Can we Contain Covid-19 without Locking-down the Economy?

can we contain kovat 19 without locking down the economy this is a question and I do care about this article because Shai shalosh warts is one of the bigger names in machine learning theory so it was interesting for me to see what he and his collaborator here had to say about the kind of outbreak and the strategy to contain it so contain maybe isn't the right word made that they asked I think the way that they asked the question is how are we going to survive this the best and so this in no

means is an endorsement by me I'm not a medical professional please just view this as a commentary and an explanation of what they are saying I'll give my opinions along the way of course so they identify three different models for handling the spread of: 19 and we'll start with the third one because they argue for the first one and this builds more suspense so they say there is countrywide lockdown right until the spread of the virus is under control they say could take anywhere from weeks to months it is the safest route but it does not prevent a second wave from occurring now of course if you have people let's say these are people right then the thing is everyone just stays in their stay in your house right everybody right until it's kind of gone now they say correctly there's a risk of a second wave because only a single infected person because there's no immunity still has the potential of creating another epicenter so they don't consider this option next option is called containment based selective quarantine which means find all the positive cases and put them in quarantine so let's say we go here and you know we let you roam around freely but we know we can test people and we know that some of them are positive so we simply tell them to stay at home right now this depends a lot on how well you can test people or and it also depends on what they claim the contagious time interval we know that there are people that are contagious without showing symptoms so unless you can test every single person all the time this is likely to not really help a lot now there is various data from various countries that actually shows it can reduce the load but they basically argue against that because there are these contagious people and you can never test fast enough or accurate or thoroughly enough and then they say there is risk based selective quarantined which means what it means that so some of these people are going to be at risk and in this case we obviously mean old people so old people and I draw them with a cane not because all people aren't fit just because they have better tastes in canes so and then there are young people and they run a smartphone with tik-tok and what we are going to say is that you youngsters you're not really at risk from this so you go out you sneeze on each other you go about your life normally and you old people basically stay at home until all the young people have immunity so we ramped up the cases and then it flattens out eventually in the low-risk population and at that point there is enough herd immunity right all these people are now immune so that the old person here even if they now go out again they won't catch it because everyone's already had it so they argue for this particular strategy and or at least they analyze this particular strategy now I have to say at the beginning that the core assumption here is that this quarantine of the high-risk people you can do basically in a perfect way so the assumption here is that you are able to perfectly quarantine all the high-risk people and that's the level of infection in the low-risk population has no influence on the level of infection in the high-risk population and in my opinion I simply don't believe that you can build this quarantine I think even these old people they need food sometimes the nursing home need staff so even they even if they can reduce their contact to the outside world they cannot fully be sheltered and that means the more infections you have in the low-risk population will have in the high-risk population so I think the fundamental core assumption of this model is quite flawed that being said let's analyze it so we assume that all the high-risk people none of them is going to get sick because they all stay at home so the math in this paper is

actually pretty basic so we'll go through it a bit more detailed so we'll understand the core argument so they introduce the following quantities M here M is the low-risk population right this is the population size viii or new let's call it new here is the probability so that's the probability that if you are sick you need to go to the ICU right so sick means simply you have the virus and ICU means that it's the symptoms are so bad that you need help from the medical system in order to overcome the disease right so if we multiply so if you multiply the population size by the probability that if you get sick you need to go to the IC what do we get a worst case scenario so the basically the authors here and I find this is the good part of this analysis they really they don't rely on kind of pandemic dynamics epidemiology exponential growth and so on they simply consider the worst case so MD here if you multiply these two numbers what does that mean that is the number of severe cases severe meaning you need ICU cases if everybody gets sick if all get sick at the same time right same time so this is the work so let's say we all go out the lowest population and we all sneeze in each other's faces as much as we can and we just all get sick at the same time then this here is the number of people going to the ICU right and if this so they introduced this quantity B here B is the number of beds in the ICU if is larger then the worst case severe cases right then we are safe so that's the argument basically it's not that we are safe it is no one will die from lack of an ICU bed right which is kind of the lever we have as a population if you assume everyone's gonna get sick anyway and so on so if the number of beds is larger than the worst case number of ICU patients we are safe that's at least how they define safe all right so that's their premise now what are they going to do they're going to find a quantity where they can bound this thing so they're going to find a bound and upper bound on the number of severe cases and if this upper bound is lower than the number of beds then they can say with we're safe with this method say they said see this is a worst-case analysis under their

assumptions all right so I said they don't resort to any kind of epidemiological dynamics they simply estimate this thing from current numbers then introduce two more quantities here P star and K now K is the current number of severe cases right so this is kind of an analogue to this thing here so these two are connected this is the current number of severe cases and this up here is the total possible like the worst case number of severe cases in the future likewise P star here is the percentage of people the percent of people that are sick right and they claim correctly this is unknown so if we could test everybody who is sick not severe just sick and up here this has no connection because of course you can imagine here another factor times let's call this P plus or something which is the number of people who are sick in the worst case which of course in our worst case scenario is one so that's why they don't include it here so this is the current percentage of sick people so this here is a percentage and this here is an actual number keep that in mind alright now if we do some basic reformulations here if we take this P star and multiply that by you see than

the in this corner here multiply it by the total size of the population right we get the number of people who are currently sick right is the percentage of current sequence this is the total size of a population to get the number of people who are currently sick if we take that in the denominator and put K here which is the number of people who are currently severe then we get an estimate of this quantity new so remember what new is the probability if you are sick you go to the ICU that's all the ICU means your severe right so these are the current number of sick people and severe people this gives you an estimate of if you are sick right what's the probability that your severe now they argue that this number it doesn't change it independent so that this quantity here is a constant so the probability that if you are sick from this virus you go to the ICU doesn't change over time so we can estimate it with current numbers right which is a pretty you know a reasonable thing to assume that this stays constant unless the virus mutates or something so we know the total size of the population we know the current number of severe cases you can make an argument about that so do we really know the current number of severe cases because there's an exponential growth involved this might be difficult to estimate and they say the same thing so they say this is

the only time where they reference the dynamics of the situation it grows at an exponential rate so what we can do is we can take a worst-case upper bound they say to be on the safe side perform a worst-case analysis so they instead of taking K they add this confidence interval on it that is based on concentration inequalities so they don't use K they use this K tilde here which has two additional summons here that is supposed to be an upper bound with confidence at least one over Delta here and this you can set for example to be 0. 05 that gives you a 95% sorry a 95% confidence that this is a upper bound on that now this comes from some concentration found and there are certain assumptions behind this upper bound here which I don't know enough about to critique them here I'm going to assume they're reasonable if they are not then of course that is an additional point of criticism of this work all right so instead of using K here we are saying we're on the safe side and we use this K tilde so we know this as well now the unknown quantity of course is this thing here P star what is the percentage of people that are currently sick right yeah so that the goal is now to find that so they say okay if we plug in this upper bound of K tilde then with this probability we can upper bound this quantity nu which is exactly you know what we wanted because we need to upper bound MD that's what they say here so since at the top we saw that M times nu equals MD and we want to upper bound MD so we can rearrange this thing this if we plug in these two together we see that the M cancels out and we can upper bound MD by this quantity here the upper bound on the current severe cases divided by the percentage of the current currently sick people so again they're reformulate and they

plug in they this of course needs to be smaller than the number beds so they plug this in here and they say now what we have to do to see is if this quantity is larger than this quantity of two quantities we know then we are safe now again our goal is going to be to find a quantity that lower bounds P star but up but is larger than this quantity here and they do this via hypothesis testing they call this quantity here they call it he tilde and they do a hypothesis test for classic statistics where they ask is P star significantly larger than P tilde if that's the case then we're safe if not we're not and how do they do that they say okay we have the population right I did draw this at one point let's go back there we have the population

here right and what we can do is we can just go out and uniformly test people I just randomly select people know this is an old person though people stay at home so we randomly select people to test and their test results come back and this one this one's healthy this one's healthy this one's not healthy right and so we have four tests and out of the four one was positive can we work out a hypothesis test from that so can we decide whether P star is probably much larger than P tilde or not and the answer is yes because this is a uniform sample you can work out using classic statistical tools whether or not you can reject a null hypothesis or not and they actually work this out and they do give a number here and that's

this so they say if we test n which is for let's say four and a half times this quantity B divided by K so the number of beds divided by the upper bound on the current severe cases so we test for point five times this many people right then if we find at least ten positive cases or more then with a probability of 95% we know that the risk based model is safe right so the more of course the more infected people you find in this case the better because that means because the number of severe cases stays constant at any given time it means that a lot more people are infected that means the probability that you were going to become severe is lower that way that's why it says at least so again you go out you test n people and according to this formula plug in the numbers here for your current situation if you find at least 10 people then with the probability of at least 95% you know that this model is safe cool and this is done using you know classic statistical testing hypothesis testing literature so I think that is a pretty cool result but

I do severely criticized the underlying assumption which is that you can perfectly enforce this quarantine of course if you can't it means that there is a direct correlation between the number of sick people in your low-risk population high-risk population which means that more of the high-risk population are going to get infected as well which again means that your number B of ICU beds is going to drop severely because they have a higher hospitalization rate which makes your entire model that we developed down there less valid because now this used to be a constant in the model it's now no longer a constant it's sinking and the worse it gets the more it's sinking and yes so that may make what you initially thought was a safe model into a very non safe model very quickly and that doesn't include all the high risk people that are going to be in danger additionally because you can't enforce the quarantine all right so this was my take on that take it for what it's worth and I wish you a healthy pandemic bye-bye