00:00
Hello, in this video,
we will tackle that
all important topic
of modeling.
00:05
It's a complicated topic
and a scary one,
but the basics are pretty easy
to understand.
00:10
In this video, we'll talk about
the SIR and SEIR models.
00:15
The assumptions, and weaknesses,
and strengths
underlying those models.
00:18
We'll talk about the contact rate
and the recovery rate.
00:21
These very important
quantities that we use
to compute
the reproduction number.
00:26
We'll introduce
the idea of Farr's law,
which you might have heard of.
00:29
Farr's law tells us
that the rise of an epidemic
resembles the fall of an epidemic.
00:34
And lastly,
we'll finish off by
talking a little bit about
the difference between
forecasting models
and dynamic models.
00:42
I hope you enjoy.
00:44
So, the basic transmission model
for propagating infectious diseases
is called the SIR model.
Sometimes SEIR.
00:55
S stands for Susceptible,
Infected, and Removed.
00:58
Sometimes the Remove
is called Recovered.
01:01
Either way, it's SIR.
01:02
And we sometimes put
an E in there for Exposed.
01:07
The SIR model was developed
by Ross and Hamer
in the early 20th century.
01:11
It's the foundation for most
of the new epidemic models
that have some more nuances,
and bells, and whistles.
01:19
But the SIR model
has some assumptions.
01:22
Most importantly, that there is
a large closed population,
like the population of a nation
with closed borders.
01:30
This is because the SIR model
bears a strong similarity
to some of the fluid flow models
in engineering and physics.
01:40
Where various compartments
are observed
to have fluids flowing
in between them
within an enclosed body.
01:47
Another assumption
underlying the SIR model
is that no new people are added.
01:52
So, no new kids are born
in this population.
01:56
And no people are immigrating
from other parts of the world.
02:01
And there are no natural deaths.
02:03
So the only people
who die or are removed
from this population do so
because of the disease
being studied.
02:10
And no one is emigrating either.
02:12
They are all staying in
this population.
02:16
We also assume that recovery
from the illness confers immunity.
02:22
So, once you get it,
you can't get it again.
02:25
And we assume that
people in the population are
homogeneously distributed
had engaged in mass action.
02:32
What does that mean?
It means that everyone
in the population
has an equal probability
of interacting
with everybody else
in the population,
which we know is not
how the real world works.
02:43
Most people have a
set number of contacts
and are unlikely to see anyone
beyond those contacts.
02:49
But for the purposes of modeling,
these assumptions can be allowed.
02:56
Initial challenge when thinking
about infectious disease modeling,
is to differentiate between
linear growth and
exponential growth.
03:04
And to my mind,
this is the greatest challenge
for the layperson in understanding,
how infectious diseases
explode out of control?
Exponential growth
has essentially two phases:
the slow, almost linear phase
in the beginning,
and the explosive,
almost vertical phase, later on.
03:24
By the time you're in
the vertical phase,
you need to be investing
an extraordinary
public health infrastructure
to control the situation.
I liken it to investing.
03:36
If there is a bit of stock
that you want to buy,
that we know is going to be
worth a lot of money
because of exponential growth.
03:45
If you invest
a little bit of money early on,
you get a big payout
several months later.
03:52
If you wait later, until the stock
is clearly doing well,
you have to invest a lot more money
to get the same outcome.
04:00
So acting early,
allows you to act less strenuously
when controlling exponential growth
in an explosive disease.
04:10
So, it's important from
a public health perspective,
to act early and
possibly to act hard.
04:16
So early on, a disease growing
with exponential growth
may not seem as
dangerous or as impressive
as a disease growing
with linear growth.
04:26
But there comes a time
when exponential growth
equals then exceeds
the seriousness
the case growth of a
linear growth disease
as shown by these two curves.
04:40
As noted, the SIR model contains
three compartments, essentially.
04:45
They are those who are susceptible,
which is everybody
in the population,
because they've never seen
the disease before.
04:52
They all have an equal probability
of getting the disease
and no one is immune.
04:57
Then there are the infected people.
05:00
So, as the susceptible individuals
are exposed to the infection
and get the disease,
they move into the
infected compartment.
05:10
And gradually,
those who are infected
either recover or they die.
05:15
In either case they are removed
from the compartment.
05:23
So, over time, eventually everybody
in the susceptible category
will enter the infected compartment.
05:30
And over time
everyone who is infected
will either recover or will die.
05:35
That is the assumption
for the SIR model,
the flow of people from
compartment to compartment.
05:42
As I noted,
sometimes we add some additional
compartments among them, the E.
05:48
So, first we have a population
of susceptible people.
05:52
No one has seen the disease before,
no one is immune.
05:55
Then gradually, people become
exposed to the disease.
06:00
And over time, they become
infected and infectious.
06:04
Over time, they become
removed as well.
06:08
The thing about
the SIR and SEIR models
is that they don't account
for several things
like incubation period.
06:16
Or the heterogeneity of
people's interactions,
or the fact that in reality,
not everybody is susceptible.
06:24
Some people are more likely
to get diseases than others.
06:27
So, it's a very simplistic way
of thinking about
population health dynamics.
06:32
But again, a model
doesn't have to be perfect.
06:35
It just has to be useful.
06:39
So in the SIR model,
there are several characteristics,
several mathematical aspects
that we care about.
06:46
There is the function of change
of people
in the susceptible compartment,
right,
that's given by S of T,
S as a function of time.
06:56
We have I, which is the number
of infected individuals,
and I changes over time.
07:01
And R, the number of
removed individuals,
those who either
recover or die.
07:06
And that changes over time.
07:08
We can divide each of these by N.
07:10
The total number of people
in the population
to give this thing
called the fraction,
the susceptible fraction,
the infected fraction,
or the fraction of people
removed from the population.
07:24
At any given time
T however,
the sum of
s(t) + i(t) + r(t) = 1.
07:32
Why is that?
Because we have a closed system.
07:36
There is a set number of people
at all times.
07:39
And if you're not susceptible,
it means you're either infected
or you've been removed.
07:44
If you're not infected,
it means you're still susceptible
or you've been removed.
07:47
And if you're not been removed yet,
it means you're either still
susceptible or infected.
07:52
So, at all given times,
if you live in this population,
you're in one of these
compartments.
08:01
So, some more assumptions
about the ideal SIR model.
08:05
First everybody gets removed,
eventually.
08:08
It's enough time goes by.
08:10
Everybody gets moved
through the machine
and either recover or die.
08:15
The duration of infection is the
same for everybody in this model.
08:19
Clearly, that's not true
in real life.
08:21
But again, a model doesn't
have to be perfect, just useful.
08:25
And once recovered
from infection,
the person is immune and
can no longer infect anyone else.
08:32
They cannot get the disease again.
08:36
Only a fraction of contacts
with the disease
actually cause infection
in these models.
08:41
And as I mentioned,
the system is closed.
08:43
Meaning that the total of those
in the susceptible compartment,
those in the
infection compartment,
and those in the
recovered compartment
does not change.
08:55
So, here's a sample population
of 500 people.
09:00
And the red curve shows us
the number of people in the S,
or Susceptible category.
09:10
As you see, everybody starts out
susceptible
and over time that compartment
dwindles to zero
as they become infected.
09:20
The green curve shows
us the infection curve.
09:23
The number of people in
the infection compartment.
09:27
And that's really the curve
that most people look at
when they look at the SIR model,
because it shows us the path of the
disease through the population.
09:39
So at Time 0,
nobody is infected.
09:41
But as more and more susceptible
people become infected,
that I curve increases,
until finally it starts to diminish
because not enough people
are still susceptible.
09:53
This is a natural infection curve,
by the way.
09:56
There's no vaccine
at work here.
09:58
There's no interventions being apply
at any point in time.
10:01
As a result,
this decline in infection
is caused by people becoming immune
or resistant naturally over time.
10:12
And this blue curve,
that's the recovered curve.
10:15
Zero is how many people have
recovered or removed at Times 0
because no one's got
the disease yet.
10:23
But over time, as more people become
susceptible and become infectious,
more people end up exiting
from the other end of this
either recovering or died
until finally 100% of the population
is in the R compartment.
10:38
Here's the example of an SIR model
made for Spain
in the early days
of the COVID-19 infection.
10:46
So, the red curve shows us
the path of infected individuals.
10:54
It was expected that infections
that is detected infections
would arise up until
the end of April
and then start to decline
thereafter.
11:05
Because the number of
recovered people would be
meeting saturation.
11:10
But if you look at the blue curve,
the death rate was just starting
to climb as well.
11:15
So, that's how they hope
to get a handle on the disease
by putting forth an SIR model
to better understand
the timelines at play.