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SIR Model of an Epidemic

December 28, 2019


>>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

  • Reply Mathemactis help June 12, 2019 at 3:50 pm

    thanks for this helpful video

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