 to start. Recording in progress. Okay so welcome everybody to this third week of the spring college. So we'll start this cycle of lectures on quantitative viral dynamics with Joshua Weitz who is a professor at Georgia Tech in Atlanta and the visiting professor at ENS in Paris. So Josh thanks for being here and the floor is yours. Okay great thank you. Can everyone hear me? I have my mic on. I'm still gonna wear a mask but as long as you can hear me enunciate I think it'll be okay. Sounds good? All right good morning everyone and wherever people are online. So I'm Joshua Weitz. I'll be giving a series of nine lectures over the course of this week and next week and I'll try to break up my lectures into two segments. This week's lectures on viral ecology and next week on viral epidemiology more on epidemics. And in doing so this week I'm going to focus predominantly on the interactions between viruses and their microbial hosts and next week I'll focus predominantly on their interactions between viruses and human hosts. And then so doing really focus on epidemics at large scales and ways that models can help and focusing predominantly on COVID-19. Okay so that'll be next week and this week will be more on the ecological side. But since I'm new to you and I assume that every time a new lecture starts you have to adjust a little bit to the style of the person. So I wanted to give a little background introduction to myself and my group. Usually I'm based out of Atlanta, Georgia which is in the southeast part of the United States very as anyone been there or to the to the airport maybe it's a very well-known airport. Yacobo has been there a few of you. Okay and that's the building that we sit in. It's a block style building right in the center of the city and originally I have a PhD in physics but my appointment is in biology so I really sit in biology that's where I work that's where most of my interests lie and I have a courtesy appointment in physics and electrical computer engineering and I've also started a number of years ago a new PhD program in quantitative biosciences. This is a flyer from a few years ago featuring an individual from one of our first cohorts who won a National Science Foundation graduate fellowship and we've had a number of cohorts since and you can see the group size it's usually about seven to ten individuals applying both from the U.S. and internationally and it you can start this after your master's degree or after your undergraduate degree so if you're interested in this at some point feel free to reach out to me or you can look at qbios.gatech.edu but this year I'm based somewhere else I'm a visiting faculty member at the Institute de Biologie at the Ecole Normale Supérieur in Paris so if you're nearby and you want to talk about viruses just let me know and I'll be there until about early July and I'm also doing schools like this so that this is really a college and I view what I'm about to do in two weeks as an advanced school winter school it's a one-week course and I bring it to your attention because it's on the same theme quantitative viral dynamics I'm just giving one lecture there because I'm the organizer and there's a number of lecturers who are coming from all over the world to give lectures in this course so if you get interested in this topic I am hoping to release some of these lectures online so you can check out the Qlife website I didn't list it here because it's you know 75 characters long but if you look up quantitative viral dynamics across scales Qlife you should be able to find it and find the information okay great so in my group at Georgia Tech and now based here based in Paris we work on a number of areas that are interlinked as I said predominantly viral ecology and viral epidemiology and I'm going to try to use some of those real-world applications as the basis for the material that I'm going to present this week and next week okay and in doing so I just wanted to introduce some of the people that make this work just because some of you may be already in research teams maybe you're thinking about embarking on a research career and so we have folks from all over the world from Europe from Asia South Asia the US etc and we also have collaborators both at Georgia Tech but also internationally so folks like Laurent de Barbue in Paris folks in Israel and it really is sort of these international collaborations which is also the spirit of the ICTP which makes all of this work and so I wanted to go back to these research areas which as you can see here on the other side or online we tend to work on problems of the foundations of viral ecology how is it that viruses interact with their microbial hosts and if we're thinking about microbes we also have to think about microbial ecology and we have really two big application areas one into using viruses of microbes as a potential therapeutic and this is a bit of an overture I'm giving you a preview of everything that's about to come this week and we also work on trying to understand the way that viruses impact marine systems so we care about human health and environmental health and their ways that viruses intersect and hinge upon both in parallel to that we also work on infectious disease dynamics questions of how do diseases and virus in particular spread in human populations and the intersection of mathematical models with those ideas and in doing so I think we have a common approach which is we do try to seek and I'll sense who Delbrook this radical physical explanation of life which I think is my often the spirit of physics of living systems but again I'm sitting in a biology department ecology evolution it tends to be the centerpiece of what we're interested in and the methodology really is one of nonlinear dynamics of complex systems so the lectures this week and next will tend to emphasize those methods even if based on my read of the prior courses mine will be much more biological and applied and maybe even a little bit stranger for some of you depending on background and so I wanted to ask one question of the people in this audience online maybe I'll get another read how many of you have taken like a physics of biology class before okay a few oh good almost everyone has great everyone has fine well it's good I only wrote one lecture maybe I'll adjust but we'll see how it goes and so I'm hoping that these lectures will provide a gateway if you want to learn more and explore more right in particular in this area of viral ecology and evolution so let me get started because what I'm going to do now is just embark upon a series of interlink lectures and I'll explain that in a few minutes on the theme of viral ecology and then next week I'll do viral epidemiology and I wanted to start the material obviously this is not part of the material this is an ad campaign for the world's most interesting man if anyone ever see this dozeckies you can see the beer he's promoting it doesn't matter it was a silly ad campaign but I wanted to contrast it with this very boring PowerPoint slide so it's a very boring PowerPoint slide just a bunch of numbers experiments replica and obviously this doesn't have any of the bells and whistles it doesn't have animation there's no colors but I claim that this is perhaps the most interesting table in the history of biological sciences ever ever never seen a more interesting table and part of the reason I think it's interesting what let's look across at one particular row and hopefully that'll work let's look at experiment five I mean told you what these experiments are but you can see there's different replicas you're redoing the experiment again and again and sometimes you get zero sometimes you get one sometimes a hundred seven and the numbers here are the number of colonies that are being counted on a particular agar plate so in some cases you can find a hundred seven colonies sometimes zero and if you can see anything here you see that the experiments are basically non-repeatable you do it again and again with this replica and you get totally non-repeatable outcomes and now ask a different kind of question is anyone here ever done microbiology experiments before one of you okay I'll ask that individual what's your name Manuel if you did a microbiology experiment got these kind of results would you be proud to share the results with your advisor you don't think so because it is turning out different every time right so that seems like something we don't want to see happen and I know biological systems are noisy but this seems almost too noisy does anyone have an idea I've given you a hint these are bacteria colony counts has anyone ever seen this data before not not you yuckable you're not taking this course okay good this may be the right course for you all if you haven't seen this before someone in the back was there no okay not a guess so these are taken although I've inverted the orientation from this famous paper by Lurian Delbrook mutations of bacteria from virus sensitivity to virus resistance and those numbers were the number of resistant colonies against viral infection so you take a susceptible culture that should all be vulnerable to destruction lysis by viruses and yet when you put it on this bacterial plate you find that sometimes you get 107 resistant colonies so founded potentially by a single resistant bacteria and other times you don't find any and you can see that data column right there and although this may seem somewhat banal maybe even benign maybe don't care I want to point out that the Nobel Prize in 1969 was given in credit in part for this experiment and I'll explain why in a moment has anyone ever heard but this is also helping me orient and get to know you all has anyone ever heard of this Lurian Delbrook experiment before how many have how many have not okay most have not okay fine and most of you answer to I was gonna ask the third category not sure if you have or have not which is fine okay so let me explain a little bit about how this experiment worked and why actually these results are quite interesting so at the time before this experiment was conducted there was a really a fundamental not to say misunderstanding was a lack of understanding of whether or not mutations were dependent or independent of selection meaning that a mutation which is a change in the genotype of this bacteria maybe as a result of the experience of interacting with the environment or it may be totally decoupled from interacting with the environment so as if some of you have understand there's like Lamarckianism versus Darwinian evolution the idea would be in the class example the giraffe the mother giraffe stretches out its neck farther to reach the acacia leaves and be the more it stretches the more that it's offspring will have longer necks that seems again it seems preposterous but that is the notion that the mutation might actually depend on selection there was some experience and therefore whatever the difference to the genotype would have some property based on the experience the independent one would be that the mother giraffe can stretch as long as she wants for the acacia leaves and some of her offspring will have longer necks and some may have shorter necks and perhaps the ones with longer necks will be able to reach those leaves when they get older and therefore over time that variation will be selected upon so we have a notion of independent or dependent on selection okay good so how do you actually go about thinking about testing this how would we know so learning about delbert devise this experiment together to try test this idea with the with the notion that you start from a population of susceptible bacteria that begin to replicate over time starting from one to two to four and over log to event to some population size n number of bacteria and then you provide some sort of selection pressure in this case exposure to viruses that exclusively infect in lice this bacterial population should eliminate them all this bacterial population is originally susceptible and then you go and count to see how many of them have managed to survive and you do that by putting them on this bacterial plate and if they have somehow survived because they've resisted that then they should proliferate and make a visible colony that you can count and those are those numbers five and a hundred seven etc. that you saw in the earlier example if it were the case that a sub-population of bacteria acquire resistance and survive our infection meaning that the mutation depends upon selection then in some sense you're doing a coin flip experiment right and we should have a Poisson distribution because we have a probability small probability p very very small of acquiring resistance we try this many many many times because then is a very large number something like 10 to the 8th or 10 to the 9th etc. if not more and then you find a very small number of these resistant colonies however if you do this again and again here I have an example of three if I were to do this again and it depended upon selection how many would I expect the next time do you think anyone would hazard a guess I observed three this actually is an answerable question I know it seems unanswerable but if mutation depends upon selection I have an observable and I measure three and I do it again should it be a number close to three very far from three here we say someone says three yes exactly three you can it's okay you can hazard a guess somewhere close to three you want to hazard a guess is that yes for close to three or yes exactly three exactly through maybe but remember there's some chance here because if I were to do this again it may be a different set of bacteria that acquire this resistant but certainly would be a small similar number maybe four maybe seven it would be similar every time because if you think about the law of large numbers and I'll give you the exact derivation in a moment then we have some probably p then we have one minus p that it doesn't happen and essentially we have either the binomial distribution which becomes the Poisson distribution when and is very large and p is very small and we don't have much variation between them right so we expect very similar numbers let me contrast that situation if mutations were independent of selection if they were independent of selection it's possible that very early although this would be quite rare and I've made a dramatic example for effect it is possible that very early on in the proliferation process in one of the early generations there might have been a mutation from susceptible to resistant and if this mutation did not have a significant cost in this particular growth environment then that subpopulation would grow and grow and grow and I've given this extreme example where half the cells happen to be mutants and then when I exposed to the viruses the mutation is already there and this is very important if you as physicists whenever to move into biology not to think somehow that these mutations are purposeful this one was there you might never even expose those bacteria to this viral population but it was there already and then when I go and I do this plate counting experiment I find a very large number a jackpot number of these resistant colonies okay now if I were to do that again and again here let's say this was something like a hundred seven obviously this is huge this is n over two this would basically be almost unobservable with my dilution method but in this case if I re-ran this again do you think the next time it would be similar or very different I hear of a head shake what does that mean similar to very different why you have an instinct it's it's the instincts are good and I'm gonna ask questions here so don't be afraid if I go and try to engage with you and it'll make the class more fun yes it could happen at every step so maybe it happened early and the earlier it happened it seems rarer than it happened because it's very low probability but if it did it has a very big outcome if it happens later it would have a very small outcome right in fact this is the extreme limit of it happening at the very end in which there's almost no variation but because now this could happen anywhere in between we might get lucky right and find it very early or might be unlucky and find it very late and therefore we'd have very big differences between the observable this is why they did this very simple experiment grow overnight cultures let them grow up exposing the viruses count the number of resistant colonies and instead of measuring the average they look to see how different the numbers were if mutations dependent on selection the number should be very similar we should have a low variance if mutations were independent of selection the number should be very different we should have high variance and I hear I've done something where now I've formally written down the probability that we should actually observe M for a Poisson process in which each one is random and independent with some probability mu A for the acquisition probability of a mutation and this I haven't derived I assume this crowd knows how to do this but you can observe here where the number of mutations are on the xx output log scale that it should be narrow whereas if we actually simulate a process in which mutations are independent of selection we find this long tail that we can't reconcile and this is precisely what Lurie and Delbrook did was look not necessarily Manuel was that right they looked for irreproducibility or the lack of repeatability was actually the hallmark signature not the fact that the mean was low because if you don't look at the variation you might just get a different wrong estimate of the mutation rate actually looking at the variation as the hallmark signature for this fundamental biological process the fact that mutations are independent rather dependent on selection and I have to caution there's always an exception in biology is anyone aware of instances in which biology does have this lamarckian the giraffe reaching out for the acacia leaves kind of effect in bacteria or do you understand the question I'm having puzzled looks meaning I've just explained the Darwinian view with mucidation or independent selection has anyone ever heard of CRISPR cast it's this big biotech thing you know people fight over billion dollar patents fundamentally that's an acquired immune system in bacteria in which a better that survives a viral infection can pull in a piece of that genetic material so that if they're ever exposed again they start expressing these small little segments that essentially detect inside a cell fine matches and block infection and that's sort of the bacterial analog of a giraffe reaching out for acacia leaves and when doing so it's offspring have longer necks right nonetheless for the bulk of what I'm going to talk about in the bulk I think of the way you think about evolution you should think about this as mutation being independent of selection even without selection and pose we can see these pre-existing variants which explains this variation and why the variation itself was the important feature okay so let me try to again and this is my first lecture so I'm trying to set the biological stage as I explore the way that we integrate models into understanding them and I think you all can already see one of the consequences that viruses impose a strong selection pressure right mean that in the absence of this mutation to resistance these viruses were able to obliterate nearly all of these cells except for a very few rare number of mutants and obviously a mutation that confers resistance is beneficial if you're able to resist infection and license that seems good and therefore in a fundamental sense viruses induced host evolution the change the heritable change in a genotype over over generations or over time but what about the viruses so this early example shows you that if I add viruses to bacteria then we can end up getting this change in the host population we end up having a very different genotype composition afterwards than we did before so Luria did a similar kind of experiment a few years after the first one with Luria and Delbrook and just to get yourself oriented into these slides the squares are phage which are these viruses that affect bacteria the rectangles are the bacteria a solid line denotes infection a dash line denotes an evolutionary relationship meaning there's a mutation that gets you from one to the next and I think what you should try to take away from this is that if you look at the original viruses they infect these bacteria which then generate just as the process I've just showed you these resistant mutants but if you then try to get the viruses to infect these resistant mutants you can and it's called a host range expansion meaning this virus is host range which host it can affect has expanded and you can see that by the fact that the solid lines coming out of alpha prime include those of alpha but include new ones you can see to the side and this is repeated in the one below yes correct so this alpha prime can infect the same better that alpha can infect but it can also infect new better so it's host range which is the host that it is able to infect and sometimes that's knowable at least amongst the this particular set of hosts that we have can therefore expand and you can see that both for the alpha type and the gamma type in those transitions and Luria repeated this multiple times but eventually came to a dead end found a host range in which it wasn't seemingly possible in the context of this experimental setup to find a virus that could infect this particular host which suggested in some sense that there might be a mismatch or an asymmetry in the potential for evolution to drive both host and virus change and this dogma persisted for decades here's one example the covalentiary potential of virulent phage is less than that of their bacterial host meaning somehow maybe the host always have the advantage there head in the race and they can at some sense escape the potential of the viruses to also mutate and infect new host types and this also explains a little bit why viral ecology not viral epidemiology but viral ecology the study of viruses of natural host populations in particular microbial populations was not that active a field you could always presumably find these bacteria that jumped away so perhaps in nature that meant they weren't that abundant they meaning the viruses until a few years later something happened so in 1989 Berg working with colleagues started to examine the abundance of viruses using a culture independent method they will go to aquatic systems take the sample stain everything that contain DNA and look at under the microscope and count and in doing so they found pictures like this and you'll notice the scale bar so we have a lot of one micron size things presumably bacteria they're even bigger things probably micro carats and then these small circle like things which are about 50 nanometers in size and contain DNA is anyone aware of things in nature that are about 50 nanometers in size that contain DNA viruses and what they did was essentially just count these numbers of circular virus like particles because at the time certainly they weren't quite sure there are viruses called the virus like particles and based on how much water they had to add to the sample they counted the number of these VLPs these virus like particles which is called viruses and then back calculated based on how often they would see these circular particles how many viruses there should have been in this in the natural setting and what they found was 250 million virus particles per milliliter in natural waters and that's a lot that's also true if you were to come back to Trieste and go out into the water then maybe a little bit less but you would find typically about 10 million virus particles per milliliter this is a bit of a high number but still on the order of 10 million virus particles per milliliter is not an atypical number if not a hundred million per milliliter near productive coastal waters and yet you go swimming and for the most part people don't get sick because these viruses are not necessarily viruses of humans they're viruses of microbes and what is also notable is that these levels were a thousand if not 10 million times higher than previous reports so people have been going out into natural settings and measuring viruses and virus like particles they just weren't finding that many and part of the reason and the reason why I showed you those examples from Lurian Delbrook is that they would often take not everyone but some would take these samples and try to plate them on a known host maybe a Vibrio host maybe even a Nikolai host and then try to measure the number of times that they found these small developments of bacteria in them and I should just point out that when I keep talking about this is an agor plate and if you were to put bacteria on it and dilute it you would find something like this and you probably have all seen these examples of colonies right these are precisely the things that I keep talking about that Lurian Delbrook examined you can do the same thing if we take an agor plate plus bacteria so imagine now this is darker because it's full of these bacteria and what I'm going to do let me try to do this nicely is now a race and make little holes and this is actually not so different than what you would see these are called plaques these plaques are the proliferation of the phage equivalent of a colony their proliferation by viruses on a bacterial lawn and so in the past people have been using this method they would take a known host take their sample filter out things so they just had viruses do an experiment like this and try to count how many viruses they measure presuming that just like one bacteria can form a colony one virus can form one of these plaques okay but if you don't use the right host then nothing will show up and this explains why both A people might have thought that there was this less relevant viral impact but it's not necessarily the case they just may have been looking the wrong way okay so that's just one example another example is that viruses aren't just abundant they have other effects on bacterial communities when a virus infects a bacteria it lices the host cell and outcome new virus particles and also outcomes things like dissolved organic material so now you have new nutrients that might potentially be utilized by other bacteria in the system and so therefore we can think about viruses as not just agents of mortality but modulators of ecosystem functioning and here you can see one quote viruses divert the flow of carbon and nutrients by destroying host cells and releasing these continents back into this do and pool and often these are studied in ocean systems with perhaps the best studied of these viral ecological systems so we have drivers of selection agents of mortality and also ecosystem functioning but also want to point out that going back into early 2000s very little was known about what these viruses were and also explains why if you did this kind of method you might not have had success the vast majority when one looked at the metagenomes of these viruses their genetic content you found that most of them were novel and unknown to science and this is still in some sense the case that we know far less about viruses environmental viruses than we do about the very small number of study viruses that are used often in molecular studies okay so this is gives you some sense that in the late 80s into the 90s early 2000s there was a paradigm shift in terms of the interest in studying viral ecology in part because they were both so abundant and yet still so unknown in terms of what who they were and what their effects might be so what do we talk about when we talk about viruses I think for the most part if you've studied viral dynamics and certainly the last two years has told us many things but it might have been Ebola or Zika influenza HIV we usually talk about viruses that infect humans and I recognize for the last couple years we probably only been talking about that virus that will be next week's lectures just to give you some preview because of the nature of work in my group we have been doing quite a lot on COVID-19 modeling and still are having a lot to do with foundations of models and what they say and what they can say about new kinds of reactions and so I'll give you some sense of that kind of the mathematics and the non-linear dynamics involved with that next week both foundations as well as frailties so what I'm hoping to do next week is to give you a sense of what we kind of get right about these kind of models and also what we get wrong and what COVID-19 revealed that we get wrong but I will also try to show ways in which these same kinds of approaches can motivate action taking and so a lot of what I've been doing in the past couple years along with colleagues is ramping up an asymptomatic testing program at Georgia Tech we're approaching 500,000 measurements 500,000 asymptomatic tests and measurements trying to control spread on this open campus which opened up in August 2020 back to students and also developing ways to communicate that there might be someone near you who has COVID as part of this risk assessment dashboard which has been expanded to many places including in Italy, Spain, Mexico and elsewhere so I will try to emphasize next week ways in which these basic models have gotten things right and also gotten things wrong but also have motivated action taken but these are not just the two extremes where viruses are making an impact viruses really affect organisms across the diversity of life from humans all the way across different scales including obviously plants as well as microbes and bacteria and I won't maybe belabor this interesting story and I don't know how many of you are familiar with the very hungry caterpillar you are if you if you have a if you have a niece or a nephew in my case I have children so I read this story and this may be a U.S. thing but I think it's been translated so some of you may have who's never heard of a very hungry caterpillar never heard of a very hungry caterpillar okay so I'm going to give a two minutes because I'm with you for nine days and don't worry we'll get to some technical stuff eventually but the first lecture I'm trying to introduce concepts it's a very nice story that you want to read to a small child about a very hungry caterpillar who essentially explains the life cycle how it turns into a butterfly and in the process has to eat all this stuff and what should be eating is nice green leaves but of course it's a children's book so it's eating a bunch of junk food and that's not really helping it and eventually eats the nice green leaf and then it obviously goes through its transformation that amorphous eyes is into a butterfly and it's a very nice story but unfortunately sometimes these leaves have little bacular viruses on them and when the caterpillar takes a bite of this leaf it also ingests bacular viruses which begin to replicate inside these caterpillars turning them essentially into zombies and they start to act very strange and move up to a higher portion of the plant and then they stop being caterpillars and basically become dripping oozing viral factories and eventually just slough off all these viruses and which fall down to another leaf so this is sort of the horror version of the very hungry caterpillar where eating the leaf turns out to be very bad and this promotes junk food eating etc so you can listen or read more about this very interesting story about the life cycle these bacular viruses and in a broad sense i'm trying to convince you in this first part of this lecture is there really is a planet of viruses out there to explore everything from viruses of humans to viruses of microbes and in between now there's also a role for modeling physics physicists mathematics and so what i'm going to try to do in this lecture this first series of lectures is to try to bridge and introduce different ways in which a non-linear dynamics and release systems level approach can help us understand different facets of virus microbe interactions and that's what i'll be doing this week so today i'm going to try to explain some principles of eco-evolutionary dynamics and hopefully build to tomorrow where i'll talk about dynamics in complex communities so today's lectures today and tomorrow will be linked and then i will try to explain a different modality beyond which viruses infect hosts i'm focusing today and tomorrow predominantly on this what is called lytic route by which there's a very antagonistic lethal relationship between the virus and the microbial hosts and then on thursday i will try to take these concepts and begin two series of applications one to understand in which one can use viruses as potential therapeutics obviously that's not something that we often are thinking about now we're trying to have therapeutics against viruses but for bacteriophage these viruses infect bacteria if the bacteria is a pathogen and the pathogen is potentially a multi drug resistant pathogen then it may be possible to use viruses as a way to target whether with or without or in combination with existing antibiotics to try to eliminate bacterial pathogen so i'll talk about therapeutic applications on thursday and then finally i will do more in the direction of model data integration how do we learn about what's going on in natural systems with the focus on marine systems again using these non-linear non-linear dynamics models as a chassis but trying to figure out what we got wrong by again using sort of more of a model data inference approach and that'll be friday so this week i'm really going to focus on viral ecology in a sequence of linked lectures and then next week i'll restart and reset and do my viral epidemiology lectures any questions before i set sales at work nope okay any questions from the chat nope fine okay so to get us started i want to organize my lecture today really over three basic questions and this will go from today until tomorrow based on the pace i want to try to adopt and i'm going to try to ask the question how does viral infection change microbial population dynamics right just fundamentally how can we think about viruses not necessarily as only a outcome between one virus and host but how do we move that to the population scale and then i will try to ask questions about evolution and even co-evolutionary change because we can't think about biological systems in the same way as we think about physical systems the rules that we have at the start may not be the rules that we have at the end these systems autonomously change their dimensionality during the course of their interactions right they change their properties as they interact and then i will try and i believe tomorrow given my timing try to figure out more about interaction networks and virus host interactions in complex communities where we don't have these small number of players that we might be able to realize or control in the experimental setting okay so part one how does viral infection change microbial population dynamics so to do this i'm going to go back to another intellectual course which is the mathematician Vito Volterra and the physicist Alfred Lacca who independently came up with what we now call the Lacca Volterra or Volterra Lacca depending on your preference I guess I should call it Volterra Lacca here equations that describe prey or prey dynamics and originally Volterra was inspired by a son in Lambert to the Dancona to examine fluctuations in Adriatic fisheries the fact that there seemed to be these oscillations in the amount of catch and one to understand something about the ways in which that variation might be generated endogenously meaning intrinsic to the system or exogenously because perhaps there was some change in temperature or other factors that mediated from the outside and what they both realized is that you didn't necessarily need an external driver of a system to find oscillations in that system that the interactions within the system would in and of themselves be sufficient to generate oscillations and in modern terms they proposed the following system of coupled non-linear differential equations which I assume all of you have seen before or some of you but it's okay we're going to get started and then we'll move a little bit faster in this case we have prey abundance and predator abundance and you can see it's non-linear because of the b times p term and we have something in which we have prey birth which that's a n term there's predator consuming prey that minus b n p converting it into prey biomass or predator biomass or abundance and then the death of predators in the absence of prey and if you take these system of equations and simulate them you end up getting these oscillations and the oscillations have a particular feature you should see that the prey peaks before that of the predator as the prey peaks the predator goes up driving the prey down as the prey is down the predator declines and then the prey increases again and we have a full cycle which repeats okay and just so that you all do something in these lectures and not just me I wanted to ask the question how do we anticipate the structural dynamics of this system near the fixed point so I assume again you all know what a fixed point is the system a point in the system in which both n dot and p dot are zero so you all seem to have some writing implement whether electronic or a piece of paper and what I'd like you to try to do it should only take a few minutes for you to get started as I'd like for you to try to draw out a little phase plane just in front of you and p because this will help us when we make these models a little bit more complicated and first of all let's identify the fixed points maybe together does anyone see a fixed point in this system and you just go ahead raise your hand and tell me where one is at least zero zero I think there's nothing there there's nothing there okay good we found one right so zero zero is one I haven't said anything about its stability yet are there any others okay what was it p equals a over b okay and that's something what would you need to have the other part over here okay good so I'd like you just to take a minute to refresh yourself on your own so that you're trying to get yourself oriented how would you describe the dynamics in this phase plane so everyone take a minute or so and just think to yourself how would you even sure I plotted this and you could even just try to intuit which is also okay meaning if I start somewhere here where do I go if I started somewhere over there where do I go and obviously you could think about it very close to that fixed point so if nothing else I'd like you to try and just use your intuition and draw what you think an orbit might be using an arrow to denote the flow of time everyone understand the question that you're supposed to do just take a minute and try to do it on your own y'all is this the eraser of this cloth anyone need more time you know you've written about it people are still making an effort so I'm gonna give you about 30 more seconds here and if you're listening at home wherever you are you can try this too for whomever is there in the internets who's ready for me to move ahead here and try to work through this together you all need any more time anyone no okay so first of all we found these fixed points how did that person from the front row tell us about these fixed points we set these equal to zero and you can see because I factored them here I haven't factored them that if n is zero p is zero certainly that's a fixed point and this is how you solve the other one but I want to do a little bit more than that and point out that there's something called nullclines these are where one component of the system is zero not all the components and you can see here that n will not change the prey will not change when p equals a over b so I can draw this dashed line across and this is the n dot equals zero nullcline at this level there should be no change in n so if we ever cross this line we can only go vertical right so if we're at that point where our change can only be vertical whereas there's another nullcline n equals d over c it's the p dot equals zero nullcline and if we ever cross this line our change can only be horizontal I haven't told you which way yet but it can only be horizontal likewise there's actually another nullcline when n is zero another one was p is zero just to point out you can't see them there but this one n dot is zero this is also the n dot equals zero nullcline and this is the p dot equals zero nullcline when nullclines intersect we get fixed points here's one where the n dot and p dot is zero nullcline intersect we get a fixed point the other one is right here on this corner here there's intersections but they're of the same nullcline so we don't have fixed points at those other two corners okay now we can ask the question about what we expect in terms of the dynamics in this phase plane let's imagine we were to start over at this point we can use these nullclines to tell us let's think first of all is the dynamics of n whether or not n should increase or decrease at this point okay so based on this we have a minus bp equals zero which is why we get this for n so if p is ever higher than a over b what do you think happens to n say again decreases and that's true there and that's also true there whereas if we were to start below if p is ever less than a over b obviously then it goes up that's n goes up so now i draw those arrows okay we can now use the p dot equals zero nullcline and we can see that p will be the change in predator excuse me will be zero when n equals d over c here so therefore when n is larger than this nullcline value what should happen to the change in predators when n is larger what should happen to the sign of this increases which means i can draw this arrow and this arrow and by analogy this arrow and this arrow which means that in these four quadrants i know exactly the directionality of the flow i put these two together i don't have another color chalk but i will just go that like that like that and like that and if i were to connect these together you can anticipate something like that where the flow is counterclockwise okay good but now we haven't necessarily described or figured out what's going to happen here because it's possible what's an alternative version of a counterclockwise flow by the way here i have one that makes a nice orbit it could spiral it i haven't said if maybe it goes like that right and often biological systems do converge this particular example doesn't look very convergy but that's not to say that it can't right how would we know there are a few ways to know one of which would be to take the system and linearize it around this particular fixed point and that would at least tell us whether or not we would spiral in or not right i want to do one thing before i do that but i am going to do that in a moment i want to notice that this has a particular weird property that if i take the ratio of dn dt over dp dt and i get rid of the dts you can see that we get an equation like this which are separable okay which means that i can draw out the dp to one side and the dn to the other i can write these as follows which means that if i were to integrate and see how the differential of n goes with the differential of p and integrate both sides we end up getting cn minus d log n equals a log p minus bp plus a constant this means that there's some weird conservation law in this system if i start with some combination of p0 and n0 and i add up this bizarre thing by going like that right if i were to just take cn minus d log n minus a log p plus bp i would get some constant and there all the trajectories have to be like that forever which would exclude the possibility of all of these things converging to this central point because then that constant would no longer be preserved in fact what this particular system equation says is that the initial conditions will be remembered forever which in physics is great and makes you feel very comfortable but in biology is often a sign that something's pathological and gone wrong so with just this simple little feature we can see that if i start up with a particular number of bacteria in viruses or a particular number of links in hair we would somehow always be trapped on that orbit forever and there would be an infinite number of these all with this different crazy constant okay now we can also reassure ourselves that if we'd had a small perturbation near this fixed point we would not in fact relax to it and there's another way for us to do that and i i feel i should do this because this is my first lecture and i want to make sure we're all on the same page which is i might be interested in knowing the dynamics because most of you i think wanted me to start there that's your intuition here which is to linearize the system near the fixed point so we have our fixed point that's the one we're interested in which is the coexistence fixed point and we want to see what happens but nonlinear dynamical systems are hard and if we could turn them into linear systems at least temporarily we might be able to get a sense of whether or not they go up or down or not change at all and that would help us potentially figure out between this spiral and between these orbits now there's a formal process of doing it but i just want to point out because sometimes it's lost which is that we can think of this value n of t there's a bit of an echo in this one particular location n of t is being n star plus some perturbation which is small and p of t is equal to p star plus some other perturbation i'll drop the t's in a moment this little v and therefore because these are fixed if i were to write the derivative you can also see i'm just saving board space that the derivative of n is equal to the derivative of u and the derivative of p is equal to the derivative of v okay which means that if we go back to this original system of equations we can write this as u dot equals and v dot equals and in fact there's just too much echo there so i'm going to move over here does anyone know do you need this anymore or can i erase this i think you all got this and it'll show up in a slide in a moment this puts me closer to where i want to be and hopefully a little bit away from that echo point okay so what i can now do is begin wherever i see an n to replace let me go back and just write it in that way wherever i see an n i can just think of it as the fixed point plus the perturbation and the same with the value of p so i can write this as n star plus u a minus b p star plus v and what i can do that is group the terms together so i can see for the an i have a n star and eventually i'm also going to have a minus b n star p star that's going to show up as well i'm going to have plus a u and i'm also going to have minus b p star u and finally i'm going to have minus b n star v i've missed one thing if you're paying attention what have i missed i think you know the chance of making algebraic mistake here is high but so far i haven't made one i don't think what have i missed say it out loud the end one this u i have it there i think i've got it i thought you were going to tell me this u and the v i haven't hit them together so somehow intentionally i didn't hit the u and the v because i'm only interested in looking at this expansion to small order i'm assuming u and v are very small and i don't care for the moment although this is an unusual example of it for various reasons about u times v which i'm going to assume is very very small okay so what i'm going to notice here is that at the fixed point a n star minus b n star p is zero which means that i end up getting something like a minus b p star minus v b n star okay so i have a system plus and i'll make an explicit higher order terms of u squared u v v square turns out i only have the u v here but i'm going to eliminate those higher order terms which i'm not going to worry about so i've taken this non-linear dynamical system and now you can see i'm at the start of a linear dynamical system the same thing for v dot i can write this as and i maybe now will avoid doing the whole expansion but i can do the whole expansion and i can write this as c n star p star minus d p star which i know is all going to go away to zero plus c p star u plus v c n star minus d odds of making a mistake here very high but i think i've avoided it okay so i can end up writing this all together u dot v dot in terms of a matrix that you should identify as the jacobian which i assume who here's assume everyone here has heard of a jacobian before i'm going to assume if that's not the case you don't have to tell me no but maybe come afterwards and chat with me because i would like you to know what that is and i can't do all that today and i don't want you to have to put your hand out for that one but if you feel like that you can either talk to me or send me an email send yacoboan email we can give you a little primer if you want to get a refresher but you can see here that what we end up having is the equivalent and now if i were to write this as f of n comma p and this is g of n comma p all i've done is take the derivative of f with respect to n the derivative of f with respect to p the derivative of g with respect to n and the derivative of g with respect to p and in doing so we have a system that originally was nonlinear but we've turned linear and i'll write it here and now i'm going to plug it in at the value of the fix point right because i have to evaluate at this particular context this a minus bp is precisely the condition for our equilibrium so i have a zero there and cn star minus d is precisely the value at the equilibrium point so we also get a zero there and on these sides we have b with n star which is just d over c and here i have c times p star which is ca over b now it turns out that when you have linear systems like this we expect things to grow decline or somehow oscillate exponentially so we posit that there should be some sort of onsets that goes like e goes like e to the lambda t u goes like e to the lambda t same with v there should be some exponential rate of growth or decay or perhaps stasis in a particular sense and if this is true the only way for this to be the case is that the determinant of this Jacobian minus lambda times the identity matrix is equal to zero but we have a strange feature the trace is zero which means that if we were to plug in rj and put these minus lambdas these eigenvalues across our diagonal we end up getting lambda squared minus this stuff is equal to zero in other words is equal to let me make sure everything cancels a d which means that lambda is plus or minus i times the square root sorry minus a d a d do I miss a sign somewhere I must have missed the sign somewhere in my madness where did I miss my sign yes this one no where am I where am I yes this is one that's why good that's why that didn't look right but I corrected the sign there it was a never was going to make us algebraic mistake somewhere I'm surprised I got this far so we end up getting something that has no real part but has an imaginary part which means that rather having a small perturbation growing magnitude which would imply spiraling out or some sort of saddle or both real parts being negative which would have implied as you suggested a spiral in we have something in which the magnitude doesn't change and in fact technically speaking you should always check if your first order expansion doesn't work you'd have to go to higher order but we have already done it because we already solved it before but I erased it doesn't matter this crazy formula which I'm going to show in a second which says that if you have a perturbation near the fixed point that perturbation would not grow exponentially or decline exponentially it would stick onto this closed orbit okay and I assume you're going the next speaker will also be talking about dynamical systems and going through some of these concepts maybe in different ways and in terms of chaos and Fabio will be talking about that but still this concept of looking near a fixed point for the behavior of a system is going to permeate what we're going to do today and tomorrow and to many really most of the lectures okay so what we have done together in this little series is identify that we have this counterclockwise flow but it doesn't converge rather a linear stability analysis which we just did yields a pair of purely imaginary eigenvalues real part zero which means we have a conservative system these aren't limit cycles in the sense that they if we have an orbit if we were to perturb away from it it is not an attracting orbit right so it is not isolated it is next to an infinity of other orbits because of this pathology in the system okay so I took a long way but I still wanted to make sure we're sort of on the same page with some of these basic mechanisms if we start with the classic Locke-Voltaire model we end up getting this conservative system that does not have a limit cycle but once you start to introduce certain features and that's some of what we're going to do in a moment with this virus microbe system you find these generic features that prey peaks before the predator they tend to be quarter phase lag in terms of the oscillations and that is why they appear counterclockwise in this plane we have the prey peak followed by the predator peak followed by the prey trough followed by the predator trough and then the prey peak again and that cycle is the thing that repeats okay I bring this up and this was a long diversion I'm pulling myself out of that particular alley but explaining why we went in there that the study of virus in microbes has essentially opted much of the both mentality and formalism of predator prey dynamics the idea is that the phage are the predators and the bacteria are the prey in 1961 Alan Campbell explained this concept of a simple predator if a virulent phage meaning this phage that can only infect and kill the bacteria are mixed together then this would happen and in some sense these early models and I'm going to explain some in a moment use these same ideas and I'll explain them more in details in the next few slides what are these models the one thing I want to just preface this in terms of looking at ecological systems is I think often again in physics there's this notion that if we have a set of equations that's the real world we analyze them we find conclusions and therefore we've learned something about the real world Herbert Levine gave a lecture at Georgia Tech a number of years ago when he said he worked in high-energy physics and he interacted with experimentalists I don't know what he said twice a year because he just he had his equations if he solved something then they were going to be borne out and then he switched into condensed matter he had to visit more frequently then he got into biological physics maybe every week and he said he got into cancer he had to talk to them almost every day because these equations are not sacrosanct so we use them as a means to engage and look at experimental data and they give us a way of thinking about these systems here's an example of a three-part system and I'll explain the images in a moment in which we have resources for the bacteria to grow on and viruses that grow on the bacteria and let me explain each term I'll give you a schematic in a moment we're envisioning a context in which we have this open resource vessel that has constant volume incomes resources with a density r0 at a rate omega which is why we have this term there everything is being flowed out because we want to make sure the vessel doesn't overflow minus omega r minus omega n minus omega v then the bacteria can grow on the resources which you can see as the minus gamma rn in that upper right portion converting with some efficiency epsilon into new bacterial biomass and likewise the viruses interact by collision with bacterias we have this minus phi nv it depends on both the concentration of bacteria and the concentration of viruses and then have a burst size beta so for every infected cell we get beta number of virus particles out in a lysis event okay good if you take this system and you start as I do with a bacteria and resource only system which is here at the top what you can see is that if you add viruses the viruses rapidly shoot up in this little model and there are these oscillations that here go down because they're not limited anymore by the resources in fact the viruses are controlling their density and because the viruses are controlling their density there's actually more resources in the environment whatever the limiting sugar content might be glucose, maltose, etc if you project this in the phase plane I know it's a three-dimensional system I'm just going to look in two you can see again these counterclockwise cycles for the same reasons though these happen to spiral in okay so we take this locker voltera model and we can start to use it as the basis for thinking about virus host population dynamics but obviously I'm missing some pieces here any questions about this model that I've put up I'm going to give a couple other examples in a moment yes what are all the terms mean sure resources bacteria viruses these are the change of the expected time so it's a coupled system of non-linear differential equations and it's three dimensions omega is the outflow rate and the inflow rate so everything leaves that is inside this chemistat the resources the host and the viruses just because we need to make sure we have constant volume and we're constantly putting in new resources so think of this as sterile media coming in without bacteria or viruses it has a density of whatever the sugar content is of r0 and this it becomes plus omega r0 so in the absence of any of these other features so if I'm just making media all I would have is a sterile media there's no better and it would be eventually in the chemistat r equals r0 would be my fixed point but it would not be stable and since only Manuel has done a bacteria experiment before you would imagine if I put a little bacteria into a vat full of resources they would increase right let me just explain the rest of the terms and I'll take your question so this therefore is the reduction and you can use more complicated forms for the consumption that involve saturation this one is unsaturated and this converts with some epsilon rate from this carbon into the prey either biomass or abundance this is the lysis term the reason why it shows up twice is because when a virus infects then we've removed one virus particle from the media but outcome beta and I've made it explicit because sometimes there'll be an internal state in which case the release might come later yes epsilon is the conversion rate from the uptake of resources into new bacteria and what was your other question the number of viruses that come out from each lysis events what we call the birth size Fabio did you have a question a supply of resources from the what yes so this is an example let me see if I have it here I might have switched the order and I'll go back what I'm imagining here we have a reservoir of resources for this simple example in which the resources are being pulled in at this rate omega into this chemist out vessel and then it's all going and pulled out right so this is often the case even in in open flow systems even in a marine system where you imagine there's upwelling there might be resources coming from someone other environment and so that also might be an explanation for why we have that but in these chemist that models this is a controlled rate of inflow from a determined reservoir okay in fact I'll just stay on this slide since I'm here this model really combines and maybe this makes it more apparent we have this media reservoir where we're pulling in resources things are happening in a controlled temperature environment where it's being shaken so we can think of it as homogeneous and on top explains a little bit more about the details of how a virus infects and exploits a host cell these viruses are passively diffusing in the environment they come in contact with bacteria inject the genetic material the viral genetic material into the cytoplasm of the host it begins to replicate and take over first to a control process of taking over the host cell machinery replicating viral genetic material let's think about this DNA packaging that into and self-assembled capsids and through a time process through a release of a hole in which makes a hole in the inner membrane and a license which makes a cut in the cell wall outcome all these viruses and so this beta you can think of as the number of viruses that pop out and it can be a number that can be as small as 10 or 15 or 20 oftentimes 50 100 if not more okay questions about this concept okay this does suggest though that maybe the model we had before was a little bit off because the model before says that as soon as we have infection we have lysis so here I've removed the resource layer for a moment just to focus in on what the dynamics might look like if we have an explicit infected class here we have a model in which we have susceptible bacteria infected bacteria and virus particles these cells increase in abundance in some logistic like fashion in the absence of viruses they would reach some saturating level you'll notice here I still have these minus omegas we're still imagining that everything is being flowed out but I'm not worrying about the resources right now it's implicit in this value K it would be the level at which bacteria would get to some carrying capacity within this chemist that environment and it would depend on flow rate and R0 and other things including the efficiency we have infection which removes virus particles from the media but instead of immediately creating new virus particles we now increase infected cells these infected cells lice at some lysis rate eta producing beta virus particles and just so I'm oriented if the cells are licing at a rate eta how long on average is the period in which they're infected one over what one over eta yeah and what is the typical time at which they're infected here's some time you're saying that's how long they're infected on average I think that's what you just told me what is the typical time if I had to stop watching could measure after absorption in this little representation where should I put my typical time meaning if I have some distribution where would it peak no beta is the birth size so who here thinks it should peak near one over eta who here thinks it should peak over here down near zero who here has no idea what's going to happen in this model which you can raise your hand and that's okay fine okay good okay good terrific wonderful let's just think about a process in which we have a probability per unit time of lysis in a unit time of eta which means the probability of lysis in a little time dt is equal to eta delta t right the probability that I don't lice in this time is one minus eta delta t the probability that I lice between time tau and tau plus delta t must therefore be I don't lice between any of the time zero and tau which means I don't lice in the time delta t I don't lice in the next delta t and I don't do that tau over delta t times I don't lice in any of those little small increments I'm breaking up my time tau to tau plus dt and I've broken them up into a whole bunch of increments and I'm trying to ask what's the chance I lice at that point I shouldn't have lice beforehand this is the probability I don't lice in a very small amount of time I must do that n times where n is tau divided by the little time increment times eta delta t this raised to tau over delta t is just e to the minus eta tau times eta and I'll make a little differential dt right I've just taken if you think about what your Taylor expansion is e to the minus eta delta t the delta t's cancel I get e to the minus delta tau which means if you have a rate of leaving this is just like a radioactive decay problem I have a fixed rate of leaving what is my residency time it turns out it should be exponentially distributed with mean one over eta the rate right so this is the probability of lice but it has a very strange feature it says that the typical time the modal time is actually zero it's instantaneous right whenever you see these rate processes this will be leveraged quite a bit in my lectures you should think of them as saying you can think of it as the analogy of a stochastic process which is happening at some probability rate per unit time and therefore if I just have a first order process my residency time in that state should be exponential okay does that help I thought some people said had no idea does this make more sense than it did a second ago when I just tried to ask that question good I'm seeing nods we're learning on day one which is great now that was a side thing but I want to then pull it back into this if you take this system of equations including this explicit infected class and you look then at the number of bacteria and the number of viruses you get true limit cycles you get end up getting this counterclockwise dynamics so they're exactly the reasons that Locke-Volterra found them except now instead of having a conservative system you have one in which the initial conditions don't matter that much and you end up converging to this limit cycle either from the inside or the outside and you can see that in the red lines and the blue show you that they're not that they are in fact isolated there's not an infinite number of these closed orbits okay now you might not like this model for the reason I just described in fact if you're a microbiologist you really might not like this model because you know this is not the case you know that it takes some time for viruses to assemble and to DNA to replicate and you probably think that the timing should look more like this or even maybe sharper this is one extreme limit this exponential the other extreme limit would be a delta function there is a trick to turn this kind of model into this kind of model it's called a linear chain trick which I'll just mention as an aside which is that if you ever are building these kind of little systems you want to change the structure of the shape of the residency time instead of just having one infected state you could have two where the rate goes twice as fast the mean time stays the same but now you've made something that looks gamma like or you could make three and the rate goes three times as fast or four four times as fast and eventually you're going to shrink down so that you'll keep moving instantaneously for one state to the other and you'll get to a delta function limit when you have very very very large numbers of states or I could take the same system of equations and now I've just put the boxes in red so you know what I've made different and what I've made different is instead of having the viruses released immediately I've written this tau notation to denote the fact that x tau is equal to x of t minus tau so anytime you see a subscript tau you should think of it as a time delay this says that the viruses get in now minus phi nv they create infected cells but infected cells lice based on how many new infections occurred tau ago and that's why you get this beta times the adsorption rate times this collision tau ago and you'll notice you have e to the minus omega tau the reason why you have that e to the minus omega tau is that in the intervening time period some of these infected cells were washed out of the system how many e to the minus omega tau because the loss rate of infected cells is omega so for the same reason I just explained this we have to add an e to the minus omega tau here let's think of this works in the right limits if tau were zero we don't lose any that becomes one and it just becomes our original model or basically we don't even have an infected cell class anymore they're instantaneously released if tau gets very long then none of the cells that were almost essentially none of the cells that were infected a while back are present anymore to burst they've already been washed out of the system any questions about this this is called a time delay differential equation or delay differential equation it is a bit nasty I'll get your question just one second because we have to have initial conditions that are essentially an infinite number of initial conditions because in order to even start this model we have to know the dynamics between zero and negative tau before obviously we can set those to zero but it becomes a way of representing a more complex system of equations yes here and then on the audience go ahead right okay good let me explain that and I'll take the question this is the time from infection to lysis and when I have something like an i-class that has i dot minus eta i this is the lysis we end up getting an exponential distribution with mean one over eta like that this is a biological process there's nothing wrong with this if it was radioactive decay but in fact in biology it takes time to make these virus particles in reality there's probably no ability to make a virus particle maybe for the first in the case of many phage of E. coli for the first five or 10 minutes after infection you're just not going to find any virus particle the timing is often controlled so that if I were to look at the probability it would be much more likely to find it near the modal time so that the mean and the mode coincided rather than having the mode and the mean not coincide so biologically bio reasons it takes time to build stuff it should be more like that this is an extreme version of that kind of distribution which implies a delta function a narrowing of that distribution question from online yes so there is a question from the chat on the original setting so why did you remove the resource R and how do you include it if you look in my book I explained how to take a fast approximation but in some sense if you were to set the R dot is zero you can make a quasi stationary or an adiabatic approximation in which the amount of resources is controlled by the host if you were to plug that value back in you end up getting an equivalent value for K which depends on some of these microscopic parameters so it just also to illustrate this it becomes a little bit unwieldy but essentially it has some of the same features not entirely there are some subtleties there so it's a good question here I'm just making an alternative simplification I'll give you the reference to the book as well but you essentially essentially set that R treat as a fast variable set it to zero yes I wanted to give the two extremes next week I'll actually talk about something called generation interval distributions which we'll talk about intermediate cases and it won't actually be Gaussian that won't be a good choice because it has a tail on the wrong side of zero but gamma would be good or an Erlang distribution which is really what these are right and I kind of gave you the example of how to do it by extending the number of eye classes and that's precisely how to do it so I want to just maybe finish with one last point and then I'll wrap up today and I'll continue this lecture tomorrow is it an urgent question because I wanted to explain this yeah can you explain which yes the only new one is tau which you should think of as 1 over 8 it is the latent period yeah anytime you see the subscript it means a delay different a delay I know I'm supposed to finish at 45 so can I do I have time to do one last thing because I wanted to logically connect and I won't go farther than this okay let me do one last thing when can these viruses invade it would seem that it should always work the viruses in fact host they should always be able to proliferate you could imagine a technique and maybe we'll do it to start class tomorrow or if you're interested you can do on your own we can imagine a system in which we start near the virus free equilibrium and part of the reason why Manuel I wanted to use these simple systems I can make the point without doing too much algebraic work and in fact I'm even not going to do much algebra here I'm going to do something more intuitive you can imagine taking that system finding the fixed point linearizing the system and checking for stability and you would find that the system in which we don't have viruses is sometimes unstable which means viruses invade and sometimes it's stable viruses are added but they don't invade and proliferate and it would be a bit algebraic and you might not find it that satisfactory but you should try it anyway to confirm what I'm about to explain I want to give you an intuition a physical intuition as to the conditions that relate to these parameters to give them a little bit more life imagine we start with a single infected cell this is a cell this little squiggle denotes a virus genome inside okay what are the fates of this infected cell fate one it could be washed out right because if you see there are two things that can happen to infected cells they can burst or they can be washed out it could then burst and I do that like that where I now generate beta number of new virus particles what is the chance that it washes out before lysis inside this reacting vessel we have two rates one is omega one is eta they're competing what is the chance that the washout happens first so we have a rate omega and a rate eta this is lysis I think you more or less have it but let me just say that this is washout if you're flipping coins I have two probabilities where I have two rates in which two processor are happening the probability that one happens before the other is just the ratio of one rate versus the total rates so here we have omega over omega plus eta and here's eta over omega plus eta that's the probability we go down this path we generate beta virus particles each of these particles itself has two fates it could be washed out and array omega here's this so I can think of this as the eta rate and the omega rate here's omega this one can absorb to a new cell it absorbs at a rate phi n phi n star because originally we only have the bacteria if it does that we've made a new infected cell we've made a whole loop from infected cell to another I have to apply that to every one of these viruses because initially we can think of the bacterial population as fixed therefore I can ask the question how many new infected cells do I get starting with one infected cell well I should get eta over eta plus omega of the time I get a lysis event otherwise I'm adding zero if I get a lysis event I multiply that by beta because I make beta new virus particles of those virus particles only phi n star over phi n star plus omega find homes find new hosts this probability times the number of virus particles times this probability of finding a new cell per virus particle should be greater than one if this product is greater than one it means that initial infected cell leads to more than one infected cell on average that keeps happening and you have a proliferation you can also think about it from the perspective of the virus particle if I start with one particle does my loop end up having more than one virus particle on average in which case I would say what's the chance that you find a host cell for that one virus particle what's the chance you can survive in the infected cell to lead to lysis and you generate beta new virus particles whether you think about infected cells loops or viral loops you end up getting the same criteria and since I'm almost done with time you might want to try this on your own this process of linearizing and checking for stability will also lead to a condition in which you either get a stable fix point or an unstable saddle when this is true but this is more fun because it has all the biological intuition and none of the algebra so I've basically done all the work of calculating the trace and the determinant using the trace determinant diagram and yet that's all there is to okay and now you have a physical sense of what's going on whether from the perspective of cells infected cells leading to one or more infected cells or one virus leading to one or more infected viruses if initially in the uninfected case in the disease free equilibrium we generate more than one infected cell more than one virus particle if we start with one that process repeats repeats repeats and we get an exponential growth which corresponds to viral invasion which also means traits that favor invasion include bigger birth sizes higher ability to absorb faster lysis rates all things considered all of these things will favor interaction and also pointing out that the answer will depend on how well the bacteria are doing if there aren't that many bacteria around even if you're potentially a great virus in terms of your viral life history traits that the environment stinks for your host it stinks for you too okay I think that's as far as I can get today and I think my next slide is a whole new section and I think I'll start with some of the oh actually no can I can do the let me just do the last ones because I'm finished with part one I want to do because I haven't shown you any data yet but I want to show this last little bit it's only one more minute I'll be done right on time and I'll start with this tomorrow as well in the late 90s people built these chemostat experiments and looked at these same kinds of oscillations and I'll start with this tomorrow showing that you can get endogenous oscillations and chemostats without any external change and here you can see the virus host oscillations are changing on the order of two or more orders of magnitude over about a seven or eight day experiment and they have that same feature that the virus are peaking after the host peaks to the extent to which this data is resolved and this is all being driven by the interactions between the viruses and the bacteria so to summarize and this is part one these original models of virus host dynamics presume a simple predator-prey like relationship very similar to Lockevolterra in these models we expect there to be these virus peaks followed excuse me host peaks followed by virus peaks leading to the host decline followed by viral decline and it's the endogenous interactions that drive it we can use simple principles of invasion to understand ways in which the life history traits make a difference so that not every virus that can infect a host can proliferate in a particular environment in other words the invasion is not inevitable and tomorrow I'll pick up and begin to expand this beyond populations into evolution and beyond so that's it for today thank you so we have time for questions if any from the chat no yes there is a question in in these two last models we just saw we could infer that we could find limit cycles to our system my question is do we have only a single one limit cycle or could we get more than one I see in these systems we get a unique limit cycle that's a good question I understand it yeah in some other systems I get that there could be other more arcane things happening but this system has a single adsorbent limit cycle other questions okay so we can take our break for 15 minutes and we'll be back at 11 for the first lecture by Fabio Shocconi