05:22,970 Whereas, if you started out in a population 1267 01:05:22,970 --> 01:05:25,950 with just two females and they were reproducing 1268 01:05:25,950 --> 01:05:32,290 hermaphroditically, then you end up with-- whatever-- more. 1269 01:05:32,290 --> 01:05:33,660 8. 1270 01:05:33,660 --> 01:05:35,650 So you can see that you get a factor of 2 1271 01:05:35,650 --> 01:05:38,096 in the rate of exponential growth of the population. 1272 01:05:38,096 --> 01:05:38,596 Yeah? 1273 01:05:38,596 --> 01:05:41,054 AUDIENCE: Well, it seems to me the question is not why sex, 1274 01:05:41,054 --> 01:05:42,932 but it's why males. 1275 01:05:42,932 --> 01:05:44,670 Right, I mean-- 1276 01:05:44,670 --> 01:05:46,010 PROFESSOR: Yes. 1277 01:05:46,010 --> 01:05:52,150 So I think this is the most extreme cost of sex. 1278 01:05:52,150 --> 01:05:55,400 But then, also, you can think about even just 1279 01:05:55,400 --> 01:05:57,730 horizontal gene-transfer among bacteria. 1280 01:05:57,730 --> 01:05:59,740 It's a costly behavior in some way or another. 1281 01:05:59,740 --> 01:06:02,010 It's not as costly as this. 1282 01:06:02,010 --> 01:06:05,670 But if you're going to think about bacteria 1283 01:06:05,670 --> 01:06:08,440 in their competence state, when B. subtilis kind of pulls 1284 01:06:08,440 --> 01:06:10,630 in DNA, it stops dividing. 1285 01:06:10,630 --> 01:06:12,930 And then, it enters a state where it reels things in. 1286 01:06:12,930 --> 01:06:17,530 And it can pick up DNA that may be harmful in some cases. 1287 01:06:17,530 --> 01:06:23,042 So there are various costs for, if you want to call it, 1288 01:06:23,042 --> 01:06:24,250 horizontal gene-transfer sex. 1289 01:06:26,970 --> 01:06:29,100 So the key thing of sexual reproduction 1290 01:06:29,100 --> 01:06:33,020 is it somehow is the sharing of the DNA. 1291 01:06:33,020 --> 01:06:36,730 And I'd say that this can either be relatively low cost 1292 01:06:36,730 --> 01:06:37,739 or relatively high cost. 1293 01:06:37,739 --> 01:06:39,530 But this is the most extreme version of it. 1294 01:06:39,530 --> 01:06:44,560 And I'd say, as a species that reproduces 1295 01:06:44,560 --> 01:06:46,520 with obligate bi-parental sex, then 1296 01:06:46,520 --> 01:06:52,770 I'd say that, not only humans, but almost all animals 1297 01:06:52,770 --> 01:06:55,500 have this form of reproduction. 1298 01:06:55,500 --> 01:06:57,420 I would say it's sort of surprising, 1299 01:06:57,420 --> 01:06:59,130 given that this is a huge cost. 1300 01:07:01,484 --> 01:07:02,900 But it's not that all species that 1301 01:07:02,900 --> 01:07:05,480 engage in any sort of gene-transfer 1302 01:07:05,480 --> 01:07:07,190 bear this large of a cost. 1303 01:07:07,190 --> 01:07:09,100 But they bear more modest costs. 1304 01:07:09,100 --> 01:07:13,410 AUDIENCE: Well, although, other species do-- even 1305 01:07:13,410 --> 01:07:15,310 with this [INAUDIBLE], it's like having 1306 01:07:15,310 --> 01:07:18,668 multiple children per sex. 1307 01:07:18,668 --> 01:07:21,126 PROFESSOR: You're saying that they can just have more kids. 1308 01:07:21,126 --> 01:07:21,980 AUDIENCE: Yeah, I mean-- 1309 01:07:21,980 --> 01:07:23,620 PROFESSOR: Yeah, although the basic statement is still true. 1310 01:07:23,620 --> 01:07:26,349 Let's say that these females all have three kids. 1311 01:07:26,349 --> 01:07:27,640 They get to grow exponentially. 1312 01:07:27,640 --> 01:07:29,210 But then, these guys grow faster. 1313 01:07:29,210 --> 01:07:31,793 And ultimately, there's going to be competition for resources. 1314 01:07:31,793 --> 01:07:33,575 And these ones will still win. 1315 01:07:33,575 --> 01:07:34,200 AUDIENCE: Yeah. 1316 01:07:34,200 --> 01:07:35,674 PROFESSOR: So the question is-- you 1317 01:07:35,674 --> 01:07:37,340 can imagine you have a population that's 1318 01:07:37,340 --> 01:07:40,480 reproducing sexually-- if one female has a mutation that 1319 01:07:40,480 --> 01:07:43,744 leads her to start reproducing parthogenetically, 1320 01:07:43,744 --> 01:07:45,410 that mutation you'd expect should spread 1321 01:07:45,410 --> 01:07:47,900 throughout the population very rapidly. 1322 01:07:47,900 --> 01:07:50,550 And indeed, there are these cases of, for example, 1323 01:07:50,550 --> 01:07:53,950 sharks held in captivity where a female held 1324 01:07:53,950 --> 01:07:56,370 in captivity for years eventually gives birth 1325 01:07:56,370 --> 01:07:58,925 to daughters. 1326 01:08:02,780 --> 01:08:05,100 So this sort of virgin birth is possible. 1327 01:08:05,100 --> 01:08:07,275 I'm not aware-- well, I'm not going to talk about-- 1328 01:08:07,275 --> 01:08:08,280 [LAUGHTER] 1329 01:08:10,730 --> 01:08:15,520 But in some animals, it is at least possible, I'll say. 1330 01:08:15,520 --> 01:08:18,870 So then, you have to ask, well, what 1331 01:08:18,870 --> 01:08:22,220 kind of selective advantage could sexual reproduction have 1332 01:08:22,220 --> 01:08:25,460 that could possibly compensate for this so-called twofold cost 1333 01:08:25,460 --> 01:08:27,727 of sex or twofold cost of males? 1334 01:08:30,910 --> 01:08:36,120 And what is the argument that's made in this paper? 1335 01:08:54,220 --> 01:08:54,720 Anybody? 1336 01:08:58,828 --> 01:08:59,780 Yes? 1337 01:08:59,780 --> 01:09:03,588 AUDIENCE: The recombination favors genetic diversity. 1338 01:09:03,588 --> 01:09:04,910 PROFESSOR: That's right. 1339 01:09:04,910 --> 01:09:07,880 This recombination is favoring genetic diversity. 1340 01:09:07,880 --> 01:09:10,260 So there are a number of different mechanisms. 1341 01:09:10,260 --> 01:09:13,399 I'd say that at the heart of this idea of the Red Queen 1342 01:09:13,399 --> 01:09:17,330 hypothesis is-- let me see if I can find 1343 01:09:17,330 --> 01:09:21,060 the actual quote for you guys. 1344 01:09:21,060 --> 01:09:21,700 Maybe not. 1345 01:09:24,229 --> 01:09:26,840 So it's from a Lewis Carroll novel, 1346 01:09:26,840 --> 01:09:28,290 Through the Looking Glass. 1347 01:09:28,290 --> 01:09:31,845 The quote was something-- run-- oh, shoot. 1348 01:09:37,370 --> 01:09:39,740 Never mind, I can't remember what the quote was. 1349 01:09:39,740 --> 01:09:44,010 But the idea is that sexual reproduction 1350 01:09:44,010 --> 01:09:46,830 may allow a population to adapt against, say, 1351 01:09:46,830 --> 01:09:49,830 changing environments more rapidly. 1352 01:09:49,830 --> 01:09:51,580 And this has a couple of reasons. 1353 01:09:51,580 --> 01:09:53,910 Because you generate genetic diversity. 1354 01:09:53,910 --> 01:09:56,565 You don't have the same clonal interference effects 1355 01:09:56,565 --> 01:09:58,650 that we talked about earlier. 1356 01:09:58,650 --> 01:10:00,780 So if you have asexual lineages, then, 1357 01:10:00,780 --> 01:10:02,640 if you have two beneficial mutations, 1358 01:10:02,640 --> 01:10:04,960 they cannot both fix. 1359 01:10:04,960 --> 01:10:06,647 So the more fit version is going to fix. 1360 01:10:06,647 --> 01:10:08,480 And then, you have to wait for the next one. 1361 01:10:08,480 --> 01:10:11,360 Whereas, in sexually reproducing populations, 1362 01:10:11,360 --> 01:10:14,240 those genes can spread throughout the population, 1363 01:10:14,240 --> 01:10:16,390 sort of as genes, rather than being tied 1364 01:10:16,390 --> 01:10:18,007 to a particular individual. 1365 01:10:18,007 --> 01:10:20,590 So that means that, if you find yourself in a new environment, 1366 01:10:20,590 --> 01:10:23,140 it may be the case that sexual reproduction can 1367 01:10:23,140 --> 01:10:25,670 allow for the population to adapt more rapidly. 1368 01:10:25,670 --> 01:10:28,671 But on one hand, you might say, well, the environment's always 1369 01:10:28,671 --> 01:10:29,170 changing. 1370 01:10:29,170 --> 01:10:31,030 So that can always favor the sexually reproducing 1371 01:10:31,030 --> 01:10:31,550 populations. 1372 01:10:31,550 --> 01:10:33,680 But then, there's a feeling out there 1373 01:10:33,680 --> 01:10:35,760 that maybe that's not enough, in the sense 1374 01:10:35,760 --> 01:10:37,800 that the environment is not changing 1375 01:10:37,800 --> 01:10:39,520 rapidly enough and dramatically enough 1376 01:10:39,520 --> 01:10:43,880 to force the population to reproduce sexually as compared 1377 01:10:43,880 --> 01:10:46,270 to asexually. 1378 01:10:46,270 --> 01:10:49,040 And so, the proposal from the Red Queen Hypothesis 1379 01:10:49,040 --> 01:10:52,060 is that the constantly changing environment 1380 01:10:52,060 --> 01:10:55,870 is a result of co-evolution between hosts 1381 01:10:55,870 --> 01:10:57,930 and their parasites. 1382 01:10:57,930 --> 01:11:01,650 Because parasites are always trying to target common host 1383 01:11:01,650 --> 01:11:02,180 genotypes. 1384 01:11:02,180 --> 01:11:04,490 Because they can spread on those. 1385 01:11:04,490 --> 01:11:06,140 So the parasites are evolving. 1386 01:11:06,140 --> 01:11:08,860 And then, the host populations or genotypes 1387 01:11:08,860 --> 01:11:11,570 are kind of being chased away by those parasites that 1388 01:11:11,570 --> 01:11:12,860 are targeting them. 1389 01:11:12,860 --> 01:11:16,120 So the notion is that parasites, as a result 1390 01:11:16,120 --> 01:11:18,800 of this co-evolution, can be the source for the constantly 1391 01:11:18,800 --> 01:11:23,440 changing environment that may be driving the evolution of sex. 1392 01:11:23,440 --> 01:11:23,940 Yeah? 1393 01:11:23,940 --> 01:11:26,910 AUDIENCE: And this, proportional with [INAUDIBLE], bacteria 1394 01:11:26,910 --> 01:11:30,870 also get parasites [INAUDIBLE]. 1395 01:11:30,870 --> 01:11:32,899 PROFESSOR: That's right. 1396 01:11:32,899 --> 01:11:35,800 AUDIENCE: So-- 1397 01:11:35,800 --> 01:11:38,440 PROFESSOR: Yes, so is the question is why-- 1398 01:11:38,440 --> 01:11:40,530 AUDIENCE: Why only in multi-- 1399 01:11:40,530 --> 01:11:42,260 PROFESSOR: Right. 1400 01:11:42,260 --> 01:11:45,690 So I can give you my best guess on this. 1401 01:11:45,690 --> 01:11:49,090 First of all, not everybody agrees that the Red Queen 1402 01:11:49,090 --> 01:11:52,200 Hypothesis is the true explanation, 1403 01:11:52,200 --> 01:11:55,730 if a true explanation even exists. 1404 01:11:55,730 --> 01:11:58,120 But within this framework, you definitely 1405 01:11:58,120 --> 01:12:04,030 want to try to explain why it is that large life forms seem 1406 01:12:04,030 --> 01:12:06,350 to have a lot of obligate sexual reproduction. 1407 01:12:06,350 --> 01:12:08,711 Because this is a very strong pattern that you see. 1408 01:12:08,711 --> 01:12:10,960 And I guess what I would say is that there's certainly 1409 01:12:10,960 --> 01:12:17,620 a correlation between physical size and generation time. 1410 01:12:17,620 --> 01:12:19,490 And generation time tells you something 1411 01:12:19,490 --> 01:12:22,130 about the typical time scales over which you can evolve. 1412 01:12:22,130 --> 01:12:23,920 So my sense of this is it just maybe 1413 01:12:23,920 --> 01:12:29,190 that, in general, large animals, by their nature, 1414 01:12:29,190 --> 01:12:32,552 will evolve rather slowly as compared to their parasites. 1415 01:12:32,552 --> 01:12:34,760 And so, that means that they are the populations that 1416 01:12:34,760 --> 01:12:41,200 are most in need of speeding up their evolution. 1417 01:12:41,200 --> 01:12:44,180 And it's hard to know how convincing that argument 1418 01:12:44,180 --> 01:12:44,680 should be. 1419 01:12:44,680 --> 01:12:48,640 But-- yes? 1420 01:12:48,640 --> 01:12:52,922 AUDIENCE: What's the difference in reproductive productive time 1421 01:12:52,922 --> 01:12:55,916 between phage and bacteria? 1422 01:12:55,916 --> 01:12:57,912 Like, how fast does phage-- 1423 01:12:57,912 --> 01:12:59,120 PROFESSOR: Right. 1424 01:12:59,120 --> 01:13:03,230 So bacteria can-- as we've discussed, 1425 01:13:03,230 --> 01:13:05,870 division times are order hour. 1426 01:13:05,870 --> 01:13:11,410 And phage-- when you get a phage infection, what happens 1427 01:13:11,410 --> 01:13:14,250 is that the phage you can infect as a single phage. 1428 01:13:14,250 --> 01:13:21,130 And then, they will divide within the bacterial cell 1429 01:13:21,130 --> 01:13:24,986 and then burst out as a population of 100, 1430 01:13:24,986 --> 01:13:26,890 200-- typical of phage. 1431 01:13:26,890 --> 01:13:30,800 And I think that that might take four or five 1432 01:13:30,800 --> 01:13:32,240 hours, is kind of my sense. 1433 01:13:35,000 --> 01:13:37,180 And then, they go off and they find new bacteria. 1434 01:13:37,180 --> 01:13:42,181 So in that sense, it's a little bit faster than the bacteria, 1435 01:13:42,181 --> 01:13:42,681 I'd say. 1436 01:13:49,070 --> 01:13:52,320 So can somebody say what the core experiment was that they 1437 01:13:52,320 --> 01:13:53,670 did in this experiment? 1438 01:14:15,538 --> 01:14:17,029 Yes? 1439 01:14:17,029 --> 01:14:25,975 AUDIENCE: [INAUDIBLE] and one time, 1440 01:14:25,975 --> 01:14:29,951 it was reproducing asexually when they always 1441 01:14:29,951 --> 01:14:31,442 produce sexually. 1442 01:14:31,442 --> 01:14:33,001 And the other time, it was dividing. 1443 01:14:33,001 --> 01:14:34,000 PROFESSOR: That's right. 1444 01:14:34,000 --> 01:14:40,232 So this worm, C. Elegans-- sort of a millimeter in size-- 1445 01:14:40,232 --> 01:14:42,440 and there are going to be three different conditions. 1446 01:14:42,440 --> 01:14:51,270 One is the wild type that can out-cross, can have males mate. 1447 01:14:51,270 --> 01:14:54,510 Then, there's the obligate out-crossing, 1448 01:14:54,510 --> 01:15:00,310 which means they have to mate with males. 1449 01:15:00,310 --> 01:15:03,950 And then, there's the-- what did they call it? 1450 01:15:06,759 --> 01:15:07,550 --obligate selfing. 1451 01:15:13,505 --> 01:15:14,005 OK. 1452 01:15:18,870 --> 01:15:23,146 And then, what they do with those worms? 1453 01:15:23,146 --> 01:15:25,122 AUDIENCE: They put them right back 1454 01:15:25,122 --> 01:15:31,121 to [INAUDIBLE] more typical. 1455 01:15:31,121 --> 01:15:32,120 PROFESSOR: That's right. 1456 01:15:32,120 --> 01:15:37,360 So then, there's a bacterial pathogen, Serratia marcescens. 1457 01:15:37,360 --> 01:15:41,610 And they're going to have these three different conditions. 1458 01:15:41,610 --> 01:15:45,300 For the SM, the bacteria, they're 1459 01:15:45,300 --> 01:15:51,520 either going to allow for co-evolution, where 1460 01:15:51,520 --> 01:15:54,670 they take the bacteria from each of the infections 1461 01:15:54,670 --> 01:15:55,940 and propagate. 1462 01:15:55,940 --> 01:16:00,180 Or they're going to do the no evolution, where you just 1463 01:16:00,180 --> 01:16:03,060 compete against the ancestor. 1464 01:16:03,060 --> 01:16:04,190 And there's also a control. 1465 01:16:04,190 --> 01:16:05,720 So there's co-evolution. 1466 01:16:05,720 --> 01:16:08,544 There's this no evolution of the bacteria. 1467 01:16:08,544 --> 01:16:10,960 And then, there's also control, where there's no bacteria. 1468 01:16:17,246 --> 01:16:18,745 And what was the most striking thing 1469 01:16:18,745 --> 01:16:20,120 that they saw in this experiment? 1470 01:16:28,825 --> 01:16:32,290 AUDIENCE: I guess I would say that the obligate-selfing 1471 01:16:32,290 --> 01:16:33,990 population died in evolution. 1472 01:16:33,990 --> 01:16:35,020 PROFESSOR: That's right. 1473 01:16:35,020 --> 01:16:39,140 So if you allowed the bacteria to be evolving, 1474 01:16:39,140 --> 01:16:44,659 against this obligately selfing population, 1475 01:16:44,659 --> 01:16:45,575 this killed the worms. 1476 01:16:48,740 --> 01:16:50,430 And what was the other thing that 1477 01:16:50,430 --> 01:16:55,060 was, maybe, very striking about their experiment? 1478 01:16:55,060 --> 01:16:56,530 Yeah. 1479 01:16:56,530 --> 01:17:00,889 AUDIENCE: The worms that did the out-cross or self increased. 1480 01:17:00,889 --> 01:17:01,430 [INAUDIBLE]-- 1481 01:17:01,430 --> 01:17:03,510 PROFESSOR: Yeah, so this was kind of amazing. 1482 01:17:03,510 --> 01:17:06,130 So they saw, as a function of time, 1483 01:17:06,130 --> 01:17:10,400 if you look at out-crossing-- the rate of mating 1484 01:17:10,400 --> 01:17:15,340 with the males-- this started out at, like, 0.2. 1485 01:17:15,340 --> 01:17:19,130 And then, in the presence of the co-evolution, it went up. 1486 01:17:19,130 --> 01:17:22,510 And it goes up to, maybe, 80%. 1487 01:17:25,760 --> 01:17:29,470 And for co-evolution, it stayed up high. 1488 01:17:29,470 --> 01:17:35,250 Whereas, if the wild-type worms continued 1489 01:17:35,250 --> 01:17:38,780 to be just challenged by the ancestral bacteria, 1490 01:17:38,780 --> 01:17:39,890 it initially came up. 1491 01:17:39,890 --> 01:17:41,324 But then, it came back down. 1492 01:17:41,324 --> 01:17:42,490 So this is the co-evolution. 1493 01:17:45,340 --> 01:17:51,950 And this is the ancestral SM. 1494 01:17:51,950 --> 01:17:54,205 So there was a sense that that wild-type population 1495 01:17:54,205 --> 01:17:57,040 had initially evolved to out-cross more. 1496 01:17:57,040 --> 01:17:59,040 But then, once it had solved this problem of how 1497 01:17:59,040 --> 01:18:01,249 to handle the ancestral Serratia, 1498 01:18:01,249 --> 01:18:02,790 the out-crossing rate went back down. 1499 01:18:05,450 --> 01:18:07,550 So we are pretty much out of time. 1500 01:18:07,550 --> 01:18:09,869 But I don't know if you guys noticed 1501 01:18:09,869 --> 01:18:11,160 the last sentence of the paper. 1502 01:18:11,160 --> 01:18:15,405 It is amazing that they got this through the publication 1503 01:18:15,405 --> 01:18:15,905 process. 1504 01:18:18,810 --> 01:18:21,100 All right, so they say, "taken together, 1505 01:18:21,100 --> 01:18:24,150 the results demonstrate that sex can facilitate adaptation 1506 01:18:24,150 --> 01:18:25,700 to novel environments. 1507 01:18:25,700 --> 01:18:27,620 But the long-term maintenance of sex 1508 01:18:27,620 --> 01:18:30,396 requires that the novelty does not wear off." 1509 01:18:30,396 --> 01:18:31,870 [LAUGHTER] 1510 01:18:31,870 --> 01:18:36,310 It's one of those things that you read and you think-- OK. 1511 01:18:36,310 --> 01:18:38,570 So I will leave that sentence with you. 1512 01:18:38,570 --> 01:18:43,670 And then, we'll meet on Tuesday, OK?