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In this video

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what I really want to do is

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to start giving you an intuition

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for how we go from a compartmental model,

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represented as boxes and arrows,

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to a mathematical description.

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And as a start we're gonna be thinking

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about discreet dynamical systems,

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meaning they're discreet both

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in terms of types

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of parts in the system.

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So in terms of a discrete number of boxes,

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but also discrete in time.

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So this week we're gonna look
at mathematical description

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and also how to simulate these processes.

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To keep with the theme
of the previous video,

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for now let's stay

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within the realm of disease model.

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We're gonna look at a simpler model.

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Which is called the SIS model.

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Meaning that unlike the SIR,

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here we're only gonna have

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two different boxes.

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And with flow of density of population

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that go both ways.

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So we're still gonna
have our susceptible box

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for people not currently infectious,

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and our infectious box

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for people currently infectious.

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The key difference with the SIR is

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that the second arrow

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on the one at the bottom here,

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which corresponds to recovery,

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people leaving the infectious state,

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doesn't go to a new immune box

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but goes back to the
susceptible population.

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So this could be something
more like the common cold

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where when you recover

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you might not have long-lasting immunity

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and you could get infected again,

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sometimes just a week later.

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Right?

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So there are different ways

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that people would put weight

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or describe these arrows.

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We could look at each arrow

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in terms of the mechanism

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that's really taking place.

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So in terms of infection
really is that we go

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from a pair SI of individual,

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so one susceptible,

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one infectious,

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to two infectious individual.

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And that might occur at
some rate, beta for example.

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And in the case of recovery,

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then we simply go from
one infectious individual

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to one susceptible.

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And that might occur at some rate, gamma.

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This is different

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from what I used last week.

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And in a minute I'll
explain the difference.

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Really the key part here is that

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when we're gonna start reading papers

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that use compartmental models,

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there's no clear standard

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on how to represent
these boxes and arrows.

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So you have to try and understand

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what the authors really mean

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by their representation.

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So let's imagine

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that we know some initial state

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of the system.

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Right?

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Like these models,

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they all have the same recipe.

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You define your compartments

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and arrows.

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Then you initialize them to some value.

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So let's say that we know

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the fraction of the population S of t,

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that is susceptible at time t.

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And I of t, meaning infectious at time t.

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And these could be,

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as I said,

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fraction of the population.

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So between 0 and 1

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or they could both be

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between 0 and N

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for a total population of N.

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So let's say that they are numbers

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of susceptible and infectious individuals.

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Then to get a dynamical system

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in discrete time.

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What we want to ask is,

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"If I go from time t

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to time t plus 1,

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how many susceptible
individuals do I have?"

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Well I know who is susceptible

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at time t.

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And the ones that are still susceptible

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are those that have not been infected

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by any of their infectious contact.

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Right?

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So every single susceptible
individual here,

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remember that there's no space

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in these compartmental models.

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So every susceptible individual is

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in contact with all I
infectious individual.

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So if I have 100 infectious individual

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that's 100 infectious contact.

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And the only way for me
to remain susceptible,

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is for all of those infectious contact

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to not transmit the disease.

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So if we call beta,

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the probability that an
infectious individual transmit

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to a susceptible individual.

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The probability that they don't,

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is simply 1 minus beta.

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If there's a 10% chance

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that it transmit,

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there's a 90% chance that it doesn't.

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And to stay susceptible,

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I need all of them

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to not transmit the disease.

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So if I have two contacts,

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then that's 90% chance of not transmitting

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from the first one,

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90% chance from the second one.

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That's 90% times 90%

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of transmission from neither.

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And if I have a number 100,

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then that's 90% to the 100.

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So that's what I have here.

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And I also have a positive term.

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So one way to look at this is that

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when you have these arrows

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that flow from one box to the next,

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you should always have a negative term

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for the flow that leaves the box.

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So here there's a negative term

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for the people that
don't remain susceptible.

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You should have a positive term

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when it enters the box.

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So I'm losing some

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through transmission here,

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corresponding to flow of water,

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or density,

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or number of people

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out of my susceptible box.

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And I'm gonna gain some
here based on recovery.

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So if this is my infection term,

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I'm gonna have some recovery term

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which is gonna correspond to

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every infectious individual.

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And with what probability did they recover

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in one time step?

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It's simply gamma, right?

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So if they do recover,

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then they flow out of the I box

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and into the S box.

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And then we could write the same thing

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for I of t plus 1.

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But see here,

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I don't have any arrows

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out of my boxes leading

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to the outside world, right?

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I have a closed population.

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So I know

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that for all time,

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S plus I must always equal N.

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There's nowhere to hide.

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There's nowhere to go

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for these people except S and I.

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So I don't even have to
write this other equation.

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I simply know

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that I at time t plus 1

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is gonna be equal

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to N minus S

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at time t plus 1.

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'Cause S and I always have to sum to 1.

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And that's another key insight here,

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is that even though we have two boxes

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that doesn't mean

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that we have two degrees of freedom

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or two important variables

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in our system.

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Because it's a closed population,

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we know there's a conservation

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of population,

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I plus S equals N.

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And that's a constraint

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that we can use to reduce the complexity

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of the model,

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and fix one variable

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as a function of the other.

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So in this case,

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we have one constraint,

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conservation of population.

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Two compartments.

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So we have two compartments
minus one constraint,

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one degree of freedom.

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That's one way to think about this.

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So it's really a one dimensional system.

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The other thing I wanna talk about

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in this video

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before we discuss how
to write this equation

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in greater detail during
the discussion group,

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is the fact that we've made
some approximations here.

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Right?

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I'm not...

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For example,

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in my recoveries,

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really you might have wanted to write,

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"Well what is the probability
to get one recovery?

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Two recovery?

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Three recovery?"

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And list all possible cases.

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Here really,

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this is just gonna be the average number

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of recoveries

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that you might expect.

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Let me make that a little clearer

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for you all.

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I'm gonna create a new page.

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So I have this SIS model.

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And let's say that we have
a very simple population

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N equal 2,

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right.

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Only two people.

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It's not gonna be a
very interesting model.

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Well,

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I of t

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in any discrete simulation

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could only take

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one of three values.

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There's either no one infectious,

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one person infectious,

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or two people infectious.

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Really,

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in the mathematical description,

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it's gonna be able to go freely

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between 0 and 2

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because we're not limiting,

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in those previous equations,

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we're not limiting the transition

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to discrete values.

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So you can have 1.5 infectious individual.

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And what would that mean, right?

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What does it mean if here I have S of t,

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a number of susceptible

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that might be discrete,

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let's say by chance is equal to 50,

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but I increase it

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with these recovery events

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by 1.5, right?

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So now I'm saying that
there's a fractional number

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of infectious individual.

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What does that mean?

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Well consider it this way,

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let's just focus on recoveries.

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And if I of t is equal to 0,

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well the only thing

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that can happen is nothing.

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I can't have any recovery

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if I don't have infectious individual.

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If I of t is equal to 1,

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well now I can get 0 recovery

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or 1 recovery.

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Right?

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This is gonna occur

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with probability

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1 minus gamma.

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'Cause gamma is my recovery probability

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per time step.

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And this is gonna be

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with probability gamma.

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I'm gonna have one recovery.

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If I of t is equal to 2,

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well now have three possible cases

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with probability,

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1 minus gamma squared,

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I'll have no recovery at all.

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So if both infectious
individuals are unlucky

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and they both choose to not recover

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or the system chooses to
not have them recover,

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I'll have no recovery.

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And with probability

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00:11:13,590 --> 00:11:14,910
2 gamma,

289
00:11:14,910 --> 00:11:16,680
1 minus gamma.

290
00:11:16,680 --> 00:11:18,710
So this 2 is...

291
00:11:19,650 --> 00:11:21,090
No N choose K

292
00:11:21,090 --> 00:11:23,611
where it's 2 choose 1.

293
00:11:23,611 --> 00:11:24,600
(instructor laughs)

294
00:11:24,600 --> 00:11:25,800
So there are two ways

295
00:11:25,800 --> 00:11:27,870
to get only one recovery.

296
00:11:27,870 --> 00:11:28,950
And in both cases,

297
00:11:28,950 --> 00:11:31,110
I need one of the sick individual

298
00:11:31,110 --> 00:11:32,130
to not recover

299
00:11:32,130 --> 00:11:34,110
which is with probably 1 minus gamma,

300
00:11:34,110 --> 00:11:35,130
and the other to recover

301
00:11:35,130 --> 00:11:36,030
with probability gamma.

302
00:11:36,030 --> 00:11:37,470
So I have two ways

303
00:11:37,470 --> 00:11:38,340
with probability gamma,

304
00:11:38,340 --> 00:11:39,173
1 minus gamma,

305
00:11:39,173 --> 00:11:40,863
each to have one recovery.

306
00:11:43,050 --> 00:11:44,550
And I'll get two recoveries

307
00:11:44,550 --> 00:11:46,650
with probability gamma squared.

308
00:11:46,650 --> 00:11:47,820
Well, that's pretty simple.

309
00:11:47,820 --> 00:11:48,653
The reason I'm going

310
00:11:48,653 --> 00:11:50,400
through this right now is

311
00:11:50,400 --> 00:11:53,640
that I wanna highlight one thing,

312
00:11:53,640 --> 00:11:56,223
which is the average of all these cases.

313
00:11:59,040 --> 00:12:00,840
And throughout this course

314
00:12:00,840 --> 00:12:02,670
when we think about averages,

315
00:12:02,670 --> 00:12:04,770
I know we all see it differently depending

316
00:12:04,770 --> 00:12:05,603
on our background.

317
00:12:05,603 --> 00:12:06,510
So I just wanna highlight

318
00:12:06,510 --> 00:12:09,300
that an average is always the sum.

319
00:12:09,300 --> 00:12:10,173
And by the way,

320
00:12:12,060 --> 00:12:14,250
this capital sigma simply means sum.

321
00:12:14,250 --> 00:12:16,710
So you're going over all possible outcomes

322
00:12:16,710 --> 00:12:17,850
in this case.

323
00:12:17,850 --> 00:12:19,780
And you're summing a quantity

324
00:12:21,330 --> 00:12:23,103
times a probability.

325
00:12:24,960 --> 00:12:27,270
And that's how you calculate
the average of that quantity

326
00:12:27,270 --> 00:12:30,123
over a probability distribution for it.

327
00:12:31,980 --> 00:12:34,350
So here's how these
averages work out, right?

328
00:12:34,350 --> 00:12:36,360
So in the first case,

329
00:12:36,360 --> 00:12:38,520
the average is simply gonna be 0.

330
00:12:38,520 --> 00:12:40,170
Here.

331
00:12:40,170 --> 00:12:41,790
Because my quantity is 0

332
00:12:41,790 --> 00:12:43,200
with probability 1.

333
00:12:43,200 --> 00:12:45,000
So that's not super interesting.

334
00:12:45,000 --> 00:12:45,903
Let's look at this other case.

335
00:12:45,903 --> 00:12:47,790
I of t equals 1.

336
00:12:47,790 --> 00:12:50,643
Well with probability 1 minus gamma,

337
00:12:52,500 --> 00:12:53,720
I have a quantity 0

338
00:12:53,720 --> 00:12:55,110
of recovery.

339
00:12:55,110 --> 00:12:57,273
And with probability gamma,

340
00:12:58,680 --> 00:13:02,040
I have quantity 1 of recovery.

341
00:13:02,040 --> 00:13:04,503
So that's an average of gamma.

342
00:13:06,480 --> 00:13:07,313
Okay?

343
00:13:08,400 --> 00:13:09,750
But that's gonna be fractional.

344
00:13:09,750 --> 00:13:10,583
Right.

345
00:13:10,583 --> 00:13:11,416
In the second case

346
00:13:11,416 --> 00:13:13,620
with I of t equal to 2,

347
00:13:13,620 --> 00:13:15,090
I'm gonna get again,

348
00:13:15,090 --> 00:13:16,830
1 minus gamma squared

349
00:13:16,830 --> 00:13:17,850
this terms disappear

350
00:13:17,850 --> 00:13:19,533
'cause it's a quantity 0,

351
00:13:20,730 --> 00:13:22,410
plus 2 gamma,

352
00:13:22,410 --> 00:13:25,320
1 minus gamma times 1.

353
00:13:25,320 --> 00:13:26,340
So that's the probability

354
00:13:26,340 --> 00:13:27,960
of having one recovery

355
00:13:27,960 --> 00:13:30,000
plus 2 gamma squared.

356
00:13:30,000 --> 00:13:32,280
Then you can work out the math.

357
00:13:32,280 --> 00:13:33,113
Right.

358
00:13:35,457 --> 00:13:36,600
And this term here

359
00:13:36,600 --> 00:13:38,370
that I'm highlighting

360
00:13:38,370 --> 00:13:41,040
you're gonna get 2 gamma
minus 2 gamma squared.

361
00:13:41,040 --> 00:13:43,840
So the gamma squared term are
actually gonna cancel out.

362
00:13:46,590 --> 00:13:47,760
And what you're gonna be left

363
00:13:47,760 --> 00:13:51,003
is simply this 2 gamma quantity.

364
00:13:52,800 --> 00:13:54,390
So this is simply this average

365
00:13:54,390 --> 00:13:55,590
that we had before.

366
00:13:55,590 --> 00:13:56,423
Right?

367
00:13:56,423 --> 00:13:58,470
No matter how many infectious individuals,

368
00:13:58,470 --> 00:14:00,660
no matter matter how
many cases I go through,

369
00:14:00,660 --> 00:14:02,400
this term is always simply going

370
00:14:02,400 --> 00:14:04,773
to be gamma times I of t.

371
00:14:05,893 --> 00:14:07,320
So what we're doing here

372
00:14:07,320 --> 00:14:09,780
with this approximation
is following the average

373
00:14:09,780 --> 00:14:11,340
of all possible future

374
00:14:11,340 --> 00:14:13,080
for this system.

375
00:14:13,080 --> 00:14:15,780
And often the reason why
I'm describing this is

376
00:14:15,780 --> 00:14:17,220
that I just wanna highlight

377
00:14:17,220 --> 00:14:18,610
that this approximation

378
00:14:20,520 --> 00:14:23,560
along with others that we're gonna see

379
00:14:27,795 --> 00:14:29,040
are often called,

380
00:14:29,040 --> 00:14:30,760
mean-field

381
00:14:34,110 --> 00:14:35,613
approximations.

382
00:14:40,170 --> 00:14:41,545
And what this means is that

383
00:14:41,545 --> 00:14:42,378
I of t

384
00:14:42,378 --> 00:14:43,211
and S of t,

385
00:14:43,211 --> 00:14:45,330
now describe the average state

386
00:14:45,330 --> 00:14:47,100
of all possible future.

387
00:14:47,100 --> 00:14:49,140
So the average future of a system.

388
00:14:49,140 --> 00:14:50,040
So one way to look

389
00:14:50,040 --> 00:14:50,940
at this is that now,

390
00:14:50,940 --> 00:14:52,620
whenever you're writing these equations

391
00:14:52,620 --> 00:14:54,060
and you're considering, you know,

392
00:14:54,060 --> 00:14:56,670
one susceptible individual in the system,

393
00:14:56,670 --> 00:14:59,610
that individual is not actually connected

394
00:14:59,610 --> 00:15:02,493
to I of t individuals here.

395
00:15:03,390 --> 00:15:06,090
But connected to the average

396
00:15:06,090 --> 00:15:07,510
or mean-field

397
00:15:08,460 --> 00:15:11,640
of all possible states for that future.

398
00:15:11,640 --> 00:15:13,590
And that's kind of a fuzzy thing

399
00:15:13,590 --> 00:15:15,300
but really instead of thinking of it

400
00:15:15,300 --> 00:15:18,750
as one susceptible individual connected

401
00:15:18,750 --> 00:15:21,120
to some number of infectious individuals,

402
00:15:21,120 --> 00:15:22,713
which is gonna be discreet.

403
00:15:24,810 --> 00:15:28,260
What we have with a
mean-field approximation

404
00:15:28,260 --> 00:15:31,740
is one susceptible individual connected

405
00:15:31,740 --> 00:15:33,060
to some weird,

406
00:15:33,060 --> 00:15:34,560
I'm trying to draw a field here,

407
00:15:34,560 --> 00:15:36,120
some cloud, right?

408
00:15:36,120 --> 00:15:38,943
So some diffused
description of the system.

409
00:15:39,834 --> 00:15:41,943
That's one way to think about this.

410
00:15:42,900 --> 00:15:44,160
But really what it means is

411
00:15:44,160 --> 00:15:46,770
that we're only following
the average state

412
00:15:46,770 --> 00:15:47,910
of the system.

413
00:15:47,910 --> 00:15:49,170
So that's what we often mean

414
00:15:49,170 --> 00:15:51,300
by these mean-field approximations.

415
00:15:51,300 --> 00:15:53,070
But that's as simple as this.

416
00:15:53,070 --> 00:15:54,660
So these equations would be enough

417
00:15:54,660 --> 00:15:56,340
to write a little code

418
00:15:56,340 --> 00:15:58,293
that would iterate these equations.

419
00:15:59,580 --> 00:16:00,810
The readings for the textbook

420
00:16:00,810 --> 00:16:02,250
for this week go through

421
00:16:02,250 --> 00:16:05,250
how to analyze discrete
dynamical systems like this,

422
00:16:05,250 --> 00:16:06,083
instead.

423
00:16:06,083 --> 00:16:07,620
And the next few videos,

424
00:16:07,620 --> 00:16:10,080
what I wanna focus on is how

425
00:16:10,080 --> 00:16:13,490
to simulate discrete computational version

426
00:16:13,490 --> 00:16:14,850
of this model.

427
00:16:14,850 --> 00:16:17,250
And eventually how to go from there

428
00:16:17,250 --> 00:16:20,040
to continuous dynamical system.

429
00:16:20,040 --> 00:16:21,640
So I'll see you in the next one.