 So, how come everybody, this is another version of our ECB COVID-19 webinars, and today we are welcoming Tom Britton, Tom is a professor at Stockholm University, he's the first mathematician and now I see a number of epidemiologists, usual fair microeconomists, and we thought it would be very important since we are trying to forecast how the epidemic will affect us. Tom has written one of the handbooks on this and has prepared for us a presentation on an application to something like COVID-19, so we're very much looking forward to that, and please send your questions via the chat button and keep on the mute button so we can listen to Tom. So with that, over to Tom. Okay, thank you very much. Can you just, can someone confirm that you can hear me now so I know it's working? It's working Tom. Okay, thank you very much. Okay, so the plan for me is to give a 45 minute presentation, let's call it a popular version of how mathematical models for infectious diseases are constructed, and I will of course try to make some connections to COVID-19, and then you will be able to ask questions. I'm not sure exactly what you want to know, I of course I tried to put this in the talk, but it's good to have questions later on. So before I start, let me just say that I'm not an expert on virology nor am I an expert on practical aspects of prevention policies, but what I do know is models for infectious diseases, and I've been working in this area for 30 years. I got my PhD 25 years ago on these type of modeling, and I've been working on it mainly for the rest since then. So I have sort of three sections of this talk. The first is the real world, which I guess you are interested in, and then we go back to the modeling world where I have been working mainly, but I'm also done some applied work, and then we go back to focus on COVID-19. So here you see a reported number of cases of COVID-19 over time. There is a first peak here, which corresponds to the outbreak in China, and then there is a much bigger peak here corresponding to Europe and the US mainly. And this is not the only type of graph that looks like this. This is quite often the case if you have a new outbreak of a new disease. It looked very similar for Ebola. It looked similar when you had the new swine flu pandemic in 2008-2009. So this is a common feature. So when you see this type of things and you're a modular, or even if you're not a modular, the type of questions that you might ask, okay, so here is what we see. How many will get infected? When will the peak of the outbreak be? How about effects of different preventive measures? What effect will that have? And another important one is, is there any other type of data that would be good to collect in order to improve precision in our statements? And here's another figure of reality. So this is called an endemic disease. And in fact, this might be the situation for COVID-19 in a few years from now. Hopefully by then we have a vaccine and better treatment, but there are a lot of endemic diseases circulating on the globe. This one is called rotavirus. I don't think it's that harmful. It is for children, but for adults, most people don't even notice that they have it. They have a little cough and that's it. And this is reported number of cases. And of course, if you see this thing, I've been working a little bit on this data material. That's why I show it then. One question that's important. How about underreporting? This is what is reported. How much is underreported? Can we say anything about that? If a vaccination program is introduced, how many are necessary to vaccinate in order to stop this endemic situation? So those are typical questions arising for endemic diseases. There is another endemic disease, or what used to be endemic. It's measles, considered to be the most highly infectious disease around. It has an R-naught of about 15. I will come back to what R-naught means. And here we see what happened with measles reported number of cases in the UK. And this is long time ago, because now we have a vaccine, so there are hardly any outbreaks at all. But this is UK. And this is Iceland. And we see that in UK, measles was around all the time. And there are peaks coming up and down. But it's present the whole time. Whereas in Iceland, it's not present the whole time. And the reason for that is that Iceland is a smaller community. So there is not enough influx of susceptible individuals to keep it going. But later on, when some visitor comes to visit Ireland, there is a new outbreak. And here the peaks are actually sorry. So here the peak is every year. But for measles in UK, the peak was every other year. So why is that? Well, a simple explanation is that if there is a peak now, then everyone gets infected. And most of the transmission is in schools as opposed to COVID-19. And the next year, which is here, when school starts in September, it's only the first graders that have not been infected earlier. So that's not enough to have a new epidemic outbreak. So we are still above herd immunity after one year, whereas after two years immunity has partially waned. But in particular, there have been new born children entering school. So that's enough for a new big spike. Okay, that was a bit of reality. I will come back to reality for COVID-19 later. But let's now go into modeling. So I thought I would present one of the simplest models just to get the feeling for how a mathematician works. So the simplest model is of course not very realistic. But let me emphasize, even though I will not describe them, there are much more realistic models. But let me also admit or acknowledge that all model, all mathematical models for anything is a simplification of real world. That is the essence of mathematical modeling to simplify the complicated real world and try to make some conclusions. The hope is that the simplifications play a minor role so that the conclusions from the simpler modeling is valid also for the more complex reality. But that needs to be sort of verified or confirmed. Anyway, what assumptions do we have? Let's assume that there is no prior immunity. That actually is true for COVID-19. Let's assume that all individuals behave similarly. So that people are equally susceptible. And if they get infected, let's assume that they're equally infectious. And probably the most unrealistic feature of all is supposed that everyone meets with everyone at equal rates. So that is called uniform mixing. So there are no social structures, very unrealistic, but that's the simplest model. And then we consider this disease, which is called an SIR. So at first, you're susceptible. If you get infected, you are infectious for a while, and then you recover. And when you are in the recovered state, you're considered to be immune. COVID-19 is among this class. It is unclear how long the immunity lasts, but surely it lasts for at least a few months. So you will not get re-infected after a few weeks. So SIR is probably the most common mathematical model. And to start off, let's also assume that we don't change our behavior during the course of the epidemic. This is, of course, not true for COVID-19, but probably it is true for the seasonal influenza. We don't really change our behavior during an influenza season. So here is the model. We start with the population size N. And we assume that everyone is susceptible except one individual. And we think in terms of discrete generations, so you might think of week one, week two, week three. And the model is as follows. When we start, we have one infected individual and the rest being susceptible. Then the next week, this initial guy infects each and every one of these independently with some small probability P. So the model has one single parameter and that is P. And then suppose the result is that this first individual infects by chance maybe three new individuals. Then week two, there are three people infected and those three people infected, they infect each and every one of these remaining susceptibles with probability P. So viewed from a susceptible point of view, in order not to get infected in the second week, you must avoid infection from all three of the infected people. And then this goes on. So maybe next week there are seven infected and maybe the week thereafter, there might be 18 fewer people here. They will have moved to here. So there will be fewer. Each infected person will infect fewer and fewer since there are fewer susceptibles around. This is an effect of the model but this is also true in reality. Once immunity builds up, there are fewer new infections per infected individual. So the model goes on until at some week there is no new infections and then the epidemic stops. So the model has one single parameter that is P and of course there is community size. More important than P or let's call it an equivalent parameterization is to not consider P but to consider the average number of new infections caused by one typical infected individual during the early phase of an outbreak. That is the definition of the basic reproduction number and I'm sure that most of you nowadays know what the basic reproduction or at least have heard about it. I think three months ago very few people in the world knew about the basic reproduction number but now it's quite well known. So consider this model I said suppose we have a small community of let's say 1000 individuals and suppose this transmission probability to a specific individual is 0.0015. What is then R0? So here is R0. You have a small probability to infect each one but there are many individuals so the average number of people that you infect is the probability multiplied by how many that are around. So that is 1000 multiplied by 0015 and that is 1.5. So for this choice of P and this community size the basic reproduction number equals 1.5 and 1.5 is a common value for influenza whereas for COVID-19 a common estimate of R0 is 2.5 so on average you infect 2.5 individuals in the beginning of the COVID-19 outbreak whereas for influenza in the beginning assuming everyone is susceptible it is estimated it's believed to lie around 1.5. Okay that was the wrong direction so here we go oh here is a misprint but anyway so a question is of course how okay so this is the model we know what R0 is then the question is how many will get infected another question might be okay how does incidence and prevalence vary over time so those are the two typical questions and then of course what happens if we insert preventive measures let's start first of all with a final number infected and that's called a final epidemic size so now we forget time and we look what happens at the end of the outbreak so what I've done is I've simulated this model that I described above not once but I've simulated 10,000 times and on the next slide there is a histogram of not one simulation but of all these 10,000 simulations and what I have stored is only the final number infected for each of these 10,000 simulation so each simulation results in maybe the first simulation resulted in just three people getting infected the next simulation might have resulted in 595 people getting infected and so on so I've done this for two choices of R0 one is the first one is when R0 is 0.8 and the second one is when R0 is 1.5 so here comes the histogram for R0 equal to 0.8 so here we see that when R0 is 0.8 we only observe smaller outbreaks so there is no big outbreak how about when R0 is 1.5 well then there are still some small outbreaks but there are also some big outbreaks the perhaps most surprising thing that there are no outbreaks here and no outbreaks here all of them are either concentrated here or here if I would have done the same thing with a much larger community having the same R0 there would still be a spike here and this one would be much more spiked so it would of course if it was 10,000 it should say 6,000 here and 10,000 here and then the distribution would be much more spiked so in essence there are only two things that can happen either you end up here or you end up here and this is when R0 is larger than 1 if R0 is smaller than 1 we will only have small outbreaks and why is the value 1 so magic well maybe you believe me that if you're on average in fact less than one person there is no chance that such an epidemic can take off whereas if R0 is bigger than 1 there is a chance that it takes off okay so now I thought I would actually explain there is actually a formula that can derive what this fraction is so there is a there is a formula and I will try to explain it so if you tell me what R0 is I can tell you how many that will get infected in case we have a major outbreak so let's call this size tau so tau you should think of this it's a Greek letter you could have called it X if you preferred but this is an unknown and we want to see how what is the fraction getting infected at the end of the epidemic if this is the fraction then this is the number getting infected and remember that this small transmission probability multiplied by n was equal to R0 so R0 divided by n is the same as p and this is the transmission probability so here comes the long equation so tau is the p the fraction getting infected so 1 minus tau is the fraction not getting infected and the fraction not getting infected that's more or less the same as the probability not to get infected and the probability not to get infected if you remember the model what is the probability that an individual does not get infected well in order not to get infected you must avoid infection from each of the people that did get infected they did their experiment to everyone including you and they transmitted it to you with probability p so the probability to avoid infection from one of those people is 1 minus p infection from you must avoid infection from all of those that have been infected and that is n times tau because this is the number of people getting infected so now I replace p by R0 divided by n so we have this equation and I'm sure that some of you have not taken much math then you just have to trust me that this expression here is equal to this expression when n is large so we end up with an equation 1 minus tau is equal to e to the minus r tau so this is the equation it looks a little bit odd but anyway if you give me a value of R0 I can numerically solve what tau must be it it's not explicit because you see I have tau here and tau here or if you prefer x most people are used with x as the unknown it will say 1 minus x is equal to e to the minus r naught x it has x on two places so it's not explicit but numerically the computer solves this in a millisecond so what comes now is a plot of how many that get infected what fraction that get infected as a function of r naught so here is this so we see that if r naught is smaller than one no one gets infected there is no possibility of a big outbreak whereas if r naught is 1.5 which was my numerical example earlier you follow this line you see you end up here then you go horizontally you see you end up if I think I remember correctly the solution when tau when r naught is 1.5 the solution is 50 0.58 and we see that this agrees quite well with the simulations and for covid 19 which is of the order 2.5 it suggests that about 90 percent would get infected let's go back to our assumptions well first of all the most important observation maybe was that when r naught is smaller than one there will be no major outbreak and the results I just explained was based on three assumptions one was that there was no immunity in the community either natural or by vaccination this is true for covid 19 but not for seasonal influenza another assumption was that we had the homogeneous community people were equally susceptible uniformly this is of course not true for any disease but the perhaps surprising effect the surprising result is that these heterogeneities that are present in reality turn out not to have such a big difference doesn't affect things very much and the effect they have is nearly always that fewer will get infected so as a rule of thumb if you have a more realistic model you would see 10 to 20 percent fewer infections and the other the final assumption was that there was no changing behavior during the epidemic okay so now I will show the same type of plot for the situation that I showed earlier but also assuming that there is heterogeneity and assuming and another situation where there is initial immunity so there will be three curves one is the original model one is for one specific type of heterogeneity and the other one is where there is no heterogeneity but there is initial immunity so here is the original curve and this is the curve where there is heterogeneity so we see it is lower than the blue curve but not huge difference so this maybe this this is just one type of heterogeneity but if this would fit to COVID-19 we can read off here and we end up here so then somewhere slightly less than 80 percent would get infected and probably you have heard this that without preventive measures around 80 percent will get infected by COVID-19 there is a huge difference however if there is initial immunity either natural or by vaccination for instance if half of the community are immune then the basic reproduction number has to be larger than two for anything to happen and if we have R0 equal to 2.5 we see that very few will get infected so immunity plays a big role heterogeneity plays a smaller role and then let's take a closer look at R0 one way to describe R0 is to factorize it into a product of three quantities you can think of it as a probability the probability that there is transmission given that you have what is called a contact the other factor is how frequent you have contacts with others and the third one is the duration of your infectious period what is meant by contact depends of course with depending on the type of disease for aerosol born or a water droplet the type of transmission a contact is considered well it's loosely defined as someone that you are in proximity to for more than a couple of minutes so someone that you speak with for a few minutes that is one way to define a contact here if you're interested in sexually transmitted infections of course then it's a sexual intercourse so this depends on the type of disease you're interested in and there is no sort of strict for COVID-19 there is no of course no strict context there is an approximation involved here in a way one way one advantage with this type of distinction of the different parts of R0 is that you when you think prevention you want to reduce the reproduction number you can aim at reducing this this or this for instance how do you reduce the probability of transmission given a contact well one for COVID-19 one is a face mask another one is hand washing if you're interested in sexually transmitted reduces the transmission probability dramatically the rate at which you meet people you can reduce that by avoiding public transport avoiding public events also quarantining people as is done currently in many European countries that of course dramatically reduces the rate at which you meet other people and the length during which you are infectious that is harder to to change but if if you do a lot of testing and you test people positive and then you quarantine them that is a way to reduce the effective in infectious period so there are lots of different way to prevent things and if you do that and suppose that the combined effect of all the preventive measures is that you we managed to reduce the original basic reproduction number by some factor a so if you reduce this by 50% then you reduce this by 50% then a is reduced more than 50% of course so then we get the new reproduction number which is called the effective reproduction number and that is the original one multiplied by the remaining parts so if you have removed a fraction a of all the all the effective contacts then the remaining one is the remaining one is is 1 minus a times r0 and for similar reasons as before if we have reduced it we will have no big outbreak if the new reproduction number is smaller than one or if we do this during the outbreak if we insert preventive measures during the outbreak the epidemic will fade off rather quickly if the the overall effect of our preventive measures is such that the new reproduction number is below one and that is the same thing as saying that the overall effect must exceed this quantity so that is why people want to know what r0 is because r0 tells us how much we have to reduce the transmission in order to stop the outbreak herd immunity is another term that has been popular the last few a few months and here is an explanation to it suppose that we have a disease and community for which we have a given value of r0 let's take 2.5 as an example this means that everyone sorry this means that when someone gets infected you infect on average 2.5 individuals if everyone is susceptible but what is then the reproduction number once 30 percent have been infected and what is the reproduction number when 65 percent have been affected well if the original reproduction number is 2.5 once 30 percent have been infected you only infect 70 percent of your contacts so the new reproduction number is the original one multiplied by the fraction that are still susceptible so that's a new number which for this example is 1.75 whereas if 65 percent were infected the new reproduction number is 2.5 multiplied by the fraction susceptible then which is 35 percent so then the new reproduction number is 0.87 and this is smaller than one so then we will have no then the epidemic will stop so if once 30 percent have been infected the epidemic still grows but it will grow slower and once 65 percent have been infected the epidemic is the border when this is uh becomes smaller than one well that is actually easily showed that this is at 60 percent or more generally herd immunity is obtained when the fraction infected exceeds this value 1 minus 1 upon r0 so whenever this is the case transmission will fade out or if it hasn't started it will uh if we do this before it has started there will be no outbreak and how about herd immunity once preventive measures are put in place well suppose we have these preventive measures with overall effect a as we did before then the disease will fade out once the fraction immune exceeds this number because this was the new effective reproduction number if you remember from the previous slide okay so here is uh it's not exactly the same figures I had before but this is also COVID-19 another figure so suppose we observe this I've tried to motivate why it's important to know what r0 is so then the question is okay this is what we observe what is then r0 well unfortunately it is not possible to say what r0 is by only observing this figure why is that well this growth rate here depends on two things it depends on r0 but it also depends on something called uh the generation time and this is something which is very closely related to the incubation time or the incubation period so you can imagine if you have two diseases one with very short incubation period and another with the long incubation period for instance COVID-19 which has an incubation period of let's say four days and ebbola which has an incubation period of two weeks then of course if they have the same r0 COVID-19 will grow much quicker than ebbola will because it takes another two weeks before transmission takes place so this growth rate depends both on the generation time as well as r0 and the bigger the r0 the bigger growth rate so small r is quite often used for the growth rate so bigger r0 means bigger growth rate whereas for the generation time is the opposite the shorter generation time or the shorter incubation period the quicker growth rate you have so this means that there are several combinations of the reproduction number and the generation time that give rise to the same growth rate so therefore you need some additional information in order to estimate what r0 is you need to know something about the generation time and that is possible to obtain for example using contact tracing that is when you try to find out who people were infected by and then they can see okay the infector was infected on day five and then that person infected a new one on day 12 so that means the generation time for that individual was seven days so if you do this systematically investigating then you have information about the generation time and then you can also estimate r0 so that is how the estimates of r0 usually are obtained okay so what I said up until now was about the end of the outbreak but of course now we are in the middle of the outbreak so then we're interested in what happens through time so on the next plot I will show two figures I'm sure you have seen a similar type of figures but what I'm plotting is the fraction of infectious individual over time related model to the one I had not exactly the same but closely related and I have two figures one is when r0 is equal to three and the other one is let's assume that we managed to reduce spreading from three down to two so we have reduced the factor a here is 33 percent remaining transmission is two-thirds of what it used to be and from the final size expression that we had earlier we know that if this is the case if you look at that blue curve I had earlier then about 92 percent would get infected whereas if the effective reproduction number is two I looked at this curve and it will say that 78 percent got infected so this reduction has some effect on the final fraction getting affected I would say it's not a huge effect in both situations we have quite a lot of people getting infected however there's a bigger difference here is when we did nothing and here is when we reduce transmission by one-third we see that the peak is close to reduced by 50 percent this is very good news for the hospitals that need to take care of the newly infected not all but some of the newly infected so this will be much tougher on the health care system than this will also there is a delay which is also good for the hospital so that they get time to prepare this time axis you should make take no it does not it's not days a week just forget about the time axis so the total number infected in these two situations is not a big difference the a much bigger difference is that the peak is reduced dramatically which is very good for the health care system so now let me end by a few words about the corona outbreak as I said common estimates of r naught lie in the range between 2.2 and 2.8 I've also seen bigger estimates than 2.8 but these this is the most common range let me remind everyone that everyone realizes that r naught depends on the disease but it also depends on the community so there are differences between communities for instance we have seen that it has spread at a much higher rate in southern Europe as compared to northern Europe before the preventive measures were put in place this has to do with some differences in the in the societies so r naught depends both on the disease and the community that's a very important message I would like to bring to you anyway the conclusion is that if we did absolutely nothing somewhere between 70 and 85 percent will get infected and how many that get infected of course depends on how we act both how the the government acts but also how we act as individuals if we have no prevention at all we behave just as a ordinary day life there will be many infected people and the outbreak will be short in time and after that we will have heard immunity if the other if we take the other extreme we do complete lockdown there will be very few infected and but after that or after that it we even that either have to have this for a very long time or we can gradually relax it and then there will be more infections coming on later you can have it for long and I hope that some good treatment appears or that the vaccine appears and of course there is a range of situations in between here so you can call them they are often in the literature literature called mitigation and you might summarize that that not stop spreading but reduce spreading and of course you can do this not in one way but you can reduce it much or a little and also this is quite often combined with trying to protect risk groups to reduce the case fatalities and with this situation you might depending on how much you reduce spreading you might obtain herd immunity after a longer period of time as compared to this situation and then what happens after the outbreak is over well coronavirus will not disappear from the globe in a long time is my and most people's belief so most likely we will have seasonal we will talk more about have you had the seasonal corona rather than had the vaccine and probably better treatment so here is a description of an imaginary city this is assuming homogeneously mixing so it's not good for a big country but consider a bigger urban area so suppose it starts here sometime in February and suppose that at some stage let's say in mid-march one country does nothing then or one city does nothing then you have this huge outbreak quick outbreak and then it stops down another country takes some preventive measures so the peak is much lower and slightly shifted to the right and then it fades off a third country has higher reduced measures preventive measures and it looks like this my and this yellow orange one would correspond to to a complete lockdown let's say so then it drops down quickly and fades out very quickly and how what will be the effect of this in the long run well if this was the curve explains how many new infections there are per day here we have the accumulated number of infected people so this will grow up or it will start with percent so the situation where nothing was done we have about 90 percent getting infected where there were some preventive measures but not that many around 80 so we see that we end up in different situations depending on how much preventive measures we put in and we see that herd immunity if or not is 2.5 is here so we see that lockdown and even the other ones with a strong partial reduction end up below herd immunity so if you end up here and in particular if you end up here if you really if you relax your preventive measures there will be a new outbreak in this situation there will be a very very small additional outbreak but if you suddenly reduce your preventive measures or restrictions here you will have a rather big new second wave I think this is my second from last slide and the timing seems to be fairly good okay I only talked about the very simplest model there are of course much more advanced modeling that many groups do in Europe the imperial group is the most well known and probably the best group for this what do they do well they put in age structure that is in particularly important for the if you want to say effects on the healthcare system and if you want to say something about case fatalities then you need to keep track of the age structure they also have social structure for instance they in their modeling they have households assuming that you transmit that much higher rate within households and they have people going to children going to school and adults going to workplaces allowing for higher transmission there they also have spatial aspects and traveling aspects both between countries but probably more important is traveling commuting within the country and then they look at different different types of preventive measures and they try to estimate what effects these have with such more complicated model it's very hard to say anything analytically so what typically is done is that they assume a certain some features here and then they simulate many outbreaks to see what happens there is the more different structures you put into a model the harder it is to give numerical values to all these things how do you know how much more transmission is within and between households how do you know the effect of school how do you know how much you transmit within workplaces so here you have to make a lot of assumptions on the quantitative effects of these things so that's one part that makes this analysis very hard but i would say that the hardest part is when you predict the future because no one knows how people will behave in the future nor do we know what the restrictions will be in the future but even if we did know about the restrictions people can behave differently so that is the hardest part to predict the future i think there is a lot of discussion about the mortality for this covid-19 and mortality is usually defined as the probability to die if you get infected there is another quantity called the case fatality risk and that is a probability to die if you have been identified as a case but that is a very different thing and this is the important thing for the driving force of the epidemic there's high uncertainty here i've seen estimates lie between 0.2 percent and even higher and this is meant by a representative fraction of the community but in reality most countries try to reduce the risk of infection in the risk groups and that will of course reduce the mortality even more and also there are big differences between countries would be my guess even if i'm not an expert but first of all the health care system has different quality in different countries maybe there is much more smoking in certain countries which i'm sure that would affect the case for the risk of dying so there is a lot of uncertainty in mortality and then let me say the first wave the first wave is what i'm describing here this is the first wave and if we relax assumptions here there might be a second wave so for the first wave in europe now i think i think countries are somewhat out of phase but i think in nearly all countries we have passed by now we have passed the peak of transmission uh the peak of transmission maybe not the peak where the health care system have a highest load but the peak of transmission i believe that is as it has passed if we keep the current restrictions on but if they are relaxed we will of course have another peak probably but if the current restrictions are kept i think we have passed the peak of transmission in nearly all countries of europe and it and transmission will be much lower towards the end of may however what happens after this first wave will be different in different countries in in those parts either countries or parts of countries where covid-19 have hit hit the region very hard those parts there will be maybe not exactly herd immunity but at least very close to herd immunity so for instance lombardy i think in the summer they will have close to herd immunity in lombardy i would be my guess whereas other parts that have not been hit very hard for instance the nordic countries in particular those except sweden norwin finland for instance there will be hardly no immunity and if there is little immunity in a country and or if the restrictions are suddenly lifted simultaneously then there will be a strong second wave if there is partial immunity the wave will be smaller if the restrictions are lifted gradually the second wave will also be smaller but on the other hand it will also be longer than and another important thing is that this very simplified modeling and i haven't talked about spatial aspects and it differs between countries but also within countries i think the immunity levels will differ for instance in i think in most of europe transmission is much higher in the bigger cities so after the outbreak immunity will be higher in the cities as compared to smaller towns and countryside when it comes to predictions i think the only way forward is to use models however when it comes to the current situation which i think is very important a thing i think people should ask how about immunity now how big is the immunity in different parts of europe now then you can try to say something with models but there is always a lot of uncertainty whereas testing is more reliable in particular if you test random samples of the community so one thing that surprises me is why is there not more testing going on in random samples that way you will get a good idea of immunity test one is called a serological which tells you someone has antibodies but if you're currently infected you don't have antibodies so that you can test by something called a pcr test to see if you currently have viruses so one thing that surprises me is why are countries not doing these tests on random samples to get any information about how many are currently infectious and how many have been infected earlier another my end comment is that sooner or later we have to lift the restrictions however i think there will be a new normal situation in the sense that we have all experienced this scary disease and i think we will have changed our behavior in a long-term perspective maybe we will forget in a few years but for now i think we will change our behavior permanently i think there will be less kissing and hugging in particular in southern europe even when all restrictions are lifted and less handshaking probably less hugging in northern countries that we do quite often so i think this permanent change will have reduced the reproduction number permanently as compared to what it was with that i would like to end my talk and i see that i talk slightly longer than i anticipate is sorry for that many thanks to me there's a big echo all right i tried so there are a couple of questions that came in can you hear me yes okay so one is a sort of more technical question on the first part where you showed that you can either have no outbreak or a very large outbreak so this sort of by modality of the epidemic size yes is that something that is sensitive to the assumptions you put into the model or is this a general result i would say that it is a general result so here we have it so exactly so this is for the simple model that i have whereas in a more complicated model maybe the peak will be here rather than here but this is why you talk about minor outbreaks and major outbreaks so for any situation it would not vary very much so if we did nothing we would have pretty much the same outbreak size in italy as we did in uk as we would in sweden some small differences because we're not exactly the same but it would not be the case that 10 assuming no preventive measures it would not be the case that only 10 got infected in one country and 80 in the other country so the feature of this model but i think it's also a feature of reality thank you then there are a bunch of questions around multiple waves right so you first showed what would happen in a first wave and then there's a particular trade-off that would sort of say well you want to go on lockdown but then you also may prevent building up herd immunity and then you talked about what it may also be second waves and indeed this is sort of what epidemiologists are telling us will happen so is there sort of an optimal way of approaching this as a country is there a trade-off are some countries basically going or for lockdowns that are too severe too long period of time and they will then end up actually in a worse situation down the road or is this all dictated by what the health system can ultimately manage i mean how do you approach this from a modeling perspective yeah i mean from a modeling what is optimal that i think should be left to the politicians but it is clearly the case that if you have a lockdown well if you have a lockdown early on let's say i mean if you have a lockdown very late for instance in lombardia the lockdown transmission had taken place quite for quite a while in lombardia before the lockdown so i think you they will be close to herd immunity anyway but if you have a lockdown early on which for instance norway in finland has i don't know all that much about the other countries but if you have a lockdown early on there will be very few impact infected people and then of course when you relax they prevent the restrictions then they are at risk of having a new big outbreak so i think i mean it could be that lombardia they have had such a tough situation now this this spring but it could be that in the summer they are have herd immunity which is a positive side effect of their terrible spring i don't know if i responded to your question well yeah yeah so i was very much thinking and some of the colleagues as well about sort of sweden versus norway what's sort of the better approach and what dictates ultimately what you you choose there seems to be something in the middle that is suboptimal that where you go for sort of half big kind of lockdowns you don't build up the immunity you need to have this have big lockdowns for a long period of time multiple times in a row since this model is pointing more in the direction of sort of more extreme measures one way or another otherwise either you go for immunity or you go for really strict lockdowns yes yes so let's say sweden and norway i know a little bit about those countries so this is norway and my guess is that sweden is somewhere here and we have much more case fatalities than norway have or we don't so currently people get sick at the rate which our healthcare system manages we have they have a tough period but they manage but we have much higher case fatality than norway but i was in a in some type of panel meeting with london school of economics yesterday and then they showed plots of consumption reduction in consumption and i think sweden had reduced consumption by 28 percent whereas norway had reduced consumption by 60 percent so of course the more you look down the more it hurts society but on the other hand you save lives so there is a very tough trade-off there how much do you lock down you save lives but on the other hand you hurt society and at some stage you might save covid 19 lives but in the long run you might there might be lives spilled for other reasons they are discussing in i've heard in countries that if this economy breaks down totally there the number of suicides will increase so the overall the net effect might be negative if you lock down too much but what is a good balance that i prefer not to say the answer to i understand um then maybe a last question by the way there are a lot of very positive comments like excellent talk wonderful introduction to the topic so i can take that with you so let me add one thing this of these of these curves that are here and you can of course imagine that they're cursing between but the black and the red curve are definitely not among the optimal solutions that is clear so it is sort of in the lower range where the optimal solution is but which of these that i don't prefer not to say but clearly if you do this the the health care system will break down and you will have additional deaths both because you are more infected but also additional death because the health care system doesn't breaks down thank you thank you um so um since r not a such an important number i would have for you is how reliable are all the estimates of r not that we hear from various public authorities politicians in germany there is a is a particular institute and and also because it seems that we are interested in re as you said because r not changes over time yes so for instance in germany miss murka has gone on live saying now r not is about point mine how reliable are these estimates based on your experience because they very much model dependence yes you're right so r not i mean r not is an average you one must remember that keep that in mind that it's sort of an average so um if you consider a country as a community then my guess is that the people have much more contacts with each other in the big cities as compared to the countryside so the average number of contacts among people that live in cities is is lower in the countryside so with this simplification to talk about one single value of r not so i mean r not is bigger in cities and smaller in rural areas and probably differs between different countries due to cultural reasons as well so it is a simplification to talk about one single r not uh like that okay thank you i guess a sort of related question i had in mind was whether we can trust the estimates that these authorities are coming up with um i mean do they do do the people that produce these numbers know how to actually calculate r not i would well you mentioned some politicians uh yeah well that is this r e rather the over time yeah someone said now r re is 0.8 there that is hard but in general if countries have some feeling of if the number of infections is it in are we having more and more infections per day or are we having fewer and fewer infections per day that is possible to to meet by testing and if the number of infections new infections per day is decreasing that means that r e is smaller than one whereas if it is increasing it means that the effective reproduction number is bigger than one so that i think one can rely on but the the exact value if it is 0.9 or 0.8 i wouldn't trust that very much but if you see the plot the reported case the number of reported cases over time that will still give an indication if it is growing the reproduction number is bigger than one if it is decaying the reproduction number is smaller than one but this is not r not is it it is the current uh what i called r t and what is some kind sorry what i called r e and what is some in other places called r t as a function of time okay thank you for clarifying so many thanks on behalf of all of us here at the ECB tom we will share your presentation with all the colleagues that had signed up but we didn't have space for them because this technology doesn't allow for more than 30 people to to listen in so that's why we have the video for them and with that thank you very much and i hope you stay healthy there in sweden i would do my best