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So now we understand the diffusion model,
we want to move on to something called the

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SIS model. Now the SIS model looks pretty
much exactly the same as the diffusion

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model, exact for now we allow for the
possibility that after someone's become

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infected, they can move back into the
susceptible fool, pool. So they can become

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recovered in a sense, but then they're
also susceptible again. So this would be

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for something like the flu, where after
you're cured of the flu, the flu mutates

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and you can become infected again. It
wouldn't work for something like measles

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where once you've had the measles, you're
no longer gonna get the measles. So here's

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how our model looks. It looks exactly like
our model before. There's some number of

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people that have it at time t+1, and that's
the number they have it at time t, right,

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plus the people, the new people who get
it, right. And this depends on somebody

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who has it meeting somebody who doesn't
have it, right. And this is the

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transmission rate and this is the
number of contacts. But all we do to this

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is we subtract off this minus aWt. Well,
what is this? These are the people who

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become cured, right? Who no longer have
the disease and they're gonna go back into

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this pile N. Maybe people who could get
the disease anew. So I would think it's

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sort of like throwing in a rate of people
sort of getting better. Well now here's

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where this model is sort of more interesting,
'cause in the diffusion model something gets

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spread. So it starts out slow,
it goes faster, then it tails off.

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But here, what happens is, while people
are getting sick, at some, at some other

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rate some rate "a", they are getting better.
And if people get better faster than

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people get sick then what's gonna happen?
The diseases ain't gonna spread. So this

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model's gonna produce a tipping point,
so. Let's simplify things a little bit if

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we rearrange terms, what we get is an
equation that looks like this. Now how did

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I get this equation? All I did was I
pulled this Wt term out, right, and I

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crossed out some N's. Let me show you how
that works, right. So up here I've got,

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let's just focus on this part, I've got N c
tau times W over N times N minus W over N

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minus aW, okay? And so what I do is I
say, well you know what, let's cross out

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this N with this N. That's fine, 'cause
those N's go away. And then I've got a W

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here and a W here. So I'm gonna just pull
out that W. And then I've got c tau. N

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minus W over N minus a. Alright? That's
what I've got. And if I look over here,

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that's what I've gotta get. c tau N minus
W over N minus a. So this is just a simple

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application. So when you run models is
useful to be good at algebra. [laugh] If

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you can do lots of algebra, you can
simplify things. Well, why do you want to

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simplify things? Here's why. Look, suppose
we're on in the disease. So that means Wt

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is really small. So that means that N
minus Wt over N is gonna be really close

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to 1, because basically a very small
percentage of people in the population

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have the disease. So now if I look at this
thing, I can say well you know, Wt plus

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1 is really equal to Wt. Wt plus 1 is
equal to Wt plus Wt times c tau minus a.

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So, this thing is gonna spread if c tau
minus a is positive and it's not gonna

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spread if c tau minus a is negative. So
this is gonna come down to: Is c tau minus

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a bigger than zero? Right? Or is c tau
bigger than a? Or another way to write this

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is: Is c tau over a bigger than one? All
right, and this leads to what's called the

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basic reproduction number. We let R0
equal c tau divided by a. If c tau divided

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by a is bigger than one, the disease
spreads, right? Because that means that Wt

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plus one equals Wt plus something
positive. Right? If R0's less than one,

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right, in other words, if c tau over a is less
than one, then Wt plus one equals Wt plus

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something negative and the disease dies
off. So this R0 is called the basic

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reproduction number and what it basically
tells you is: Does the disease spread? But

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notice here, we've got at tip,
right? R0 less than one, no disease

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spread. R0 bigger than one, disease
spreads. So, let's take some real

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diseases. Diseases like measles, mumps,
the flu. The R0s are fifteen, five and

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three. That's why these are real diseases.
There might be a ton of diseases out there

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that have R0s less than one and we're never gonna
hear about them. Why? Because they don't

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spread. Now again, for things like measles
and the mumps, once you get them you don't

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fall back in the population. So, there's a
model you use their called the SIR

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model, very similar to the SIS model.
But for the flu, right, you fall back into

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the pool, so you can think of it like this
SIS model. Now it's interesting here,

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from these basic reproduction numbers we
can say, you know, that past the tipping

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point, the disease is gonna spread. But
let's think about this figure. Why do we

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construct models? Well, bunch of reasons,
right? But one reason is to design

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policies. You could say, how do we stop
these things from spreading? Well, one

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obvious answer is vaccines. Well, then my
question is, how many people do you have

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to vaccine [sic]? Well, turns out the model will
tell us. So let's think about it this way,

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right? There's, let V be the percentage of
people that you vaccinate. So there's this

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basic reproduction number R0. That's
like the rate in which things spreads

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through the population. Well if I
vaccinate some percentage of the

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population, that's just gonna to reduce,
right, the basic reproduction number by

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that fraction. So if half the people were
vaccinated, R0 is effectively divided by

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two. Right? And if 75 percent of the
people are vaccinated, R0's gonna be

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divided in effect by four. Right? So
you've only got one fourth of the

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population. So the question is, how many
people do you have to vaccinate as a

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function of R0? Well, in some sense if we
vaccinate like V people, it's like we've

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got a new r0. Right? Little r0, which is
just the big R0 times the percentage of

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people in the population that aren't
vaccinated. So what we can do is we can

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say, we want R0, big R0 times one minus V
to equal one. Actually it's going to be

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less than one, right? Well we can multiply
this out, we get R0 minus R0 V, right? Has

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gotta be less than one, right? So we can
bring this over together so I'm going to

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get R0 minus one. We need to be less than
R0 times V, so that means we need V to be

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bigger than one minus one over R0, right?
So what we get is, we get this equation

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that says: This is how many people we need
to vaccinate. So let's go back. Remember

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the measles were fifteen, the mumps were
five. How many people do we need to

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vaccinate to prevent the measles from
spreading? Well that's just one minus one

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over fifteen, which equals fourteen
fifteenths. So we need to vaccinate

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fourteen fifteenths of the population
against measles if we want the measles not

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to spread. For the mumps, right?, we
need one over one-fifth. Right, we only

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need to vaccinate 80 percent of the
population. Well here's why the model's so

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useful. Again, this model's isn't exact in
working out all sort of things like

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networks, and changes in contract, contact
structures, and different locations and

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stuff like that, but still, what this
tells us is, which is really important, is

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depending on how variant the disease is,
you have to change how many people you

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vaccinate in a pretty, you know,
understandable way. Now the other

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interesting thing there's a tipping point
on vaccines right? If we vaccinate

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75% of the people, and we
needed to vaccinate 80% of the

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people, but guess what? It's not gonna
work. The 25% who don't get

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vaccinated are all gonna get the disease.
But if we'd have vaccinated 81% of the

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people, then the 19% of the
people who don't get vaccinated, they're

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still gonna be protected, because the
disease isn't gonna spread. 'Kay, so the

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cool thing about this model is it's given
us a tipping point, R0. It's also given us

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a policy, and this policy is interesting in
the sense that like it's of no effect,

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really, vaccination has no effect really,
other than the people you vaccinate,

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really, there's no population level effect
until you get to the threshold. Once you

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pass the threshold, then that next person
to get vaccinated, in a sense, right?,

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makes the whole rest of the society immune
because the disease can't spread. So, step

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back a second. In a diffusion model,
right, where we got that nice sort of S-shaped

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curve, there's no tip. In the SIS model,
there's this R0, which is the tipping

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point, right? So this is value one. So
there's no disease if R0's less than one.

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There's disease if R0's bigger than one.
What we do by vaccinating people is in

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effect reduce R0. So that's the SIS model.
In fact in the context of disease you can

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even think of it, though, in terms of the
spread of information that we talked about.

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And it's an interesting model in the sense that
it does also generate a tipping point. Another

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non-linear model, and it's a model where we
get this, you know, threshold phenomena.

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Less than R0, no, R0 less than 1: no
spread; R0 bigger than 1: everybody gets

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the disease. Okay. Not a happy thought.
But let's move on. Thank you.