>>Instructor: The SIR

Model is a simple model for looking at epidemic

spread of an infection across a population. It’s termed SIR because, well, it’s a three state dynamical system. The susceptible population, the infected population, and the recovered population. And as the arrows suggest

in this dynamical system, the states transition from

susceptible to infected when someone becomes

infected with the disease. And then after some period of time an infected person will

transition to the recovered state, and that’s what this arrow represents. And so this is a three

state dynamical system. It’s not linear. It’s actually non-linear. But let’s think about how

we would go about modeling this basic system. Well because it’s a three state system we have to write our three

differential equations. We’re going to have one for S, one for I, and one for R. We’ll start out with the

transition from I to R simply because that’s a much

easier idea to understand than the S to I transition. The number of people from

I that transition to R will be proportional to how many people are in that current state. And so if we were to

write out the equation for the number of people recovering, number recovering, that would look like something where there’s a constant parameter, some proportionality constant, times I, the number of people

in the infected population. It’s worth noting that in this model there’s no births or deaths, right? The population stays static. That means if N represents

the total number of people in the population, that’s equal to S plus I plus R. So this is number people. And so the number of people

that recover at any given time, for any given time step,

is given by gamma times I. What that means is that

for any instantaneous time the change in I is going to be influenced and going to be decreased by this amount, because that represents

the number of people that are leaving I and going into R. So this is going to be minus gamma I. I think I’m going to put

this a little further over to give us more room. Minus gamma I. And similarly if people are exiting I, if the rate of change, if I is loosing people by gamma I amount, then R must gain the same amount of people in order for this equation to hold true, for the number of people in

the states of the system, across the states, to be conserved. That means the rate of change

of R must be adding gamma I. And we’ll just leave it

here to keep it symmetrical. So my gamma I people leave I and enter R, so minus that plus this. That’s the change that’s

represented as people recover from the infection. Now let’s look at this transition here, the number of people that go

from susceptible to infected. Well this depends not

on a single variable. Unlike the situation here where the number of people

transitioning out of I into R is only dependent on one variable, the I, the number of people in I. This is a first order parameter, right? A first order kinetic term. And if you have modeled

any differential equations for molecules this would

be the reaction rate where you only have one molecule depending on the rate of the reaction. But that is not the case here. To go from a susceptible

person to an infected person, you have two things that are interacting. You have to have an infected person bumping into a susceptible

person transmitting that disease from the infected to the susceptible. That we’re going to say

is the number of people that are acquiring the disease. So number newly infected. And we’re going to write

that as a parameter. It’s going to be some

proportional concept as always. So there’s going to be

some scaling factor, we’re going to label that as beta, and that’s just historical

reasons, that’s what it’s called. And then we’re going to

recognize that this is a function of the population of susceptible

people and infected people. The more susceptible people or the more infected people there are, the higher the number of new

infections that will occur. So this is just a second

order interaction term where you multiply these

two states together. And this represents that interaction. Now if you’re keeping an

eye on the dimensions here, these rates of changes are all in people because we have S plus I plus R. These are just number

of people that change, leaving or exiting. If this number is positive, people are joining the susceptible state. If this number is negative, people are leaving the susceptible state. Same with I and R. If this is positive, the new, the infected group is getting larger. And if this is negative then the infected population

is getting smaller. But if you look at these dimensions here, beta’s just a constant

’cause it has no dimensions, S is a number of people,

and I is a number of people. So this expression is in people squared. And that is not the

units of these equations. So in order to make this make sense, we have to divide this term by N because if you perform the

dimensional analysis here you get people times

people is people squared, divided by people, and

now you’re back to people. So that’s why this

expression is beta SI over N. And so if this is the newly infected, if this is the term for

the newly infected people, guess what. The I population, the I state, is going to grow by that amount. So we’re going to put that over here and say beta SI over N. Similarly if I is growing by that amount, then guess what. The susceptible population must fall by that equally same amount. And this negative term is describes the rate of change for S. That’s it, this is the model. This represents how a single

indexed case individual, or the number of people

in I is one initially, when and everyone else, N minus one, because there’s zero

recovered people in that case. And N minus one people are susceptible, how an infection will spread

through that population to go from susceptible

to infected to recovered. That’s the basics of the SIR model.

## 1 Comment

thanks for this helpful video