 We are one minute in advance to the official starting of today's lectures, 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 that are three dots. And you can find the button raise and and again I'll give you the possibility to talk about the question. If you are following from the from YouTube and 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. My name is a professor of ecology at the Polytechnic of Milan. He has a research spend many topics in the quantity ecology, epidemiology and environmental modeling. And today's getting the first of three lectures on this is called. 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. Let me share my screen. And, okay, I see that now there are 79 participants. Okay, great. Actually, I'm a professor of ecology, I'm married to now because I'm retired, also I'm still teaching. And today, the topics today, sorry, and also the other two lectures, the general topic models of disease ecology. 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 spread of the 19. So first of all as professor ecology, let me remind you that porosity is basic ecological interaction. First of all, sorry, I missed one slide. Now, first of all, okay, today I will go through the basics. And then Monday, December 7. I will deal with macro parasites and in particular with schistosomiasis and 9 December, COVID-19. Part of my lectures are coordinated with Professor Andrea Hernandez 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 with Cambridge University Press. Okay, so today we are going to speak about basics and as I told you before, first of all, as professor ecology must remind you that porosity is a basic ecological interaction. Sometimes people ask me, but why you're a professor of ecology and you're interested in COVID-19 and schistosomiasis, how come? And I must remind them that parasites are everywhere. You have parasites of plants, parasites of animals, non-human animals, any human animals, and so on and so on. So porosity is a basic ecological interaction and that is central in a way to the problem of diseases even to the problem of human diseases. So first of all, let me tell you that actually, porosity is it's very important even for non-human, or you can say that in many cases many population, even wildlife population are regulated by porosity. This is a very famous example. Many years ago, my good friend Pete Hudson and then the 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 when the population 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 forth. So there is a large evidence that porosity is regulating population. So 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 and mazawa and these domestic cattle were bringing Rinderpeth and this is the result. Currently, Robert Koch in 1897 found a vaccine so they started vaccinating and so they can eradicate Rinderpeth from Southern Africa, but then only recently they can eradicate Rinderpeth everywhere. And the Rinderpeth were 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. 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 they're not very much different from 2018 and 2019, the infectious diseases were about eight millions. Now to give you an idea of the importance of how hard the time is, I mean the time in which we are living is that COVID-19 has already claimed in less than one year 1.49 billion deaths. 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 not found because they do not test that. Okay. Now, again, being an ecologist, I'm also interested in understanding, oh, sorry, what did they do? What is that? I understand there is a, do you see that red scratch on my screen? Yes, I think you use the pen on the screen. I didn't have to use the pen. Okay, I'm sorry. Okay, so if we go to another problem for the ecologists, and that is the fact that now globalization in a way and the land use change and climate change, impacting on everything, but in particular they're also impacting on diseases. So for sure, habit of change, and 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 seeds, 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 a habitat which was not good for vectors now in countries that actually were not favorable to vectors, but 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. There's a 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 it is Professor Posi or being here in Italian originally, I mean, Fauci we would say, and he's the counselor for COVID-19, and he was already pointing out that there are a lot of emerging and reemerging infectious diseases around the world, and many of them, 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. 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 I would advise not only humans who are susceptible but also civets for instance. Oh, I'm sorry, whenever I'm using my mouse to point out something with red scratch appear. So and then the bird flu, for instance, waterfall and Ebola, again, various bats and so on and so on. Okay, Jacob, 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 now you see. If there is annotate, perhaps you have, if you go to view option on zoom on top, you should probably see annotate. Yes. But I didn't use annotate. Let me stop annotate maybe. Mouse. Okay. Okay, now it shouldn't. I can move the mouse. I cannot get rid of that scratch unfortunately. Okay. I'm sorry. Okay, thank you very much. Sorry for this. Okay, now let's go deeper into into the problem and also into the problem of modeling. Okay. First of all, a big distinction, which was made many, many, many years ago, basically by Anderson and me, both me and Roy Anderson. In terms of modeling is the difference between micro and macro parasite. Micro parasites are typically viruses and bacteria. And they have a short lifetime with respect to the lifetime of host of their host. And the way you can neglect the dynamics of the, the macro 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 distinguishes acceptable. In fact, they are so on we will see that. So you can see the parasites instead you see an example there, they have a lifetime which is comparable to the lifetime of their course. So you can include, and you must actually include the dynamics of the of the parasites. So micro parasite would be the object of my next lecture, where we will see the model of just as my is. Okay, now if you consider now micro parasitic diseases, first of all, we should distinguish because you must use different models, the transmission pathways of micro parasitic disease. So first of all, you have direct airborne or sexual transmission that typically the cold and measles SARS COVID-19 influence and so on. And then a vector born diseases, the diseases need a vector without that vector, the disease will not be there. And then typically your malaria or dengue Zika. And in many cases, they are transmitted by mosquitoes or by flies and so on. And then there are other more diseases. This, in this case, propagules are transmitted via contaminated water so you bring contaminated water for instance to get cholera, or rotavirus. Okay. Other diseases that we will not treat environmental diseases, you mean that the propagules can stay in the environment for the long time so you can get infected by contacting those propagules with very long, long time so typically anthrax for instance or tetanus that. And then sometimes we also have vertical, vertical transmission from mother to their progeny. So for instance HIV can be transmitted from the mother to children or hepatitis B and C. We were mainly daily with direct vector born and water born diseases, I will not talk about environmental diseases and I will never consider vertical transmission. Just here to let you understand instead the life cycle of macro parasites and then we will leave the topic to the next lecture. This is a very simple life cycle, usually macro parasites, the adult stage of the adult reproducing stage of the macro parasite is inside the host. So look at that pig on the left, it is ingesting eggs or the macro parasite that eggs will develop, we go to the long and then we go to the guts and then the adult 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-infect. Another cycle that's important because for instance schistosomias here, which I am going to speak about is that kind requires a second host. So typically for instance in fasciolopsiasis, the 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 mericidium, but that mericidium must have these nails in order to stay inside the snail and create the other stage, the circaria stage and then the circaria 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, river blindness and so on. Other than 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, there are 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 point 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 Ronaldo in his third lecture. Okay. And 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. Because this was very basic and probably many of you already know the topics. I think we can move forward then. 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 Ronaldo. That into a simplified network, a graph, where you have nodes and where you have arcs and arcs, 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. You might have 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. For instance, in cholera, which is going to be described by Professor Ronaldo, but also in stochastic matrices, what 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 approximation, locating in the cities or the villages, and the same node where you have the nodes for the hydrologic transfer for the hydrologic network. In other cases, like this example, this is the model schistosomalis 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 matrix by using mobile phone, but then the antennas are not located only in villages, for instance, and then you have water points. So in water points is where the people can get infected. So in that case, you have multiplex network. Okay. In that case, in each node, you need a local model. Okay, suppose that now all the connections are cut, you got all the connection, 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 in the network. 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. If 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 infecting exposed infectious recovery and so on. So micro parasitic models, we 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 use real numbers. You 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. 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 with now with the real real models. I will start with micro parasitic models with direct transmission. And the simplest model is the one where you can see just the susceptible people, people who are not infected by might be getting infected. Oh, sorry. And then you have another compartment which is the compartment of infected and infectious people. 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. 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 these are these arrow is not there. 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. So you have 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 certainly part of 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 S-I model because in Anarchy, it is a very good example to start with. The basic S-I 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. And you have a disease in a bed 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. You have a certain infection rate and so susceptible people can get infected. 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 with that you, they make a test and they know that you have malaria per week per week that's the incidence prevalence is instead of fraction. It is the ratio between the infected people and the total number of people susceptible plus infected. A big and important distinction is in terms of infection rate. Now, what is the infection rate is the probability per unit time that one susceptible gets infected. Now, 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. Now, 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, an infected guy, then you might not get the disease. So you must multiply by a probability of becoming infected and infectious. So we would suppose that it is constant, but in reality, in reality, it depends on the behavior. So for instance, if you wear a mask, and then you will not get COVID-19. Okay. But it is very difficult, especially if you wear an FP2 mask. So, but anyway, let me suppose that it is a constant and so it is 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, 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 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 a sexual intercourse. Okay, you are alone. So, even when you consider sexually transmitted diseases, the contact rate must go to zero when it goes to zero. And on the other hand, even with airborne transmission, even if you go in, say, in a very crowded underground, you're anyway surrounded by no more, say, the 10 people. Even airborne transmitted disease can actually saturate to a maximum rate. Okay. Now, let me start now with density dependent transmission and what I got my confusion grow. So let me consider it. A simple case in which I is proportional, the infection rate is proportional to capitalize the number of infected. So you have a very simple multiplicative term. I'm sorry. When I move the mouse sometimes, you know, switch from one slide to another slide. If 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 that this population, if there is no infection would actually grow in an exponential way. If you now introduce the possibility of infection, what comes out that if you study now these no linear equations and I think that you have a tutorial on the linear analysis. So these are the, the, you see the eyes of clients. Okay. And you see that now there is a known preview or equilibrium. This one. And therefore, the main message is that disease can regulate the population a population that will grow exponentially does not grow actually exponentially. The disease is introduced. Now another result was 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. And diseases that are very lesser, a very low prevalence, fortunately, so, so for you to consider evil evidence is very, very less of disease prevalence is low, fortunately. Okay, that's not being that of course you can go into a more realistic model. It's one where the susceptible cannot grow in indefinitely if there is no disease that cannot grow exponentially. I hope that you're familiar with the logistic model. It is a model where the population would grow out what we call a carrying capacity. Okay. 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 previous equilibrium. No susceptible no infection. Okay, population, not there are no infection population goes to the carrying capacity so that second equilibrium are 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-previous equilibrium. However, that's very important message. If you go to the expression of that non-trivial equilibrium, you'll find out that although these eyes of client will anyway intercept the red curve which of course extends also to negative numbers, negative numbers do not make any sense. 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 mu plus alpha plus gamma divided by beta. You see that for instance, suppose that we increase we consider a disease with a larger alpha into you and you have the same carrying capacity same population that another disease. So these eyes of client will move. And now you have a situation where there is no intersection. Now, this is an example. I don't know if it was a tutorial on the analysis of a transcritical bifurcation. Now, in that case, the 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 this way by introducing what what 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 carrying capacity. Now in demography, it is the average number of daughters produced by mother in the course of its lifetime. So here you have mother infections 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 infectious for such a time and beta K, now this susceptible to carrying capacity, so it is a healthy population. Beta times K is the number of susceptible infected per unit time in a disease free population by one infectious individual, but that guy would stay infectious for this time. And this is exactly the average number of secondary infection caused by one primary infection and if he's larger than one, then the disease can increase. Otherwise, the disease cannot increase. So the disease cannot become endemic. So if are not more than one, the disease free equilibrium stable, you have a transfer by application and the disease cannot become endemic. That's the question now, because that's a very important concept. Any question on. There are not 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 so free to support now you can see a more realistic contact rate. Well, it is very simple instead of beta times K, since the contact rate now is hand divided by Delta plus and you simply have beta K divided by Delta plus K multiplied by the residence time. Okay. That's not an interesting thing. Suppose you have what we call a frequency dependent disease for instance, or syphilis, and you don't get syphilis by going to the underground. So now this delta is really very small that they close to zero. So, if it is close to zero, you see that. Okay, now, delta cancels and your beta K divided by K, you end up with the another not which is approximately beta divided by new plus alpha plus gamma, but it does not depend on the carrying capacity. And a basic message in this case and even in the previous case with density dependent that are not is larger if K is larger. It is easier to get any airborne transmission in New York than in a day. In the wildlife of China. Okay, or clearly. Okay. 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. The times one remains in infection so bring them for AIDS, that you cannot age by going around. 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, you would visit with recovery are not the expression. So you replace this gamma, which is the rate at which you become susceptible game with actually raw the rate at which you recover and go into the recover compartment. Now an important message, however, for the remainder of the force and of the lecture, that it is possible to generalize these machinery for network models. It is not just to ordinary differential equation three ordinary differential equation and so on, but you have a system for instance ordinary differential equations, where each local model is connected to another local model by an accurate possibly a multinational multi multiplex model, but you can still use no linear analysis bifurcation theory and the important concepts are dominant eigenvalues or the spectral radius of appropriate matrices. And in this way, as possible to introduce generalized reproduction numbers. So for instance, the general reproduction numbers that with my, which might be the spectral radius of an appropriate matrix. Okay. So to decide whether the disease can become endemic or cannot become an endemic based on that analysis, but the that say that the basic idea and 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. Since we are switching here. If you wonder how to question on our note. Oh, great. So there is one question by so read in the zoom chat asking is not time independent. Okay, now. That's very good. A good question. Okay, now by our not please note the zero and not. The basic, the basic, okay, basic reproduction number. So it means that you can see the healthy population. Nobody cares about that disease. Then the infection appears. And, and there is no treatment. People are not wearing masks and so on and so on and so on. Okay. And then you must decide whether that disease will actually become endemic or not become endemic, possibly spread. Okay, so that the basic reproduction number. And now I know that it is very popular another reproduction number which is called our tea tea. So our tea is the average number of secular infection caused by one primary infections at time T, when the disease that had already spread into the population. But now are people are being treated are 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. In the basic reproduction number, the prevalence of the disease is initially one. Infected well, if you're exceptional, they want infection, you're infection, you can approximate. Okay, so then beta might be time dependent. Today, which is the number of susceptible people at the very beginning might be replaced by the number of the density of prevalence of susceptible people after a while. Also, now, so consider also recovery. 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 our tea so the number of secondary infection. Not at the very beginning that the basic reproduction number, but at time T. Okay. I answered the question. No, there's second question. No. Sorry, there is actually a question from YouTube. We booty is asking a broader question how, can you explain how 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. Oh, I mean, okay. Well, okay, there's a problem with stochasticity. That's true. Okay, that's very, very good. Okay, okay. Yeah. Oh, this is everything is based on a ordinary differential equation model where these as and these are treated like real numbers, but in reality, if you consider for instance, initial initial infection. 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 infectious here, we are using an approximation. In mathematical terms, we should say epsilon with epsilon is more. 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 curriculum, number of infected people's curriculum. So that's a real number. So 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 these kind of are not. If the one is the threshold for or not for the disease to become endemic, meaning that the only stable equilibrium in the long run will be the disease-free equilibrium. So even if, oh, I'm sorry. Oh, I'm sorry. So suppose that now we are at carrying capacity. And then you put a small number of, sorry, a minimum state, because they should not point here. They should point to kept it up. OK. I'm sorry. I think I made a mistake in creating this graph. OK. 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. Not this graph. In reality, no, I'm sorry. There's not this one. OK. It should be OK. OK. I'm sorry. That was a mistake. No, not this one. OK. I think you understand now. So in that case, even if you introduce a few infected people, then the population will go to the disease-free equilibrium. That is the sense. OK. So true, there might be stochasticity. And second, of course, you have some infection initially, even if R0 is more than 1, but that infection will die out. OK. Even in the deterministic model. I hope I answered the question. OK. Can I go on for two more questions? But perhaps it's better to ask them at the end. OK. 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 Pacinini in 1954. And you see that here, the basic eco-pidemiological 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 disease model is one where now you simplify the logistic growth by thinking of a demography which is close to carrying capacity age, where you can linearize away the logistic model around the carrying capacity. Let me go to 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. OK. And then that's supposed 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 1 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 current capacity is larger than a certain amount. Now, you can transform that inequality into the usual expression for R0. Why? Now, you see, number of these beta times age 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. OK. Sorry. Beta times s is per unit time per bacterium. So they must introduce the mean residence time of bacteria and water, 1 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 1 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 1, the disease can establish. If it is more than 1, the disease cannot establish. OK. I will make my slides available. I had also a slide for vector war diseases. But my time is over. So better to stop here. And I'll stop sharing and ask if there are some final questions. Yes, let's say a couple of questions. So 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. OK. They can see. I start sharing the screen a bit again. Let me go. So, well, even here, even if you can see the basic waterborne disease model, raw is the recovery rate. So the larger the recovery rate and the smaller the residence time away. OK. So if you have a smaller residence time, of course, R naught goes down. Well, so for instance, if you can see your 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 of the infectious compartment is larger. The same if you go, OK, to, for instance, directly transmitted diseases, you see that again, if you consider a logistic model with the recovery compartment now in evidence, you see that again, you have 1 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 Kavi. So if you identify people, you isolate them. Oh, well, even if they do not need to be hospitalized, then you isolate them. So they are infected, but they are not infectious. OK. So, OK, that is true. There are other questions? Yes. Another one. There was one, again, on the previous part, when you were talking about the transcritical bifurcation as it is with R0. Yeah. Is it possible for your model to have other bifurcations like acetylnode bifurcation where they are 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 acetylnode bifurcation. But you can have a more general acetylnode bifurcation. You can have hop bifurcation. You can have a subcritical, a supercritical hop bifurcation. So for instance, with models of schistosomiasis, that is possible. And you can also have, at least in the possible instance, how to get corrected. So you might have phytonbone cascade and so on. That is possible. But these are the very basic model. And you don't have that kind of bifurcation. So in fact, whenever people speak about R0, RT, RT is smaller than 1, OK, OK, OK. 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, yes, that's true. That is possible. Yes, thanks a lot. You're welcome. Great. So is there any other question either here on YouTube? OK, 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. OK, so thank you all. Thanks a lot. Thanks a lot. And so see you on Monday. Thank you very much. And stay unexposed. 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 on the screen. Thanks.