 Great, I think that in a few seconds, people should join back to the main room. And we can start again. So just the usual announcements, if you just joined us. So if you are following from YouTube, you can ask questions in the chat. I'm sure you have heard this many times. And if you are on Zoom, you can either use the raise and button or ask a question in the chat. And I read it for you. So I think that everyone is back in the main meeting room. So please, Marino, if you can share your screen, you can start. Thank you very much. OK, good afternoon to everybody. So let me share my screen and then optimize screen view. OK, here it is. OK, so this is my third lectures. OK, and the topic I'm going to speak today is unfortunately fashionable, and that is COVID-19. Again, I stress that in a way, my lectures and Professor Andrea Rinaldo's lectures are coordinated in a way. But COVID-19, however, is not included in the book that Andrea Rinaldo has been advertising, and I have also been advertising. OK, so very briefly the summary. First, I will talk about the context and the main epidemiological characteristics of COVID-19. Then I will talk about our model that we developed for Italy, which is a spatially explicit model. And then I will illustrate the model results up to the end of March 2020, which was the first month, well, for one month and a half in Italy. And then I will show what might happen after the lockdown, might have happened after the lockdown. And I will talk about the scenarios and the possible containment measures and then a few conclusions. Now, this is something I have already shown to you. I want to stress again that if you look at the statistics of communicable diseases, infectious diseases around the world, you see that the share of infectious diseases has been shrinking in a way along the years. Of course, the number of deaths has been increasing because the population of the world has been increasing. But if you look at the share, the red share of communicable and also maternal and maternal nutritional diseases, the share has been shrinking. And so around 2017 and probably the same for 2019, about 8 million. And now if you look at the statistics of COVID-19, it is about 1.55 million deaths. You know that it is possible to link to Johns Hopkins University site and day by day, you see the said statistic of global cases, which is probably a large underestimation of the real global cases. Global death is also possible, possibly an underestimation, but not such a large underestimation at the global cases. And so we are now to a figure of more than one point 5 million deaths, which is really, really a very large share, which means that COVID-19 is really a very important disease. In a way, it is not like other infectious diseases that might ravage the globe every year like influenza. It is much more important. Now, let me stress that what I am going to present is actually a teamwork, a team of people with whom I've been working for a long time and who are located in different universities and in different places. Although I must admit that they share one common feature. They are Italian anyway. Now, let me connect to the lecture by Professor Rinaldi on the next Thursday 11th, if I'm not wrong. To say that we have a lot of experience with spatial temporal dynamics. And that is something that Professor Rinaldi already stressed in the past lecture. And so for instance, he will show the progress of cholera and IET and you see data on the left and model on the right. Just to let you understand that there is a spatial signature in many, many, many cases. That is something that you should account for. And that is common to many diseases. For instance, the Spanish flu, probably you may remember that the Spanish flu was probably the largest pandemic, I mean, with the exception of COVID-19. I mean, if we go back in time, the Spanish flu claimed more lives than the First World War. And actually, my grandfather was detained in a concentration camp in Austria in Kathnau because he was an Italian citizen. And unfortunately, he was in the Austrian-Hungarian Empire in 1915. So he was sent a concentration camp. But he didn't die there because that's a lack of food or anything. He died of Spanish flu. And if you look, for instance, at the spread of Spanish flu in the United States, you see how fast it was. So it started in the Eastern coast and then very rapidly in a few weeks it spread to the whole United States. Now, let me describe the characteristics of the COVID-19. First of all, the pathogen of COVID-19 is SARS-CoV-2. So why two? Because it is the same family that actually caused the SARS. So it is an RNA virus. It has a crown-like appearance. It's actually due to spikes on the surface. So corona in Latin and in Italian means crown. So it's a beta coronavirus like SARS-CoV-1. And again, to stress the ecological importance of what I am going to say, it is a zoonosis. And the hosts are several species of bats and rodents and particularly, probably, pangolin, maybe other rodents. And what are the main characteristics of the disease? Now, first of all, the main way the virus spreads is by respiratory droplets. Among people who are in close contact with each other. Now, let me show something. Now, this is an example of the aerosol emission when you breathe. Now, in this case, this is a study which was conducted together with a very famous orchestra. I'm very fond of classical music. Sorry for that. So it is the symphony orchestra. There's Bayerian runes. And here, you see a way for visualizing the droplets and the aerosol that one can emit when singing or playing an instrument. Now, more or less, you can get the virus via contaminated surfaces. But it is possible. So I should be careful about that. Now, another main characteristic that we know now. Now, let me provide a few technical terms, which are shown here. So if you consider one individual who gets infected, he, at first, is not infectious. This is called the latent period. You may remember when we described micro parasitic models that we were also talking about susceptible exposed infected recover, exposed, latent or exposed. This is the same term, which means the same thing that you are infected but not infectious. Then at a certain point, you will become infectious. But that does not mean that you show symptoms. Symptoms, actually, in general, show after a while. Now, this period is called incubation. And then you become, sorry, then you become symptomatic. Now, of course, when you're infectious, you can infect someone else. And again, the infectee will have a latent time. Incubate the disease, become symptomatic, necessarily become infectious. Now, we call generation time the interval of time between the moment when the infected was infected and the moment when the infectee was infected. We call serial interval instead the time between when the infectee became infectious and when the infected becomes, sorry, the time when the infectee gets symptomatic and the time when the infected gets symptomatic. This is called the serial interval. Now, the generation time, the average generation time, which under some independence hypothesis is equal to the serial interval, is about six, seven days. So it means that the time scale for the infection is the order of one week. That is important. So from one infecting and one infected, in the average, there's a week interval, one week interval, six days, six, seven days. As probably all of you know, the asymptomatic fraction is quite high, larger than 50%. It may vary between countries. You probably know that it is more likely that old people show symptoms than young people. So the asymptomatic fraction might be higher in countries where there are a lot of young people and, let's say, smaller in countries like Italy with a lot of old people. Now, one important message is that the maximum infectiousness occurs during the pre-symptom transmission. So that little time when you end to be latent and you are already infectious, but you've not developed symptoms, that's clear for many studies. So it is about five days after being infected. Then what happens? That a pre-symptomatic, actually, it should be that we should call this period post-latent. Because a pre-symptomatic, some of them will never develop symptoms. So we'll remain asymptomatic, while some of them will become symptomatic. That's important for what I'm going to say later. Now, another thing is the mortality. Mortality is about 1%, 2%. Of course, it depends upon condition, upon the sensory system, upon the fact that there are many old people and, of course, the old people have a higher mortality and so on and so on. But we know now that, more or less, it's about 1%, 2%. And that might be compared, for instance, to mortality from influenza. It is something like 10 times larger. So there are two combinations that make COVID-19 in a way the perfect epidemic. The epidemic that, unfortunately, all the epidemiologists and these ecologists were waiting for. A large fraction of symptomatic people and the mortality, which is not so low as to make ourselves not very worried, let's say. But on the other hand, it's not as high, like, for instance, able to deplete the infected reservoir, the infected and infectious reservoir. Because you may remember that when I introduced even the simple SI model, I was stressing that the fraction of that the prevalence of infected people and also the R0 were, in a way, decreasing with the mortality rate of the disease. So it's sort of an intermediate mortality, let's say. And a large fraction of asymptomatic make this pandemic so aggressive. And then, again, there is a patient's signature and this disease at the global level. So I'm going to show you this map and this video that shows how it all started, let's say, in China and then it reached France and in Germany and then Italy and then the whole Europe and then, of course, started spreading to the Philippines and Korea and then it reached the United States and then it reached Canada, South America, Africa, Australia, and so on and so on. So again, it is clear that the reason is very large, important part played by space, by how the disease develops in time and space. And therefore, it is very important to take into account space. And in fact, that was, in a way, hit our interest because we had been studying many diseases in space and time. And when the disease reached Italy, and that was at the beginning of January, it is now clear that it was January, maybe even December. But that was not clear until the end of February. And if you look again at the development, the spatial temporal development of the Italian epidemic, this is March, March 15, March 16, March 17, and so on and so on. So there is a clear spatial signature of this epidemic. And that's why we decided to work as soon as we can very rapidly on building a model for the spatial temporal spread of COVID-19 in Italy. And that actually was then materialized in this paper, which appeared at the end of April, maybe on the Procedure of National Academy of Sciences. And then I'm going to illustrate to you. Now, first of all, the epidemiological compartments. Now, the subscript I indicates the nodes, the nodes of the network that we're going to consider. And then we have the compartments of susceptible, the compartment of exposed. But then we had to introduce, with respect to the usual single compartment of infectious, three compartments, because we have the pre-syntomatic infectious, the symptomatic infectious, and the asymptomatic or mildly symptomatic infectious in each location, I. And then, of course, if you are infectious and symptomatic, you might be hospitalized. Or if you're not too symptomatic, let's say you might be quarantined, or maybe you might die without even being hospitalized, or you are hospitalized and you might die or you might recover. And if you're asymptomatic, usually you recover. OK, so that the basic, let's say, engine of our model will hold these compartments. And here are the equations. And basically, again, the core of this local model in each node is actually the force of infection. Now, first of all, we can assume frequency dependent contact rates. Now, it should be clear that in an unpopulated area, then you should assume density dependent contact rate. But that's not the case of Italy. Wherever you are in Italy, basically you are in a populated area. And therefore, it is reasonable to think that every day you cannot have more than a certain number of contacts. So usually people have about, say, 10, 12 close contacts every day. So you can assume frequency dependent contact rates. Now, so at the denominator, you have the sum of susceptible, exposed, et cetera, et cetera. So the total number of individuals. And then at the numerator, you have the people who can infect. So the pre-syntomatic, the symptomatic infection, and the asymptomatic infections. And they might have a different transmission rate. Different transmission rate. Because for instance, we were expecting from other studies that the transmission rate of pre-syntomatic might be higher than those of asymptomatic or symptomatic infections. So that's really the core. Now, if you go to the spatially explicit model, now you have the local model at each node, say the province of Milan, where I live, or the province of Padua, where Professor Ronaldo lives. But then these nodes are connected. Of course, they are connected by mobility. There is mobility within each node. And there is mobility connecting the different nodes. So now the first of the infection when you go to the spatially explicit model is much more complicated. And in a way, you have to now introduce those mobility matrices I was talking about when I showed to you the model of schistosomiasis in the Senegal. And so the probability that individuals who are susceptible, pre-syntomatic, et cetera, et cetera, move from side i to side j and contact individual people from side i who usually live inside i will contact individual inside j, and so on and so on. We even considered the probability that one individual living inside i and one individual living inside j will actually meet inside k. And one is infected, and the other one is not infected. And so you have a close contact. And the one who is not infected becomes infected. And then, of course, we have again this transmission rate, which depends on the stage, pre-syntomatic or asymptomatic infections or asymptomatic infections. And in principle, might also depend on the site because you might think that there are different behaviors in different sites, and so on and so on. So this is the general structure of the spatially explicit model. Now, for this spatially explicit model, you can calculate that basic index, which is very much utilized, and that I have introduced to you for very simple model, which is the generalized reproduction number, which is, however, called now generalized because it is generalized to a model with a network. So again, let's consider the initial phase with no containment in force. So we can calculate the basic reproduction number. And to do that, we can, as usual, start from the disease-free equilibrium and the spatial model, the no infection anywhere in all over Italy. And then we introduce a little bit of infection. Initially, the susceptible prevalence is 1. And then again, at r0 equal to 1, we have a transcritical bifurcation. So you can run a bifurcation analysis on the model. And so you can find a transcritical bifurcation, which occurs at r0 equal to 1. Another equivalent way is to use the next generation matrix, which was introduced by Dickman, Esther Beck, and Matt. And one can show that r0 is the spectral radius of the next generation matrix. And in practice, this next generation matrix is actually built from the connection of the matrix CS, representing the contact probabilities. And then you can recognize some specific times. 1 divided delta p is the residence time in the pre-syntomatic compartment. 1 divided by eta plus gamma a plus alpha is the residence in the symptomatic infectious. 1 divided by gamma a is the residence time in the asymptomatic infectious. In this case, because you can get infected from pre-syntomatic or from asymptomatic, actually you have to sum these matrices, find the spectral radius of the sum of those matrices, and that will provide the generalized reproduction number. Actually, when you calculate, when you do the stability condition of the disease-free equilibrium and you go through the bifurcation analysis, you are considering the Jacobian of that complicated system at the disease-free equilibrium. And the dominant eigenvalue of the Jacobian is the initial exponential increase rate. On the other hand, elements of the next generation matrix describe the main roots of infection. And the dominant eigenvector of the Jacobian, which is the unstable manifold of the disease-free equilibrium, provides the initial geographic distribution of infections. And in fact, this is the way that we got this beautiful picture, which is courtesied by two of our co-authors, Enrico Bertuto and Stefano Micholi, which represent the main pathways, the main roots of infection in Italy at the beginning of the infection. And you can see Milano, Curin, Rome here. OK. Now, of course, we have to calibrate the model. And here you see a logarithmic scale, semi-logarithmic scale, days. And the first patient in Codogno in February 19, and then the development of the disease up to the end of March. And these are the data that we've been using to calibrate the model. Now, the parameter estimation procedure, I want to devote, say, one or two minutes to that. First of all, when you have to calibrate the model, which is quite complicated, quite complex, there is always a trade-off between proximity on one hand and realism. So you cannot use too many parameters or too few parameters. So first of all, we add mobility from an ancient census at the municipal level. But we upscale mobility to the second administrative level, 107 provinces and metropolitan areas. The epidemiological parameters are not space dependent. In the first paper that we wrote, the transmission parameters, beta, pre-syntomatic, beta, infectious, symptomatic, and beta asymptomatic, are not space dependent. And then, of course, we had to take into account that there were containment measures. So we should expect that these transmission rates would decrease. So we consider a sharp decrease within two days after the measurements announced on February 24 and March 8. And so we consider also the step reduction. Then, OK, we add red areas and go into that. Of course, there was a reduced fraction of traveling people. How could we account for that? Through mobile applications, data that were collected by some colleagues, from voluntary mobile data collection. And then there's an important thing, true spread of the disease from some initial foci. But there was not only one initial focus, although it was clear that the main infection foci were located in the northern part of Italy. But, however, there were other foci. So we had to estimate also, in a way, the initial condition in the different in each province and also the starting time of the epidemic. Also, we made another decision. The number of cases is not reliable. What is more reliable in terms of statistics is the number of hospitalized people. Unfortunately, the number of deaths and the number of patients discharged from the hospitals. So we use the Bayesian framework. We gave priors for the parameters. We sample the posterior parameter distribution via the Markov chain Monte Carlo algorithm. And, OK, technically, we use the likelihood according to negative binomial distribution. And, OK, sorry. And here are the results that I show for the whole Italy. These are the hospitalized people. These are the number of deaths, but also for the hardest regions. Now, it should be clear that these results are shown with reference to regions for convenience only, not because regions are isolated from one another. Because that's a common problem with many other problems, that they focus on the date of that region as if that region were disconnected from other regions. And the same is true for countries at the global level. But, OK, that's another thing that we should discuss. And here you see a pictorial representation of the spread. These are the data. These are the calibrated model at the level of the second administrative level. And here is a fancy projection made of my simulation at the municipal level. But of course, OK, it is just, in a way, a fancy simulation. Now, one important thing of having a model that it is possible to make retrospective scenario. Because one question that we asked, were the containment measures effective? And the answer is yes. And you can estimate how effective they were by using the model. Because suppose, for instance, that no restriction had been taken. Without any restriction, so no change of the transmission rates, no change in the mobility, and of course, no change in the people's behavior, that would have been the number of, no, sorry, sorry. Let me first describe hospitalized cases. That would have been the number of hospitalized cases. And then you can consider another scenario, February restriction, but no March restriction. And that is the second scenario. Now, this is the reality. So we had about 40,000 hospitalized people, a huge number, sufficient enough to send all our hospitals into big problems. But the number of averted hospitalized cases were about 180,000. So we would have had so much more without any containment measures. Another thing that you can estimate is the people who were infected. Because the people that you measured, the number of cases, the number of cases that you saw in the Johns Hopkins site, of course, these are the cases that are discovered. People taking a swab and the swab being positive. Or, let's say, they're using this antigen, antigen test. OK. Oh, I'm sorry. This is happening again. OK, thank you. OK. So you can estimate how many people were infected and possibly infectious at the end of March, whether they were about 700,000, 10 times more than the official number of cases. But if the containment measures had not been taken, then the accumulated number of infected cases would have probably run to 6 million cases. So you see now that a model is really useful in this way, because you can also estimate variables that are not measured directly. So main epidemiological result. We calculated the basic reproduction number. It is about 3.60, well, very similar to 3. Let's say 3.22.8, something like that. That is a common number all over the work. We add a confirmation that the presymptomatic are extremely infectious because we estimated that the beta, the transmission range of the presymptomatic, and in order to fit the Italian data, that beta had to be much larger than the transmission rate of the asymptomatic or the asymptomatic infectious. However, there was a large negative correlation between the fraction of asymptomatic cases and their transmission rate. OK. So that's a problem. Containment measures had reduced the transmission rate by 45%, not enough at that time to make the reproduction numbers smaller than 1. That will be achieved later. And again, the model allowed the estimation of inapparent infections and prevalence of susceptible, prevalence of infected. I'm sorry. Then after that, we went on. Italy came out of the lockdown on May 3. So we started thinking, what is going to happen after May 3? And of course, everywhere in Italy, too, there was a concern about the economic consequences of enforcing a lockdown. And of course, the lockdown is a mess in the social and the economic terms. So we wanted to understand what might happen after the end of lockdown. Now, of course, the first thing we had to do, we had to recalibrate the model between March 25 and May 3, the end of the lockdown, which we did. And so we updated the model. And here, again, you see at the end of, sorry, at the end of every beginning of May, the data, the cumulative hospitalization and the model. And then we also calculated the transmission reduction and transmission down to, say, 0.30, 0.4, depending on the regions and provinces. OK. And in fact, you can see here, I'll call this again. I think you should stop annotating somewhere. Annotate. Let me stop annotating. Where is it? I think if you have the annotation, you can do clear. And it should clear. Because I don't see my mouse now. There's something wrong. This light looks nice, though. So there is no sign or... OK. Now, in fact, I can now move my mouse to annotate. Let me go on anyway. So here you see the calibration. And then we try to have future scenarios. So first of all, we showed that there was a further decrease of transmission after March 25, the transmission rate, fortunately. And then we tried to estimate some scenarios. So the blue line is the baseline scenario. Suppose that the transmission rate after the end of lockdown stays the same as it was during the lockdown. Or green and purple represent scenarios with transmission rates that increase by 20% and 40%. Now, why should they increase? Well, because, of course, mobility increases the end of the lockdown. On the other hand, mobility increase might be effectively mitigated by the personal protective equipment. Now, people are free to go around, but they wear masks and so on and so on. And so let me show how effective that might be. So your personal protective equipment. Oh, I can show you. Now, the barrier. The researchers say that two and a half meters distance to the neighbor to the front and one and a half meters to the side is best with a plexiglass washer in between and constantly blowing. Masks also provide security. These gates, there's a vertical mark and a barrier. OK. Then, of course, you might consider social and physical distancing. Then, again, you might implement, again, a lockdown. That's more on a large spatial scale. And then one important containment measure, of course, testing, tracing, quarantine, hospitalization, isolation of people. And then, of course, you might have a combined implementation of all of these containment measures. So we sent the paper. We had reviews. And of course, the reviewers, first of all, they say, well, you all know that review takes time. And first of all, they say, oh, it might be nice if you could update the model. So to answer the reviewer, we updated the model up to June 15. And from here, you can see that Italy was actually quite, let me say, virtues that, in a way, Italy, although the lockdown had been relieved, were following the blue line, so the baseline scenario, as if the transmission rate remained the same as they were during the lockdown, while with the exception, maybe, of Lombardy, the highest populated region in Italy, where, by the way, they happened to live. We also ran a sensitivity analysis to understand whether the fraction of asymptomatics, which is very unclear, in a way, was, say, 10%, 25%, 50%, sorry, the fraction of symptomatic, in this case, 10%, 25%, and 50%, that is a famous study in Boyle, Ghana, blue. And it turned out that 25% is the most likely scenario, because that was confirmed by the estimated zero prevalence of the infected statistic, which was available in Italy on July 15. And for instance, it came out that Lombardy 7.5% July 15 had been infected with, more or less, support the green line, the 25% that we were considering as a possible scenario. Of course, another containment measure that you can do is testing and tracing on a large scale. And again, what we did, we estimated, for instance, the daily percentage of exposed and pre-syntomatic people that should be isolated. Now, it's very difficult to isolate exposed, of course, because in many cases, when you were exposed, the swab test is not yet positive. So it's probably more realistic to consider the infection generated by symptomatic cases and trying to trace, to test the symptomatic case, then look at the context of the symptomatic cases, positive symptomatic cases, and then look at the context of the context, and so on. OK, just to finish, one of the reviewers asked, are there possible scenarios after June 15? And OK, so what about the problem? Suppose that there is immunity loss. So within scenarios, when at September 1, the transmission rate were back to the initial transmission rate, March, and without immunity loss. I would say that it is now clear that immunity lost, at least for a few months. Not clear yet that there's not enough evidence, and hopefully we can hope that it is long. So these were the scenarios that we ran, let's say, without immunity loss, the one you see here, for Italy, for Lombardy, and this is what is going on. We were assuming that there would have been a new lockdown on October 1. Instead, the lockdown has been implemented one month later, and that is what is going on now. To conclude, so a spatial model, at least according to our opinion, including mobility, is fundamental. If we want to project in real time the epidemic trajectories, at least at the very beginning, when it might be useful to implement immediately containment measures in order to stop the spread of the disease. In these ways, it is possible to estimate the demand for critical care, the hospitalization you should expect in order to avoid the big problem for a hospital, to estimate how much testing and tracing we should do. On the other hand, integrated models are also necessary. Age structure should be included. Social contact structure should be included. Many other models that have been developed around the world, and even in Italy, also consider age structure and social contact structure. But unfortunately, I must say that they do not include space. Unfortunately, lack of available data is a problem. They are not made public. So in some cases, and in our case too, there was insufficient spatial granularity for some compartments. And if we look at the problem really at the global scale, then it is clear that there should be nested models from global. I mean, all the world, like for the problem on climate change, for instance, where you have global modeling and then nested, you have country models and then nested regional model so that you can fine-grain the strategies, because these strategies might differ from location to location. And one thing that is unfortunately common to many countries around the world, that although the data were made available in a relatively short time, and you have seen the Johns Hopkins University site and so on and so on, however, in Italy and in other countries, most of the data were not made available to the scientific community. The scientific community were requesting a wide availability of these data in many cases. This data only part, I would say, a small part of this data, of course, anonymized data were made available and that unfortunately. Well, I'm sorry, it takes such a long time. I'm sorry, I'm going to stop here. No, I mean, it was very interesting. There is no reason to be sorry. Thank you very much, Marina, for the very nice lecture. So we have time for questions. So there are a few in the chat. So one from Miguel Rodriguez. In the local model is the parameter H, which I think represent the hospitalized individuals, capped by the maximum hospital capacity in each region, or is it assumed to be unlimited? No, I don't remember age. No, age. I don't remember what I indicated with age. In the local model, let me see. Oh, the hospitalized, OK. The hospitalized people. No, no, no, no. OK, the hospitalized people are very numbered. It is a number of people who are in the hospital at a given time. Yes. So of course, it was capped by the hospital capacity, by definition in a way. Although, you know, the people were hospitalized and put everywhere, everywhere, though it was real and mass. And it is now clear that at that time, many people were actually dying, even if they were not without even being taken to critical care units. I guess this question alludes to the fact that the admission, for instance, to ICU depends on the occupancy of ICU, so the criteria change with the capacity. Then there is a question by Sylvia asking, what is the interpretation of the large negative correlation between the fraction of asymptomatic cases and their transmission rate? Yes, OK. The interpretation, as follows, that if the number of asymptomatic asymptomatic cases is large, very large, then the transmission rate can be, in a way, lower because it is the product, let's say, of the beta A and the transmission rate of A and the number of asymptomatic. Because that will be, let me say, the viral load that goes into infecting the susceptible people. Now, a good thing of Bayesian modeling, of Bayesian statistical approach is that you cannot only estimate the confidence of each parameter, but you can also look at the correlation matrix. And in fact, if you inspect the correlation matrix, then you find out that some parameters are not so much correlated to the other parameters, which is a good thing. It means that they are well estimated. Now, if you see negative or positive correlation, that means that the model is not so parasymonious, in a way. And that's why we were concerned, and that's why we made a sensitivity analysis in the second paper with respect to the fraction of symptomatic and the fraction of asymptomatic people. And then the serological test confirmed that the fraction 25% of symptomatic and 75% of asymptomatic or mildly symptomatic so that they don't even care they don't go to the hospital, they don't take to up that thing and so on. It's reasonable, at least for Italy. Great. So there is a question about vaccines. Good news. Of course. Yes. So after 2004-2001. And we hope that we are doing that. And Ronaldo, I think, is still joining maybe the no, no longer. I am here. I'm here. I'm here in Marino. Oh, you're here. Yeah. So the question is. Yes. Yes. Yes. We are working on that. Yes. So can this model be used to optimize the special distribution of vaccination campaign? Or do you think that it is of small importance relative to the demographic groups? Well, of course, there are some things that, that pertain to the common sense. And actually these are the rules that in a way would be enforced by the European Union to distribute the vaccine. You first vaccinate people in the hospitals, medical doctors, nurses, and so on and so on. And then of course you can anyway optimize the vaccination campaign. Given some constraints. So the constraints are the rules of the European Union happens through Europe. Now the. The number of a batch of vaccines. And of course the number of. Well, of course. Well, the idea, if I may. Well, they say 280 for Europe. Something like that. No, but the question is whether even a batch of vaccines and set of rules, finding out the best distribution in space and time of that batch is still in a thing, which are precisely what we're working on now. So there is a question by Luca that says that, well, he really enjoyed your talk and he found very interesting. The Bayesian approach. And he's asking, according to your model, which parameters are the ones that are the best distribution in space and time of that batch is still in a thing which are precisely what we're working on now. And it's not trivial. It's not. Great. So there is a question by Luca. That says that. Well, he really enjoyed your talk. So what parameters are the one that change the model results the most? Well, certainly. Let's say the, the beta of the pre-syntomatic. Make a lot of difference. You know, I told you that there is evidence that is experimental evidence that this pre-syntomatic are. Very much infections. And that's why we decided to include a compartment of pre-syntomatic. But it's not our invasion. There were other people who had done that. And, but we found that that was fundamental, but then we didn't fix the transmission rate of the pre-syntomatic. And it naturally came out from calibration. That the beta of the pre-syntomatic must be quite large. Compared to the beta of the asymptomatic and the infected. Symptomatics. And again, it came out of calibration that the other two betas, apart from that problem of the negative correlation of the beta of the asymptomatic are similar in terms of the automatic, which makes sense. Even from studies on the viral shedding, so viral shedding is actually decreasing after you get the symptoms. Or if you are asymptomatic and you are followed during the asymptomatic development of your disease, viral shedding is similar, kind of similar to that of symptomatic. But it is the pre-syntomatic with the highest viral shedding. More or less, of course, then there are a lot of individual variations. I don't know whether I answered, Luca, whether I answered your question. Let's see if answering the chat. So is there any other question? Yes. The question is answered. If not, well, next week we are going to have, actually, one of the roundtables, they should appear soon in the program if they are not yet appeared. One of the roundtables is going to be precisely about COVID and the pandemic. And Marino Gatto will be one of the panelists. So there will be more discussion about it and more possibility to interact about the... Yes. The most important topic of 2012. So thank you very much, Marino, for the lectures. Thank you all. Okay, so I'll see you at the roundtables. Yes. So thanks also, Andrea, for staying with us. Thank you. And so now we are going to break in the breakout rooms. Oh, sorry. There is actually a question that I didn't see. I don't know if Andrea... Marino is still here. Please ask the question if you... Yeah. Please, I am asking, can the generation generalize reproductive number of influence on the generation train of the model function queue? Is it possible? Sorry. You know, at my age, I cannot hear very well. He understood that better than me. I think it's better if you type it because the communication is a little bit disturbed. I think you have a 49 probably. Okay, okay. I can type it right. Yeah, thank you. Okay. Marino, I think that Monday was asking about generalized production number applied to the COVID model. Yes, but again, you know, which kind of question... I don't know. You want to know more in mathematical terms? No, I think it was a very precise question. I think he's typing it, so... Is he using chat? Yes, I think he's typing it. Because I don't think... I thought it was possibly right. Okay. I can ask a question. Please scan the generalized... Okay, so if the generalized reproductive number have influence on the generational trend of the model function queue? The model function queue. Now, what is queue? Wait a minute. Well, I'm sorry, but what is the queue function? Because I thought I didn't remember it, I don't see any queue function in my presentation. What do you mean by... Oh, quarantine. On the generational trend of the model function queue, the quarantine. So it's not function, you mean a compartment. So this... Yeah, the quarantine compartment. No, it is just the opposite. That the rate at which you quarantine people influences the... or not, of course. Because if you quarantine people, so there was... Now let me share again my screen, maybe. Okay. Okay, so there's a... You see a rate at which you quarantine people that are symptomatic infected. Because at the beginning, at least in the first paper, you know, at that time, Italy was not able to discover any asymptomatic. So it was a miracle if they did swap testing for the symptomatic. So you are symptomatic and then you can quarantine because the medical doctors might decide that you do not have enough symptoms to be hospitalized. And so you introduce a rate. And of course the larger the rate and the better it is in terms of decreasing or not because these people are quarantined and they are no longer infectious. At least they are not so infectious. They are still infectious in the household, of course, unless they can isolate. They have a very large apartment and in the family, you can isolate from one another, okay? But anyway, in any case, you are constrained to stay at home and so you cannot go around and infect people. So the larger that and the better in terms of the are not. So, of course, if you quarantine a lot of people, that is a containment measure that will have a positive effect on the reproductive number. I hope that was clear. Okay, I see that you're doing so. Okay, so let me stop. Yes, sharing. Okay. Man is satisfied. He's satisfied. So thank you very much Marina again for answering all the questions. So we will take a five minute break before Jonathan Levin question. And if you're following from YouTube before the next lecture is not going to be live stream, but we'll be back on YouTube for the lecture of Daniel Segre at 5pm. So we're going to be split in breakout rooms. See you in five minutes.