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PROFESSOR: So now,
let's begin talking
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about the spread of
disease, initially focusing
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on a population of individuals.
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So the type of modeling,
which we'll be discussing,
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is sometimes referred to
as compartmental modeling
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because a population is divided
into different compartments.
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Such as, for example, a set
of susceptible people that
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have yet not been infected,
a set of infected people,
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and finally a compartment
of recovered people.
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Additional compartments
could be added, for example,
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for different age groups or
sub-populations, for incubation
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prior to infection
by the disease,
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and, for example, death and
separated from recovery.
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This type of modeling based
on population compartments
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was introduced by
Kermack and McKendrick
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in the simplest case of what
is called the SIR model, where
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S is the number of
susceptible, I number infected,
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and R the number of recovered
individuals in a population.
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And the dynamics
take a simple form
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of a set of coupled nonlinear
Ordinary Differential Equations.
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So first of all,
the rate of change
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of the number of
susceptible people
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is minus the transmission
rate beta times S times I
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because SI base refers
to how many pairs
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of susceptible and individual,
or infected persons there are,
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and beta is the transmission
rate per such individual
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pair.
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And then, next, we look at
the dynamics of the number
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of infected persons.
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So that starts by a conversion
from susceptible to infected.
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So beta*S*I is the rate of
producing new infected persons.
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And then, we introduce
another rate, constant gamma,
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which is the removal rate.
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So this could be
people are removed
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from the infected compartment
either by recovering
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or, potentially, we could
lump into that, dying.
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So we finally complete
the balance here
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by writing the number of
recovered, changing as gamma*I.
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So basically, we
have a model here
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with three compartments
and two rate constants.
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Now the important
aspect, really, here
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is the rate of change of the
number of infected individuals.
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So I'll write that
equation over here, which
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is dI/dt equals, and
let's factor out gamma
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and write the prefactor
here on the rate
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as (beta*S/gamma-1)
times gamma*I.
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And if we look now
at early times,
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then the number of
susceptible people
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is approximately equal to the
initial number, at t=0.
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So that's essentially the
size of the entire population,
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typically.
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And so we would then write this
as beta*S0/(gamma-1),
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times gamma*I.
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So now this is just a
formula that at early times
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gives you an
exponential increase.
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So we would then find that I
grows like the initial number
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of infected persons.
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It could be, for
example, just 1 times
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e to this factor here times t.
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And I'm going to write it
in a certain way, which
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is (R_0-1)*gamma*t.
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So this is the early times.
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And then we have put
a little squiggle
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here to indicate that's
the initial growth
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rate or the initial dependence,
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where R_0 is beta*S_0/gamma.
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And this is called the
reproductive number
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of the disease, or
of the epidemic,
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because we can see here
that if R_0 is bigger than 1,
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then we have an
exponential growth
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of the number infected persons.
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So then we essentially
have an epidemic
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starting from an
initial index case,
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or some set of
cases, numbering I_0.
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Of course, if R_0 is less than
1, then we have no epidemic.
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In other words, there may be
an infected person or two,
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but the number will
exponentially decrease,
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and there won't be any growth.
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So the reproductive number
is an important concept
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in epidemiology that comes
directly from these models.
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Related to that is the concept
of herd immunity, which
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is the point where enough
members of the population
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are immune that the
epidemic starts to die
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out and eventually disappear.
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Let's make a plot of the typical
predictions of the SIR model.
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So as a function of time, the
number of susceptible people
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starts at some value
S_0, and it decreases.
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Initially, the number
of infected persons
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starts at some small
number I_0, which
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might even just be one index
case, and as we showed here,
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it exponentially increases.
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The number of
recovered starts at 0
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and increases as
well, with some delay
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given by the recovery time.
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And what we then see is that
as the number of susceptibles
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comes down-- let's look
at this equation here.
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We can write this as
(beta*S-gamma)*I.
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So initially, (beta*S-gamma)
is positive.
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It starts, in fact, at the
value (R_0-1)*gamma.
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But eventually, as the number of
susceptible people comes down,
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there's a certain point
where this factor goes to 0.
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And that would be leading
to dI/dt equal to 0.
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So, in other words, a maximum of
the number of infected people.
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So at some point, there's
going to be a value where
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this will turn around.
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So dI/dt is equal to zero.
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And where that happens will
be at a certain value of S_0,
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which we'll just put--
of S, I should say.
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I'll call that S_h for the
value of herd immunity
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in the susceptible number.
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And that is when
this factor
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beta*S-gamma is equal to 0.
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Or in other words, S_h
is gamma/beta.
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OK, and once we
get to that point,
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then now dI/dt is
going to change sign
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and will only be negative.
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So the number infected
will only be decreasing.
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Notice S is strictly
decreasing because this
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is a negative rate here.
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So S only continues
to go down, which
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means that the prefactor
here is always negative.
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So the number of infected
people now, at this point,
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must also necessarily
continue to go down,
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and ultimately the
number of susceptibles
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will, from that
point, tend towards 0.
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The number of recovered
will tend, of course,
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with some lag to S_0.
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So this is the
number of recovered.
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This is the number
of susceptible.
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And the number of
infected, of course,
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also goes to 0,
something like this.
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And ultimately, in the
long run, the decay--
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because S is going to 0,
dI/dt is minus gamma*I*t.
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So the final rate of drop
here is -gamma*t.
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So basically, the
recovery rate is really
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dominating how quickly people
are being converted, basically,
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from, to create the
recovered and remove
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the infected population.
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So the final result here that
comes from these models, which
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is quite interesting,
is to ask, what
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is this fraction of
the population that
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needs to become
immune, or what is
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this sort of threshold
of susceptible
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in order to achieve
herd immunity?
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So if we look at what is
this S_h over the initial?
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So how far do we
have to go down?
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Well, that would be
gamma/(beta*S_0),
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and you'll recognize
that that is
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nothing more than the inverse
of the initial reproductive
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number.
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So essentially, herd
immunity is reached
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at a value of the susceptible
fraction becoming 1/R_0.
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And this is an
interesting prediction
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of this very simple
model, which is
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that more infectious
diseases that
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have a very high value of
R_0, for example, smallpox,
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leads to a situation that,
when the epidemic finally
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ends, the number of susceptibles
when you've reached herd
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immunity, well, when this
starts to turn around,
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is actually quite low.
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So, in other words,
you have to go very far
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in infecting the population
to start to end the epidemic.
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Conversely, a disease that
has a small value of R_0,
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even COVID-19, might
be a value of say 3.5,
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then this fraction here
might not be so [low].
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It might be, maybe,
only say, 20% or 30%.
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Where you can start to see then
herd immunity being reached
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and the number of infected
people going down dramatically.