 that I'd like to introduce the next lecturer. So it's my pleasure to introduce Mercedes Pascual. Mercedes is a professor in the Ecology and Evolutionary Department of the University of Chicago and an external faculty of the Santa Fe Institute. Her research focuses on modeling and theory of the Ecology and Evolution of Infection Diseases, combining, in my opinion, a unique way, highly conceptual theoretical question with highly applied ones. And today is giving the first lecture on the Ecology and Evolutionary Dynamics in host pathogen systems and data set. So Mercedes, thank you very much for being virtually with us. Yes, what can I do? Here I am. OK, thank you. Let me share the screen. OK, let me just, OK. I see Jacopo here in my screen. It works well. Yes, I don't see myself. That's better. OK, so yeah, thank you, Jacopo, and Matteo, and also Simon, and the organizers. So I will give this first lecture. And then I change a little bit my program. The third lecture will not be on this interplay of Ecology and Evolution. And I'll mention a little bit more about that at the end. So but today, I'd like to follow up from what all you have heard about communities and talk about assembly and coexistence, but no longer in the context of species, but pathogen strains within a population and their interaction with a host. So this will concern the interplay of ecology and evolution in host pathogen systems. And it will also be about assembly when we consider ecological interactions, but now as a function of explicit traits. And finally, there is a conceptual connection to what you heard yesterday from Jonathan on stabilizing competitions and success and I would like to emphasize frequency-dependent selection. My real goal is to set the stage for what I want to tell you tomorrow about our work, rather than so anyhow, this is intended as background to tell you really about what I will call hyperdiverse systems in next lecture. So oops, why is it that I can't go forward? Jacopo? I don't know. Try to... Let me try again. Okay, good. Ah, thanks. So let me start with a quote from Simon, of all people Simon Levine. This is from his paper on complex adaptive systems, where he wrote the fundamental problems in the study of any... Why is it now that the fundamental problems in the study of any complex system are understanding what maintains diversity and how the existence of diversity affects system dynamics? And he mentions it is these three features, heterogeneity and its maintenance, frequency dependence and modularity that complicate the picture and that will occupy most of my attention today. So I will basically be telling you about frequency-dependent interactions and frequency-dependent selection, as you will see. And when we talk about, of course, diversity in ecology, we almost have two large camps. One that emphasizes the sort of glissonian view that Stefano told about you, about that emphasizes stochastic assembly and demography, the processes, the birth death processes of extinction, immigration. And, of course, we have there the neutral theory of Steve Hubbell. In the other camp, we have approaches, perhaps traditionally more based on deterministic systems, but not necessarily that emphasize interactions, ecological interactions. And I have here specific ecological interactions that were differences between the species really matter. And, of course, this is very different from neutral theory where we basically have an equivalence of species in these kind of birth death processes. We heard from Jonathan yesterday about the many hypotheses, the many possible mechanisms and explanations for coexistence that lie here. And, of course, one of them involves this limiting similarity through niche partitioning character displacement, et cetera. And I like now to get back to this picture you saw yesterday to emphasize that besides help us organize these different mechanisms, they also tell us about the different traits that underlie these interactions. In the axis that was called niche differences, the traits really confer, as we heard, an advantage to the rare, but also disadvantage to the common. So this is the axis where we have frequency-dependent interactions, frequency-dependent selection when we start looking from the perspective of evolution. The other axis, which was called yesterday, fitness differences really relates to traits that confer absolute fixed advantages. And what I will talk about today will lie on the x-axis here. So I will consider that the strains vary only in their frequencies dynamically, but not really on the y-axis. And, of course, the realm of the y-axis for pathogens concerns the reproductive number r0. So really there we have this, of course, the exclusion of the pathogen with the lower r0. And there are mathematical studies of those kinds of interactions. Now, you can tell me why am I talking here about competition, perhaps it's obvious to some of you. I like to remind everybody, and especially those that are not interested in pathogens, but like ecology, that infectious diseases are consumer resource systems. They are essentially natural oscillators where the infections are the consumers and the resource are the hosts. And when we do not consider any strain differences in the typical susceptible infected recover system where individuals acquire immunity and that removes them from the pool of resources, we have essentially this generalized competition for the resources. When we have specific, when we have now specific immunity, we will have something more interesting. And, of course, the traits that matter here are the variant surface antigens, the molecules that the immune system recognizes. So one advantage of thinking about competition between strains is that we really know what are the traits we should be looking at. So what are the traits that underlie competition for hosts? And this is an example in influenza, in this diagram where we have, of course, the well-known lycoproteins that are on the surface of the cell, in particular hemagglutinin, and this variable protein is the major antigen of influenza. There is a second. So when we hear about H2N2, H1N1, we are referring to the big types of these, essentially the big classification of the subtypes of influenza. Now, you see here to the right, H3N2, and a typical tree of the seasonal influenza, H3N2, where the colors, essentially, well, this is, of course, a tree from the genetic sequences, but what you see with the colors here are the different clusters corresponding to different phenotypes. So these are seen by the immune system as distinct, and this can be really tested with assays, and so we know that there are these clusters. Now, what is interesting about this tree and has led to a lot of interesting theory is not the high diversity, the opposite, the limiting diversity at any given time. So you have here a dynamics of replacement with short branches, and it has puzzled people, why is it that given this very fast evolution of the virus in theory, we don't have the branches persist, why don't we have much more diversity of H3N2 globally? This led to interesting theory and essentially to a field, not by itself, not just that question that was introduced by Brian Grenfell and others in terms of its name, phylo-dynamics, where we look at essentially the influence of population dynamics on the structure of the trees and then how does the structure feedback onto the population dynamics? I like to briefly mention in passing two well-known papers in this area just to emphasize that there are studies that have relied on computational models that track both genotype and phenotype, essentially the genetic sequences of the viruses and then how they map to the phenotype, essentially to the traits I'm talking about, this antigenic variation on the surface of the viruses. And this is one paper by Neil Ferguson and others in Nature, where essentially they noted that of course the trees tend to be extremely explosive in terms of diversity and it is not easy to get these short branches. The explanation they provided for the short branches was that of short term, what they call strain-transcending immunity. So this is not, this is generalized immunity if you get infected by any strain, you are protected, you are no longer a resource and therefore this is a generalized competition that works in this regard. An alternative explanation was provided by Katya Cole and well, I'm in this paper too. And this was a paper that relied on this idea of a complex genotype phenotype map, the idea that we know that in flu, although the genotype varies with time in a continuous way, the phenotypic change of the virus is punctuated. So this is an observation, empirical observation from a paper by Derek Smith and colleagues, but what Katya proposes in this paper is essentially an interesting genotype phenotype map that I will not describe in detail, but I want to mention because this returns later where essentially, and this map is implemented with the idea of neutral networks and this was sketch here in a commentary on the paper where essentially the nodes of these networks of these networks are sequences, the different colors correspond to sequences that map to different antigens and you see that mutation moves you around, that's innovation, so you can move around and not change color in a sense. Once you change color, you have a great advantage so you have a selective sweep, then you explore again this new one and you go on and on. She showed, we showed in this paper that essentially this continuous mapping between the genetic distance and the phenotypic one allow you to have this limited effect on diversity. But again, these are complex systems very complex systems that rely on knowing something about the genotype to phenotype map and the computational models implement these ideas in some complex code. But I like to get back to simpler models after this somewhat brief introduction, but because before phylo dynamics, there was a large field I'm going to call strain theory that look at the competition of strains and for hosts and questions of diversity. And here I will mention the work of Suna Tragupta and focus on work now that does not consider the genotype at all, but focuses directly on the traits that is the antigenic identity of the pathogens. And it's interesting in this sort of seminal paper by Suna Tragupta and Day to see that they were motivated by malaria and the question of malaria in very endemic regions and the question of if there is such high transmission and people get infected so much, will it ever be possible to control malaria to vaccination? So we can return to these questions tomorrow when we get to malaria, because that's truly a very hyper diverse system. Now, the early models, I should say there is, as I said, a large literature and I'll provide at the end of this talk some references for those of you who are interested. There are a series of more analytical studies on the dynamics, analyzing moles with multiple strains. But I like to focus here on moles that in particular consider combinations of traits. So phenotypes that are essentially combinatorial in nature, right? Because we don't have, so this is not like, okay, strain I, strain J, right? The identity of the strains will emerge from a pool of variation and from a combination of traits. And I think that is important and I'll return to that tomorrow to give you a sense for why. And here we can choose to go with a discrete or a continuous space and I'm going to start with a discrete trade space. So you have to visualize here the representation as a pathogen having a set of I'm going to call them sites or loci. Each of those represents an antigen. You may find in the literature the word epitope, that's the part of the molecule that the immune system recognizes. And for each of these antigens, we have a finite number of possible variants as represented. So essentially a strain is a combination of these variants. So for example, we may have three epitopes or three antigens, one with three variants, another with four, another with five and then a strain is a combination. These are not linked, they are not physically linked but we will see that they can become dynamically linked. So one of the possible outcomes if you write models for these kinds of systems and here I have sketched the simplest possible case of well, essentially two antigens with two variants each. So there are four possibilities and what happens under some parameter regimes is that of course, as you let the model go, the ones that have some overlap will be selected against because they are at the disadvantage and the model goes towards this state that is this discordant state in which there is no overlap between the coexisting types. So we have hosts now that are immune to Y and B so they can get infected by this virus XA and the other way around. And I have this here just to let you see that essentially the niches that have emerged are these groups of hosts with different histories with different memories. And so it is these different groups of hosts that are the niches that emerge from this frequency dependent, essentially advantage of the rare disadvantage of the common. And you can see here that this depends strongly on the intensity of competition. We will see the equations in a moment if there is very weak effect of cross immunity. So if I'm infected by another kind, my memory does not influence what happens next. Of course, there is no strong structure. At the limit of very strong immune selection we get this emergence of niches with this stable discrete strain structure of the strains that have limited overlap. Now, when we write these equations, I have borrowed, I want to show you a little bit the equations that Gupta and colleagues wrote just to show some equations. And I also like the way they simplify the notation of the system by defining the state variables in a particular way. I have also borrowed this from a methodological paper they wrote after they had proposed all these ideas because this comes from, this comes a company with an R package that embeds some C code. And this is called Mantis. If you like to write and play with these kinds of things you can go to this paper and download that code. But what you can see here is that we have in this two by two systems, the state variables, the Zs here, Zax, is the proportion of the population that has seen, has been infected and has memory to strain Ax. More interesting is the definition of the variable Y, sorry, W in which you basically take the union with all those other essentially memories, right, that have some overlap with Ax so that either have A or X. And you do that for every pathogen there is a typo here in the blue this should be Bx, not Ax, but you see the picture. So just to show you that this simplifies greatly the notation because now for a given, of course there would be additional equations for other types, but you can see here in particular the equation for Y, which is the infected. Sorry, Jacopo, if I'm not mistaken the sort of typical epidemiological moles have been shown, right, by Marino, right? Yes, yes, Marino has discussed that. He has gone further, obviously further but here you can see, we have Z, W and this you can see how easily we can now write the equation for the infections, which this Y are the infected and you can see that we have okay beta, the transmission rate, then the infected with Ax, but now we have these two kinds of susceptibles, the one minus W have never seen Ax so they are fully susceptible while the W's minus those that have full protection have a reduction in their susceptibility, this is the cross immunity and this parameter gamma has the intensity, is the intensity of the competition or this cross immunity because you can see essentially if gamma is one, right, you basically any exposure will protect you completely even if you haven't seen both types, if you just have seen one of the types. So in this initial paper, one of the initial papers in science by Sunetra and colleagues, you can see here that gamma, if you look at the bifurcation diagram on the bottom, that of course this intensity of competition or cross immunity gamma is very important to the dynamical regime you see. Of course, if gamma is weak enough, you see no strain structure, if you move to the right, completely to the right, we get to this, what they call a discordant set, which is like the sort of complete limit, no overlap between the essentially, the steady state here. And then in the middle, we have the either periodic or chaotic cycles with some replacement. And this picture is a little busy, but just to emphasize those transitions, if you look in the center for one of these variable Ws, the bifurcation diagram, moving from the left to the right, you get essentially a steady state with everybody there. You go to these more cyclic or chaotic dynamics and finally completely to the right, one of the discordant sets. Of course, there are multiple discordant sets and which one you reach will depend on the, where you start and as we will see the history of this. Now, I provided before some explanations for the limited diversity of influenza, H3N2. And in a paper in PNAS 2006, this model that I just show you was revisited as an explanation, an alternative explanations to the one I gave you, essentially this limited set of possible strains, would lead under some parameters in this cyclic or more chaotic regime to something resembling influenza. Now, that's when we got together with Suna Tragupta in reaction to that paper, because when you look at those models in particular, in the deterministic version, obviously there is no stochastic extinction, but more important in these models, there was no explicit evolution. And by that I mean, there was of course, the change in frequencies, and you can call that evolution, but there was no explicit mutation, there was no explicit innovation. So we started with all the pool of variation in the system and what happened was that, of course, the abundances, the relative proportions went up and down when you have the discordant sets, you basically get some dominant coexisting ones, but in the fluctuating regime, you can always be rescued from these low levels, right? Nothing goes completely extinct. So we wanted to see what happens when we essentially take these models and really look at evolution. And again, there is a literature, an analytical literature and I'll get to that later, but here we don't want to separate the time scales of evolution and ecology or evolution and epidemiology because, well, essentially for these pathogens, that separation does not necessarily apply. So we will be talking about mutation and I should remind you that since we don't want to track genotypes and we don't want to deal with genotype phenotype maps, we are going to basically go, call mutation, a phenotypic change. So I may go from one epitope to another variant of this epitope in the pool of possible variation, but we are not going to track genotype. And this does not preclude us from looking at the tree. We will have individual basemoles, stochastic individual basemoles, where we can track the infection history of each host in time. And because we do that and the risk of infection then given a contact will depend on your immune history, whether you are protected, how much cross immunity you have, we can in fact, we don't need to reconstruct the genealogy in the way trees are built because since we know the whole history, we can sample it and basically we know the parent of any virus. So we can construct the tree computationally as we go along. So in that sense, we don't really need to track the genotype. So this was done in this paper by Zindor and colleagues. It was also done in a paper by Trevor Bedford. I'll show you in a moment. I'm just to give you a few more details about this here on the right. As I said, we can follow the tree. So we can also measure diversity, genetic diversity, for example, by going back from two viruses back to a common ancestor and the time to the common ancestor is a measure of that genetic diversity. On the ecological side, rather than having this cross immunity given just that you have seen one of the epitopes, we mold that a bit differently. We said the risk of infection will depend on the fraction of the epitopes. So the fraction of these traits you have seen before. And of course, here you see that a parameter sigma as we move sigma down here, the intensity of competition. So how important it is to have seen a certain fraction goes up. I did place down here a quantity that tells you whether we are seeing some positive or negative selection in the system. We adapted an index from McDonnell-Crogman to look at selection. Here, we basically count or compute the mutation rates on the trunk of the tree compared to the rate of mutations on the side branches. So the side branches are those that will go extinct. As you see here in black, from the end of the simulation, you can trace back the trunk. Those are the ones that have become fixed in the system. And so this ratio will give you, essentially when it is bigger than one, we have positive selection. When it is below one, you have negative selection of mutations. So the main result I want to show you is here. Remember that perhaps I was not clear enough in the work before, in the purely ecological work without explicit mutation, the parameter that was very important for the bifurcation diagrams was the intensity of competition. Here, I want to show you the effect of the speed of evolution of the mutation rate on the emergent structure and shape of the tree. So we start on the left by comparison with essentially the completely neutral case where there are no antigenic mutations. So we still get the tree, but all are equivalent. So this is the neutral tree. As we move to the right with mutation a bit higher, in this particular example, we see this regime of positive selection with the index I just described clearly above one. And this short branch is a short diversity, genetic diversity. This is the flu-like regime, the H3N2 regime. Now if we move to the right, and now the antigenic mutation is even faster, we get a more interesting situation, at least if you are interesting in diversity. I have to say I have cut the tree here just for the purpose of the figure. These purple and orange branches joined in the past. So they are two coexisting set of branches. So you see what we would call these discordance sets where now mutations may come in but get selected against because essentially the niches in the host have been established at least temporarily by these coexisting types. So we see here that these different dynamical regimes are very influenced by the speed of evolution and that to achieve the coexistence of discordance trains. So the emergence of niches in this system, evolution needs to be fast enough to explore trade space. And for this essentially, yes, equivolutionary dynamics to assemble niches with limiting similarity. That may be a bit counter-intuitive. If we go a little bit faster, or I mean, if we go faster, of course at some point we lose the pattern we are going to have essentially just a random pattern with variation on top of the tree. Oops, did I write on my slides or did you write on my slides, Jacopo? I'm not sure, but now I don't see the error. I just was wondering if you are interfering. So I will stop in a moment to see if there are temporary questions, but I wanted to mention that all this does not depend on of course having a discrete space. And we had a discrete space that was limited. I mean, here in the paper by Zinder, you will find an application to flu where we simulate with the tropics, the northern hemisphere, the southern hemisphere. So we try to see whether we can get the flu-like dynamics with realistic parameters. But we consider essentially five epitopes and each has a certain number of variants. So there is a limited space and that limited space has been emphasized in the past. I like to show you that in continuous straight space with this no limit to the different types, you can get essentially the same dynamics. This continuous straight space has a very strong empirical connection motivated by the very elegant work of Derek Smith and colleagues looking at data on the virus. And what you see here on the right is really what is called an antigenic map. So you can look at distances in a map that tells you how similar these viruses are. This is not like just an imaginary space. This is derived from data. We have for these viruses and this has been used for a long time to produce vaccines. Essentially, you can test one virus against another for how much cross-protection it gives you. There are these immuno-assays than in ferrets that allow you from those tables then to define a distance. And the very interesting idea of Derek Smith was to define that distance from those tables and then ask how many dimensions do we need to describe H3 and 2 influenza? And they found that you need a few dimensions with two dominant ones. And you can see in this plot that there is indeed one dominant one that corresponds to this tree with very short branches, right? And you are moving from 68 on the top down with a little bit of fluctuation in a second dimension but it's very one dimensional corresponding to this limiting diversity of flu. In this model, so this is a model by, let's see if I have, I'll have the reference later by Trevor Bedford when he was in my lab with Rambo and myself. This model now lives in a continuous and dimensional space. Here I'm going to start with two dimensions and now a phenotype is just a vector in that space. So we can look at what happens in this kind of representation. Importantly, the mutation here is where the trick of this model comes in a sense. So the mutation occurs at the rate. This is a stochastic model. It's an individual based model as before. And when a mutation occurs, then the phenotype, you choose the direction at random but the size of the mutation was chosen from a gamma distribution. And this was based on some empirical data. The idea of a gamma distribution comes here from the important, the parameters we chose. Most of these steps are going to be small but eventually you'll have a bigger step. And this is essentially a phenomenological representation of what Katya Cole did with this, with this genotype phenotype map represented here in this simpler way. So that is essential. And I like to just show you that this implementation recovers all the characteristics of flu. We have it here with northern hemisphere, southern hemisphere and the tropics because there is differences in reality in those parts of the world. And of course, the virus travels around. You can see the replacement dynamics. You can see the short branches of the tree and on the right in B, you see the two antigenic dimensions. The virus moves predominantly in one direction. And in the bottom, we just added the noise of the size that is typical of these kinds of measurements. But that's more or less what we should see. You can ask yourself, did it matter that we started in two dimensions? It doesn't seem to matter if you run this mall and you start, for example, with 10 dimensions, what emerges for the flu-like parameters of H2N2 is also just this one, largely one-dimensional trajectory shown here by showing the principal components of this antigenic space, where you see that, for example, PC1 against PC2, you are moving in time to the right, essentially, mostly along one dimension. And you see that, well, anyhow, in other projections. So these dynamics did not depend strongly on assuming two dimensions. Now, before I ask whether there are questions, for this part of the talk, I'd like to refer you, for those of you who are interested, to this paper by Trevor Bedford and colleagues in Nature, which followed, and which shows these actually are trees from data. So they are very interesting empirical trees. And I want to show you that, of course, as usual, the connection between these structure and dynamics, there are many, well, there are other factors that matter besides the speed of evolution and the intensity of the cross-protection. Here, we have the observed trees, we've reconstructed from sequences for H2N2, now you are familiar with it, except the colors here have to do with their geographical location. You have H1N1, another subtype within influenza A, and then two types of the influenza B that are less prevalent. And you see that the trees vary. A very interesting thing here is that H1N1 and the other flu viruses are known to evolve. Genetically, they mutate as fast, but phenotypically, they are not capable of changing as much. So the traits that matter to competition in H1N1 are not changing as rapidly as H3N2. What this paper then proposes is that, of course, then these changes, essentially, what is happening is that you are reducing the overall, the general pool of cost, right, faster for H1N1 because the virus is not moving away fast enough and therefore you are moving more towards what I would call neutral dynamics, but in a way that interacts here with the geography because the new types are coming from mostly from East Asia, South Asia, Southeast Asia. And so what you have is this new, in the mold that is presented in this paper, you also have this more detailed immigration patterns. And to show you the complexity of this, if essentially you are reducing overall susceptibles more, you are reducing the age of infection towards children farther and the children travel less. So you are having this interaction of travel, age of infection and intensity of competition. I mentioned that because there is a parallel work in phylo dynamics that really goes into inference. And when you hear about community phylogenetics and all the efforts to use community patterns in species, to say something about the underlying forces that are at play, I would like you to have a pitch for essentially inference. So you may have your hypothesis, but you may have to connect your molds to the data, the machinery for doing that in viruses where we have sequences and phenotypes as a function of time exist. And there is a lot of interesting work at this interface with statistics. So just to show you that here the demography, the competition and the speed of evolution are interacting to shape the trees. So what inference we can make, what explanations we have for the trees and what do they tell us in terms of how to intervene, how is it maybe to control the evolutionary drift of the virus, all of that requires this sort of going back and forth with the data. Okay, I'm going to take a break here for questions. Yes, there is one question by Miguel Rodriguez. Yes, this is really interesting in the models. For example, in the neutral tree that you show from the cinder paper, what is the source of the oscillations? What causes the rate of infections to oscillate? Yeah, this is an interesting question. And here, what is the source for, we have seasonal transmission in this model. Now, and I assume this is what is causing it here because it's very regular, but I would say that we would also have in these neutral models because of the equivalence, because of the complete equivalence, we would have also the possibility of cycling because you are basically eating up the susceptibles and having to wait for those to be replenished by births. So in the typical, if you take the paradigmatic studies of nonlinear dynamics in disease, you have of course a tendency of disease systems to oscillate. That tendency is essentially a damp cycle going to equilibrium. And then when it interacts with seasonality, you can get more complex patterns because you have two cycles interacting. So we could have even in the neutral case, in fact, measles, measles which has been studied and is the sort of more, as I said, paradigmatic studies of what we understand in epidemiology, measles is largely neutral because measles is a virus and this is why the vaccine works so well. Measles mutates but it cannot change antigenically. It cannot effectively change antigenically. So it's mutating but it's neutral because once you are infected, so essentially every virus is competing, it's a form of generalized competition that is neutral. So it's interesting to think about those parallels just because if I'm a host, I got measles or I got vaccinated, I'm removed. I'm no longer a resource. Right, and in the paper later on, when you simulate these dynamics and show that you can reconstruct the similar dynamics from the real data, in those simulations, are you imposing that seasonal? Yes, they are, yes. And I have not gone enough into those details in some simulations like here we have, I have to remember here, here we have seasonality in this paper. We also have a figure with the three, the simpler geographic structure, right? The tropics where there is, because the very interesting thing is that seasonality influences persistence. So away from the tropics, the seasonality is stronger. And for example, persistence in the northern hemisphere or the temporary regions is not high. And so essentially a lot of the persistence is in the tropics and there is also a typical movement from the north to the south. And so there is the very interesting, there has been a lot of inferential work on looking at the origin. Where is the trunk? So essentially where is the trunk of these trees in the world, right? Because that tells you the ones that the mutations that eventually become fixed, where do they come from? And it's very interesting to see where is the trunk, where do these mutants, these effective mutants come from? Thank you. So you can see the trunk right in the different places. And yeah, it's kind of interesting. There is another question from Gabriel. Yeah, I was wondering about the result you showed for the high dimensional phenotype. I was wondering, so is it because of the specific way that flu works that you're assuming that there's immunity towards past strains that it's always 1D? Like why can't we see like limit cycles, oscillations? Very interesting, yeah, good question. There is a nice picture, Trevor, but for this magnificent work. In fact, I should point you to all to the page he has. His websites, he has developed something called Next Flu which monitors on real time, does predictions of the next virus on real time and so on. So he has a picture of that landscape, that evolutionary landscape in this paper and you see exactly what you said. What happens is that the virus, there is this sort of wave of immunity behind it and it sort of has to move forward. But it's interesting if in this mall, and this is a toy mall, of course, if you mutate a bit faster, not that much faster, you will bifurcate and you will have coexisting, you know, the regime I showed you before in which you don't get replacements but you get coexistence of types. And so probably, of course, reality always has a mixture of hypotheses. And so there is a generalized immunity that was proposed, the generalized immunity that was proposed by Neil Ferguson as the original mechanism that gives you the short branches is also probably at play helping with this, preventing those bifurcations, right? And in this mall, you can get it, but of course you have to have the speed of evolution. Otherwise the system will bifurcate and will find niches and coexistence and then you get this coexisting regime. But what about the oscillations? Can you ever get like the immunity chasing the virus? Yeah, this is a very interesting question because in this paper that I like to go back here, sorry, here at the bottom I like to point you to, in this paper by Riker and colleagues in PNAS, they did propose that this mall, essentially this mall from strain theory that Sunetra had proposed earlier was an explanation for flu. And that's this discrete mall where you have this essentially a limited number of combinatory possibilities. And so in theory, what you should see at some point is the system cycling back, which has never been seen for H3N2. So of course in their paper, they argued that some interesting, there are some interesting issues about how we get the data from the assays, these immuno assays and which virus we test against past viruses, right? And whether we have seen, there is a question, have we long enough in time to see the cycles or are we sampling in the right way to see the cycles? I would say most people will feel more comfortable. I don't know where reality is. I don't think that this limited pool of variation is an important hypothesis for explaining the patterns. Now, how, whether an assumption of a completely infinite space like here is reasonable, it's hard to tell. So far, so far we have not seen any of these viruses we turn in a cyclic fashion. Yeah, interesting. Great, I don't see any other question, so. Do I have time, 10 more minutes? Yes. Yes, people must be very tired after this long day, but let me do this because I, yeah, I like to do this. So before I go on, I wanted to just stop here for a moment and point out again for those of you who are really tired of now listening to about, I mean, this lecture about strains, but there are very strong conceptual analogies to what we would see in ecological systems if we start essentially modeling this frequency dependent competition. Essentially along the Chesson axis of niche differences. And this was done in this paper. I'd like to direct you to this paper by Schaefer and Van Ness in PNAS where they have a niche axis. It's a one-dimensional niche axis. This is essentially a version of the MacArthur and Levine's model, but with evolution. So you have time running down and you start with these pieces that can mutate and are competing as a function of distance. And the only thing I like to notice at the end is that what persists in this discordant state here is these clusters, these clusters. So these are species and they are not one species at the distance of another species. It is these clusters of similar species away from another cluster of similar species. Very similar to the model I just showed you in continuous space for flu except that this one was done in one dimension. These communities do not talk to each other. I'll mention tomorrow further, well, sorry Thursday, some further analogies. Here I wanted to leave you with an example. We'll do this quickly off another virus because you can say, well, flu motivated all this work but is there evidence for the discordant regime in nature? Essentially the emergence of these kinds of niches. In the work by Gupta, Caroline Bacchi and so on, they talk about the meningococcus virus. You will see some empirical evidence. I like to mention here these other viruses. This is a rotavirus, the main cause of diarrheal disease in children and mortality due to diarrheal disease worldwide. This is a virus with, it's a double standard RNA virus. It has on the surface on this sort of nice picture that they drew here. We have two types, two main, two or three but let's say two main determine and molecules that determine the G types and then the P types. So those are different parts of the capsule of the virus. And I like to show you that if you look at here the genealogy of the G types, you'll see this coexistence. This is the global data set that exists for this segment. Essentially the segment is the segment that encodes the G type. If you do these trees, you find these coexistence that goes back to very, very, very long time. And if you look at the population dynamics, you also see coexistence here on the right, on the top with some fluctuations. And of course some new types that are coming like the green because this virus is of zoonotic origin and we get new viruses that come from animals into humans and introduce. So we have not just mutation but we have novelty through immigration. And I just like to point out that we have, we did an individual based model with three of these proteins just to mention that now the risk of infection again depends on distance but we have two components here in this formula. One is generalized. So if you have been infected before you are protected and that is believed to be the strongest protection against these virus. We added some specific protection with the Sigma specific depending on the fraction of these three types you have seen before. And this specific immunity is much believed to be much weaker in this virus. Now, this virus is messy because I said you get immigration but you also get reassortment. So these segments you get essentially recombination in these viruses, it's called reassortment. So you get this genetic mixing. So can you get any structure in the presence of this kind of genetic mixing that is essentially you, it's a force against this, just the simple lineages through mutation. I just want to tell you that in all of these, you see here an example where there is no reassortment the colors, I'm just stacking up the abundance of the different viruses in colors. The colors represent a different combination, a different strain, the red strain invaded and you see that we on the bottom, we are basically look at the number of shared segments. If you are on just sharing segments in the diagonal you are purely discordant and you see that these introductions get you some overlap temporarily and then the system reorganizes into the discordant states. If you get reassortment, of course you will get only partial discordance but this force, this frequency dependent competition is still leaving a strong signature of partial discordance. And I like to show you this wonderful, there is not enough data for these virus, there is nothing compared to flu but I wanted to show you this because I think it's an incredible pattern possibly of niches in nature. If you take the global data set that exists and take, you map it to the percentage of samples here on the left that have a certain G and a certain T type. So the red colors are the most abundant, the blue, the least abundance. You see that we have a lot of G1, P8, some G2, P4. So it looks like there is not that much overlap between these domain and things but you can say, well, maybe, maybe this is just because of their frequencies and it has nothing to do with a force acting like a negative frequency dependent selection. So on the right, I'll show you what is the probability of essentially is if you look at what you would expect from the frequencies, the global frequencies of the corresponding Gs and Ps, do you have a positive or a negative deviation and how significant it is? You can do this with randomizations, you can also do it with certain statistical tests here with randomizations in red. In red, we have the very, the sort of the deviations that are positive, right, that under P value and you see that of course we have much, we have much more G1, P8 and G2, P4 than we expect at random but more interestingly, some of the later arrivals like G6 or G12, they have come in and of course there is a lot here on the left of them associated with P8 because there is a lot of P8 around but if you look at the deviations, they are deviating positively towards places that were not occupied by these dominant types. So there is somehow this movement of the system to limit similarity. Maybe I could end there or I can show you a little bit more but I don't want to, you know, maybe I can end there. How am I doing with type? Maybe I'll go to the conclusion. So we are five minutes past. Yeah, so let me just conclude. I just wanted to show you that because it is a nice picture of this discordant state possibly in nature. And this makes the point that we can get these non-overlapping repertoires, in this case, even with weak cross immunity and that this frequency dependent selection can remain this effect of what we would call niche differences sense to Chesson but in a context of evolution can remain apparent despite this mixing by reassortment and despite immigration. Of course, depending on the relative strength here of these processes. So under these data shows patterns consistent with niche partitioning, this emergent niche partitioning but what are we missing? And I like to get to that tomorrow. I will not touch much on functional differences or barriers to reassortment essentially how does this access interact with the other access in Chesson which is a very interesting idea. But we haven't explored a lot. How do we compare the patterns to what we expect under neutrality which is a very important question. And because, and I have been talking about limiting diversity. I really want to talk about diversity and I want to talk about hyper diversity. So tomorrow I will present a system where the diversity of the pool of variation is so large that you will say, well, there can be any structure here and I hope to surprise you by showing you that we may be in a regime that is the complete opposite of neutrality and where specific interactions are incredibly important and are seen in the patterns we see in nature and that to see those patterns because we have recombination we are going to use not trees but networks. And networks will allow us to look at the structure that emerges from the underlying processes. So this is where I'm going to go tomorrow to hyper diverse systems. Hopefully connecting with the more analytical work that Daniel Fisher will show us. I think there are some conceptual connections to that and here are some papers for those of you who may be interested in some more mathematical aspects of competition between strains and in particular at the end some attempts to use population genetics to explain the patterns I just show you for flu. Thank you very much. Thanks a lot Mercedes for the very nice lecture. So we have a few more minutes for questions. If any, please use the raise hand feature. Okay, I don't see any. So otherwise, I mean, we will be back with Mercedes tomorrow, right? Yes, I don't have the program. No, I think Thursday. On Thursday, yes. I don't have the program. Yeah, I should say Jacob for the third lecture will not be anymore on diversity or whatever because as we discussed, since yes, this was about quantitative approaches in ecology. I like to talk a little bit about climate urbanization and vector-borne diseases. So different aspects from diversity which of course that fits more with what other people have been talking but I like to go a little bit there. Yes, absolutely. But to say that if there are more questions there will be, if question arises there will be time to answer them and this video will be available on YouTube. So you can really watch it.