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And then I, infected, and then susceptible again.

Â So, the idea here is you can recover. So you, you can catch something you

Â become infected so, you're, you're susceptible, you could catch it.

Â You get it, then you recover and this is something which you catch overtime, so it

Â might me something that I erase a, a virus from my computer, I'm susceptible

Â again. And I can catch em when new one comes, I

Â catch it again, I erase it and so forth. So, I go back and forth from this process

Â of you know, realizing that I have a virus, getting rid of it, and then

Â catching it again later in time. Okay, so the, the key thing is, is nodes

Â are going to move back and forth over time and you know, you can think of this

Â as, as I might be changing my mind over time.

Â and various things, but we'll look at the basics of it.

Â [COUGH] So, nodes are in these two states, infected or susceptible.

Â The probability that you get infected in the simplest version of this model, is

Â proportional to the number of infected neighbors, with some rate, let's say v

Â great than 0. and we'll add in a spontaneous epsilon so

Â that you can catch things, as in the bass model.

Â And then you get well in any period with, at some rate, delta.

Â So, this is like the bass model, except here you can actually reverse yourself

Â and get well, and that's going to happen with a, a rate delta greater than 0.

Â And let's let rho be the percent of the population that's infected any point in

Â time. So, what then I want to do, is make

Â predictions about rho as a function of the network and, and these other

Â parameters of the model, okay. So, what we're going to do, is start with

Â a simple version where all the in, individuals players agents in this

Â society, the node are going to mix with even probabilities.

Â So, you random meet one person per unit of time, and that's just going to give us

Â a large Markov chain, and we can do calculations on that.

Â And the steady state distribution is just going to be one in which the, the change

Â of this infection parameter rho with respect to time is 0.

Â So, the simplest version of this model is one where there's not actually an

Â explicit network structure, it's just a completely random process.

Â And this looks a lot like the bass model in its basic form, and then we'll bring

Â in network structure on top of this in just a few minutes.

Â So, let's start with the simplest version.

Â so what's the change in the infected population over time?

Â Well, you can only become infected if you're not infected yet, so you're

Â susceptible. So, this is the susceptible size of the

Â population. Then you catch it from a given individual

Â with v times rho. And epsilon is this spontaneous rate.

Â So, this looks lot like the bass model did.

Â Basically the same function form is the bass model.

Â But we're also going to do this. We're going to have people re, recovering

Â over time. And so we'll look for steady state so,

Â out of those who are infected. They recover at some rate delta.

Â And so, all put together, you're gaining new infection at this rate, and losing

Â infection at this rate. And in order for this to be in steady

Â state, these two things are going to have to balance.

Â The new infection rate's going to have to balance against the, the number of people

Â who are recovering for a period of time. And so if you solve this equation then

Â you get an expression for what rho looks like as a function of the rest of the

Â parameters. in, in this setting.

Â Okay. So, there we've got a simple equation and

Â a simple solution. And now we're looking for a steady state

Â and what we've done is we've enriched this bass model essentially.

Â To have a recovery part which then allows for a steady state distribution, which is

Â going to be different from everybody becoming infected.

Â So, if we if we let epsilon go to 0 and then we solve this basically we end up

Â with two solutions. One is that nobody's infected, nobody

Â gets infected. And then the other one, the more

Â interesting one is that rho is equal to 1 minus delta over v,

Â So, if this turns out to be greater than 0.

Â So, if, if delta is bigger than v, basically what does that mean?

Â That means that the people recover so fast that this thing will never really

Â take root. But if delta's smaller than v, so you can

Â catch things faster then you can recover from them, then rho can be positive.

Â And basically the smaller delta is and larger v is, the larger rho is going to

Â be. So, rho is increasing in v and decreasing

Â delta. and it only has this positive, solution

Â as long as, delta is less than v. Right.

Â So, so we have this simple solution and, you know, very, very simple steady state

Â here. So this now hasn't brought in the network

Â structure at all. So, this is like the bass model, but now

Â with the recovery rate. And we end up with a solution here, which

Â makes sense as, as, as long as, delta is less than v.

Â Okay. So, we've, we've got, an infection at

Â least when, delta is less than v, where it's going to stay at some level for low

Â recovery rates, which can lead to large infections.

Â and, what we haven't brought in yet is where's the network, right?

Â So, this is uniformly at random interaction, we're missing the

Â heterogeneity degree, we're missing local patterns.

Â And what we're going to do, is, is we're going to start by just bringing this in.

Â And bringing in local patterns and explicit network structures is going to

Â be a lot more difficult without doing simulation.

Â And so what we'll start with is, is just taking a look at how we might bring in

Â the fact that some people are going to have more interactions per unit time than

Â other individuals, okay. And so exploring the, the dependence of

Â this one, the degree of distribution is what we're going to do is start by having

Â a random matching process. Where each different individual might

Â have a different degree, and their degree is just going to tell you how many

Â matches per unit of time they're going to have.

Â Okay? And what we're going to keep track of now

Â is the fraction of nodes not just overall which are infected, but also as a

Â function of a degree. So, it might be the people that have

Â three interactions per unit of time have a higher infection rate than people who

Â have two interactions per unit of time and so forth, okay.

Â And another thing we're going to keep track of is,

Â If I'm meeting a random person in the population.

Â So, I, each period, I'm meeting some number of people, my di.

Â So, say this is four, I'm going to meet four people per unit of time.

Â what's the chance that any one of those four people is infected?

Â And theta's going to be that fraction, okay?

Â Now what's going to be important, is the fraction of people over all that might

Â have something in their population, is not going to be the same as the fraction

Â of people I meet. Because I'm more likely to meet people

Â who are meeting lots of people. So, some people have lots of

Â interactions. Those are the people I'm more likely to

Â meet. Those are also the people who are more

Â likely to be infected. Okay?

Â So, so that's the process that's going on.

Â Okay, so how are we going to deal with this?

Â let's deal with it, again this is this random matching process.

Â So, let's let P of d be our degree distribution.

Â So, this is the fraction of nodes that have degree d.

Â And when I think about what's the probability that I'm going to meet

Â somebody, in terms of this random process, where we're all randomly

Â matched. I'm much more likely to meet somebody

Â with high degree. And in particular given that the high

Â degree people if somebody has ten meetings per unit of time they're

Â going to have to meet ten people. Somebody that has five meetings per unit

Â of time is only going to meet five people.

Â The person with ten is going to be twice as likely to be met by somebody as the

Â person with five meetings. So, the people with more meetings are

Â going to be easier to find, and the likelihood of meeting a node of degree d

Â is going to be directly proportional. To their degree compared to the average

Â degree. So we look at the fraction of those

Â people in the population but we have to re-weight that by what's their relative

Â degree compared to the average degree in a population.

Â Because that's going to determine how many meetings they have and how easy it

Â is to find them, when you're bumping into people in the population.

Â Okay, so that's an important thing, and that's a critical thing for understanding

Â contagion processes more generally. We've already seen it once earlier in the

Â course, and you know, this is important in, in trying to understand that fact of

Â the, the operation of this SIS model. Okay.

Â So, if we want to calculate the fraction of infected people I'm likely to meet,

Â well, this is the likelihood that I'm going to meet somebody of degree d.

Â This is how likely they are to be infected, and then we're just going to

Â sum across d's, and that gives us a theta.

Â Okay, so we have an expression now for theta, and we're going to have you solve

Â for this expression and see what, what it gives us.

Â So, this is the fraction of infected neighbors, random partners.

Â If we look at steady states, steady states are going to tell us for each

Â difference degree, we have to have the change over time of the infection rate of

Â different degrees all going, being 0. So, what we end up with is the infection

Â rate for each different type being 0. And what we know, what is those infection

Â rates look like for different types, and so we can then set that equal to 0.

Â What does that infection rate look like for different types?

Â 12:02

And the chance they get affected by one of them is, is v.

Â So, this is the, the rate at which there going to gain infections, and then they

Â get better at a rate delta. So they recover.

Â They, they get rid of their computer virus.

Â they recover from whatever cold or they had.

Â so, so here we've got a situation where we've got an expression now that involved

Â our theta and we can solve this for each rho of d.

Â This has to be equal to 0. So, a steady state sets us equal to 0.

Â And basically that tells us that the rho of d for any given d, is going to be

Â proportional to this expression over here, lambda theta d over lambda theta d

Â plus 1. Where lambda is v over delta.

Â So, whats the relative rate at which you get infected compared to the rate at

Â which you get better? so generally for this infection to take

Â hold, this expression is going to be something which is bigger than 1.

Â And so here we've got something which sort of captures the, well, it doesn't

Â necessarily have to be bigger than 1 in instances where we have two different

Â degree distributions. But so this, this is going to be a very

Â useful parameter, and this happens because when we solve through its only

Â the relative rates of v and d that matter, and not the absolute rate.

Â So, if we double both of these, and the expression is already solved we'll get

Â the same solution, so, These are scaling together and it's, it's

Â relative rates at which you get infected and recover that matter, not the absolute

Â rates. Okay.

Â So, solving this equation we've got this expression now, for rho of d.

Â We can plug that back in to our expression for theta.

Â And then we end up with theta equals a function of theta, which now depends only

Â on the primitives of what's the degree distribution.

Â what's the infection relative infection of compared to recovery rate in this.

Â and those are the only expressions that n are in.

Â So, we've got expected degrees and so forth, and now we've gotta solve this for

Â theta. Okay, so we've got theta as a function of

Â h of theta which is where h of theta is, this big expression over here.

Â so we've got some function of theta. and basically we're, we're going to look

Â for a fixed point of this expression. Okay?

Â So, now we've boiled everything down, this model to a simple equation that one

Â can solve. Okay.

Â So, solving this equation we, we know that theta has, is some expression which

Â is a function of the Ps and so forth. generally this isn't going to be easy to

Â solve. We can solve it by simulation, it's a

Â nonlinear equation. It depends on the expected degrees and

Â the full degree distribution and the lambdas.

Â but what we can do, is do some fairly easy comparative statics.

Â And say, let's suppose that we increase the probability of higher degree nodes,

Â what's that going to do to theta? what happens as we change the expected

Â degree, just what happens as we change lambda.

Â So, we can do different comparative statics and begin to understand how these

Â things work. Okay, so in particular we want to ask

Â what H of theta looks like, and how it depends on these different.

Â Parameters, the degree distribution the expected degree and so forth, so solving

Â this would depend on making sense out of these.

Â So, let me just go through how this looks in terms of the solutions.

Â So, basically when we go back, to this we're trying to, to look for thetas,

Â which solve this equation. We want theta equal to H of theta.

Â H of theta is going to be an increasing in concave function, and we'll talk about

Â that in little bit. So, H of theta is increasing in theta and

Â its concave function. And in particular when we look at looking

Â for thetas and H of theta, that intersect.

Â One possibility is is theta 0, so if nobody's infected, then nobody gets

Â infected in this model. And so one steady state is no infection,

Â and once you eradicate something, it just doesn't come back.

Â Another possible steady state is a positive one, but it's going to depend on

Â what this H function looks like. So, it could be that this H function is

Â so shallow, that there is no positive solution.

Â It could be the steeper function and that concavity will give us a positive

Â solution to it. So, understanding whether this works, in

Â terms of having a, a nonzero steady state in this, is going to depend on the

Â properties of this H function. Which depend on the dis, degree

Â distribution the relative infection rate and so forth.

Â So, even though there's a little bit of technicality here, the intuition's are

Â fairly simple. basically more degree higher degree

Â nodes, more interactions, higher infection rates in terms of the v

Â compared to delta. Are going to lead to higher H's, which

Â are going to lead to higher steady states, and more infection in a

Â population, okay? So, we're going to take a look at that in

Â some detail next. that'll be our next look in, in, in more

Â detail at the diffusion process. So, we'll solve out the SIS model for,

Â for explicit expressions in different settings.

Â And then look at what we can say about comparative statics.

Â