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Unlike the other models, in our infection models, we'll see state transition across

more than two states. It's not just flipping from one to the

other states. We're going got see three states, for

example. And they are, variants with even more,

four, five, states. And also, instead of looking at discrete,

how we look at continuous time through the more convenient mathematical

representation of differential equation. We'll first look at population-based

infection models which should have been brought up in the last lecture because it

doesn't involve topology. And then in the next segment of the video

and the last one of this lecture before the advanced material part, we'll look at

the topology's impact together with a case study.

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So the state transition diagrams can be visualized as follows, we'll talk about

three variants of the infection model, starting with the simplest one.

S stands for susceptible, I stands for infected and R stands for

recovered population. Okay?

We'll be looking at also the percentage of the overall population rather than the

absolute values. And in the stage transition diagram we see

that if you are susceptible to an infection disease, this is where that

infection model started, originally, thus these particular choice of words you have

a chance of being infected. And the rate of that happening is

represented by a beta. And if you're infected, then we have a

choice of deciding what to say in the model.

One model, the simpler ones, says that then you just stay infected forever.

Another model says, well, you may actually be able to recover.

Or at least go back to susceptible with a certain rate gamma.

And the third one says, you actually- truly recover and become, immunized from

this infectious disease with a, a certain rate gamma.

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So let's start with the very first model okay.

Just word of caution that the beta and the gamma rates of transition in these

diagrams actually denote diff-, slightly different concepts.

For beta, we're representing the rate of getting infectious disease that's

proportional to the product of the susceptible and infected population.

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And therefore, the equation we'll write down, the French equation, is Ds at time

t. Dt is -beta, S of T and I of T. Again, S

of T, and I of T, either susceptible and infected percentage of the population at

time T, continuous time, so that we can use differential equation and integration

to solve, the model, okay? So, the rate of change of S of T is

basically -beta, one -beta.. Because you're leaving from this subset

both to, to the worst state or the infected state, okay?

And at, as I mentioned, that the beta is the rate based on the product of how many

people are susceptible, how many people are infected in the product form in the

assumption of this model. We can, of course, also write down,

What happens to the I population? The rate of change of that is a positive

beta, because we're moving into that state.

S of T, I of T. Now two obvious observations, first.

We normalize the overall population, so at any time S of T and I of T are sum up to

one, or 100%. And therefore naturally D-S-D-T plus

D-I-D-T is zero. That's why these two are exactly just

negative of each other. Second observation is that as you can

visually see already without any algebraic representation, this is going to suck all

the population eventually will be 100% infected.

Well that's why this model is too simplistic, but it's a good starting

point. And indeed we can solve this differential

equation. And skipping the solution differentiate

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equation which is now the subject of this course.

And you can easily verify by differentiating the answer, this correct

answer. We have, the following result.

Okay? The infected population I of T follows the

following shape and initialization, I of zero.

And exponential growth into the beta T, okay?

And then divided by initialization condition times E to the beta T.

And this is what people call a sigmoid function.

In particular, represented as a logistic growth is actually coming out of very

standard economic modeling of these kind of dynamic systems.

And we can plot it on a graph over time. T, both S and I.

The S population clearly goes down to zero, because it has to add up with I to

one. And the I population started out from some

initialization condition very close to zero, all the way, up to, 100%.

Okay? And this is the inflection point, below

which is concave, above which is convex. Above which is concave.

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Now, we'll say that, model doesn't quite make sense.

Right? If you are infected maybe you will be able

to come back to this susceptible state. Rather than staying in infected state

forever. And once you provide.

This path of state transition with a right gamma as you can guess, that at

equilibrium of time goes infinity, you won't have 100% of people in the I state

because there's always some people moving back.

Just want to look at it clearer where moving in and moving back balance each

other out. And this moving back is really just people

from infected state of being recovered and therefore it only multiply I of T, it does

not multiply S of T. It's not just the susceptible people

coming into contact with infected people. And therefore the overall equation now,

the rate of change of susceptible population has two component.

The first term is people coming back in from the infected state, gamma times I,

minus people getting infected which is beta times the product of S and I because

you need susceptible people to come into contact with infected people.

Now of course you can also write down DI of T, DT the rate of change of that, which

is really just minus the above expression because I and S must add up to 100% all

the time. Okay.

Now against skipping the duration of the differential equation solve or we see that

we again have a closed form expression. For example I of T turns out to B, one

minus gamma over beta, a constant weight in front of this expression, it's a ratio,

C sum constant. That depends on the initial condition I of

zero, S of zero, we just don't care to write them down at this point.

Because it's the shape and the rate of the growth that we would like to highlight.

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So, now, the growth pattern clearly now depends on beta versus gamma.

Okay. And we're going to denote this ratio, beta over gamma, as a sigma.

This is so-called the basic reproduction number.

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Sometimes is denoted by other symbols, With some subtle differences that don't

concern us at this point. Okay?

A standard, a typical trajectory can be visually represented as follows.

Applauding the S population here over time.

And then the infected population. You can see that it turns out, in this

case, it's about 90% infected, Ten percent remains acceptable as the

equilibria. For a particular pair of beta and

[UNKNOWN], and gammas. Now, we'll later come back to this basic

represen-, reproduction and, number gamma, in, the next model.

But if you look at this model, you see that if beta is less than gamma, or if the

sigma is less than one, then infected population will actually go down

exponentially. Because this exponential will be a

negative exponential. Whereas on the other hand, if beta is

bigger than gamma in other words, that sigma is bigger than one, okay, then we

say that the infected curve will go up. But it won't go up to a 100% necessarily.

So this is the case for this. If I have to plot this case, then the I

curve instead of going up like that, will be going down exponentially.

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In another words if you are recov-, if you can recover fast enough, then you won't

even have a rise in the I population. So now we go to the third infect-, the

last infection model of influence. So here instead of just looking at two

states, we have three states now. Okay?? S goes to infected.

Once you're infected that with a certain rate of gamma, you actually can become

truly recovered. Not only you're not infected, you're no

longer susceptible anymore. Got immunization from this recovery

process. And therefore we have to write down three

differential equations. This rate is minus BS times I, just as

before. The contact of susceptible and infected

population with a rate beta. Then, the infected population's rate of

change, is, you can see, visually, beta coming in.

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But then also gamma going out. That part, and the recovered population's

rate of change over time is simply gamma. Times the infected population, time T, is

coming in. Now you can try to solve this [UNKNOWN] of

differential equations. It turns out that we don't have close run

solution to the resulting equations. But we can observe the following

qualitative behaviors. Again, this sigma, basic reproduction

number, beta over gama, plays important role here.

In this particular model is actually sigma times the initial susceptible population

percentage as of zero at the onset of this infection that plays the threshold levels

role. If sigma S of zero is less than equal to

one, then affected population actually decreases.

Okay, the initial value S times the space of reproduction number is not large enough

to cause increase of infected population. That's all good.

However, as in all the pandemics, this will be large enough and then you will see

IT infected population going up, Okay. How high does it go.

Well, that depends on a few factors, including the initial conditions.

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Now S is always a decreasing function in this numerical plot of a particular

example. You see S going down.

Because this is a one-way arrow. Eventually everybody drinks out of S, and

moving into recover. So eventually recovery reaches 100%.

But in between, during the transient, before the equilibrium, you see a large

spike of this infected population, in this case going to two-thirds of the entire

population before it eventually dies down. This is a very typical pandemic infected

population. Hopefully, not as big in terms of absolute

scale as this graph shows. Now, of course, we have assumed there's no

death or birth, in particular there's no mortality rate of people in the infected

state. For deadly diseases then people will die

out of this with the mortality rate is, say one%, then this means the area under

this curve is one percent of people that will die through this transient stage

before eventually people are fully recovered and immunized.

So, so far we have assumed that, the impact is on the entire global population.

So, if somebody is in Europe and you are in South America, you can also get

infected, obviously for certain influence models such as infectious disease itself,

that's hardly realistic. So, we have to now take into account the

impact of topology.