WEBVTT 00:00:00.060 --> 00:00:02.500 The following content is provided under a Creative 00:00:02.500 --> 00:00:04.019 Commons license. 00:00:04.019 --> 00:00:06.360 Your support will help MIT OpenCourseWare 00:00:06.360 --> 00:00:10.730 continue to offer high-quality educational resources for free. 00:00:10.730 --> 00:00:13.330 To make a donation or view additional materials 00:00:13.330 --> 00:00:17.212 from hundreds of MIT courses, visit MIT OpenCourseWare 00:00:17.212 --> 00:00:17.837 at ocw.mit.edu. 00:00:20.975 --> 00:00:22.350 PROFESSOR: So today, what we want 00:00:22.350 --> 00:00:25.402 to do is we want to talk about two related topics. 00:00:25.402 --> 00:00:26.860 The first is going to this question 00:00:26.860 --> 00:00:28.670 of the evolution of virulence and how 00:00:28.670 --> 00:00:31.830 to model host-parasite interactions more broadly. 00:00:31.830 --> 00:00:34.580 That's kind of modeled on chapter 11 of Martin's book. 00:00:34.580 --> 00:00:38.040 We'll focus on the first half of it for the discussions today. 00:00:38.040 --> 00:00:41.320 This is in the context of when a given host can only 00:00:41.320 --> 00:00:46.320 have one strain of the parasite or virus or whatnot 00:00:46.320 --> 00:00:49.890 inside that body. 00:00:49.890 --> 00:00:51.630 The model presented in Martin's book 00:00:51.630 --> 00:00:56.590 is very similar to classic models in epidemiology, which 00:00:56.590 --> 00:00:59.750 are the so-called SIR type models, 00:00:59.750 --> 00:01:02.130 where you divide up the host population 00:01:02.130 --> 00:01:05.200 into whether the they are sensitive-- i.e., 00:01:05.200 --> 00:01:09.360 non-infected-- infected, or resistant. 00:01:09.360 --> 00:01:11.120 And then, we'll draw the parallels 00:01:11.120 --> 00:01:15.030 of how we get from the model that you read about in Martin's 00:01:15.030 --> 00:01:17.990 book to the classic SIR models. 00:01:17.990 --> 00:01:21.450 But in both of these cases, the fundamental parameter 00:01:21.450 --> 00:01:24.690 that drives these things is this R0 parameter. 00:01:24.690 --> 00:01:29.989 It tells us about the expected number of new cases 00:01:29.989 --> 00:01:32.280 that will result when you introduce one infected member 00:01:32.280 --> 00:01:33.664 into the population. 00:01:33.664 --> 00:01:37.460 AUDIENCE: Sorry, will you be taking about super-infections? 00:01:37.460 --> 00:01:39.280 PROFESSOR: Only a little bit but, I 00:01:39.280 --> 00:01:40.920 would say, depending on time. 00:01:40.920 --> 00:01:43.530 But if you're interested in the super-infection discussion 00:01:43.530 --> 00:01:46.156 more, we can talk about it after class, maybe. 00:01:46.156 --> 00:01:46.655 All right. 00:01:49.609 --> 00:01:51.150 And so, for the second half of class, 00:01:51.150 --> 00:01:52.733 what we're going to do though is we're 00:01:52.733 --> 00:01:56.320 going to talk about the possible evolutionary benefits of sex. 00:01:56.320 --> 00:01:59.700 And in particular, we'll talk about this hypothesis, which 00:01:59.700 --> 00:02:02.110 is one of the reigning hypotheses for why it might 00:02:02.110 --> 00:02:05.640 be that sex is as widespread as it is, which is the Red Queen 00:02:05.640 --> 00:02:10.970 hypothesis, from Lewis Carroll's novel. 00:02:10.970 --> 00:02:13.730 And we're going to discuss this paper that you guys read about, 00:02:13.730 --> 00:02:15.188 "Running with the Red Queen," which 00:02:15.188 --> 00:02:18.290 I think has a nice discussion of this debate 00:02:18.290 --> 00:02:20.130 and, then, some nice experiments looking 00:02:20.130 --> 00:02:23.880 at experimental coevolution between the C. elegans worm 00:02:23.880 --> 00:02:27.420 and it's infecting parasite, which is a serratia bacterium. 00:02:30.190 --> 00:02:31.880 Any questions before we get going? 00:02:37.770 --> 00:02:42.390 OK, what I want to do is start by discussing 00:02:42.390 --> 00:02:45.630 this model in Martin's book. 00:02:45.630 --> 00:02:49.860 But also, there's a little bit of this philosophical question. 00:02:49.860 --> 00:02:53.600 Any time, that you are modeling, you always 00:02:53.600 --> 00:02:55.869 had to make decisions about which of the details 00:02:55.869 --> 00:02:57.910 you want to try to model and which of the details 00:02:57.910 --> 00:02:59.290 you do not want to model. 00:02:59.290 --> 00:03:01.750 And depending upon the situation, 00:03:01.750 --> 00:03:03.890 it may be that some assumptions are more or less 00:03:03.890 --> 00:03:05.510 appropriate than others. 00:03:05.510 --> 00:03:11.920 Now, in the model that Martin wrote down-- well, 00:03:11.920 --> 00:03:15.150 we'll try to figure out what the assumptions are here. 00:03:15.150 --> 00:03:18.140 So we have what you might think of 00:03:18.140 --> 00:03:20.840 as some sensitive individuals. 00:03:20.840 --> 00:03:24.240 Plus, the infected individuals are going 00:03:24.240 --> 00:03:26.590 to interact at some rate, beta. 00:03:26.590 --> 00:03:30.230 So this is how the sensitive become infected. 00:03:30.230 --> 00:03:33.300 And it results in, now, two infected individuals. 00:03:36.020 --> 00:03:37.920 Now of course, each of these individuals 00:03:37.920 --> 00:03:41.810 will, say, have some lifespan or die at some rate. 00:03:41.810 --> 00:03:47.200 All right, so the sensitive or uninfected individuals 00:03:47.200 --> 00:03:48.840 die at rate u. 00:03:48.840 --> 00:03:53.290 Whereas, infected individuals, others-- an increase 00:03:53.290 --> 00:03:56.658 in the death-rate, described by some virulence, v. OK? 00:04:01.817 --> 00:04:03.230 OK. 00:04:03.230 --> 00:04:05.136 Now, the model as written-- what's 00:04:05.136 --> 00:04:06.760 going to be the fate of the population? 00:04:11.870 --> 00:04:13.774 AUDIENCE: Everyone all dies. 00:04:13.774 --> 00:04:14.730 PROFESSOR: Yeah. 00:04:14.730 --> 00:04:16.579 Everyone's going to die, right? 00:04:16.579 --> 00:04:18.279 And that's even true in the-- it's not 00:04:18.279 --> 00:04:22.910 even that the population's dying as a result of the infection. 00:04:22.910 --> 00:04:26.216 Because even in the absence of any infected individuals, 00:04:26.216 --> 00:04:27.340 you just have people dying. 00:04:27.340 --> 00:04:31.270 So you need to have some way of keeping the population going, 00:04:31.270 --> 00:04:33.080 so you can study it perhaps. 00:04:33.080 --> 00:04:36.830 What we're going to assume is that sensitive individuals, 00:04:36.830 --> 00:04:39.770 or uninfected individuals, will enter the population just 00:04:39.770 --> 00:04:40.650 at some rate, k. 00:04:44.330 --> 00:04:50.120 Now, in terms of the philosophical question, 00:04:50.120 --> 00:04:52.680 in the beginning of the chapter, Martin 00:04:52.680 --> 00:04:55.820 talks a bit about this question of microparasites 00:04:55.820 --> 00:04:58.460 versus macroparasites. 00:04:58.460 --> 00:05:02.425 And I can somebody remind us, what's the distinction? 00:05:02.425 --> 00:05:06.695 and For what kind of parasite might this 00:05:06.695 --> 00:05:07.790 be intended to model? 00:05:12.560 --> 00:05:13.080 Yes? 00:05:13.080 --> 00:05:20.460 AUDIENCE: Well, what I got from it is that microparasites 00:05:20.460 --> 00:05:27.348 are on the order of single-cellular organisms, 00:05:27.348 --> 00:05:34.256 generally things that have much shorter reproductive steps, 00:05:34.256 --> 00:05:35.679 I guess. 00:05:35.679 --> 00:05:37.220 They reproduce a lot more frequently. 00:05:37.220 --> 00:05:38.768 PROFESSOR: That's right. 00:05:38.768 --> 00:05:40.184 AUDIENCE: Whereas, macro-parasites 00:05:40.184 --> 00:05:43.642 are like the [INAUDIBLE] or the tapeworm or something, 00:05:43.642 --> 00:05:48.088 which would not necessarily reproduce 00:05:48.088 --> 00:05:49.570 a lot inside the host. 00:05:49.570 --> 00:05:50.710 PROFESSOR: Mm-hm. 00:05:50.710 --> 00:05:51.210 Right. 00:05:51.210 --> 00:05:53.690 And I would say, that just given this distinction 00:05:53.690 --> 00:05:56.420 between the microparasites, that might be viruses and bacteria, 00:05:56.420 --> 00:05:57.836 as compared to the macroparasites, 00:05:57.836 --> 00:06:00.720 that are things like tapeworms and so forth, it's 00:06:00.720 --> 00:06:03.950 not obvious from that that you would have two 00:06:03.950 --> 00:06:05.520 different modeling frameworks. 00:06:05.520 --> 00:06:11.050 But what is the argument that is made in Martin's book? 00:06:11.050 --> 00:06:13.030 Or can you think up an argument for why 00:06:13.030 --> 00:06:15.270 it is that it might be this kind of model you would 00:06:15.270 --> 00:06:16.561 want to use for microparasites? 00:06:19.145 --> 00:06:19.726 Yes. 00:06:19.726 --> 00:06:21.101 AUDIENCE: Because we don't really 00:06:21.101 --> 00:06:22.568 care about-- he mentioned something 00:06:22.568 --> 00:06:24.035 that the microparasites reproduce 00:06:24.035 --> 00:06:26.480 in large numbers in infected individuals. 00:06:26.480 --> 00:06:29.169 So we don't have to keep track of the internal state 00:06:29.169 --> 00:06:31.380 of someone that's infected. 00:06:31.380 --> 00:06:32.510 PROFESSOR: That's right. 00:06:32.510 --> 00:06:35.440 And some of it's, maybe, even a historical thing. 00:06:35.440 --> 00:06:40.590 There might be huge numbers of viruses-- a flu virus or so-- 00:06:40.590 --> 00:06:42.440 in an infected individual. 00:06:42.440 --> 00:06:44.430 And in some ways, maybe, the number 00:06:44.430 --> 00:06:48.409 of viruses that is in that host is not the most relevant thing. 00:06:48.409 --> 00:06:49.950 And it's certainly would be much more 00:06:49.950 --> 00:06:51.730 complicated to try to keep track of that. 00:06:51.730 --> 00:06:57.635 And so, if you can get meaningful predictions-- rather 00:06:57.635 --> 00:07:00.260 than keeping track of the number of viruses, say, in each host, 00:07:00.260 --> 00:07:03.570 instead you just put the host into different classes-- 00:07:03.570 --> 00:07:06.199 sensitive and infected, for example. 00:07:06.199 --> 00:07:07.990 Later, we'll talk about what happens if you 00:07:07.990 --> 00:07:10.150 have a resistant type of class. 00:07:10.150 --> 00:07:13.730 But the idea there is that there's, maybe, even also 00:07:13.730 --> 00:07:15.550 some separation of time scales. 00:07:15.550 --> 00:07:16.830 Because you get infected. 00:07:16.830 --> 00:07:22.376 And kind of quickly, you're just sick and may be infective. 00:07:22.376 --> 00:07:23.750 But at some rate, you get better. 00:07:23.750 --> 00:07:26.120 And it's not that you'll necessarily 00:07:26.120 --> 00:07:28.890 gain very much by keeping track of the precise number 00:07:28.890 --> 00:07:30.450 of viruses in the host. 00:07:30.450 --> 00:07:33.050 Of course, this is ultimately an experimental observational 00:07:33.050 --> 00:07:34.840 question of whether this sort of model 00:07:34.840 --> 00:07:36.552 provides you inside that you're going 00:07:36.552 --> 00:07:38.510 to need to make sense of these diseases, right? 00:07:38.510 --> 00:07:40.995 AUDIENCE: And then it also seems like your method 00:07:40.995 --> 00:07:43.977 of transmission of macroparasites 00:07:43.977 --> 00:07:45.480 can be very different. 00:07:45.480 --> 00:07:46.480 PROFESSOR: That's right. 00:07:46.480 --> 00:07:48.510 So the mechanism of transmission depends 00:07:48.510 --> 00:07:52.310 very much on the disease that you're studying. 00:07:52.310 --> 00:07:55.200 And the macroparasites, in many cases, 00:07:55.200 --> 00:07:58.260 they're transmitted not from direct interactions 00:07:58.260 --> 00:08:02.450 between the hosts, but through the environment or something 00:08:02.450 --> 00:08:02.950 else. 00:08:07.120 --> 00:08:09.140 It's also, perhaps, just worth pointing out 00:08:09.140 --> 00:08:14.260 that parasites are just a ubiquitous aspect of life. 00:08:14.260 --> 00:08:19.290 So you can name an organism, and you can pretty much 00:08:19.290 --> 00:08:21.560 be guaranteed that there's going to be some notion 00:08:21.560 --> 00:08:23.650 of a parasite on that organism. 00:08:23.650 --> 00:08:26.860 And there can be multiple layers of this. 00:08:26.860 --> 00:08:30.590 So we certainly have many parasites. 00:08:30.590 --> 00:08:34.630 We're infected by many viruses and bacteria and other things. 00:08:34.630 --> 00:08:37.830 But bacteria-- we think of them as being very small-- 00:08:37.830 --> 00:08:42.289 they're also preyed upon by these phage, which 00:08:42.289 --> 00:08:46.842 is a parasite that targets specifically bacteria. 00:08:46.842 --> 00:08:48.800 So it's not just that it's an incidental thing. 00:08:48.800 --> 00:08:52.570 But these are really viruses that have evolved specifically 00:08:52.570 --> 00:08:55.920 to divide in bacteria. 00:08:55.920 --> 00:08:58.560 And we didn't really talk about this very much. 00:08:58.560 --> 00:09:02.890 But one of the classic models for cooperation and cheating 00:09:02.890 --> 00:09:05.690 is based on what you could think about as 00:09:05.690 --> 00:09:10.120 some sort of parasitic sub-population within phage. 00:09:10.120 --> 00:09:12.110 So this is a classic paper by Lin Chow where 00:09:12.110 --> 00:09:14.480 he showed that if you evolved phage and bacteria 00:09:14.480 --> 00:09:18.290 in a condition where many phage infect a given bacteria, then, 00:09:18.290 --> 00:09:22.394 you can evolve what you could think of as cheater 00:09:22.394 --> 00:09:23.560 strategies or cheater phage. 00:09:23.560 --> 00:09:26.270 Because these are phage that maybe 00:09:26.270 --> 00:09:28.680 can't reproduce on their own, but have shorter genomes 00:09:28.680 --> 00:09:32.350 and can out-replicate the normal phage. 00:09:32.350 --> 00:09:35.550 So if both of these end up in a single bacterial cell, 00:09:35.550 --> 00:09:38.560 then these cheater phage can spread 00:09:38.560 --> 00:09:40.810 by taking advantage of, say, the replication machinery 00:09:40.810 --> 00:09:42.060 from the rest of the phage. 00:09:42.060 --> 00:09:44.740 So in some ways, you might call that some sort of DNA 00:09:44.740 --> 00:09:46.172 parasite or so. 00:09:46.172 --> 00:09:48.630 So there's really parasites in many, many different levels. 00:09:53.965 --> 00:09:55.280 AUDIENCE: [INAUDIBLE]. 00:09:55.280 --> 00:09:55.905 PROFESSOR: Yes? 00:09:55.905 --> 00:09:58.330 AUDIENCE: [INAUDIBLE] ultimately think about is, k, 00:09:58.330 --> 00:10:01.260 is this really the number-- 00:10:01.260 --> 00:10:04.050 PROFESSOR: Yeah, you know, I would 00:10:04.050 --> 00:10:08.470 say that the k is, in some ways, not a very satisfying feature 00:10:08.470 --> 00:10:09.120 of this model. 00:10:09.120 --> 00:10:12.670 Because it makes it feel that the model is very special, 00:10:12.670 --> 00:10:13.170 right? 00:10:13.170 --> 00:10:15.265 AUDIENCE: But I mean, in a natural population 00:10:15.265 --> 00:10:19.740 and rate at which people are born, is that the same change? 00:10:19.740 --> 00:10:21.460 Or-- 00:10:21.460 --> 00:10:23.060 PROFESSOR: Yes. 00:10:23.060 --> 00:10:25.320 That was the example I was going to give. 00:10:25.320 --> 00:10:31.170 It's not clear-- of course, it requires somebody 00:10:31.170 --> 00:10:33.370 to give birth to kids, right? 00:10:33.370 --> 00:10:36.705 So in that sense, modeling this as a constant number per unit 00:10:36.705 --> 00:10:39.660 of time-- which is what we're doing-- this rate k, 00:10:39.660 --> 00:10:40.730 is a little bit funny. 00:10:40.730 --> 00:10:42.480 Because then, what you'd really want to do 00:10:42.480 --> 00:10:44.970 is say, oh, maybe it's these guys that 00:10:44.970 --> 00:10:47.130 give birth at some rate or so, if you really 00:10:47.130 --> 00:10:48.420 wanted to be accurate. 00:10:48.420 --> 00:10:50.470 So I'd say that this is, in some ways, just 00:10:50.470 --> 00:10:53.142 a mathematical simplification so that we can get 00:10:53.142 --> 00:10:54.350 at the heart of the dynamics. 00:10:54.350 --> 00:10:57.020 And what we'll see is that, in these SIR models, 00:10:57.020 --> 00:11:00.180 you don't invoke anything like this. 00:11:00.180 --> 00:11:03.510 But rather, what you do is you assume that, at some rate, 00:11:03.510 --> 00:11:07.430 infected individuals don't just die. 00:11:07.430 --> 00:11:09.500 But they become resistant. 00:11:09.500 --> 00:11:12.560 And then, maybe later, they become sensitive again. 00:11:12.560 --> 00:11:14.956 So you need some way of being sure that it's not 00:11:14.956 --> 00:11:19.080 the case that everybody just always dies. 00:11:19.080 --> 00:11:21.700 So in some ways, this is more mathematical convenience. 00:11:21.700 --> 00:11:23.589 And the basic conclusions end up being 00:11:23.589 --> 00:11:25.130 very robust to these sorts of things. 00:11:32.420 --> 00:11:34.250 All right, so in these models, it's 00:11:34.250 --> 00:11:37.180 always good to be clear about how 00:11:37.180 --> 00:11:40.280 we go from this framework to something that is more 00:11:40.280 --> 00:11:42.870 of a differential equation. 00:11:42.870 --> 00:11:44.500 And what we can do is we can think 00:11:44.500 --> 00:11:49.920 about these uninfected individuals, i.e. 00:11:49.920 --> 00:11:51.605 the S-es as compared to the infected. 00:11:55.060 --> 00:11:57.335 And we're going to have these guys be x. 00:12:01.680 --> 00:12:05.040 And this S and I. 00:12:05.040 --> 00:12:08.670 So the way that the x will be changing 00:12:08.670 --> 00:12:11.135 is that we're assuming that there's always 00:12:11.135 --> 00:12:12.510 some influx of individuals, which 00:12:12.510 --> 00:12:15.035 could be birth or migration or something else, 00:12:15.035 --> 00:12:17.537 that are just always entering. 00:12:17.537 --> 00:12:19.120 But then, there's going to be two ways 00:12:19.120 --> 00:12:21.670 that x is going to decrease. 00:12:21.670 --> 00:12:25.780 One is that there's just a death rate that is resulting 00:12:25.780 --> 00:12:27.627 in the absence of infection. 00:12:27.627 --> 00:12:29.085 But then, also, there's going to be 00:12:29.085 --> 00:12:30.110 some rate of infection, which is going 00:12:30.110 --> 00:12:31.470 to be proportional to beta. 00:12:31.470 --> 00:12:33.960 So this is the simplest way that you 00:12:33.960 --> 00:12:37.890 can imagine capturing this element 00:12:37.890 --> 00:12:42.870 that the infected individuals can transmit the infection 00:12:42.870 --> 00:12:45.090 to the sensitive individuals. 00:12:45.090 --> 00:12:50.290 So we're modeling them as a well-mixed population, 00:12:50.290 --> 00:12:51.780 just like in chemical reactions. 00:12:51.780 --> 00:12:54.500 And somehow, the rate of infection 00:12:54.500 --> 00:12:58.080 is proportional to the frequency that they hit each other. 00:12:58.080 --> 00:13:02.810 Certainly, the simplest kind of model you can imagine. 00:13:02.810 --> 00:13:06.250 Whereas, the infected individuals-- well, 00:13:06.250 --> 00:13:09.530 we're going to have an increased rate of death. 00:13:09.530 --> 00:13:12.180 So this is a simplified way to write it. 00:13:17.120 --> 00:13:20.230 So this is really that there's a minus, u plus v times y. 00:13:20.230 --> 00:13:23.130 So this is just the death rate. 00:13:23.130 --> 00:13:27.530 But then, any individual that leaves the sensitive class-- 00:13:27.530 --> 00:13:31.460 this minus beta xy-- enters the infected class. 00:13:36.120 --> 00:13:37.860 This makes a lot of sense, I think. 00:13:37.860 --> 00:13:42.310 Now, the question is, can we make sense of what's going on? 00:13:42.310 --> 00:13:49.215 Now, you saw in your reading what this R0 parameter was. 00:13:49.215 --> 00:13:50.960 And you should always remember that it's 00:13:50.960 --> 00:13:54.970 defined as this thing of, if you introduce 00:13:54.970 --> 00:13:57.460 one infected individual into a population 00:13:57.460 --> 00:13:59.540 of sensitive individuals, what is 00:13:59.540 --> 00:14:03.430 the mean number of new infections that you get? 00:14:03.430 --> 00:14:07.730 And it makes sense that the key thing is whether that R0 00:14:07.730 --> 00:14:09.026 is greater or less than 1. 00:14:09.026 --> 00:14:10.400 Because if it's greater than one, 00:14:10.400 --> 00:14:13.204 that leads to this exponential explosion 00:14:13.204 --> 00:14:14.370 of the infected individuals. 00:14:14.370 --> 00:14:17.500 It doesn't mean that everyone's going to die, necessarily. 00:14:17.500 --> 00:14:18.580 We'll get into that. 00:14:18.580 --> 00:14:20.760 But if R0 is less than 1, then you 00:14:20.760 --> 00:14:23.710 expect that infection to die out. 00:14:23.710 --> 00:14:27.380 So why is it that if R0 is greater than 1, 00:14:27.380 --> 00:14:29.130 and you introduce one infected individual, 00:14:29.130 --> 00:14:33.149 it doesn't necessarily lead to a wipe-out 00:14:33.149 --> 00:14:34.190 of the entire population? 00:14:42.860 --> 00:14:44.822 Or maybe it does. 00:14:44.822 --> 00:14:46.030 This model is a little funny. 00:14:46.030 --> 00:14:47.988 Because you always have an individual entering. 00:14:51.581 --> 00:14:52.080 But-- 00:14:52.080 --> 00:14:54.415 AUDIENCE: If the [INAUDIBLE] is really virulent, 00:14:54.415 --> 00:14:56.872 then only infected individuals die before-- 00:14:56.872 --> 00:14:57.580 PROFESSOR: Right. 00:14:57.580 --> 00:14:59.984 So if it's very virulent, then the infected individuals 00:14:59.984 --> 00:15:00.650 may die quickly. 00:15:00.650 --> 00:15:01.930 And this gets into this question of there 00:15:01.930 --> 00:15:03.804 may be some trade-offs in terms of virulence. 00:15:03.804 --> 00:15:07.190 And we'll talk more about, then, the evolution of virulence. 00:15:07.190 --> 00:15:12.820 But there's a wide variety of classes of models, 00:15:12.820 --> 00:15:16.020 not just the ones where you have k entering. 00:15:16.020 --> 00:15:20.461 But the question somehow is-- just because R0 00:15:20.461 --> 00:15:21.960 is greater than 1, that doesn't mean 00:15:21.960 --> 00:15:25.667 that the entire population will necessarily become infected. 00:15:25.667 --> 00:15:27.750 Because we have this idea of an exponential growth 00:15:27.750 --> 00:15:32.740 of the infected population if R0 is greater than 1. 00:15:32.740 --> 00:15:35.330 So why is it that it's not necessarily 00:15:35.330 --> 00:15:40.780 going to happen that the entire population is-- well, 00:15:40.780 --> 00:15:43.411 this question's a little bit ill-posed in this model. 00:15:43.411 --> 00:15:47.259 AUDIENCE: Is it because the number of sensitive individuals 00:15:47.259 --> 00:15:49.320 becomes very low at some point and-- 00:15:49.320 --> 00:15:50.320 PROFESSOR: That's right. 00:15:50.320 --> 00:15:52.010 And I think this is the basic intuition. 00:15:52.010 --> 00:15:56.440 As more and more members of the population become infected, 00:15:56.440 --> 00:15:58.610 then that could, in principle, reduce 00:15:58.610 --> 00:16:04.760 to the possibilities for new individuals to be susceptible. 00:16:09.770 --> 00:16:15.310 And I'm using susceptible and sensitive interchangeably. 00:16:15.310 --> 00:16:18.990 And so eventually, this exponential growth 00:16:18.990 --> 00:16:20.951 of the population can be limited in some way. 00:16:20.951 --> 00:16:22.700 We'll maybe look at this a little bit more 00:16:22.700 --> 00:16:23.960 in this SIR model. 00:16:23.960 --> 00:16:25.636 Because it's more clear. 00:16:25.636 --> 00:16:29.890 AUDIENCE: So we basically collect a rate. 00:16:29.890 --> 00:16:31.404 R0 is a rate, right? 00:16:34.258 --> 00:16:34.758 [INAUDIBLE] 00:16:34.758 --> 00:16:36.240 PROFESSOR: No. 00:16:36.240 --> 00:16:38.440 It's a number. 00:16:38.440 --> 00:16:40.894 It's the expected number of new infections 00:16:40.894 --> 00:16:43.185 that you get when you introduce one infected individual 00:16:43.185 --> 00:16:45.681 into the population. 00:16:45.681 --> 00:16:46.180 And we're-- 00:16:46.180 --> 00:16:47.801 AUDIENCE: It's like, period. 00:16:47.801 --> 00:16:51.730 So there's not, like, per-unit time or-- 00:16:51.730 --> 00:16:55.030 PROFESSOR: It's a number, period. 00:16:55.030 --> 00:16:57.940 AUDIENCE: OK, so if you have R equal to 1, 00:16:57.940 --> 00:16:59.386 you expect that when you introduce 00:16:59.386 --> 00:17:01.073 an infected individual into a population 00:17:01.073 --> 00:17:04.051 of sensitive individuals, you will get one other infected 00:17:04.051 --> 00:17:04.550 individual. 00:17:04.550 --> 00:17:04.710 PROFESSOR: That's right. 00:17:04.710 --> 00:17:07.240 So that's kind of the neutrally-stable situation. 00:17:07.240 --> 00:17:09.516 If R0 is 1, then you add one infected individual 00:17:09.516 --> 00:17:12.099 and you expect to get one other one, so you have a random walk 00:17:12.099 --> 00:17:14.970 and-- well, in general, it will randomly 00:17:14.970 --> 00:17:16.221 go extinct, eventually. 00:17:16.221 --> 00:17:16.720 Yes? 00:17:16.720 --> 00:17:19.672 AUDIENCE: What is the lifetime of the infected-- 00:17:19.672 --> 00:17:21.530 PROFESSOR: So it depends. 00:17:21.530 --> 00:17:23.560 And so, that's what we're going to do right now 00:17:23.560 --> 00:17:27.339 is see if we can reconstruct what this R0 is equal to 00:17:27.339 --> 00:17:28.860 in this model. 00:17:28.860 --> 00:17:30.720 AUDIENCE: Another quick question-- 00:17:30.720 --> 00:17:34.060 what's the distribution of the-- so-- 00:17:34.060 --> 00:17:34.690 PROFESSOR: Yes. 00:17:34.690 --> 00:17:35.340 AUDIENCE: --a deterministic role-- 00:17:35.340 --> 00:17:36.360 PROFESSOR: Yes, exactly. 00:17:36.360 --> 00:17:38.317 So this is very interesting. 00:17:38.317 --> 00:17:40.400 The question is, what is going to the distribution 00:17:40.400 --> 00:17:44.160 of the number of infected individuals? 00:17:44.160 --> 00:17:45.920 And in this model, we're assuming 00:17:45.920 --> 00:17:49.690 that every infected individuals is the same. 00:17:49.690 --> 00:17:51.590 So we should be able to-- I wish that we 00:17:51.590 --> 00:17:54.580 just, on the wall somewhere, had our five different standard 00:17:54.580 --> 00:17:56.860 probability distributions, so that we could always 00:17:56.860 --> 00:17:58.980 go back to them. 00:17:58.980 --> 00:18:01.180 So the question is, in this model, 00:18:01.180 --> 00:18:04.486 if you introduce one infected individual in the population, 00:18:04.486 --> 00:18:06.360 how many new infected individuals do you get? 00:18:06.360 --> 00:18:08.800 R0 tells you about the mean. 00:18:08.800 --> 00:18:11.270 But will you always get the mean? 00:18:11.270 --> 00:18:13.270 No. 00:18:13.270 --> 00:18:15.160 So let's all think about this for 10 seconds. 00:18:15.160 --> 00:18:17.500 And we will verbally yell out what 00:18:17.500 --> 00:18:19.250 we think the distribution will be 00:18:19.250 --> 00:18:21.610 of the number of new infections from a single infected 00:18:21.610 --> 00:18:22.110 individual. 00:18:24.755 --> 00:18:25.255 OK? 00:18:31.960 --> 00:18:32.870 Verbally, ready! 00:18:32.870 --> 00:18:34.146 Five, (WHISPERING) four-- 00:18:36.730 --> 00:18:37.480 AUDIENCE: Poisson. 00:18:37.480 --> 00:18:39.722 AUDIENCE: Piosson. 00:18:39.722 --> 00:18:41.384 AUDIENCE: Exponential 00:18:41.384 --> 00:18:43.550 PROFESSOR: All right, everybody thinks it's Poisson. 00:18:43.550 --> 00:18:44.549 Why would it be Poisson? 00:18:47.670 --> 00:18:50.539 AUDIENCE: Because you have a rate-- 00:18:50.539 --> 00:18:51.622 AUDIENCE: It's not a rate. 00:18:51.622 --> 00:18:53.598 AUDIENCE: It's not? 00:18:53.598 --> 00:18:55.862 AUDIENCE: The initial population is much larger-- 00:18:55.862 --> 00:18:56.570 PROFESSOR: Right. 00:18:56.570 --> 00:18:58.510 OK, so the idea is that we imagine 00:18:58.510 --> 00:19:00.490 we're some infected individual. 00:19:00.490 --> 00:19:04.770 And there's some rate that we are infecting others. 00:19:04.770 --> 00:19:09.110 Now, if I ask you the question, how many individuals will I 00:19:09.110 --> 00:19:13.990 infect in the next 10 days? 00:19:13.990 --> 00:19:17.030 Or we could do 21 days if you guys like that. 00:19:17.030 --> 00:19:19.760 So all right, if I ask, how many do I 00:19:19.760 --> 00:19:23.010 infect in the next 10 days? 00:19:23.010 --> 00:19:24.680 Now, I'm assuming that I stay alive. 00:19:24.680 --> 00:19:26.349 But let's say, assuming I stay alive, 00:19:26.349 --> 00:19:28.140 how many do I infect over the next 10 days? 00:19:28.140 --> 00:19:33.750 That's going to be distributed as-- that is Poisson. 00:19:33.750 --> 00:19:36.280 But the question you're asking is a different one. 00:19:36.280 --> 00:19:39.440 You're asking, what is going to be 00:19:39.440 --> 00:19:41.690 the distribution of the total number of new infections 00:19:41.690 --> 00:19:43.710 that I cause? 00:19:43.710 --> 00:19:48.130 And this is precisely the same situation 00:19:48.130 --> 00:19:50.930 that we've analyzed lots and lots of times. 00:19:50.930 --> 00:19:52.155 What does it look like? 00:19:52.155 --> 00:19:53.580 AUDIENCE: Geometric distribution. 00:19:53.580 --> 00:19:54.205 PROFESSOR: Hmm? 00:19:54.205 --> 00:19:55.480 AUDIENCE: Geometric. 00:19:55.480 --> 00:19:56.750 PROFESSOR: OK, yes. 00:19:56.750 --> 00:19:57.910 It's going to be a geometric distribution. 00:19:57.910 --> 00:19:58.470 But why? 00:19:58.470 --> 00:20:01.890 AUDIENCE: Well, because you can have an infection and then 00:20:01.890 --> 00:20:02.640 another infection. 00:20:02.640 --> 00:20:04.050 And so then, it's, like, multiples of ten. 00:20:04.050 --> 00:20:05.050 PROFESSOR: That's right. 00:20:05.050 --> 00:20:08.530 So we have an infected individual. 00:20:08.530 --> 00:20:10.120 There's two things that can happen. 00:20:12.930 --> 00:20:14.600 He's going to die at some rate. 00:20:14.600 --> 00:20:21.560 And there's this other rate, which 00:20:21.560 --> 00:20:25.367 is going to go as beta times x or so, 00:20:25.367 --> 00:20:27.200 telling us about the rate of new infections. 00:20:27.200 --> 00:20:28.908 And we want to know, how many times do we 00:20:28.908 --> 00:20:34.310 go around this loop before we degrade, or die, or something, 00:20:34.310 --> 00:20:35.849 disappear from the population? 00:20:35.849 --> 00:20:37.140 Does this look at all familiar? 00:20:37.140 --> 00:20:38.170 AUDIENCE: Mm-hmm. 00:20:38.170 --> 00:20:38.460 PROFESSOR: All right. 00:20:38.460 --> 00:20:40.145 Were you guys the same students that 00:20:40.145 --> 00:20:41.345 were here for the first half of the class? 00:20:41.345 --> 00:20:42.240 AUDIENCE: Ha. 00:20:42.240 --> 00:20:43.038 PROFESSOR: Yeah? 00:20:43.038 --> 00:20:43.538 No. 00:20:43.538 --> 00:20:47.031 AUDIENCE: And so, R0 is like the [INAUDIBLE]? 00:20:47.031 --> 00:20:52.021 It's like the number of-- 00:20:52.021 --> 00:20:54.017 AUDIENCE: It's the mean number for bursts. 00:20:54.017 --> 00:20:56.530 AUDIENCE: You haven't evaluated when the population-- 00:20:56.530 --> 00:21:00.342 PROFESSOR: So R0 is the mean size of protein bursts, 00:21:00.342 --> 00:21:02.050 in the context of this other model, which 00:21:02.050 --> 00:21:03.258 was-- what was the situation? 00:21:05.471 --> 00:21:07.720 And this one's a really good one for you guys to know. 00:21:07.720 --> 00:21:08.719 It's going to be useful. 00:21:13.230 --> 00:21:17.970 So we saw this exact model in the context of gene expression. 00:21:17.970 --> 00:21:20.381 And what was the situation that we-- 00:21:20.381 --> 00:21:21.880 AUDIENCE: Production of [INAUDIBLE]. 00:21:21.880 --> 00:21:22.877 PROFESSOR: Production of-- 00:21:22.877 --> 00:21:24.160 AUDIENCE: --proteins from a single mRNA. 00:21:24.160 --> 00:21:26.610 PROFESSOR: Yeah, the production of proteins from a single mRNA. 00:21:26.610 --> 00:21:26.840 Right? 00:21:26.840 --> 00:21:29.130 Because remember, we had this thing where we had the mRNA. 00:21:29.130 --> 00:21:30.850 And we said, oh, well the mRNA is going 00:21:30.850 --> 00:21:32.450 to be degraded at some rate. 00:21:32.450 --> 00:21:35.732 But also, it's going to be translated at some rate. 00:21:35.732 --> 00:21:37.190 So the distribution number of times 00:21:37.190 --> 00:21:39.149 that it's translated before it degrades 00:21:39.149 --> 00:21:40.190 is going to be geometric. 00:21:40.190 --> 00:21:42.500 Because we go around this loop some number of times. 00:21:42.500 --> 00:21:46.050 All right, so this is the same thing. 00:21:46.050 --> 00:21:52.150 And the paper that I put as supplementary reading, 00:21:52.150 --> 00:21:54.870 by Jamie Lloyd-Smith? 00:21:54.870 --> 00:21:57.200 I always get his name and Jamie Lloyd Wright-- right? 00:21:57.200 --> 00:21:58.075 Something-- mixed up. 00:21:58.075 --> 00:21:59.241 But yeah, Jamie Lloyd-Smith. 00:21:59.241 --> 00:21:59.770 Smith. 00:21:59.770 --> 00:22:05.050 So he was studying the dynamics of infections 00:22:05.050 --> 00:22:07.110 when you have this thing where there's 00:22:07.110 --> 00:22:11.100 intrinsic variation in, say, the infectivity of an individual. 00:22:11.100 --> 00:22:13.380 Because here, you get this geometric distribution, 00:22:13.380 --> 00:22:17.010 even though all the individuals are, in principal, identical. 00:22:17.010 --> 00:22:20.140 Now, the question is, if there's some distributions of, say, 00:22:20.140 --> 00:22:23.650 infectivities, then you'll get an even broader 00:22:23.650 --> 00:22:26.585 distribution of resulting number of new infections. 00:22:29.470 --> 00:22:31.670 So there's this classic thing of Typhoid Mary. 00:22:34.390 --> 00:22:38.346 She's was a nurse-- OK, now I don't remember the story. 00:22:38.346 --> 00:22:38.846 Yeah? 00:22:38.846 --> 00:22:39.840 AUDIENCE: She was a cook. 00:22:39.840 --> 00:22:40.589 PROFESSOR: A cook. 00:22:40.589 --> 00:22:41.400 Oh, a cook, nurse-- 00:22:41.400 --> 00:22:43.733 AUDIENCE: [INAUDIBLE] so she cooked for a lot of people. 00:22:43.733 --> 00:22:46.120 PROFESSOR: OK, so she was somehow resistant to typhoid. 00:22:46.120 --> 00:22:48.852 But then, she was cooking for other people and, so then, 00:22:48.852 --> 00:22:50.060 caused a bunch of infections. 00:22:50.060 --> 00:22:50.560 Is that--? 00:22:50.560 --> 00:22:51.366 OK. 00:22:51.366 --> 00:22:52.740 Yeah, so this would be an example 00:22:52.740 --> 00:22:54.810 of a very infective individual that's 00:22:54.810 --> 00:22:58.590 beyond the assumptions in this model. 00:22:58.590 --> 00:23:00.800 And as you can imagine, if you have variations 00:23:00.800 --> 00:23:04.490 in this infectivity, then what it does, for a given R0-- 00:23:04.490 --> 00:23:07.080 so if you fix R0, then you have a broader distribution 00:23:07.080 --> 00:23:07.720 of infectivity. 00:23:07.720 --> 00:23:11.270 What it means is that a larger fraction of the infections 00:23:11.270 --> 00:23:12.310 will go extinct. 00:23:12.310 --> 00:23:16.826 But those that get going will be explosive. 00:23:16.826 --> 00:23:18.700 If you're curious about these sorts of ideas, 00:23:18.700 --> 00:23:21.380 you should look at this optional reading paper 00:23:21.380 --> 00:23:24.697 that I put out there. 00:23:24.697 --> 00:23:26.280 All right, so I just want to be clear. 00:23:26.280 --> 00:23:33.221 This is geometric number of new infections, distribution 00:23:33.221 --> 00:23:33.970 of new infections. 00:23:39.550 --> 00:23:40.050 Yes? 00:23:40.050 --> 00:23:43.410 AUDIENCE: So is this just one cycle? 00:23:43.410 --> 00:23:45.350 Like, one-- 00:23:45.350 --> 00:23:49.320 PROFESSOR: So we're talking about the number of infections 00:23:49.320 --> 00:23:51.915 that result when you just add one infected individual 00:23:51.915 --> 00:23:52.685 to the population. 00:23:52.685 --> 00:23:55.222 AUDIENCE: OK, so then, you don't think about, afterwards, 00:23:55.222 --> 00:23:56.972 what happens to those infected individuals 00:23:56.972 --> 00:23:58.064 and if they infect-- 00:23:58.064 --> 00:24:00.230 PROFESSOR: Well, we are not yet thinking about them. 00:24:00.230 --> 00:24:03.460 Although, in this case, those are also geometrically 00:24:03.460 --> 00:24:04.260 distributed. 00:24:04.260 --> 00:24:06.690 But what you expect is that the mean of those things 00:24:06.690 --> 00:24:07.620 will change. 00:24:07.620 --> 00:24:10.970 Because the number of susceptible or whatnot 00:24:10.970 --> 00:24:14.300 individuals-- that's going to change. 00:24:14.300 --> 00:24:17.200 So the mean number is going to change. 00:24:17.200 --> 00:24:19.381 But the distributions will still be geometric. 00:24:19.381 --> 00:24:21.786 AUDIENCE: But in total, that won't be geometric anymore. 00:24:21.786 --> 00:24:23.619 Because if we're looking at the total number 00:24:23.619 --> 00:24:27.077 of infected individuals after learning about the geometric-- 00:24:27.077 --> 00:24:27.660 PROFESSOR: No. 00:24:27.660 --> 00:24:31.100 So just because each of these sub-steps is geometric 00:24:31.100 --> 00:24:34.910 does not mean that you end up with a geometric distribution. 00:24:34.910 --> 00:24:40.977 Indeed, let's say that i put in 20 infected individuals 00:24:40.977 --> 00:24:41.810 into the population. 00:24:41.810 --> 00:24:43.990 And I ask, what's going to be the distribution 00:24:43.990 --> 00:24:46.750 of the number of infections caused immediately 00:24:46.750 --> 00:24:47.770 from those 20? 00:24:47.770 --> 00:24:49.030 That's going to be what? 00:24:49.030 --> 00:24:51.010 [INTERPOSING VOICES] 00:24:51.010 --> 00:24:54.020 Yeah, and for 20, it's going to be basically Gaussian. 00:24:54.020 --> 00:24:58.414 OK, well if I said 100, well definitely Gaussian. 00:24:58.414 --> 00:25:00.330 It's a gamma distribution that looks very much 00:25:00.330 --> 00:25:04.062 look a Gaussian, in that case. 00:25:04.062 --> 00:25:05.837 All right? 00:25:05.837 --> 00:25:09.240 All right. 00:25:09.240 --> 00:25:11.610 So we've been talking about the definition of this R0. 00:25:11.610 --> 00:25:15.495 But of course, we should figure out what it is. 00:25:15.495 --> 00:25:18.900 We want R0 is equal to-- and what 00:25:18.900 --> 00:25:26.262 I'm going to tell you is that there's a 1 over u plus v. 00:25:26.262 --> 00:25:27.720 But then, there's some other terms. 00:25:27.720 --> 00:25:30.330 And I've unfortunately lost my notes. 00:25:30.330 --> 00:25:33.479 So you guys are going to have to help me figure this out. 00:25:33.479 --> 00:25:35.020 And what you're going to do is you're 00:25:35.020 --> 00:25:36.603 going to take advantage of your cards. 00:25:36.603 --> 00:25:38.390 And again, put things in the numerators 00:25:38.390 --> 00:25:39.884 and the denominators, corresponding 00:25:39.884 --> 00:25:41.800 to how I'm supposed to fill out this equation. 00:25:44.039 --> 00:25:46.580 You can start thinking about it while I give you the options. 00:26:01.800 --> 00:26:02.820 OK. 00:26:02.820 --> 00:26:05.809 So I guess you could recapitulate this 00:26:05.809 --> 00:26:07.600 by just putting B and C, although maybe you 00:26:07.600 --> 00:26:09.772 need it more than once. 00:26:09.772 --> 00:26:10.480 Do what you will. 00:26:10.480 --> 00:26:13.404 Do you understand the question? 00:26:13.404 --> 00:26:14.820 There's going to be something else 00:26:14.820 --> 00:26:15.790 I'm going to put right here. 00:26:15.790 --> 00:26:16.756 And I want to know-- there are going 00:26:16.756 --> 00:26:18.290 to be somethings in the numerator, somethings 00:26:18.290 --> 00:26:19.081 in the denominator. 00:26:22.120 --> 00:26:24.380 I'm going to give you 30 seconds to think about it. 00:26:24.380 --> 00:26:26.714 Because it's important to be able to reason 00:26:26.714 --> 00:26:27.630 your way through this. 00:27:13.786 --> 00:27:16.136 All right, do need more time? 00:27:16.136 --> 00:27:18.080 Yep, OK. 00:27:18.080 --> 00:27:19.680 I'll give you another 15 seconds. 00:27:46.021 --> 00:27:48.420 All right, let's go ahead and vote. 00:27:48.420 --> 00:27:49.370 Ready? 00:27:49.370 --> 00:27:54.040 Three, two, one. 00:27:54.040 --> 00:27:55.320 All right. 00:27:55.320 --> 00:27:55.820 I like it! 00:27:55.820 --> 00:27:58.064 We're really looking quite nice. 00:27:58.064 --> 00:27:59.480 So there's a claim that it's going 00:27:59.480 --> 00:28:08.336 to be AD over B. We should be writing a beta k over u. 00:28:08.336 --> 00:28:10.770 All right. 00:28:10.770 --> 00:28:12.880 Can somebody explain how they got there? 00:28:21.710 --> 00:28:22.957 Yes, please. 00:28:22.957 --> 00:28:29.747 AUDIENCE: Well, it's going to be-- these two-- 00:28:29.747 --> 00:28:30.580 PROFESSOR: OK, yeah. 00:28:30.580 --> 00:28:31.640 This helped, right? 00:28:31.640 --> 00:28:34.390 OK, good, perfect. 00:28:34.390 --> 00:28:38.392 So this thing is what? 00:28:38.392 --> 00:28:41.242 What is this term here? 00:28:41.242 --> 00:28:43.034 AUDIENCE: That's the death rate. 00:28:43.034 --> 00:28:43.830 PROFESSOR: Right. 00:28:43.830 --> 00:28:48.095 Which means that the one over it is the expected lifetime 00:28:48.095 --> 00:28:49.970 of an infected individual. 00:28:49.970 --> 00:28:53.360 So the definition of R0 is, you put an infected individual 00:28:53.360 --> 00:28:57.027 into a population of susceptibles. 00:28:57.027 --> 00:28:58.610 Now, we want to know, OK, well there's 00:28:58.610 --> 00:29:01.860 a expected lifetime of this infected individual, which 00:29:01.860 --> 00:29:02.915 is given by this. 00:29:02.915 --> 00:29:04.540 And then, we have to think about, well, 00:29:04.540 --> 00:29:07.081 what's the rate that we're going to be infecting individuals? 00:29:07.081 --> 00:29:10.850 And that's going to be beta times x. 00:29:10.850 --> 00:29:15.590 But what we want to know is, is x before we add any infection? 00:29:15.590 --> 00:29:19.251 And without any infection, then we just have a rate of entry 00:29:19.251 --> 00:29:20.250 and, then, a death rate. 00:29:20.250 --> 00:29:22.067 So it's just k over u. 00:29:22.067 --> 00:29:22.567 OK? 00:29:25.850 --> 00:29:28.960 Now, the key thing in all of these epidemiological models 00:29:28.960 --> 00:29:33.190 is whether this R0 is greater or less than 1. 00:29:33.190 --> 00:29:36.340 And that's going to tell us whether the disease becomes 00:29:36.340 --> 00:29:40.920 endemic or not, whether, at steady state, 00:29:40.920 --> 00:29:46.140 we have a population of infected individuals. 00:29:46.140 --> 00:29:52.060 So R0, greater than one, means it's in an endemic population. 00:30:05.530 --> 00:30:09.884 Now, in this model, we can then ask-- 00:30:09.884 --> 00:30:12.060 [LAUGHTER] 00:30:13.551 --> 00:30:16.036 AUDIENCE: That really is really funny. 00:30:16.036 --> 00:30:17.530 AUDIENCE: That is the hard part. 00:30:17.530 --> 00:30:18.405 PROFESSOR: All right. 00:30:18.405 --> 00:30:22.300 So do you like A, donuts, B, cupcakes, or C, carrots? 00:30:22.300 --> 00:30:24.074 [INTERPOSING VOICES] 00:30:24.074 --> 00:30:24.990 Yeah, it's quite nice. 00:30:24.990 --> 00:30:28.130 Does anybody have any notion of why somebody might 00:30:28.130 --> 00:30:30.570 have-- so Sam likes cupcakes. 00:30:30.570 --> 00:30:32.000 That's good to know. 00:30:32.000 --> 00:30:36.130 Yeah, they're so pretty that I actually feel bad erasing it. 00:30:36.130 --> 00:30:37.661 OK, well, we'll leave it up there -- 00:30:37.661 --> 00:30:38.160 [LAUGHTER] 00:30:38.160 --> 00:30:41.090 --for a little bit longer. 00:30:41.090 --> 00:30:43.240 Everybody can just smile, because they know 00:30:43.240 --> 00:30:45.348 that the drawing is back there. 00:30:45.348 --> 00:30:46.782 [INTERPOSING VOICES] 00:30:47.740 --> 00:30:49.530 All right. 00:30:49.530 --> 00:30:54.880 So in this model, the fact that R0 00:30:54.880 --> 00:30:56.520 is this parameter that tells us about 00:30:56.520 --> 00:30:58.894 whether there's going to be an epidemic and then, indeed, 00:30:58.894 --> 00:31:01.640 whether later the disease will be endemic, 00:31:01.640 --> 00:31:04.240 does that already tell us that R0 is 00:31:04.240 --> 00:31:05.785 what's maximized by selection? 00:31:08.980 --> 00:31:10.231 No. 00:31:10.231 --> 00:31:10.730 Right? 00:31:10.730 --> 00:31:14.020 What was the key thing that Martin does in the chapter 00:31:14.020 --> 00:31:16.260 in order to try to understand something about what 00:31:16.260 --> 00:31:17.520 strain is selected for? 00:31:36.430 --> 00:31:37.140 Yes? 00:31:37.140 --> 00:31:42.475 AUDIENCE: Are you thinking of when he introduces [INAUDIBLE]? 00:31:42.475 --> 00:31:45.020 PROFESSOR: Right. 00:31:45.020 --> 00:31:48.160 Yes, although, the part of the chapter your thinking about is, 00:31:48.160 --> 00:31:49.582 I think, the second half. 00:31:49.582 --> 00:31:51.040 It's talking about super-infection, 00:31:51.040 --> 00:31:53.331 where there's all these different types, and craziness, 00:31:53.331 --> 00:31:54.110 and so forth. 00:31:54.110 --> 00:31:56.800 But the initial insight about what 00:31:56.800 --> 00:31:58.580 is going to be selected for comes 00:31:58.580 --> 00:32:00.688 from a simpler model than that. 00:32:04.680 --> 00:32:07.340 So you don't have to think about all those many parasites 00:32:07.340 --> 00:32:09.910 and those triangles and all the craziness. 00:32:09.910 --> 00:32:12.228 Instead, there was a simpler model that-- yeah? 00:32:12.228 --> 00:32:14.618 AUDIENCE: If you have multiple parasites, 00:32:14.618 --> 00:32:18.560 then study states only that one parasite exists. 00:32:18.560 --> 00:32:19.560 PROFESSOR: That's right. 00:32:19.560 --> 00:32:22.240 And so, what he does is he just writes down 00:32:22.240 --> 00:32:28.180 this model of where he now allows two different parasites 00:32:28.180 --> 00:32:29.860 to be spreading the population. 00:32:29.860 --> 00:32:32.380 And at the beginning-- well, we'll write it down. 00:32:32.380 --> 00:32:33.890 And we want to be clear. 00:32:33.890 --> 00:32:36.080 There's a very important assumption that he makes. 00:32:36.080 --> 00:32:42.400 What he's going to find is that selection maximizes R0, 00:32:42.400 --> 00:32:43.988 the basic reproductive ratio. 00:32:47.749 --> 00:32:49.165 And it's going to be in this model 00:32:49.165 --> 00:32:50.560 that I'm writing down now. 00:32:50.560 --> 00:32:52.530 So there's this x dot. 00:32:52.530 --> 00:32:53.950 And things look very similar. 00:33:17.806 --> 00:33:18.350 All right. 00:33:18.350 --> 00:33:20.349 Now, the question is, what is the key assumption 00:33:20.349 --> 00:33:22.580 that we're making in writing down these equations? 00:33:36.606 --> 00:33:37.105 Thought? 00:33:40.869 --> 00:33:42.910 AUDIENCE: One cannot have more than one type of-- 00:33:42.910 --> 00:33:43.909 PROFESSOR: That's right. 00:33:43.909 --> 00:33:47.710 A host cannot be infected by more than one type of strain. 00:33:47.710 --> 00:33:49.830 So that's very, very important. 00:33:49.830 --> 00:33:52.514 So there's no super-infection, as they say. 00:33:59.554 --> 00:34:00.970 And depending on the disease, this 00:34:00.970 --> 00:34:04.170 could either be a better or worse assumption. 00:34:04.170 --> 00:34:07.330 But then, what is it that is defining these two 00:34:07.330 --> 00:34:08.150 strains then? 00:34:08.150 --> 00:34:09.483 In what ways are they different? 00:34:13.919 --> 00:34:15.889 AUDIENCE: In virulence. 00:34:15.889 --> 00:34:19.489 PROFESSOR: Their virulence is different. 00:34:19.489 --> 00:34:21.430 And what is virulence again? 00:34:21.430 --> 00:34:23.514 AUDIENCE: How likely it is that it could kill you. 00:34:23.514 --> 00:34:24.513 PROFESSOR: That's right. 00:34:24.513 --> 00:34:26.159 It's the additional mortality that 00:34:26.159 --> 00:34:29.717 is caused by being affected by that strain. 00:34:29.717 --> 00:34:31.550 Is that the only way that they're different? 00:34:31.550 --> 00:34:32.092 AUDIENCE: No. 00:34:32.092 --> 00:34:32.675 PROFESSOR: No. 00:34:32.675 --> 00:34:33.683 What else is different? 00:34:33.683 --> 00:34:34.116 AUDIENCE: The infectivity. 00:34:34.116 --> 00:34:35.600 PROFESSOR: The infectivity, right. 00:34:35.600 --> 00:34:38.850 So the betas are also different. 00:34:38.850 --> 00:34:41.190 And it's imported to note that in this model-- 00:34:41.190 --> 00:34:42.750 it's a very simple model-- but we're 00:34:42.750 --> 00:34:46.360 allowing the strains to be different in these two ways. 00:34:46.360 --> 00:34:49.090 And I think that it's very intuitive 00:34:49.090 --> 00:34:53.080 to just say that, oh, well, you want to have a larger beta, 00:34:53.080 --> 00:34:55.000 all other things equal. 00:34:55.000 --> 00:34:57.650 Because you would like to spread. 00:34:57.650 --> 00:35:00.480 But I'd say, maybe, it's not as obvious what 00:35:00.480 --> 00:35:02.160 happens in terms of virulence. 00:35:02.160 --> 00:35:03.730 And then, of course, if you think 00:35:03.730 --> 00:35:07.200 about these two parameters-- in any biological context 00:35:07.200 --> 00:35:08.510 they may be coupled. 00:35:08.510 --> 00:35:13.176 In which case, things are more subtle. 00:35:13.176 --> 00:35:18.016 AUDIENCE: So how reasonable is something of something 00:35:18.016 --> 00:35:20.440 of [INAUDIBLE] on the strain? 00:35:20.440 --> 00:35:22.972 PROFESSOR: Yeah, I think that this depends on the disease. 00:35:22.972 --> 00:35:23.888 AUDIENCE: [INAUDIBLE]. 00:35:23.888 --> 00:35:27.372 Is it that you're trading virulence and new response 00:35:27.372 --> 00:35:27.872 and-- 00:35:27.872 --> 00:35:31.740 PROFESSOR: Right, so that's somehow the argument. 00:35:31.740 --> 00:35:38.510 And I'd say that, depending on whether you're 00:35:38.510 --> 00:35:42.380 thinking about the host at the level of an organism or a cell, 00:35:42.380 --> 00:35:45.920 then this would correspond to very different worlds. 00:35:45.920 --> 00:35:49.680 So certainly, in the context of viral infections 00:35:49.680 --> 00:35:51.870 and individual cells, there are various mechanisms 00:35:51.870 --> 00:35:55.020 where, if one strain gets in, then other strains 00:35:55.020 --> 00:35:57.960 have trouble getting in. 00:35:57.960 --> 00:36:00.920 Or if they do get in, they can't do anything. 00:36:00.920 --> 00:36:03.446 So I think, this really depends on the biology 00:36:03.446 --> 00:36:05.821 of the situation, whether this is a reasonable assumption 00:36:05.821 --> 00:36:06.321 or not. 00:36:20.242 --> 00:36:22.200 So I'm not going to go through all of the math. 00:36:22.200 --> 00:36:26.150 Because it ends up being a little bit involved. 00:36:26.150 --> 00:36:31.610 But the condition for the mutual invasibility of the two strains 00:36:31.610 --> 00:36:33.300 is a little bit subtle. 00:36:33.300 --> 00:36:37.130 So I do want to talk about that a little bit. 00:36:37.130 --> 00:36:39.930 And so, in general, in this model 00:36:39.930 --> 00:36:53.900 there's some equilibrium, x star and y star. 00:36:53.900 --> 00:36:57.677 And indeed, there will generally be damped oscillations 00:36:57.677 --> 00:36:58.510 to this equilibrium. 00:37:03.470 --> 00:37:05.480 So this is in the model where there's just 00:37:05.480 --> 00:37:09.460 a single strain that's described by some beta 00:37:09.460 --> 00:37:12.740 and some v. Of course, you can also 00:37:12.740 --> 00:37:18.640 think about the equilibria, E1 and E2, that would result when 00:37:18.640 --> 00:37:24.050 you have-- so E1 is what would happen at the equilibrium 00:37:24.050 --> 00:37:30.550 when you have x star evaluated for the particular parameters 00:37:30.550 --> 00:37:32.850 of strain one, i.e. 00:37:32.850 --> 00:37:36.810 evaluated for beta1 and for v1. 00:37:36.810 --> 00:37:39.120 x star-- and that's also y star. 00:37:39.120 --> 00:37:41.120 Whereas, you would have a different equilibrium, 00:37:41.120 --> 00:37:46.030 E2-- so this is a x star and a y star-- 00:37:46.030 --> 00:37:49.110 evaluated at beta2 and v2. 00:37:49.110 --> 00:37:50.890 Do you understand what I'm saying? 00:37:50.890 --> 00:37:52.390 So these are the equilibria that you 00:37:52.390 --> 00:37:54.645 would have if this was the only strain that 00:37:54.645 --> 00:37:57.350 was present in the population or if this was the only strain 00:37:57.350 --> 00:37:59.620 in the population. 00:37:59.620 --> 00:38:01.880 Now, it's not obvious that those-- the equilibria-- 00:38:01.880 --> 00:38:04.010 are the result when you have both strains 00:38:04.010 --> 00:38:05.670 present in the population. 00:38:05.670 --> 00:38:07.560 But that's what we want to try to figure out. 00:38:10.510 --> 00:38:12.895 But certainly, if you only added strain one 00:38:12.895 --> 00:38:15.520 and you didn't add strain 2, you would come to equilibrium, E1. 00:38:20.300 --> 00:38:25.745 So the question is, if we start out at equilibrium E1, 00:38:25.745 --> 00:38:27.370 how is it we can determine what happens 00:38:27.370 --> 00:38:32.040 if we now add an infected individual, 00:38:32.040 --> 00:38:34.510 but infected by strain two? 00:38:43.730 --> 00:38:46.895 So what we want to know is-- strain 2 can invade. 00:38:52.799 --> 00:38:54.340 And really, what we're thinking about 00:38:54.340 --> 00:38:58.190 is a situation-- these are the equilibria if it's 00:38:58.190 --> 00:39:03.155 the case that R1 is greater than 1 and R2 is greater than 1. 00:39:03.155 --> 00:39:05.100 Because in some ways, it's clear what happens. 00:39:05.100 --> 00:39:09.430 If both of the R1 and R2 are less than 1, then what happens? 00:39:12.995 --> 00:39:14.120 AUDIENCE: It would die out. 00:39:14.120 --> 00:39:15.160 PROFESSOR: They both die out. 00:39:15.160 --> 00:39:17.368 If one of them is greater than 1, one is less than 1. 00:39:17.368 --> 00:39:20.660 Then, right, the strain that's below 1 00:39:20.660 --> 00:39:23.230 goes-- so the only interesting or the only the only 00:39:23.230 --> 00:39:25.460 non-obvious question, somehow, is what happens 00:39:25.460 --> 00:39:28.800 if they're both larger than 1? 00:39:28.800 --> 00:39:30.465 What that means is that, for example, 00:39:30.465 --> 00:39:32.590 if you had a population of susceptible individuals, 00:39:32.590 --> 00:39:36.030 and you had one infected by strain one, 00:39:36.030 --> 00:39:38.040 one infected by strain two, then, 00:39:38.040 --> 00:39:40.420 in the deterministic differential equation limit, 00:39:40.420 --> 00:39:43.470 they would both spread. 00:39:43.470 --> 00:39:45.939 So they're both exponentially growing. 00:39:45.939 --> 00:39:47.480 Does that already tell us that you're 00:39:47.480 --> 00:39:51.380 going to have coexistence of the two strains? 00:39:51.380 --> 00:39:52.116 No. 00:39:52.116 --> 00:39:54.240 It means that they're both exponentially spreading. 00:39:54.240 --> 00:39:56.800 But who knows what's going to happen later? 00:39:56.800 --> 00:39:59.250 And indeed, what you can show is that only one of the two 00:39:59.250 --> 00:40:00.250 strains is going to win. 00:40:00.250 --> 00:40:03.280 And it's the strain with the higher reproductive ration, 00:40:03.280 --> 00:40:05.810 the higher R0, or in this case, here. 00:40:08.291 --> 00:40:10.457 So how is it that we can determine if strain two can 00:40:10.457 --> 00:40:13.510 invade equilibrium one? 00:40:13.510 --> 00:40:15.795 It's if-and-only-if something. 00:40:15.795 --> 00:40:18.170 Does anybody remember what this condition ended up being? 00:40:30.988 --> 00:40:34.470 AUDIENCE: y2 dot at E1 has to be positive? 00:40:34.470 --> 00:40:39.010 PROFESSOR: All right, y2 dot at E1 has to be positive. 00:40:39.010 --> 00:40:39.540 Yes. 00:40:39.540 --> 00:40:41.370 And this ends up being equivalent, I think, 00:40:41.370 --> 00:40:43.190 to what he writes. 00:40:43.190 --> 00:40:44.390 So let's write. 00:40:44.390 --> 00:40:45.770 I'll write and tell you. 00:40:45.770 --> 00:40:49.540 So the way that he wrote it, is that it's y2 00:40:49.540 --> 00:40:52.761 dot, with respect to y2. 00:40:52.761 --> 00:40:55.260 But I think that this is really equivalent to what you said. 00:41:00.310 --> 00:41:02.850 Because you said, OK, y2 dot has to be something, right? 00:41:02.850 --> 00:41:04.632 But then, y2 dot evaluated what? 00:41:04.632 --> 00:41:06.090 And you would say, evaluated at E1. 00:41:06.090 --> 00:41:11.244 But then, at E1-- what is y2 dot evaluated at E1? 00:41:11.244 --> 00:41:12.170 AUDIENCE: 0. 00:41:12.170 --> 00:41:13.890 PROFESSOR: 0, right? 00:41:13.890 --> 00:41:17.280 Because at E1, there is 0 y2. 00:41:17.280 --> 00:41:19.865 So then, of course, y2 dot is 0. 00:41:19.865 --> 00:41:23.340 So it's almost what you want, but not quite. 00:41:23.340 --> 00:41:25.640 So this condition, which looks very weird, 00:41:25.640 --> 00:41:30.210 is really saying that, if we are at E1-- so there's 00:41:30.210 --> 00:41:34.770 no infected type-two infections-- what we want to do 00:41:34.770 --> 00:41:36.840 is add a little bit of y2. 00:41:36.840 --> 00:41:39.360 And then, we want to ask, what is y2 dot? 00:41:39.360 --> 00:41:41.000 And this derivative, evaluated at E1, 00:41:41.000 --> 00:41:44.080 somehow allows you to do that. 00:41:44.080 --> 00:41:46.220 And do we want this to be greater than 0? 00:41:46.220 --> 00:41:48.060 Or less than 0? 00:41:48.060 --> 00:41:49.661 We're going to do verbal answer. 00:41:49.661 --> 00:41:50.160 Ready? 00:41:50.160 --> 00:41:52.100 Three, two, one. 00:41:52.100 --> 00:41:52.850 AUDIENCE: Greater. 00:41:52.850 --> 00:41:53.750 PROFESSOR: Greater, right. 00:41:53.750 --> 00:41:55.400 Because if it's saying you add a y2, 00:41:55.400 --> 00:41:56.710 you want y2 to start growing. 00:41:59.610 --> 00:42:01.450 So you want this thing to be greater than 0. 00:42:01.450 --> 00:42:03.870 And this looks really crazy. 00:42:03.870 --> 00:42:05.370 But it's actually pretty easy to do. 00:42:05.370 --> 00:42:07.286 Because you take the derivative of this thing, 00:42:07.286 --> 00:42:08.240 with respect to y2. 00:42:08.240 --> 00:42:11.070 And you just get the thing in parentheses, right? 00:42:11.070 --> 00:42:12.700 But you evaluated it at E1. 00:42:12.700 --> 00:42:15.400 So it's beta2. 00:42:15.400 --> 00:42:23.954 And this is x star at E1, minus u, minus v2. 00:42:23.954 --> 00:42:25.620 And this thing has to be greater than 0. 00:42:28.910 --> 00:42:34.340 But this is x at E1. 00:42:34.340 --> 00:42:38.710 That's u plus v1 divided by beta1. 00:42:38.710 --> 00:42:47.650 So what we have is beta 2, then, u plus v1 divided by beta1. 00:42:47.650 --> 00:42:50.220 And this thing has to be greater than-- we'll move this over 00:42:50.220 --> 00:42:53.830 to the other side-- u plus v2. 00:42:53.830 --> 00:42:57.680 OK, so this is not so horrible, right? 00:42:57.680 --> 00:43:00.610 There's a u plus v1, beta1, right? 00:43:00.610 --> 00:43:04.839 I just want to make sure I write-- (WHISPERING) u plus v1. 00:43:04.839 --> 00:43:07.130 (NORMAL VOICE) OK, we have to these in the denominators 00:43:07.130 --> 00:43:07.671 now, somehow. 00:43:10.730 --> 00:43:13.550 So we'll put divide by both of these things. 00:43:13.550 --> 00:43:18.250 So we have a 1 divided by a u plus v2. 00:43:18.250 --> 00:43:21.350 1 divided by u plus v1. 00:43:21.350 --> 00:43:23.820 So we put these things in the denominator. 00:43:23.820 --> 00:43:26.550 And now, the beta1 is going to come up here. 00:43:26.550 --> 00:43:30.550 So we have a beta1 and a beta2. 00:43:30.550 --> 00:43:33.420 And there still is a greater-than sign. 00:43:33.420 --> 00:43:36.250 Did I screw anything up yet? 00:43:36.250 --> 00:43:36.750 Maybe? 00:43:36.750 --> 00:43:37.920 No? 00:43:37.920 --> 00:43:38.420 All right. 00:43:41.240 --> 00:43:43.020 Now, we can do something wonderful, 00:43:43.020 --> 00:43:45.100 which is we can just multiply by k divided by u. 00:43:51.918 --> 00:43:53.490 And so what does this say? 00:43:56.822 --> 00:43:58.730 AUDIENCE: R2 is greater than R1. 00:43:58.730 --> 00:44:00.790 PROFESSOR: R2 is greater than R1, right? 00:44:03.359 --> 00:44:04.900 And what were we trying to calculate? 00:44:07.671 --> 00:44:09.170 How did we get started on this math? 00:44:12.044 --> 00:44:13.960 AUDIENCE: What can [INAUDIBLE] the strains. 00:44:13.960 --> 00:44:15.418 PROFESSOR: Right, we wanted to ask, 00:44:15.418 --> 00:44:19.000 strain two can invade the equilibrium one, 00:44:19.000 --> 00:44:31.457 if and only if R2 is greater than-- now, 00:44:31.457 --> 00:44:32.540 that's interesting, right? 00:44:32.540 --> 00:44:34.164 So that's saying that, if you start out 00:44:34.164 --> 00:44:37.290 with the endemic strain one and you add the strain 00:44:37.290 --> 00:44:39.590 two in there, it's going to be able to invade 00:44:39.590 --> 00:44:42.000 if it's R2 is greater than R1. 00:44:42.000 --> 00:44:43.720 That's not obvious. 00:44:43.720 --> 00:44:45.600 Because R2 and R1, that was telling us 00:44:45.600 --> 00:44:47.317 about a different situation. 00:44:47.317 --> 00:44:48.900 That was telling us about what happens 00:44:48.900 --> 00:44:52.520 when you add those infected individuals into a population 00:44:52.520 --> 00:44:54.260 of susceptibles. 00:44:54.260 --> 00:44:57.590 But this also ends up telling us about what 00:44:57.590 --> 00:45:00.860 happens when the strains are competing against each other. 00:45:00.860 --> 00:45:05.302 So this is so simple that it feels trivial. 00:45:05.302 --> 00:45:07.260 I'm sure that if you understood it well enough, 00:45:07.260 --> 00:45:08.093 it would be trivial. 00:45:08.093 --> 00:45:11.540 But you'd have to think about it more than I have. 00:45:11.540 --> 00:45:13.110 So I think this is surprising. 00:45:13.110 --> 00:45:15.900 Now, you can also ask the same question, 00:45:15.900 --> 00:45:26.010 which is about whether strain one can invade strain two. 00:45:26.010 --> 00:45:27.570 And that is kind of the same thing. 00:45:27.570 --> 00:45:32.840 So it all comes down to the orderings of R2 and R1. 00:45:32.840 --> 00:45:38.390 And what you find is that, if this condition is satisfied, 00:45:38.390 --> 00:45:42.140 strain two will drive strain one out of the population. 00:45:47.996 --> 00:45:50.904 AUDIENCE: What if they strains are the same? 00:45:50.904 --> 00:45:52.320 PROFESSOR: So if they're the same, 00:45:52.320 --> 00:45:55.050 then you can get coexistence. 00:45:55.050 --> 00:45:58.920 But as Martin says, it's non-generic, 00:45:58.920 --> 00:46:02.380 in a sense that it's kind of a coincidence, 00:46:02.380 --> 00:46:04.810 or it's of measure 0, the situations 00:46:04.810 --> 00:46:10.960 in which the two will have the same R parameters. 00:46:10.960 --> 00:46:12.674 But of course, you could imagine-- 00:46:12.674 --> 00:46:14.840 just like we talked about all this neutral evolution 00:46:14.840 --> 00:46:16.770 business-- things are nearly neutral 00:46:16.770 --> 00:46:19.730 and so forth, da-da-da-- you can kind of invoke similar ideas. 00:46:19.730 --> 00:46:22.310 Because you can imagine that, as these two become closer 00:46:22.310 --> 00:46:23.810 and closer to each other, it's going 00:46:23.810 --> 00:46:27.165 to take longer and longer for the more-fit strain 00:46:27.165 --> 00:46:28.880 to out-compete the less-fit strain. 00:46:31.820 --> 00:46:36.310 Now, even though this is such a simple condition 00:46:36.310 --> 00:46:41.560 and the R parameters have such a simple physical origin 00:46:41.560 --> 00:46:45.260 or mathematical origin, the actual expression for the Rs 00:46:45.260 --> 00:46:46.880 is kind of complicated. 00:46:46.880 --> 00:46:48.829 But again, it's a combination of all 00:46:48.829 --> 00:46:50.620 of these different parameters, whether it's 00:46:50.620 --> 00:46:52.860 the beta1, B1, or beta2, v2. 00:47:06.190 --> 00:47:08.205 Which board do you think is least useful? 00:47:13.530 --> 00:47:16.079 Well, I don't want this one anymore. 00:47:16.079 --> 00:47:17.870 But in particular, we want to say something 00:47:17.870 --> 00:47:21.510 about the evolution of virulence in this model 00:47:21.510 --> 00:47:24.577 and what the expectation is. 00:47:24.577 --> 00:47:26.160 So what we're going to do is I'm going 00:47:26.160 --> 00:47:29.180 to give you some different situations, in terms 00:47:29.180 --> 00:47:34.220 of how the infectivity depends upon the virulence. 00:47:34.220 --> 00:47:36.390 And then, you guys will get to tell me 00:47:36.390 --> 00:47:38.185 how the virulence will evolve. 00:47:55.520 --> 00:48:24.720 Now, this is virulence, v. All right, so the first situation 00:48:24.720 --> 00:48:30.230 is if beta as a function of the virulence-- so 00:48:30.230 --> 00:48:35.250 the infectivity as a function of virulence is, we'll say, 00:48:35.250 --> 00:48:37.110 some beta0. 00:48:37.110 --> 00:48:40.570 So first, the question is, if the infectivity does not 00:48:40.570 --> 00:48:47.940 depend upon the virulence, where will virulence go to? 00:48:51.750 --> 00:48:55.460 You have equation, various places that'll help you. 00:49:07.270 --> 00:49:09.170 Do you need more time? 00:49:09.170 --> 00:49:09.670 No? 00:49:09.670 --> 00:49:14.820 All right, ready, three, two, one. 00:49:14.820 --> 00:49:16.860 OK, we've got many A's. 00:49:16.860 --> 00:49:17.460 That's great. 00:49:21.690 --> 00:49:25.550 So if it's the case that the parasite, 00:49:25.550 --> 00:49:28.360 regardless of how rapidly it's killing the individual, 00:49:28.360 --> 00:49:31.220 has the same rate of getting to other individuals, 00:49:31.220 --> 00:49:37.220 then we want to make the R0 to get as large as possible. 00:49:37.220 --> 00:49:40.134 So then, we make v go as small as possible. 00:49:40.134 --> 00:49:42.050 That's the mathematical thing you can look at. 00:49:42.050 --> 00:49:43.350 And why does this make sense? 00:49:51.093 --> 00:49:52.592 AUDIENCE: You're killing people off, 00:49:52.592 --> 00:49:54.089 and you're infecting most people. 00:49:54.089 --> 00:49:55.090 But-- 00:49:55.090 --> 00:49:55.940 PROFESSOR: Yeah. 00:49:55.940 --> 00:50:00.480 So from the standpoint of the parasite, 00:50:00.480 --> 00:50:03.190 in this situation where the infectivity doesn't depend 00:50:03.190 --> 00:50:05.080 on the virulence, well in that case, 00:50:05.080 --> 00:50:06.570 you don't want to kill your host. 00:50:06.570 --> 00:50:09.530 Because the longer that the host lives, 00:50:09.530 --> 00:50:11.380 the more other individuals you can infect. 00:50:14.060 --> 00:50:19.310 And this is the basic statement underlying the statement 00:50:19.310 --> 00:50:22.960 you often hear that a well-adapted parasite does not 00:50:22.960 --> 00:50:25.236 harm its host. 00:50:25.236 --> 00:50:26.860 And I think this is one of those things 00:50:26.860 --> 00:50:29.620 that you can write down a simple model 00:50:29.620 --> 00:50:32.390 and convince yourself it should be true. 00:50:32.390 --> 00:50:34.360 You can maybe find a few case studies 00:50:34.360 --> 00:50:36.180 where it seems to be true. 00:50:36.180 --> 00:50:40.179 But then, you always have to be careful about how strongly you 00:50:40.179 --> 00:50:41.720 should believe such a statement based 00:50:41.720 --> 00:50:43.110 on those kinds of evidence. 00:50:43.110 --> 00:50:46.500 Because it's a very simple model. 00:50:46.500 --> 00:50:49.770 It's making an assumption that is very likely not true 00:50:49.770 --> 00:50:53.130 and, actually, many assumptions that are very likely not true. 00:50:53.130 --> 00:50:54.710 And there are counterexamples. 00:50:54.710 --> 00:50:57.590 So Martin talks about malaria, which 00:50:57.590 --> 00:51:00.110 humans have had for millions of years, 00:51:00.110 --> 00:51:05.450 and still causes a lot of problems. 00:51:05.450 --> 00:51:06.950 And so you might imagine, what would 00:51:06.950 --> 00:51:11.510 happen if the infectivity is a function of the virulence? 00:51:11.510 --> 00:51:13.960 Maybe it's just proportional to the virulence. 00:51:13.960 --> 00:51:15.850 And this kind of would make sense. 00:51:15.850 --> 00:51:19.590 Because you say, oh well, let's imagine that the number 00:51:19.590 --> 00:51:25.560 of viruses in the host-- that somehow, the virulence 00:51:25.560 --> 00:51:27.070 is proportional to that number. 00:51:27.070 --> 00:51:30.182 And also, the infectivity is proportional to that number. 00:51:30.182 --> 00:51:31.640 In that case, infectivity will just 00:51:31.640 --> 00:51:38.330 scale linearly with v. It's kind of another reasonable world. 00:51:38.330 --> 00:51:42.540 So in this situation, where does the virulence evolve to? 00:51:42.540 --> 00:51:44.700 We'll think about this for 10 seconds. 00:52:01.100 --> 00:52:01.660 Ready? 00:52:01.660 --> 00:52:03.480 Three, two, one. 00:52:11.130 --> 00:52:13.780 All right, so now we actually have votes that 00:52:13.780 --> 00:52:16.010 are pretty distributed around. 00:52:19.310 --> 00:52:21.880 OK, I'm just going to write down the expression. 00:52:21.880 --> 00:52:23.720 Just because I think we can do it quickly. 00:52:23.720 --> 00:52:27.399 So now, the R0 is going to be given by-- we're 00:52:27.399 --> 00:52:28.690 going to be thinking about a 1. 00:52:28.690 --> 00:52:31.120 There's a u plus v down here. 00:52:31.120 --> 00:52:33.730 But beta-- we're going to put beta here. 00:52:33.730 --> 00:52:39.173 This is an a times v, and then a k over u. 00:52:39.173 --> 00:52:41.140 Right? 00:52:41.140 --> 00:52:46.161 So if we plot this as a function of v, what does it look like? 00:52:46.161 --> 00:52:47.862 AUDIENCE: It monotonically increases. 00:52:47.862 --> 00:52:49.320 PROFESSOR: Monotonically increases. 00:52:49.320 --> 00:52:52.420 This is a Michaelis-Menten type form 00:52:52.420 --> 00:52:58.680 where the R0 is a function of v. It starts out 00:52:58.680 --> 00:52:59.885 linear and, then, saturates. 00:53:04.170 --> 00:53:05.770 Oops, this is not v. This is R0. 00:53:08.540 --> 00:53:14.706 If we want to maximize R0, then v goes to infinity, right? 00:53:20.060 --> 00:53:22.430 There was one other model that Martin talked about, 00:53:22.430 --> 00:53:25.745 which was the situation if the infectivity itself. 00:53:25.745 --> 00:53:28.550 It had a kind of Michealis-Menten type form. 00:53:28.550 --> 00:53:32.390 And we still have an a there. 00:53:32.390 --> 00:53:38.630 So it's some a,v c plus v. So this would be the situation 00:53:38.630 --> 00:53:41.360 where, initially, the infectivity increases with 00:53:41.360 --> 00:53:41.870 virulence. 00:53:41.870 --> 00:53:46.110 But beyond some virulence, you no longer 00:53:46.110 --> 00:53:47.484 get an increase in infectivity. 00:53:47.484 --> 00:53:49.650 So for example, if it's the case that every time you 00:53:49.650 --> 00:53:51.884 sneeze on somebody you're going to infect them, 00:53:51.884 --> 00:53:54.300 then it doesn't matter if you double the number of viruses 00:53:54.300 --> 00:53:54.966 you have. 00:53:54.966 --> 00:53:56.590 You've still saturated the infectivity. 00:54:00.700 --> 00:54:07.400 Now, I'll just tell you that, in this case, 00:54:07.400 --> 00:54:11.540 there ends up being some intermediate infectivity 00:54:11.540 --> 00:54:14.850 that you evolve to, which comes here. 00:54:14.850 --> 00:54:16.750 And indeed, this makes sense. 00:54:16.750 --> 00:54:31.590 As c goes to infinity, so if c is very large, 00:54:31.590 --> 00:54:34.217 then it just looks like this. 00:54:34.217 --> 00:54:35.550 It's divided by something large. 00:54:35.550 --> 00:54:38.530 But in terms of the scaling with v, it's just linear in v. 00:54:38.530 --> 00:54:42.670 And that means that the evolved virulence goes very large. 00:54:46.575 --> 00:54:49.850 Does that make sense? 00:54:49.850 --> 00:54:50.350 No? 00:54:50.350 --> 00:54:51.320 OK. 00:54:51.320 --> 00:54:54.560 So I guess what I'm saying is that, if you're 00:54:54.560 --> 00:55:00.120 in the region where c is somehow very large, then 00:55:00.120 --> 00:55:02.080 you end up with the infectivity just being 00:55:02.080 --> 00:55:04.700 proportional to the virulence. 00:55:04.700 --> 00:55:07.580 Because the virulence, the v here in the denominator, 00:55:07.580 --> 00:55:09.510 doesn't really contribute very much. 00:55:09.510 --> 00:55:12.076 And that means that it's going to be the same kind 00:55:12.076 --> 00:55:13.200 of functional form as this. 00:55:13.200 --> 00:55:14.670 Except if it's a divided by c. 00:55:14.670 --> 00:55:16.950 But it's still proportional to the virulence. 00:55:16.950 --> 00:55:21.772 And that leads to a larger and larger evolved virulence. 00:55:21.772 --> 00:55:24.240 OK? 00:55:24.240 --> 00:55:24.890 Yep? 00:55:24.890 --> 00:55:25.390 Yes? 00:55:25.390 --> 00:55:29.471 AUDIENCE: Is it possible to say something 00:55:29.471 --> 00:55:32.894 in actual infection, what the data is? 00:55:32.894 --> 00:55:33.872 Is it virulunce? 00:55:33.872 --> 00:55:34.880 Or can you [INAUDIBLE]-- 00:55:34.880 --> 00:55:35.546 PROFESSOR: Yeah. 00:55:35.546 --> 00:55:37.296 AUDIENCE: --other than just these verbal-- 00:55:37.296 --> 00:55:38.600 PROFESSOR: Yeah, right. 00:55:38.600 --> 00:55:43.160 So I think, from real data, I think people do argue about it. 00:55:43.160 --> 00:55:52.180 But I think it seems to be there is some sense that it 00:55:52.180 --> 00:55:54.020 does plateau. 00:55:54.020 --> 00:55:58.460 But it's an increasing function of v, but sub-linear. 00:55:58.460 --> 00:56:00.729 And the question is how strong of a statement 00:56:00.729 --> 00:56:01.520 you can make there. 00:56:01.520 --> 00:56:03.769 Because also, in many cases, there is super-infection. 00:56:03.769 --> 00:56:07.460 And super-infection tends to lead to even higher virulence 00:56:07.460 --> 00:56:10.510 than what you would expect from this. 00:56:10.510 --> 00:56:14.340 Because you're competing against viruses in the same host. 00:56:14.340 --> 00:56:19.570 So you have to out-compete the other parasite in the host, 00:56:19.570 --> 00:56:21.340 as well as go on. 00:56:21.340 --> 00:56:23.260 And you don't pay the full cost associated 00:56:23.260 --> 00:56:25.210 with keeping the host alive. 00:56:25.210 --> 00:56:26.710 So when you have super-infection, 00:56:26.710 --> 00:56:29.981 there's this notion that the better strategy, 00:56:29.981 --> 00:56:31.480 from the standpoint of the parasite, 00:56:31.480 --> 00:56:34.336 can be to just evolve very high virulence. 00:56:34.336 --> 00:56:35.710 So you're going to kill the host. 00:56:35.710 --> 00:56:37.340 But you can get out quickly. 00:56:37.340 --> 00:56:41.170 So then, you can imagine that the less-virulent strain 00:56:41.170 --> 00:56:44.380 was kind of stranded in that host that died. 00:56:44.380 --> 00:56:46.770 So from that standpoint, you can think about, 00:56:46.770 --> 00:56:48.650 in some cases, for a parasite, evolving 00:56:48.650 --> 00:56:52.620 low-virulence is somehow like some cooperative behavior. 00:56:52.620 --> 00:56:56.810 Because you're keeping this host alive. 00:56:56.810 --> 00:56:59.010 You're using the resources in a wise way, 00:56:59.010 --> 00:57:01.240 so that you can infect other individuals. 00:57:01.240 --> 00:57:03.356 But then, that kind of strategy is 00:57:03.356 --> 00:57:05.230 susceptible to these cheating strategies that 00:57:05.230 --> 00:57:07.340 just have high virulence and kill the host 00:57:07.340 --> 00:57:09.645 and, then, get on to a new host. 00:57:09.645 --> 00:57:11.730 Yeah, so I think there are enough of these issues 00:57:11.730 --> 00:57:16.370 that it's hard to take any of this too seriously. 00:57:16.370 --> 00:57:18.590 I would say that, perhaps, where this field 00:57:18.590 --> 00:57:22.375 has had the biggest impact is in the context of vaccinations. 00:57:30.000 --> 00:57:34.840 Because you can really measure R0 for many different diseases. 00:57:34.840 --> 00:57:36.430 And I think that's somehow easier 00:57:36.430 --> 00:57:37.846 to measure than many other things. 00:57:37.846 --> 00:57:42.700 Because you can try to do tracing of infections. 00:57:42.700 --> 00:57:45.620 So if you think about somebody that gets infected 00:57:45.620 --> 00:57:48.040 and moves to a new city or lands in a new city, 00:57:48.040 --> 00:57:50.840 you can try to figure out who they infected. 00:57:50.840 --> 00:57:55.250 So then, you can go and measure these R0 parameters. 00:57:55.250 --> 00:57:58.410 And of course, the diseases that we worry about 00:57:58.410 --> 00:58:01.810 tend to have R0s larger than 1. 00:58:01.810 --> 00:58:04.630 And how large they are tells you about how difficult 00:58:04.630 --> 00:58:09.470 the vaccination will be in order to be successful 00:58:09.470 --> 00:58:12.800 and to remove the disease from the population. 00:58:12.800 --> 00:58:15.520 So if you're curious, after class, you can come up. 00:58:15.520 --> 00:58:18.550 And there's a nice table that I have here 00:58:18.550 --> 00:58:21.550 for smallpox, measles, whooping cough, German measles, 00:58:21.550 --> 00:58:23.881 chickenpox, diphtheria, scarlet fever, mumps, 00:58:23.881 --> 00:58:24.630 and poliomielitis. 00:58:27.240 --> 00:58:34.456 Estimates of R0s-- and they kind of range 5 to 15, 00:58:34.456 --> 00:58:36.680 to give you a sense. 00:58:36.680 --> 00:58:40.820 And so, if you want to remove the disease 00:58:40.820 --> 00:58:43.300 from the population, what is it that changes? 00:58:43.300 --> 00:58:47.230 --in terms of vaccination, in order to remove the disease. 00:58:55.260 --> 00:58:55.760 Yes? 00:58:55.760 --> 00:58:58.586 AUDIENCE: In R0, then, the [INAUDIBLE] for u, 00:58:58.586 --> 00:59:02.230 as the population available [INAUDIBLE]. 00:59:02.230 --> 00:59:03.230 PROFESSOR: That's right. 00:59:03.230 --> 00:59:07.189 So by vaccinating you're removing 00:59:07.189 --> 00:59:08.980 susceptible individuals from the population 00:59:08.980 --> 00:59:11.420 and making them resistant somehow. 00:59:11.420 --> 00:59:15.670 And the R0 parameter tells you about what fraction 00:59:15.670 --> 00:59:17.670 of the population you have to vaccinate in order 00:59:17.670 --> 00:59:20.830 to remove the parasite from the population. 00:59:20.830 --> 00:59:23.230 And so, basically, you have to vaccinate 00:59:23.230 --> 00:59:29.000 a fraction, a percentage, p that's greater than 1 minus 1 00:59:29.000 --> 00:59:30.630 over R0. 00:59:30.630 --> 00:59:33.850 So as R0 gets very large, it means 00:59:33.850 --> 00:59:37.770 that you have to vaccinate, essentially, everybody. 00:59:37.770 --> 00:59:39.787 So if you have R0 of, say, five-- 00:59:39.787 --> 00:59:41.620 which is typical of many of these diseases-- 00:59:41.620 --> 00:59:44.170 it's saying you have to vaccinate 00:59:44.170 --> 00:59:46.260 80% of the population. 00:59:46.260 --> 00:59:48.084 You can never get to 100%. 00:59:48.084 --> 00:59:49.500 And that's why it's very difficult 00:59:49.500 --> 00:59:53.860 to get rid of these diseases with large R0s. 00:59:53.860 --> 00:59:58.730 And incidentally, they note that smallpox has an R0 of 3 to 5. 01:00:01.330 --> 01:00:03.960 And this always depends on the environment. 01:00:03.960 --> 01:00:06.420 But in the case where they measured, 01:00:06.420 --> 01:00:10.380 smallpox R0 is 3 to 5. 01:00:10.380 --> 01:00:12.027 And this is sort of small, as compared 01:00:12.027 --> 01:00:13.110 to many of these diseases. 01:00:13.110 --> 01:00:17.660 And this is telling us that smallpox is easier 01:00:17.660 --> 01:00:19.360 to get rid of via vaccinations than many 01:00:19.360 --> 01:00:20.360 of these other diseases. 01:00:20.360 --> 01:00:24.940 And indeed, the vaccination procedures 01:00:24.940 --> 01:00:28.882 have been more successful in smallpox than the others. 01:00:28.882 --> 01:00:31.740 AUDIENCE: Do you know what is 3, 4? 01:00:31.740 --> 01:00:32.750 PROFESSOR: I don't. 01:00:32.750 --> 01:00:35.070 But maybe in the next 20 minutes, 01:00:35.070 --> 01:00:38.580 somebody can Google this. 01:00:38.580 --> 01:00:41.450 We can estimate this right now, though. 01:00:41.450 --> 01:00:42.840 We've had, what? 01:00:42.840 --> 01:00:45.710 --five Ebola patients come to the United States. 01:00:45.710 --> 01:00:49.470 And we've gotten two or three infections. 01:00:49.470 --> 01:00:51.815 So I'll say, 3/5. 01:00:51.815 --> 01:00:53.342 [LAUGHTER] 01:00:53.342 --> 01:00:55.300 It obviously depends on the environment, right? 01:00:55.300 --> 01:00:57.040 AUDIENCE: Yeah. 01:00:57.040 --> 01:00:57.665 PROFESSOR: Yes. 01:00:57.665 --> 01:00:59.623 AUDIENCE: I mean, I guess that was my question. 01:00:59.623 --> 01:01:01.449 You're not going to take someone with Ebola 01:01:01.449 --> 01:01:05.097 and throw them in New York City and just measure how 01:01:05.097 --> 01:01:06.235 many people they infect. 01:01:06.235 --> 01:01:06.690 PROFESSOR: That's right. 01:01:06.690 --> 01:01:07.145 AUDIENCE: So-- 01:01:07.145 --> 01:01:08.060 AUDIENCE: That would be [INAUDIBLE]. 01:01:08.060 --> 01:01:09.005 AUDIENCE: Like-- 01:01:09.005 --> 01:01:09.921 AUDIENCE: [INAUDIBLE]. 01:01:09.921 --> 01:01:12.327 PROFESSOR: Yeah, but the thing is, you 01:01:12.327 --> 01:01:14.160 don't have to do anything that's so immoral. 01:01:14.160 --> 01:01:15.630 Because what you're interested in 01:01:15.630 --> 01:01:18.309 is the R0 for individual in an actual environment 01:01:18.309 --> 01:01:19.850 that they're actually going to be in. 01:01:19.850 --> 01:01:24.210 So this is a situation where the doctor comes back from Africa 01:01:24.210 --> 01:01:27.170 after working with Doctors Without Borders. 01:01:27.170 --> 01:01:29.704 In this day and age, he knows that he has to watch out 01:01:29.704 --> 01:01:30.620 for a fever, da-da-da. 01:01:30.620 --> 01:01:32.910 And then, if he gets a fever, he calls in. 01:01:32.910 --> 01:01:34.440 And he's brought to the hospital. 01:01:34.440 --> 01:01:36.240 And that's the world that we're interested in of what 01:01:36.240 --> 01:01:36.950 the R0 is. 01:01:36.950 --> 01:01:40.050 It's not the world in which there's fevers everywhere 01:01:40.050 --> 01:01:43.470 and nobody knows. 01:01:43.470 --> 01:01:45.280 So the R0 in the United States is 01:01:45.280 --> 01:01:50.108 going to be much lower than the R0 somewhere else. 01:01:50.108 --> 01:01:51.056 Yeah? 01:01:51.056 --> 01:01:53.599 AUDIENCE: Is there enough of R0 for a disease that you 01:01:53.599 --> 01:01:54.848 don't transmit between people? 01:01:54.848 --> 01:01:56.952 You get malaria of the plague-- 01:01:56.952 --> 01:01:57.910 PROFESSOR: Yeah, right. 01:01:57.910 --> 01:02:03.024 So I think that you could try to generate a similar kind of R0 01:02:03.024 --> 01:02:03.940 for those to diseases. 01:02:03.940 --> 01:02:05.990 Although, it's going to be very muddled, 01:02:05.990 --> 01:02:08.540 in the case of where you have all the vectors and so forth. 01:02:08.540 --> 01:02:13.830 Because it's not even clear-- yeah, 01:02:13.830 --> 01:02:16.360 I'm hesitant to say too much. 01:02:16.360 --> 01:02:18.736 Because I don't know anything. 01:02:18.736 --> 01:02:20.155 AUDIENCE: According to Wikipedia, 01:02:20.155 --> 01:02:23.315 it's like 1.5 to 2.5. 01:02:23.315 --> 01:02:24.190 PROFESSOR: For Ebola? 01:02:24.190 --> 01:02:24.550 AUDIENCE: Yeah. 01:02:24.550 --> 01:02:25.216 PROFESSOR: Here? 01:02:25.216 --> 01:02:26.522 Or in-- 01:02:26.522 --> 01:02:31.076 AUDIENCE: It says the 2014 West Africa aggregate. 01:02:31.076 --> 01:02:33.876 PROFESSOR: Ah, so this is in West Africa then. 01:02:33.876 --> 01:02:34.750 AUDIENCE: Apparently. 01:02:34.750 --> 01:02:35.458 PROFESSOR: Right. 01:02:35.458 --> 01:02:38.460 Yeah And it has obviously spread exponentially, 01:02:38.460 --> 01:02:40.779 which means it had to have been larger than 1. 01:02:40.779 --> 01:02:43.320 And I think it's important to remember that, just because you 01:02:43.320 --> 01:02:47.130 have a chart with a bunch of R0s, this is not set in stone. 01:02:47.130 --> 01:02:50.640 Public policy, hygiene, and everything changes this. 01:02:50.640 --> 01:02:52.850 And we'd like to drive it down. 01:02:52.850 --> 01:02:54.258 Was there a question in the back? 01:02:54.258 --> 01:02:56.620 AUDIENCE: I just was going to say about two. 01:02:56.620 --> 01:02:57.995 PROFESSOR: Same thing, about two. 01:02:57.995 --> 01:03:02.170 OK, so that means that we could actually, in principle, 01:03:02.170 --> 01:03:03.950 vaccinate against Ebola. 01:03:03.950 --> 01:03:07.580 We'd only have to get over 50% of the population vaccinated 01:03:07.580 --> 01:03:08.230 in West Africa. 01:03:08.230 --> 01:03:11.940 And then, we can maybe make it so it cannot spread and become 01:03:11.940 --> 01:03:12.750 an epidemic. 01:03:12.750 --> 01:03:17.434 Of course, we need to have a vaccine first. 01:03:17.434 --> 01:03:19.046 AUDIENCE: [INAUDIBLE]. 01:03:19.046 --> 01:03:20.670 PROFESSOR: So I think I'm going to skip 01:03:20.670 --> 01:03:22.350 the discussion of SIR models. 01:03:22.350 --> 01:03:24.670 Because you are going to play with some of them 01:03:24.670 --> 01:03:27.140 in the context of your problem set. 01:03:27.140 --> 01:03:30.180 And if you just Google SIR, you can find it. 01:03:30.180 --> 01:03:31.712 And a very similar kind of intuition 01:03:31.712 --> 01:03:32.920 that you get from this model. 01:03:35.895 --> 01:03:38.270 Because I do want to spend the last 15 minutes, at least, 01:03:38.270 --> 01:03:39.728 talking about the evolution of sex. 01:03:39.728 --> 01:03:41.850 Because it is an interesting topic. 01:03:41.850 --> 01:03:47.126 And I think the paper is a nice discussion of it. 01:03:54.920 --> 01:04:00.540 So can somebody say why it is that this is a puzzle at all? 01:04:11.408 --> 01:04:13.813 Yeah? 01:04:13.813 --> 01:04:15.854 AUDIENCE: In almost all the situations we imagine 01:04:15.854 --> 01:04:19.806 and things that can be [INAUDIBLE] introduce much 01:04:19.806 --> 01:04:21.501 faster, even violating the [INAUDIBLE]-- 01:04:21.501 --> 01:04:22.500 PROFESSOR: That's right. 01:04:22.500 --> 01:04:25.634 AUDIENCE: --80% [INAUDIBLE] right? 01:04:25.634 --> 01:04:29.130 PROFESSOR: So sex is costly, and in particular, if you have 01:04:29.130 --> 01:04:33.130 this obligate bi-parental sex. 01:04:33.130 --> 01:04:37.795 In particular, there's the so-called twofold cost 01:04:37.795 --> 01:04:38.295 of males. 01:04:41.670 --> 01:04:44.200 Because you can imagine comparing these two 01:04:44.200 --> 01:04:47.660 populations, one of which has both males and females. 01:04:47.660 --> 01:04:51.984 And one of them is just, maybe, reproducing asexually, 01:04:51.984 --> 01:04:53.900 or parthenogenetically, or hermaphroditically, 01:04:53.900 --> 01:04:56.310 or what not. 01:04:56.310 --> 01:04:59.810 And so, if you have a male and a female, 01:04:59.810 --> 01:05:01.570 then on average, if they have two kids, 01:05:01.570 --> 01:05:03.999 you end up with another male and a female. 01:05:03.999 --> 01:05:06.040 And of course, this could be many different males 01:05:06.040 --> 01:05:06.539 and females. 01:05:06.539 --> 01:05:08.990 So you don't have to have any sibling anything. 01:05:08.990 --> 01:05:15.125 But if, every generation, each female 01:05:15.125 --> 01:05:18.610 is giving birth to two progeny, then you 01:05:18.610 --> 01:05:20.610 end up with a constant population size. 01:05:20.610 --> 01:05:22.970 Whereas, if you started out in a population 01:05:22.970 --> 01:05:25.950 with just two females and they were reproducing 01:05:25.950 --> 01:05:32.290 hermaphroditically, then you end up with-- whatever-- more. 01:05:32.290 --> 01:05:33.660 8. 01:05:33.660 --> 01:05:35.650 So you can see that you get a factor of 2 01:05:35.650 --> 01:05:38.096 in the rate of exponential growth of the population. 01:05:38.096 --> 01:05:38.596 Yeah? 01:05:38.596 --> 01:05:41.054 AUDIENCE: Well, it seems to me the question is not why sex, 01:05:41.054 --> 01:05:42.932 but it's why males. 01:05:42.932 --> 01:05:44.670 Right, I mean-- 01:05:44.670 --> 01:05:46.010 PROFESSOR: Yes. 01:05:46.010 --> 01:05:52.150 So I think this is the most extreme cost of sex. 01:05:52.150 --> 01:05:55.400 But then, also, you can think about even just 01:05:55.400 --> 01:05:57.730 horizontal gene-transfer among bacteria. 01:05:57.730 --> 01:05:59.740 It's a costly behavior in some way or another. 01:05:59.740 --> 01:06:02.010 It's not as costly as this. 01:06:02.010 --> 01:06:05.670 But if you're going to think about bacteria 01:06:05.670 --> 01:06:08.440 in their competence state, when B. subtilis kind of pulls 01:06:08.440 --> 01:06:10.630 in DNA, it stops dividing. 01:06:10.630 --> 01:06:12.930 And then, it enters a state where it reels things in. 01:06:12.930 --> 01:06:17.530 And it can pick up DNA that may be harmful in some cases. 01:06:17.530 --> 01:06:23.042 So there are various costs for, if you want to call it, 01:06:23.042 --> 01:06:24.250 horizontal gene-transfer sex. 01:06:26.970 --> 01:06:29.100 So the key thing of sexual reproduction 01:06:29.100 --> 01:06:33.020 is it somehow is the sharing of the DNA. 01:06:33.020 --> 01:06:36.730 And I'd say that this can either be relatively low cost 01:06:36.730 --> 01:06:37.739 or relatively high cost. 01:06:37.739 --> 01:06:39.530 But this is the most extreme version of it. 01:06:39.530 --> 01:06:44.560 And I'd say, as a species that reproduces 01:06:44.560 --> 01:06:46.520 with obligate bi-parental sex, then 01:06:46.520 --> 01:06:52.770 I'd say that, not only humans, but almost all animals 01:06:52.770 --> 01:06:55.500 have this form of reproduction. 01:06:55.500 --> 01:06:57.420 I would say it's sort of surprising, 01:06:57.420 --> 01:06:59.130 given that this is a huge cost. 01:07:01.484 --> 01:07:02.900 But it's not that all species that 01:07:02.900 --> 01:07:05.480 engage in any sort of gene-transfer 01:07:05.480 --> 01:07:07.190 bear this large of a cost. 01:07:07.190 --> 01:07:09.100 But they bear more modest costs. 01:07:09.100 --> 01:07:13.410 AUDIENCE: Well, although, other species do-- even 01:07:13.410 --> 01:07:15.310 with this [INAUDIBLE], it's like having 01:07:15.310 --> 01:07:18.668 multiple children per sex. 01:07:18.668 --> 01:07:21.126 PROFESSOR: You're saying that they can just have more kids. 01:07:21.126 --> 01:07:21.980 AUDIENCE: Yeah, I mean-- 01:07:21.980 --> 01:07:23.620 PROFESSOR: Yeah, although the basic statement is still true. 01:07:23.620 --> 01:07:26.349 Let's say that these females all have three kids. 01:07:26.349 --> 01:07:27.640 They get to grow exponentially. 01:07:27.640 --> 01:07:29.210 But then, these guys grow faster. 01:07:29.210 --> 01:07:31.793 And ultimately, there's going to be competition for resources. 01:07:31.793 --> 01:07:33.575 And these ones will still win. 01:07:33.575 --> 01:07:34.200 AUDIENCE: Yeah. 01:07:34.200 --> 01:07:35.674 PROFESSOR: So the question is-- you 01:07:35.674 --> 01:07:37.340 can imagine you have a population that's 01:07:37.340 --> 01:07:40.480 reproducing sexually-- if one female has a mutation that 01:07:40.480 --> 01:07:43.744 leads her to start reproducing parthogenetically, 01:07:43.744 --> 01:07:45.410 that mutation you'd expect should spread 01:07:45.410 --> 01:07:47.900 throughout the population very rapidly. 01:07:47.900 --> 01:07:50.550 And indeed, there are these cases of, for example, 01:07:50.550 --> 01:07:53.950 sharks held in captivity where a female held 01:07:53.950 --> 01:07:56.370 in captivity for years eventually gives birth 01:07:56.370 --> 01:07:58.925 to daughters. 01:08:02.780 --> 01:08:05.100 So this sort of virgin birth is possible. 01:08:05.100 --> 01:08:07.275 I'm not aware-- well, I'm not going to talk about-- 01:08:07.275 --> 01:08:08.280 [LAUGHTER] 01:08:10.730 --> 01:08:15.520 But in some animals, it is at least possible, I'll say. 01:08:15.520 --> 01:08:18.870 So then, you have to ask, well, what 01:08:18.870 --> 01:08:22.220 kind of selective advantage could sexual reproduction have 01:08:22.220 --> 01:08:25.460 that could possibly compensate for this so-called twofold cost 01:08:25.460 --> 01:08:27.727 of sex or twofold cost of males? 01:08:30.910 --> 01:08:36.120 And what is the argument that's made in this paper? 01:08:54.220 --> 01:08:54.720 Anybody? 01:08:58.828 --> 01:08:59.780 Yes? 01:08:59.780 --> 01:09:03.588 AUDIENCE: The recombination favors genetic diversity. 01:09:03.588 --> 01:09:04.910 PROFESSOR: That's right. 01:09:04.910 --> 01:09:07.880 This recombination is favoring genetic diversity. 01:09:07.880 --> 01:09:10.260 So there are a number of different mechanisms. 01:09:10.260 --> 01:09:13.399 I'd say that at the heart of this idea of the Red Queen 01:09:13.399 --> 01:09:17.330 hypothesis is-- let me see if I can find 01:09:17.330 --> 01:09:21.060 the actual quote for you guys. 01:09:21.060 --> 01:09:21.700 Maybe not. 01:09:24.229 --> 01:09:26.840 So it's from a Lewis Carroll novel, 01:09:26.840 --> 01:09:28.290 Through the Looking Glass. 01:09:28.290 --> 01:09:31.845 The quote was something-- run-- oh, shoot. 01:09:37.370 --> 01:09:39.740 Never mind, I can't remember what the quote was. 01:09:39.740 --> 01:09:44.010 But the idea is that sexual reproduction 01:09:44.010 --> 01:09:46.830 may allow a population to adapt against, say, 01:09:46.830 --> 01:09:49.830 changing environments more rapidly. 01:09:49.830 --> 01:09:51.580 And this has a couple of reasons. 01:09:51.580 --> 01:09:53.910 Because you generate genetic diversity. 01:09:53.910 --> 01:09:56.565 You don't have the same clonal interference effects 01:09:56.565 --> 01:09:58.650 that we talked about earlier. 01:09:58.650 --> 01:10:00.780 So if you have asexual lineages, then, 01:10:00.780 --> 01:10:02.640 if you have two beneficial mutations, 01:10:02.640 --> 01:10:04.960 they cannot both fix. 01:10:04.960 --> 01:10:06.647 So the more fit version is going to fix. 01:10:06.647 --> 01:10:08.480 And then, you have to wait for the next one. 01:10:08.480 --> 01:10:11.360 Whereas, in sexually reproducing populations, 01:10:11.360 --> 01:10:14.240 those genes can spread throughout the population, 01:10:14.240 --> 01:10:16.390 sort of as genes, rather than being tied 01:10:16.390 --> 01:10:18.007 to a particular individual. 01:10:18.007 --> 01:10:20.590 So that means that, if you find yourself in a new environment, 01:10:20.590 --> 01:10:23.140 it may be the case that sexual reproduction can 01:10:23.140 --> 01:10:25.670 allow for the population to adapt more rapidly. 01:10:25.670 --> 01:10:28.671 But on one hand, you might say, well, the environment's always 01:10:28.671 --> 01:10:29.170 changing. 01:10:29.170 --> 01:10:31.030 So that can always favor the sexually reproducing 01:10:31.030 --> 01:10:31.550 populations. 01:10:31.550 --> 01:10:33.680 But then, there's a feeling out there 01:10:33.680 --> 01:10:35.760 that maybe that's not enough, in the sense 01:10:35.760 --> 01:10:37.800 that the environment is not changing 01:10:37.800 --> 01:10:39.520 rapidly enough and dramatically enough 01:10:39.520 --> 01:10:43.880 to force the population to reproduce sexually as compared 01:10:43.880 --> 01:10:46.270 to asexually. 01:10:46.270 --> 01:10:49.040 And so, the proposal from the Red Queen Hypothesis 01:10:49.040 --> 01:10:52.060 is that the constantly changing environment 01:10:52.060 --> 01:10:55.870 is a result of co-evolution between hosts 01:10:55.870 --> 01:10:57.930 and their parasites. 01:10:57.930 --> 01:11:01.650 Because parasites are always trying to target common host 01:11:01.650 --> 01:11:02.180 genotypes. 01:11:02.180 --> 01:11:04.490 Because they can spread on those. 01:11:04.490 --> 01:11:06.140 So the parasites are evolving. 01:11:06.140 --> 01:11:08.860 And then, the host populations or genotypes 01:11:08.860 --> 01:11:11.570 are kind of being chased away by those parasites that 01:11:11.570 --> 01:11:12.860 are targeting them. 01:11:12.860 --> 01:11:16.120 So the notion is that parasites, as a result 01:11:16.120 --> 01:11:18.800 of this co-evolution, can be the source for the constantly 01:11:18.800 --> 01:11:23.440 changing environment that may be driving the evolution of sex. 01:11:23.440 --> 01:11:23.940 Yeah? 01:11:23.940 --> 01:11:26.910 AUDIENCE: And this, proportional with [INAUDIBLE], bacteria 01:11:26.910 --> 01:11:30.870 also get parasites [INAUDIBLE]. 01:11:30.870 --> 01:11:32.899 PROFESSOR: That's right. 01:11:32.899 --> 01:11:35.800 AUDIENCE: So-- 01:11:35.800 --> 01:11:38.440 PROFESSOR: Yes, so is the question is why-- 01:11:38.440 --> 01:11:40.530 AUDIENCE: Why only in multi-- 01:11:40.530 --> 01:11:42.260 PROFESSOR: Right. 01:11:42.260 --> 01:11:45.690 So I can give you my best guess on this. 01:11:45.690 --> 01:11:49.090 First of all, not everybody agrees that the Red Queen 01:11:49.090 --> 01:11:52.200 Hypothesis is the true explanation, 01:11:52.200 --> 01:11:55.730 if a true explanation even exists. 01:11:55.730 --> 01:11:58.120 But within this framework, you definitely 01:11:58.120 --> 01:12:04.030 want to try to explain why it is that large life forms seem 01:12:04.030 --> 01:12:06.350 to have a lot of obligate sexual reproduction. 01:12:06.350 --> 01:12:08.711 Because this is a very strong pattern that you see. 01:12:08.711 --> 01:12:10.960 And I guess what I would say is that there's certainly 01:12:10.960 --> 01:12:17.620 a correlation between physical size and generation time. 01:12:17.620 --> 01:12:19.490 And generation time tells you something 01:12:19.490 --> 01:12:22.130 about the typical time scales over which you can evolve. 01:12:22.130 --> 01:12:23.920 So my sense of this is it just maybe 01:12:23.920 --> 01:12:29.190 that, in general, large animals, by their nature, 01:12:29.190 --> 01:12:32.552 will evolve rather slowly as compared to their parasites. 01:12:32.552 --> 01:12:34.760 And so, that means that they are the populations that 01:12:34.760 --> 01:12:41.200 are most in need of speeding up their evolution. 01:12:41.200 --> 01:12:44.180 And it's hard to know how convincing that argument 01:12:44.180 --> 01:12:44.680 should be. 01:12:44.680 --> 01:12:48.640 But-- yes? 01:12:48.640 --> 01:12:52.922 AUDIENCE: What's the difference in reproductive productive time 01:12:52.922 --> 01:12:55.916 between phage and bacteria? 01:12:55.916 --> 01:12:57.912 Like, how fast does phage-- 01:12:57.912 --> 01:12:59.120 PROFESSOR: Right. 01:12:59.120 --> 01:13:03.230 So bacteria can-- as we've discussed, 01:13:03.230 --> 01:13:05.870 division times are order hour. 01:13:05.870 --> 01:13:11.410 And phage-- when you get a phage infection, what happens 01:13:11.410 --> 01:13:14.250 is that the phage you can infect as a single phage. 01:13:14.250 --> 01:13:21.130 And then, they will divide within the bacterial cell 01:13:21.130 --> 01:13:24.986 and then burst out as a population of 100, 01:13:24.986 --> 01:13:26.890 200-- typical of phage. 01:13:26.890 --> 01:13:30.800 And I think that that might take four or five 01:13:30.800 --> 01:13:32.240 hours, is kind of my sense. 01:13:35.000 --> 01:13:37.180 And then, they go off and they find new bacteria. 01:13:37.180 --> 01:13:42.181 So in that sense, it's a little bit faster than the bacteria, 01:13:42.181 --> 01:13:42.681 I'd say. 01:13:49.070 --> 01:13:52.320 So can somebody say what the core experiment was that they 01:13:52.320 --> 01:13:53.670 did in this experiment? 01:14:15.538 --> 01:14:17.029 Yes? 01:14:17.029 --> 01:14:25.975 AUDIENCE: [INAUDIBLE] and one time, 01:14:25.975 --> 01:14:29.951 it was reproducing asexually when they always 01:14:29.951 --> 01:14:31.442 produce sexually. 01:14:31.442 --> 01:14:33.001 And the other time, it was dividing. 01:14:33.001 --> 01:14:34.000 PROFESSOR: That's right. 01:14:34.000 --> 01:14:40.232 So this worm, C. Elegans-- sort of a millimeter in size-- 01:14:40.232 --> 01:14:42.440 and there are going to be three different conditions. 01:14:42.440 --> 01:14:51.270 One is the wild type that can out-cross, can have males mate. 01:14:51.270 --> 01:14:54.510 Then, there's the obligate out-crossing, 01:14:54.510 --> 01:15:00.310 which means they have to mate with males. 01:15:00.310 --> 01:15:03.950 And then, there's the-- what did they call it? 01:15:06.759 --> 01:15:07.550 --obligate selfing. 01:15:13.505 --> 01:15:14.005 OK. 01:15:18.870 --> 01:15:23.146 And then, what they do with those worms? 01:15:23.146 --> 01:15:25.122 AUDIENCE: They put them right back 01:15:25.122 --> 01:15:31.121 to [INAUDIBLE] more typical. 01:15:31.121 --> 01:15:32.120 PROFESSOR: That's right. 01:15:32.120 --> 01:15:37.360 So then, there's a bacterial pathogen, Serratia marcescens. 01:15:37.360 --> 01:15:41.610 And they're going to have these three different conditions. 01:15:41.610 --> 01:15:45.300 For the SM, the bacteria, they're 01:15:45.300 --> 01:15:51.520 either going to allow for co-evolution, where 01:15:51.520 --> 01:15:54.670 they take the bacteria from each of the infections 01:15:54.670 --> 01:15:55.940 and propagate. 01:15:55.940 --> 01:16:00.180 Or they're going to do the no evolution, where you just 01:16:00.180 --> 01:16:03.060 compete against the ancestor. 01:16:03.060 --> 01:16:04.190 And there's also a control. 01:16:04.190 --> 01:16:05.720 So there's co-evolution. 01:16:05.720 --> 01:16:08.544 There's this no evolution of the bacteria. 01:16:08.544 --> 01:16:10.960 And then, there's also control, where there's no bacteria. 01:16:17.246 --> 01:16:18.745 And what was the most striking thing 01:16:18.745 --> 01:16:20.120 that they saw in this experiment? 01:16:28.825 --> 01:16:32.290 AUDIENCE: I guess I would say that the obligate-selfing 01:16:32.290 --> 01:16:33.990 population died in evolution. 01:16:33.990 --> 01:16:35.020 PROFESSOR: That's right. 01:16:35.020 --> 01:16:39.140 So if you allowed the bacteria to be evolving, 01:16:39.140 --> 01:16:44.659 against this obligately selfing population, 01:16:44.659 --> 01:16:45.575 this killed the worms. 01:16:48.740 --> 01:16:50.430 And what was the other thing that 01:16:50.430 --> 01:16:55.060 was, maybe, very striking about their experiment? 01:16:55.060 --> 01:16:56.530 Yeah. 01:16:56.530 --> 01:17:00.889 AUDIENCE: The worms that did the out-cross or self increased. 01:17:00.889 --> 01:17:01.430 [INAUDIBLE]-- 01:17:01.430 --> 01:17:03.510 PROFESSOR: Yeah, so this was kind of amazing. 01:17:03.510 --> 01:17:06.130 So they saw, as a function of time, 01:17:06.130 --> 01:17:10.400 if you look at out-crossing-- the rate of mating 01:17:10.400 --> 01:17:15.340 with the males-- this started out at, like, 0.2. 01:17:15.340 --> 01:17:19.130 And then, in the presence of the co-evolution, it went up. 01:17:19.130 --> 01:17:22.510 And it goes up to, maybe, 80%. 01:17:25.760 --> 01:17:29.470 And for co-evolution, it stayed up high. 01:17:29.470 --> 01:17:35.250 Whereas, if the wild-type worms continued 01:17:35.250 --> 01:17:38.780 to be just challenged by the ancestral bacteria, 01:17:38.780 --> 01:17:39.890 it initially came up. 01:17:39.890 --> 01:17:41.324 But then, it came back down. 01:17:41.324 --> 01:17:42.490 So this is the co-evolution. 01:17:45.340 --> 01:17:51.950 And this is the ancestral SM. 01:17:51.950 --> 01:17:54.205 So there was a sense that that wild-type population 01:17:54.205 --> 01:17:57.040 had initially evolved to out-cross more. 01:17:57.040 --> 01:17:59.040 But then, once it had solved this problem of how 01:17:59.040 --> 01:18:01.249 to handle the ancestral Serratia, 01:18:01.249 --> 01:18:02.790 the out-crossing rate went back down. 01:18:05.450 --> 01:18:07.550 So we are pretty much out of time. 01:18:07.550 --> 01:18:09.869 But I don't know if you guys noticed 01:18:09.869 --> 01:18:11.160 the last sentence of the paper. 01:18:11.160 --> 01:18:15.405 It is amazing that they got this through the publication 01:18:15.405 --> 01:18:15.905 process. 01:18:18.810 --> 01:18:21.100 All right, so they say, "taken together, 01:18:21.100 --> 01:18:24.150 the results demonstrate that sex can facilitate adaptation 01:18:24.150 --> 01:18:25.700 to novel environments. 01:18:25.700 --> 01:18:27.620 But the long-term maintenance of sex 01:18:27.620 --> 01:18:30.396 requires that the novelty does not wear off." 01:18:30.396 --> 01:18:31.870 [LAUGHTER] 01:18:31.870 --> 01:18:36.310 It's one of those things that you read and you think-- OK. 01:18:36.310 --> 01:18:38.570 So I will leave that sentence with you. 01:18:38.570 --> 01:18:43.670 And then, we'll meet on Tuesday, OK?