Hi folks. So, let's talk a little bit more about
getting solutions in the SIS model. So, as we had talked about, in terms of
finding the steady state, we have an equation which relates the fraction of
people that you're going to randomly meet who are infected.
To an expression which involves the degree distribution and parts of that
degree distribution, as well as parameters indicating the relative
frequency of infection compared to recovery.
So what we need to do, is see what this H looks like.
And then figure out what the steady state solution looks like.
So again, as we pointed out, H is going to be a function where, if we're
finding we want to find thetas that equal H of theta then basically if we've got a
situation where H prime at 0 is less than 1 and it's a concave function.
There's going to be no steady state and otherwise, we'll find a positive steady
state. So let's take a look at this function in
a little more detail. So we want to take derivatives of this
with respect to theta to figure out what its shape looks like.
And if we take the first derivative of this with respect to theta that's quite
easy. And in fact we'll find that this
expression is greater than 0 and therefore we we end up with a, an
increasing function. So you can take the derivative here.
And verify that this is the expression you get, and indeed, what we end up with
is a positive function. and then if we take the second derivative
of this function then we get a, an expression, which is less then 0, so here
I'm not going to explicitly solve for the derivative.
You can go through and, and take those derivatives.
it's relatively straight forward, these are just polynomial functions.
Take the derivatives, the first derivative is positive, second derivative
is negative. and so here, we've got a function which
is going to be increasing and strictly concave, and that's going to tell us that
the picture that I was showing you before is an accurate picture.
And then the question is, where is this going to hit?
and have an intersection. So are we in a situation where if H prime
at 0 is bigger than 1, then initially, it's going above the 45 degree line
again, it's given as strictly concave. Either we're going to have a steady state
all the way up at 1, so we'll end up hitting up here, or it'll cross somewhere
and have a, a non-zero steady state. Or if H prime 0 is less than 1, then
basically, there's no possibility of sustaining a, an infection, the only
theta solution's going to be 0. And that's going to be in a situation
basically where the degree distribution is putting weight on very low degrees,
and the lambda's fairly low, so you don't have much infection that's going to be a
situation where H prime has a fairly low derivative.
So if we look at H prime at 0, we can plug in 0 for theta and then see what
this thing looks like. And if we calculate H prime at 0, it
looks out to be lambda e of d squared over e of d, so it's looking at the
relative expectation of the square of the degree compared to the expectation of the
degree, and weighting that by lambda, where we recall lambda's looking at the
relative infection rate compared to the recovery rate.
So we have those expressions and so the theorem then we get that the conditions
for a steady state process of of the SIS model to have a non-zero steady state.
Is going to be if and only if, this land is large enough, and what it needs to be
larger than is e of d compared to e of d squared.
Okay, so you need the infection recovery rate to be high enough relative to the
average degree divided by a second moment of, roughly think of variance there, so
large enough lambda is, is going to give us a steady state that's non-zero.
Now the interesting thing here is increasing the variance, if you do a
mean-preserving spread. So you increase the variance.
That makes it easier to satisfy this equation, right?
So in a situation where we keep E-d constant and increase E of d squared,
then we're going to be in a situation where basically, what we're doing is
spreading out the degrees. But that makes higher degree nodes, which
are going to have. serve as hubs and be conduits for
infection and that aids in, in the spread of, of the infection.
And allows us to have a steady state that's non-zero.
So, if we think about this condition, then we can plug in what we know for
various different models. And for regular network, what does this
turn out to be? Well in a regular network, everybody has
the same degree. So the expected degree is just whatever
that degree is. the expected degree squared is just the
expected degree of squared. So, in this case for regular network, E
of d squared, is just equal to E of d. Everybody has the same squared.
So, in that case then you just need lambda to be bigger than 1 over the
expectation. So, in that case, the larger the expected
degree, the easier it is to satisfy this, so that, that makes sense, and, and also
the larger lambda obviously the easier it is to sustain a positive steady state.
For an Erdős–Rényi random network if you work through what E of d is and E of d
squared are for, a Posone random network, then you end up with a situation where
the e of d squared is just equal to e of d times 1 plus e of d, so in that, in
that model, then we end up with in this case, lambda with 1 over 1 plus the
expanded degree. If you work in a power-law network.
so if we have, say for instance we work with one where the density function looks
like c times d to the minus gamma. Then what we end up with is if you do a
calculation say, integrate that and, and look for the variance e of d squared
actually becomes infinite. And if that becomes infinite, then this
whole expression becomes 0, and so we end up with lambda greater than 0.
And so, it's you, you basically always have a non-zero steady state.
And what's happening here is, in a power-law network, at least in the, in
sort of the limit, if you have a very large network, you're going to have very,
very large degree nodes. They're going to interact and, and always
become infected, and carry the infection through the society.
So, in that setting you end up putting weight on the tails.
And sufficient weight on the tails that you always sustained an, an infection.
Now, if you, you, you , if you do a power-law where you actually truncate the
distribution and have some maximum degree, then you won't quite find this.
But you'll find that that that expression converges to 0 as you let that, whatever
the maximum degree you allow in the society, to go to infinity.
So, so in the limit you always have a non-zero steady state in that model.
And so, basically what we find is the, the presence of hub kinds of nodes helps
substantially in sustaining non-zero steady states.
So the idea here, is these high degree nodes are more prone to infection.
They serve as conduits. Higher variance allows more such nodes,
and that enables infection, and we see that directly in the theorem.
So this is one of the kinds of insights that comes out of the SIS model.
Which is a useful insight and, and made explicit in this particular model.
And it also then allows us to compare degree distributions, showing that if you
have the same mean but you're increasing the e of d squared, then it's easier to
satisfy these conditions. Okay, so, so that shows some insights
that we get out of the SIS model. next will take a little more, a close
look. So what this did, is allow us to know
when it is that we get a non-zero steady-state.
We can also ask questions about how large that steady state is and, and whether we
can do comparative statics in that. And that's not going to be exactly the
same kind of answer as just when there exists one, which was just looking at
that derivative at 0. And more generally we can, we can go
through and try and solve this model, and say something about, what's the average
infection rate in the society?