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Welcome back.

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So this video is meant to provide you

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with two simple examples of how

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to use the modeling
recipe that we just saw.

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Of course, as I build a model,

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I need to use certain modeling tools.

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So I'll focus this video
on our first type of models

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which are called compartmental models,

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so examples of compartmental models.

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And this goes back to video
two on the modeling recipe.

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So the second video for this week

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where I gave the example that
when we look at a system,

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and we wanna decide which are our parts

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and what distinguishes them,
it's sort of like trying

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to tidy up a room where
you wanna put items

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in boxes that makes them
easy to find in the future.

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So you want, you know, all
the toys to be with the toys,

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all the books to be with the books

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and compartmental models are really just

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a formalization of this idea.

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So compartments are different boxes.

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And then we wanna put different parts

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of our system in there.

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So the way these examples

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are gonna look is always the same.

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We're gonna literally draw boxes

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and then draw the interactions

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between the parts as
arrows between the boxes.

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So let me just formalize
this a little bit.

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So compartments

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will be boxes

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where the assumption

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of a homogeneous behavior hold.

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So anytime you hear about
compartmental model,

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that homogeneous behavior is assumed

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within a given compartment.

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So every part, every
element that we're gonna put

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in the same compartment are assumed

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to behave in the exact same way.

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And different compartments
then are just meant

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to formalize this intuition

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of heterogeneous behaviors in the system.

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And then everything
else, the model itself,

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or the mechanism

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will simply be arrows

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that define flows of the parts.

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So all of the interactions will be arrows

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that tell you how elements flow

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from one box to another

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as their behavior change

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or their state changes,
something like that.

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So it's boxes

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and arrows.

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That's really all this
first tool is gonna be.

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So first, our first example
will be something really,

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really simple, people walking
in and out of a building.

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This could also be to be
more historically accurate,

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could also be a drug compound

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injected in an organ.

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I'm saying this because
a lot of the theory

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and the process of compartmental modeling

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was invented as part of pharmacokinetics.

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So this study of how drug
compounds or molecules

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can be injected into an organ,
diffuse around the body.

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So then the different boxes
would be different organs

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and flows between them

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would be obviously different bloodstreams.

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This is very similar actually here.

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Let's stick with the example

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of people walking in
and out of a building.

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But really, everything
that we're gonna do here,

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could be applied to drug compounds.

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And I'll try and keep the analogy going

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throughout this, but just for now.

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So all we care about really is this.

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So we have this human
population, that's our whole.

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The only space we care about

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is sort of binary here which is rare.

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All we care about is whether people are

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in this building of interest
or not really, so in or out.

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That's all we care about.

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And time seems like it's
left for us to decide,

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so it could obviously be discrete.

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So maybe if we have data
about how many people entered

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the building every hour,

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we could use hour increments for time.

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It could be discrete, or
it could be continuous

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if we know sort of the rate

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at which people enter and leave.

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It doesn't matter too much apparently

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Parts will be people

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distinguished

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by whether they're

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in or out of that building.

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In or out.

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And then interactions between the part,

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I'll just leave that unknown for now.

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Okay, so then, the idea
behind compartmental modeling

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is to represent this intuition
in just a simple schematic.

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And we're gonna be doing this time

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and time again with different systems.

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So then, we're
distinguishing our population

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and whether it's in or out.

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So there's actually two
different conditions

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that we could implement here.

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And it depends on whether
we want an open population

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or a closed population, which
is a step in this definition

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of the whole that I
just like skipped over.

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So with all that, assume,
let me erase this here.

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And then the step that is here
for how we're gonna draw this

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as a compartmental model is

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whether we're dealing with an open

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or a closed population.

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Okay. So what we said
is that all that matters

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is whether people are
in the building or not.

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So in the open population,

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essentially, we're only
gonna model the building,

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and we're gonna say we don't
know what's going on out there.

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It's a scary world.

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So we're gonna have flows
of people coming in,

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but we don't know where
they're coming from.

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Whereas the closed population
is gonna have two boxes,

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two compartments for within the building

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and people outside of the building.

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And that's gonna be slightly different

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in terms of how we model that.

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So one single compartment,
one single box here,

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usually I wouldn't
label things explicitly,

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but this is people in the building.

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And then the way this behaves

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is simply that you have
some people coming in,

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let's say, at rate alpha.

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Within this building, we're
gonna have a population

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that we're gonna call N of T.

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So every compartment always
has one variable associated

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with it that is allowed to change in time.

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So here, I'm gonna use N for number

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of people in the building.

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People are coming in at rate alpha

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and then people are gonna go out

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and assuming at rate beta.

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And I'm assuming that the more people

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there are in the building,

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the more people come out just simply

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because if no one's in the building,

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no one should be coming out.

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So having this rate, this arrow here,

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be proportional to N of T, just make sure

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that I don't get negative
numbers in my building.

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Well, also, this is pretty simple.

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You can already imagine
how this would behave

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as if initially there's
no one in the building

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through time.

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The number of people in the building

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would go up initially
at rate alpha, right?

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This which is a linear increase in time,

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so N of T increases
with time at rate alpha.

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So that's the slope
here, the initial slope,

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then the slope starts slowing down because

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as people come in,
people start walking out

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at this rate beta and eventually,

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it's just gonna settle out.

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Because beta times N of
T will be equal to alpha.

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So it's really just the flow

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of the parts through different states

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is what these compartmental models

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are supposed to represent.

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In a closed population,

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it would be slightly different

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because we could have the inside

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of the building and now,
let me draw a nicer square.

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The outside of the building,

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lemme start over here.

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Nice square for
compartment one, (laughing)

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like too bad for the
automatic shape, okay.

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Okay, so we have one
compartment for the inside,

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let's say, and one for the outside.

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And now as I said, in a closed system,

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no arrows come directly from the outside.

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We're modeling a complete universe.

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So we're gonna have people
that are outside that go

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in the building and people
that are in that go outside.

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And then the key part here is that

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because we have a closed system,

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we have a fixed population,

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there's a total of let's call them NT

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for the total number of
people in the population.

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And that's not a dynamical variable.

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That's just like 100 people.

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That's all there is in this
little imaginary universe.

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So inside, we still have
N of T and then outside,

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we would have NT minus N of T people,

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so that the sum of people inside

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and outside are fixed at all time.

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And now, we could have, we could
imagine that we have a rate

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at which outside people choose to go in,

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which is again, now proportional

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to this NT minus N of T,
number of people outside.

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So let's say, we call
that NO for outside of T.

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Here, we would have now, alpha NO of T

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and again, beta N of T here.

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And this model will be
slightly different, right?

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So by choosing open or closed systems,

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we've built some approximations

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or some assumptions about this population.

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In subsequent videos,
especially next week,

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we'll see like how to go

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from these little schematic cartoons

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to actual mathematical equations

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or how to model these systems
using stochastic codes.

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But most of which we care
about is in designing

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the cartoons of boxes and arrows.

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So let's do a more involved example

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before we end this video.

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So simple disease models, right?

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So I think I already said it here.

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We care about a human population

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and what's gonna be important

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is whether people are sick or healthy.

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So the whole is a human population

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Oops, population,

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and the classic model
that are still mostly used

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to this day to forecast COVID,

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we're gonna assume what I call
a fully connected population

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which really you could
think of as either everyone

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is either that there's no space,

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everyone is in one point,

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or you have a fully connected network.

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I like to think of anything,
everything as network.

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So that's our space, right?

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So if you have different people,

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well, everyone is connected to everyone,

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like I would just be drawing
a really big mess here.

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But that's essentially
what we're dealing with.

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Anyone can infect anyone.

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Okay, let me remove my mess.

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And time,

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here, again, I'll leave it free for here.

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It doesn't matter too much.

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It really matters once we
start writing down equations

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or, but often we talk in terms

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of per-contact transmission rate.

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So we could assume a discrete time systems

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where at every time step,

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some contact sucker in the population.

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The parts are going to be people again

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distinguished by their
epidemiological state.

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So we're gonna have, let me
do, in the simplest case,

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we would have what we
called susceptible people,

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meaning healthy and susceptible.

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They could potentially be infectious,

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and that's usually just called S.

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And we're gonna have
infectious individuals,

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usually called I.

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So in epidemiological
compartmental models,

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what I'm about to draw would
be a potentially a SI model.

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So we have two boxes.

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Box for how many people are susceptible,

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and a box in red here for how
many people are infectious.

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And then I'm assuming a
closed population, let's say.

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So I'm gonna add here, in my whole,

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I sort of skipped a step that's important.

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So I'm gonna assume a closed population.

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If I'm modeling something
like COVID on the UVM campus

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for this week, that might
be a fine approximation.

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If I'm modeling COVID
over the next 10 years,

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then I wanna have some
birth-death process.

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Maybe some new students coming
in, students coming out.

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So then there's a lot of exchange

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with the rest of the world.

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So really, this closed population here

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is a matter of timescale.

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If we're thinking about
the campus over a week,

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it's fine to think it's closed.

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Okay, so how is this gonna work out?

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We're gonna have S of
T variable associated

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towards susceptible and I
of T variable associated

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with our infectious individuals.

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Now, we need to think at
what rate do susceptible,

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so people that could be
infected do become infectious?

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Well, a little bit like
people leaving the building,

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we're just gonna assume a
rate at which the event occur.

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So let's call it beta

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and like people in the building, again,

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if there's no susceptible individual,

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no one can leave, right?

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00:15:46,770 --> 00:15:50,313
This isn't something
flowing from the outside,

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so I'm gonna put a S of T

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because the more
susceptible people I have,

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the more I imagine that they
will get infected, right?

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00:15:59,160 --> 00:16:02,910
Disease incidents is usually
this per-capita kind of thing.

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So to get the actual
number of new infections,

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you multiply it by the total
susceptible population S of T.

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Well, similarly, you can
make the argument that

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if no one's infectious, no
one's gonna get infected, right?

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And similarly, if you have 10
times more infectious people

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00:16:18,450 --> 00:16:20,820
on campus, you get 10 times more contact

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00:16:20,820 --> 00:16:22,140
with infectious people,

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00:16:22,140 --> 00:16:24,303
10 times more likely to get infected.

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00:16:26,610 --> 00:16:30,633
So this rate really here
is this non-linear term,

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00:16:31,526 --> 00:16:35,280
beta times S of T times I of T,

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00:16:35,280 --> 00:16:37,350
and I call it a non-linear term

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00:16:37,350 --> 00:16:39,120
because we have a closed population.

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So we have a constraint here

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00:16:40,410 --> 00:16:42,513
that I'm gonna write in the margin.

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That S of T plus I of T is equal to NT,

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00:16:49,320 --> 00:16:50,850
same as in our previous example.

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00:16:50,850 --> 00:16:53,580
We have a total population that is gonna

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be fixed in time, not going to change.

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So really, I could have written
I of T as NT minus S of T

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or S of T as NT minus I of T.

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It doesn't matter too much.

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00:17:05,670 --> 00:17:09,480
But if you do that, what you
realize is that this rate here

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00:17:09,480 --> 00:17:13,440
in the middle, beta times S
times I is really something

315
00:17:13,440 --> 00:17:17,520
like beta times S times NT minus S.

316
00:17:17,520 --> 00:17:19,800
So you do have some S square term

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00:17:19,800 --> 00:17:21,600
that's gonna create some
interesting behavior

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00:17:21,600 --> 00:17:23,010
when we start studying

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00:17:23,010 --> 00:17:25,293
the dynamical behaviors of these models.

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00:17:26,250 --> 00:17:28,590
So that would be a simple SI model.

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00:17:28,590 --> 00:17:30,360
It's kind of ridiculous in practice

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'cause here, we're assuming
there's no interaction

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00:17:32,760 --> 00:17:35,700
out of the I of T compartment.

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00:17:35,700 --> 00:17:37,530
So people are infectious forever.

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00:17:37,530 --> 00:17:40,030
So the only way this model would end is

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00:17:41,490 --> 00:17:43,800
with the population fully infectious.

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00:17:43,800 --> 00:17:46,050
In practice, we know
that people do recover

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00:17:46,050 --> 00:17:47,490
from most disease.

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00:17:47,490 --> 00:17:52,297
So we could imagine having an arrow back

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00:17:53,400 --> 00:17:54,990
to the susceptible population.

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00:17:54,990 --> 00:17:56,820
So that could be something
like the common cold,

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maybe, when you do recover,

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00:17:58,650 --> 00:18:00,183
you can get it again.

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00:18:01,380 --> 00:18:02,430
And then this would happen,

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00:18:02,430 --> 00:18:05,043
maybe at a rate, gamma times I of T.

336
00:18:06,810 --> 00:18:10,140
And now, we would have what
we call the classic SIS model.

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00:18:10,140 --> 00:18:12,210
It's SIS because individuals go

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00:18:12,210 --> 00:18:15,213
from susceptible to
infectious to susceptible.

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00:18:19,230 --> 00:18:21,330
If we wanna think about COVID,

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00:18:21,330 --> 00:18:25,500
hopefully, the situation
looks actually more like this.

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00:18:25,500 --> 00:18:26,880
We would have a third state

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00:18:26,880 --> 00:18:29,640
that we would call recovered individuals

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00:18:29,640 --> 00:18:32,223
and then infectious
individuals would go here.

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00:18:33,360 --> 00:18:36,030
And I think you're starting to see why,

345
00:18:36,030 --> 00:18:39,370
and I just added a state
here, recovered in black

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00:18:42,330 --> 00:18:44,100
which I'm calling R.

347
00:18:44,100 --> 00:18:46,740
I think you're starting
to see why I say that.

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00:18:46,740 --> 00:18:49,350
When you look at these cartoons,

349
00:18:49,350 --> 00:18:50,910
once you've seen enough of them

350
00:18:50,910 --> 00:18:52,350
or done enough of them,

351
00:18:52,350 --> 00:18:54,960
you really start understanding what is

352
00:18:54,960 --> 00:18:56,550
the question people are trying to answer.

353
00:18:56,550 --> 00:18:58,290
What assumptions are they making, right?

354
00:18:58,290 --> 00:19:01,320
If you see just the two
boxes for a disease model,

355
00:19:01,320 --> 00:19:03,600
you know you gotta have healthy and sick.

356
00:19:03,600 --> 00:19:05,250
So you're assuming that people like,

357
00:19:05,250 --> 00:19:07,110
go back from sick to being healthy.

358
00:19:07,110 --> 00:19:10,050
If you see a third box, you
know what they're assuming.

359
00:19:10,050 --> 00:19:12,180
Looking at the mechanisms, right,

360
00:19:12,180 --> 00:19:15,750
is really helping you
see the different flows.

361
00:19:15,750 --> 00:19:17,763
If we add an open population,

362
00:19:19,080 --> 00:19:20,670
let me just cross this out, right?

363
00:19:20,670 --> 00:19:22,830
If we add an open population,

364
00:19:22,830 --> 00:19:25,380
then maybe we would have a migration

365
00:19:25,380 --> 00:19:28,653
of susceptible individual
here at some rate alpha.

366
00:19:29,700 --> 00:19:31,200
And doing so, would tell you

367
00:19:31,200 --> 00:19:33,300
that I'm either thinking
on a longer time scale.

368
00:19:33,300 --> 00:19:35,100
I'm thinking about a
town and people coming

369
00:19:35,100 --> 00:19:37,650
into town over, you know,
months or something.

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00:19:37,650 --> 00:19:39,700
It does tell you about what I care about.

371
00:19:40,590 --> 00:19:43,770
So I'll leave it at that for this video,

372
00:19:43,770 --> 00:19:45,630
like constructing these models

373
00:19:45,630 --> 00:19:47,280
and drawing these boxes is something

374
00:19:47,280 --> 00:19:48,810
we're gonna do for a long time.

375
00:19:48,810 --> 00:19:49,800
And I think this week,

376
00:19:49,800 --> 00:19:52,470
we're just gonna run
through a lot of examples,

377
00:19:52,470 --> 00:19:54,780
trying to see how we map our assumptions

378
00:19:54,780 --> 00:19:57,930
to actual assumptions and approximations

379
00:19:57,930 --> 00:20:01,560
to actual models as boxes and arrows

380
00:20:01,560 --> 00:20:03,993
with these compartmental models.

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00:20:04,920 --> 00:20:07,143
So I'll see you in class
for the discussion.