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In the last video we saw to go

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from a discrete SIS model
to a continuous model.

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And here what I'd like

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to show is the power of
the continuous formulation.

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Just because once we write it

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as differential equation,
models are way easier to solve

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in a way because we unlock
the power of calculus

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and of a lot of mathematics.

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The SIS model is also interesting

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because it's the simplest
model we've seen so far.

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We've discussed some more complicated ones

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without conserved population

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and with a lot more than two states.

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But this one is interesting
because we took the complexity

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of disease spread and
reduced it to two boxes.

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Meaning we have a human
population that we put

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into two boxes that we call S and I,

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depending on whether
individuals are susceptible

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or we put them in the S box or infectious

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we put them in the I box.

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Then we had two arrows for two mechanisms.

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There's a mechanism for infection.

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So as individuals go to "I"

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and mechanisms for recovery.

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To write down the equations,

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we did a lot of approximations
and idealizations.

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We assume, you know a
fully connected population,

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a mean field approximation.

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We saw how the time derivative

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of S which I'm often gonna write S dot,

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this is a notation from Newton,

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which I like 'cause it's
just a little simpler.

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The dot just means time derivative of S.

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Could be simply written as minus beta SI

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by assuming very small time steps,

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gamma of I, here it's just an average.

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We're tracking the average
state of the system.

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That's what this differential
equation is doing.

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If you look at different places,

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we're gonna be some reading,

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doing some readings about
some of these models.

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You might also see stuff like

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minus beta SI over N plus gamma I.

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This is just a different formulation,

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really it is beta over N is
just using different units.

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Beta over N is a like
shifted transmission rate.

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And the question is whether
your beta is formulated

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as a per contact, in which
case you don't need the over N

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or as an overall transmission
rate in which case you you do.

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So there's different ways

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and again, I'm showing this just

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because I wanna stress
that there's no one way

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to write an equation for
some of these models.

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It depends on what units you're using,

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what approximations you're doing.

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But they're not more valid
one than another really.

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We also saw, once you
go through the process

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of writing those equations
what's useful is to think

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of it as writing one equation

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per compartment per state in the system.

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And then you have some negative terms

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for every mechanisms that leaves.

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So every arrow that leaves
your box needs a negative terms

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and that's this minus beta SI here.

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And every mechanism that enters your box,

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anything that flows in the system

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needs a positive term in your equation.

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And so we could also write the
"I" dot equation very quickly

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by realizing that everything
that leaves S has to enter I.

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So I need a plus beta SI term

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and everything that
leaves "I" was entering S.

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So my gamma "I" is now
negative 'cause it's leaving.

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There's another way to realize

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how easily you can write this equation

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for "I" dot given S dot, which is the fact

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that we here have a conserved population.

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We assumed a closed system,

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meaning that there's no arrows flowing in

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or out of our boxes into
the rest of the universe.

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So it's a closed system.

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So I have S plus I equal
to N for all times.

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And if I take the time derivative of this.

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Well I get S dot plus I dot,

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the derivative is the linear operator.

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I can just sum my derivatives.

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I get N dot which really is the derivative

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of a constant with zero.

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The number of people

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in the system is not changing in time.

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Again, there's nothing flowing in or out.

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So this gives me that S dot

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has to be equal to minus "I" dot.

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So I know I can just flip all terms.

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The reason why I'm highlighting
this different perspective

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on it is because whenever you
have a conserved population

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even if you have like 20 different states,

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you know that the sum of all
your differential equations

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should be equal to zero

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'cause nothing's leaving
the system or entering it.

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And so that's a very useful sanity check

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to make sure that you wrote your equation

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in a correct way is

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whether do they respect
your conserved population

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do they sum to zero?

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So that's useful.

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The formulation in terms

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of differential equation
is great to do things

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like flow diagrams, which
we did for discrete models.

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Really those differential equations were

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in calculus were often
just like given equations

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and we're told to solve them.

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What they mean is, you know

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what is the speed of the
system at any given point.

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So these S dot and "I" dot are
plotting this flow diagram,

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give me what S is, I'm
gonna know what "I" is,

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it's gonna be N minus S

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and I'm gonna know what
my speed is in the S space

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and the I space and it's very
easy to plug those diagrams.

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They're giving me the
direction of the system.

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And often what we're
gonna care about are is

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where is this system gonna end up?

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And what does that mean is

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when are S and I gonna stop moving?

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So that's what we call a fixed point

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or an equilibrium point.

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The term equilibrium
makes a little more sense

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in classic physics model.

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If you think of the pendulum

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which is damped and
eventually stops moving

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now it's at equilibrium, it's fixed.

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Here it's sort of a what we
call a dynamical equilibrium.

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So the quantities S and
I are gonna stop moving

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eventually they're gonna
become fixed in time.

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But underlying this there are still people

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getting infected and recovering.

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It's just that the number

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of expected infection is
gonna manage the number

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of expected recovery so
that S and I are fixed.

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So how do we find those fixed points?

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Well, it's the coordinates S
and I such that our derivative

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in time is equal to zero,
things are not moving.

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So I'm looking to solve this equation.

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Well, and if I go really quickly,

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I can cancel out my "I",

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I just have minus beta S
plus gamma equals zero.

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So that gives me S star.

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I'm often gonna use star to
denote dynamical equilibrium

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and equal to gamma over beta.

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So I can do a little thought experiment.

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Where does my system end up?

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This S star for different diseases

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with a whole range of different
beta from zero to infinity.

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Well, at infinity I'm gonna
start very close to zero, right?

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Gamma over beta goes to zero
and beta goes to infinity.

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And that makes sense.

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People are always gonna recover

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but if they get infected infinity fast,

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they're not gonna stay in
the state for very long.

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So the larger beta is the closer I am

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to this S star equal zero

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and then it's gonna do something like this

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and blow up as better approaches zero.

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Well that's the problem
because I know that S

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has to stay between zero and N.

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So there's a point here where
my solution is crossing N

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and that's a big problem.

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So really the problem is
that I went too quickly

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when I canceled out my "I" terms.

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I is not a constant here,

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it's another variable in the system.

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It's just that I know its
value, it's N minus S.

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So if I rewrite my equation like this,

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I'm highlighting the fact that
I expect a quadratic term.

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I have a plus beta times S squared.

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So it's a quadratic equation.

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So I know that I have two
solutions to this state

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and the other solution is
actually really easy to see.

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So I'm gonna write this as S1 star

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with my gamma over beta,
but I have another solution

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which is S2 star, which is equal to N.

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And this is matched with I2 star solution

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equal to M minus S equal to zero.

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So that's my disease free state

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and of course that's a solution.

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Of course, if I don't have
anyone infectious in the system,

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I know that I'm a steady state.

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There cannot be infections
without infectious individual

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and there cannot be recovery
without infectious individuals.

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Okay, so now I have this interesting point

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where my two solutions cross

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and what I wanna know is
really what's going on,

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which state is my system going to go to

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if I run a simulation of it

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or if I integrate my
differential equation?

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Both of these approaches we're
gonna see in the next videos.

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But here I wanna try
and make a prediction.

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What's state, the red state
which is bad or the blue state

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which is good without disease,
is my system going to go to?

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So let me just take a picture of this here

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and we're gonna do what we
call a stability analysis.

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I don't know why my picture didn't work

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Just, I'm looking at S star, oops,

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as a function of different betas

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and I have a S1 star
equal to gamma over beta

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and a S2 star in blue

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equal to N.

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Okay, well what's stability
analysis means is I want

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to put myself in one of those fixed points

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and asked if I move away from it,

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which direction am I gonna move in.

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And so if I'm at S equal
N which is a fixed point

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and I introduce a little bit
of infectious individual,

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is my derivative gonna be positive?

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Meaning that I go like
down from the blue state,

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do I go back up to the
good disease free state

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or do I go back further
down to the red line.

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Stability is gonna help us understand

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what the impact of introducing
a disease is gonna be.

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So what I like to do in
this case is to go back

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to our differential equation.

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And I am gonna ask if
I introduce the disease

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for every infectious
individual that I introduce

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what is gonna be the
rate of change of S dot?

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So really I'm looking at S dot over I.

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So this is a useful perspective to take

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and this is minus beta S plus gamma.

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Okay.

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So if I'm at the fixed point S2 star,

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this is minus beta N plus gamma.

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And that thing can be
greater or smaller than zero.

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And I want to know which
is it for a given disease.

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So often people are gonna
summarize this equation,

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this question really as the
basic reproduction number.

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And if you've followed some of the meatus

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especially early on in the COVID outbreak,

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you've probably heard this phrase before.

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So my question here becomes
is this number R0 which

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in this case is equal
to beta N over gamma,

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is it greater than one?

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So if it's smaller than one,

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so if beta N over gamma
is smaller than one,

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that means that this
equation here is positive.

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So that happens, let's
say if gamma's too big,

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beta N over gamma is
gonna be smaller than one.

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Or if beta is too small.

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So if beta is too small, what do we have?

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We start at the blue state here,

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we move a little bit away from it,

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we go back up 'cause
that's beta is too small.

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This our equation is gonna be positive.

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Our S dot over I is gonna be positive.

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If beta is greater and is big enough

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such that beta N over gamma
gonna be greater than one,

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then our S dot over I
is gonna be negative.

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And as we move away from
the disease free state,

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we collapse down to the red curve.

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So this line here is where
our R0 is equal to one.

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That's where our two solution meet.

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And that's often called
the epidemic threshold.

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Meaning that we can use
this little model to try

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and determine whether
our disease is gonna take

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over the system or not.

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And that's a simple analysis.

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We look for fixed point, we
look for their stability.

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It's nothing too complicated.

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But if you can go through it,

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really this is at the core,

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this is state-of-the-art disease modeling.

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It's at the core of our
understanding of diseases.

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That's why early on people were so focused

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on measuring this basic
reproduction number, R0,

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there are movies about it.

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So really we're already,

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in tools that are super useful

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in helping us understand the
complexity of the real world.

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If we care about the time
evolution, like, okay

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I know I'm gonna go to the red
state, I wanna know how fast,

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what is my evolution, my
time series gonna look like?

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Then we might turn to
different tools, then analysis.

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We might look at numerical integration

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which we're gonna see next.

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If we want to look at
the impact of randomness

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then we might look at
numerical simulations.

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And we're gonna see that too.

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But we really can't
underestimate how important it is

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to just look at the differential equation

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and try to get some mathematical insights

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from their basic structure.

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So that's a full story.

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That's how we understand
disease spread to this day.

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And I hope it's useful
in illustrating the power

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of modeling even when we
use simple toy models.

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So I'll see you on the next ones

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for some more numerical tools.