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Good evening, everybody. Hope you can hear me well. Yes.

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Excellent. So I'm as I said, I'm going to talk about scaling, but actually, I'm not going to talk about that type of scaling.

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I'm going to talk about dynamical scaling and how we actually use that to be able to understand population dynamics.

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As I said in my background, as is in zoology, I'm actually trained as an ecologist, so I'm going to talk a lot about ecology.

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But we also, as I said in my group,

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we have lots of diverse interests across the breadth of the life sciences where we've been interested in thinking about applying mathematics.

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And we're going to I'm going to sort of delve into those sorts of things.

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And I'm also going to do a little experiment halfway through or thereabouts where some of you will get to eat chocolate.

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So you have to be fast. When I need six volunteers, the people who put their hands up fastest will get fed chocolate.

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So I am going to is are we going to do that? And I'm going to start at the very end. And it's another little game that we used to solve.

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So I explore some ideas in mathematics and biology.

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So this is what I was just saying is very, very, very common in biology.

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We think about scaling in in a static way. So here, just like this.

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Some sort of crab with a sort of a pincer.

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And we can relate it to the size of the body, the size of the crab, to its the size of its pincer.

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And often those are plotted on a log scale and they show that this is very classic al symmetric scaling.

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And in this case that I just nicked from the web, we can just see there's actually a grey.

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The bigger the crab, the much bigger it is.

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Pincer is in relation to its body size.

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So this idea of scaling in this elementary scaling in biology is very common, but it's a static thing.

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We tend just to measure species.

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We can go out and we can collect crabs so we can measure the length of the car pieces and we can measure the length of their pincers.

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And we can look to see how that how those relationships play out.

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Well, I'm a population dynamics artist, and I'm interested in thinking about how things change through time and space.

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But tonight, just about moment, mostly about time. And we want to think about how we do that.

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How do we scale across these different types of dynamics?

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How do we take into account the fact that things may operate on different temporal or spatial time, time or space spatial scales?

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So I'm going to go through about three things, three different types of ways that we approach this.

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And I'm going to so explain the mouse and I'm going explain the biology behind some of these with some examples.

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I'm going to explain some of the maths and at the end I'm going to show and talk about the implications where this becomes really,

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really important and why we think it's important and all the stuff that we do in in extending our ideas of maths and biology to the real world.

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So I'm going to talk about slow, fast dynamics.

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I'm going to talk about this functional response around this point here where we get to surprise interactions.

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That's where we get to eat chocolate.

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This point here, I'll talk about how we can how we can use approximations, and I'll talk about some stuff that we've been doing.

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And so thinking about how we can understand bipolar disorder and mood fluctuations,

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I'm going to finish off with some implications and some very contemporary stuff that we do in terms of biological control.

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So slow, fast dynamics. What I mean by that, well, I'm going to I'm thinking, okay.

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I'm thinking about these sorts of systems. So this is a sign of our regular system.

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If any of you ride horses, I know it is about horses. This plant is pretty nasty to horses.

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It causes lots of things that make them sick and poorly. It's a very it's a very common plant in the UK landscape.

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It's a accident. It's an either an annual short lived perennial plant or long live perennial plant.

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It has these very colourful and yellow flowers, but it's stuffed full of very nasty things that may make horses and other things sick.

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This is the Cinnabon moth. It's as a as a caterpillar. It feeds on this this particular plant.

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It only feeds on this particular plant. And it's able to sequester, capture up those those those chemicals.

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And under that, these caterpillars boast that the caterpillars are distasteful and the the moths are distasteful.

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This is very pretty. It's on the way out in about the middle of May, through to the end of July.

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And it's a pretty common sight, a pretty common moth. We're interested in the dynamics.

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How do what sort of population dynamics, how does this moth and plant interact over time?

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And one of the things that we can do is we can write down an equation to capture that sort of dynamic.

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So so this might be one way that we might do that. So I have resources are changing through time.

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So the change in resource through time on the left. And then I've got some of the biology that describes that, that sort of dynamic.

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So Lambda is going to be some input rate of resources into the system.

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And then I have some consumption by the moths which I.

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Usefully called G and they're consuming this resource at right alpha.

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And the other thing that will happen is if the resource doesn't get eaten, it will just decay away.

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And then there's some coupling and I'll come back to that coupling later in the talk.

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But we're going to couple the. So what gets eaten by the what gets eaten by them also gets converted into new, important new moth.

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So this is the birth rate of the moths, if you like. And, and then the only thing that's going to happen to them is that going to die.

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And we can do lots of analysis on this sort of model.

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We can really think about, you know, what, when the population dynamics would be stable, when they might show cycles, I was quite involved.

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And we really want to think about, well, actually, if we're thinking about scaling all those things,

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all those things operating on the same sort of temporal scales, and if not,

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can we get round that other some approximations that we can take to think about that?

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So one of the things we can do is we can say, well, actually, maybe the resource dynamics are operating faster.

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Maybe this plant is just going through its lifecycle a lot faster than the moth.

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And I can't really drop my tummy and pop my head at the same time. But you get the idea this hand is going faster than this.

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And so the resource dynamics are going faster than than the moth dynamics.

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And the idea then is that the moth only experiences on average what the resources are doing.

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So we can then approximate those resource dynamics with a little bit with with a little bit of, with a bit of math.

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So we're going to assume that actually, on average, those resource dynamics are not changing.

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The D, the d r by d t is set at zero.

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So if we do that, I can then sort of set that first equation equal to zero and I can send re-arrange it just algebraically for R.

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They're making a couple of assumptions in that and doing that. Firstly, as I said, the resource dynamics are going faster than the math dynamics,

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but the little things are going to make an assumption about is the loss rate, this decay rate of the resource.

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So we're just going to see that the models are really good eating their resource and not so good and they eat lots of it.

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So there's very little left around so that actually the resource consumption rate is actually exceeding these loss rates.

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And if we do that, we can make this some very simple approximation. And so if we make this rearrangement to the average,

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the average resource dynamics are all now a given by lambda divided by this this the rate of consumption and the number of that around.

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And that makes pretty good sense. Basically, if there's lots of resource flowing into the system,

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that means the resource is going to increase and as the numbers of of the month increase than the amount of resources going to go down.

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So it captures in some and some sense what might happen to the resource up our average amount of resource that's around in the system.

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But we're not really interested in that.

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I'm trying to think about what would we do when we when we plug that into the to the second equation to do with to do with the to the moth.

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So I what I'm going to do is going to replace this term here by the approximate that by that approximation.

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So if I do that, I'm going to take this average result some place replace the all in in that second equation.

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And it's quite we get we get a much simpler expression in terms of the moth dynamic.

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So I can basically say I can cancel these two terms and I'm left with lambda minus new G.

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So basically so the. So in this in this slow, fast dynamic, I have the amount of resource flowing into the system.

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I know the only thing the other thing that's happening is that the moths are just dying.

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Just a nice, simple equation. So nice, simple differential equation.

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And on on the right hand side of this and I'm pleasing my schoolteacher very much

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because she or she used to tell me that I should show my workings a lot more. What we can do is that we can solve that equation.

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I can go from this equation to knowing what was this function actually look like for all time?

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So how many generalist moths, how many moths would be around at each point in time?

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So I could work through some calculus? Is this this is this equation written small?

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And here is my solution. At the end, it's a bit of calculation.

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You need to know how to take the integral of one over G and something else and it all falls out.

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Basically, I have this sort of function at the end. Okay, so that as well it may mean something to somebody, but we can do some things with this now.

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We can I can draw it on the board or we can just do some thought experiments about what

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happens for different values of T and the best values of T to pick are zero and infinity.

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So if we set t, if we set T equal to zero and we work through this and everything else is just we work through to see what happens.

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What how many generations do we expect to be around? At times zero.

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It turns out to be zero. So we are down at the origin of a graph on an x y graph.

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We're down in that. So when there's no time, there's no generate, there's no models around.

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If we set T to infinity,

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well basically this then approximate this this term here then is essentially zero and we're left with one -0 lambda over you, lambda over anew.

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And what that basically tells me is that there's then there's a fixed state.

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So if I know the value of the resource input, which is lambda and I know this decay rate, then I should be able to say how many generations around.

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But a very cool thing about that, this approximation is that if we setting t to infinity and infinity goes on forever,

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then I go from zero to some sum fixed constant value a ratio between London and a new.

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And that's really cool because that's telling me that the dynamic should be pretty stable, right?

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So for all time beyond zero days, this graph is going to go up and it's going to stay flat for all time.

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And that tells me that actually with this approximation, my slow, fast dynamics,

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I should expect to see resource dynamics being the most dynamic, being relatively stable.

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Why is that? Well, there's an ecological process, so the resources are flowing into the system and it's called the donor control.

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That's a bit of a jargon term,

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but it basically means that the resources are flowing into the system and then modulating the dynamics between the moth and the resource.

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And they're giving rise to these very stable dynamics. So the next thing we have to ask is, well, how true is that?

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Well, this is where data gets in the way and we have the moss.

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So it's a pretty pictures of the moss.

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And this is the Time series that was collected from the 1970s to the early 2000s of the cinnabar moth interaction.

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So I've got the log numbers on the on the Y axis and that on time, every year,

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every year this person went out based in the Netherlands, just just south of Leiden.

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It's a very nice habitat. They're counting the number of regular plants and the number of moths on those plants over that time.

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So and we see that the black line is the moths and the red dotted line is the plants.

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Okay. It doesn't really show that sort of that sort of expected dynamic.

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So we've got to think why why might that not be the case?

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And we'll get onto that very shortly. But it's actually sort of like brings in thinking about other bits of of biology and maths.

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And we want to think about what we don't see that nice stable interaction, not nice stable dynamics of the moss.

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And why is that? Well, we can we can appeal.

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We've got an idea about the sorts of dynamics that we have in these these time series that we have,

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allow us to be able to do some statistical analysis.

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So I can ask basically what determines what numbers in this place in the Netherlands and how is it dependent?

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Is it dependent on the market? Is it dependent on the resources? And how do those two things interact?

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So the black line is the as I said, is the is the most numbers.

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And I'm asking basically, how do those numbers change over time from one time steps?

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And what are the drivers?

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How are the numbers in the previous time step important or two time time steps back really important in driving these sorts of population dynamics.

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And we see two things. One, we see that the models affect themselves.

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So they compete, if you like, for access to this plant in some way.

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But there's also a dependency of the plant.

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There's actually a link that that that the idea that they are specialists and they do need this plant and that plant that drives their dynamics.

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But it's more interesting than that. It isn't actually the number of plants that are around in the last year.

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It's the number of plants to around two years back that drives these sorts of dynamics of the moth.

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And we can do the same thing for the plants. We can do the time series and statistics on the plant so we can say, how do the plant numbers change?

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And it turns out, well, plants affect themselves both now.

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So the number of plants now are affected by the numbers that were around last year and the numbers are around and two years back,

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but also affected by the number of moths that are around as well.

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So we really there's a really dynamic interaction there between the plants and the moths and the moth in the plant.

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And this approximation,

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while it's really good in theory to solve doesn't it doesn't really hold when we think when we compare to data because of this inherent link.

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And if I just flip back really quickly, is this is this idea here,

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this idea of this can this this response and this link between what gets eaten and what gets and what translates into new,

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new moths in the next in the next time step. So that brings me naturally, really naturally onto this idea.

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We call that link in biology, in ecology a functional response.

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If you're a biochemist, you'll know it as a Michalis mentum function.

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It takes so many different words or phrases to explain what this what this is,

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but it's a link between two different states of the system and it's a really great way and we'll get into it surely about how you can,

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how you can approximate individual behaviour and individual performance into dynamics.

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And I want to sort of explain that is with a completely different bit of biology,

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and that is to do with the the blood stem cell system, the haematopoietic stem cell system.

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So we're really thinking about how we can how the the individual response is

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needed by we needed to make a blood system are dependent on what's around.

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It's called the demand controls it's called the demand control system.

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And your blood system is just fantastically fascinating.

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In your bone marrow, you have stem cells that are constantly making new blood cells.

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And they can make two broad types of of stem cells.

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They can make two types of blood cells.

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The lymphoid cells going down this sort of root are all the sorts of things that make your immune system, make it work.

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And the other ones are those or myeloid cells which go on to make of a whole set of red blood cells.

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And we just think about we just kind of stylise that sort of interaction into thinking about myeloid cells,

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those that make red blood cells and lymphoid cells that make immune system.

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We're not. But there's immense amount of complexity in terms of the different types of cells that can be made.

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And obviously when it goes wrong, it goes wrong and it can make people very sick.

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And we're trying to understand how we can use a mouse to actually understand the dynamics of this,

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this really incredibly diverse system and how we can use these scaling as these functional spot response scaling to understand that.

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So what we can do is that we can stylise it by going from that to a very complicated bit of biology to sound like.

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It's like saying that I could write, you know, just write out in terms of just a schematic diagram.

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So basically we were thinking about transition. So you can go, how do these how do these stem cells give rise to new stem cells?

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How do they give rise to progenitor cells? And how do those progenitor cells give rise to those myeloid or those red blood cells or white blood cells?

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And what tends that what the sort of ideas behind this that sort of so reach back to

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the 1960s we really just thought about two types of transitions that could happen.

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So this this this particular little blob represents a stem cell.

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And the things that can happen in the next time step in the next event are that I get two new stem cells,

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or I can go from a stem cell to a progenitor cell that's going to give rise to a red blood cell or a white blood cell.

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But actually, the system's a lot more complicated than that. Like, if these are the man control system, it's basically your body.

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Say, I need more. I need more red blood cells or I need more immune cells to fight off this disease.

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There has to be other things going on and we can sort of extend that out.

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So a stem cell could say there's also like an emergency response that says, actually,

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this stem cell now needs to not only it needs to sort of like die and die and make

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tuned you to new progenitor cells that then would make red or white blood cells,

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depending on how the stem cell could die,

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the progenitor cell could differentiate even further and make two new two different types of progenitor cells, two more progenitor cells.

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Because that's what happens in these in these stem cell systems,

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the progenitor cell could pass away or it could differentiate into something that's really specialised,

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making a platelet or a T cell or whatever it might be.

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So we can really think about these sorts, this sort of system in many different steps, so many, many different sort of event, event driven steps.

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And by that, I mean that in any small interval of time, only one of these is allowed to occur.

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And the way that we I'll show you what I mean by that shortly.

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But what we can do is that we can we're getting toward some more mass. We can sort of start to put some some detail on to that.

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So here's the stem cell, here's a progenitor cell and has a differentiate itself.

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So a red or white blood cell. And these are sort of intermediate cells.

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So I have some background that affects whether the stem cells are going to give birth or they're going to die or what controls their their activity.

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But the important thing to note here is this these these arrows, this one.

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So where we have this this this particular these two particular these two particular functions,

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this negative signal, that's the demand control that we're interested in.

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We're really thinking about what is it that how do we link this,

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what's needed from the body in the peripheral blood to to tell the stem cells what to do?

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Because they don't really know.

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They're just going to just say they're just going to keep making things and they need to be told what to do and they need some way of linking.

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And that's where that functional response actually comes in.

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So we've got a progression of births, if you like, of progenitor cells, births of of differentiation, differentiate themselves.

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But then when they've differentiated, we need to be able to understand,

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we need to be able to counter this idea of this negative feedback, this negative signal.

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It says, make more or make less, do something, basically.

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We can capture that again with a set of ordinary differential equations, all of that biology and are more complicated than the ones I showed earlier.

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So again, we've got this body t that's the change in stem cells through time.

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I've got deep body. T That's the progenitors through time.

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And Dee dee dee dee is the change in differentiated cells and they've got bits of biology that capture this.

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They're actually the ideas of what we expect that happens in a stem cell system.

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So this first term, this term here, this term here, depending on where you see, my point is just telling us that the stem cell has a niche.

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It has a limited amount of size that it can occupy your. Your bone marrow is limited in space, the space it occupies.

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And that means that the number of stem cells that can live in that space is going to be limited.

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And that's set by K. We call that the carrying capacity in ecology, but it's just a fixed amount of space that the stem cells can live in.

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And then then those things that can happen, they can they can give birth to new stem cells.

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All they can do. They can change these. They make progenitor cells, which is this term here.

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This is the second picture I had where a stem cell which replaces a stem cell and also makes a

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presenter cell or this one here where there's a two says there's a more emergency signal that says,

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make me two. That's that. That's the use of stem cell.

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Need to now make me two progenitor cells. Okay. And then there's and then there's lost terms that's decay term to death terms of stem cells and death,

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terms of progenitor cells and transfer of progenitor cells into into differentiate yourselves.

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But it's these two functions that are really critical, this function response at linking these dynamics,

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linking the individual performance so that to the to the population dynamics of the system.

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So if I if I use some numerical optimisation methods, I can I can iterate that model.

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So here's the number of differentiate yourself through time. The red line is the deterministic dynamic.

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So if I just run that model is just as a is an all set of a coupled order and the differential equations, I end up with this red line.

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As I said, it's a stochastic system, so it's a random system.

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So if I make this event driven, if I make those, if I make each of those transitions an event,

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I can only operate in a small, small amount, a small time stab.

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Then I get a whole raft I can. So look at this of repertoire of dynamics that I see out of this.

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What do I expect from this system,

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given that that is that it is these random to knowing whether stem cell is going to make another stem cell or a progenitor cell.

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And while this captures those actually the variability that we could see,

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it shows us is bounded in some sense that actually the system doesn't go off to infinity and it doesn't go to zero.

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There's actually some bounds. So we can run this many times and we can we can do some fancy maths and we can quantify the variability

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that we see in this in the number of differentiated cells produced by any single one stem cell.

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And this is just a spatial plot, if you like, of just a heap liposome.

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Well, here's the number of stem cells, so of around about ten, and here is the number of differentiate ourselves on average around about 40.

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But we see some spread around that that function response is, you know, not demand control is not perfect, actually.

251
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And I'll come on to this. Surely there's we need to understand in a little bit more detail this idea of homeostasis.

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And this idea of feedback. So how this demand feedback plays out and how it can can occur in all three Broadway's so really thinking about.

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The number of lymphoid saw a number of cells that are improved for blood.

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So we'll say the number of and the number of sort of immune cells that are around.

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And we think, well, what proportion of myeloid cells should be produced given this number of lymphoid cells?

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Well, the first one is could be completely independent. These two things in the skull could just operate independently of each other.

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So we just have the number myeloid cells. It doesn't really care how many immune cells, how many immune cells.

258
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So then I'm just going to make more red blood cells because that's what I set up to do.

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All that could be some relationship and proportional to need.

260
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So is the is that is that as there's more lymphoid cells, there's there's a need for more myeloid cells.

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So it could be non-linear, it could actually start to increase and then plateau out.

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We need to be able to understand that. We need to understand how this is so peripheral blood.

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How this peripheral blood feeds back is information to the stem cells.

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And this is the crux of it. You can write down a functional response.

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We can write down a mathematical expression that says, how do we link the the the proportion,

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the number or the proportion of cells that need to be made to what's already there?

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And we start by doing that in homeostasis, where the system is not being perturbed.

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So we think of this row H as the number of myeloid cells or the number of lymphoid

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cells that are being produced in homeostasis to maintain the body in a fixed state.

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And it's given by. So how many should I produce is in proportion to how many of that.

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This is the ratio of myeloid cells that are there to lymphoid cells that are there.

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And then there's some parameters, alpha and gamma,

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that we want to be able to understand what they look like and what values should they have that give us a homeostatic state.

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Well, if you want to know what values well, we can. So if this is homeostasis, that's a fixed state.

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We know what that is because it's homeostasis. It's not changing over time.

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So I can then sort of rearrange this expression and say, well, what values of alphas give me this?

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What values? What values of alpha should I have that give me this homeostatic state?

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And what values of gamma also give me this homeostatic state. And it turns out there's a whole family of them.

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They all look like this. There's a whole set of values.

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So here's the values of alpha and the values of Gamma at every single point on that line represents homeostasis.

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So there's a whole different value. There isn't one set of alpha, and there's not one gamma value that's going to tell me that this is homeostasis.

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It's just this whole family. Every value along this line is a point where we identify a home that will give us a homeostasis.

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Give us homeostasis.

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Well, that's pretty cool, because actually what that means is I can pick of I can pick a value of alpha and gamma and actually ask,

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what will the functional response actually look like?

286
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It doesn't actually look, you know, how how many so what proportion of myeloid lymphoid cells should I produce, given how many are there?

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Basically, from each of those choosing two of those choosing an alpha and an associated gamma gives me a whole set of family of of outcomes to this.

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So the functional response is it demand control? Is it that is in proportion to what's already there or is it independent of what's already there?

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Really depends on the parameter values that we have to all along that line.

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We can get a whole set of different dynamical outcomes from this, and I'm about to explain what I mean by that.

291
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And when we talk about predator prey interactions, but the but we have a whole family,

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a whole set of different shape functions that relate individual performance,

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the demand required so of what's in peripheral blood to what the stem cells, what those stem cells should actually do, which is pretty cool.

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And I'm going to get on to talk about predator prey interactions and I'm going to talk about the same sort of thing.

295
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So the same sort of thing plays out. We're really thinking about how do individual predators, moths, eating plants or whatever it might be,

296
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polar bears eating seals or dinosaurs eating small rats or whatever it might be.

297
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How do they how does that how does that behaviour of eating those things translate

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into a population dynamic outcome that affects these things through time?

299
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So notice that differential equations. Oh, no, no, not differential equations, but similar to the ones I introduced before.

300
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So I have different letters, but the same sort of biology. We're really good at this.

301
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In biology it does changing things with letters and not really changing much.

302
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But we've got predators and prey.

303
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So n is my numbers of prey and P's, my number of predators and damned by d d invites the change in imprinting p by d t is the change in predators.

304
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And I got some biology on the right hand side. So I've got a birth rate. Oh, I've got a consumption rate by predators on prey.

305
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I've got now I've got a conversion rate, so I'm converting dead prey into new predators, and then I've got a death rate of predators.

306
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So let's just scale a lot back and think, well, what sort of dynamics should I expect to have if the predators aren't there?

307
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So let's set pizza zero and see what happens here by six.

308
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So we've got DRM by D, T, R and basically what sort of dynamics do we expect to see?

309
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So again, we can do some maths this time. I'm not showing my workings, but I am.

310
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Show you a picture that shows the sorts of dynamics we get. We expect if our is positive.

311
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So there's positive birth as the R is greater than zero,

312
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then the population of rabbits in this case just will grow exponentially unbounded and they'll keep doing that right.

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So that's all that's going to us. All we expect to see in this is predator prey.

314
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And in this in this interaction, when the predators are not around,

315
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the population of rabbits, in my case will just continue to expand through time forever.

316
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Well, that's not going to really happen, because what's going to happen is that the predators are going to come out and start consuming rabbits.

317
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Basically, I'm going to think, well, actually, we really want to understand how does it as I said,

318
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it was really thinking about how does the individual predator eating a rabbit,

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which is a behavioural response, the predators are going to find a rabbit and catch it and eat it.

320
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How does that translate into population dynamics?

321
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Okay. So we are going to work out what this functional response looks like and do some analysis.

322
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So here's the opportunity. I have six bags of chocolate.

323
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You don't know is how many chocolates are in each bag. And will I like to do this sort of experiment?

324
00:29:05,710 --> 00:29:09,640
It's a little exercise if you do. So I need six volunteers to put your hands up.

325
00:29:09,640 --> 00:29:15,340
I will pass you a tray and I. And I'll pass your tray and a bag of chocolate, and we'll work through this.

326
00:29:17,070 --> 00:29:28,930
All right? Oh, I can guarantee the trays came from the cafeteria, and they're still warm, so they're clean.

327
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But I also bought clean pieces of paper as well, so feel free.

328
00:29:34,900 --> 00:29:39,160
I'm just going to put those there, remember? Just put those I remember behind our trays.

329
00:29:39,160 --> 00:29:42,970
First, take a tray. Who else can I. Can I give you a tray?

330
00:29:43,300 --> 00:29:48,410
Excellent. Oh. Can I give you a try?

331
00:29:49,950 --> 00:29:52,950
Can I give you a shake? Thanks. All right.

332
00:29:53,160 --> 00:29:56,940
Anybody on this side want to try? Excellent. Well, yeah, they're still wet.

333
00:29:57,070 --> 00:30:00,240
Well, that's disgusting. I'm sorry. You're going to get a wet tray.

334
00:30:00,630 --> 00:30:03,720
Sorry. I've got one more. I'll go over this side of the room.

335
00:30:03,960 --> 00:30:10,980
Bear with me a moment. Somebody over here is interesting enough.

336
00:30:11,850 --> 00:30:16,020
Excellent. Take your tray. It's a bit wet. Sorry. I'll be back with some chocolate in a moment.

337
00:30:19,160 --> 00:30:22,560
Right. All right. Right.

338
00:30:24,950 --> 00:30:28,700
Now, remember, I gave my trace back a piece of paper. Thank you.

339
00:30:29,150 --> 00:30:33,620
And I'm going to give you a random bag of sweets. You've probably got about 25 in there.

340
00:30:33,830 --> 00:30:39,470
You need to open them and tip onto the piece of paper. So I'm going to be a piece of paper and a bag of chocolates.

341
00:30:39,740 --> 00:30:42,800
I want I need you to do when you have a piece of paper to open the bag.

342
00:30:44,000 --> 00:30:48,500
So guess how many chocolates you have. I'll try and tell you. And just spread them out on the piece of paper.

343
00:30:49,070 --> 00:30:53,630
Probably about five or ten in there. What's your opinion?

344
00:30:54,890 --> 00:30:58,130
Somebody's got to get really lucky because one of these bags has got about 100 sweets in them.

345
00:30:58,950 --> 00:31:02,400
Oh, it's going to be a good. Oh, I'm sorry, Doctor.

346
00:31:02,420 --> 00:31:06,050
You got five. I'm sorry. Don't worry. It could get worse.

347
00:31:06,530 --> 00:31:10,760
You go back in there. Where else is my tray?

348
00:31:11,330 --> 00:31:18,050
Oh, one here. Excellent. So it's put about ten or 25 in there.

349
00:31:19,820 --> 00:31:26,660
Okay. Sorry. I have one more to do over here.

350
00:31:28,100 --> 00:31:32,070
You get the bag of Eminem's that's left in here. Well, you can't use them all.

351
00:31:32,090 --> 00:31:37,490
I'm afraid I only need you to take two out of the bag, but you can share them with the people that are around you if you like.

352
00:31:38,030 --> 00:31:42,650
So there's the chocolate, and here's the back of them. So just take two out and put them on the piece of paper.

353
00:31:43,040 --> 00:31:48,800
So if you've opened up your. So what we're going to do is we're going to do this idea of a functional response.

354
00:31:49,400 --> 00:31:56,389
And the idea is that you to you are going to be predators and you're going to find your prey on the M&Ms that you've cast out on your piece of paper.

355
00:31:56,390 --> 00:31:59,900
So take them out onto the piece of paper. If you don't know the person next to you,

356
00:31:59,900 --> 00:32:07,220
it's a great opportunity to introduce yourself and just help get them to spread them at random so you can't see where they are.

357
00:32:07,580 --> 00:32:15,050
Because what we do, we have to close our eyes. You're going to take your your your pride claw, which is actually your index finger.

358
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This is your index finger. I have to say this every time this is your index finger and this is what you're going to use to search for your prey,

359
00:32:22,850 --> 00:32:26,000
not your hand, your index finger. Okay.

360
00:32:26,210 --> 00:32:33,050
What are you going to do is you're going to close your eyes and you're going to tap on the board on the piece of paper when you find a mate.

361
00:32:33,320 --> 00:32:37,040
Please don't eat it just yet. Put it on the side because we need to know how many you've got.

362
00:32:37,400 --> 00:32:42,080
So if you've all got a good idea of how many you've got on the paper. Yeah, how many you've got to start with yet.

363
00:32:42,500 --> 00:32:46,800
You've got ten. Somebody's got 100. Yeah.

364
00:32:46,920 --> 00:32:50,170
Somebody got 50. Who's got 50? Stick your hand up.

365
00:32:50,170 --> 00:32:53,670
He's got 50. Somebody must have quite a lot out there.

366
00:32:54,360 --> 00:32:59,010
Yeah. Yeah. So. So you've got some idea about how many you've got out there, right?

367
00:32:59,410 --> 00:33:03,209
Okay, so we're going to do this. So what? The idea is that you're going to close your eyes, but it's not for a minute.

368
00:33:03,210 --> 00:33:07,350
You're going to see how many you find. Okay. All right. We're going to do is going to do some analysis.

369
00:33:07,350 --> 00:33:12,510
I'm going to see what how your behaviour of tapping relates to population dynamics.

370
00:33:12,900 --> 00:33:16,560
Okay. So very happy with that. Yeah. Cool. Go happy at the back.

371
00:33:17,900 --> 00:33:21,650
Yeah. Yeah. Everybody happy? I need to hear everybody's happy. Wave if you're happy.

372
00:33:22,040 --> 00:33:26,170
Yeah. Go. I'll have you go watch Ireland.

373
00:33:26,290 --> 00:33:29,600
Have you got what? Can you take me a minute? Yeah.

374
00:33:29,690 --> 00:33:33,560
Yeah. Well, I know. Really. Excellent. Okay. Everybody ready? So you going to close your eyes?

375
00:33:33,580 --> 00:33:37,060
Index fingers, people. Hold on. Let me see an index finger.

376
00:33:37,570 --> 00:33:40,650
Index fingers. Index fingers. Thanks.

377
00:33:40,780 --> 00:33:44,370
Okay. Close your eyes. Ready? We're going to talk for a minute.

378
00:33:44,380 --> 00:34:28,990
Go. How we do.

379
00:34:34,300 --> 00:34:41,340
10 seconds. Excellent. All three.

380
00:34:43,260 --> 00:34:46,630
What? Stop. Okay, please stop.

381
00:34:47,050 --> 00:34:50,110
Hopefully the person who found two managed to find them all.

382
00:34:50,800 --> 00:34:56,420
Yeah. Excellent. That's a good start. Okay, so we're going to do the analysis now you found.

383
00:34:56,440 --> 00:35:00,010
So I just need you to count up how many you found. So somebody had five.

384
00:35:00,520 --> 00:35:04,570
How many did you get? Well done. You're a good predator. Excellent.

385
00:35:04,660 --> 00:35:08,510
Top of the chain. Excellent. Somebody had ten on the board.

386
00:35:08,530 --> 00:35:12,250
How many did you find? Three. You're not so good.

387
00:35:12,840 --> 00:35:16,260
At least you got them to eat for the rest of their lecture, even if I'm boring. So that's quite good.

388
00:35:16,530 --> 00:35:20,650
Somebody had a 25 on the board. How did you two get?

389
00:35:21,550 --> 00:35:25,570
22. Did you use your index finger or a whole hand? Excellent.

390
00:35:25,750 --> 00:35:29,290
Good one. Always good to know. 22. All right.

391
00:35:29,540 --> 00:35:33,599
Somebody had 50. 3838.

392
00:35:33,600 --> 00:35:38,010
Wow. My God. This is going to be interesting. All right.

393
00:35:38,070 --> 00:35:42,060
So and so. The person in the back. 42. 42.

394
00:35:42,090 --> 00:35:46,110
Excellent. All right. Let's see how this works.

395
00:35:47,400 --> 00:35:52,550
So what we're going to do. I am going to. Quickly make sure I've done this right.

396
00:35:53,660 --> 00:35:59,690
Oh, there we go. Always an issue when it doesn't like the numbers, but, hey, he's got a bit cranky.

397
00:36:01,010 --> 00:36:07,220
Can I go to work? Okay, it's not gonna work. So, what, we. Okay, I'm just going to fudge the numbers, but you'll see just a little bit.

398
00:36:07,250 --> 00:36:13,010
Just to make it look. Let's see if we can make this. Just to give you an idea of what happens.

399
00:36:14,290 --> 00:36:18,520
So the two things that two things that go, the two types of responses that go on.

400
00:36:18,530 --> 00:36:23,890
So I have a picture somewhat here. So you guys were really great.

401
00:36:24,010 --> 00:36:27,910
Without a shadow of a doubt, my analysis doesn't take into account the fact that you could be linear.

402
00:36:28,090 --> 00:36:34,540
The fact is that what we have on this graph sorry, the font is not great on a but on the x axis I have the number of M&M that are

403
00:36:34,540 --> 00:36:38,980
on the board and on the y axis I have the number that were caught in a minute.

404
00:36:39,310 --> 00:36:44,080
And what you guys have done is managed to find whether you have 100 on the board.

405
00:36:44,080 --> 00:36:48,159
You found half of them when you found when you have 50 on the board, you got quite a lot.

406
00:36:48,160 --> 00:36:51,250
I know the idea is that that response is linear.

407
00:36:51,250 --> 00:36:56,110
It will go the I've I've fudged the points because we're trying to fit a curve through this.

408
00:36:56,290 --> 00:37:03,850
We you would have a nice linear relationship there of the number court and the number the number on the board and the number that you caught.

409
00:37:04,420 --> 00:37:09,720
What tends to happen when we do this and we replicate and then obviously I can only do this with six of you if you all

410
00:37:09,730 --> 00:37:14,620
were able to do this and we were in a little bit more of a controlled environment when we were doing this experiment.

411
00:37:14,620 --> 00:37:20,259
What would happen is that over time, even when you had how many did you find when you had 100 on the board?

412
00:37:20,260 --> 00:37:23,350
40 to 42. Right. So you don't find them.

413
00:37:23,350 --> 00:37:29,680
All right. You find about half of them. And the really important thing there is that actually what we're doing is we're linking the behaviour,

414
00:37:29,890 --> 00:37:36,760
this idea of, of, of, of foraging for the prey to the population dynamics.

415
00:37:36,760 --> 00:37:41,140
And what we expect to see is that as you increase, as you increase on the X axis,

416
00:37:41,440 --> 00:37:46,419
you can't actually get many more than about 40 out of 30 or something out there.

417
00:37:46,420 --> 00:37:50,499
So it doesn't there's a finite amount of time that you have to actually be able

418
00:37:50,500 --> 00:37:55,270
to take these Eminem's off the board and put them on the side or as a predator,

419
00:37:55,600 --> 00:37:59,290
kill your zebra and convert that zebra into a new lion.

420
00:37:59,530 --> 00:38:04,599
That takes time. You have to eat the thing. You have to digest it. You have to you have to give birth to new lions.

421
00:38:04,600 --> 00:38:08,350
That's a has a time dependent response and it has this particular shape.

422
00:38:08,860 --> 00:38:17,200
Okay. So as the number increase, as I have numbers up on this axis at the moment against the numbers on the on on the on the board.

423
00:38:18,040 --> 00:38:21,580
But if I think instead of having numbers, if I have the proportion,

424
00:38:21,790 --> 00:38:25,780
what happens with this graph is that actually down at the low end, most things get called.

425
00:38:26,230 --> 00:38:32,740
You know, that might be counterintuitive, but actually if there's a small number of prey, most of them are going to get captured.

426
00:38:33,070 --> 00:38:37,070
But when there's a large number of prey. They don't all get caught.

427
00:38:37,080 --> 00:38:40,410
And as the number as the number increases, they continue to.

428
00:38:40,740 --> 00:38:44,430
They continue to they continues to go down.

429
00:38:44,640 --> 00:38:51,060
So we have this negative relationship. So the proportion eaten against the number on the board goes down.

430
00:38:51,270 --> 00:38:59,310
And that basically means out here, when we have lots of prey, then we expect to see many of those prey.

431
00:39:00,550 --> 00:39:06,490
Giving are escaping predation and are able then to do whatever they do give birth.

432
00:39:07,000 --> 00:39:09,580
So we expect to see that actually when there's a large number of prey,

433
00:39:09,850 --> 00:39:15,759
this when there's when the small number of prey are being caught, lots of them will be able to increase in the absence of predation.

434
00:39:15,760 --> 00:39:18,850
As we talk and as we showed earlier, they all increase exponentially.

435
00:39:19,690 --> 00:39:24,040
So what is that? What does that what is it how does that translate to my moth dynamics?

436
00:39:26,170 --> 00:39:31,329
So I have this sort of idea, I have this sort of relationship that actually when there's when we have this dynamic,

437
00:39:31,330 --> 00:39:38,049
when they have this functional response between behaviour and dynamics that give rise to this idea that when there's lots of them around,

438
00:39:38,050 --> 00:39:41,050
very few, proportionately, very few of them are going to get eaten.

439
00:39:41,350 --> 00:39:45,070
Lots of them will escape. Predation means that they can grow exponentially.

440
00:39:45,310 --> 00:39:49,879
And then what happens is that the predator catch up is a numerical game. So there's more prey around.

441
00:39:49,880 --> 00:39:52,900
There's going to be eventually more predators around. And that's going to turn over.

442
00:39:53,110 --> 00:39:55,240
And we get these predator prey type cycles.

443
00:39:55,480 --> 00:40:01,390
And that's what we see in this system, basically, that we see the again, this is the the moth numbers through time.

444
00:40:01,480 --> 00:40:05,590
The black line is the most the morphs, the red dotted line is the plants.

445
00:40:05,590 --> 00:40:10,059
And we see this cyclic dynamic playing out. And that's because of the relationship.

446
00:40:10,060 --> 00:40:15,010
The link between the scaling between individual performance,

447
00:40:15,310 --> 00:40:20,710
how a predator or a moth in this case performs, and how that translates into the population dynamics.

448
00:40:21,340 --> 00:40:25,120
And that's pretty cool. It's a pretty cool way of which linking, you know, different,

449
00:40:25,120 --> 00:40:30,160
different aspects of behaviour operates on a different temporal scale to population dynamics.

450
00:40:30,580 --> 00:40:34,930
And when we can take that, we can translate it into these sorts of population dynamics.

451
00:40:36,950 --> 00:40:43,030
So to finish, I want to sort of like talk about this sort of like the third way that we can sort of think about mass gatherings.

452
00:40:43,040 --> 00:40:45,650
And it's a bit mathematical, but it's about approximations,

453
00:40:45,860 --> 00:40:50,179
how we can take approximate how we can approximate dynamics to operate on a different time scale.

454
00:40:50,180 --> 00:40:54,740
It's like the fast, slow dynamics, but we can do it in a lot better and a little bit more rigorous way.

455
00:40:55,310 --> 00:41:03,380
And I want to also motivate this by a really useful and so really cool way in which we approach this sort of analysis.

456
00:41:03,650 --> 00:41:07,910
We are interested and a group of colleagues in Oxford and elsewhere.

457
00:41:08,150 --> 00:41:13,160
We're interested in being able to understand the mood dynamics associated with patients with bipolar disorder.

458
00:41:13,580 --> 00:41:20,570
So bipolar are very, very common mental illness that occurs in late teens, early twenties,

459
00:41:20,720 --> 00:41:25,730
and is so a lifelong chronic condition where you get episodes of mania and depression.

460
00:41:26,030 --> 00:41:28,969
But in between, those episodes of mania and depression are really,

461
00:41:28,970 --> 00:41:35,690
really rare compared to the mood fluctuations that go on on a day to day or a week to week basis.

462
00:41:36,050 --> 00:41:41,810
And what we can use there is a set of standardised psychological scoring system sets, statistic sets,

463
00:41:42,050 --> 00:41:48,440
standardised psychological scoring metrics that can be used to quantify mood, and they're called quints.

464
00:41:48,710 --> 00:41:55,310
So it's a quick inventory of depressed symptom, symptom or symptoms, and we can use that as a self-reporting system.

465
00:41:55,310 --> 00:42:05,120
People use their cell phones or the computer to answer 16 questions that give a give an answer to, and we can create an aggregated score of that.

466
00:42:05,420 --> 00:42:12,830
So this is what this is. That's what these are the scores. Those are the mood scores that we get out of patients over a period of time.

467
00:42:13,220 --> 00:42:17,299
And we can do that. We can do that on very different temporal scales.

468
00:42:17,300 --> 00:42:23,090
We can do that on a daily basis. We can do it on a weekly basis. We can do it before and after intervention analysis.

469
00:42:23,090 --> 00:42:30,470
This is just a sort of cohort experiment that we had for 14, 15 patients or so where we monitor that mood,

470
00:42:30,620 --> 00:42:38,510
which is the, the, the solid black line over a 28 day period before and after a treatment intervention.

471
00:42:38,720 --> 00:42:44,480
My colleagues I work with with this are a clinical psychologist and they use behavioural interventions.

472
00:42:44,480 --> 00:42:48,860
So cognitive behavioural therapy to treat people and help people with bipolar disorder.

473
00:42:49,250 --> 00:42:55,399
What we can do is we can ask the question is does the mood fluctuations change before or after treatment?

474
00:42:55,400 --> 00:43:00,080
So we can quantify it before they go into treatment for four weeks and we can come out.

475
00:43:00,080 --> 00:43:03,389
They come out the other side and we can characterise their mood dynamics.

476
00:43:03,390 --> 00:43:04,100
So this is what we see.

477
00:43:04,100 --> 00:43:11,659
We're able to sort of like characterise their these mood profiles and just collect them in a, in a temporal time series sequence.

478
00:43:11,660 --> 00:43:17,660
So again, it's just, it's just population dynamics, it's just population ecology, the sort of stuff I grew up doing.

479
00:43:17,840 --> 00:43:20,720
So with a very clinical relevance here.

480
00:43:21,380 --> 00:43:28,790
So what we can do is that we can sort of like pick that apart a little bit and we can ask basically how can we understand mood dynamics?

481
00:43:28,790 --> 00:43:36,470
How can we get to the mass mood, mass we call it, how can we unpick the dynamic, those things that drive mood fluctuations?

482
00:43:37,160 --> 00:43:40,819
And we've done a bunch of things. We've used a whole set of different mathematical methods.

483
00:43:40,820 --> 00:43:45,760
And I'm just going to so I'll talk to you about this one and how we can then link this to some scaling dynamics,

484
00:43:45,950 --> 00:43:49,070
how we scale between different things that may be going on.

485
00:43:49,580 --> 00:43:51,350
So again, another just so a differential equation,

486
00:43:51,350 --> 00:43:57,169
we're going to split mood up into sort of like take them, we'll take the words, we've got change in mood.

487
00:43:57,170 --> 00:44:02,390
So from one time step to the next, very small time step, I'm not going to be explicit about what that is at the moment.

488
00:44:02,660 --> 00:44:08,600
And then we can split up and we can use a total derivative approach. We can split it up into different component parts, essentially.

489
00:44:08,990 --> 00:44:12,080
So I've got some sort of like partial derivative mood respect to time.

490
00:44:12,080 --> 00:44:18,469
So that might just be baseline mood. What happens, you know, this just happens to your mood and just baseline and then we can.

491
00:44:18,470 --> 00:44:25,760
So spitting out further into something to do with fluctuations some things driving this mood fluctuations in our in our heads.

492
00:44:25,760 --> 00:44:28,460
And we've got to really think about how we understand that.

493
00:44:28,910 --> 00:44:35,690
And we can so split that we can sort of think of this the X by D, T and D, Z by T as oscillator, something that's driving that.

494
00:44:35,690 --> 00:44:42,380
So they may be innervate, inhibitory and accumulating oscillators that drive dynamics.

495
00:44:42,680 --> 00:44:47,719
And then there's all these sorts of things. The partial change in mood with respect to X is just a slope.

496
00:44:47,720 --> 00:44:54,140
It's just a regression on how mood changes with respect to the size of the the oscillator that we're

497
00:44:54,290 --> 00:45:01,400
although the the inhibitor or excitatory effect that we're trying to measure so we can write out this.

498
00:45:01,670 --> 00:45:07,170
So this differential equation in terms of the change in mood is just baseline mood that's just going to be flat.

499
00:45:07,170 --> 00:45:12,350
So some constant that's going to change over that's going to that's not going to change over time.

500
00:45:12,620 --> 00:45:20,509
And then again, these things are going to be so there's going to be fluctuations to do this oscillator and this oscillator with some with some

501
00:45:20,510 --> 00:45:26,600
magnitude associated with the parameters in front of that and what we wanted to be able to do and we can do that statistically.

502
00:45:26,600 --> 00:45:32,480
We've done that statistically. And that was what was on the previous slide is what we can try and fit these sorts of models to the Time series.

503
00:45:34,060 --> 00:45:38,210
But I want to get into a little bit more about this. And there's a lot of words and there's a lot of detail on this slide.

504
00:45:38,530 --> 00:45:47,469
And what I want to just explain is that we can we can write that this differential equation or this differential equation as a as an oscillator.

505
00:45:47,470 --> 00:45:55,770
So if thinking about things that inhibit and things that accumulate sort of like this, these dynamical fluctuations and that really cool way,

506
00:45:55,780 --> 00:46:01,840
one of the really cool ways to be able to capture that and phenomenological very broadly without too much mechanistic detail,

507
00:46:02,080 --> 00:46:07,510
is with what we call a relaxation oscillator. And a relaxation oscillator is a really cool thing.

508
00:46:07,510 --> 00:46:12,100
It's a to say it's going to oscillate, it's going to go up and down through time and it's relaxation because actually I'm going

509
00:46:12,100 --> 00:46:16,420
to spend a long time in one state and I'm going to relax very quickly to another state.

510
00:46:16,720 --> 00:46:22,480
So it's just like a friend to price cycle might be. One way to think of it is I'm going to stay in a high state and I'm going to stay there.

511
00:46:22,480 --> 00:46:26,800
And then I go to relaxed very quickly to a low state, and then I go relax out of that state to a high state again.

512
00:46:27,250 --> 00:46:35,770
And we can capture that. We can capture that phenomena logically, very broadly with this this this ordinary differential equation.

513
00:46:35,770 --> 00:46:37,450
So we can couple two things together.

514
00:46:37,720 --> 00:46:44,140
We can put some complicated nonlinear function in there, and then we can capture this idea of a relaxation oscillator.

515
00:46:44,530 --> 00:46:49,419
Well, for bipolar disorder, that's pretty cool, because then we can think about that as I see you're in a state,

516
00:46:49,420 --> 00:46:53,470
maybe you're in a high mood state and you can relax to a baseline state.

517
00:46:53,710 --> 00:46:57,370
You may be in a low mood state and you can relax to a sort of baseline state.

518
00:46:57,370 --> 00:47:01,480
So we are capturing that idea. One idea that might be one idea that we're trying to capture.

519
00:47:01,790 --> 00:47:10,390
We may also be trying to capture this idea that there's inhibition and excitatory processes going on within our heads to sort of drive the dynamics.

520
00:47:10,750 --> 00:47:14,170
We can do other things. So it's a little bit more complicated stuff on this slide.

521
00:47:14,170 --> 00:47:18,650
There's some stuff at the end. So this stuff here and this stuff here is capturing the idea.

522
00:47:18,680 --> 00:47:23,559
We can make these things stochastic so we can kick the system with a little bit of randomness each time.

523
00:47:23,560 --> 00:47:31,690
So that just to add some just to add some additional fluctuations in that that may not be captured very broadly by this by this oscillator.

524
00:47:32,020 --> 00:47:35,110
And the other cool thing we can do is that we can couple these things together.

525
00:47:35,350 --> 00:47:39,250
We can say, actually, what's happening to one isn't independent of what is happening in the other.

526
00:47:39,490 --> 00:47:44,139
So we can really make this these sources, this sort of scaling. But when we can, as I said,

527
00:47:44,140 --> 00:47:51,400
we can fit that to the Time series so we can introduce noise and we can do or we can look at all of these couplings and we can see we

528
00:47:51,400 --> 00:48:00,250
can challenge these models against the data and look at what the what the explanations might be that best describe individual patients.

529
00:48:00,250 --> 00:48:03,879
So you as an individual patient with bipolar you need to oscillators.

530
00:48:03,880 --> 00:48:07,030
Do they have to be couple? Do they have to be noisy? And it really is.

531
00:48:07,030 --> 00:48:10,720
And it really helps us understand the temporal dynamics of bipolar.

532
00:48:12,000 --> 00:48:15,710
So it's just what we do. We consult those three patients just picked at random.

533
00:48:15,930 --> 00:48:19,620
And here's the oscillators that are all oscillating.

534
00:48:19,620 --> 00:48:25,919
They don't have to oscillate. The parameter values, the parameters in the in that oscillator could actually give rise to stable dynamics.

535
00:48:25,920 --> 00:48:29,790
It's just what values they choose choose to take on when we fit them to the Time series.

536
00:48:30,120 --> 00:48:34,740
But each of them, as you can see, everybody each of these three patients has an idiosyncratic.

537
00:48:35,590 --> 00:48:39,340
So a light oscillator system. Sometimes they have to be coupled.

538
00:48:39,340 --> 00:48:42,790
Sometimes they don't. Sometimes they're not. Sometimes they have to be noisy. Sometimes they don't.

539
00:48:43,060 --> 00:48:49,180
And we can fit those to the times, as you can see how good a fit they are, which is the purple dots in this and this in this case,

540
00:48:49,450 --> 00:48:55,059
it's really good technical challenge because actually bipolar patients are pretty bad reporting every time.

541
00:48:55,060 --> 00:48:58,090
So there's lots of missing values in this and we have to be able to deal with those statistically.

542
00:48:58,390 --> 00:49:03,280
And that's all pretty cool, too. Well, we can do that. That's pretty nice. But that's not what I want to get to.

543
00:49:03,640 --> 00:49:09,880
I want to take you to where we are with all of this, because I think it's really cool because what we're trying to do is really understand.

544
00:49:10,240 --> 00:49:16,840
So I've explained this. This is my total derivative. I've got the baseline mood, I've got some parameters associated with these fluctuations,

545
00:49:16,840 --> 00:49:25,870
but we're really trying to do is to step down into heads and think about how the neuronal firing patterns drive mood dynamics.

546
00:49:26,230 --> 00:49:27,730
Well, it's a huge ask, right?

547
00:49:27,730 --> 00:49:35,680
We're actually things no neurone firing dynamics are operating on such a fast time scale compared to how fast your mood changes.

548
00:49:36,160 --> 00:49:40,360
And I like my mood, but my moves, well, I don't know, up and down at the moment, cause I'm standing here talking.

549
00:49:40,360 --> 00:49:45,639
But, you know, it changes, you know, not as fast as a my neurones are going like, oh my God, I've got to keep talking.

550
00:49:45,640 --> 00:49:49,270
They're really firing off in my head to of hopefully make me lucid.

551
00:49:49,510 --> 00:49:57,339
But what we really want to do is to link these sort of these sorts of neurone firing patterns to the to these to this population dynamic,

552
00:49:57,340 --> 00:50:02,709
this, this dynamic of mood change over time. And these are operating on different time scales.

553
00:50:02,710 --> 00:50:07,060
And the way that we're going, the way we're thinking about approaching that is actually approximating these functions.

554
00:50:07,060 --> 00:50:14,440
So these are the these two differential equations are describing the inhibitory and excitatory processes that go on in your own firing patterns.

555
00:50:14,680 --> 00:50:20,290
And we can aggregate those together and we can really think about how we how we can describe we can just take those,

556
00:50:20,500 --> 00:50:28,090
those two differential equations, these two here, and really get into the neurobiology and the theoretical neurobiology and neurobiology of that.

557
00:50:28,450 --> 00:50:31,870
Well, we want to take you up a scale, but these are operating on different time scales.

558
00:50:31,870 --> 00:50:35,200
And the way that we can do that is to approximate these things with a Taylor series.

559
00:50:35,590 --> 00:50:40,510
And a Taylor series is just a way of writing out a function in an infinite sum,

560
00:50:40,870 --> 00:50:43,840
and we can write it out in terms of what at a particular point in time.

561
00:50:44,380 --> 00:50:47,500
And we can take that function and we can just say, Well, what is it, what is it?

562
00:50:47,530 --> 00:50:51,879
What is its linear component and what is its quadratic component?

563
00:50:51,880 --> 00:50:58,420
What is the cubic component? What is its quartette component? And we can just keep adding on it forever and we can get an understanding.

564
00:50:58,420 --> 00:51:07,750
We can approximate these neurone dynamics at a particular point in the mood dynamics with this Taylor expansion, and that holds out great possibility.

565
00:51:08,110 --> 00:51:13,990
It's where we are. We're at this step, and this is where I'll take you to where we are in this sort of problem.

566
00:51:14,200 --> 00:51:15,580
I don't have any answers to that yet,

567
00:51:15,880 --> 00:51:23,590
but we think it's a really cool approach to be able to approximate the neurone dynamics in terms of using these these sorts of Taylor expansions.

568
00:51:25,190 --> 00:51:30,530
So let me finish up with another sort of implication about scaling a different one, completely different one.

569
00:51:30,800 --> 00:51:37,310
Just another little game we'll play very shortly. What I should do is sort of like thinking about bio control people in my group.

570
00:51:37,430 --> 00:51:44,240
I guess a sat around here are interested in thinking about how we control pests,

571
00:51:44,570 --> 00:51:49,790
insect pests, vectors that spread disease, how we can think about the biology,

572
00:51:50,090 --> 00:51:51,559
biological control of doing that,

573
00:51:51,560 --> 00:51:59,420
and how maths helps us understand that there's a very old method in insect control called the sterile insect technique.

574
00:51:59,750 --> 00:52:06,140
And the idea is that you would mass these insects that lower radiation, make them sterile,

575
00:52:06,230 --> 00:52:09,980
release them out into the environment, and when they make, they make no viable offspring.

576
00:52:10,340 --> 00:52:14,450
So they don't so they don't hatch or the insects die or whatever it might be.

577
00:52:14,450 --> 00:52:22,880
So there's the idea being that you're aiming to sort of like to send the population into decline because the meetings are not successful.

578
00:52:23,450 --> 00:52:28,670
Well, there are many new twists on that. And one of the latest ones is being able to use genetics to do that.

579
00:52:28,970 --> 00:52:34,550
So you can genetically modify your favourite agricultural pest or your mosquito spreading disease.

580
00:52:34,880 --> 00:52:38,090
You can change its genes so that when it goes and you must rear them again,

581
00:52:38,300 --> 00:52:43,610
throw them out when they mate with wild types, they don't make any viable offspring.

582
00:52:43,610 --> 00:52:47,450
Those those larvae may die or the caterpillars may die or the eggs may die.

583
00:52:47,840 --> 00:52:53,600
So the idea being that you're trying to suppress the population, we can do it both conventionally with radiation or with genetics.

584
00:52:53,600 --> 00:52:56,750
So the idea is essentially to suppress the population.

585
00:52:57,950 --> 00:53:03,680
There's a whole set of cool maths behind all of that, and it's to do with what we think about as optimal control.

586
00:53:04,160 --> 00:53:10,760
We can do optimisation, we can do static optimisation, we can say, actually, we want to just know what this is for some fixed values,

587
00:53:10,760 --> 00:53:15,870
but actually this is a dynamic problem and somebody said You should shock people.

588
00:53:15,870 --> 00:53:18,169
Mike So here's the shocking bit of this still.

589
00:53:18,170 --> 00:53:24,620
So this is a bit of an old poster, but actually still something like 2000 kids die every day from malaria.

590
00:53:24,830 --> 00:53:28,580
Since I've been speaking over the last hour, probably 50 kids have died from malaria.

591
00:53:28,940 --> 00:53:31,160
What we have is a completely preventable disease.

592
00:53:31,400 --> 00:53:37,219
And using these sorts of methods, these bio control methods, we have an opportunity to be able to do that on our mats.

593
00:53:37,220 --> 00:53:40,820
Really helps us think about that because actually we've got a dynamic.

594
00:53:40,820 --> 00:53:49,550
So this equation at the top DV by d t is the change in the vector numbers, the change in the number of mosquitoes over time.

595
00:53:49,910 --> 00:53:53,630
And it's something to do. There's some bursts of mosquitoes and some deaths of mosquitoes.

596
00:53:54,230 --> 00:53:59,150
And we can fact this thing in brackets is the function that says, I'm going to do this style insect release.

597
00:53:59,150 --> 00:54:05,690
I'm either going to do it genetically or I'm going to do it conventionally, and I'm going to release a number of modified mosquitoes.

598
00:54:05,690 --> 00:54:11,540
I will if i0vtv over v t, it's just going to be is going to be one.

599
00:54:11,540 --> 00:54:16,700
So it's going to cancel out. But it's actually as I increase the number of these star releases,

600
00:54:16,970 --> 00:54:22,580
this term is going to become more important and this term is going to end in a whole term on the first term,

601
00:54:22,580 --> 00:54:24,380
on the right hand side is going to get smaller.

602
00:54:24,650 --> 00:54:31,760
We're aiming to reduce the number of births, but it's not not all that what's happening in this system when we're thinking about malaria,

603
00:54:32,030 --> 00:54:35,120
that we've got people being infected, we've got infected vector.

604
00:54:35,130 --> 00:54:42,800
So we've got a dynamic situation there. We've got dynamics operating in terms of the changes in the vector numbers, being born and being controlled.

605
00:54:43,040 --> 00:54:48,619
What we're trying to do, we're trying to stop people getting sick. So actually we can think about this as an optimal control problem.

606
00:54:48,620 --> 00:54:55,490
This is a dynamic control problem in that we're thinking about actually what we want to do is we want to control this problem over time,

607
00:54:55,640 --> 00:54:59,240
a fixed period of time, we're going to discount it. That's an economics term.

608
00:54:59,600 --> 00:55:02,090
It's a way of scaling thinking in the future.

609
00:55:02,390 --> 00:55:06,620
And we're going to think about what we want to do is we want to minimise this function in terms of the number of people

610
00:55:06,620 --> 00:55:13,490
that are getting sick and the number of the so the number of these sterile insects that we have to put out there.

611
00:55:13,880 --> 00:55:19,010
So also optimisation, we need to this has to be bigger than zero, but we don't want it to be too bigger than zero because that's going to cost us.

612
00:55:19,010 --> 00:55:22,880
It's going to be really costly to do that economically. It's going to be costly to do that.

613
00:55:23,390 --> 00:55:27,650
And this is where we're able to scale both biology and economics.

614
00:55:28,130 --> 00:55:37,580
And what I'd like to do you just to finish up is just to sit on my group, on my on our on our group website.

615
00:55:38,850 --> 00:55:44,690
It's just going to work. We have a game, we can play this game and we'll just play this game because it's really it's really instructions.

616
00:55:44,690 --> 00:55:49,579
It's an ultimate optimal control game and it's so set out without any of the math detail.

617
00:55:49,580 --> 00:55:53,770
It's got lots of the biology in there. So one of the first things you need to do when you're thinking about this,

618
00:55:54,410 --> 00:56:00,980
there's this economics of biocontrol is to know the problem, know what the size of the problem is that you're trying to control.

619
00:56:01,370 --> 00:56:07,639
And the way, way that we do that is we have to estimate, well, the big what the insect, the pest population of the vector population is.

620
00:56:07,640 --> 00:56:12,140
How big is it? Well, ecologists do that with a method called the mark release recapture method.

621
00:56:12,140 --> 00:56:15,980
So you mock a bunch of insects, you throw them out, you wait a bit time,

622
00:56:15,980 --> 00:56:18,920
you go back and you collect a bunch and you work out the proportion that were marked,

623
00:56:18,920 --> 00:56:26,239
which is a mouse, and it tells you what you can use as some wacky formula and it tells you the number of of of mosquitoes.

624
00:56:26,240 --> 00:56:30,110
Oh, that was the wrong one, but. Yes.

625
00:56:30,120 --> 00:56:34,360
Go back. Did you do the record?

626
00:56:34,450 --> 00:56:36,819
No, I did not. Okay. So are you guys not going to do it?

627
00:56:36,820 --> 00:56:41,500
Because we with kids, we get them little kids, we get them to run around and they can find mosquitoes and can do all of this.

628
00:56:41,560 --> 00:56:46,930
So if I pick a number of how many mosquitoes we want in the population, there's some stuff about the biology.

629
00:56:47,500 --> 00:56:53,630
Well, this is where it becomes important. We can now try to optimise this problem and trying to make it as cost effective as possible.

630
00:56:53,660 --> 00:56:57,920
We're trying to scale the economics and the biology to make this as cost effective as possible.

631
00:56:58,160 --> 00:57:00,230
And there are two things. The two things you can control.

632
00:57:00,470 --> 00:57:06,500
Firstly, that I how many sterile mosquitoes, how many sterile insects am I going to put out there?

633
00:57:06,830 --> 00:57:12,890
And then I got to think, how long can I run my program for? So the top one is the ecology, and the bottom one is the economics.

634
00:57:13,200 --> 00:57:18,980
Okay, so on the top one, I can choose any number between one and ten.

635
00:57:20,150 --> 00:57:23,690
Excuse me. A number between one and ten to ask a good number.

636
00:57:24,300 --> 00:57:28,520
Okay, so that means I've got to pay for every wildtype mosquito I'm going to put out there.

637
00:57:28,910 --> 00:57:32,480
I'm going to release two style insects. Okay.

638
00:57:32,720 --> 00:57:40,140
How long do you wanna run your program for? You can have two, five or ten. Well, five was the first number I heard.

639
00:57:40,170 --> 00:57:43,590
That's a good number. Okay, so this we'll see what happens.

640
00:57:44,910 --> 00:57:46,049
So we're going at two graphs.

641
00:57:46,050 --> 00:57:51,120
One of the top one, we're going to do the ecology of the mosquito, the dynamics, the population dynamics of the mosquito.

642
00:57:51,420 --> 00:57:55,230
On the bottom one, we're going to see how many people you managed to save with your particular program.

643
00:57:55,590 --> 00:57:58,980
Okay. And we're going to look for a dollar figure.

644
00:57:59,100 --> 00:58:03,990
And we want that to be as small as possible. The bigger is, the worse it's going to be, essentially.

645
00:58:04,440 --> 00:58:10,110
So I'm going to start on the other thing of ecology is that I need to know when in the season I'm going to start my control.

646
00:58:10,410 --> 00:58:17,729
I'm just going to pick one of these. I'm going to see two lines, basically. And the black line on the top is what I did in the absence of control.

647
00:58:17,730 --> 00:58:24,090
And the green line is what I do in the presence of control. And you can see that actually, I can reduce my.

648
00:58:24,330 --> 00:58:29,130
So I'm doing a 2 to 1 release ratio. So two sterile mosquitoes to every wild type.

649
00:58:29,580 --> 00:58:36,150
And on the bottom, I have the number of sick people in red that in the absence of control and in the presence of control in blue.

650
00:58:36,660 --> 00:58:41,130
And it goes up and down because we're talking about mosquitoes. And they like it wet and dry or they like it.

651
00:58:41,340 --> 00:58:48,180
They don't like it dry. So there's some seasonal dynamics that are essentially what I call things about this is actually when I got to the numbers.

652
00:58:48,180 --> 00:58:53,040
Yeah, well let's talk about the dynamics is not on the top. I only suppressed my population a little bit.

653
00:58:53,050 --> 00:58:57,330
There's 2 to 1 release ratio. It doesn't actually squash the population down very far at all.

654
00:58:57,760 --> 00:59:03,650
Yeah, it's still it's only been suppressed a little bit. But actually I can reduce the disease dynamics by about half the number.

655
00:59:03,750 --> 00:59:07,140
The average number between the red line and the blue line is gone down by half.

656
00:59:07,590 --> 00:59:09,510
So that's pretty cool, right? Actually, I do know.

657
00:59:09,690 --> 00:59:15,300
So one of the things is I don't actually need to eradicate my mosquitoes from the system to start to have an effect on disease control.

658
00:59:15,780 --> 00:59:18,890
If I choose a bigger number, it will be more costly to do that.

659
00:59:18,900 --> 00:59:25,050
And actually, I could get the same sort of phenomena. And as we know, this is a well-known this is a well known fact in epidemiology,

660
00:59:25,260 --> 00:59:33,089
where if you suppress those agricultural pests or those disease vectors be a lower threshold, then, you know, you don't have to eradicate them.

661
00:59:33,090 --> 00:59:38,069
You just have to push them down below that threshold. Then you can have a big effect on the disease dynamics.

662
00:59:38,070 --> 00:59:42,750
We stop our control. After five years, we run out of cash. So that stops after five years.

663
00:59:42,990 --> 00:59:48,780
And actually then because of reinvention, there's really good documented evidence of that happening throughout the world.

664
00:59:49,080 --> 00:59:52,560
So actually we see the disease rebounding to pre control levels.

665
00:59:54,180 --> 00:59:59,120
So what happens then? The most important thing is what this dollar figure is for the better.

666
00:59:59,370 --> 01:00:02,790
The bottom. How many cases did we avert? How many cases do we avoid?

667
01:00:03,000 --> 01:00:05,440
And how much did it cost? How much was the control enough?

668
01:00:05,670 --> 01:00:11,190
What was the optimal value that we can get out of that turns out to 52 is not bad, but we can be better.

669
01:00:11,400 --> 01:00:18,330
I had a seven year old girl do this one time. She spent 45 minutes playing on this, this, this, this game, and she came out with 229.

670
01:00:18,600 --> 01:00:22,640
I've never got down to 229. So if anybody plays this game, it gets below to 29.

671
01:00:22,830 --> 01:00:28,860
I want to know because that's really is really important thinking about how we are able to blend this idea of

672
01:00:28,860 --> 01:00:34,530
economics and ecology in scaling these two things so that different products are operating on different time scales.

673
01:00:34,530 --> 01:00:38,640
You've got to think about how long you run your program for, and that could be many years.

674
01:00:38,730 --> 01:00:42,090
But these mosquitoes and people getting sick is on a daily basis.

675
01:00:42,090 --> 01:00:46,860
And as I said, many, many people still die from from from these sorts of infectious diseases.

676
01:00:47,610 --> 01:00:55,919
Okay. I I'm just about done. I think I've told you about three things, three ways that we can scale slow, fast dynamics.

677
01:00:55,920 --> 01:01:01,469
I didn't do the scaling laws that I might have expected to do, but I've done some dynamical stuff, and I hope you found it interesting.

678
01:01:01,470 --> 01:01:04,620
We can have slow, fast dynamics so we can set those things, one,

679
01:01:04,620 --> 01:01:09,780
going fast and the other we can make really cool approximations and use some very, very cool master work that through.

680
01:01:10,140 --> 01:01:17,370
We worked on the functional response for enterprise interactions in particular, and there's a ways in which we can approximate these things with,

681
01:01:18,360 --> 01:01:26,400
with particular types of functions like the Taylor series, there's really nice ways and think linking things that operate on a different time scales.

682
01:01:27,090 --> 01:01:31,229
We do a raft of things in my group, if anyone is interested. We have a great website, some of them around the room.

683
01:01:31,230 --> 01:01:33,720
If they stick their hands up, they may show their faces, they may not.

684
01:01:33,900 --> 01:01:37,740
They may run out the room thinking, Oh my God, my supervisor is embarrassing me, but that's perfectly fine.

685
01:01:38,160 --> 01:01:41,220
And I'm more than happy to take some questions. Thank you.