PROFESSOR: So now let's focus on the steady-state transmission
rate, which is really the most useful in designing a safety
guideline.
It's also the most conservative because
the transient transmission rate is always
smaller than a steady state.
So, our formula for the steady state transmission rate
is shown here, in terms of the relaxation
rate, lambda_c(r), of the aerosol concentration
in the air, and also n_q(r), which
is the density of infection quanta in the air, per radius.
So let's sketch some of the important functions
here as a function of radius and try
to get a sense of how we can maybe simplify this expression.
First of all, we've already defined
C_q, which is the integral of n_q(r),
dr. This is the critical disease parameter, which
is the infection quanta per exhaled air volume.
I don't mean to cross that out but rather just to do this.
So C_q is a very important quantity for us
and we will return to that.
That's the quality that we're going
to want to fit to disease data for COVID-19, specifically.
And this will be the exhaled infection quanta per air volume
and we'll typically want to measure
that sort of peak infectivity of an individual
in order to design the conservative criteria.
So what is C_q?
If I plot this n_q, it has a bunch of factors in it.
So it has the roughly constant assumed viral load
per liquid volume, it has the infectivity,
which we've already argued should be smaller
in larger droplets because it's more difficult for the variant
to diffuse out of those droplets once you get above, say, 5,
10 microns, if not less.
There's the droplet distribution itself,
which depends on the type of respiration
but often has a peak which is submicron
and then sort of a fairly broad tail at the higher end,
with smaller amounts of larger droplets,
and then V_d is 4 pi r^3, which
is just the volume of a drop.
So this net quantity, n_q has some kind
of peak around 1 micron or less and then a tail.
And then in the integral here, we
have the integral of n_q over r is c_q, so that's very important.
But there is these other factors, p_m and lambda_c,
or 1/lambda_c.
Each of those quantities gives us
a cutoff which makes the larger droplets less
important for this problem of airborne transmission
in a well-mixed room.
So lambda_c, as you can see, is a bunch
of constant factors except for the sedimentation rate.
So this lambda_a times (r/rc)^2,
that is the sort of radius-dependent change
of the sedimentation rate relative to the ventilation
rate, lambda_a.
So as you can see, this goes like 1 plus a constant plus,
r^2.
So as you go to large r, the inverse of that
is 1 over a constant plus r^2, so it goes to 0.
So it provides a cutoff and the scale for that cutoff
is what we called r_c, that's the critical size of a droplet,
which is just sedimentary at a rate
comparable to the ventilation rate
because, really, this is ventilation
and sedimentation which are compared when you define r_c.
In addition to that, we have (p_m)^2,
which is the max penetration, or transmission factor.
So while masks are 100%, or very efficient,
filtering large droplets which don't
fit through the fabric or the mesh,
they're not as good as filtering
smaller droplets.
So if you look at the transmission probability
when you're down well below micron,
most masks are not doing a great job filtering,
they may get 5%, 10% if you're lucky,
depending the quality of the mask.
But then it comes down because you
start to have better and better blockage of particles
by the masks.
All these factors serve to cut off this distribution so
that we're not worried about the large drops,
and we're interested in aerosols.
But it does so, here, in a way which is quantitative.
So we're not just arbitrarily saying,
as it's sometimes said in the field, that
say 10 microns or 5 microns is the limit of the aerosols,
but rather, we actually have a well-defined characteristic
size that can emerge here.
And the way we can define that is by taking the full
expression for the steady state transmission that is has
the radius-dependent terms in it and write this as (Q_b)^2/V
times--
and then we'll keep the C_q from the integral of n_r-- C_q times--
and then we'll imagine that the remaining radius-dependent
factors, p_m and lambda_c, are sampled
at a certain value, r-bar.
So what is r-bar?
Well if you know the function is p_m and lambda_c
is a function of r, there is a value
of r, which we call r-bar, which is when you actually
do this full integral, you would get that value.
So that has to be determined.
It can be done numerically but you can kind of see graphically
where it ends up.
What we're asking here is what is
the typical value of the mask penetration
factor and the relaxation time?
Well, it's going to be where the most weight is here,
keeping in mind, also, that there's
more volume at the higher side than at the lower side.
So if we look at how much activity there is,
we might want to emphasize that.
So depending on the details here, somewhere over here
is going to be r-bar.
What we're saying here is that even though our theory has
all of the radius dependents in it,
so if you know exactly the type of masks
you have and you know p_m(r) from experimental measurements,
maybe a lot about the virion and how infectious
it is in different-sized droplets,
or you studied sedimentation-- you have all these functions.
There's a well-defined r-bar at which you can just
use this simple expression in place of actually doing
those integrals.
So that's actually useful simplification.
And in addition to that, we can also
write this another way, which can be useful,
is to take the mask factor out and write it
as a quanta emission rate, lambda_q, where
lambda_q is Q_b*C_q.
This is the quanta emission rate by an infector.
So if you don't like this notion infection quanta per volume,
when you multiply by the breathing flow rate,
you're actually getting how many quanta per time
are being emitted by the infector.
And then what's leftover is another factor,
which I'll call f_d, which is something
we'll come back to later, which is what I call the dilution
factors.
If we take the breath of an infected individual
and then it ends up being diluted
into the room, the ratio of the concentration of infection
quanta, or virions in the breath compared
to that which emerges in the well-mixed room,
that's the dilution factor.
This will become important later when we look more closely
at respiratory fluid mechanics and we
look at the plumes, or clouds, of droplets
that are being emitted by a person when they're breathing,
very close to the person's mouth it's
a much higher concentration and eventually it
gets sort of swirled around and mixed in the room,
and it reaches the steady state values that we calculate.
This f_d gives you that ratio in some sense
and gives you a sense of how bad the risk is
from short-range transmission versus the well-mixed room,
so we'll come back to that.
But this is a nice simplification
for how we can think about the steady transmission
rate in terms of several key variables, which
I've boxed here.
And so we will now move on to applying this to COVID-19.