PROFESSOR: So now that we've done a very simple calculation
of C of t, the concentration of virions
in the air per volume, which is coming
from just a mass balance in a well-mixed room,
let's zoom in, now, and think about transmission
between the infected person, or an infected person
and then a susceptible person who is not yet infected.
And the transmission is occurring
by breathing the infected droplets in,
and then the virus has to get out of those droplets
and interact with the host tissues.
So if we let beta be the transmission rate,
so this is basically new infection per time,
so basically, it's the rate at which another person may
become infected.
Then how will we write this?
Well, we could write it as--
so we're, now, forgetting about the infected person.
We care about their breathing only
because it's producing this concentration C that we just
talked about.
But now we're going to focus on the susceptible person.
The susceptible person is now breathing in at the same flow
rate Qb because the breathing in and breathing out are the same.
And so Qb is the volume per time around, let's say,
0.5 meters cubed per hour with which they're sampling the air.
And the air comes in, and then C of t
is the concentration of virions per volume.
So this combination now tells me how many virions per time
they're actually taking in.
There'll be a quantity Ci, which we mentioned earlier, which
is the infectivity, and that's the probability
that an individual virion actually
causes this person to get sick and to become
infected themselves.
So this is the infectivity of an individual virion.
And then on top of that, we also want to put in--
I'll just change the color to highlight it--
the mask factor that we talked about earlier.
So that's the transmission or penetration probability
for the droplets of interest through the mask.
These quantities, many of them will depend on size,
and we'll come to that, size of the droplet,
but just as a rough approximation,
this is the starting point.
So this is the number of, basically, new infections
per time, and there is a useful notion
in epidemiology, which is that of the infection quantum.
So transmission rates are often written
as infection quanta per time, and that
is the rate at which a person, which is susceptible,
will get infected.
What we have not yet captured is if you
have a finite number of people in the room,
when someone gets infected, they can't get infected again.
So the numbers of susceptible people is changing.
So we have to model the progression of the disease
in the room, which we have not done yet.
So that's why the beta is not the number of infected people
cause eventually you run out of people to infect.
So we have to account for that later.
But a useful way to think of it is
that beta is sort of the rate at which this person is sending
infection quanta over here.
Those quanta may not actually lead to an infection
because they might already have been infected.
But if they're susceptible, that tells you
the rate at which that person would become infected.
So that's the notion of infection quanta
is essentially defined by the transmission rate beta.
Now, this infectivity is something we'll come back to.
We will actually go through the calculation for SARS-CoV-2,
but it's been estimated before to be at about 2%
from the original SARS virus in 2003.
And in fact, I will argue that it's greater than 10%
for SARS-CoV-2.
And we'll do that by analyzing spreading data with the model.
And, of course, that helps to explain
why SARS-CoV-2 has led to the COVID-19 pandemic,
and SARS-CoV-1 was not able to spread as much.
So now we have here our transmission rate.
And we can ask ourselves, this is a transmission rate, which
is time-dependent, but what if now
we calculate the steady-state transmission rate?
So the transient would be when the infected person first
enters the room.
The concentration is changing in time in the air,
but eventually, there's kind of a steady-state where there's
a balance of the production and then the flow rate
through the room of refreshing the air with outdoor air.
And in steady-state, we have the transmission rate
is going to go to a constant value,
which I'll call beta bar, and that
is given by the steady-state concentrations.
So here I'll just write it--
I'll rewrite this expression here Qb Ci Pm and then
the steady-state C, which is the production rate P divided
by the outdoor airflow rate Q.
And another way we can write that is
that remember Q we can write as lambda a times
V. So this is Qb, and in fact, let me-- well, here,
I'll write it one more time.
That's my Ci Pm capital P over lambda A V.
Now, recall that the P we had written as,
that's the production rate, also depends
on Qb, that's the rate at which the infected person is exhaling
infected air.
So that was Qb times nd, the number of droplets per volume,
Vd, the volume of liquid in a droplet,
so this nd Vd's the volume fraction of liquid.
And then we needed Cv, which was the concentration of virions
in the liquid or in the fluid.
And there's also a factor of Pm if that person
is wearing a mask.
So we put all this together, we get an important result here,
which is that the steady-state transmission rate can
be written, when I plug-in here, as Qb squared times Pm squared.
So the mask factor comes in twice
because if they are wearing masks, there's two masks.
You have a mask at the source.
You also mask at the target, and the fluid
has to go through both of those filters.
So that's one reason, as we'll see, that masks can be,
actually, very effective.
And then we'll lump all the parameters in something
I'll call Cq, which I'll come back to,
and then we'll leave lambda a V in the denominator.
So this is the main transmission rate
where I've defined this important parameter Cq which
has all the information about the specific disease,
and what is it?
It's everything else is left.
It's nd Vd, so that has to do with respiration,
so the distribution of droplet sizes
and the size of the droplets is something
that's coming from the type of respiration
that the infected person is engaging in.
Cv is their viral load, so it has
to do with the progression of their disease
and how many viruses or virions are found.
And then we have Ci, which is this infectivity,
the probability that any one of those virions
will actually infect this susceptible person
if it manages to get in there and diffuse out
of the droplets.
So coming back to this notion of infection quanta,
while beta is an infection quanta per time, which
are being kind of transmitted from one infected person to one
susceptible person, the way I've written it here
is I've reexpressed it as infection quanta per volume
of air exhaled.
So while C is the concentration of virions in the background,
there is this Cq, which is essentially the infection
quanta that are being released, and the factor of CvCi
is actually what connects those two.
In fact, sometimes we write Cq little cq CvCi,
this is infection quanta per liquid volume in a drop.
So from the mucus or material that's being released,
there is a certain concentration of infection quanta, which
is the physical concentration of the virions Cv
times the probability that if they
were to be exposed to the susceptible person's cells,
that they would actually infect those cells
and cause a transmission of the infection.
So that's another important quantity,
and this here is really the primary sort of lumped
or combined disease and physiological
parameter in the model.
So what's nice about separating this way is
that Qb is something which has to do with people's activity,
it's how fast they're breathing, and that's
something we know very easily whether they're at rest
or they're exercising.
Pm is also known if we know the kinds of masks people
are wearing.
There's various studies of transmission factors
and filtration efficiencies of masks.
And so we can put reasonable estimates there.
And then here we see the importance
of lambda a, which is the air exchange rate.
So how quickly is fresh air coming in the room?
That's a physical parameter of the room, has
nothing to do with the disease.
And V, of course, is a geometrical parameter,
the room, which is the volume.
And so all the disease aspects are kind of lumped into the Cq.
So if I want to apply this to actual spreading of COVID-19,
I have all these parameters that I
know that come from the physics and fluid
mechanics of the room.
And then, I have a single parameter Cq
that I need to obtain from an understanding of disease
transmission and looking at spreading events
in indoor situations.