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Welcome back.

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Welcome back for a third week

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of Modeling in Complex Systems.

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Last week when I left you in the videos,

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I gave a very vague definition

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of the first type of modeling

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that we started doing
using compartmental models.

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The definition I gave was something like,

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you know, compartmental models

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are reduction of complex systems

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in schematics of boxes and arrows

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where boxes represent
subpopulations of your system,

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of the parts of your system,

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where approximation of homogeneity holds.

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And this approximation
of homogeneity, really,

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I didn't really define.

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So I thought it would be
nice to revisit in this video

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in light of all the good discussions

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that we have this week, so.

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Lemme share a little bit of slides here.

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So I really wanna revisit

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this idea of compartmental modeling.

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And maybe a clearer definition

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would be really that
compartments are boxes

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where we put indistinguishable
parts of the system.

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So it's not just that we can approximate

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that their homogeneous,

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really these approximation of homogeneity

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is the idealization that we
make to get to the model.

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But within the model,

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parts that are put in the same boxes

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are literally indistinguishable.

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So if we are curious about an ecosystem,

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for example, with wolves eating rabbits

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and rabbits reproducing,

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we might have two boxes, one
for wolves, one for rabbits.

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And we're saying that all
wolves are exactly the same.

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They behave in the same way.

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They have no individual features,

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they have no memory of their past dates.

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And same with rabbits.

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So once we do this compression, really,

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what we're trying to do is to think about

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the different parts of our
system, which is heterogeneous,

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sort of by nature of complex system.

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And then we try to reduce
this heterogeneity into boxes

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where there's no heterogeneity
within the boxes,

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only across, so discrete types of parts.

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And then once we have that,

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what we try and do is
reduce all the interactions,

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all the flow of energy or information

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within that system as
arrows between the boxes.

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We can have different types of arrows.

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But really they're just a
way to formalize what we mean

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by interaction and flow of
parts across the system.

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The example that we
discussed at great length

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was inspired by, you know,
current events in the world.

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And we thought about outbreaks of COVID

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on a university campus, right?

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And then we saw the classic
disease models which say,

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well, we have this
complex human population,

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different types of people and all that,

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but at the very minimum,

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what we care about in terms
of building a disease model

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is their state regarding
this infectious disease.

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So in the classic case,
we would've three boxes,

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so Box S, I, and R to
distinguish our population

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in terms of whether people are
susceptible, meaning healthy,

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and potentially could be
infected in the future.

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Currently infected and infectious.

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So this I box.

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And then recovered.

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And we simply have two arrows,

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one from S to I, and one from I to R.

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So those two arrows, really
the way to think about this is,

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I mean, the way I like to think about it

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is that boxes are simply backed up

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that get filled with water
representing a certain density

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or fraction of the total system.

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And then mechanisms, the
arrows are simply pipes

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through which water
can flow, in this case.

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So you have a lot of water initially

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in the susceptible bathtub or compartment,

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which flows at some rate
that might change in time.

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But this pipe has a certain width

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through which water flows
from the S to I bathtub

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and that models infections in the system.

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And then the I, we get
a little bit of water

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that grows in the I bathtub

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and another pipe that flows
with a different width

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into the R bathtub, representing
recovered individual.

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The reason why I use this idea of water

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will become clearer in the
next videos for this week

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when we start thinking about
the mathematical description

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that we can get from these
systems of box and arrows.

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And then we'll see that
they're not necessarily ideal

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at dealing with discrete quantities.

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So it's often better to think of it

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as a density of water.

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Now the one thing I'll say at this point

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is that we have three
boxes that we've labeled,

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you know, susceptible, infected,

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or infectious and recovered.

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And we have pipes that
we label infection events

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or recovery events.

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And I discussed this

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with the entire class throughout the week

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but I wanted to put it on
the record in one video.

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My number one modeling
tip is to be very careful

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with the labels that we
put on boxes and on arrows.

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Putting label is super
useful to remind ourselves

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of, okay, this compartment,

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this box represent infected individual.

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And then we have an idea

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of what this means in terms of mechanisms

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and it help us think about
the mathematical description.

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But when comes the time to
discuss the result of the model,

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we have to be careful about
not putting too much weight

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on the words that we chose

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to use to label these
boxes and arrows, right?

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So if I take, you know, some other model,

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we might think of some arrows as,

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you know, brain complexity,

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or some boxes as individuals

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with more technological savviness, right?

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And then if our conclusions
read something like,

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imagine headlines read something like,

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with increases in brain complexity

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or increases in technological
savviness comes X, Y and Z.

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Well that's not quite right, right?

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Technological sevens or brain complexities

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are what our model is trying to describe,

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which really it's with this decomposition

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and all the idealization that we've made

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that comes X, Y, and Z.

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And so sometimes putting things on,

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putting things with realistic label

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can be a little misleading.

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'Cause we forget that
we're really talking about

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our model at this point
and not reality, right?

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So seeing it by the dip,

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the representation is not the real world

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and sometimes labels can
lead us to forget that.

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And that's my number one tip

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about modeling to be
careful about the labels.

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Oka, but for getting to this now,

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we have, you know, three
boxes, S, I, and R,

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and two arrows labeled
infection and recoveries.

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And this little description

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is the simplest possible disease model

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for something like COVID on campus

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where R would be, you know,
people that have recovered,

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that are either dead or immune.

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In any case, they're
not gonna participate,

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they're not gonna get infected anymore.

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So they're not gonna
spread the disease further.

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And then we discussed ways
to generalize this model.

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Really, the way we did
that was to try and list

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all the possible idealization
that had to be made

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by the modelers who came
up with the S I R model

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to go from a complex
system, a human population,

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and an infectious
pathogen spreading therein

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to this simple, like three
boxes, two arrow system.

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One idealization is that
infectious individuals

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are all the same.

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In terms of COVID, we know
some people are asymptomatic,

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some are not.

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So one improvement that we made

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was to introduce two boxes
for two different types

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of infectious individuals,

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whether they are aware,
IA, or unaware, IU,

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of their infectious state, right?

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So untested, asymptomatic
individual would be an IU

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and you can imagine that those
individuals contribute more

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to the flow of individual
from S to IA or IU

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than IA individuals.

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Meaning that IU individual
might be more contagious

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'cause they're not taking any precautions,

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they don't even know they're sick.

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Another important
generalizations that we made

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was to not only consider
recovered individuals,

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that all individuals
previously infectious.

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One reason why we might
wanna model disease

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is to be able to test
different interventions.

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While in which case, you
know, it really matters

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whether people recovered
and are now immune,

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recovered and might have
long-term consequences from COVID,

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or if they're dead, right?

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So disease individuals,

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what we wanna be able to evaluate

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is the cost of this disease burden.

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So then we split all of that
up into a disease compartment,

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a recovered compartment,

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and a long-lasting
consequence compartment.

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In terms of dynamics of the spreading,

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they're all the same,

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meaning they're not
gonna get sick anymore,

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they're not gonna infected anyone else.

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But in terms of evaluating
different outcomes,

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it really makes a big difference.

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So again, here we have
a more complex model,

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a lot more arrows.

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And mathematically, that can be annoying.

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It's gonna be harder to analyze.

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But if the purpose of our model
is to evaluate the quality

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of different outcomes with
different interventions,

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well this is essential, right?

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The final generalization
that I wanna talk about

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that was made during the discussion group

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is also to consider

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more different types of
susceptible individuals.

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So we're not all as
susceptible on campus, right?

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So if you're a remote student,

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so you're in town,

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you might get sick
through some other venue

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but you're not physically on campus,

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you're a low risk individual.

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And we might call that SL.

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If you're on campus,

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but you take all your precautions,

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you might be medium risk, SM.

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If you're on campus and you
have pre-existing conditions

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and a lot of contact,

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then you might be a high
risk individual, SH.

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And as you can see, like
it's easy, if you want.

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There are so many
idealizations that we made

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to go from the real world

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to a system of three boxes and two arrows.

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It's easy to think of
generalizations to make.

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The key part is in making
the right generalization

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that don't introduce too much more,

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too much additional
complexity in the model,

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but are required for the
purpose of the model, right?

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So if we wanna consider the
role of having remote student,

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then we have to distinguish
on these three different types

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or at least two types of
susceptible individual.

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It's all about the balance here.

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So to wrap it up,

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before we start going
from boxes and arrows

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to mathematical description,

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I'll give yet another definition
of compartmental model.

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One way to think about it

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is that it's a
decomposition representation

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or compression of a real system

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into different types of homogeneous parts.

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Really, the key thing
is we wanna acknowledge

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that per nature of complex system,

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we have a collection of parts,

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multiple interconnected parts,
and they're not all the same.

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They're often heterogeneous.

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But we wanna ask what is the minimum,

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minimum number of of different
types that I can distinguish.

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And then within those types,
then I'll start ignoring

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all individual heterogeneity
for convenience.

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And then if you can get to
this minimum description,

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then you have a good compartmental model

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that's worth going through
a mathematical description

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and solving.

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And then that's what we will be up to

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over the next few videos.

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So hopefully that clears it up.

262
00:12:03,630 --> 00:12:07,410
Really, this is the
type of of modeling tool

263
00:12:07,410 --> 00:12:09,663
that only becomes obvious
the more you do it.