So now we understand the diffusion model, we want to move on to something called the
SIS model. Now the SIS model looks pretty much exactly the same as the diffusion
model, exact for now we allow for the possibility that after someone's become
infected, they can move back into the susceptible pool. So they can become
recovered in a sense, but then they're susceptible again. So this would be
for something like the flu, where after you're cured of the flu, the flu mutates
and you can become infected again. It wouldn't work for something like measles
where once you've had the measles, you're no longer gonna get the measles. So here's
how our model looks. It looks exactly like our model before. There's some number of
people that have it at time t+1, and that's the number they have it at time t,
plus the new people who get it. And this depends on somebody
who has it meeting somebody who doesn't have it, right. And this is the
transmission rate and this is the number of contacts. But all we do to this
is we subtract off this minus aWt. Well, what is this? These are the people who
become cured. Those who no longer have the disease and they're gonna go back into
this pile (N). Maybe people who could get the disease again. So I would think it's
like throwing in a rate of people getting better. Well now here's
where this model is more interesting because in the diffusion model something gets
spread. So it starts out slow, it goes faster, then it tails off.
But what happens here is that while people are getting sick, at some other
(a), they are getting better. And if people get better faster than
people get sick then what's gonna happen? The disease isn't going to spread. So this
model's gonna produce a tipping point. Let's simplify things a little bit.
If we rearrange terms, what we get is an equation that looks like this. Now how did
I get this equation? All I did was I pulled this Wt term out and I
crossed out some N's. Let me show you how that works.
Let's just focus on this part. I've got N c tau times W over N times N minus W over N
minus aW. Let's cross out
this N with this N. These N's go away. And then I've got a W
here and a W here. So I'm gonna just pull out that W. And then I've got c tau. N
minus W over N minus a. Alright? That's what I've got. And if I look over here,
that's what I've gotta get. c tau N minus W over N minus a. So this is just a simple
application. So when you run models is useful to be good at algebra.
If you can do lots of algebra, you can simplify things. Well, why do you want to
simplify things? Suppose we're in the early stage of a disease. That means Wt
is really small. That means that N minus Wt over N will be really close
to 1, because only a very small percentage of people in the population
have the disease. If I look at this thing, I can see Wt plus
1 is really equal to Wt. Wt plus 1 is equal to Wt plus Wt times c tau minus a.
This thing will spread if c tau minus a is positive and it won't
spread if c tau minus a is negative. So this is gonna come down to: is c tau minus
a bigger than zero? Or is c tau bigger than a? Or another way to write this
is: Is c tau over a bigger than one? All right, and this leads to what's called the
basic reproduction number. We let R0 equal c tau divided by a. If c tau divided
by a is bigger than one, the disease spreads, right? Because that means that Wt
plus one equals Wt plus something positive. Right? If R0's less than one,
right, in other words, if c tau over a is less than one, then Wt plus one equals Wt plus
something negative and the disease dies off. So this R0 is called the basic
reproduction number and what it basically tells you is: Does the disease spread? But
notice here, we've got at tip, right? R0 less than one, no disease
spread. R0 bigger than one, disease spreads. So, let's take some real
diseases. Diseases like measles, mumps, the flu. The R0s are fifteen, five and
three. That's why these are real diseases. There might be a ton of diseases out there
that have R0s less than one and we're never gonna hear about them. Why? Because they don't
spread. Now again, for things like measles and the mumps, once you get them you don't
fall back in the population. So, there's a model you use their called the SIR
model, very similar to the SIS model. But for the flu, right, you fall back into
the pool, so you can think of it like this SIS model. Now it's interesting here,
from these basic reproduction numbers we can say, you know, that past the tipping
point, the disease is gonna spread. But let's think about this figure. Why do we
construct models? Well, bunch of reasons, right? But one reason is to design
policies. You could say, how do we stop these things from spreading? Well, one
obvious answer is vaccines. Well, then my question is, how many people do you have
to vaccine [sic]? Well, turns out the model will tell us. So let's think about it this way,
right? There's, let V be the percentage of people that you vaccinate. So there's this
basic reproduction number R0. That's like the rate in which things spreads
through the population. Well if I vaccinate some percentage of the
population, that's just gonna to reduce, right, the basic reproduction number by
that fraction. So if half the people were vaccinated, R0 is effectively divided by
two. Right? And if 75 percent of the people are vaccinated, R0's gonna be
divided in effect by four. Right? So you've only got one fourth of the
population. So the question is, how many people do you have to vaccinate as a
function of R0? Well, in some sense if we vaccinate like V people, it's like we've
got a new r0. Right? Little r0, which is just the big R0 times the percentage of
people in the population that aren't vaccinated. So what we can do is we can
say, we want R0, big R0 times one minus V to equal one. Actually it's going to be
less than one, right? Well we can multiply this out, we get R0 minus R0 V, right? Has
gotta be less than one, right? So we can bring this over together so I'm going to
get R0 minus one. We need to be less than R0 times V, so that means we need V to be
bigger than one minus one over R0, right? So what we get is, we get this equation
that says: This is how many people we need to vaccinate. So let's go back. Remember
the measles were fifteen, the mumps were five. How many people do we need to
vaccinate to prevent the measles from spreading? Well that's just one minus one
over fifteen, which equals fourteen fifteenths. So we need to vaccinate
fourteen fifteenths of the population against measles if we want the measles not
to spread. For the mumps, right?, we need one over one-fifth. Right, we only
need to vaccinate 80 percent of the population. Well here's why the model's so
useful. Again, this model's isn't exact in working out all sort of things like
networks, and changes in contract, contact structures, and different locations and
stuff like that, but still, what this tells us is, which is really important, is
depending on how variant the disease is, you have to change how many people you
vaccinate in a pretty, you know, understandable way. Now the other
interesting thing there's a tipping point on vaccines right? If we vaccinate
75% of the people, and we needed to vaccinate 80% of the
people, but guess what? It's not gonna work. The 25% who don't get
vaccinated are all gonna get the disease. But if we'd have vaccinated 81% of the
people, then the 19% of the people who don't get vaccinated, they're
still gonna be protected, because the disease isn't gonna spread. 'Kay, so the
cool thing about this model is it's given us a tipping point, R0. It's also given us
a policy, and this policy is interesting in the sense that like it's of no effect,
really, vaccination has no effect really, other than the people you vaccinate,
really, there's no population level effect until you get to the threshold. Once you
pass the threshold, then that next person to get vaccinated, in a sense, right?,
makes the whole rest of the society immune because the disease can't spread. So, step
back a second. In a diffusion model, right, where we got that nice sort of S-shaped
curve, there's no tip. In the SIS model, there's this R0, which is the tipping
point, right? So this is value one. So there's no disease if R0's less than one.
There's disease if R0's bigger than one. What we do by vaccinating people is in
effect reduce R0. So that's the SIS model. In fact in the context of disease you can
even think of it, though, in terms of the spread of information that we talked about.
And it's an interesting model in the sense that it does also generate a tipping point. Another
non-linear model, and it's a model where we get this, you know, threshold phenomena.
Less than R0, no, R0 less than 1: no spread; R0 bigger than 1: everybody gets
the disease. Okay. Not a happy thought. But let's move on. Thank you.