 Okay. Okay. So welcome everybody to this second week of the school. Thank you very much for being here. So today we are going to have a different format compared to the ones we are used to from last week. So today is my pleasure to introduce the lecturer of today with Jordi Vasconte. Jordi is a professor of ecology and evolutionary biology at the University of Zurich in Switzerland and his research is mostly focused on ecological network and in particular on motoristic networks. So you have, as you know, Jordi Vasconte has uploaded two recorded lectures that were available on the website starting last week. And this session of today is a Q&A session. So it's an opportunity to discuss with Jordi the material presented in the lectures and I think ago also a little bit beyond that. So thank you very much, Jordi, for recording the lectures and for being with us. Thanks to you, Giacopo. It's really a pleasure and welcome everybody. And so now we are open to questions. So if you have any question on either of the two lectures, please use the raise and button of Zoom or write it in the chat or if you're following from YouTube, you can write it in the chat of YouTube. So we have a hand raised by Silvia. So Silvia, if you want to ask the question, please unmute and ask. Yeah, can you hear me? Yes. So I have a question on the first lecture concerning how the nested structure emerges. In fact, like in general, one could hypothesize that the structure could emerge either from the ecological dynamics or from the evolutionary dynamics, right? And on Friday, we heard the lecture by Stefano Alesina that was talking about community assembly. And we learned that there will be a structure in the interactions only if there was a structure in the original pool of species. So this would suggest that the nested structure can only emerge from the evolutionary dynamics. So could you comment on this and is there any of your works where you dealt with how the nested structure can emerge? Yes, that's a great question. Thanks for posing that. I think that as we've learned more about these ecological networks and the suite of mechanisms that are compatible with the structure, we tend to move from an initial focus where people were picking up a particular mechanism to considering like simultaneous mechanisms that can potentially be at war here. So early on, right after the first wave of studies describing the structure of these networks, people start focusing on particular mechanisms. For example, one of those is species abundance. So people realize that if most abundant species tend to be more available by chance, it's more likely that species would tend to interact with these more abundant than less abundant species. So this kind of neutral approach become one important mechanism in generating networks. At the same time, other people were focusing on other aspects. For example, phylogenetic signal is well known that there is phylogenetic signal, meaning that species close in the phylogeny tend to play similar roles in the network of interactions. What this is suggesting is that past evolutionary history may be important in understanding contemporary patterns of network built up. So for a while, as it tends to happen in science, it seemed like there was a little bit of a competition between specific mechanisms. I think where we are standing today is in a place where we recognize that there's a suite of mechanisms. There's not only one mechanism. So ecology certainly plays a role because species abundance, it's important in order to explain these patterns, but most likely it's not enough. And it's true that this coexist with this evolutionary signal with trade matching, for example, which is another set of explanations that emphasize that trades are the currency that explains interactions. And therefore, for example, one trade would be the length of a pollinator stone. Another trade could be the length of a plant's corolla, flower's corolla. And therefore, whether or not there's a matching between trades may be key in order to explain those interactions. So what I think it's now, the state where we are is in trying to understand, not proving that this is the mechanism or that's the mechanism, but in trying to weight the relative importance of a series of mechanisms, always taking into account the phylogenetic structure. Because in these comparative studies, one cannot forget that species are not independent units, but they form part of this process of related species coming from an ancestor origin. So that's where I think we are right now. I think that evolutionary mechanisms certainly are important, what you were emphasizing. But there are other set of mechanisms like ecological ones. And I don't think there's nothing wrong on that. I think that as with many other things in science, a pluralistic view is most likely going to be at work. So yes, I would highlight evolutionary mechanisms, trade mechanisms and neutral or species abundance mechanisms. If I can ask you just a little clarification on this, like when you mentioned the importance of abundance, so are the species that are most generalist typically the more abundant ones? Yes, that's something that people realize from an early stage. And if something, the debate was, I mean, are species more abundant because they are more generalist or the other way around. So that's really a bit where the discussion was. But from very much the early stage of these studies on ecological networks, and in particular these mutualistic networks, it was clear and particularly I'm thinking about the Great War by Diego Vázquez in Argentina and colleagues that abundance was really important. The thing though is that, for example, going back to our own work, we had a paper laid by a former student, Abek Rizna, that proved with a simple model that while abundance is important, when you combine abundance with trade, matching, the fit to the data is even higher. Great. So Silvia, you wanted to follow up? No, I'm fine. Thank you very much for the answer. So there is a question from Washington Taylor. Yeah, hi, thanks. I thought your lecture was very interesting. You found some nice, clear and simple ways of starting to address some really interesting and deep questions. I had actually two somewhat unrelated questions. Maybe I'll throw both of them at you and you can choose which one or address them whichever you want. So one is you focused on climate as a driver of extinction events and as kind of a primary thing throughout your lecture. But as I'm sure you're very well aware, most of the things currently driving species extinction are other things like habitat loss, human use of ecosystems, pollution, invasive species and things like that. So the first question is whether those would have a similar impact or whether there are some climate specific signatures in what you were describing. And then the second question is when you describe these essentially two-level networks, of course, in a real ecosystem, there's all kinds of very subtle other species playing niche roles that are mediating interactions and maybe playing key roles in all kinds of places in the system. So I'm wondering whether that you've explored whether there's any sense in which these networks you have are robust against intermediate species that are involved in the system going extinct or how that plays in the missing pieces that you don't have in the network. So those are sort of two questions. Yes, these are very great questions. Going back to the first one, you're totally right. I mean climate change is just one driver of global environmental change, the other being habitat fragmentation or nitrogen deposition and so on. While our focus was here, probably the reason we're focusing on climate change is for historical reasons because from early on there was this attempt of bridging between these somehow distant approaches, one the network approach and the other the climate change ecology, right? So to some extent focusing on climate was a consequence of these two different approaches. But there's been a previous work by other people who focus on habitat transformation. Some of this work is theoretical using models of habitat loss, habitat fragmentation and looking at what's the rate and shape of network collapse. And I would say I would expect finding the same type of qualitative results. I don't think those results were quite specific about climate in terms of the rate and shape of collapse, in terms of finding this phylogenetic signal or having a moment where the rules of the game, so to speak, change and then we're focusing different species from the phylogenetic tree. So I would say those are quite general results, although to be totally honest, only a subset of these questions have been addressed using other drivers of climate change. So here I'm kind of telling a little bit my gut feeling. For example, the one is for sure similar is this kind of abrupt collapse, this idea that the consequence as we are moving through this axis of global environmental change, the thing that early on nothing seems to change too much up to a point where suddenly there's a collapse. That's the kind of result that different people have seen when looking at different drivers. The other ones I'm not so well aware of studies, the ones looking for example at how functional diversity is eroded or evolutionary history is eroded. But my gut feeling would be that these are quite, have to be quite general consequences. So I would expect a similar kind of signal. In relation to the other question, that's very interesting. It's true that one should expect having this kind of, if you want, gist on a species, species that play a major role in bringing the network together. And that's a piece of research that has focus on that, but in my view it's more a static one. For example, research looking at the modularity or compartmentalization of these networks, this research tends to look not only at this tendency to be organizing modules, but also at the role that different species play. And in particular, this role by a few species in bridging across different modules in being the sort of the glue that keeps this module together. Now if I understood your question correctly, you were asking whether some studies have focused on what happens when these species disappear. And I can think of a study that was looking at a genetic diversity of some of these species that emphasized this thing that once you remove one of these species, you can have a major change because suddenly you can have a network that previously was more or less cohesive and now you have a collection of different networks. Great, thanks. Those are great answers to both questions. And on the second question, I guess one, if I could just go a little further on that. So I guess one of the things I was also asking about there was you have this database of interactions between different species in these different habitats. And I guess I'm wondering, probably there are important species that were missed in each of those databases. So for instance, you may have 73 species in the given area, but there may be another 30 species that really play a keystone role. So part of the question is, even if you missed some of those key species, and those were somehow just encoded secretly in the interactions, are the results that you're getting robust against, say, replacing the network with one where you imagine that you only know about a subset of the species and then test the same hypothesis? Yes, you're totally right. And that's the kind of, I would say that encapsulates a series of criticism that was about using these large database studies where, you know, I mean, it was a nice attempt in the sense of looking at generality, but there are tradeoffs and the part of the consequence of that. And I think it's a fair point, part of the consequence of people who were more critical about our work and the work by many others for the same sake was that each of these networks has been compiled by a different author or a different team spending different time or using slightly different methodologies. And therefore, in the same way that when looking at studies on species diversity, it's very clear now pretty much everybody knows and understands that we should use rarefaction curves. We were not at this stage and that arise some questions about the value of this generality. Now there's been, as the field mature and these kind of studies become a little bit more mature, there has been already a subset of those that started using similar methodologies and in particular use these rarefaction curves. And this allows two things. First, it allows to focus on the smaller subset of networks that have been sampled enough, so have a similar level of sampling and then focusing on those, but also like looking at how these properties may change across a gradient in sampling and there are properties that may vary quite a lot, but some of these properties do not vary that much, like in particular that nested structure I was emphasizing during that particular first talk, each one that is a little bit like the different builds of an onion, right? Essentially, you have this core of generality species and then these few generalities. Thing is that if you sample more, you start having a longer tail of a species and normally those species tend to be specialist and to be less common species, but also they tend to attach to the most generality. So depending on the type of dimension of structure you want or perspective, these may not depend that much on the level of sampling, but in general, I think that's a poor answer and I think that now what we should try to do is every new study try to have a sort of rarefaction curves and people can do that in the same way that we look at how many different species we have when we sample 100, 500, 1000 individuals, we can do the same for example with a number of interactions and oftentimes we have enough of a sample that people consider a little bit of an asymptote. Great, thanks a lot. My pleasure. Great, so the next one in line is Alfonso. So Alfonso please mute yourself. Hello everybody, thank you for your great lecture and my first question is can the mutualistic networks account for weighted interaction between plants and insects and in that case, how much the distribution of these weights might change the results or the number of species, the maximum number of species supported by the network or the change of extinction in case that, in the case of non-random extinction like the extinction that you talked about in the lecture. And my second question, I just come up with that question recently, is that is there a relation in like generalist species tend to be more abundant? My question is related with the species abundance distribution, if there are a relation between the species abundance distribution of a community and the network structure. Very good, very good set of questions. Let me start by the second one. The answer is yes and some of these approaches we were referring to a few minutes ago in terms of like checking the relative role of a different mechanisms in explaining network structure. One of those, this kind of a neutral approach was using these observed skew species abundance relationships and then for the plants on one hand, for the animals on the other and then assuming that the probability of drawing an interaction is going to be proportional to the product of these species abundance. So it's kind of simultaneously taking into account both the skew distribution for both sets. And when one does something like that, you come up with a network of interactions that tends to be similar than the one we observe in nature. So people would tend to think that kind of supports the idea that neutral processes, species abundance is certainly important. Again, as I said, when you have a model that takes into account these and other factors, you can get an even better fit, which tends to support the idea that there's not only one mechanism, but most likely a suite of mechanisms. So you are totally right, species abundance and the particular empirical distributions is something that may be very important and has been empirically used in order to come up with the expectation of these network of interactions and then matching that expectation with the observed one. I don't know if that answered your question. Yes, I think that answered my question. Lovely. Then in regards to the first one, you are totally right. I think my talk, you probably realized that, I mean, when I give this talk on mutualistic networks, now it's a talk that spans now 20 years. So early on, when we were mainly looking at the structure, I was going through different levels of structure and focusing on interaction and strength. As more results were packed, I tend to reduce the focus on the structure and just focus on that particular dimension, the nested one, but you are totally right. And although early on, the first set of papers were looking at binary data, and that refers to, for example, this nested pattern I was talking about, but also these connectivity distributions, whether they are, they follow over law or a truncated power law or an exponential that was like the kind of things people in network research were doing at the same, at the time. I mean, a few people, again, were critical about that, and they would say that, okay, all these results may be meaningful without like considering, embracing the fact that there may be a huge variability in the wave, in the strength of those interactions. And actually, there was a few studies that were looking at the structure, but using weighted data. In that big repository, I mentioned during my talk, right now, there's almost half of these networks that contain information not only on who interacts with whom, but on the relative weight of this interaction. Oftentimes, this is the surrogate of frequency of interactions is used, or number of fruits removed, for example, things among those lines. So there's this kind of information and some of these studies describing network structure were focusing on that component. For example, they were looking at the dependence of an animal in a plant, like and they were focusing, or they were highlighting this idea of asymmetry in the interaction, which can be also observed in binary data. One of the results of an STD pattern is this asymmetry in the sense that specialists tend to interact with the most generalists. Now, when you look at weighted network, you can see that this also happens in a pairwise scale. So like, for example, a plant that depends very much on an animal for its pollination, normally you encounter that the animal depends very little on that particular plant. So some of these results, you can still see when you move from binary to weighted networks. And other results only make sense or only kind of tools or approaches when you have a weighted network. So overall, yes, you have this kind of information. You can address new questions that you could not with a binary data. And a few questions you can check with both. And I would say you tend to find like similar patterns. Great Alfonso, you have a follow up or? No, I think that answers were really nice. So thank you very much. Thanks to you for the question. Great. So next in line is Violeta. Yes. Hi. Hello. Thank you, Jordi. Hello. Yes, my question is about structural stability. More precisely, when you place the mutualistic empirical network in the figure whose axes are next in this and the mutualistic trade off there. Okay, for me, Crystal clear, how do you firm the nest of the network? You just measure it with some algorithm. But what about the mutualistic trade off? I mean, how have you inferred this mutualistic trade off? Because then if you see the equations that generalized the couple of equations there, the parameter space is multidimensional and very, very wide. So I suppose that you just chose certain parameters, but can we be sure that the about the general generality of the results when doing that? Yes, in terms of how one does this, on one hand, you have for these weighted networks, you have empirical information. Essentially, that's information that relates the degree how many species, a focal species interact with that may be two, five or 10 species, but you also have information on the weight of each one of these interactions. So this is information that you have empirically from the network. So that means that you can determine where the point is in that figure. In terms of how to explore, it's very much similar than with nestedness. I mean, with nestedness, you take the network as it is, and then you randomize that with a series of assumptions. Normally, you tend to preserve total number of plants or the number of animals, total number of interactions and approximately how many interactions each species has. With the trade, obviously a bit the same, you could have like a randomization where you maintain the degree of each species, right? You maintain like the number of interactions they have, but you can shuffle the weights of each one of those interactions. So that would allow you to explore the axis in parameter space. But anyway, that axis is also related with the intrinsic weight of the species and also the competition parameter. That axis, it's not related to anything else because it's the way you define it. You define an axis like being only a value of nestedness, and then it's only structural and you can change it. What is related to these other parameters, and in particular to growth rates, which this was the variable we were looking at, is in terms of the measure of structural stability. So on one hand, you have parameters that define the structure of the network, and then you use growth rates as a way to quantify how much variability in those growth rates the model can cope with before one or more species is driven extinct. Okay. There's a little bit of an uncoupling. Some of these variables of network structure are just used to show how much variability you could have. And then for each level of variability, the demographic parameters are used in order to explain the range in growth rates, in this case in particular. So you choose like the center of this domain, the center where this domain of feasibility where all species can coexist. There's ways by which you can focus on that center. And then you start perturbing growth rates, changing growth rates, increasing, increasing, increasing up to the point where one or more species disappears. So there's a little bit of a decoupling on how you treat these different different parameters. Okay. Sorry, that's the last question. But when you are varying the range of the intrinsic growth rate there in your equations, you must fix the other betas and gammas. So maybe it happens that for those 13 parameters of gamma and betas, your intrinsic growth rate behaves in some way, but maybe there's a tiny region where they don't. So is that actually important or not? Yes, you're totally right. And to be, I mean, to make a long story short, essentially that's for a specific values of the other parameters. One, what one can do is then repeat the analysis for another value for each one of these parameters. So this gives you an idea of how robust results are for variation in the other parameters. But to be honest with you, essentially that's more like, okay, you fix the other parameters and then you focus on variability on growth rates. And yes, you cannot rule out the possibility that there may be some combination of parameters where now you could have a slightly larger or a smaller range of conditions compatible with feasibility. You are right. It's a very complex problem just because of the dimensionality in parameters. So to some extent, you fix some of the parameters and then you focus on growth rates. So if you want, it's a partial account of the much more complex variability in parameter space. Okay. No, but thank you. Thank you very much. Very interesting. Great. Thanks a lot for the questions. So the next in line is Martina. Hi. Hello. And thank you for your lectures. So one is, I have two questions. One is a clarification. The other one is more general. So maybe I'll start with the clarification. So when you were talking about the extensions that were driven by climate compared to the secondary extensions that are driven by interactions, you had those phylogenetic trees and with the circles. And you were saying that you can predict the, say the direct extensions with the geographic location, whether you predict the others with the traits. And I was wondering, maybe it's completely... Network ID, sorry. Just to be precise, the single most important variable in explaining co-extinction is what we call network ID, so a property of the network, yes. Okay. And I was wondering whether, okay, if you have interactions, you're supposed to, you have them in the same location. So I was wondering how can you get these, say, switching the first predictor? Essentially, the way is by using these generalized linear models where you can have different factors. One is geographic location. The other, it's a factor called network ID, so a property of the network, which it's unique and it's not affected by the others. And then through this kind of a statistical approach, you can you can wave the relative contribution of one of these variables accounting for the other ones. So it's a way by which you can focus on the relative role of geographic location while keeping into account, if you want, keeping fixed the role of network ID. That's one component. And the other one is or the complement to that is by using what's called a Caikis information criteria, because other things being equal, I mean, the more variables you have in the model, right, the better it gets. But you have to penalize it. To some extent, it's an artifact or just a corollary of having more variables. So the idea is to really focus on the variables, right, that explain the variability in a model taking into account the number of variables or the dimensionality of the model itself. So that's a kind of statistical approach to come up with these kind of results. So essentially, it's statistically, you can do that. Even when you have several variables, these kind of models allow you to focus on one variable at the time and kind of give you the relative relevance of that variable while keeping the others constant and also the interaction. Because sometimes interaction between factors or variables, oftentimes not just that variable x or variable y are important, but sometimes you can also find a significant interaction between these two variables. And so given these models, so the network ID is in the fixed part or in the random part? That's a good point. To be honest, now I don't remember if we treat it as a random factor. This I cannot remember now. No, it's fine. I can read the paper, probably. Yes. I mean, I'm sure the details are there. Because that's a very interesting thing and it's not trivial. The way people who are really knowledgeable about these models, the way they treat these factors as either a random or a fix, that depends very much on the structure of the question, but also a little bit the constraints or the limitations of the data. So again, if you are interested, the details are there. I just do not remember now. Okay. And the other one is more general. So you find this very nice relationship between nestedness and biodiversity. And do you know what happens if you have modularity on the x-axis? That's a very good point. We've not really checked that in the context of that framework. So I cannot give you a solid answer to that question just because we did not check. Obviously, one could think about these two dimensions of network structure not being totally independent, but somehow related. And although it's not perfect relationship, people tend to assume that the higher the nestedness, the lower modularity, that's not necessarily the case because that depends on a level of connectivity. So below that threshold of connectivity for low-connectivity, you actually find a positive relationship. You find the networks which are more nested. They are also more modular. But when the network is well connected, you find this kind of in their relationship. So one could conclude that because of that, you would find the opposite sort of trend. Okay. Thank you. My pleasure. Great. So Martina, okay. No, I think okay. So we can move on with Sri Rama. Thank you for your talks. Am I audible? I hope I'm audible. Hello. Hi. Yes, sorry. I couldn't hear you very well. Sorry. Am I audible? Okay. Okay. So actually, I have a general question. Suppose if you take a generalist predictors or anything, they have some referential structure. So there's something like optimal forages. So what happens is every time the network structure changes because they can prefer one prey or another prey, so the network becomes a dynamics. So how this dynamic network can be modeled generally? That's an excellent point. You are totally right. And one of the limitations of the last part of my talk, talking about these models of climate change, is that we were not taking into account that. So essentially, in this kind of approach and also the approach by many others, the thing is that once a species runs out of resources or the fraction of resources disappear, those species have a higher probability of being driven coextine. And we know that things are a little bit more complicated because as you already point out, there is a traffic flexibility. The fact that some species, whenever their favorite prey items are not abundant enough, they can shift to another one. So I mean, to my knowledge, the very first person who brought this forward was Mikio Kondo, a theoretical ecologist based in Japan that in a paper in Science, that's probably now 15, about 15 years ago, proved that this flexibility can certainly shift. He was focusing on the relationship between stability and complexity. It can be shifted. So when you have a traffic flexibility, you can see that more complex food webs tend to be more stable. Also in the context of mutualistic neighbors, people in Fernanda Baldovinos, it's a good example, have shown that this relationship between stability and complexity can largely change through that traffic flexibility. So I would say that the sort of models that we and others have used, ignoring traffic flexibility would be a sort of worst case scenario. Whenever you have a traffic flexibility, things become a little bit better. But I would say that some of the qualitative results, for example, the existence of these tipping points are still there, only that you can shift the tipping point to higher values of habitat loss or species extinctions and things like that. I think now the question, though, is how to really bring a biological informed model of network rewiring. This is what we're trying to do now in the lab with Marilia Gallarsa. And I think that, again, that phylogenetic signal can be key here. The fact that, okay, although there's a potential for rewiring, oftentimes this is not going to be random. So any species will most likely not have the possibility to shift to any other item. But we think that it may be a good starting point to consider phylogenetic signal, meaning that if one species runs out of resources, most likely may shift towards resources that species close in the phylogeny depends on. So I think that would be a good way to start introducing these rewiring a little bit in a more biological informed way than just assuming every species has a probability to rewire with any other species randomly picking the community. Can I add one thing? Generally, most of this optimum for raising is something, suppose they're not able to see that if they are not preferring, they're not, the feeding may not be giving the required growth or anything, they will divert to another species. So that means the something like, suppose if they're seeing that we are having a lot of energy, we are spending in time of predation, but we are not getting the required energy to our growth, then we have to shift for another species. So how can you do this? That is what my question, I mean, I don't know, I'm able to properly convey that or not, because I'm not a biologist, I'm a mathematician, I'm working in this area. So yeah, I think it's a very good point. I mean, to be honest, we know very little about that. And the reason is again, because these kind of studies have gone quite independent from each other. A little bit, I was emphasizing that independence between network research and climate change research, right? And that was a little bit of the rationale for us trying to bridge them. Another big gap exists between these models of or these approach of ecological networks that tend to be quite static and this optimal foraging, which obviously emphasized a dynamic component. I think it would be a very interesting direction to try to bridge those. And for example, try to see what would be the predictions, what would come up out of these basic ideas of optimal foraging. So for example, allowing like a few species of animals to forage, forage in a given landscape, and then trying to see how out of these basic rules of optimal foraging, what kind of network structure would arise. I think there's a little bit of that, I seem to recall, a paper probably by Morales and perhaps Diego Vázquez as well. I may be wrong, but I think these are the authors who tried to do that. And that was certainly very, very interesting. But I think there's there's lots of room to kind of expand this kind of bridging between optimal foraging and network structure. Thank you. My pleasure. Great. So we are going to get very high pace here. So there is another question by Ankit. Hi. Hello. Thank you for this very broad survey of mutualistic networks. So I had a question regarding like which is somewhat related to May's result of stability in large complex ecosystems. So there obviously he looks at like entirely random networks. But let's say if you talk about large mutualistic networks, where you have some asymmetry between the number of plants and the animals, like let's say there are very few plants, but many animals, which like depend on these plants. So in that case, like, is there a theoretical result of like stability in the same sense as May or like, yeah, or like, is it difficult to like define that? Because I guess in such a setting, you would have a lot of negative interactions instead of like just totally random interactions. What you mean by negative interactions, non random just non random random negative interactions, but they don't sum up to zero. I mean, like the interactions of the entire matrix. Yes, that's a very good point. You're totally right. I mean, Bob's May great paper has to be has to be seen as like a baseline expectation. So some some of the criticism the paper had actually did not arise because of the paper itself, but our, our faulty way of interpreting that paper. So we could not interpret that May said that most complex communities have to be unstable. Rather, what he said is that our baseline expectation, if communities were randomly organized, is that there's a limit to complexity. And therefore, I think that paper was extremely influential in shaping the field in many directions. One of those was just asking ourselves what may be those mechanisms or these dimensions of structure that can help reconcile being complex and being stable. So yeah, there's our our our original work on trying to bridge between these network structure and stability that was our science paper in 2006. We tried to do that in in following a very similar approach than than Bob May with lots of limitations. I mean, that was our first attempt. And therefore, we had to simplify a lot things. And but what what I want to emphasize in the context of your question is that, yes, you come up with an equation for the linear stability and feasibility condition for both being feasible and linearly stable that very much resembles Bob May. Essentially, you have in Bob May, you have that the average strength of interactions has to be lower than an amount amount that involves number of species and connectivity. So what you have here, it's a similar thing. But instead of having like the average strength of interactions, you have the average product of the strength of interaction of the animal and the strength of interaction of the of the plan. And this is what has to be less than an amount. And that we that allow us to to predict that what's relevant in these motoristic networks is that either you have species that depend very little on others or when one species depends a lot on a second, that second this depends very little in the one. So even when one term is large, if the other is very, very small, the product still remains, remains small. So that would be an example of a very similar kind of criteria than Bob May, but it had some interesting variability and that variability allow us to start like thinking about how these dependencies of a plan and an animal and the animal on that plan have to be rearranged to keep stable communities. Yeah, thanks. I'll also look at your paper. Thank you. Great. So is there any other question? I don't see anyone with the raise and in the participant list. And no one no question in the chat. But we had a 50 minute very intense question session. So I think that if no one has a question, we can move forward. Great. Well, I think it was this was a sort of experiment for us to have these play record sessions, a plus question, but I personally think it worked very, very well. And I'd like to thank to thank Jordy for being with us and for answering all the questions, as well as recording the lectures. Thanks. Thanks, Jacopo. And thanks, any, every one of you, I think you come up with extremely good questions. And I think a proof of that is that these are the questions that we are encounter when trying to publish papers. So in that regard, I think that you are thinking very well. So I've really enjoyed and had a great time. So I'd like just to add that feel free to email if some of these ideas start developing or you have further questions. So just drop an email and it would be my pleasure to keep discussing some of these ideas. And best luck to every one of you. Hopefully you have a nice school and a great career. Thank you. Thanks. Thanks, Jordy, also for your availability. And so what we're going to do now before the next lecture, which will start in about 20 minutes, is that we're going to split again in breakout rooms. So feel free to chat with whoever you are randomly assigned to. If you have the version of Zoom 5 or later, I remember you that remind you that you can also switch breakout rooms. So if you find someone you want to chat with or a friend you have not seen for a while, you can also do that. So we'll see each other back to the main meeting rooms in 22 minutes. Thank you very much. Welcome everybody. So if you are following from YouTube, we are starting in about one minute. In the meanwhile, we are waiting for the participants following on Zoom to come back from the breakout rooms. So they should be joining back in about 30 seconds now. So just a reminder for those following from YouTube, if you want to ask questions, you can use the chat on YouTube and I'll read the question from, I collect questions and read them from you. I think that everybody should be back when I finish this sentence. Yes, okay. I think everybody is back to the main meeting room. So just as a reminder, I mean, I'm sure you are pretty, you know, pretty well these few rules. So if you want to ask a question, you can either post it in the chat or use the raise and button under participants three dots raise hand. So please do that if you want to ask questions and I'll make sure you have an opportunity to do so. So before we start with the the second lecture of today, I'd like to remind everybody that tomorrow at 345 p.m. Italian time, we are going to have another Q&A session as the one that just finished with James O'Dwyer. So there are two lectures by James O'Dwyer that are available on the school website. So please watch them in advance and come to tomorrow's session with the question. So that's the end of all the announcements I had to make. So I'd like to welcome again Marino Gatto who is giving the second lecture out of three. So please, Marino, a few thanks for being with us and if you want to share slides, please. Thank you very much, Jacopo, and good afternoon to the Italian as usual. Good evening and good morning to the other people from all around the world. So let me start by sharing my screen presentation. So this is actually the second part, models of disease ecology. And as you may remember, today is devoted to the general topic of macroparasites, a brief introduction to the basics. And then I will illustrate our work on schistosomiasis in Senegal. So you may remember this slide where I introduced micro and macroparasites, the difference in terms of modeling. And macropharasites with respect to micro are characterized by lifetime, which is comparable to their host's lifetime. So their dynamics cannot be neglected. Also, of course, they are larger macro parasites. So you can actually count them in a way in many cases. So you can look at the load of the parasites inside each host and count the number of parasites. Now, the life cycles I am going to consider. Well, the one of the left is the simple cycle. For instance, the roundworm or nematode. And you see that that pig is ingesting the eggs. And then the eggs will develop inside the pig. And then they will become adult. And they would reduce more eggs. And then these eggs will be defecated in the environment. And then the infection goes on that way with another pig eating the eggs and so on and so on. On the right, you have a more complex life cycle, which is actually also the life cycle of schistosomiasis, which I am going to speak later on. Because in that case, you have two hosts. So you have the human host, or in this case, the cattle. And in that case, it is different. There is a stage called the circarial stage. And this stage will penetrate in general the skin of the host. So we will get inside the host. And then, again, the adults will develop inside the host. And then they will reproduce inside the host, the main host in a way. And then eggs will be produced. These eggs will actually hatch and produce another stage, which is called myrosidium. And this myrosidium will actually infect another host, a snail, in this case, in the case of fasciolopsis muskete. And so then there are other stages inside the snails. And then finally the snails will release the myrosidia and the circaria. And then the cycle will go on. But without the snails, the digits cannot establish. And on the other hand, without the cattle, without the human host, the digits cannot establish a snail either. So they are necessary. Both hosts are necessary. And we will first start off, obviously, from the simple life cycle. And then we will proceed to the more complex life cycle. So first of all, I told you that in a macroparas session, in many cases, you can count them. And here you see, for example, for instance, the perch, and this is a tapeworm inside the perch, and you can count the burden. So some of the perches have zero parasites, some have one parasite, some have two parasites, and so on and so on. Okay, now you can do the same with a completely different parasite. In this case, it is a fly, a stinging fly. And these are the reindeer. And of course, in this case, the number of parasites roast is much larger. And so, well, what you do, you do a histogram. And again, you see that there's some reindeer without any parasite. And then, okay, you've been the number of parasites in your histogram. Here is instead a starling, and in that case, it is a nematode, and also it is a nematode for the frogs. And you see that the histogram of the parasite burden is quite different. So, I mean, the shape, the kind of shape, and in some cases, you see, for instance, typically in the case of frog, what we call an over dispersion with a few holes carrying a lot of parasites and many holes without any parasite. Okay, so that typical structure of the word in many cases. So it would be nice to find a way to statistically describe the burden. And well, the first thing you might think about is the, for instance, a simple binomial distribution, where are the number of parasites and you have P, the probability of having a parasite, hosting a parasite. And, well, you know, the binomial distribution is correct, right, actually by under this, what do we call under dispersion, because if you consider the mean and the variance, then the variance is smaller than the mean, smaller or equal. It is equal to the mean. When you go to the limit for the number of trials going to infinity, and the probability of hosting the parasite going to zero, so that n times p converges to a constant. And then you have a Poisson distribution. In the Poisson distribution, the mean is equal to the variance. But actually, in many cases, you don't have variance equal to the mean, but you have a variance which is larger than the mean. That's why usually the most appropriate distribution is the negative binomial. It is more flexible. Of course, you have one more parameter with respect to the binomial or, if you like, and the Poisson distribution. And it is this parameter K, which is a parameter of clumping. And the smaller is K and the larger the over dispersion. In fact, you can prove that the variance is equal to the mean plus the square of the mean divided by this parameter K. So you see that if this parameter is very large, practically you have Poisson distribution with variance equal to the mean, then with K equal to five, you have an aggregated distribution. And you may remember that, for instance, well, I say this one is kind of aggregated, K equal to one of these Ks. And well, for instance, K is more than one, you have a highly aggregated distribution. So it is very popular, let's say, to use a negative binomial distribution as a flexible distribution with just two parameters. If you adjust these two parameters, you can reasonably fit the parasite load. And that would be useful for the simple model I am going to show to you. So it is a simple model where you have a simple cycle where you have the host number of the host density, if you like, number for instance of pigs per square kilometer or number of deer per square kilometer. And P is the allowed parasite number or density, if you like. Of course, the parasite burden is P divided by H in the average, the total number of parasites divided by total number of hosts, and that will give you the average burden. But actually, each host might have a different, might host a different number of parasites, zero, one, two, three, et cetera. So now let me also introduce the number of free living stages, for instance, larvae or eggs, if you like. And then we can write down two differential equations, one for the host and one for the parasite. And then the one for the host will be the birth rate minus the death rate times the host, no disease. But if there is a disease, then what happens? Well, let's first suppose that if you carry a lot of parasites, your mortality is larger. So let me call alpha the additional mortality goes by one parasite. So if you carry I parasite, there is an extra mortality alpha I. Now, if you consider the whole population and you consider the distribution PI probability that one host carries I parasites, you see that this is actually the average mortality and then you multiply by H and then you have the dynamics for the host. As for the parasites, well, each host might, you know, ingest a certain number of larvae, then the parasite will have their own, let's say, intrinsic natural death rate. But there's another thing, anytime a host dies, also all the parasites that are carried by the hosts will also die. And so you have to consider more mu plus some alpha I PI and that it should be included in this equation. Now notice that here you multiply by I PI because whenever a host dies, all the I parasites that are hosted by that host will also die. Okay, so that's the equation. Now, okay, you can, we can now calculate the parasite load. Well, PI divided by H is the mean. And then you may note that if you develop this term now, okay, it involves also the, sorry, the mean of the square of the I square. Well, so you may remember that this is the square of the mean plus the variance. Now, if we assume negative binomial, then the square of the mean plus variance can actually derive from the formula that I showed to you before. And so it turns out that it is P divided by H plus K plus one divided by K. Remember the clumping parameter. And when the clumping parameter is low, then there is a lot of over distribution time P squared divided by H squared. So to make it short, you can get Anderson and May's model provided you also introduce a static equation for the larvae where if there are many adult parasites, they will produce a lot of larvae. But on the other hand, if there are a lot of hosts around, they will ingest larvae. And therefore the number of larvae in the environment will be lower. Actually, you can deduct this kind of, find this kind of static relationship. If you also add another differential equation for the larvae dynamics, and you assume that the larvae dynamics is so fast that in practice, you can use the slow, fast approach. And then, okay, you get a static relationship for the larvae. Now, if you plug everything in, you get a celebrated Anderson and May's model, which in practice is a simple system of two differential equations, H and P, which is closed under the hypothesis that the larvae describe this relationship and that the distribution of the parasite burden actually follows a negative binomial distribution. Now, if you study the system of differential equation as usual, for instance, by drawing the eyes of clients or linearizing whatever you want, then you find out that, again, you can define a basic reproduction number. And in a way, the recipe is always the same. So one divided by m plus mu plus alpha, this is the residence time in the infectious stage. And this actually is the number of, say, parasites that are ingested in unit time. Okay. Now, as usual, r naught equal to one marks a transcritical bifurcation because, you know, these green eyes of client can be shaped in this way. And this is, of course, the case of r naught larger than one. So there is an endemic equilibrium, which is stable. But then, of course, you can also be shaped in this way that eyes of client, in that case, r naught is smaller than one. So, again, we see a simple transcritical bifurcation with r naught equal to one marking the boundary when you switch and they, you have a bifurcation at r naught equal to one. Now, what is interesting that you might now say, well, that is true if the parasites are going to affect the mortality of their hopes. That is a very interesting study. And maybe you remember that I introduced that show the red grouse to you at the very beginning of part one. And my good friends, Andy Dobson and Pete Hudson, now they observe that actually the red grouse have an oscillatory behavior. Okay. And they carry this intestinal parasite, tricostroendular stainless. Now, how is it possible? How is it possible to describe such behavior? Because in this case, you do not get any permanent oscillation. Now, what they observe actually is that the parasites do not affect so much the mortality. They affect the fertility, the reproductive success of the red grouse. So, let me now introduce that kind of hypothesis. And you see that in this case, mortality is not affected by the parasites, but it is the fertility new, which is actually decreased. And the larger the number of parasites that one host carries and the larger the decrease in fertility. And then, of course, the parasite will die from their own mortality, but they will also die when the host dies. And the host dies with mortality mu, which is the interesting mortality. Now, if you go to these equations, these equations actually appear in a way simpler with respect to the previous one. Notice that here you don't have to assume any negative binomial. Okay. You still have to assume that the larvae are described by the static relationship. When you study this very simple system of equation, what you get is something like that. And again, you have, you know, this is an isocline, and this is the other locus with the other isoclines. And fine. Okay. Now, the expression for are not this one. Now you don't have alpha, which was the mortality in use by the parasites host. And, you know, what is interesting, if you look at the number of hosts at the equilibrium. Okay. Now, if are not is larger than one, you have anyway, an intersection. Of course, are not might be smaller than one. If are not smaller than one, this isocline is actually placed here. So you don't have any intersection, any intersection. So you don't have a known trivial equilibrium. So are not less than one. And you have only the disease free equilibrium. As usual, and are not equal to one, there is a transcritical bifurcation. When these H star is exactly equal to K, then you have transcritical bifurcation. But what is interesting that if H star is actually smaller, much smaller than K, and you can actually prove that, that when it is smaller than K divided by two, you have a hop bifurcation. So this equilibrium is no longer asymptotically stable, it becomes unstable and surrounded by a limit cycle. So you see here, what I told you that you can have hop bifurcation in this case. Now note that K, the carrying capacity of the density of the host, as usual, is influencing are not. So the more dense the population, the larger the carrying capacity and the larger are not. So you can make are not larger than one. And you have an endemic disease because of, okay, so the epidemics can reasonable easily establish when the population is very high, think of cat raising or pig raising in a farm. And so they're there. So of course, it's easy to get a disease there. But if you look at the H star, H star does not depend carry capacity. So are not can be larger than one, but it very much depends on the parasite fertility. So for increasing parasite fertility, you see H star is decreasing and therefore you have first a transcritical and then hop bifurcation. Okay, now it's time to go to schistosomiasis. No, first, let me stop and ask whether there are questions regarding this introduction. So there are currently no questions in the chat, but if anyone, yes, there is a question by Alfonso, please, Alfonso. My question is related with the if there are ecological explanation behind the fact that the parasite burden is distributed like a negative binomial variable. Well, no, I would say that, well, as far as I know, maybe I'm wrong. It's mostly an empirical an empirical remark that the negative, we know that that that that that in most cases you have over dispersion. And so the negative binomial is let's say the simplest distribution that can describe over dispersion. Well, well, let's say that in any way like the story of the super spreaders that, you know, that like people that are super spreading the same way in you might have supercharged supercharged hosts. Well, it also depends on your immune system, of course, because of course, when you count the parasites, the adult parasite that you count. And there are ways either you sacrifice the host and go and see how many parasites it is carrying okay, or you purge, if it is an intestinal parasite you purge. So clearly, the adult also depends on the reaction of your immune system. So if your immune system is very, very active, so you might ingest a lot out of eggs, but the immune system is actually recognizing that there is a something going wrong, it is react. And well, we know that the immune system in the different individuals is pointing in a very different way. So I don't know whether this is an explanation. I would say that anyway, it's mostly empirical observation you are you have many different okay cases like the one I showed to you at the very beginning, okay. And you say, well, can I find something which is so flexible as to describe all this possible cases? In fact, you see that you go from K equal to six, to K equal to one, to K equal to 0.35, to K equal to 0.38. And you so with a negative, I normally succeed in describing that all of this reverberation of okay, I hope I answered your question. Yes. Yes. I think that this, I have another question and it's related with the we are going to talk about another models that account for different stages in the life cycle of the parasite more explicitly or? Yes, because my is this. Okay. You mean that? No. I mean, schistosomiasis you see even more. So I'm starting the simple life cycle one on the right on the left. And then now we are proceeding to the actual life cycle of schistosomiasis where you have two different hosts. Okay. And then you can have even more complex parasite life cycles with three hosts and well, okay. I think that Professor Rinaldo, for instance, might speak up proliferative kidney disease in his lecture where the life cycle is even a bit more complex. Okay. Thank you. Okay. So can I proceed to schistosomiasis then? I think yes. Yes, there are no other questions. Yeah, please go ahead. So schistosomiasis is actually affecting many parts of the world, mainly sub-Saharan Africa, a little bit in Middle East and Far East and South America. It affects more than 700 million people, more than seven countries and at least potentially because they live in endemic areas. More than 200 million people affected worldwide. And every year there are several tens of thousands of deaths that might be ascribed to schistosomiasis and 90% of global infections are found in sub-Saharan Africa. Now, sorry, before going into that, now let me first of all show the schistosoma life cycle. And it is very similar to the one I showed to you for fascialopsis. Well, consider humans and then humans are actually infected because they simply contact infested water, infected with circaria. So the circaria can actually penetrate through the skin. And then the adult parasites will develop inside humans. Okay. And they mate, actually, so you have male and female, so you need a pair of actually parasites. And these will actually produce schistosome X. And the schistosome X will develop into myresidia. This myresidia will infect snails of different genera beyond folaria, bulinus, oncomelania. So oncomelania is typical of schistosomiasis in the far east. And also the schistosoma is a little bit different. So you have schistosoma japonica, schistosoma manzoni, et cetera, schistosoma hematomium. And the humans mainly will suffer urogenital or intestinal problems. Usually it is not deadly disease per se, but anyway, it can contribute to lethality very strongly. You can be infected several times. It's not that if you get the disease that you will not get the disease again. You can get the disease. So you can get infected and reinfected. The treatment is simple, but for poor countries, although it is simple, it might be expensive. And in practice, you have to take a very few praticuanto. Now, we have mainly started the problem of schistosomiasis in Senegal and Burkina Faso. We started that a few years ago with the team of Professor Rinaldo and also with our friends in Stanford, and then other French people working in Senegal. I will mainly talk of Senegal today. So first of all, let me show the local model that you can make. And then we will proceed to consider a more complicated model where you have a network. And first of all, what's important that here you have the mixture of the two approaches. You can recognize the negative binomial approach of being the mortality due to the parasites carried by the hosts. But now you couple that with the snails. And in the case of the snails, you can treat that as a micro parasitic disease. So you divide the snails into susceptible snails, exposed snails. So these are infected but not yet infectious and infectious snails. And so then you come out with a five differential equation where you have a number of human hosts, a number of adult parasites, the density of susceptible snails, the density of exposed snails, and the density of infectious snails. Now, I'm sorry. And then again, you can make an approximation that we made, the approximation we made in this paper, that the number of circaria, that the circaria are very fast, very fast dynamics. So you can suppose that the circaria, the number of circaria is simply proportional to the number of infectious snails. And the same for mericidia, that mericidia is simply proportional to the number of adult parasites. And if you do a bifurcation diagram, okay, you find out that after all, what you can get is something which is very similar to what I showed to you. But for the case in which the parasite was affecting the reproductive success. In this case, no, the parasite is actually affecting the mortality of humans. But in this case, in this more complicated case with the schistosomiasis model, then you have transcritical bifurcation and house bifurcation. You can study that in a two-parameter space with human infection rate and the snail infection rate of two parameters. So in this case, you do the bifurcation study with pet to both parameters. And again, you get a transcritical bifurcation to increase the human infection rate and increase the infection rate and go through it. Transcritical bifurcation, you further increase both infection rates and then you have hop bifurcation with this kind of limits. But the most interesting case is when you consider now a more realistic, well, actually, this is partially realistic, meaning that the value of the parameter would tune on the Senegal and Burkina Faso case. But the most challenging case came out for us in Senegal when this challenge D4D by Orange and Sonatal data for development was launched. And in this case, Orange is the mobile phone provider, mobile phone connection provider. And they put an analyzed data on phone calls available to scientists and ask them, okay, choose a problem of social importance that you might want to solve using our data. And then we decided to use those data for developing a model for schistosomiasis in Senegal. You see the schistosomiasis, you can find urogenic schistosomiasis a little bit all over Senegal. And especially in the areas, especially in the rural areas, clearly, you're more subject to schistosomiasis when you live close to water. So agricultural areas, you're more exposed to schistosomiasis. Now, so when you consider the network structure of the model, what do you need? Well, you need a highly solution population density that's available by a geographical information system. Then human mobility fluxes that have been made available in a way by starting the phone calls in year 2013. Then people living in rural settings and rivers, these are mostly ephemeral rivers. And then of course, the data on the prevalence of urogenic schistosomiasis. Now, first of all, we had to study human mobility from cell phone data. So it's big data. There are about nine million solatile mobile phone users. And, well, at the beginning, we were not given nine million, actually nine million users, but a smaller sample. And then because we were winning the challenge for health, actually, later on, they provided us with nine million, really nine million mobile phone users, not to name, of course, they are anonymized and they're collected from one year. And so, of course, by algorithm, you can actually deduct mobility in a way. I don't go into the details. You can find the details in the written in our paper. But it's not very much used. Well, of course, now remember one thing. These are mobile. These are not smartphone in general. So they don't have GPS global position system. But you know, you know, where the position of the people by knowing the antenna to which they are connected in a certain moment. Okay. Now, these are the results of study mobility. So for instance, it's very clear there are two big festivals where the Senegalese go to two cities. Oh, I'm sorry. The Grand Magal de Tuba and the Kazula job. And indeed, the very precise period. Okay. And it's all, you can really find them. So, for instance, this is mobility from San Luis region that we have studied, which have to hear. And of course, most of the people stay home with the region mobility. But then, okay, they can move to other to other to other departments. And here you see the gamut of Tiva, Tiva one, Kazula job. And the grand Magal de Tuba. Okay. So we are rather confident that the we can find mobility in these ways. Now, if you look at the model, it is similar to the local model I showed to you. But it's even more complicated. Because now we are also modeling, sir, Kariya and Mira city up in each location I, and then not only that, but the host in location high can they are also divided into a host carrying zero parasite, a certain number of percent, the maximum number of parasite. So actually, you know, it's more complicated with host having zero parasite, and then be infected and getting one more person and so on and so on. And then, okay, the core, in a way, I'm sorry, the core, of course, is the human mobility matrix because the disease is spread by people moving, and people moving and therefore they can have adult parasite and then release the mericidia somewhere else where they go, or they go somewhere else, get infected in the place where they do not leave usually and then come back and infect their home place. So, okay, so it's complicated now because the force of infection and the rate of freshwater contamination will depend on that matrix Q. And so you have Qij and Qji, I go to J, or coming back from J and going to I. Okay, so exporting or importing the disease. So, you can now round that model, of course, part of the parameters are actually known in a way measured, and some of these parameters have to be estimated. And so here are the results, the calibration is performed against the reported prevalence in each region, each region, and here is in a way the fit, these are the prevalence data, these are prevalence calculated by the model. Of course, it worked perfectly to stay on the 45 degrees line, but anyway, it's a reasonably good fit. Now you can do a sensitivity analysis with respect to the mobility of people, by mobile people, we mean the percentage of people that might move away from their own, from their own region. Of course, most of the people stay in their own region, and one of the 14 regions in which you can divide Senegal, administrative region. And you see that the prevalence corresponding to the mobility that we estimated is actually more or less the minimum, and then the average parasite burden is about seven, seven parasites per human, that's the average parasite burden. Now, when you have a model like that, then you can say, well, what can I do? Can I prevent the disease in some way? And there are different intervention strategies that you can think of. So, first of all, you might have so-called wash strategies, water sanitation and hygiene, okay? And then information, education, communication strategies. So you say, children, please be aware, don't go play in that river in that canal, because that canal by being fasted by snails, snails will release their car and the car can penetrate your skin. Or if you go there, wear boots for instance, it's not difficult to think that children wear boots and gloves, but anyway, okay. And then you can distinguish between what we call untargeted strategies. So you try to ameliorate sanitation everywhere in Senegal, or you can have targeted strategies. That might be prevalence targeted, where the prevalence is larger than you put more sanitation or risk targeted. So for instance, this that is a rural and water discourse there, maybe depending on that. And you see that for wash strategies, it is better to have targeted strategies. While for the information, education and communication were rather intuitive, untargeted strategies, those strategies that are aimed at informing people all over Senegal in a way are better in terms of reducing the average and the maximum prevalence. Now give me five more minutes to say that then we have also started with people, friends at Stanford, who carry a program on the region of San Luis, which is located here near the Senegal River, the other part of Senegal, at the border with Mauritania. And that we are carrying on that program is also going to be financed by Polytechnical and Milano. And there are a lot of people actually collaborating, a tool which also spent a period with us in Polytechnical, that went back to and then master students. And then for instance, the epidemiologist who is actually doing the work in Senegal, in the Indian law, and Lamin, and so on. Oh, okay. Okay, so by the way, that is Professor Casa Grande in Senegal. You know, he's wearing gloves and boots, of course. And this is Lamin, I think, I'm not sure. Okay, they're wearing boots and gloves because it'd be very dangerous if you go there because you see the smooth nails, these are the smooth nails that are releasing the sulcaria. Now in this case, we went more in details because you see there are villages, okay, the triangles, phone antennas, but they're not all located in villages. Some of them are located in between the villages. And then the sample points, and then the water point, the water points. So, you know, we developed a more complicated, only completed model, where you have several connection matrices, because you have some matrices describing the probability that the sulcaria or the muricidium moves between any two water points using the ideological network, which you didn't use for the whole Senegal, actually. The antenna to antenna mobility matrix, then another matrix describing the village to antenna movement, well, not really movement, but you see you have to decide that that village is actually connected to that antenna or another antenna. And then another matrix describing the proximity of water points to antennas. Okay, so it's a complicated, so we introduce all that. And then we, so this model now includes several transportation mechanisms, and we found a reasonably good agreement with prevalence data in people. Unfortunately, prevalence can be very high. And all over, especially along this lake and also the canal and then river, it can mean, you know, in some places the children might have, the 80% of the children might have blood in their urine. So it's really, you know, a big problem in rural areas. And I hope that in this way we have small contributed to the fight against schistosomiasis. And well, I think that's the end of my time. But of course, if there are questions I'm willing to answer. And of course, I would like to answer your questions. Yes, so there is indeed a question from the chat. But how do you account for the infection occurring somewhere else, but reported in the patient home? For instance, infection happening in San Luis, but reported in the car, in the car? No, the infection that are reported, the infection was reported at home. Because these are actually, if I remember, the so-called sanitary department. And these are the regions. So, of course, usually you get sick at home. I mean, if you travel, then, okay, if I make that approximation. So you report home, but then if you go to the hospital, of course, you can get, well, you see, you are, each host stays in a location, I, which is the home. And the home actually, how do you find the home of a phone user? Well, usually, most of the time, the phone calls at evening, most frequent phone calls at evening are usually attributed to home. So you can say, one of those nine million users, the home is this one. And then usually, he or she gets sick at home. Okay, so they're attributed to home, but they can get the infection somewhere else. So they might go somewhere else, be infested by certain areas. They come home. And, well, usually they release mirasidia in their own water body or sewage system. Okay, so that's the approximation. Yes, there is a partially related question, which is, in the last part, how do you account for under-reporting, under-reporting in prevalence data? Well, okay, the data on prevalence were directly collected by Gilles Riboux. Okay, so it was careful, called by Gilles and Lamine. Gilles and Lamine were actually conducting their own, at least in Saint-Louis. They were conducting their own campaign and looking at the prevalence. So the prevalence I have shown to you, this one is actually the prevalences that were mentioned. So we are rather, well, hopefully under-reporting is not high. Any other questions? Yes, there is no other question in the chat. If anyone has any questions, please raise hand with the tool you are, I'm sure, now familiar with. Okay, I don't think there is any other question. Everything is very clear or very obscure. There is another question about how, I try to interpret it, how did you infer mobility from the phone record? Okay, well, first of all, as I told you, okay, now there are very, very many algorithms around and you would reduce mobility from phone record. Now, of course, if these were smartphones and the GPS were on, the global positioning system, that would be an easy way to live. In this case, unfortunately, you don't have always to use a smartphone, but you can use a normal, or maybe you have a smartphone, but you don't switch on the GPS, the global positioning system, because you don't want to be located any minute of your life. So in this case, you know, it is the antennas that we know. So we know that one of the users at a certain precise moment was connected to a certain antenna. And of course, the density of the antenna is not the same all over Senegal, and not all over Italy or over France or everywhere. Because of course, there are more antennas in urban settings and fewer antennas in a rural setting. So for instance, if you go to Saint-Louis, these are the locations of the antennas. So you see that sometimes they are close to a village, and sometimes they're not even close to a village. Also, it might be possible that there are many antennas at the border with Mauritania. Okay, that probably because they are trying to not have people connected to antennas in Mauritania. Okay, or something like that. Anyway, now, so first of all, there's a problem that you have to attribute antennas to villages, for instance, okay, you can do that by usual algorithms that are also used by hydrologists, where they have to reconstruct, for instance, the rain precipitation and so on. Okay, so for instance, they use the polygon method or something like that. Okay, first problem. Then you can attribute home to the place or say the antenna and then the village where you connect most frequently in the evening. You get that assumption. Okay, and then of course, you can reconstruct, for a certain user, you can construct the different antennas through which the user is going at different times. Okay, so home and then these guys stay usually here. And then, well, I will see that corresponds with the gramma gal di tuba, he's there. Okay, because they're using the phone and that phone is not connected to the home antenna, but it is connected to the antenna closed to the gramma gal di tuba. Okay, and then here she will come back home. Okay, and therefore, this is the way that you can reconstruct for instance, or this. And then of course, we, because and then we took the averages to describe the yearly, the yearly mobility in a way, but you can do more than that. You might run the model, not an yearly basis, but also model and daily basis. I don't know whether it would make sense because it would be too huge and the epidemiological data are not so detailed. Okay, I'm sorry, I think it's time for the next speaker. Yes, I think we are perfectly on time. So thanks again to Marino Gatto for giving this fantastic lecture. So next lecture is going to be about again models in disease ecology, but applied to COVID-19. So it will be on Wednesday. That's right, yeah. Yes, let me check on, yes, it will be on Wednesday to 30 Italian time. So thanks again Marino and what we're going to do is to take a short break and we are going to be divided in breaking rooms and we're going to start again in 15 minutes with the lecture by Jonathan Levy. Thank you very much. Okay, can you give a link to this lecture now to the participants? Goodbye. Bye-bye.