 synchronization of epidemic oscillations induced by social network control, the supervisors were Daniel DeMartino, Fabio Caccioli. So please, Luis, if you are going to start, go ahead. Thank you. Yeah, actually, I'll let the girls start. Oluakimi, are you there? I'm here. Good day. Thank you for this privilege and opportunity to be able to see. Thank you. We, Project 16, are here to present a project titled synchronization of epidemic oscillation induced by social network and we are being supervised by Daniel and Fabio. We'll be guided by the following outline. Mathematics being the universal language of nature and the foundation of all the natural sciences and engineering as historically being used to gain realistic insights into the transition dynamics and control of imaginary, imaginary infectious disease. And this is a public interest. Fred and Carlos 2001 and Anderson and May 1991, the Van Epidemiology to be the study and analysis of the distribution, determinant and control of health and disease condition in the blind population. And this has been a cornerstone of public health research since the 19th century. Epidemic model generally assumed that the population can be divided into different classes of compartments. This depends on the stages of disease according to Anderson and May 1992 class. One of the simplest two-state compartmentalization in the epidemic model is the SIR model and the SIS model. In the SIR model, we have the susceptible population, the infected class and the recovery class. This is the way where the susceptible class being interact with the infected class, then the rates at which the infected class recover from a particular infection. And here in the equation two, we have our beta, which is the transmission rate. We call it transmission rate is that the rates at which the susceptible is being infected with a particular disease. Then we have the gamma here to be the recovery rate at this, at which the infected class being recovered from a particular disease, which is an equation three. Then here we have our in equation four from the equation three and two, we have a reproduction number to be beta over gamma. The reproduction number generally is the number, is the production number is the average number of secondary infections, which caused by an infection individual during is our entire period of infectiousness. Then here we have our reproduction number to be greater than one, that means we have an endemic state. Biologically, this means that if reproduction is greater than one, the infection we perceive, that means each newly infected individual, we spread the disease to at least one susceptible individual on average. And conversely, if we have reproduction number to be less than one, then the epidemic, the disease can still be managed, that means on average an infected individual will spread the infected to at least less than one infected, I mean individual at the period of the infectious state. Then I will allow my next presenter, Olajima Keoludu, to take calls through the networking, the model fitting and the epidemic case stereotype. Thank you. Thank you. Go ahead Olajima Keoludu. Yes, please go ahead, we'll listen to you. Thank you. The infection rate on networks, where the infection rate on this model depends on social matter, on the physical or social matter, that is how much is equal to beta over gamma, multiplied by the k square over k, where k is the probability of choosing a round on modes with k connections. From figure two, we can see an example of two different networks. We expect an epidemic to spread faster on the right network. Why? Because they are closely limited. Because if the network on the figure, as in the notes, the notes are closely limited in the right hand figure on figure two. So that means that by controlling the KAs, by controlling the connectivity of the network, one can also control the spreading of the infection. Once the connectivity of the infection is being controlled, then the spread of the disease can be controlled. Epidemic oxidation induced by feedback. Epidemic models in networks with feedback, they are the infection states to the network structure. From figure three, this is an example of a school of friendship network by natural 2015. From the figure, you can see at figure A, you can see that the notes are closely limited. So that means that infection can spread faster because of the network. The notes, they are jam-packed, they are closing each other, so infection can spread faster. And when the notes begin to split, you can see from the big part, when the notes begin to split, you can see that the rate of transmission reduces. On the C part, the notes are distorted, they are not distorted. That means at the C point, the infection rate reduces. And at the D point, you can see that they are totally dispersed. That's so that once the network can be controlled, that means when we stimulate lockdown, when there is no relationship, when there is no interaction between people, the infection cannot be spread properly. In this figure four, we have feedback control impact on an SIR model. On the left, you can see that there is no feedback model. So there is no control, there is no control, but there is no oscillation. But on the right, you can see that we have three oscillations on the plots, on the figure on the right. That means that these are oscillations. And this shows that there is a closed path or there is a trajectory. Once you have a trajectory, that means that when our notes are less than one, when there is a trajectory, you can see that there is a limit circuit. This is an epidemic way from the region of this is an epidemic. When there is a trajectory, then you can be fitted. This can be fitted into real data. It shows that once we have seen the margins of a trajectory, then they show that you can be fitted in real data. Nice piece. So now that we are treating oscillators, are these oscillators really coupled? Is there any type of synchronization between those oscillations that were shown? What are the relationships between the logic region? What is the synchronization between them? That is the main reason for this research. So we are going to, for the phase reconstruction, we are going to use the e-over-transform. Transform, that is example in the figure of six. Y of t equals the h of f of t. That is the formula we are going to look at with data. We are going to extract the phase for each country and compare. The figure of c is the example of e-over-transform, a very formal theoretical given structure. This is done by General Pivosti in 2020. If trajectories are closed, the phase can tell you that they are different as little and are synchronized, but not exactly where their trajectory is going to be. So the idea behind this is to find the trajectory and know how to synchronize to their enemy. So I will leave the room for my next presenter to continue. Yeah, so thanks a lot, Moki. Good afternoon, everyone. The girls briefly introduced you to the background that we are working at, and now I'm going to present you what we have actually done. So our first task was to extract daily deaths from our raw file containing data sets of the state of the COVID-19 outbreak for countries in the world. We can see on figure seven left panel an example of data we are working with. In purple, we have our signal e-g daily deaths over the course of the pandemics on the country of Ghana, as example. On the same panel, green dots represent the signal related e-over-transform obtained from directly applying equation six, previously the presented by Elijah Moki. This e-over-transform directly here. So when you look at phase space of these two variables, y as a function of x, we can indeed see that this seems to approach a closed trajectory on the phase space. This result is important. So I'll just take a few more moments on this slide to explain this a little further. So Lajmoke spent her last couple of slides convincing you that both the proposed model by Daniela Fabio and the COVID-19 epidemic data seems to perform a closed trajectory on the phase space. This is important because now we can treat each country as dots in this phase space, each of which running on its own closed trajectory. So these trajectories are not necessarily the same. They are not the same, but the countries are running on these trajectories and we'd like to know if these runners are somehow synchronized in this trajectory. And if we found that this is a yes answer, then why this happens? So our next step is then to reconstruct the phase from the signal by supposing that a point in this phase space can be written by a simple complex number. And then the phase will be simply given by equation seven. So going further, we then want to compare the distracted phases for different countries. To do this, we define a distance between phases as in equation eight, where we integrate the models of the difference between two phases over the time series. So for now on, we are working with countries in pairs. So here I have two signals that are different from this one. This is a spear signal, the daily depths over the days, and this is the distracted phase. So when you simply look at this phase on the left panel, we can see that these countries are more synchronized than on the right one. The wave seems to play a role on each other. And this seems to, the dynamic seems to be similar. So when we see countries that are more synchronized, we can see that this area in purple is much smaller than on the right case. On the right case, the case of Afghanistan, the waves are unsynchronized clearly. And the area in purple is larger. So now we have a measure of how unsynchronized these epidemic waves are. So with this relation in mind, we want to build a synchronization matrix between the analyzed countries, where elements of this matrix are given by exactly by equation eight. We expect then if this matrix have some kind of box structure, or can be built in some kind of box structure, then there will be some relation, some synchronization between particular countries, not all, but between some particular countries. So this is our first attempt to build this matrix. On this first attempt, I tried to previously group the countries that are plotting. So the first eight countries are from South America, the middle four are from Africa, and the last eight countries are from Europe. So we then followed the previously discussed procedure to obtain the matrix that I showed to you here. So we can clearly see that there is a box structure that can be built. And more than this, we can see that topological structures seem to play a role on this result. Also, we can see that this is more important in Europe and Africa than on South America, for example. So topological structure seems not to be the only thing that causes this synchronization to happen. So to understand this further, we extended our procedure to 164 countries available to us. So left panel shows us the raw matrix where columns lines are organized alphabetically, while right panel shows our first attempt to clusterize this matrix, now using a cluster algorithm, obviously. So when you look at this proposed clusters on right panel on the world map, I mean, looking at each country on which cluster it is, we can see that this is not yet the optimal structure that we want. So this is an ongoing work that we intend to keep collaborating on. And we extended this to our next objective. So our next objective is now to relate in some way this box structure with migration or connectivity between two populations. So we then propose a model to explain how specific countries and not all of these countries are synchronized and some more than others and other less. So we suppose now that there are two previously independent networks, each of each one with its own ongoing epidemic. Then at some point in time, you introduce connections between these two networks. So that at each time, a node will try to contaminate its own network. It also has a smaller chance to infect some node of the adjacent population. So we simply make few connections between two previously unrelated networks. So one would expect that given enough time and connections, eventually this epidemic waves would be somehow synchronized. So on a simple model like this, we have two clear limits, a lower limit where there are no connectivity between two populations and leading to a system with two completely independent networks. And an upper limit where the two networks become a single network with an average degree of connectivity equals to K internal. So this slide shows our results for the model proposed. Trigla-11 shows the syntax code it generated. While left panel, the model for an average of zero connections between populations. Right panel shows us the same simulation, but now it's an average of one connection between the populations. We can clearly see that they're on the right panel, the waves are more synchronized than on the left one. So that area in purple that we were discussing would be much bigger on the left panel than on the right panel. This is impressive on figure 12, where we can see that the higher the average networking to the degree, the lower the distance between phases that the measure that we proposed for the synchronization matrix. So our model seems to bring to the table the mechanism behind the synchronization between two somehow connected populations, be it via migration, proximity, airflow, etc. This process is not well understood yet. So just to conclude, we have so far elaborated a synchronization matrix for a worldwide data for the COVID epidemics, and also have shown that this matrix can be decomposed in a block structure. We also proposed a mechanism for this synchronization via migration or connectivity between two populations. For the future, we intend to optimize the clustering algorithm so we can in some way minimize the distance between countries and then estimate the synchronization parameter for each group of synchronized countries. In this way, we'll be able to show that instead of let's say 164 independent epidemic waves, we could treat 10 independent epidemic waves in the world, which would be a nice result to present. So just yesterday, Daniela told me that we already have some preliminary results for optimized cluster. And perhaps if you have enough time and you want, he can tell us a little about them. So this is it. I hope you guys enjoyed. This was an amazing experience to all of us. We want to thank everyone responsible for this school to happen and also our supervisors, who are always helpful to us. So thank you very much. Thank you, Luisola, Jumoke, and Oluwakemi. I hope I pronounced the names correctly for this very interesting presentation and also the congratulations to you and to your supervisors. We have time for questions. Please, as usual, just directly unmute your microphones and go ahead, please. Yeah, I would have a question, perhaps. I'm not an expert on signal processing altogether, but I understand that you have used the Hiba transform to extract the phase, right? Exactly. This looks interesting, and I haven't thought of the Hiba transform at all. I mean, I would have directly used probably a power spectrum to check for paradisities. And I'm wondering whether I'm on the wrong track here or what really the advantages are of using the Hiba transform here, say, compared to a power spectrum, I wish you could also extract paradisities from a signal. Yeah, perhaps it's just one way of treating the problem, perhaps the approach may get similar results. I mean, we just wanted to find some kind of synchronization, some measure to estimate the synchronization. I believe that any other measure, perhaps, would be sufficient to do this. Maybe Daniel and Fabio want to complement my answer here. Yeah, I mean, one idea is that these are, I mean, the mode says this can be nonlinear oscillators. So if you do the Fourier transform usually is something to use. And so the power spectrum used for linear equations. So the Hiba transform for us is somehow a way to reconstruct the derivative that for noisy, when you have a noisy data, so it's, we are using it in a neuristic way, like the reference we gave to the Picozki article. Yeah, it's the problem when you use Hiba transform with nonlinear oscillators, the power spectrum, it's complicated. Now you had told this before question of peaks. So I think maybe the Hiba transform is better for nonlinear. So it's a rather new technique, but in this context, I mean, as you're referencing Akari Picozki, who apparently has used it recently. So is this my understanding correct that this is a new interesting technique? I mean, Hiba transform has been around for a long time, I suppose, right, but to apply it in this particular context, then appears to be rather recent, right? Well, I never seen, yeah, maybe we are original. I mean, the Hiba transform might be using it for instance for Magneto and Chefologram data. They are used in neuroscience, no? For instance, computational neuroscience, but the computational neuroscience, we have a lot of weights. You have millions of them. I mean, you record a whole night of a subject or even some minutes with the frequency of hundreds of hertz. So here instead, you have two, three waves. So that's why we have also a model in which we can make a benchmark and the model with the model we can produce all the waves we want. I mean, at least with the model, the SIS. Okay, thanks a lot. Thank you, Daniel. And thank you, Daniel. And by the way, ciao, Daniel. I hope you are fine. More questions, colleagues? No more questions? If so, shall we thank the speakers for this very nice presentation? Thank you very much, guys. I think we are actually in the end. So let me stop the recording.