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Hi folks. So, let's talk a little bit more about

Â getting solutions in the SIS model. So, as we had talked about, in terms of

Â finding the steady state, we have an equation which relates the fraction of

Â people that you're going to randomly meet who are infected.

Â To an expression which involves the degree distribution and parts of that

Â degree distribution, as well as parameters indicating the relative

Â frequency of infection compared to recovery.

Â So what we need to do, is see what this H looks like.

Â And then figure out what the steady state solution looks like.

Â So again, as we pointed out, H is going to be a function where, if we're

Â finding we want to find thetas that equal H of theta then basically if we've got a

Â situation where H prime at 0 is less than 1 and it's a concave function.

Â There's going to be no steady state and otherwise, we'll find a positive steady

Â state. So let's take a look at this function in

Â a little more detail. So we want to take derivatives of this

Â with respect to theta to figure out what its shape looks like.

Â And if we take the first derivative of this with respect to theta that's quite

Â easy. And in fact we'll find that this

Â expression is greater than 0 and therefore we we end up with a, an

Â increasing function. So you can take the derivative here.

Â And verify that this is the expression you get, and indeed, what we end up with

Â is a positive function. and then if we take the second derivative

Â of this function then we get a, an expression, which is less then 0, so here

Â I'm not going to explicitly solve for the derivative.

Â You can go through and, and take those derivatives.

Â 2:34

and have an intersection. So are we in a situation where if H prime

Â at 0 is bigger than 1, then initially, it's going above the 45 degree line

Â again, it's given as strictly concave. Either we're going to have a steady state

Â all the way up at 1, so we'll end up hitting up here, or it'll cross somewhere

Â and have a, a non-zero steady state. Or if H prime 0 is less than 1, then

Â basically, there's no possibility of sustaining a, an infection, the only

Â theta solution's going to be 0. And that's going to be in a situation

Â basically where the degree distribution is putting weight on very low degrees,

Â and the lambda's fairly low, so you don't have much infection that's going to be a

Â situation where H prime has a fairly low derivative.

Â So if we look at H prime at 0, we can plug in 0 for theta and then see what

Â this thing looks like. And if we calculate H prime at 0, it

Â looks out to be lambda e of d squared over e of d, so it's looking at the

Â relative expectation of the square of the degree compared to the expectation of the

Â degree, and weighting that by lambda, where we recall lambda's looking at the

Â relative infection rate compared to the recovery rate.

Â So we have those expressions and so the theorem then we get that the conditions

Â for a steady state process of of the SIS model to have a non-zero steady state.

Â Is going to be if and only if, this land is large enough, and what it needs to be

Â larger than is e of d compared to e of d squared.

Â Okay, so you need the infection recovery rate to be high enough relative to the

Â average degree divided by a second moment of, roughly think of variance there, so

Â large enough lambda is, is going to give us a steady state that's non-zero.

Â Now the interesting thing here is increasing the variance, if you do a

Â mean-preserving spread. So you increase the variance.

Â That makes it easier to satisfy this equation, right?

Â So in a situation where we keep E-d constant and increase E of d squared,

Â then we're going to be in a situation where basically, what we're doing is

Â spreading out the degrees. But that makes higher degree nodes, which

Â are going to have. serve as hubs and be conduits for

Â infection and that aids in, in the spread of, of the infection.

Â And allows us to have a steady state that's non-zero.

Â 5:17

So, if we think about this condition, then we can plug in what we know for

Â various different models. And for regular network, what does this

Â turn out to be? Well in a regular network, everybody has

Â the same degree. So the expected degree is just whatever

Â that degree is. the expected degree squared is just the

Â expected degree of squared. So, in this case for regular network, E

Â of d squared, is just equal to E of d. Everybody has the same squared.

Â So, in that case then you just need lambda to be bigger than 1 over the

Â expectation. So, in that case, the larger the expected

Â degree, the easier it is to satisfy this, so that, that makes sense, and, and also

Â the larger lambda obviously the easier it is to sustain a positive steady state.

Â For an ErdÅ‘sâ€“RÃ©nyi random network if you work through what E of d is and E of d

Â squared are for, a Posone random network, then you end up with a situation where

Â the e of d squared is just equal to e of d times 1 plus e of d, so in that, in

Â that model, then we end up with in this case, lambda with 1 over 1 plus the

Â expanded degree. If you work in a power-law network.

Â so if we have, say for instance we work with one where the density function looks

Â like c times d to the minus gamma. Then what we end up with is if you do a

Â calculation say, integrate that and, and look for the variance e of d squared

Â actually becomes infinite. And if that becomes infinite, then this

Â whole expression becomes 0, and so we end up with lambda greater than 0.

Â And so, it's you, you basically always have a non-zero steady state.

Â And what's happening here is, in a power-law network, at least in the, in

Â sort of the limit, if you have a very large network, you're going to have very,

Â very large degree nodes. They're going to interact and, and always

Â become infected, and carry the infection through the society.

Â So, in that setting you end up putting weight on the tails.

Â And sufficient weight on the tails that you always sustained an, an infection.

Â Now, if you, you, you , if you do a power-law where you actually truncate the

Â distribution and have some maximum degree, then you won't quite find this.

Â But you'll find that that that expression converges to 0 as you let that, whatever

Â the maximum degree you allow in the society, to go to infinity.

Â So, so in the limit you always have a non-zero steady state in that model.

Â And so, basically what we find is the, the presence of hub kinds of nodes helps

Â substantially in sustaining non-zero steady states.

Â So the idea here, is these high degree nodes are more prone to infection.

Â They serve as conduits. Higher variance allows more such nodes,

Â and that enables infection, and we see that directly in the theorem.

Â So this is one of the kinds of insights that comes out of the SIS model.

Â Which is a useful insight and, and made explicit in this particular model.

Â And it also then allows us to compare degree distributions, showing that if you

Â have the same mean but you're increasing the e of d squared, then it's easier to

Â satisfy these conditions. Okay, so, so that shows some insights

Â that we get out of the SIS model. next will take a little more, a close

Â look. So what this did, is allow us to know

Â when it is that we get a non-zero steady-state.

Â We can also ask questions about how large that steady state is and, and whether we

Â can do comparative statics in that. And that's not going to be exactly the

Â same kind of answer as just when there exists one, which was just looking at

Â that derivative at 0. And more generally we can, we can go

Â through and try and solve this model, and say something about, what's the average

Â infection rate in the society?

Â