 Great, so we are one minute in advance to the official starting of today's lecture, so welcome everybody, it's a pleasure to have you again here. So before I introduce the next, the first lecture of today, I'd like to remind you a few rules to interact with the speaker. So if you're following on Zoom, you can post your question in the chat and I'll read it for you, or you can raise the end going to participants, there are three dots, and you can find the button raise end, and again I'll give you the possibility to talk and answer the question. If you are following from the from YouTube, you can post question in the chat there and again I'll read that for you. So I'm very happy to introduce the next lecturer, Marino Gatto. Marino Gatto is a professor of ecology at the Polytechnico of Milan. He has a research, spent many topics in quantitative ecology, epidemiology and environmental modeling, and today's giving the first of three lectures on disease ecology. So thank you Marino for being with us. Thank you very much, Jacopo, and welcome everybody. Good afternoon for the Italians, and then good morning for, let's say the western part of the world, and good evening for the eastern part of the world. So I will share my screen now and start immediately because time is running. So let me share my screen and okay I see that now there are 79 participants. Okay great, from everywhere. So actually I'm a professor of ecology, a maritus now because I'm retired, also I'm still teaching at Polytechnico de Milano, and today the topics, today, sorry, and also the other two lectures, the general topic models of disease ecology. So today basics, then macro parasites, and then finally COVID-19 in the third lecture, and you see a simulation of our model here on the left, on the spread of COVID-19 in Italy. So first of all as professor of ecology, let me remind you that paracities is basic ecological interaction. Now first of all, sorry, I missed one slide. Now first of all, okay today I will go through the basics, and then on Monday, December 7, I will deal with macro parasites, and in particular with schistosomiasis, and on 9 December COVID-19. Part of my lectures are coordinated with Professor Andrea Rinaldo's lectures, and much of the material, I would say everything, with the exception of COVID-19, can actually be found in this book which just came out of Cambridge University Press. Okay, so today we are going to speak about basics, and as I told you before, first of all as professor of ecology, must remind you that paracities is a basic ecological interaction. Sometimes people ask me what, why you're a professor of ecology, and you're interested in COVID-19, and in schistosomiasis, how come? But then I must remind them that parasites everywhere, you have parasites of plants, parasites of animals, non-human animals, any human animals, and so on and so on. So paracities is a basic ecological interaction, and they are central in a way to the problem of diseases, even to the problem of human diseases. But first of all, let me tell you that actually, paracities, it's very important, even for non-human, or you can say that in many cases, many populations, even wildlife populations, are regulated by paracities. This is a very famous example. Many years ago, my good friend Pete Hudson and Andy Dobson came out with this data, where you see the breeding hands of the red grouse and parasites, this kind of parasite that you can find, they are interesting, and you see that one of the populations is going down, and the parasite load is going up, and then the parasite load is going down, and the population is going up, and so on and so there is a large evidence that paracities is a regulating population, and there's another very famous example that of the Rinderpeth pandemic in Western Africa. Rinderpeth was introduced by Italians, you know, at that time, Italians were dreaming of being a colonial power, and so they introduced domestic cattle in Masawa, and these domestic cattle were bringing Rinderpeth, and this is the result. Unfortunately, Robert Koch in 1897 found a vaccine, so they started vaccinating, and so they could eradicate Rinderpeth from Southern Africa, but then only recently they could eradicate Rinderpeth everywhere, and the Rinderpeth was not only affecting the domestic cattle but also the wildebeest and the zebras and so on and so on, just to give you an idea of how everything is connected. Now when we come to humans, of course the kind of diseases that are in a way, let's say, more interesting for ecologists are infectious diseases, and if you look at the global death, because you see now that most of the diseases are non-communicable diseases, others are due to injuries, but there is a big chunk of diseases that are communicable, that are infectious, and in fact, if you consider, for instance, statistic in 2017, but if they're not very much different from 2018 and 2019, the infectious diseases were about eight million, and just to give you an idea of the importance of and the how hard the time is, I mean the time in which we are living, is that COVID-19 has already claimed less than one year, 1.49 billion deaths, and I think many of you are familiar with the Johns Hopkins site where they show the number of deaths, the number of cases every day, so we are about now to 1.5 million global deaths, and this is certainly an underestimation probably because the global case is certainly an underestimation, this is the global cases that are actually found, but there are many, many cases that are around, they are not found, because they do not test that. Okay, so now again, being an ecologist, I'm also interested in understanding, oh, sorry, what did they do? What is that? Yeah, I don't understand, there is a, do you see that red scratch on my screen? Yes, I think you use the pen on the screen, so yes. I need to activate the pen, okay, I'm sorry. Okay, so if we go to another problem for the ecologists, that is the fact that now globalization in a way and the the land use change and climate change, they're impacting on everything, but in particular, they're also impacting on diseases. So for sure, habitat change, agricultural development, for example, which is a good thing on one way, but on the other hand, it is providing more and more water bodies, for instance, for some vectors of disease, and then of course, globalization of trade and travel allows viruses to spread everywhere very quickly. And if you consider climate change, certainly it is impacting the habitat, so it is making habitat which was not good for vectors now in countries that actually were not favorable to vectors, but it also it is also impacting on non-communicable diseases, clearly think of heat waves and desertification and so on. So these are anyway, global problems and these global problems are heavily impacting also on diseases in college. Now this is the map that I also usually show to my students, and it is a paper which appeared in 2004 in Nature, and you see that I specifically read the name of the last author because this is Professor Posi, or being here in Italian originally, I mean Fauci we would say, and he's the council of COVID-19, and he was already pointing out that there are a lot of emerging and re-emerging infectious diseases around the world, and many of them, using more recent statistics, are zoonoses. Now zoonoses are actually due to pathogens which are usually hosted in non-human animals, but they can be transmitted to humans, and they are shown in red, and you see how many of them are around. You see here a picturesque representation of some of these diseases, and you see for instance the West Nile virus, and if you can read that the usual animal reservoir is various birds, especially robins in the United States, and SARS, the original animal reservoir was bats, and then of course it's not only humans who are susceptible, but also civets for instance, or I'm sorry, whenever I'm using my mouse to point out something this red scratch appeared. Okay, so and then the bird flu, for instance, waterfall, and Ebola, again various bats, and so on and so on. Okay, Yaakov, can I get rid of that red scratch? I'm not sure how to get rid of that. I don't know, because usually, I think if you press the, so probably if you go, because usually I should use, if I use annotate, but I'm not annotating, I don't know what happened, because when I use zoom with my students, I can move my cursor and maybe no, you see, wow. Because there is annotate, perhaps if you go to view option on zoom on top, you should probably see annotate. Yes, but I didn't use annotate. Well, let me stop annotate maybe. Mouse, okay. Okay, now it shouldn't, okay, so I can move the mouse. I cannot get rid of that scratch, unfortunately. I'm sorry. Okay, thank you very much, Yaakov. Sorry for this. Okay, now let's go deeper into the problem and also into the problem of modeling. And okay, first of all, a big distinction which was made many, many, many years ago, basically by Anderson and May, Bob May and Roy Anderson, in terms of modeling, is the difference between micro and macro parasite. So, micro parasites are typically viruses and bacteria, and they have a short lifetime with respect to the lifetime of host, of their host. So, in a way, you can neglect the dynamics of the micro parasite load inside the host, also because it would be almost impossible to count all the parasites inside a host, and in any case, their dynamics is quite rapid. So, what you do usually, you do another approach, you use compartments for the host and distinguished susceptibles, in fact, and so on, we will see that. Micro parasites, instead, you see an example there, they have a lifetime which is comparable to the lifetime of their host. So, you can include, and you must actually include the dynamics of the parasites. Okay, and now micro parasites would be the object of my next lecture, where we will see a detailed model of schistosomiasis. Okay, now if you consider now micro parasitic diseases, first of all, we should distinguish, because we must use different models, the transmission pathways of micro parasitic diseases. So, first of all, you have direct airborne or sexual transmission, that typically are the cold, the measles, SARS, COVID-19, influenza, and so on. Then you have vector-borne diseases. These diseases need a vector, without that vector, the disease will not be there. And then typically, malaria or dengue, Zika, and in many cases, they are transmitted by mosquitoes or by flies and so on. Then, water-borne diseases, in this case, propagules are transmitted by contaminated water, so you bring contaminated water, for instance, to get cholera or rotavirus. Okay. And then there are other diseases that we will not treat, environmental diseases. You mean that the propagules can stay in the environment for a long time, so you can get infected by contacting those propagules with very long time, so typically anthrax, for instance, or tetanus, that type. And then sometimes we also have vertical transmission, so from mother to their progeny, and so, for instance, HIV can be transmitted from the mother to children or hepatitis B and C. We were mainly daily with direct vector-borne and water-borne diseases. I will not talk about environmental diseases and I will never consider vertical transmission. Just a hint to let you understand that the life cycle of macroparasites, and then we will leave the topic for the next lecture. Here is a very simple life cycle. Usually macroparasites, the adult stage of the macroparasite is inside the host. Look at that pig on the left. It is ingesting eggs on the macroparasite that eggs will develop. We go to the langa, then we go to the gut, and then the adult whores will reproduce, and then they will produce eggs, and the eggs will be defecated into the environment, and the same pig or another pig can get infected and re-infected. Another cycle, that's important, because for instance, schistosomiasae, which I am going to speak about, is of that kind, requires a second host. Typically, for instance, in fasciolopsiasis, again, the adult are inside the host, then they produce eggs. The eggs are shed into the aquatic environment. They develop into a stage which is called myrosidium, but that myrosidium must have these nails in order to stay inside the snail and create the other stage, the cercharia stage, and then the cercharia will actually swim into the water. They can also penetrate the skin or be ingested, and the cycle goes on. Vector-borne diseases, just to give you an example of very many kinds. Malaria, typically, chagas disease, sleeping sickness, fever blindness, and so on. And then water-borne diseases, cholera is probably the most famous water-borne disease. Here, for example, the country is reporting cholera in this five-year period, and then you have many other typically diseases which involve diarrhea, and many of these diseases, unfortunately, are a leading cause of death among children, typically among children under five years of age. And so the death of diarrhea diseases are fifth among the leading causes of death. Now, water-borne diseases and cholera, this is the topic that will be dealt with by Professor Renaldo in his third lecture. Today, I will mainly introduce the basics to you. Now, are there any questions after this brief introduction or not-so-brief introduction? Not in the chat and not on YouTube, so if anyone has a question, please don't hesitate to write it in the chat or to raise hand and ask it in person. Because this was very basic and probably many of you already know the topics. I think we can move forward. Okay. Great. Now, today I will go into the basics, but let me remember that actually the aim is to go into more complicated models than the models I am actually introducing today. And that the common spatial setting of this model will be networks. And the next lecture you're going to have today is about networks. So the basic idea that you, for instance, for water-borne diseases, you might describe the hydrology of body water. But you can transform, and that would be the topic of the next lecture by Professor Renaldo, that into a simplified network, a graph where you have nodes and where you have Rx and Rx, of course, are the connections. Now, there are more complicated models, for instance, if you consider a different kind of transmission, not only due to water, then you might have a graph which is not a tree, but a more complicated graph. And you may introduce connectivity metrics, that might be stochastic matrices or stuff stochastic matrices and so on. And not only that, in many cases, you have more than one network to describe the disease. So, for instance, in cholera, which is going to be described by Professor Renaldo, but also in stochastic matrices, when I think about, these are diseases that are connected to water. So, you have the hydrologic transfer, for instance, but you have also human mobility. And so, and in that case, you have a double network. And in some cases, you make the nodes coincide for the two networks. This is sort of an approximation, locating in the cities or the villages the same node where you have the nodes for the hydrologic transfer for the hydrologic network. But in other cases, like this example, this is the model Schistosomai's redeveloped for Senegal. In that case, you have water points, you have villages. And that's interesting, because we have four antennas, because we can actually develop the human mobility connectivity metrics by using mobile phone, but then the antennas are not located only in villages, for instance. And then you have water points. So, and water points are where the people can get infected. So, in that case, you have multiplex networks. Okay. And then in case, in each node, you need a local model. Okay, suppose that now all the connections are cut, you've got all the connections, you can see only one village, and you try to develop a model for the disease in that village. And then you will connect the nodes and the network. Today, I'm going to speak about the local models. And please, I think that, forgive me if you already know, because that would be very basic. But on the other hand, I think it's good to have very basic notions. Okay. So, when you consider local models, possible approaches are easier, you might have compartmental models. So, for instance, we would see that for a micro parasitic diseases with direct transmission, you can distinguish between susceptible, infected, exposed, infectious, recovered, and so on. A macro parasite, or a way might consider the parasite load. Usually, in these cases, what do we use ordinary differential equations? But you can also develop stochastic, stochastic models. Typically, when, for instance, you have only a few cases, and then you cannot make the approximation that you use real numbers. You must use integer numbers, so five infected people. Okay. So, when you go to, of course, small numbers, stochastic effects are very important. Another possible approach, but I will not use that, is through distributed infection periods. So, you don't consider just compartments, but you use, for instance, the age of infection as a continuous variable. In that case, you must use partial differential equation or the integral differential equation. But that is an approach I will not use, and I think, and not even Professor Ronaldo will use, I think, that kind of approach. Okay. So, let's start now with the real models. And I will start with microparasitic models with direct transmission. And the simplest model is the one where you consider just the susceptible people, people who are not infected, but might be getting infected. Oh, sorry. And then you have another compartment, which is the compartment of infected and infectious people. And then, and then there is no immunization. So, these infected people will practically have no immunization and go back to being susceptible at a certain rate. Then a more realistic model is the one way you consider the recovered people. So, these people are recovered and are immune, at least for a while, then they might lose their immunity and go back to being susceptible. Or they might have a permanent immunity in that case, that this, this arrow is not there. Or finally, you must distinguish between infected people, but not yet infectious, and people who are infectious. So, this is, for instance, the typical case of COVID-19. So, in COVID-19, you have some people who are exposed. They are not yet infectious. That is going to last about five days in the area. And then you get infectious. And then you can infect, of course, the susceptible. Then you can recover. You're certainly part of the people recovering COVID-19. We don't know whether they get some sort of immunity. For sure some immunity. We don't know how long it is going to last, maybe one year, maybe two years, maybe three years. Okay. And these are called SEIR models. Now, let me go into the simplest SEIR model, because in any case, it is a very good example to start with. So, the basic SEIR model, you have susceptibles. And these susceptibles can reproduce. So, you have a birth rate. Well, even the infected people might reproduce, but let's suppose that when you're infected and you have a disease, well, let's say that you will not reproduce. It will be in a bad and you don't have time to reproduce. Okay. So, let's make the hypothesis that this is not there. Then there is a certain mortality near the natural death rate from other causes other than the disease. And then you have a certain infection rate. And so, susceptible people can get infected with this infection rate. And then you might have that the infected people recover, but then they go back to being susceptible because there is no immunity. Or it is very, very short. When you go to the infected people, of course, this rate will go into the infected people, and then the infected people might die from other causes, die because of the disease, or recover and go back to being susceptible. Now, a very important distinction is related to the infection rate. But now before doing that, let me introduce some terminology. Incidence. Incidence is a flow, is the flow of newly infected. So, remember, it is not a number, it is a number per unit time. So, you might have, so the number of positive slides of people that you, they make a test and they know that you have malaria. Per week, per week, that's the incidence. Primalance is instead of fraction. It is the ratio between the infected people and the total number of people susceptible plus infected. Now, a big and important distinction is in terms of infection rate. Now, what is the infection rate? It is the probability per unit time that one susceptible gets infected. So, if you examine that probability, you see that it is actually the product of three different things. So, first of all, the contact rate. In order to be infected, you need to contact people. So, you have number of contacts per unit time given a certain number or density and of individuals, both susceptible and infected. But of course, only if you meet the infected people, you get infected. So, you multiply by the prevalence. But then, even if you meet an infected guy, then you might not get the disease. So, you must multiply by a probability of becoming infected and infectious. Now, we will suppose that it is constant, but in reality, it depends on the behavior. So, for instance, if you wear a mask and then you will not get COVID-19, okay? Or it is very difficult, especially if you wear an FP2 mask. So, but anyway, let me suppose that it is constant, then use a parameter that can vary. Then, what makes the difference in the contact rate? So, the contact rate, number of contacts per unit time might depend, as a first approximation, if you consider so-called density dependence, might be proportional to the density of people. So, if you stay in an environment with a lot of people surrounding you, the contact rate will be higher and just the opposite. But, for instance, if you consider sexually transmitted disease, you don't get a sexually transmitted disease by going in the underground and being surrounded by people, okay? So, in that case, this is called frequency dependent. I is proportional to I divided by the prevalence because the contact rate, in a way, is kind of constant. So, with density dependence, I is proportional to capital I, that is called the law of mass transmission. And with frequency dependence, the infection rate is proportional to the prevalence. Now, both assumptions are unrealistic because if you are in a desert, you can have anyway a sexual intercourse, okay? You are alone. So, even when you consider sexually transmitted diseases, the contact rate might go to zero when it goes to zero. And on the other hand, even with airborne transmission, even if you go in a very crowded underground, you're anyway surrounded by no more, as I say, that 10 people. So, even airborne transmitted disease can actually saturate to a maximum rate, okay? Now, let me start now with density dependent transmission and when I go in my enthusiasm growth. So, let me consider now a simple case in which I is proportional, the infection rate is proportional to capital I, the number is affected. So, you have a very simple multiplicative term. I'm sorry. When I move the mouse sometimes, you know, I switch from one slide to another slide. So, you have a term like that which goes into here. Now, let me suppose that to give you the idea of the ecological importance of the diseases that this population, if there is no infection, would actually grow in an exponential way. Now, if you now introduce the possibility of infection, what comes out that if you started out these no linear equations, and I think that you have a tutorial on no linear analysis. So, these are the, you see the isoclines, okay? And you see that now there is a known previous equilibrium, this one. And therefore, the main message is that disease can regulate the population. A population that would grow exponentially does not grow actually exponentially if a disease is introduced. Now, another result with very important result is that the prevalence of the disease decreases with the mortality alpha due to the disease. That's not a very important message. So, the diseases that are very lesser have very low prevalence, fortunately. So, so for you consider Ebola, for instance, very, very less of disease prevalence is low, fortunately. Okay, that's not being met. Of course, you can go into a more realistic model. It's one where the susceptibles cannot grow indefinitely if there is no disease that cannot grow exponentially. I hope that you are familiar with the logistic model. It is a model where the population would grow to what we call a carrying capacity, okay, if there were no infection. Now, you introduce now an infection. And what comes out is that if you do a nonlinear analysis of that nonlinear model, you now have three equilibria. The trivial equilibrium, no susceptible, no infection, okay, population not there. Or no infection, the population goes to the carrying capacity, so that's a second equilibrium. Or third possibility, there is an infection and that infection actually creates a third equilibrium. Okay, so this one is what we call the disease-free equilibrium. And this is a non-trivial equilibrium. However, that's a very important message. If you go to the expression of that non-trivial equilibrium, you find out that although this isocline will anyway intercept this red curve, which of course extends also to negative numbers. Negative numbers do not make any sense if you cannot have a negative number affected. So if you go into the mathematical expression, it turns out that that mathematical expression makes sense, meaning that this guy here is larger than zero, is no negative, only if this, of course, if capital K is larger than nu plus alpha plus gamma divided by beta. And in fact, you see that for instance, suppose that we increase, we consider a disease with a larger alpha. And you have the same carrying capacity, same population, then another disease, consider another disease with a larger alpha. So this isocline will move. And now you have a situation where there is no intersection. Now, this is an example, I don't know, Jacobo, whether it was a tutorial on no figure analysis of a transcritical bifurcation. So in that case, this non-trivial equilibrium will actually disappear, let's say, after it's not true, it will go down here and become unstable, and the disease-free equilibrium will become stable. So you can use linearization and value criteria. But basically, when the non-trivial equilibrium is no longer feasible, you have the transcritical bifurcation. Now, you can write this condition in an equivalent way, which is very important, because it introduces one basic notion, which is now very popular, the basic reproduction number. You can write the inequality in these ways by introducing what? What do we call the basic reproduction number? And you can interpret the basic reproduction number. By the way, it was introduced by demographers one century ago, but this concept can be actually translated into epidemiology. It is the average number of secondary infection caused by one primary infection in a healthy population of caring capacity. Now, in demography, it is the average number of daughters produced by a mother in the course of its lifetime. So here you have mother infection and daughter infection, exactly the same concept. And why? Well, it's very simple. One divided by mu plus alpha plus gamma is the mean, it is a time, and it is the mean residence time in the infectious compartment. So the infected state infects for such a time. And beta k, now this susceptible to caring capacity, so it is a healthy population. Now beta times k is the number of susceptible to infected per unit time in a disease-free population by one infectious individual. But that guy would stay infectious for this time. So this is exactly the average number of secondary infection caused by one primary infection. And if it is larger than one, then the disease can increase. Otherwise, the disease cannot increase. So the disease cannot become endemic. So if R naught is more than one, the disease-free equilibrium is stable, you are transferring by application, and the disease cannot become endemic. Are there questions now? Because that's a very important concept. Any question on R naught? There are no questions. Wow. Now there are two possibilities that everything is very clear or everything is very obscure. Hope that the first option is the right option. Okay. Now, you can go to more realistic model. Self-reason, suppose now you consider a more realistic contact rate. Well, it is very simple instead of beta times k, since the contact rate now is n divided by delta plus n, you simply have beta k divided by delta plus k multiplied by the rest of the time. Okay. But now there is one interesting thing. Suppose you have what we call a frequency-dependent disease, for instance, a syphilis, and you don't get syphilis by going to the underground. So now this delta is really very small, let's say close to zero. So if it is close to zero, you see that, okay, now delta cancels and you have beta k divided by k. You end up with an R naught, which is approximately beta divided by nu plus alpha plus gamma. So it does not depend on the carrying capacity. So you see a basic message in this case and even in the previous case with density dependent that R naught is larger if k is larger. So it is easier to get any airborne transmission in New York than in, say, in the wildlife of China, okay, or clearly, okay. While if you have other diseases, sexually transmitted diseases, you see that k is not playing a role. What is playing a role is the probability of the beta probability of getting infected and the times one remains in infection. So, for instance, for AIDS, you cannot age by going around, but you can remain infected for a very long time, actually. And then, of course, you might go into more complicated model, but also complicated, where you also consider recover. And so, for instance, okay, with this recovery, R naught is the expression. So you replace this gamma, which is the rate at which you become susceptible to gain, with actually raw the rate at which you recover and go into the recover compartment, but it's particularly safe. Now, an important message, however, for the remainder of the course and of the lecture, that it is possible to generalize these machinery for network models, where you don't have just two ordinary differential equations, three ordinary differential equations, so on, but you have a system, for instance, ordinary differential equations, where each local model is connected to another local model by a network, possibly a multiplex model, but you can still use no linear analysis, bifurcation theory, and the important concepts of dominant eigenvalues or the spectral radius of appropriate matrices. And in this way, it is possible to introduce generalized reproduction numbers. So, for instance, the generalized reproduction numbers, which might be the spectral radius of an appropriate matrix. Okay. And so, to decide whether the disease can become endemic or it cannot become endemic, based on that analysis. But let's say that the basic idea in the way is the same. Now, to finish, just give me five minutes, and then I will be finished. I want to briefly introduce, without going into the details. Marino, for instance, we are switching here. If you have two questions on our note. Oh, great. Okay. So, there is one question by Zoret in the Zoom chat asking, is R0 time independent? Okay. Now, that's a very good question. Okay. Now, by R0, please note the zero and the not. This is the basic, the basic, okay, basic reproduction number. So, it means that you can see the healthy population. So, nobody cares about that disease. Then the infection appears. And there is no treatment. And people are not wearing masks and so on and so on and so on. Okay. And then you must decide whether the disease will actually become endemic or not become endemic, possibly spread. Okay. So, that's the basic reproduction number. And now I know that it is very popular. Another reproduction number, which is called RT. So, RT is the average number of secondary infections caused by one primary infection at time P when the disease had already spread into the population. But now people are being treated. People are being isolated. People are taking precautions and so on and so on. Not only that, but also the prevalence of the disease is not one. Because in the basic reproduction number, the prevalence of the disease is initially one. Nobody is infected. Well, to the exception of one infection, few infections. You can approximate. Okay. So, then beta might be time-dependent. K, which is the number of susceptible people at the very beginning might be replaced by the number or the density of prevalence of susceptible people after a while. Also, now, so consider also recovery. So, there might be people who recover and if they are treated, the rate of recovery is going up. Okay. And so on and so on. So, in that case, you can introduce what is called RT. So, the number of secondary infection, not at the very beginning, that's the basic reproduction number, but at time T. Okay. I hope I answered the question. No, there's a second question, Jacopo. I don't remember. No. Sorry. There is actually a question from YouTube. Vybuti is asking actually a broader question. How can you explain how a disease does not become endemic? A disease does not become endemic? Yes. I'm not sure if it is a question about interventions or is a question about the stochasticity involved in the... No. I mean, okay. Well, okay. There's a problem with stochasticity. That's true. Okay. That's very, very good remark, Jacopo. Of course, everything is based on an ordinary differential equation model where these S and these I are created like real numbers. But in reality, if you consider, for instance, initial infection, now you should consider a stochastic model where the number of infected people is an integer number, actually. So it is true. Whenever we say one infection here, okay, we are using an approximation. In mathematical terms, we should say epsilon with epsilon of small. I mean, it's more appropriate to think of S and I in terms of density. So say number of people per square kilometer, number of susceptible people's number of infected people's square kilometer. So that's a real number. I cannot really think of one infected guy per square kilometer. No. We mean a very small density of infections initially. So it is true that in order to understand whether that disease can actually become endemic, it might be necessary to use a stochastic approach at the very, very, very, very, very beginning. But if we make a deterministic approximation, then this kind of R naught is okay. The one is the threshold for R naught for the disease to become endemic, meaning that the only stable equilibrium in the long run will be the disease pre-equilibrium. Okay. So even if, oh, I'm sorry. Okay. So suppose that now we are at carrying capacity. And then you put a number, a small number of, sorry, an endemic state, because they should not point here. They should point to, oh, okay. I'm sorry. I think I made a mistake in creating this graph. Okay. So in reality, so when this isocline is here, so I made a mistake in your life. So I'm sorry. I think I should, I think I can correct this. No. In reality, no, I'm sorry. It's not this one. Okay. It should be, okay. Okay. I'm sorry. That was a mistake. No, not this one. Okay. I think you understand now. So that, in that case, even if you introduce a few infected, a few infected people, then the population, the population where we go to the disease-free equilibrium, that is the sense. Okay. So true, there might be stochasticity. And second, of course, you have some infection initially, even if R not is more than one, but that infection will die out. Okay. Even in the deterministic model. I hope I answered the question. Okay. Can I go on for? There are two more questions, but perhaps it's better to ask them at the end. Okay. So now let me go on. Just to introduce the model of what are both diseases. In a way, that is important. Actually, you see, in a way, the best known disease is cholera. The pathogen was actually discovered by Filippo Pacciini in 1954. And you'll see that here, the basic ecoepidemiological model is the one where you introduce another compartment that is bacteria in the aquatic habitat, because you do not usually get infected by contacting infected people. Well, it can happen sometimes. But you get infected by, for instance, drinking contaminated water. And so the infected people will actually contaminate the water. So there will be bacteria in the aquatic habitat. And then the susceptible will actually get infected because they drink contaminated water with bacteria. So the basic water mode is this model is one where you have now you simplify the logistic growth by thinking of a demography, which is close to carrying capacity age, where you can linearize and weigh the logistic model around the carrying capacity. Let me go that carrying capacity age. And then you have a model which is similar to the one that you saw before with the difference that now the susceptible people get infected because they get into contact with bacteria. So this third, we go into the infected compartment. Okay. And then that suppose that some of these infected actually recover and there is a permanent immunity. So the recovered people will stay recovered forever, which is not true for color, by the way. And then you might have a mortality due to the disease. And then these infected people will contaminate the water, for instance, by defecating at a certain rate. And then, of course, bacteria can die. So they can stay in the environment for a certain time, which is one divided by Delta. Now, again, you can study the equilibria of this model. And it comes out that you have a non-trivial equilibrium. But this non-trivial equilibrium is again feasible only if the age with the carrying capacity is larger than a certain amount. Now, you can transform that inequality into the usual expression for R0. Why? Now, you see, number this beta times H is the number of susceptible people infected per unit time per bacterium. Because if you go to the population, these beta is per unit time per bacterium. Okay. Sorry, beta times S is per unit time per bacterium. So they must introduce the mean residence time of bacteria and water, one divided by Delta. And then the mean residence time is the infectious compartment. So the infectious will actually contaminate water for a certain time, which is one divided by new class of R0. Then they will produce bacteria. This bacteria will stay in water for a certain time. And then the number of susceptible to unit time per bacterium is beta times H. And so you get the usual number of secondary infection produced by one primary infection. So if it is larger than one, the disease can establish. If it is more than one, the disease cannot establish. Okay. I will make my slides available. I had also a slide for vector war diseases, but you know, my time is over, so better than stop here. And I'll stop sharing and ask if there are some final questions. There are, yeah, let's say a couple of questions. So there are, there is one again by Zoré about the, it was asked a few minutes ago, and it was whether we can control the disease by changing the parameter row. Now, I don't remember the notation. Okay. They can see. I start sharing the screen again. Share. There we go. So, well, even here, even if you consider the basic waterborne disease model, raw is the recovery rate. So the larger the recovery rate and the smaller the residence time away. Okay. So if you have a small residence time, of course, R naught goes down. Well, so for instance, if you consider cholera, a simple way of making people recover is to hospitalize them and hydrate them. So clearly, if the recovery rate is larger, then the mean residence time, the infectious compartment is larger. The same if you go, okay, to, for instance, directly transmitted diseases, you see that again, if you consider a logistic model with the recovery, the recovery compartment now in evidence, you see that again, you have one divided by mu plus alpha plus raw, which is the mean residence time in the infectious compartment. So if this raw is large, the time that you are infectious going down. And the same, of course, that is true for COVID. So if you identify people, you isolate them. Oh, if they do not need to be hospitalized, then you isolate them. So they are infected, but they are not infectious. Okay. So, okay, that is true. There are the questions. Another one. There was one, again, on the previous part, when you were talking about the transcritical bifurcation with R0, is it possible for your model to have other bifurcations like a saddle node bifurcation? Yes, yes. Not for this model, not for this model, not for this model, but it's certainly possible. Now, transcritical bifurcation is actually a critical case of saddle node bifurcation. But you can have a more general saddle node bifurcation. You can have hop bifurcation. You can have subcritical, supercritical hop bifurcation. So for instance, with models of schistosomiasis, that is possible. And you can also have at least the possible equal instances of color to get corrected. So you might have feigenbaum cascade and so on. That is possible, but these are the very basic models. And here you don't have that kind of bifurcation. So in fact, whenever people speak about R0, RT is smaller than one. Okay, okay, okay. That's a very simplistic approach. In a way, you can have much more complicated, much more complicated problems. You can have instability, for instance, and so on. That is possible. Yes, thanks a lot. You're welcome. Great. So is there any other question either here on our YouTube? Okay, so thanks a lot to Professor Marino Gatto for this fantastic lecture. So what we're going to do now is to split and break out rooms while we are waiting for the next lecture, which is going to be actually a tutorial. And Marino Gatto will give another lecture, I believe, next week, if I remember by half the program. Yes, yes. And it will be about macro parasites and schistosomiasis. Yes, so I'm sure that if you have questions and any questions come up, you can of course ask it at the beginning or during the next lecture. So now let's take seven minutes. Okay, so thank you all. Thanks a lot. And so see you on Monday. Thank you very much. And stay unexposed. Great, so let's take this break. Let's take this opportunity to chat with others informally in the breakout room or stretch your legs and take a break from the screen. Thanks. Okay, great. We are about to start. So in a few, one minute, I would say all the people in the breakout rooms will join back in the main room. So while we're waiting for them, I remind you that if you are connected, just connected to YouTube, you can ask, you can of course watch all these lectures and live stream live there, but you can also use the chat to ask a question and I read them for you. So let's wait about 30 seconds for all the participants to join back and we'll be able to start. Okay, great. So everybody should be in. Thanks again for being here. So what we have now is, well, let me remind again the rule. So please, if you have a question, post it in the chat or raise hand in the, using the Zoom feature. And let me introduce the next speaker. So the next speaker is Arsham Gavazie, who is a PhD student at the University of Trento and for the senior Bruno Kessler, who works on network science, statistical physics and complex systems in general. And today is giving a tutorial on complex networks. So please, Arsham, thank you very much for Hello and welcome to this tutorial on complex networks. This would be a very brief introduction to network science, a science that has application to a broad range of disciplines. So no matter you are from a sociology background or physics, biology, economy, ecology, probably at some point in your future career or academic life, you will encounter networks. And it's better if you have the basic backgrounds to understand what's going on in network science. So first, I would like to thank the organizers for inviting me as a tutor. I'm Arsham. I'm a PhD student at Trento University. And I really hope that this lecture would be something of value to you. And also you enjoy this time that we are having together. Let's go a little bit into the details of this talk. This is the content. First, I'm going to talk about complex systems with you. You all know complex systems. Complex systems are everywhere around you. You just need to learn how to identify them and how to characterize them by their important properties. Then we move to structure and discuss the structure of these wonderful systems. And after that, we can talk about the basics of networks because we model the structure of these systems as networks. We talk about the features of real-world networks. We consider only two or three important features for this short tutorial, although there's a lot more to learn about the common features of these large-scale networks. After that, I elaborate a bit on the concept of node centrality, which is just to answer the question that which node is the most important one in the network. You have a network. You want to rank the nodes according to their importance. How to define importance? We will talk about it, but it's ill-posed question. You can define the importance in different ways based on what problem you are going to solve with these definitions. After that, we move to network formation. What processes are underlying the network growth, network formation? How these networks emerge from the smaller-scale interactions between the nodes? At the end, I will provide a very short glimpse at the problem of network robustness, which is a very important problem, because as you might guess, all systems in nature are in danger of damage, and it's important to know if a system is robust or not. This is a very brief look at the contents of this talk. I'm going to introduce two references for the network science. These books are loved across the world, so it really depends on your taste, which of them you would choose. Network science book by Barabasi is probably easier to understand, so if you are from a non-mathematics background, I mean, if you are not a physicist or computer scientist or mathematician, perhaps you will prefer Barabasi book. Also, there is a network science introduction by Professor Newman from Michigan. This book is a little bit harder to grasp and contains a lot of details that are probably more interesting to people from mathematical backgrounds, so it really depends. It's a preference to choose between them. We are going to talk about complex systems. So, complex systems by definition are large collections of entities that interact in non-trivial ways, and they are characterized by their emergent properties, which means that their properties cannot be understood if you take the system into parts and study their parts. You should always consider the system as a whole and take a systemic view to understand what's going on in these systems. And these systems appear across disciplines, so you can see them in sociology, economy, physics, or biology, and perhaps that's why a famous scientist like Stephen Hawking has said that this century is the century for complexity science, so probably you are learning something really important. Some examples of complex systems are shown in this slide, like the brain, that is probably the most miraculously complex system in the world. It contains a collection of neurons, so there is a large collection of neurons that are interacting in different ways using different neurotransmitters, electrochemical signals, and from this system emerges consciousness, memory, and other properties that you cannot learn by studying neurons in isolation. The same holds for our societies, so we are exchanging information in forms of text and voice messages, and our larger scale behavioral patterns are not yet fully discovered. We cannot claim that we understand human behavior. Also, there is transportation systems, biological systems, like protein-protein interaction networks or treated or prey interactions in nature, and also map of airlines, they all show complex systems, and you should have really a systemic view to be able to say something about these systems and to understand them at some point. All these systems exhibit structures. What do I mean by structure? If you take a look at this network of neurons, this bunch of neurons connected together on the left side, you see that the connections between them are not random. Each neuron is connected to a specific set of other neurons, and they are exchanging information, and there is something like a structure that is determining their relationships with one another. The other example is, for instance, Twitter. You can see in the middle a network of people that are exchanging information on Twitter or Facebook. You are friends with a certain number of people, and you are disconnected from the rest of the world. There is a structure and a pattern of connectivity around every individual, and this structure brings people together and restricts the flow of information between them. The third example is a network of animals. This is the predatory prey relationships between different animals. As you can see, fox doesn't eat mice. It eats rabbit, and the rabbit eats carrots, but it's not that everyone is eating everyone. There is some patterns and regularity there, and this system has a structure. These structures are often modeled by networks. So just to tell you briefly, neural networks or neural networks just show you how these neurons are connected together. Social networks are just representing the structure of a network like Twitter, and also FoodWeb is there to represent the asymmetric way animals are interrelated in a predatory prey relationship. From here, we are safe to move into the basics of the network, so I'm going to introduce a number of very, very basic properties, very, very basic concepts by which you will be able to get a clear view of the structure of complex systems. So a network is basically a set of nodes and links. Nodes are those blue circles that you see. They represent the entities or units or components of a complex system. For example, in brain, they show neurons or brain regions. In social networks, they show people. In FoodWeb, they show animals. And then links show how two nodes are interrelated. So if node one is in some way connected to node B, then you put a line between them, and it defines a link. There is different terminologies in this domain because the science is old, and there is social networks. There is graph theory. There is network science. There is complex networks. So depending on the background, people use different terms. In physics, we almost always say node and link. In mathematics, people usually say vertices and edge. In social science, people say actor or connection. People use these words interchangeably, but they are all referring to the same thing. So they are really the same thing. I'm putting it there because many people get confused by reading texts on network science. The important point of this slide is that networks represent the structure. Without networks, we miss a systemic view. And also the number of links attached to each node is important. And we call it degree. Degree is the number of connections each node has. It will come important in the next slides. One distinction you need to be able to make is between asymmetric and symmetric networks. So in an asymmetric network, the connections between nodes can be asymmetric. It means that maybe I am your friend, but you are not my friend. So there is a one-sided relationship between us. And this is an example for a social system, but in many other domains, you can see asymmetric networks. And you need to use directed networks to represent asymmetric connections. As you can see on the left side, section A, we are observing arrows going from nodes to nodes. So for example, let's take a look at node D and B. Node D is connected to node B, but node B is not connected to node A and D. So this is a type of asymmetric relationship between the two nodes. While in the network represented in section B, you can see that there is no arrows, there is only lines. So every node is in a bi-directional relationship with every other node to each it is attached. So we have directed and undirected networks. Another concept is that as I told you in the last slide, the number of connections each node has defines its degree. Here, we can have two different definitions of degree. You can take a look at the connections inward and call it in-degree, and you can take a look at the connections outwards and call it out-degree. For example, let's take a look at node B in the directed network. As you can see, there is two arrows entering node B, so the in-degree for node B would be two. And there is one arrow emanating from node B, which defines its out-degree. So the out-degree for node B is one. Of course, for undirected networks, we don't have out-degree or in-degree. There is only degree. Examples can be food web. In food web, animals often are either prey or predator. So their relationships between every pair of animal can be better shown by an arrow rather than a line. But then in WhatsApp, you are either in contact with someone in a bi-directional way or you are not. It's very low probability that you are connected to someone who doesn't respond at all to your text. So in WhatsApp, probably you want to consider the structure to be undirected. There is weighted networks and binary networks, and this is important to learn the distinction. It really depends a lot on how much information you have about the system. So if you know that, for example, in a social system, Fred and George are friends, but you don't know how much friendship there is. You can't compare the friendship between Fred and George with other people. You only know that they are friends. You define the network to be binary, so you make a link between Fred and George and every other people that you know to be friends. And for sure, for people that are not friends, you don't draw the line. There is no link. It defines a binary network. Invaded networks in contrast, you know how much friendship there is between two people. So for example, you measure how many text messages are exchanged between Fred and George. You see that it would be like 10 messages per day. You compare it to other people and you decide whether the weight of connection between Fred and George is high or is low. And then you get a weighted network in which there is more information compared to the binary version. To represent them, people often use the thickness of the links, so a link with higher weight is represented thicker, and the tinier links show the weaker connections between pairs of nodes. This is the distinction between weighted and binary networks. Another important definition is the definition of paths. So imagine that you are on a certain node of a network and you want to navigate your weight to another certain node. So let's take, for example, node A and node B. And to do that, you need to jump from the node on which you are, go through a link and land in a neighboring node and continue it. So you move through a sequence of links and you eventually reach node B. So paths are just sequences of links, and they can be of different length depending on how many links you have passed to get to your target. So you can choose a very long path, like a huge number of links to reach node B, or you can just go through the shortest possible path that connects node A to node B. Perhaps one can argue that the shortest one is the most efficient, whether it is really a transportation network, when you want to change the flights to get from New York to Sao Paulo, and you want to get to your target in the fastest way, or it can also be neurons in human brain that want to exchange information, and it would be probably much more efficient if they choose the shortest path to communicate. So finding the shortest path connecting every pair of nodes becomes something of importance. And there's a lot of algorithms to calculate and compute the shortest path between every pair of nodes in a network. It's not easy, so you need to find an efficient algorithm, because especially if your network is very large, finding shortest path has become very time consuming and computationally costly. Another important concept is transitivity. So as you can see on the right, there is a network of unconnected nodes. The second picture from the right is showing a connected pair A and C while B is isolated. The open triad example is where B is connected to A and C, but A and C are not connected together. And there is the closed triad. This is basically a triangle. And this is very important because again, in a social network example, you can think of yourself being friend with Fred and George, two other person, and you want to calculate the probability that if you are friend with both of them, they are friend with one another. So you are shaping the triad, the closed triad. But this is very important because it shows how densely connected people are around you. To measure it on a large network, you only need to count the number of triangles or closed triads and divide it by the maximum possible number of triangles in that network. So it gives you, for example, 60% or 0.6. Then you say that my transitivity in this network is 0.6. And you can compare it with another network and obtain that the other network has 0.8 transitivity. You can conclude that in the second network, the connections around people are more dense. They are more likely to be in closed triads, like tree body interactions and physics. So we move to probably one of the most important yet very simple concepts, which is modularity. This is a network. You can see that there is two groups of nodes, two communities that are almost separated from each other, while they are very connected inside within each of these communities. The red community represents the people with more tendency towards conservative parties. This is America, so it is showing the Republicans. The blue party, of course, is Democrats. And you can see that people inside the party tend to make connections with similar people, with people of similar ideas and similar political thinking. So as you can see, there is two modules formed in this network. There are algorithms that allows you to find the modules in the network, find the communities, find how strong these communities are, meaning that how separated they are from other communities. This is not compulsory, that a network has two communities. So a network can have a lot of communities. This is just an example because the political system in America has two poles generally. So that's why you see two communities here. Again, there is a lot of algorithms. There is no consensus that each of these algorithms are better, but you can use these algorithms to find the community structure of your network. Now, we can talk a little bit about the features. Dr. Gavasi, can I ask a question? Sure, sure. Let me just share the video and stop sharing here. Okay, so I'm here. Yeah, so I wanted to ask you about the difference between transitivity and connectivity. And what? The difference between transitivity and connectivity. Okay, so connectivity is just the number of edges you see in the network, the number of links divided by the maximum possible number of links. Okay, yet the transitivity is about the triangles. So connectivity is about the interaction between a pair of nodes while the transitivity is about a triangle, three nodes. So this is of course in a very dense network, they two are high simultaneously, but then in a sparse network, it can be different. So in a sparse network, you can have high transitivity while low connectivity. So how would you interpret high transitivity but low connectivity? So how do I interpret it? I'm not sure if I correctly get the question, but anyways, around every person, there is a community of densely connected people, but perhaps globally, if you think of network, it's the connections between different communities is low compared to a new model or a typical network. So I have to probably, Dr. Greely, should I answer all the questions or should I put? So if you want, you can ask, you can answer now. So I remember the participants that if they want to ask the question. Okay, okay. So probably I will move ahead, but at the end, we will have 15 minutes to There is a question about the slide that you just showed. So if you want, I can ask it now. Okay, okay, let's go. It's really, I think a question. So you showed this plot about the two parties in the US and there were some yellow links. Okay, so there is other parties in America. And I think they are libertarians. I'm not sure, but they belong to other parties. But they're, you know, the density of them is really low compared to the two big parties. Great, I think that answers the question. So please go ahead and again remind the participant to use the raise and button to Okay, okay. So I will go ahead with the talk. The real world networks, because in real world, we are observing networks for two decades. And now we can say that these networks are showing unusual and extremely interesting characteristics. They are similar in one way or another. I'm going to discuss three features that is that are common among a lot of networks. So the first feature is a small wordness. As you can see on the left, there is a network with regular pattern of connectivity. So you can really see that there is a symmetry between every pair of node. Each node is connected to its neighbor and to its second neighbor, but not connected to any other nodes. This is a regular network. You can get the pattern by a loop. One point is that there is no difference between curved links and straight links. They both are telling the same. It's just a matter of representation. On the right side, on another hand, there is random network. As you can see, there is no pattern of connectivity. Everyone is randomly connected to everyone. And it's chaos there. In the middle, you see the small word network. So there is the pattern. There is the regularity. But there also you can observe some irregularities, some randomness. And this randomness is represented in terms of long-range connections connecting B-stand nodes of the network. This is the topology of the small word networks. They are between regularity and randomness. They have a lot of important properties that are good for complex systems. For example, they have high transitivity, which is related to the number of close triads, as I told you in the last slides. And the average shortest path connecting pairs of nodes is relatively low. So this would be easy to navigate your way from each of the nodes and reach another one. And that's why these networks are often considered as efficient. They are observing multitude of systems from brain networks to social networks. And the model of the small word has been used to justify many observations people have done on real-world systems. So small wordness is characteristic of real-world networks. There is heterogeneity. It means that in real-world networks often there is a lot of nodes having only a few connections. As you can see, most of the nodes in this network have only one connection or two. And then there is a very tiny minority having a lot of connections, as you can see. This is a node A and C in this example, having each of them have about eight or even more links connected to them. So they have high degree. And that's why we are calling them heterogeneous networks or scale-free networks. So these heterogeneous networks are reminiscent of what we observe in economy, as you can imagine or you know probably the distribution of wealth between people is somehow power law, meaning that it's heterogeneous. There is a tiny minority having a lot of money while most of people in the world are poor. So the distribution of degrees in these networks are important. They follow a power law distribution. If you are a physicist or a mathematician, you probably know what a power law is, but it's a signature of the scale-free behavior in the system or scale-free systems. That's why we call these type of heterogeneous systems scale-free. So a network can be heterogeneous, but not be scale-free. But let's forget this distinction now. And from now on, I will call heterogeneous networks like these scale-free networks. The other element is hierarchy. As you can see here in this picture, there is five communities. Each of them consists of five nodes that are interconnected in a dense way. And these communities are just all of them connected to the community in the center. So the community in the center has somehow access to all other four communities. But the communities on the peripheral part are not having the same access. That's why we call this network a hierarchical modular network. And there is the element of control in these networks. So the top module probably can control some properties or some information processing going on in the lower modules. So this is the third feature of complex networks that I was intending to share with you. I hope that you have now an idea of the common properties of some of the common properties of real-world networks. And now we are probably safe to go to the topic of centrality. So what is node centrality? You have a network, and it's a mess. Believe me, when you look at the networks in your computer, you cannot really say many things based on your intuition or just the visualization you need to compute and calculate things. So you have a network, and you want to understand what is the most important node in the network? Or what is the most important connection or link in this network? And you want to find the ranking of nodes based on their importance. This is a very abstract network represented here. It can be a network of neurons, individuals, proteins, and you want to know which of them are the most influential, important, or whatever different definition of being critical for the system you have in your mind. Also, you might ask which synapse in a neural network is the most important or which social connection or which biological interaction is the most important link, defines the most important link in my network. And to answer this question, you first need to define the relative importance. How do you define it? So this is something that's worth thinking about. And it really depends on the type of the system, the type of question you have in your mind. But I'm going to give you some of the most important yet simplest definitions of node centrality. Probably you have guessed this one because it's quite trivial. You want to know what node is the most important. You calculate the degree of each node. And you would say that the node having the most number of connections or the highest degree is probably the most important one. Here in this example, I highlighted the node with red. So you can see the highest degree centrality node is node J. And you can really make the case that these nodes are very important, like in social networks, imagine someone having millions of friends and connections. So they can have really an impact on how society thinks about different things. The same example happens in protein-protein interaction networks and brain, and also any other complex system that you can possibly think of. So degree centrality is a way to rank the node and say that, okay, this one is more important than the other one. The second definition of centrality I'm going to discuss here is closeness centrality. So you calculate the average distance of each node from other nodes. To do that, you need to compute the shortest path connecting each node to any other node and average the length of these shortest pathos. And you will find one of the nodes that has a very low average compared to others, or the one that is simply closest to the rest of the networks. And you would say that this node is probably the most important according to closeness centrality. In this example, this is node P to really calculate it. You need a computer or you need to really write pages of calculation for this network, even though the size of the network is relatively small. So there is algorithms, as I told you before, to calculate the shortest path. And there's algorithms to derive the closeness from the shortest path. Another way to define the centrality is really to find the middle guy. So imagine that you are working in a physics department in a university, and you want to have some collaboration with biologists from another department. And to do that, you usually find the middle guy, the guy who works between the two departments. Maybe he is a physicist, she is a physicist, or he or she is a biologist. It's not important. The only important point is that he is the middle guy. He knows everyone from both departments. He can introduce you to many people. He can speak the language of both departments. So the middle guy is really important. Also, in brain, if you think of two brain areas that are trying to exchange signals, there probably is a middle region that the signals should travel through it and it should pass the signal from one region to another. So the betweenness of nodes in network is important. And as you can see here, it's quite graspable, usually that node H has high betweenness. It's connecting or less two patches of nodes. And this is another way that you can think of the importance of nodes in a network. So here we are moving to the subject of network formation. I'm going to discuss two theoretical models of how a network can be formed in nature. And I'm going to tell you why these processes are important. What are the mechanisms underlying the growth of networks in nature? They are very complicated and probably much different from these two theoretical ones that I'm going to introduce. But the theoretical models are providing some insights into the system. This is the random connection. This is a very simple model. You can see on the left side, there are, there is a lot of nodes that are totally disconnected. Then you add links with probability. So you add link with probability zero. There is no links added. So the nodes are disconnected. In contrast, on the right side, you can see that all nodes are connected to all other nodes. So you are in a regime where probability of connectivity is equal to one. Then moving from the left side to right side, you are increasing the connectivity probability and you will see that there would be different patches of nodes that are connected inside, but they are not connected outside. They are not connected together. And the number of these patches would decrease as you increase the probability of connectivity. And at some point, there will appear a large giant component, a component that is very big and it's integrating all the network together and the network would be formed after that point. The point at which the giant component emerges is called the critical probability has been shown by PC. And the size of the largest connected component in the system is a measure of how this phase transition from disconnectedness to integrity happens. As you can see in the plot on the bottom, this is the size of the largest connected component. You are increasing the probability of connectivity. But firstly, there is not a real change in the size of the largest connected component. The size of the largest connected component is basically the number of nodes you can see in the largest connected component. So this is not really zero. It fluctuates, but this is approximately zero before critical probability. And then after critical probability, you can find that the giant component appears and then integrates the whole network. So this is only a formation model based on adding links with probabilities and from it, you see the interesting behavior of the giant components. The other model is preferential attachment. And this is very similar to the idea of reach gets richer. So you start from one or two nodes, you add nodes, each of the nodes that is added in one specific step gets connected to the existing nodes in the network with probabilities. It is more probable that the newly arrived nodes get connected to the node with the highest degree. So you can see that the nodes that has already the highest degree will increase its degree very fast. And you end up in a heterogeneous network where there is a tiny fraction of nodes with a lot of connections, while most of the networks, most of the nodes in the network are having only one or two links, as I told you in the last section. In this network, in this visualization, the size of nodes is taken to be proportional to their degree. So the number of connections each node has determines the size of the node. As you can see, there is five very big balls in this network. It means that they are hugely connected. And you can see a large number of very small dots representing how heterogeneous these networks are. From this preferential attachment, you can analytically derive the probability distribution function for degree. So you can really show that degree is distributed in a power law way. Again, another emphasize on how the scale free the networks in nature can be. People who are familiar with power laws from a statistical physics or statistics will get what I say. It's not a hard concept. You can really follow it from Baro Bossi book on the topic of network growth and preferential attachment. You will find the full derivation of how the power laws are obtained mathematically and what they mean. But I suffice to this for the moment because I don't have a lot of time. So the last part of this talk is about robustness. And robustness is the study of how robust or tolerant the network is against failures inside or external attacks. So complex networks, because they are always in touch with nature, they can be attacked. And there can be errors and possible failures within them. And the network, if it's not robust, it will dismantle into pieces and the system will die. System wouldn't maintain its function. And it's very important to see how robust the network is. So the procedure considered to model the damaging networks is just to remove notes. So you either randomly or according to an algorithm, you select notes from the network, you remove them one by one, as you can see from the image on the top left, the network is whole. And then you remove the note around which there is a green circle. And then in the image on top right, you remove another bottom left, you remove the third one. And the bottom right shows the network after removing these three notes. As you can see, the network was connected in the first place, but it is now dismantled in the fourth place into one, two, three, four, five disconnected components. And the size of the largest components here is just one, two, three, four, five, six. So the largest connected component contains only six notes in the fourth step, while in the first step, it contains probably more than 20 notes. I don't take the time to count all of them, but as you can see, the size of the largest connected components shrinks as you remove the notes and as the network dismantles into pieces. So this gives you a criteria to measure the robustness of networks. Here is the example of a lattice, a regular lattice. So every node is connected only to four of its neighbors in a very regular way. And then the size of the network, I don't know how much it is, but it would be probably around 300 or 1000. So, and then you remove the notes randomly. So you make a code to remove the notes randomly, and you will study how many disconnected components there is in your network. What is the size of the largest connected components? Of course, in the beginning, the size of the largest connected component is equal to the size of the network because everyone is connected to everyone. There is no different patches. There is no disconnected components. Then the number of disconnected components grows and the size of the largest connected component shrinks. As you can see, there is a phase transition again around fc, the critical fraction. So you are removing about 0.4 of the total links. It means 40% of the total links. And you see that the size of the largest connected component goes to zero. So the network totally dismantles. There is no connections in the network after that. This is very important so you can see that the lattice is not very robust against random failures, against random removal of notes. But in other types of networks, this is Internet that has a scale-free topology. So the structure of Internet is really scale-free or heterogeneous for sure. You can see that if you follow the green light, you can see that the network dismantles after removing 90% of notes. So this is very, very shocking. These networks are extraordinarily robust against random failures, against random removal of notes. As you can see, the largest connected component has a large size even after removal of 50% of the notes. And this is weird. So the take-home message here would be the real-world networks are probably robust against random failures and random attacks. While I'm going to get a bit into the details and I'm going to introduce you to the targeted attacks. So you can use the notion of centrality that I introduced in the last sections. And you can rank the notes based on every centrality of your preference here. The degree centrality has been chosen, meaning that each note is important according to the number of connection it has. And then you remove the note according to that ranking. So if this is degree centrality, you are basing your attacks on, then you're firstly remove the high, the note with the highest number of connection, then you remove the note that is second with respect to the number of connections. And you go ahead. The purple line shows the response of system to target attacks. As you can see, you are removing around 10% of the notes and the system collapses. If this mantles into pieces, there is no network after these attacks. This is very important. So this is again a heterogeneous network with parallel degree distribution. And we see the green light shows the random failures. So the scale-free networks are very robust against random failures, but they are not at all robust against targeted attacks. And it depends on the type of study that you are doing. You might want to consider the random attacks. You might want to consider the targeted attacks. You might want to base your attacks based on different types of centralities. So these are the ways in which you can explore the robustness of networks in different situations. I'd like to finalize the talk here. Thank you very much for listening to me. I will be ready for Q&A. Disclaimer, I'm not an ecologist. I'm not a biologist, but I'll be happy to answer your questions about network science. And also, I will try my best to answer your questions from other domains, from ecology, biology, sociology. But no promise. Thank you very much. Thanks. Thanks a lot, Arsham. Thank you very much for this very nice, broad, but yet compact introduction of a huge field. Thank you. Perhaps I'll start asking some questions that are in the Zoom chat. And please, if you have any other, people have any other, they can raise hand or ask them in the YouTube chat if they're following from YouTube. So there is one question which I think is very broad, which is, is hierarchy always associated with modularity? Okay. So you can basically imagine networks that are not at all modular, but they are hierarchical. And there are examples of that in nature. But when I, when I'm thinking about complex systems like brain or societies, modularity is always there, hierarchy is always there. And it's common that these networks are not homogeneous. So that's why I picked these properties. So basically there can be networks with only hierarchy. Great. So there is another question by Deepak. Why is the central network, why is the central network called Small World? Okay. So this is based on a very old experiment in the field of social networks. So long before physicists entered the complex network domain, there was Milgram, a scientist who did a very famous experiment, you can search it on the Wikipedia, but it basically showed that your distance from any other person in the world is very small. Like you are connected to every other person with only a few number of other connections. So this is basically the shortest path between you and other people are very small. And the model that I showed you, that was something between the regular and the random network, this model was there to show how this is possible, how the small wordness emerged in the world, how we are really connected and our distance is very short. Great. Thanks a lot. So there is a question by Pablo. Please unmute yourself and you can ask it. Hello. Thanks for this talk. It was really interesting. You mentioned a derivation of the probability of the degree of a node when the network is constructed with preferential construction. And this is in the Barabasi book, but I looked it up and I can't find the chapter on network formation. I don't know if Okay. Professor, is there any place I can share documents with people that are interested in some? Yes. I mean, you can send it with, if it is something that you have in your hand now, you can share it in the chat. Okay. No, I don't have it now, but of course it's there in the book because this is the central claim of Professor Barabasi that networks are parallel. So this is probably in that book, but for sure there is papers. Yeah. So we can share material, so links to paper, not the email, on the website. Great. So just send an email. Sure. Great. So there is, we have time for a few more questions. So there is a question by Dionessa in the chat. So what insights do we usually make from the distribution of between a centrality in the network versus the network robustness? And is there any analytical way to get the critical FC robustness? Okay. So the critical FC can be obtained for a specific topologies. And this is a very interesting topic to discover if you are a physics fan and you like the networks. And the first question, I didn't get it. So what is the distribution of betweenness? The first question is what insights do we usually make from the distribution of betweenness centrality in the network versus as opposed to the network robustness? Okay. Okay. If I get the question correctly, please tell me if I'm mistaken. But if I get the question correctly, it's because I didn't really explain how to calculate the betweenness centrality. So between the centrality, you again have to calculate the shortest path and you will calculate how many shortest paths cross a link. That's how you generally say that the link is in between other nodes. So there is many shortest paths going through that specific link. This is how you can say that the betweenness of a node is high or low. And then according to betweenness centrality, if you base your targeted attacks on betweenness centrality, this is one of the, you know, best strategies of attacking the networks. It's probably, probably the best among the at least classical ways of attacking the network. And it dismantles the networks very fast, very quickly. Great. So is there any other question? So there is another question in the chat, which I'm not sure, but I'll read it. So it's can we combine network analysis with Markov chain by taking into time? So I guess it's related to temporal networks, but I'm not sure. Okay, okay. So I will take this time to briefly talk about the dynamical processes on networks, because this is, you know, the subject I'm working on. So you can, you can have these structures like imagine the neurons connected to each other, but you don't know how information travels from one node to another. Okay. So you can calculate the shortest path, but it's not really the way that electrochemical signals flow in the network. That's when you couple the network with a dynamical process. It can be a random one. It can be continuous diffusion. It can be synchronization and the nodes, you know, exchange information in terms of these dynamical processes. That's a very huge topic. And I really enjoyed it. And I chose it to be my, you know, master's and PhD thesis. Great. Any other question? You're welcome. This was a reply to a thank you in the chat. Yeah. Great. If not, I think I'd like to thank you to thank Arsham very much again for this very nice tutorial, which will be available on YouTube for the next generations. And before we go into the break rooms and we take a break from the lectures, I'd want to share a couple of information about next week. So let me share the screen. So probably, you know, I mean, this is okay. This is the program page. And you will see that there will be, there is a Monday slot with the Jordan Vasconte on actually mutualistic networks. And on Tuesday with James O'Dwyer on cooperation, resilience and stability. So these two slots are Q&A and the lectures are pre-recorded. So if you go to the program and you click on the material video, you can access the two videos with the lecture. So the message is when you have time, please watch the videos before the lecture and come on Monday with questions to ask to Jordan Vasconte. And the same applies to James O'Dwyer about cooperation resilience and stability. So these are going to be two pairs of exciting lectures. So please watch them in advance and take this very nice opportunity to ask questions to the speakers. The videos will also be shared on YouTube and the Q&A will be live streamed on YouTube. So if you're watching this on YouTube, you can access to all the material and interact with the speaker using the chat on YouTube. So with that, thank you very much. We'll now go, thanks again Alsham. We'll now go to the breakout rooms and we'll be back at 3.45 Italian time for Joshua Wright's last lecture. Okay, great. So I think we can start to close the breakout rooms so people can join back in the main Zoom meeting. So while we are waiting, it takes about one minute to close the breakout room. So while we are waiting for them to join, let me remind to the people that are watching this live stream on YouTube that if they want to ask a question, they can do that in the YouTube chat and I'll read it on your behalf. So yes, in about 20 seconds all the participants should be back in the main Zoom meeting room. I think everybody is back. So we have now the third and last lecture by Joshua Wright. Before he starts, just to remind you, a tip now to ask questions. If you have a question, please post it in the chat or use the raise hand feature of Zoom. I'll post in the chat the instructions to use that feature once again. So please, Joshua, if you want to share the screen, you can start the presentation. Thank you very much. Okay, perfect. Thank you, Jacoble. Can you hear me as the sound louder? It's perfect. Good. Let me then share and hopefully you can still see me, see my slides and hear me. Yes, I do. Okay, well welcome back folks to this third lecture. I'm trying to make each connected but also stand alone in case people pop up and just see this one. So this will be the third lecture on virus micro-dynamics, spanning principles, ecology, and therapeutics. And because therapeutics was the last in that sequence after the Oxford comma, I will be focusing on therapeutics today. And just to again remind folks, I started earlier in the week with some principles of predator-prey like interactions between virus and microbes and consequences from population to evolutionary dynamics to even co-evolutionary dynamics. And then in the Tuesday lecture began to go in a slightly different direction confronting these paradigms by looking specifically at outcomes of infection at the cellular scale that don't necessarily lead to lysis in the production of more virus particles, but instead potentially long-term associations in which the fates of the virus and the host become entangled. And today I will revisit again the lytic paradigm in a specific context and applied context where we'll try to leverage what we know about bacteriophage, bacteria interactions with the purpose in mind that is to try and improve treatment of multi-drug resistant infections. And obviously that's clearly motivated by a real world application. And to focus on this real world application, I would remind folks, many of you probably don't need reminding that in addition to SARS-CoV-2, there are a lot of other hazards around and one of the biggest ones that continues to be a problem globally is the emergence and spread of multi-drug resistant bacterial infections. And the CDC categorize these in different levels and the levels have to do A with the extent of harm of the pathogen, but also the extent to which we are running short of therapeutics, including last line therapeutics or therapeutics that themselves cause harm when given. And these here are some of these listed strep groups in pseudomonas originosa, multi-drug resistant staph aureus, niacir, gonorrhea, Cdiff, and so on. And the issue here spans obviously one not only impacts now, but looking forward to the extent to which there are those who are concerned, one of the most prominent, I would say, statements on this topic is this O'Neill report looking at deaths attributed to antimicrobial resistance AMR every year now compared to other majors of causes of death. And the challenge, of course, looking forward 30 plus years is the danger that will be associated with the spread of these multi-drug resistant infections that would then be untreatable and estimates that rival even that of cancer in terms of mortality. And in terms of some of the research spending, you can see at present there just has not been the same prioritization within the NIH. Obviously that is changing, but more needs to be done to really to move this field forward. And part of this is that for a very long time, no major new types or class of antibiotics were developed. There were some recent discoveries that provide some optimism with respect to discoveries based on environmental microbes, including texabactin. But again, the crisis here is a serious one already and is bound to get worse unless we really start to confront and take new approaches to address how to treat antimicrobial resistant infections, particularly those that are multi-drug resistant. And one of the ways that people have begun to reconsider is to use phage. And I reminded you earlier in the week or explained to you, if you hadn't heard before, that phage are highly abundant in natural systems where there can be tens of millions, if not more, virus particles per milliliter in natural systems, whether in water, aquatic system, soil, and other microbiomes. And these can become the basis for candidates, a natural reservoir, as it were, of potential treatments. This study and kind of broad scope description of virus host impacts is described in this book by Carl Zimmer. But my point here on the right is that there's this diversity of potential treatments that if we can harness, maybe use productively to try to treat these infections. And in fact, this is increasingly what has happened. A few years ago, this is now five years ago, there was a first multi-center clinical study of phage therapy in serious burn victims. And the idea was to try not just as a compassionate use case, but do an industry standard clinical trial to look at both tolerance and effectiveness, not just does it not do harm or is it tolerated, but it actually is effective in treating antibiotic resistant infections. And this was meant specifically on burn wound patients so that the presumption here, if the outset was there, these could be colonized by either Coli or Pseudomonas originosa, and there would be a control and would be compared against the usual treatment and to see if the phage treatment showed a benefit. Now, the problem about a year later was there was a series of delays, size and scope. And in fact, there just weren't enough participants included in the trial to get the kind of power they needed to determine efficacy. And part of that was because the design of the trial presumed that they were going to use a cocktail of phage against a single type of pathogen, E. coli or Pseudomonas. And of course, what is unfolding in many of these wounds is that these are complex communities. And so they weren't able to actually do have the right inclusion criteria and only a dozen plus of the intended hundreds plus patients were included. You can see that even if that were to work in those 15, you just wouldn't have the power to make those calls. Nonetheless, even though this was a bit of a setback, there have been a number of promising developments. And those are often through compassionate use cases. These are two well-known examples. On the left, Tom Patterson, who was in a coma in near death and was treated with a phage cocktail. And this was a collaborative effort that was using essentially information on phage of Acidibacter baumani as a way to treat this infection. And this has, again, been well documented. You can read about this particular story. His wife has described quite a lot of the story as well. And a similar story, this is again by Carl Zimmer. And you can see here on the bottom right, this is Paul Turner from Yale. And you can hear more about his work also online in slides where he, again, literally these viruses fished out of a lake, may have saved a man's life in advanced science. These phage found in natural systems, which were able to infect and lyse pseudomonas originosa. And I'll just talk a little bit more about this particular system and the rationale were used to treat this fistula. So there was a lung associated infection. And again, that was another example of a compassionate use case. And Paul and his team, including Ben Chan have done this multiple times now, successfully treating individuals using pseudomonas phage. This has led to initiatives that go beyond compassionate use cases, but to the, to the scale of institutes, clinical trials, and even really institutes dedicated to phage therapy, including treatments, not just pseudomonas and pseudobacter balmani. But here's an example for my go back to him from Graham Hatfield, Schooley, Helen Spencer and colleagues. So if you can look in the past few years, you'll see an increase in interest, a significant increase in interest with respect to using phage as the basis for treating these multi drug resistant infections in a diversity of pathogens. But of course, if we go backwards in time to the origins and just coming past really almost the centennial anniversary of the discovery of phage from the very outset, Felix Darrell and colleagues were thinking about therapeutics as one of the potential uses of this new discovery, right? Bacteria phage phagos meaning to devour this organism, this virus that was able to devour bacteria. And here's an example from his book after being assured that no harmful effects attended the ingestion of the sugar bacteria phage, this treatment was applied for therapeutic purposes to treat dysentery. And so from the very outset, Felix Darrell and colleagues thought about phage of course, then over time this was supplanted by the widespread availability of antibiotics, but we now face new challenges. And the critique has been to some extent with respect to phage therapy, that there's not been some transformative development or technology that means it's open season. Now, of course, the need is greater because of the emergence of multi drug resistant infection. The question is, is there been a development that could say that the chances of using this effectively have increased? And I would argue that the answer to that is actually yes, there has been developments, not only on the genomics or engineering side to improve the, let's say the production and characterization of phage that are available for treatment, but even the principles really ecosystem aware principles. And I think this is why it really fits in to this workshop because these are really ecosystem aware principles that are now being used to try to improve treatment. And one of them has been developed by Paul Turner and colleagues. And I've given you three references here in the bottom left in case you're interested in following. And the idea there is that a virus, and I'll get back to this at the end of my talk, a virus and an antibiotic can be used together. There's a particular class of efflux pumps in which essentially bacteria pump out antibiotics so that they are not affected by them or continue to divide and proliferate. Yet phage can absorb to these efflux pumps and uses as a means to gain access and entry to the cell. So if you apply both at the same time, you can see how this could provide an evolutionary trap. If on the one hand the bacteria, some bacteria were to mutate and block this efflux pump so phage can't get in, but then the antibiotic can't get out. And if they don't mutate that particular efflux pump, well then sure the antibiotics can get out, but the phage can get in. And this is precisely the rationale by using antibiotics in these particular efflux pump targeting phage together that Paul Turner, Ben Chan and others have managed to push this forward. The other idea I'd like to talk about with respect to synergy, again an ecosystem where synergy is our work, collaboratively with Laurent de Barbue and colleagues, to think about phage not as the only actor in the system, but rather as part of an ecosystem in which there critically are immune cells. And I will try to explain today how phage may not necessarily be the sole sterilizing agent as it were of bacteria in the system, but rather are working synergistically with the immune system where we can think of them as working synergistically with the immune system to eliminate bacteria. And if we think about that, that transforms the scope in which bacteria phage therapy may be more effective and also gives us some indication in which it may not be effective. So what I'll try to do today is really span this scale from models to mice and talk about roots towards a modern immunophage therapy. And I will do this in three parts again with the interest of trying to have a self-contained lecture. I know for those of you who were here at the prior two, each time I make this first part a little shorter, but I will just keep this in mind as the motivation for new principles underlying immunophage synergy going beyond just phage-bacteria interactions, but including immune components, and then talk about work on curative treatments of otherwise fatal respiratory diseases using bacteriophage and immunomodulated mice as well as return to some of the issues involved in antibiotic phage synergy. Okay, so in this first part, as I've described earlier in the week, the paradigm for these lytic phage-bacteria interactions originated in many ways with Campbell in 1961, but also Bruce Levin in the late 70s, the idea that there's a prey, the bacteria in the predator, the virus. And this means that we can think about the dynamics arising in terms of some predator-prey cycles in the absence of that temperate mode, which I talked about on Tuesday. In these models, to remind folks, we have some carbon source, some nutrient, some bacteria, some virus. There's going to be new resources coming in, resources going out. These resources will be taken up, leading to division of the cells, infection, and lysis. And you see these oscillations. And again, the key point here for phage therapy is if we want to get rid of the bacteria, it's important to keep in mind that in these kinds of models, we don't necessarily expect, although there could be large excursions, we don't necessarily expect generically that the virus that can kill one cell will eliminate the population, but rather will drop down the population density, but coexist with it. And that's the point of the slide, that there will be these cycles, but it doesn't lead to the joint extinction, it leads to coexist, and it will be at a lower density. Of course, we have to consider the fact that there's more microscopic mechanisms, and there can be delays between adsorption and lysis, but even in those cases, again, we don't find that the addition of phage and the absence of other factors need to lead to the elimination of the host, but in contrast, could lead to coexistence and oscillations. So we have this virus that we'd like to add in the phage therapeutic sense, but from an ecological perspective, thinking about these as prairie dynamics, if we've eliminated this temperate phage route, we expect there to be coexistence and likely to be oscillations. As I've also shown, these oscillations can be problematic, not only because it means the host hasn't been eliminated, but also because evolution can happen and resistant host can emerge. And so this initial phage that you might have wanted to use as a therapeutic, sure it'll be effective on some, but maybe not all of the hosts that one is targeting. So the reason I bring this up today as part of this talk is because if we think about trying to use phage as a therapeutic, it does provide a counterpoint to some of the standard assumptions of phage therapy. Yes, viruses can kill individual cells. Sure. But that doesn't mean they eliminate entire host population. In fact, we should expect they may coexist with host populations and may even leave the evolution of resistance to the loss of top-down controls. So instead of controlling at some low densities, we may be back exactly where we started, where now we have a phage-resistant bacteria and we have lost the efficacy of this candidate therapeutic. And you can read more about this as I said in my book for some of the context that goes into why these principles hold. The field is aware to some extent of these issues, and there's been an approach to respond to them. And that approach are cocktails, not the kind of things you might have at 6 p.m. in Milan in normal times when you're having a spritz or whatever it is one has in the aperitivo, I believe it's called, which would be a delight to have and share. But these are different kinds of cocktails. These cocktails involve a combination of bacteriophage, each with a different sort of feature. And you can see a schematic here on the upper left. The idea might be that if there are target bacteria, maybe some of them get stopped on the outside, some of them get stopped on the inside, but there may be enough coverage here that the bacteria are going to be laced by this collection of diverse phage. And if you go to Georgia, the other Georgia in the former Soviet Republic, these are actually available as OTC phage, over the counter phage. You can buy them at a pharmacy. There's been some question marks vis-a-vis what is exactly in some of these OTC phage? Is it just phage or maybe also antibiotics? And there's been effort certainly to engineer phage to have different characteristics, for example, different targeting receptors and even different contents, in other words gene content, so that they can deliver different, whether it's toxins or other features that may lead to the lysis of target cells. So this is good news in some respect, but we have to keep in mind that some of the concerns of standard phage therapy still remain even with cocktails. Yes, cocktails may kill more, but not all and there may be trade-offs with coverage because you're making a choice about which particular phage to use and if you have in some sense a constraint of the total density of phage that you're able to combine. Moreover, we see natural systems that are diverse, that are co-existing with host populations and it's true theoretically, as I described in my Monday lecture, that there can be coexistence, albeit just amongst more diverse communities. So you may be using this complex cocktail. We just may have a situation where now there's complex coexistence rather than a single oscillatory dynamic and yet there still can be this problem of evolution. Now it may be slowed, but there's now in some sense just a more complex Luria Delbrook experiment where there can still be that kind of escape. Okay, so that in some sense points us in the direction of new principles and I'll begin to explain those in a moment. Are there any questions at this stage? I don't imagine there'll be so many given I'm covering things, but with a particular purpose in mind. So there is no question in the chat of Zoom or in the YouTube chat, but if anyone wants to raise hand and ask a question, I think it's a good moment to do that. I imagine there'll be more questions after part two, once we've said that. Okay, so let me just keep going here. So this kind of gives us a direction to be somewhat skeptical about bacteriophage therapy, but I'll try to explain why other things are happening that may give us more confidence. So let me give you a contrast. Everything I just said basically implies like, why is this working? Why should this even work yet? Certainly with the mirroring model within mice and obviously there are these compassionate use cases that seem quite promising. There have been controlled studies and this is one by Laurent de Barbieu from about a decade ago in which mice are infected with an acute respiratory model of pseudomonas originosa. You can see here notably these have fluorescent modes so they should actually visualize the spread of pseudomonas originosa and the control versus in the phage treated mice and you can see the difference here in terms of intensity and on the right you can see the outcomes in terms of survival between what happens when you don't use phage one to 10 ratio and here as many phages there are bacteria or 10 times as many. And there's a contrast here between no mice surviving and all of them mice surviving. So clearly this is working at some level this actually works despite everything I just told you. So I think this should raise questions and raise questions about principles by which what may hold for a phage bacteria system the absence of other players may fundamentally be altered in the context of an immune system. And in fact Bruce Levin, Jim Bull proposed an early model to try to explain some of the differences and their ideas are shown here where you have bacteria which can be infected and laced by phage leading to more phage. These bacteria may stimulate an immune response and the stimulator immune response may therefore inhibit the growth of bacteria through immune killing. This is the schematic and now you're comfortable with these kinds of equations. Here we have bacteria, viruses, the immune response there's an implicit resource model here which I've left out so there's cell division, infection and immune killing. Note the virus particles are leaving from the outside but then being regenerated cow later right when a burst beta are produced and there's also viral decay and we see here we also have immune stimulation so there's more bacteria leads to more immune response and this immune response directly leads to killing. So this was the model they proposed and they made these claims that when there was an active immune response then there was a market difference from the case in which there was not and I'll try to explain the difference here. Here we have in the absence of phage a case in which you have susceptible and resistant populations and here we have only the immune response and what you can see is that here there's a threshold they imagine that would be a critical bound leading to the mortality of this particular organism and their claim is when you add phage the system goes up but it never crosses this critical bound and the phage aren't some sense responsible for the elimination of these sensitive bacteria. Although this is a good idea and it is very promising there's also somewhat of a problem here. First of all you'll notice that in this case without phage the bacteria were eliminated after about 12 hours but when you add the phage it actually seems to take longer to eliminate the bacteria and the other issue is that if I were to move this threshold just a little bit then in fact even in this case we might still have a process that generically you don't need phage to eliminate the bacteria in this first model so why do we even need this in the first place this model seems to imply that the immune system always works which clearly is not the case. So we tried to modify this Levenbull model and extend it in two key ways. First of all by implying that the immune stimulation has a capacity it can't just keep growing without bounds first of all that can itself cause damage but also there's a limit to the extent to which the immune response can be stimulated and the other part is that bacteria can initiate density-dependent defenses whether through quorum sensing or biofilms or virulence factors that can evade the immune response so even a stimulated immune system may not be able to eliminate bacteria once the bacterial density gets high enough so you see this term new term in the denominator. So these two red terms here these negative feedback loops this first one is carrying capacity and here's a density-dependent response what happens in this model right so it's the same model more or less as the Levenbull model except with these two additional biological features first of all if we were to get rid of the immune system just ask what happens when we add bacteria and phage together we see coexistence as we expect with these kind of predator prey models on the other hand if we eliminate the phage that unlike the Levenbull model which bacteria are eliminated here in fact bacteria increase and get to a point where they reach a steady state the immune response is on but can't eliminate the bacteria in other words there's an infection and yet when all are combined what you can see is that we have these dynamics between phage and bacteria the immune response is turning on and eventually actually the bacteria does get eliminated only when both are together so neither alone can do it but together they can and what you can also see here notably is that the phage disappeared in some way our local extent before that of the bacteria so you can even understand here that in the end it is the immune system in this model that fundamentally eliminates the bacteria rather than the phage and what the phage are doing in some sense are dropping the densities of bacteria to a level that can be controlled by an immune response so this is what we call immunophage synergy it's the elimination of bacteria through this tripartite dynamics and just to point out here that this doesn't always work but notably it can work in a larger regime that we initially expected here results for the final state of bacterial density and phage densities as a function of the decay rate meaning fast decay of viruses inside the host and slow rates and the absorption of phage to bacteria so on this upper left we have long lasting phage that are very good at absorbing and lysing cells and down here on the lower right these are rapidly decaying phage that don't do a good job and as you can see that in this bottom section we expect and find both theoretically and then via simulations that we have bacteria and the phage can't get a foothold and are eliminated when you see both colors that means there's a coexistence regime and initially we expected this only to work in this upper left hand wedge but we actually found a larger regime where the bacteria were eliminated and therefore also the phage but that's okay and this is through a dynamic mechanism that these oscillations in abundance can actually lead to opportunities in the troughs where the immune system can eliminate the bacteria and then the phage as well so we have a large regime and parameter space where we expect that immunophage synergy is possible and just to give a synopsis here in other words there's a bacterial introduction which proliferates the immune system responds but is unable to clear it the addition of phage whether because it breaks up biofilms or just directly reduces densities means there's a decrease in density and with a decrease in density the immune system can overcome the bacteria also leading to elimination of phage okay so those are the key ideas that make this different that really thinks of bacteriophage therapy as part of an ecosystem and thinking about feedbacks in that system so maybe now there may be questions before I move on to the later parts yes there is one question from the chat from Sylvia who asks with the respects to antibiotic therapy does phage therapy have the potential to be more bacteria specific and cause less side effects on the good microbial community right that's one of the key benefits of using phage it also is the one of the key negative parts of using phage because of the specific nature of phage rather than broad spectrum antibiotics it means that you have to make sure that the phage can target the specific bacteria that one has another reason why people use cocktails but it should not have the same side effects as targeting other cells so it is both good and bad news with respect to use great any other question yes there is one question from Ankit please hi so I was just wondering like since you're talking about coexistence and elimination like do you also sometimes consider systems where you might have a very large number of bacteria species and yeah and it's maybe different yeah let me try and say two is first of all we are looking at acute infections and part of the reason my group has been focusing on acute infections I think frankly the the challenge of using bacteriophage to eliminate a complex multi species community is going to be harder that's really steering complex networks nonetheless in this next part I will address what happens when there are resistant mutants and how do you deal with the fact that this is not just one bacteriotype but actually there can be susceptible and resistant types so in that sense I will address it but in the broader sense that remains a big challenge of how does one in some sense use phage either to steer or influence or control complex bacteria communities and that would be relevant for example to gut communities or even some of these surface communities we're focusing on these acute infections purposefully and I would say that even in the treatment for example of bacteriophage I'll note though Paul Turner is working in in CF and treating CF patients and then absolutely there are complex communities but there tends to be a disproportionate impact of pseudomonas originosa so there can still be significant benefits so you know those compassionate use cases are in situations where the community is more complex even if they're targeting a subset thanks yeah great any other question I'll keep going yeah oh there is one with the concern that the elimination of bacteria uh okay with the concern that the elimination of bacteria would be more effective with the synergistic involvement of phage and of the immune system how will the bacteria elimination work in immunocompromised that's that's a great question because if you look at the title of the slide that's exactly what I'm going to talk about so that person whoever that was we can let's get to it okay okay so this third part we're going to address precisely the question that this individual raise was just what happens in immunodeficient or immunomodulated systems so let's go and try to deal with that so this goes back now a few years everyone here recognizes Liverpool if you've ever been to Liverpool you'll notice that they've modified the central church to have this giant phage capsid on top you might not have noticed that this was just for the display and this the theme here is really I get by with a little help from my friends because this became not just a theory project we're really a collaborative theory project spanning both the US and France one of the first presenters on this session was Duane Roach formerly a pastor at the time but now recently professor assistant professor at San Diego State University and they were doing in peril unbeknownst to us an experimental study of phage therapy efficacy and immunomodulated mice precisely the point we've just raised in other words taking a multi drug resistant pseudomonas originosa which is that same system that I showed you from Laurent in 2010 which causes acute pneumonia which is fatal in mice and treating with pack p1 which was shown as I showed in that 2010 study to prevent fatal acute pneumonia when given at least that one to one or ten to one levels in vivo and again I've shown you this before here we have these readouts hours past post treatment of the control versus the phage therapy treated mice showing the effectiveness of phage therapy here we have complete difference in terms of survivorship no survivorship in the control group 100% in the phage group here we have radiance measurements which get to be at levels of detection after one two or three days and remain there whereas the saline treated mice then via the compassionate treatment then are not extended because they show distress and all die after 24 hours however using the same phage in a different mouse model and here I'm only denoting this word anti-gr1 which I will get back to later what you can see is that we have the original treatment and then we have a treatment and the phage treatment in this modulated mouse and in all cases even though one is using this phage that was effective in the immunocompetent mice it is no longer viable in this modified system here with radiance that look no different post infection again the two hours notes the fact that phage are added two hours after infection just to contrast these are days and this is hours post infection okay and although this is not good news obviously for these particular mice this is good news with respect to the idea any and advancing principles of trying to be more effective of how and when to use bacteriophage and the challenge of course here is that we have to then bridge the gap between these in vitro models and in vivo outcomes so we initiate a collaboration with Dwayne with Laurent and their team and we had these discussions and we figured well you know we our models are basically predicting this we I had a poster without any experimental data at the time they had experiments without any theory we got together and chatted and we began to adapt it to the particular absorption rates and birth sizes and other details of the system and this is what happened at first here is in our model two hours after the sense of bacteria are increasing close to immunity is responding we add phage and if you can see there's like a cliff and clearly something has gone wrong it's a miracle cure so in our models initially it looked like a miracle cure even with phage and bacteria you didn't even need the immune system and the reason is that at the absorption rates that they expected and the birth size and lane periods and so on it seemed that this many bacteria would immediately lead to clearance and clearly that's not the case so there was a gap here in terms of time scales and the models that we developed in a well-mixed chemostat that is not exactly how things work in the lungs of a mouse that if a bacteria is killed on one lobe by a phage that doesn't mean the phage immediately gains access to bacteria in the other lobe the other issue of course is that we have to think more carefully about the immune side can we diagnose the basis of this which are the effector cells that might be involved in this synergistic clearance so the first thing that we revisited was this notion of a linear attack rate and in some ways it's inspired by work in epidemiology where it's well known that heterogeneous mixing can lead to non-linear interaction rates we began to think about modifying this in some sense force of infection this adsorption rate not based on linear contact rates but non-linear contact rates due to heterogeneous mixing and also another problem here is has to do with phage saturation that locally there may be elevated levels of phage but if they are infecting the same bacteria they don't get to kill it more than once and in a linear model in some sense they do and so here we also took into account another wage in which there may be a non-linear response between phage density and adsorption and killing due to phage saturation and I'll refer to these as HM and PS for these different functional forms we then took this model which we have phage and bacteria and an immune system but instead of having this linear term here we used this heterogeneous mixing model or a phage saturation model and found instead of having time scales of basically instantaneous elimination it took about a day using the same parameter relay parameter sets for bacteria to be cleared here you can see it in these two models we also did the same thing with resistance and here I think there becomes another insight which is throughout this I have phage only targeting these sensitive bacteria phage sensitive vector on the other hand there can be resistant bacteria and the question is how does phage how do phage and immune affect ourselves together lead to the elimination when the phage cannot eliminate these infect and lyse these resistant cells the outcome as you can see is elimination of both and the rationales as follows which is that there can be the proliferation of resistant bacteria but because phage are targeting eliminating sensitive bacteria that the stimulated immune response can then in the absence of these large numbers of sensitive bacteria can eliminate the much smaller numbers of resistant bacteria and someone drew a red line on my slide which is really cool I don't know how that happened but you can see then that this then is controlled not by the phage but rather by immunity and so this also gives an insight as why this is very different than the version I explained before that by making these models explicit you can see that there can be despite the emergence of resistance as long as there is this notion of an ecosystem and immune cells a chance for phage therapy to work even if we don't target every particular resistant mutant there's still a chance that this can however we don't quite know which of the immune effector cells are going to be responsible for this particular synergy and one of the ways in which they probe this was to take different types of immuno modulated mice and use the same system and one of these is called my 88 minus which is immunity activation deficient mouse in other words the signaling is deficient so you can see here an example of survival and radiance and here we have wild type saline case but here are these immuno deficient mice in which when you add phage the survivorship is no longer 100 but drops down and what you can see on radiance is that they seem to be the same the phage looks like it may be improving things but then there's a reversal and that reversal in our models we also expected to happen because the phage are eliminating the sense of the bacteria but resistant bacteria are increasing and because the immune spawn system is present but it's not responding correctly then there's the uncontrolled proliferation of resistant bacteria and in fact that's precisely what they found when they actually looked and isolated these bacteria at the end they were all resistant to the phage so just in the model as in the system it's the proliferation of resistant bacteria in the absence of immune control that leads to the failure of therapy which then points to innate effector cells and I'll just point out that one of the interesting stories that they explored was examining the potential for phage therapy to work in the absence of innate and adaptive limbs of sites and it turns out it could so the synergy is not with innate lymphoid B cells or T cells and obviously the adaptive part may be less surprising but the fact that it wasn't with the innate lymphoid cells is promising and also suggested that the synergy was likely to be with neutrophils and that is in fact what I showed you in that first system when you have these neutral panic or depleted neutrophil depleted mice then yes in fact the addition of phage does not lead to survival and you see the uncontrolled proliferation of bacteria so again as I said pointing to the synergy between neutrophils and phage as being an essential alliance required for effective therapy and I'll make one more point here before wrapping up this section which is that you can also use this as a prophylactic in the sense that in a few experiments they added phages four days in advance and then added the bacteria and still it was effective at preventing infection and fatalities of the mice 100% of these pretreated mice survived and none of those treated with saline and we also expected this to happen the reason why we examine this case is because in certain circumstances there was concern that the turnover or decay of phage would be so fast and in blood treatments that shown to be very fast but they had done the work just looking at the decay which seemed to be on the order of nine to ten hours which meant that a large dose given four days before there was still going to be enough around in terms of the decay of phage for this to be effective as it was and the other interesting point here is that it doesn't seem to have at least the addition of these phage to lead to differential production of cytokines in other words there wasn't an immune response that might directly target the phage on its own right which would also be problematic and that would maybe limit the efficiency of phage therapy at the moment as you you can see I'm thinking about phage and bacteria phage and immune system synergizing with respect to elimination of bacteria and adverse impacts on phage by the immune system could obviously limit that but there's no evidence of significant priming of host immunity so maybe that's a good place to stop as well just in case I have a few few last slides I think I'm mostly on time yes so there are a couple of questions so one is from the Zoom chat from Lehan who is asking the following so biologists have been using vectors to do genetic manipulations is this one of the ways to design phages to target specific bacteria? Right so there's all sorts of different ways if you want to modify bacteria crisper system phage delivery systems and so on in this particular case we're really thinking about phage not as a means to modify you know as a way to deliver something but rather to kill and proliferate so this nothing I'm saying today would prevent interesting work in those directions but really thinking of phage as agents of mortality here of course you could also think of using a phage derived product like a license as an agent of mortality and people are using license also as whether pretreatments or therapeutics but I'm not investigating a particular case of using phages delivery vectors in this step that would be interesting but it's not the scope of this particular work Yes I think he has a very related question which is to get the new sets in action can one use phages to make bacteria express some peptide that are already known as pathogens to immune cells? So I would just say that we're exploring some of those things now I don't have any of those I now understand what the person's going at that has been of interest to us to do something more than just killing in light of this synergy and we don't have any results to share yet but yes that is certainly of interest to us. Yes then there is a question from the YouTube chat so Giulio is asking would it be possible to apply ideas from control theory optimal control stochastic control to this system such as in the immune phage synergy model? Yes and we have a paper out this year in the bulletin of mathematical biology which does that I didn't include that because I had I was going to run out of time and I included a different study that came out in M systems but if that individual is interested you can look at Juanlin Lee at all we've collaborated with Uri Warty who's a control theorist here at Georgia Tech to do precisely that and I'll just make a comment that the challenge always in optimal control is that we have a sense of timing but we all can worry about misbefuscation of the model and so in that paper what we did is try to take the lessons from our optimal control results and use them to guide something that could be done in practice which is the delivery at the outset of what we think is the right dose in combination and we're also looking to try to convert that more into a feedback control case by taking data not just at the outset and projecting forward in time but also taking new measurements and responding appropriately so we're actively working on that as part of an NIH grant that funds the study and you can see our first work in that direction again in Lee at all with myself and Uri Warty and Joy Nung in the bulletin mathematical biology 2020 great if there is not any other question I think you can go to the next oh okay there is one question in the zoom chat by Sreerama is competition and or cooperation play any role in controllability I assume that person is referring to maybe competition or cooperation between bacteria maybe if so I certainly would think that this goes back to my point about using phage to treat multi component infections right multi species infections and then absolutely we need to be thinking about those kinds of interactions here with in a species if we have systems particularly cases like CF or that have a longer time to develop biofilms then if I think of that as a cooperation mechanism within bacteria of the same species then also yes we have to think about ways to address issues of emergent properties of populations that may then lead to recalcitrance or making it different more difficult to treat with phage if phage can't penetrate those biofilms or somehow overcome those collective defenses I only have a few minutes left so maybe I'll go to this fourth part is that okay so let me now go in one direction and again I I didn't add the optimal control work but wanted to get back to this example from Paul Turner and just elaborate a little bit more on it which is again to recall that there are pseudomonas originosa that use these antibiotic efflux pumps so in other words if an antibiotic is present they can be pumped out increasing survival but yet this phage on k-1 uses these efflux pumps in some sense as a surface receptor to inject genetic material into the host of course there could be a mutation which means that some pseudomonas may have lost this efflux pump or at least do not have the surface receptor so that now they can no longer pump out antibiotics but of course phage can't get in so in these two examples and there may be a continuum of expression here there may be at least on the archetype side antibiotic sensitive types which are also phage resistant and antibiotic resistant types which are phage sensitive and this is precisely the rationale behind the dual use of antibiotics and phage together and this is what it was reported in really a pioneering study by Benjamin Chan at all with Paul Turner as the senior author leading the study and describing it here there's a caveat though that concerned us that there may be issues about targeting the wrong strain so for example if there is a phage sensitive inoculum at the beginning then using phage may work whereas if the initial community is largely phage resistant right by antibiotic here in this particular case I haven't had the antibiotics then even all the things I said before about immunophage synergy may no longer work because if it's the bulk of which are resistant bacteria they may rise to such high densities that we don't get the benefits of them being driven down so this is just to point out that once we have sensitive and resistant cells then targeting really matters and we think this also matters with respect to the this antibiotic phage joint treatment so what we did is take our original model and instead of just having phage and immune cells and a sensitive and resistant type we also are going to have including antibiotic and I'll label these two bacteria with A and P meaning A is sensitive to antibiotics and P is sensitive to phage right so now we have a specific kind of mutation which just doesn't make it resistant but now becomes sensitive to the use of antibiotic and we wanted to explore what might happen in these kinds of models one of the things that we realized initially is that a combination therapy might restore efficacy to mistargeted phage therapy so if the inoculum was initially phage sensitive and one adds phage and that's going to work but if it's an antibiotic sensitive inoculum and you use phage then of course you're just going to get the emergence of these BA types however the combination therapy here in both cases there's a background antibiotic applied can lead to elimination in both so our first point here is that we think that in this class of systems clearly this phage antibiotic combination is going to be better irrespective of what the initial breakdown of that population is right whether it tends to be more on the sensitive to bacteria phage side or to antibiotics the other thing that I think is important is it the inclusion of immune responses we think again is an implicit and hidden part of what makes this effective so we went and explored the outcomes of combination therapy with immunity and you can see in both cases we get an elimination but in the absence of an immune response we end up getting again this persistence of phage and bacteria with some background lower levels of these antibiotic sensitive bacteria they're being reduced by the use of antibiotics but the phage and antibiotics even though they're working synergistically can't clear things because phage that intrinsic relation between phage and bacteria means that we end up getting generically coexistence rather than elimination so we tried to put this all together envisioning that we have variation of the inoculum type here all antibiotic sensitive all phage sensitive the level of antibiotic and here is the m ic concentration the minimum inhibitory concentration and what i'm showing you here are bacterial densities only with antibiotics which you end up getting is except in this very corner case essentially the elimination of the antibiotic sensitive type but the persistence of bacteria that are sensitive to phage when you add immune responses so if there's an active immune response there's some regime in which there's clearance but there's still a large regime in which there's not when antibiotics and phage are added together but not having the immune system you can see that in all these cases we don't expect elimination despite the synergy although densities are driven lower there still are persistent levels and when you use a lot of antibiotics then instead of having the antibiotic sensitive bacteria you get largely that phage sensitive bacteria when you don't use much antibiotics below the m ic you get antibiotic sensitive bacteria so in other words you're selecting for the type but not eliminating but notably when all of these players come together we expect a large regime including in the sub inhibitory concentration levels where there's robust elimination of the bacteria and only when you basically don't use much antibiotics at all you end up selecting for these antibiotic sensitive types right so the point here is that the combination of antibiotics and phage is good and it's particularly robust when there is an active immune system so the conclusions here i hope i've explained this tripartite model of phage immune bacteria dynamics it really is an ecosystem aware approach and thinking about phage as part of an ecosystem made also redirect our attention to how to think about the development of therapeutics that in vivo now shows that the curative success is not just dependent on the phage but also on the immune response that a fade-neutrophile alliance in terms of is maybe necessary for therapy and that was revealed with these immunomodulated mice studies and also synergy can help resolve the resistance problems because the immune response can eliminate both susceptible resistant pathogen and we're working now on generalized symmetry to include commensals i didn't talk about that today as well as antibiotics and also keeping in mind that once we are aware of these kinds of feedbacks it may be that trying to get candidate phage we should be thinking about ways in which phage can make it easier for the immune response to work rather than thinking of the efficacy of killing on its own and with that again just want to thank collaborators here pointing out i've talked about this m systems paper here at the end but also there is that other bolden math bio paper which focus on control theory and additional thanks to our experimental collaborators at pastore as well as to the Turner group at Yale for their collaboration on this theory analysis of their fascinating antibiotic and phage evolutionary tradeoff mechanism and with that i'll be happy to take a few last questions great thanks a lot Joshua for this very nice lecture so we have time for questions i don't see any question in the chat either on zoom or youtube if you have any please either write it or raise your hand yes there is a question by Martina please um hi Joshua is always nice to hear you to hear lectures i it's a bit of a bit of a leap i think because there is all this idea that let's say the right quantity of phages might promote bacterial diversity and in this sense we might can so in a in a probably far away future do you think we could use phages to as probiotics phages as probiotics yeah okay and instead of bacteria is probiotic but using say phage in some sense i think i guess it depends on how we're going to think about the term probiotics right meaning do you want them to be residents because maybe if we add phage that are hanging out eliminating the wrong bacteria that could be good but maybe you mean something else like you'd actually like them to become residents which in fact uh like a for the gut microbiome in the sense that you need a diverse community and what so so there's an interesting paper a few years ago and it's a shame that i i'm dropping the name of the first author but the last author is mark young who was published in pns uh on a healthy gut viral which looked at the relationship between in some sense the diversity of phage and outcomes with respect to health finding that they were related and i think there are many questions left to be answered about mechanism is that essentially a reflection of healthy gut microbiome where we see phage of those microbes or is it actually the inclusion of phage that is leading directly to health benefits i don't think we know the answer to that but i think the fact that you're raising it in some talks i've made a schematic i wish i had it here of a yogurt with extra phage because you know that when people will go to the store and pay lots more money for activa and other back let's say called a bacteria infused probiotic yogurt right but right now the market for if you told people i'm put extra viruses in your yogurt i don't think i'd have a good market share maybe that will change with time i don't have that slide here today where i've created my fake campaign but you're raising an important point and maybe with time that's in fact what we'll be doing but i think we still have a ways to go okay thank you great any other question hi i have a question yes please sorry i'm gonna start the video hi thank you very much for the nice lecture my question was about always diversity i think in part you replied before like of course you were talking about like an acute face of the illness so not a big specific diversity of the of the microorganism to be killed by phages or by the immune system but what about like the intraspecific diversity that might be caused by the like the evolution of the microorganism of its own uh different and differential like um uh sorry defense against phages like and that the model then yeah so in some of my earliest work in this space and thank you for the question one of the first problems i worked on also with simon levin is co-evolutionary dynamics arising from precisely the mechanism that you describe in which phage in some sense catalyzed diversity which then leads to catalysts of more phage and so on and we've worked in other cases showing that you can have co-evolutionary dynamics and diversification stimulated by these interactions in the context of therapy that's i mean that's fascinating from a fundamental sense but they may not be the right outcome you want from a therapeutic sense as you can see here already even if there was a pre-existing phage resistant mutant right in the event that there was this immune response that got eliminated so in some sense selection operated because of your addition of phage and led to a proliferation but it could be controlled however in a broader sense there's also something that we've been working on and i'll see if i can while speaking find it for you but there's a new paper on bio archive by justin myers group talking about co-evolutionary phage training and i will it's something that we've been thinking about from a theoretical perspective i don't know if i can write to everyone in the chat uh at once maybe i'll send it to jacobo he can write to everyone in the chat uh with it but basically the idea is that it may be possible to look forward in time finding what kinds of phage might arise because of the emergence of new mutants but then go back and apply those at the beginning to stop the emergence in a therapeutic sense so i think we always have to keep in mind i mean these co-evolutionary systems are so cool and fascinating we want to understand them but in a therapeutic sense they may not be our friend per se so we have to think about maybe ways to leverage and that's one that we've been very interested in exploring great um yes this is a question that is there is a question in the chat that is related to a question that was already asked and agar is wondering whether we can engineer phages so that it will facilitate a better recognition of the bacteria by the new system so the answer to that is we've been working on that for quite some time and in principle uh probably yes in practice which we're working on maybe harder and maybe others are also working in this space but that's clearly one that as you can see implied by my the work here where there may be reasons to think about phage not just as agents of mortality but also as synergistic elements potentially even adjuvants and in that case it does reshape the way that we might want to engineer phage as was pointed out in order to have better ecosystem effects not just killing or lytic activity in the absence of that ecosystem these are all and i would just say since we're almost at time last time just to thank you for everyone in the audience for these questions obviously if people want to follow up i'd be delighted uh and also to point out that this field i've tried to introduce topics in which the story isn't totally written yet that story needs to be written by all of us so i hope that maybe some of you might be interested in what's happening here and work in your own universities um to try to advance these this sort of ecosystem and the eco-evolutionary approaches to understanding viruses and their microbial host so thank you thank you very much Joshua for these very great lectures and it was really great thank you very much for the involvement in this and well thanks again to everyone for attending the school we will meet each other again tomorrow at the usual time i guess uh yes no a little bit later than usually 2 3rd italian time with the first lecture by andreina so thank you again josh and thank you everybody for being here thanks thank you Joshua thanks all thanks again take care thank you sir