 Material I may not I saved my PDF. I don't think I posted yet the slack. So sorry I think I need to still do that. Is that correct? I didn't post it yet No one saw a giant PDF file on slack Did you know? Oh The slides I did remember to post them. Okay. I've been doing so many things. I came remember fine If they're not there, then I will repost it. It's okay Rather than just wrap up with yesterday's lecture. I mean, I think you got the spirit of it I went through an arc yesterday of Trying to introduce some basics of how Epidemic spread works some of the concepts of speed Strength and then size and then went through an unpack What is it that theorists and theory can do that's useful during a pandemic and Tried to emphasize really public interventions Underlying those public interventions were principles those principles are helpful, but we can't view them as sacrosanct We can't view them as things that can't be changed and we need to update our Understanding in light of both data and and obviously it's an emerging infectious disease. There are going to be differences Today though, I wanted to reestablish the foundations in a bit slower more deliberate fashion Some of this may be review Nonetheless, I want to make sure we're on the same page That'll probably take me about half the class maybe a little bit longer to make sure I do that And then I'm going to begin to extend the basic models in different directions So I have a sort of choose your own adventure path today, but I think I know which adventure path I want to take it when you were kids Did you have these books where you could choose which page to turn to was that an Italian thing too or in other places? Do you know I'm talking about you you had a book and if you choose this you turn to page 75 or if you do the other you Turn to page 32. Do you remember these books? Okay, fine So it's a little bit of that spirit today Where I had a bunch of modules and I can go but that's too complicated to choose your own adventure So I think what I'm going to do today is the foundations and then I'll talk about one aspect in which I go away from some of these simplified models Which is? Considering the impacts of heterogeneity. How does heterogeneity change the very rules and principles that I'm about to explain Tomorrow then I do have a couple options one of them Which I'll definitely do is expand on this notion of generation intervals And then I think I'll probably also talk about behavior tomorrow in some way So that allows me to hit on the foundations and some of the frailties of these models, but also address fixes Just methodologically so that we're all on the same page Today I'll be using For the first part mostly sort of standard nonlinear Dynamics methods linearization and so on Emphasizing intuition though in the second part of the class I hope to extend a little bit to show you how there's a structured model And in fact you could in fact think of these eventually as PDEs I won't do so much there turns out you can do some nice analysis in other ways and then tomorrow I will use some Integral differential frameworks to try to understand generation intervals, so we'll see how that goes So hopefully you learn a few methods as well Thursday and just to keep everyone on track today. I'm obviously here tomorrow I'm here, but then I leave right after lecture tomorrow Because of my flight back on Thursday Thursday's lecture will be a laboratory and just to remind everyone I would like not would like I'm asking anyone who's taking this for a grade to send me back there Jupiter notebooks before class on Thursday I will create some sort of link for you to deposit so that you can put it there because I don't need 37 messages on slack. It's the last thing. I just need them all in one place And so I will put together some Dropbox link or Google form where you can deposit it Okay, so I haven't explained or I haven't figured having to do it But I will figure it out and I'll get that to you any questions before I begin just wanted to get some Housekeeping things in order so then Thursday. I'll go over the laboratories that you did go over some of the solutions And I will also give you a new laboratory and then we won't be able to get to the end of it I will share the solutions afterwards so even though I'm not with you You will have the references for how you do that kind of work and will emphasize a stochastic model of epidemics in the laboratory Okay, I think I've given enough preface questions No good Let's not necessarily good, but that's how it goes. Let's start. So what I wanted to start with again today are Epidemic basics and I'm going to gradually build up in some complexity. So take you know Take it easy. It'll it'll be fine. And then maybe it'll get more complicated. Maybe not I want to begin by reminding folks That typically one writes these standard SIR models as the following Where there's some infection rate There's a recovery rate and We get this sort of Expectation for dynamics But I want to first unpack where this comes from in the first place. How do we interpret each one of these parameters? Okay, so we should think At the start, you notice we have this beta. We have a population And we have all these Individuals in this population and maybe one of them is infected and we want to know what happens and So we can think of this in different ways We can think of both the susceptible individual or the infected individuals as having See contacts per unit time and let's think of it from the perspective of The infectious individual and really depends on where I want to put this divide by n factor But they have see contacts Make some contact rate of Those contacts S over n will be susceptible of those contacts with susceptible people P some probability of infection Given contact, okay So we have this notion That the susceptible individual or the infected individuals contact around people if we take the perspective of the infectious individual There's some chance that they've run into a susceptible person at the beginning It's almost certain that they have because everyone else is susceptible and there's a chance of transmission Which means that we could have something like For each I individual see S over np new Infections or time Which we write as beta I S over n and we identify beta as Cp, okay, so I'm just trying to unpack a little bit of what happens and so this is then a model in which we have a Population that's fixed and we may be transitioning Certain elements, so we're going to make the further assumption that we ignore in this time scale Births and deaths so that there is a constraint on the total number Okay, so if we have this model we have this constraint now We understand a little bit of where that beta comes from we end up getting this end this population size in the mix so we would like to Rewrite things and soon after just this next little section here I'm going to rewrite things so that we think of them as fractions. I Note though, however That when we do our laboratory on Thursday, I'm going to get back to this notion of individuals So I wanted to remind you that these are actually individuals Because if we want to build a stochastic simulation, we have to think do we have a hundred individuals or a thousand or ten thousand And how many actual infectious individuals is going to make a difference in terms of how quickly the next infection event happens, right? Nonetheless, we can rescale all of this and I'll Do a standard thing? I'm being very deliberate here to start So I'm going to think of that little tilde and then eventually I'll get rid of the tilde and we'll never think of the tilde again I just want to do this once Which is the fraction of individuals that are susceptible infectious and recovered and so you see wherever I see an S I need to replace it with an S tilde and an N Right because we have the fraction and the total So therefore I should be able to write Hopefully I don't screw this up. It's not looking at that side Okay, something like that. That looks like I didn't mess it up as you can see eliminate eliminate eliminate eliminate I Have ends everywhere that I can get rid of Does everyone see that? Good, so now I can rewrite this as Simply S tilde dot is minus beta S tilde I tilde I Tilde dot is beta S tilde I Tilda and R tilde dot minus gamma I Is gamma I tilde and we'll never speak of the tildes again We're just going to call those SNI. I'm going to use them as fractions from now on But I just wanted you to see that in fact you can reduce this from this Model in terms of a population into one in which we think about fractions Okay Is this clear how I did this in sequence of moves and you're used to doing such moves And in fact we could do even other moves which is to make this dimensionless in terms of time Rescaling out the time and in some ways if we rescaled out the time is something happening. Oh Yes, high-quality chalk. Thank you Excellent. I'm not just the last person to use the prior chalks Okay Good, so now we have The model that we're going to play with and maybe I'll just put it over here So I don't get rid of it. Okay. I'm just going to keep that there as a reference That in this rescaled model because I only need to keep track of Two things not three things all I need to know is that we have infections which go into the eye category and then they recover I don't need to know are because I've s plus i plus r is one Okay So I'll hold on to that and now I'm going to go back and say what is our initial condition our initial condition in this space of The fraction of individuals who are infected and the fraction of individuals who are susceptible and so on is 1 comma 0 Everyone is susceptible. No one is infected and Then we can ask the question what happens what happens when We have a small perturbation We introduce a very very very very very small fraction of infectious individuals Maybe just one so epsilon is order of 1 over n and is a big number epsilon is very small Okay So what we know to do this is a nonlinear system of differential equations What we know we're supposed to do formally and obviously I've given you an intuition But I want to do the formal stuff a little today just to make it coincide as we should linearize Near fixed point and We know that the procedure there is to linearize them on their system calculate the Jacobian find the eigenvalues and See if any of them are positive and if they are that implies that the infection would take off so We can calculate this Jacobian by examining those equations and Notice that if we take the derivative of the first that f if I call the first one f of s comma i and the second one g Of s comma i Then I could take the derivative of the specter s and I'm left with minus beta i and here minus beta s and here beta i and Here beta s minus gamma Which we need to evaluate at the fixed point one comma zero Which becomes zero minus beta Zero beta minus Gamma now in order to find the eigenvalues then formally speaking I've brought my own method today to wet this thing See the experience in such matters Efficiently erase anyone know can you look at that? I'm just starting to erase here I have these notes written down as well, so I also will give you a PDF copy of the notes It won't be on it won't be horizontal. It'll be vertical, but nonetheless you get it. I'm gonna keep going unless anyone has questions about this Okay, I'll race half at a time then move to the next so you can see That what we need to be doing is calculating the determinant of J minus lambda times the identity matrix Setting that equal to zero and finding the values, but you don't need me to do that you can read them off Right because if I put a minus lambda minus lambda there this zero means that I can just read off The eigenvalues from the diagonal, okay? So what are they? It's clear that lambda one is Zero and lambda two is beta minus gamma Okay Now the question is how do we interpret these eigenvalues? All right keeping in mind that we think that this is supposed to be some small perturbation Where this is some small it's some small perturbation us and you I So we have some small perturbation that is linearized, and so it's growing exponentially or not growing exponentially or decaying exponentially so what are the interpretations of these two things remember that we made a perturbation and We think something may Change perhaps positively or negatively something is zero. What does the zero mean? Does I know it means not changing so there's no the perturbation doesn't change But what is that perturbation that wouldn't change? S. Okay, so let me go back and Let me maybe draw the following S one one I line Can't go Here's your initial position What is the perturbation? That's going To lead to no change you said s what do you mean by s? I think you're right But I just want to know what you said along the axis so if we did One minus Epsilon and stayed at zero rather than putting that Epsilon there There would be no change. What does that really mean? We can really think of this as one minus Epsilon Zero Epsilon in the full system susceptible infected recovered How do we do that biologically, how do we just move the s into the our category of vaccine, right? So in some sense this zero eigenvalue is Vaccination when we vaccinate we're moving The system along this x-axis which turned out to be very important later right now I'm dealing with the small perturbation, but you can see I could make arbitrary large perturbations And it also wouldn't change the system because I still have i equals zero So it's true for small this one has this feature I know there's when I went to see Tosca also had an obstructed view so feel fee to move around There's a whole bunch of unobstructed views in the front row Appreciate that my tail is in the front row. He has an unobstructed view Okay, if you want to move to the front row, that's fine So we can also think about this In terms let's go back to this particular case Where if we wanted to actually see and calculate the eigenvector associated with the zero eigenvalue We could go back and ask If I take J, which you see is zero minus beta zero no, yes beta minus gamma and subtract a zero eigenvalue from it and say where Does this happen? Right, I've subtracted the zero and multiplied by zero on the right side You can see that I have to hit something which is going to be seeing this side unless this value Has a zero in it. So it's something with the zero the only way to satisfy this Equation for some eigenvector is if that second component is zero, okay? So I'm just giving you this sort of algebraic way, but really the intuition is that this is a vaccination Okay That's one So this is a The vaccine route What about B? The lambda is beta minus gamma which you see obviously we've already talked about this We have a condition And this is lambda two Where was lambda one? Right is zero. We already did that you can see this is just gamma of Beta over gamma minus one which is just equal to r naught minus one over the characteristic time of being infected Okay When the transmission rate exceeds the recovery rate you get takeoff So we have a threshold criteria and the threshold criteria can be written in terms of the basic reproduction number We have a time scale because this also we think of this as the speed and This as the strength This basic reproduction number will characterize how strong the disease Transmissibility is at the individual level if it's greater than one that we have a positive speed and we have takeoff and this perturbation Obviously, I'm moving things along this axis and then something is going to happen and it will cue so close there That's not realistic and somehow we know that it's going to land Somewhere here, but of course it might have landed there It might even land it over there. It won't land over there in the corner For various reasons that I will unpack again today even though I talked about it and just to remind folks and This may be something I'll do later in the week But we have this recovery rate gamma Which means that if I ask you what's the probability that you're still infectious after some time? T and here's T Here's zero You're definitely still infectious at the start and this is an exponential whose mean time is ti You can see that and in that period. There's a sort of a rate And so I need to multiply because that's a Constant rate then you can see why if I have a rate of transmission beta Right. I'm going to have on average beta times the period of new infections Which is why we have this R naught and which is why when that's greater than one we get spread I'm filling in this in all the details for which I sort of went through quickly and hopefully now This is making sure you get it questions about this part in the back question Probability that you're still infectious after time T The question was for those watching at home. What is that y-axis? I know it's probably if still being infectious after time t Given a recovery rate gamma. It's exponential Tomorrow I might talk about different kind of generation interval distributions where first you go through an exposed period Then you go through an infectious period. So the probability of being infected not infectious, but the probably being that you're still infected May not necessarily have this shape anymore, and there's certainly the probably infectious doesn't have that shape right So if you have to go through two stages Then clear at the beginning the probability of infectious which is different than being infected Is probably near zero and only later it goes up, right? So and then for if I did the probability Anyway, you can see why they could have different kind of shapes. Let me do that more tomorrow. So We have that as our initial criteria all that tells us is whether or not There is some sort of takeoff and If there is takeoff Then some sort of dynamic unfolds okay, and We expect the dynamic with time This is population Fraction and here one here zero I'm going to draw this is going to be s and here I and here are And I have tried to make these two gaps the same Because s plus i plus r is one and at the end there are no infected the epidemic is over therefore whoever Hasn't been infected, right? There's still a few fraction. That's the gap between 1 minus r is whoever's left at the end Okay, I talked about this yesterday, but I want to now unpack it again slowly and deliberately We showed there was a takeoff The criteria for whether this happens as opposed to decline is whether or not the transmissibility is greater than the recovery rate The transmission rate greater the recovery or in other words are not greater than one There is another condition here where I dot is zero You will notice at this point still a lot of people get infected because it is not precipitously I Equal zero it's I dot equals zero and I'm going to explain this lecture why the herd immunity threshold is a very important concept if we vaccinate and it's a very dangerous concept if We let natural infections Do their thing? really for two reasons one of which is That natural infections come with consequences and second of which the herd immunity threshold In an active infection is not the end of the epidemic whereas it is in concept and theory when there's a vaccine of vaccination So when you pass it one way or another you have very different outcomes, which I will try to elaborate on How can we find this point? Well? We recall because I wrote it over there beta si minus gamma I is equal to zero which implies when s of t is equal to gamma over beta or one over r not then We are at a point where the number of susceptibles has been depleted so the herd immunity threshold happens when we have susceptible bull depletion the amount of susceptible depletion that we need depends on the intrinsic Strength of the disease the stronger disease the more susceptibles must be depleted before we reach this point So for example far not as to we need to get to a half Right are not three one-third four one and so on So in other words the septal's have to get lower and lower and you can also think about this a different way Because if I were to think About this speed and not set it to zero, but write it as gamma beta over gamma Sorry gamma I beta s of t minus one You would see that I could write R effective is r not times s of t Everyone see that equivalence So really what this is saying is that we start with this basic reproductive number in time because of susceptible depletion Although on average and I've already erased it, but remember I say I contact with a certain number of people a fraction or susceptible That's reducing the effective Strength of the disease over time and when that effective strength gets to one We're at a replacement value and if we're at a replacement value Then we no longer go up and soon after the next few people get infected we start to go down Any questions about that? I see some puzzled faces. So maybe there's a question No, okay fine Now I've explained that concept So now I want to go through and just explain again where we end up Good. We're getting almost Some good major concepts and getting the foundations here set So now we've shown if we have 1 0 and We go to 1 minus epsilon epsilon We have an outbreak When r naught is greater than 1 Which means now we have to go to 1 over r naught i star r star Mean we hit that herd immunity threshold Right, there's still the other kinds around so I'm not going to set it. I don't know why I wrote star. I H it So we get to a herd immunity threshold, it's not in equilibrium. Sorry didn't mean to imply that and Then after that with time we get to s infinity Zero r infinity. I just want to go through these sort of sequences of moves But these are the sort of landmarks in the classic kind of models and All this seems highly simplified This is precisely the sort of mindset of how some of these early models in 2020 were operating Showing these sort of single epidemic curves susceptible to pollution and also why when people saw things that went up and down they not just Political leaders who weren't trained but even some folks in this space said well went up and down so therefore We're on the other side of this And inevitably we're going to go here and state it tomorrow I will talk about ways in which we don't necessarily need to have that happen, but I need For you to even see the comparison. I need to walk through all these steps first and reinforce Okay so We now can go back to this diagram. I've already drawn and You can see that it's going to go somewhere Where is it going to go? Well, I've already made the claim that it's not all going to the same place For example, it could be that all these flows somehow got trapped and moved into the corner There's still some lingering infections and by the end no one was left susceptible In fact, you will see some articles out nowadays like why is it that some people haven't been infected? With COVID obviously vaccines is a big part that reduces likelihood But even in these simple models there even for a disease that has a relatively high or not We still expect that some people haven't been infected in part in structured populations It could be that a whole group hasn't been infected But it's more likely that in fact Someone got lucky people around them were infected and now most of the people they interact with have recovered and they're sort of Buffering the risk because it's very hard for the disease to reach them because it has to cross through all these other Interactions with people who were already either vaccinated or infected and recovered or maybe both and I should also point out that you know that it's clear for All of us now have a lived experience with COVID so I'm writing stuff on the board But you have your own internal sense and we also know I'll get into in the moment that the vaccines in some cases are not perfect All right, and also if you're young you aren't even eligible for a vaccine And if you're a parent like me and have a couple kids who go to a French public school Your whole family probably got infected at some point late last year Right, even if you were vaccinated or doubly that's I'll get back to some of those ideas That was a bit of divergence. Let me go and figure out. How do we figure out where we're gonna go? We have lift off And I asked you to do this, but I went a little quickly. I'm gonna do it again. Anyway, I realize I'm doing things twice But that's okay. Some of you this is the first time you're hearing it if I look at those equations and Then I write How much is there a change in septal for every change and infected you can see I don't know why I've decided to do it that way That's annoying It's clear that I want to do it that way so I can cancel things you can see that I should get something like minus one plus gamma over beta s Good and so you can see that I could turn this into di equals ds negative one plus gamma over beta s and I could take the integral of both sides and I would get I equals minus s plus log of s divided by r naught plus some constant and You would see that initially When s at the beginning is one and I is zero we have zero equals minus one plus zero plus a constant As I said unobstructed seats available in the first row all day Next lecture to probably gonna be available. This is my initial condition Now I want to solve for the end at the end I Is also zero which is convenient, but now s is not So you can see that if I is zero at the end we get something like s Infinity let me see how I like to write it. That's fine minus one Times r naught Equals log of s infinity for every value of r naught. There's going to be unique solution to that Which is why I drew this and not them all converging there This is the final size relationship it says there's a speed if the speed is positive or Realize to say the strength is greater than one the speed is positive. We hit the herd immunity threshold We eventually go to here and the strength is determining both whether or not it takes off at how big the disease gets But as I showed you yesterday, there's not a unique relationship between strength and speed There's a unique relationship strength and size Okay, any questions about this part fine now that I've set all of this up I can ask the question in a different way. Here's where it went and you'll also notice that this s infinity is Obviously larger Nah I was thinking it is obviously smaller it obviously insane statement I was just about to make which I caught myself because I was in my mind thinking about the recovered people This is obviously less than one over r naught, right? So we first hit the herd immunity threshold and then we went down and got even Glad I avoided that perilous mistake now I Want to talk about how we stop things This is what happens if The disease goes unchecked in a population any questions before I start to erase. This is one of the days where we go like through six boards It's one of those six boards days It's okay. It hasn't actually been that challenging. You're sort of it's just a flow right now We'll get a little more challenging in a moment Any questions before I race? Okay, yes, no, otherwise I'm erasing and I'm talking about vaccines No, it's not a very stupid question. Is it called finance finite size me Final final. Sorry. I wearing a mask the final size relationship F-I-N-A-L No, no, no, okay No, no problem Yes So we have this final size relationship and also I will for those who are at home I will I have a PDF of these notes. It's not quite what I'm doing, but it's close enough and I will share good, okay, okay We mentioned vaccines are here. Let me erase two boards and then I'll pick up in a moment Good This is what happens if things are not stopped So now let's go back to our original idea because we've figured out the herd immunity threshold We figured out this final size of The epidemic implicitly it relates to strength But we might not have just waited for that outcome Maybe we took some proactive step and there was vaccinations So let's imagine that we did this move one minus the fraction vaccinated Zero fraction vaccinated, so I'm gonna write it in the full sIR context. So you see it Will there be an outbreak here? I'll pause Will there be an outbreak? postgraduate ICTB diploma students Will there be an outbreak in this theoretical case that I've just unpacked if you're not sure tell me how you Might even figure out if there's supposed to be an outbreak I will point out that this question there's some people saying no some people saying yes My impression if I were sitting in your chairs, I would think oh this lecture hasn't been that hard that now And now I've asked the question like oh wait a second. I'm not as confident in my answer So let me it's a little trickier more subtle. I saw someone shake their head It's any was that a no in the back. You don't think there'll be an outbreak Yes, you don't think there's gonna be outbreak or no, you don't which one No outbreak. Okay, what happens if I vaccinate a very tiny fraction of individuals something like epsilon What happens if the fraction that I vaccinate is very small like epsilon? Let me explain You agree that I can write it in this si plane one one line Can't be there and This Depending on how big I make my fv Denotes how many fraction I've vaccinated? If fv is one then I go all the way down there I Haven't said fv was one. I could also vaccinate a very tiny number of folks over there Do you think there would be an outbreak if I vaccinate a very small number of individuals? As long as we don't have infected people we shouldn't have an outbreak right now an infected individual shows up in the population Sorry, I wasn't be I was implicitly assuming that yes We first vaccinate and then somehow there's an infection that shows up. Sorry. I wasn't clear Now you probably are more concerned about this outbreak if I have a small number they should maybe there is a threshold But maybe there's a threshold. Okay? So the question is it's not just that I vaccinate some people I have to vaccinate a lot of people Clearly if I vaccinate an infinitesimal number of people and there's an infection. It's at least theoretically I think you can feel that it's not Going to stop the outbreak. I'm about to show you that this suggestion of a threshold is the right intuition What I'm trying to say is that if you vaccinate a very very small number of people and an infected person shows up in this theoretical model You expect an outbreak If I vaccinate everyone it's pretty clear there won't be an outbreak by definition The question is do I have to vaccinate everyone to stop the outbreak or is there a threshold? Is there some point somewhere? Where there is no longer an outbreak so the answer is yes, let's go find it Okay so Let's now think about this Dynamic and recall That when we Remember when we did the linearization on the fixed point And we're going to start near this fixed point remember this is a fixed point of the dynamic Any value along this line is a fixed point. I haven't said if it's stable or not But it's certainly a fixed point Which means I can calculate the Jacobians and I can calculate my two eigenvalues One of them was zero by definition. I vaccinated some more people. I don't have any Change which is I think what you were assuming in terms of your notion of there's not an outbreak, but there's also a second eigenvalue and if you recall This was beta times s Initial minus gamma. I had replaced that value with one because that was our initial condition but if I vaccinate people then This becomes beta one minus f v minus gamma Right because there are fewer susceptible people which you can even think of this in terms of the probability that I run That's a susceptible person and stops being one but rather one minus f Okay, which means I can write this as gamma R naught one minus f v minus one There is now a new effective strength which has been reduced because of the vaccines the condition for an outbreak Outbreak Happens when r naught one minus f v is greater than one Or When one minus f v is greater than one over r naught or f V Yes Yes, good. No, I'm doing it right. This is the outbreak condition and our control condition is Simply f v greater than one minus one over r naught Just want to make sure I was saying the right words in the right conditional order. Thank you. You saw where my brain was going okay, if I want to control things Then I have to vaccinate and I can write this as r naught minus one over r naught And I'll write it higher control When the vaccination threat is Greater or than r naught minus one over r naught Let me now draw what that looks like if I have a Disease with a certain strength are not and I'm only going to be interested in those that would take off Anyway in the absence of vaccines and I asked what's my critical? vaccination threshold right two and three and four and one Half and I draw that and one third Draws two-thirds three-quarters Draw things here. I will get a shape like that The stronger the disease is the more that I have to vaccinate in order to stop things What does that mean though here in? terms of this diagram what it means Is that there is a value and I just also I want you to recognize one thing before I say that Does anyone feel like they've seen this term one minus one over zero r zero before in this lecture? Where have you seen one minus one over r zero before two one and a half blackboards ago? Someone is speaking from the internet. Can you repeat what you're going to say? I just couldn't hear it Just repeat it, please Or type it When we integrated Yes, but actually even before when we integrated because we got a bunch of other stuff Do you recall that there was a herd immunity threshold? When s was one over r naught right when we had reduced it to that much Which means that the cumulative number of infections Was one minus one over r naught In other words, we got to the herd immunity threshold where the infection was at its replacement value When that many people had already been infected vaccination Right, this is essentially herd immunity threshold via Vaccination it's the same thing the same value but we get it in A safer a more ethical way you could argue Because there are two things that happen there first of all At this value. Yes, just like the herd immunity threshold value, then we expect I dot to be zero But Remember in an outbreak we cross these points When we had a lot of infected people around we cross this value up here up high And then it goes down like this and goes down like this But in fact if we ever get down over here these are all stable parts of That fixed point line and these are Unstable If we don't vaccinate enough we get smaller outbreaks, but if we vaccinate enough then Not only are we at this kind of critical value this herd immunity threshold, which happens right here This is this critical vaccination point but Because our eye is so small then we don't have any of this Epidemic overshoot we don't first see a lot of infected people and then it turns around we just turn it around Is everyone that's an important concept So in other words if the disease unfolds then yes at a certain point it starts Turning around once we have had a lot of people infected, but there's even more people infected remember s infinity is Less than s herd immunity threshold In other words, there are more infections even after that point, but if we vaccinate Then basically we don't get any new infections We just stay near that line in other words the human number of infections is basically zero to first order Okay, this is almost done with some of the foundational concepts I just have a few more things to go and then I'm going to switch to heterogeneity I'm on track here any questions about this so when people were talking about the equivalency between Natural infection and natural immunity and vaccination derived immunity. I want to point out again two reasons first of all The safety and health risks of being infected with COVID-19 especially in older individuals, but for all ages so much greater than whenever Incredibly small risk and there are some people who you know have some medical reasons that they shouldn't or can't be vaccinated But you would much rather get your immunity through a vaccine You as an individual and as a population in order for us to get to that point where we have that herd Immunity threshold then a lot of people have to be sick and a lot of bad consequences Whereas if we can go this way rather than that way We end up not getting all this other infection so it has both the population individual protection helps the individual But it also helps the population as a whole it also says why you don't have to Vaccinate the whole population, but you have to vaccinate a lot for you to see the immunity benefits right this notion of a herd immunity benefit The more likely it is for disease to spread The more you have to push this to the left and why vaccine coverage, which is really what this FV is about Become so important. Okay So good we've explained those concepts There's one more board. I want to do on basic stuff. So I'm about to erase the board yet again Any questions? Otherwise, I'm going ahead And a racing okay Good We got this we got this. It's a nice diagram. Hopefully you have it because it's going away Clear off the whole board because I want to do just one More thing I'm even going to race this corner because I want to do one more thing there and Just as a reference Make sure I'm using the same thing so now I want to talk about Another sort of basic concept and then I'll move into things that I think for most of you will not be familiar But I just want to establish this last idea We have been talking about s IR Models in which there is some transmission rate beta Some recovery rate gamma But now we could add some loss of immunity alpha And it's becoming increasingly clear that this applies also to SARS-CoV-2 It certainly applies to things like cold and flu and you can think of this as either Or both the effect that the immune system has some memory, but maybe there's waning antibody levels and even though you have memory T cells and B cells still maybe it's not sterilizing you can get infected again It also could be the case that the disease itself is changing So what we ascribe to waning immunity is because now the target is different because of some sort of antigenic drift or shift And even though I'm not talking about all the multi-strain components effectively what we see is some loss of immunity if we do this Then we can write this full kind of model over there Which I'll explain in a second. I'm not sure if I necessarily need to I might skip it But I will explain a simplified model Which is imagine now that this alpha is so fast that we're even going to ignore that other component for a moment I'll explain the full. Maybe I'm just trying to figure out how I really need to do it. The concepts are very similar So I might skip imagine we take a model in which rather than having this Immune period we have beta, but when people recover they immediately become Susceptible again, and you can think of this as the limit in some sense of this alpha being very fast And we could do that too, but no, I don't know. It might be a little bit boring Nonetheless, I want to analyze this model now And just to ask the question well, what's going to happen in this case? Versus this case where we saw an epidemic when we didn't have this Something else could happen here We still can have the notion of an outbreak, but our equilibrium may not be the same anymore so let's again Do our playbook, which is we start Near 1 0 and we imagine that we introduce a small perturbation And we can then analyze the dynamics and ask well, what are the linearized system near this fixed point and You can read it off up there if actually now I did I never even wrote down the equations which that wasn't nice in me This would be a version with no immunity Sorry for not running the equations You can see that this would be minus beta I and here is minus beta s plus gamma here is beta I and then beta What yes beta s minus gamma Which we evaluate at this fixed point and we end up getting 0 0 The same sort of terms as before such that we read off lambda 1 is 0 and lambda 2 Is beta minus gamma Now this of course I have to be a little bit tricky here Because in fact this is actually only one equation because I have s plus i is equal to 1 So I'm being I shouldn't have even done this. I should have reduced to just one system of equations, but that's okay So I want to focus on this for the moment and Just point out that again you have an outbreak For the same reasons as before When are not which is beta over gamma is greater than 1 in the si model We get an outbreak, but our equilibrium is different Because now in fact they're just the same equation we have s star is 1 over r naught and Therefore i star must be 1 minus 1 over r naught. So in this situation what we have With time is Susceptibles going down Infected's going up si and the addition of these two must always be one so that we have a speed of beta minus gamma a strength of R naught and a size of r naught minus 1 over r naught and We have an endemic state Rather than an epidemic where if something goes up and down and you've probably heard a lot about endemic transitions because We clearly are not eliminating a SARS-CoV-2 anytime soon and some people talk about endemic is if it's a good thing Oh, it's endemic. It's just around. There are a lot of bad things That can stick around for a long time so I would just caution that Especially if we don't mitigate or do anything to prevent it or don't vaccinate or can't treat or so on These outcomes can also be bad. They're just perpetual They're just not something that goes up and down and they depend somewhat on the duration of immunity If we added a vaccinated class here, which people often do you can imagine I could have turned this and added a v-class and maybe some people went here and some people went there And you can imagine making more and more of such assumptions But if there's loss of immunity then generically what you're going to find is that there is a risk of reaching an endemic state There's only one other caveat. I want to add there Which is that these kind of models even if I were to add the loss of immunity tend to converge to a fixed point But let me ask something about when I write this minus alpha r term And I asked the question here's time. What's the probability that you're still immune after time t? What is that shape based on my model that I've written there? I've asked a version of this question like ten times in the in the last couple weeks And I feel like I should put that on the exam just to make sure you definitely know it If I've written such a model where I have probably per unit time of losing immunity What's the probability? I'm still immune after time t exponential e to the minus alpha T and it doesn't actually go to zero. That's not what exponentials do you get if you all recall remember I derived that I said if I have a rate of something happening then 1 minus alpha times t after raise it to a power And I get the exponential out anytime you see these first-order processes just like a radioactive decay process You have a residency time. That's exponential Does that seem realistic to you? Do you expect that most people who recover are immediately infectable again susceptible again? No, so it is possible that we could have done something like r Minus I Won't change this. Let me let me just change it. So if I have r dot is Gamma i minus alpha r it implies that I could have written like r dot is Gamma i minus Gamma i T Minus tau I could have written like a delayed differential equation. It says exactly tau ago It lasts for six months your immunity and then six months later. Boom. You're in fact. You're susceptible again because the individuals who came in Tau ago. That's the new rate in are now popping out. You're losing them at that time So because of the delay, I don't write the r because that's the current value. It's actually the rate of new Recoveries tau ago that then leave tau in the future So if I were to write this as a delay differential equation I would write it like that Which means the probability that I'm immune would go like that and then that It would be a step function downwards Does that seem realistic Also, no Probably it's something like that right some smoother thing So just to point out if you ever get into this business that what people often do Sometimes they do this and I'll talk about generation intervals more tomorrow But what they often do is say there's an s I and Then R1 R2 R3 they make a bunch of intermediate states What is called? the linear chain Trick and what you do in some sense is you increase the rate of moving from each one of these states So if I have alpha and I had n such states I would make each transition go n times alpha. So I'd go through n states, but n times is fast Preserving the duration But instead of making it like this. I would have it now the probability. It lasts a certain time would be Modal and the probably that I haven't I'm still immune would have now a curve shape Okay, so there I won't go through all this just to let you know that these choices have some biological consequences And if you make these kind of choices, then in fact you could get potentially oscillations In an epidemic model because what happens especially if there's new bursts of individuals who are vulnerable you get an epidemic wave There's recovery. There's still some infections going around But then later all out pop a whole bunch of new susceptible people and that can lead to drive another Oscillation as we have waves of new vulnerable people and that is Increasingly clear part of the story of SARS-CoV-2 as well the waning immunity is part of the story and It's not an exponentially distributed waning immune. Okay That is my effort at foundations Any questions before I start to move into some new territory for the last 30 minutes or so? Okay Good very nice We've covered a Lot of material, but I don't think we've rushed it We haven't moved too slow. We haven't moved too fast I hope it's a Goldilocks level of information. So we'll see I Want to do one more module here at the end? Yes, I've erased everything, but you have a question. Yeah in previous lectures you write that also for solving the same issue You add an exponential term a Like beat the parameter depends on yeah in the prior lectures you're referencing I think the fact that in the viral ecology model. I had a delay differential equation But there was something like e to the minus omega tau That's what you were referencing Yeah, the question is why in my viral ecology models Did I have a time delay with the needed minus omega tau term and here I did not The reason let me explain the answer to that question Then if you have another question just because people might be listening want to know in the chemostat model That delay was the latent period But it had to be discounted by the fact that the cells in the chemostat might have been lost in the intermediate period So in order to do the delay differential equation I had to reduce The amount of lysis by a factor e to the minus omega tau where omega was the lost rate and tau is the latent period here There's no way For the I haven't included a death term or any other thing So I'm assuming that everyone survives the infection and then later comes back so that term is one And that's why I don't see an exponential So this explanation will not be physically visible that's the Or physical achievable In this case where I have an SIR model then I'm thinking people Became recovered now now if for for example this period of Immunity Was super long that I worried about demographics then yes I would not just put all of the people who recovered 20 years ago back into the susceptible pool I have to worry about demography and deaths for other reasons and I have to discount it on This time scale I'm ignoring burst and death. I ignore that discounted We can talk about it later at the break if that's confusing Okay But that's why I don't include that term okay, I want to now wrap up with one choose your own adventure concept and Talk about heterogeneity and the Stuff I'm going to talk about today. You can find and I will post on slack Some of it though obviously it's not done in a pedic always at the same pedagogical ways and rose at all journal theoretical biology 2021 and that has a lot of other references, but I will be following some of that methodology and To do this I want to first start with a simplified model. I'm going to draw Just to have this in mind imagining that I take my S population and Break them down into two categories s low and s high and The L and the H denote the fact that these individuals are less likely to be infected and these are more likely to be Infected the reasons could be intrinsic. They could be behavioral. They could be a combination and so you can imagine that contacts between I and SL This might happen at some rate beta L and this beta and in fact, I'll do it beta times epsilon L Epsilon high and then I'll have recovery and I won't worry once people infected all soon They have essentially the same infection you could imagine writing down a model like this This is one way to break a symmetry to him introduce some heterogeneity and I had some lecture notes where I went through this two class model But actually find it a little bit easier to do a large number I just want to show you that if you can agree with me This is possible. I could induce three classes or four or really a very large number of classes Okay, and before I even get started I Want to point out That it's possible. I can choose the epsilon low and epsilon high so that the average Infectivity was the same in the homogeneous model Right, I could choose it so that I have a proportion of lows and highs and different Vulnerabilities that we basically seem like we should have the same dynamics But these are nonlinear systems And I just want to ask people even before I start if I were initially to have The same amount of less likely to be infected and more likely to be infected individuals and I look forward in time Do you think that in the future we'd have the same number of SL and si? Everyone understands my question. I start with 50-50 split of the SL and si In the future, do we still have a 50-50 split? You're saying some nose. Okay, so people already bullying intuition someone. I haven't heard from why no in the second row What why no simply because in the end we will have more as H than SL because the no the opposite the opposite yes correct because the people that are more likely to be infected will not be Will be infected correct so the intuition here is that if we have a higher rate of Infection then these individuals will probably be infected first Which means there'll be fewer relatively speaking here than here and Now you can imagine I could add a twist Imagine that in a world of just SLs there's no outbreak because it's actually not enough to drive the outbreak there alone There are not as less than one and here this one is greater than one What you can see then Intuitively there's a possibility that if this is depleted first the outbreak itself could stop earlier Because the individuals who are more likely to be affected get infected early leaving only these and actually leading to a slowing down Right, I haven't written any equations, but I'm just asking you to sort of think intuitively what might happen And if you want to you can try this little two-state model and prove to yourself That in fact that what's what's gonna happen, but it gets a little bit algebraically Yeah, not it's not that nice, and I want to just take that intuition and actually build it up a bit more formal Okay, so this intuition that if you have a higher vulnerability to infection the person is out more interacting more They're the ones likely to be infected first Depleting the population in some sense if people's actions are fixed. There's a lot of caveats here If they're fixed if they're not then we just resupply right so you can imagine Unfortunately the cash register during early COVID the person in the front line interacting with a lot of people if they got sick You put another person there, and you're just moving people from S low to SI all the time fueling more infections What I want to start with is just the consequences of heterogeneity Then we could talk later, and maybe even have a discussion about what that could be So what I want to now do is imagine that I take this population s and divided into categories Which I'll call epsilon and each one of these categories is gonna have an intrinsic vulnerability So that I Can write instead of the standard way I Can write something and in fact I shouldn't I'm not going to end up doing any movement So I'll just write it that way each one of these can you think of as a little bin minus beta and I'm going to assume that I've used these vulnerabilities that's just going to scale the overall rate You can deal with other sort of monotonic assumptions. I'm going to assume that this variable denotes how it scales the intrinsic transmissibility This is a subpopulation. It's interacting with infected individuals And so therefore I'm going to write something like that an infinite number of these I can imagine a finite number and then turning into a continuum Such that that the new infected Must be the integral epsilon so what I have is a system of equations that describes not just one group of susceptible individuals But a spectrum and now I want to establish a for a few ground rules here I'm going to assume that initially integral the Epsilon s zero of epsilon is equal to one so initially I'm going to assume everyone is susceptible but I'm also going to Claim that over time this Can decrease So I shouldn't think of s of epsilon as a probability distribution because it doesn't integrate to one But rather if I integrate it the norm the sort of magnitude of the probability mass in this distribution is equal to the fraction That are susceptible, so it's less than or equal to one given this I Can also do another definition just to kind of get some definitions in order I can ask the question What is the average? Vulnerability in this population which is going to scale this attack rate Well, if I want to take the average in the population Then it seems like I should take this integral of epsilon times s of epsilon But remember this is not by definition of probability distribution, so I need to do that in other words This is integral the epsilon epsilon s Epsilon over s And I will try to I know that I'm using s in two ways One is a big one. One is a little one. That's not so satisfying. I probably should call n sub s But I'm going to do it that way whenever I want it to be the distribution. I'll put it up epsilon okay So you can see that we have this distribution of Vulnerabilities or susceptibilities and we have an average vulnerability in the population Yes on time. Yes, absolutely. It's of time because this Time time they're dependent on time. They're changing in time. In fact, this is what I want you all to think about Imagine that we started With some distribution Epsilon s of epsilon and this was the initial value The question is what does the infection process do? well, I Would claim that if this were some feature unrelated To the rate of infection then all of the infection process does is drop all of this down Right, it would just move it down Because different categories or some other feature unrelated to infection infection just Removes individuals at a rate independent of this epsilon but This epsilon tells me something about the scaling of the attack rate Which means in fact These are Bigger rates. Does anyone see what I'm trying to do here? I'm trying to Multiply those arrows by epsilon and if that's the case Then over time This might evolve To something that looks Like that And if I were to draw the shadow of where it was before It's moving down and to the left which is what your intuition was when I just had the two right But here I have this full distribution Okay, everyone With me still I have written Somewhere here a system of equations Which now I'm going to write back here because I like them to stay because they're an important reference So we had this to state one. I gave you some intuition though I didn't do any of the math behind it and now I'm writing something like DS Epsilon dT minus beta Epsilon s of epsilon i This is of t t I dot is just the accumulation of these infections But it would be nice to write s dot right. It would be nice to write it like this We would like to know how are actually the susceptibles changing with time Okay, I'm going to start to erase the middle part of the board. I moved this over because I want to have it as a reference And collect stuff. So how do we do that? Everyone got this part? It's about to go away and I'll get back to that in a moment Was that a question Where was it okay good let us write down s Which is the integral? the epsilon s of epsilon such that s dot is integral d epsilon d s of Epsilon d t Which is equal to integral d epsilon minus beta Epsilon s of epsilon i Which is equal to this minus beta i the force of infection Times the integral d epsilon epsilon s of epsilon, but you should recognize this If I were to divide by s and multiply by s Multiply by one then this object is just the average vulnerability Which means that we actually get a macroscopic equation That is not just about how many individuals were infected and susceptible have but their average vulnerability to infection Here we might have started at one But now the average is less than one Right because if I say initially I'm going to assume the average is one because that's arbitrary I could always stack it into beta You can see that what's happening here is We're slowing down the infection rate because not only are there fewer people around to be infected those who remain are less likely to be infected than those who were there at the present because The infection process Sculpts this distribution. I'm using the word sculpt in terms of sculpture intentionally, right? It's changing the shape in a certain way good Now why do we know This epsilon bar if this epsilon bar were the same in time if this didn't evolve we'd be done right So how do I know epsilon bar is definitely going to go down? I'm sort of you feel it, but it would be nice to get a little bit better idea Let us think for a moment about Something that I mentioned very rapidly at the end too rapidly probably I Wanted to note epsilon D as being Or epsilon bar D as being the average Vulnerability of the next Individual To be Infected we have a population Someone's gonna be affected next. What is their? Average what is their value of epsilon and if we take across all the different possibilities we get this epsilon bar D Okay Well, we can formally write this this draw average as the integral The epsilon times epsilon times the probability that we draw something Whose vulnerability is epsilon right and now I don't have to do the bottom integral because by definition I have a probability, but of course I need to write the probability that my draw is epsilon as Epsilon s of epsilon Divided by the integral of epsilon s of epsilon and by definition therefore the probability is a distribution that's Integrates to one questions in the back the average vulnerability of the next individual to be infected This is sort of a draw so remember we have a distribution But because we're drawing this distribution not at random But with a probability that scales with epsilon I'm more likely to choose from this side So I'm trying to formally figure out what that is how big that is Okay, and Everyone sees why this is if I don't have anyone in that class I can't draw but I draw them with a scale factor Epsilon so now I can rewrite this and What I hope you can see is that if I write epsilon D bar Actually, I think I'm just gonna it's getting too low on the board. I like this sculpting No, can you still see if I get this low? It gets to be a little tricky I think I've talked about the sculpting enough so we know what that is So we can write this Average draw vulnerability this epsilon bar D as the integral e times Epsilon squared s of epsilon Divided by the integral of epsilon f of epsilon Which we can obviously Write s over s on top and bottom If we write it like that we get this is by just definition 1 over epsilon bar and this by definition is Just the average value of epsilon squared, but of course recall Variance is epsilon squared minus at average value squared. Sorry. I'm getting a little squishy there So we can write this as simply Let's see if I'm making sure I'm right. Yes average value of epsilon squared over the average value of epsilon plus Sigma squared over epsilon bar, which is just epsilon bar Plus something The draw is equal to the mean plus something related to the variance In other words as long as we have a distribution From the positive variance, right? We have a distribution that has some variability The effect will be that the draw is greater than the average Which means we really are sculpting distribution. We're not Uniformly pulling it down. We're actually pulling more and therefore if we're depleting things Are an average higher than the average we must be making the average lower and we're decreasing the average susceptibility. I want to point out That if we had an exponential distribution where Sigma squared is equal to epsilon bar squared for an exponential The average draw is two times the mean The thing that I mentioned very rapidly in my lecture yesterday That's just happens to be but for any general case We see that we're actually going to be drawing more from the right and therefore when we go to this equation We realize that this epsilon bar is also a function of time And if we could figure that out, we might be able to re-examine what this macroscopic dynamic should look like Okay, this board like number 10 today Okay, can I move to the next idea and I'm gonna scan all this and give it to you as well Can I try to I want to wrap up this one idea today before any questions because I'm about to erase stuff So just yes So from that equation from the average vulnerability of the next individual It means that it increases over time, right? the average vulnerability Of the next in fact individuals higher than the mean If I have some I need to erase something Let me sacrifice this part if I have some current distribution as This is my current distribution. This is my mean Somehow over here is where I'm gonna tend to pull The next individual could be anywhere, but it'll be tends to be to the right Which means that if I looked in the future if I've Pulled things from the right my distribution mean must go to the left with time and it keeps going to the left So the average drop probability actually keeps going down with time because I'm pulling those who are more Vulnerable than the average every time, okay Good so now We have written this which I can now erase Because now I'm interested. I've convinced you I hope that Epsilon bar is not a constant as long as there's variability it will change and actually will decrease because of this mechanism Right because I'm hitting this with some other feature which pulls the average to the right So now I actually can look at This average value and ask the question how does it change because I can't assume any more That the average vulnerability is constant Right, so we have this initial attack rate multiplied by this average this average is changing with time recall that the definition of the average vulnerability in this heterogeneous population is this Distribution weighted by Epsilon divided by the number of susceptible individuals and Therefore if I want to find the derivative with respect to time I Would write out a prime B minus B prime dot a all over B-square right and It seems like a lot of moves for me to do with ten minutes left. So can I skip them? It's okay. You trust me that I'm gonna do it right At the end of the day after you do these derivatives, and you appropriately take care of things You get minus beta I Average value of Epsilon squared minus the average squared in other words The speed at which this decreases is hit by the force of infection obviously if there's no infection We don't have a rate. We don't move and decrease But the variance is what drives it Because the more variance we have the more likely we are to pull out a big thing and then more rapidly decrease It also tells you that in a homogeneous population We don't have any change by definition because we're always taking an individual with the same vulnerability So that's cool But I want to just do one last move and then I'll wrap it up today. I Think I'm almost at time. This is a I think for should be for everyone here A new concept even if you've done SIR models before and even if you've seen the outbreak conditions in the endemic Equilibrium and so on So What have I done? I've introduced I've introduced a heterogeneous population Into an SIR like framework I've shown you that you can begin to write down the full Population level dynamics in terms of the average vulnerability, but the vulnerability is itself changing and Now you see that I would need to make some assumptions to close this Because you see there's a bit of a problem. I Want to figure out how the susceptibles change to do that. I need to know how the mean vulnerability changes To figure out the mean vulnerability changes I need to know something about the variance, but then the variance could change and I could just keep going on for and forever All right, it turns out and if you want to you can read more. I'll post the paper There are eigen distributions of this sculpting process that preserve their shape And I'm going to use one as a point of convenience There are others as well And it's interesting Outside today's scope to say that this sculpting can actually move things closer to these eigen distributions In which we know something about the relationship between the variance and the mean and if we do that and that shape is preserved We can actually close this system of equations Let us for a moment assume That we have an exponential like distribution But I'm going to just set this equal to one arbitrarily And therefore I don't have to have my pre-factor and if I take the integral goal of that overall epsilons I get one this Turns out is preserved by this process If I start with exponentials I always have exponentials because everything is being hit by this epsilon it just rescales the distribution and Therefore epsilon the variance Is just epsilon bar if this is the case Then what I see is that s dot is minus beta i s epsilon bar and epsilon bar dot is equal to minus beta i Epsilon bar squared Which means that ds d epsilon bar The force of infection is just saying how fast the system runs, but it's not telling me how I sculpt It just sets the speed of the sculpting not the sculpting shape is equal to s Over epsilon bar Which means you can see I end up getting log of s Is equal to log of epsilon bar plus some constant But I can arbitrarily set this to one initially because everyone was susceptible I say the average vulnerability which is just scaling this attack rate beta is equal to one Which means this constant is zero which means radically speaking Epsilon bar is equal to s if we have an exponential distribution Which means? s dot minus beta i s squared The heterogeneity Mentally alters the nonlinearity of the core s-arma, and what's notable about this is That if we were then to look At the i dot equation, which is now beta i s squared minus gamma i Right because now in this simplified model. I have this you can see initially We have beta over gamma s squared minus one initially the speed Will be the same because s is one and the r naught will still be beta over gamma We'll measure things initially that will seem exactly the same But yet you can see that the s herd immunity threshold Will be one over r naught to the one half Rather than one over r naught in other words if we look at At the number of people infected R naught and the number Infected at the herd immunity threshold. Here's one. Here's the classic result one minus one over r naught heterogeneity Should on average tend to decrease the size of the outbreak For the reason that I asked you at the very beginning and a few of you were willing to answer Because it sculpts the distribution removing The those who are more likely to be infected leaving those less likely to be infected and turning itself off faster This is interesting, but also dangerous in the same way that the classic s our models also dangerous if I think of the s our model it says That in the absence of heterogeneity things like an r out of three or four would lead to Herd immunity thresholds of 75 ready percent before you have overshoot, so you're getting into the 90s in terms of who gets infected Once you have this kind of heterogeneity you can drop this down significantly and if r naught gets low enough This effect could mean and lead to conclusions like I hinted at that go mess it all said would you have herd immunity thresholds of 6% or 10% or 20% which is clearly not the case. We've gone way past that Nonetheless, I want to make this very important point That when we make a homogeneity assumption, it's actually an extreme assumption with respect to the possibilities It's much more likely. There is some heterogeneity and the heterogeneity Fundamentally affects the nonlinearity intrinsic in the model I was able to solve it and put in this form because I made the simplifying assumption in general It's not that easy to do, but nonetheless it captures the idea of how having this heterogeneity actually leads to changes in the order That's all I have for today. I'm at time I'm happy to continue the conversation But tomorrow if we want to talk more about this I can start with this But then I will continue with two other modules probably one on generation intervals and one on And one on behavior See you tomorrow or we can chat at coffee break. Thanks