PROFESSOR: So now, let's start talking about airborne disease
transmission, thinking especially of viral diseases
where we expect the transmission to occur
through aerosol droplets or at least smaller droplets,
which may sediment but are going to be
suspended in the air for a significant amount of time.
So the first approximation of such a situation
is to assume a well-mixed room, meaning
that the air in the room is well mixed.
And even if there is, let's say, one person who's infected
and that infected person is breathing, talking, singing,
respiring, and exhaling infected aerosol droplets,
then these droplets start to spread around the room.
And they do so because the room is--
and we assume that the room is well mixed.
So even though, as I've sketched,
there's certainly going to be a higher concentration
of the droplets near the infected person, that
is there are significant airflows in the room, which
induce mixing.
That is partly driven by the flow of fresh air
through the room.
Typically, we may have some forced ventilation
coming through ducts with fans.
There could be open windows.
And even when a room is not open,
meaning that the windows are closed, maybe even
the door is closed, there's always
some leakage and exchange of air with the outside, which
typically happens on the order of hours.
So there's always at least some kind of flow rate.
Also, there's a movement of people in the room,
and there's also breathing itself, which basically
imparts momentum to the fluid and causes
swirling motions of the fluid.
So the situation actually is more like this
where there's these flows that are generated either
by the ventilation, by the movement
and respiration of the people, also by thermal flows.
So when you're breathing, the air coming out of your lungs
is warmer, and that tends to rise.
But, also, if it's very humid air and very warm air,
then actually the weight of all the humidity and droplets
that come with air could actually
cause some settling as well.
So you have all these different processes going on.
And your first approximation, just
assume that those lead to a well-mixed situation.
And so we will proceed to analyze disease transmission
from that assumption, and then we'll
come back at the end to consider what
would happen if there are actually fluctuations
and think about the distance from an infected person
and how that might play a role in departures
from the predictions of a well mixed room.
OK.
So defining some variables, then.
We'll let C of t be the, basically,
infectiousness of the air, if you will.
But more specifically, if we think of virus,
it'll be the virions per air volume.
So they're contained in droplets.
And later we will consider what size droplets
they may be contained in, but for now, let's
just average over everything and say there's
some concentration in the air.
We have a production rate of these infectious droplets,
which can be broken down into many terms.
So this is the production rate, so this
is the number of virions per time that are produced.
So that would be Qb is our breathing flow rate.
So that's basically the volume of exhaled air per breath
cycle, so that's how much air is being pushed out
in your breathing.
And this is typically--
ranges from 0.5 up to around 3 meters cubed per hour.
So 0.5 is a typical resting breathing rate.
And, in fact, even if you're just
kind of calmly speaking and sitting in a room breathing
through your nose, that'll be your typical breathing rate.
But if you start exercising and start exerting yourself,
then that could go up to maybe around 3.
And so that's the typical range of the breathing rate.
The next thing we need is Nd is the droplet concentration,
so this is the number of drops per air volume.
So that's sort of the number density of drops.
So if I have various visualization techniques,
I can actually see all the droplets, and I can count them.
I can say how many droplets are in a given volume.
The next thing is I need the volume of a drop.
So as I mentioned, of course there
are different drop volumes, and we'll
come back to assessing the effects of a droplet size
distribution.
But for simplicity, why don't we just take
these sort of average size drops that we discussed
coming from respiration,
And that might be a number around 1 micron in size.
And there's a volume that corresponds with that, which,
of course, is the 4/3 pi r cubed where
we assume for now only one size r for all drops just
for the moment.
So now we have the volume here, so now
we sort of what is the total amount of drops.
So the next thing we need now is Cv.
Cv will be the number of virions per liquid volume or per drop
volume, so there's certain-- so if I could take the pure liquid
in the drop-- let's say it's mucus--
and it's been coming out of your pharynx
and has sort of fragmented and taken some virions with it,
then that would be the viral load, essentially.
Sometimes that's another word for this.
And the viral load varies with time.
So when you first get infected, at first the viral load
is very low in the fluids that you're breathing out.
And then that raises up.
And during the period when you're most infectious,
which for COVID-19 and SARS-CoV-2 virus,
that time ends up being around a week or so when you--
well, within a few days, you reach the peak infectiousness.
And then when you're at the peak viral load,
this ends up being about 10 to the ninth virions
per milliliter of fluid when you're
at your peak infectiousness for SARS-CoV-2.
OK.
So just to give you a sense, but of course, a lot of times
it might be less than that.
But if you're very infected individual,
that's kind of a worst case.
And we're going to be interested in calculating
safety guidelines and probabilities of transmission,
so to be conservative, it's good to have
an idea of how big this number can actually be.
So what we just calculate here-- so,
basically, this Nd, Vd is the amount of drop
or liquid volume per air volume, so that's essentially
the volume fraction of liquid.
When we go times Cv, we're essentially
getting the number of virions in the air per volume of air.
And then Qv is the volume per time,
so this is basically virions per time.
And then one other factor that we should also
consider is what if we're actually
filtering those droplets right at the source?
And that'll be the case if you're wearing a mask.
So this is an important quantity we'll come back to.
This will be the mask penetration probability
for a droplet.
So this is-- of course is size dependent,
but we'll come back to that.
But for the moment, we're just saying it's one drop size.
And so for the size of drops of interest, we're asking,
do they go through the mask?
So 1 minus Pm is also called the filtration efficiency.
Yes.
So a very good mask might be 99% of droplets are filtered.
A very poor cloth covering might be
10% of droplets are filtered, and we'll come back
to that in just a moment.
So this here is our production rate capital P.
And already with the variables that we've written down here,
we can write down a mass balance for the virions.
So virions are being produced.
They end up in these droplets.
The droplets are being swept out of the room at a flow rate Q,
and the room has a volume V. So if I write down
just the conservation of mass, making sure that I'm not
losing any virus yet--
I'm not allowing them to stick to the walls
or do anything else just yet but just
looking at the mass balance of one infected person breathing
out.
This is, I should say, production rate per infector,
or an infected person.
So if there are more infected people,
then you'll have this production rate for each person.
They also might be at different stage of the disease,
so maybe the viral load will be a little
different for each person.
But let's not worry about such details right now.
We want to keep things general.
So for the mass balance, we write down
the total number of virions in the room
and how that changes in time.
So that'll be the concentration per room air volume, which
is well-mixed times the volume of the room per time.
So this is the change in the number of virions per time,
and that can change in two ways.
One, we have the production P. So for every infected person,
we have production P, but this will be--
I should say virions per air volume per infector.
So if we want to think about having multiple infected people
in the room, we can always just basically increase
this concentration.
A well-mixed room, it doesn't matter where people are placed.
You're just getting more and more droplets,
and it's assumed to be mixed.
So we produce at a rate P, but then the outdoor flow
is taking away droplets and, hence,
removing virions at a rate Q. So we have a Q times
C removal rate.
So this is our equation.
So let's divide through by V.
So we can write this as dC dt is P/V minus,
and then Q/V I'll write as lambda a C.
So lambda a, which is Q/V, this is the outdoor air change
or exchange rate.
So that is the rate at which the entire volume of the room
is replaced with outdoor air.
So the outdoor air is refreshing the air
in the room at this rate Q/V. OK.
And so that's what appears here.
And if you compare these two, you can see that this is dC dt.
This is C times lambda, so lambda is units of 1 over time.
It's a t.
This is also sometimes called--
the ACH is air changes per hour if you write it in per hour.
So that's a typical way that this is written.
And, in fact, while we're just talking about,
this is a very important concept.
So lambda a is around 0.3 per hour, so roughly every 3 hours
for a closed room.
So closed room or what you might call natural ventilation where
there's no attempt to deliver air to that room.
Of course, this number depends on the tightness
of the construction and whether there is cracks in the windows,
whether doors are being opened to the hallway.
So, of course, that's not a perfect number,
but that's a rough estimate how quickly air is escaping
from typical construction.
But then it can be, also, in a different range.
And this a very important parameter for the theory,
so let's pause just to look at some of the numbers.
So it's in the range-- it's typically 3 to 8
per hour for mechanical ventilation.
So this could be--
it could be open windows with fans blowing in and out,
which might give you 3 or even 6 on this number.
It could also be a ventilation system,
which is delivering fresh air to the space.
And for typical classrooms, offices, and even homes,
this is a typical range.
So for example, if it's 3, then that
would be every 20 minutes the room gets its air
to be fully exchanged.
But of course, if you have situations
where you need to have better air quality
and you have more risk of, say, transmission
of disease or passage of pollutants or contaminants,
then we need higher values.
A typical number in the United States for hospitals
is 18 air changes per hour.
And then it can be even larger.
So if you have a laboratory, which
is dealing with toxic chemicals or even,
let's say, virus and pathogens, then you
need even higher air changes.
And typical rates, then, can be as high as 20 to 30
for labs that are dealing with toxins of various types
because--
and any airborne toxins have to be quickly removed
so that if they happen to be leaked
into the air from your experiment or from your hood,
they need to be quickly sucked out.
Also, even parking lots where you
have cars in an enclosed space that are generating carbon
monoxide and other fumes, which have to be quickly rushed out,
parking lots tend to have this number around 30.
So that's a full air change of the entire room in 2 minutes.
That's a very fast flow rate.
So this is kind of the range of this lambda a.
That's, obviously, a very important parameter.
So let's now solve this equation here.
So, first of all, you can see when dC dt is 0,
then the steady state is just P over lambda a V.
But if I write Q equals lambda a V,
I can see the steady state is just P/Q.
So I can write the solution like this--
that if we're given time dependence, that the P/Q is
the steady state.
And if I started out with my initial condition,
was that C of 0 equals 0.
So let's say time equals 0 is when the infected person enters
the room and starts breathing.
And then there's a mixing process
and there's a build-up of concentration
until there's a balance between the production of virus,
virions, or infectious air and the removal of infectious air
by the ventilation.
And that gives you this ratio P/Q. But the way the relaxation
happens, though--
if you balance these two terms here,
that's just an exponential decay with a decay rate lambda
a, so e to the minus lambda a t.
So this is basically the way the concentration builds up.
If I plot this, then at a certain time here, which
is sometimes called Tres--
in chemical engineering, these kinds of models
are commonly used to design chemical reactors.
In fact, this kind of model is in chemical engineering
called a continuous stirred-tank reactor.
We don't worry about the details,
but we have a flow of some reactants
and various chemical species going into a tank.
We assume it's well mixed and then it leaves,
and this is the mass balance that we use.
And the residence time, Tres, is the inverse
of lambda a in this case.
So that is the--
as soon as you know the volume and the ventilation flow rate,
there's a typical speed going through here.
And the time that fresh air spends
in the room interacting with all the droplets and people
and then leaving, carrying some of those droplets, is this.
And in a simple model like this, it's just
an exponential relaxation at that time scale approaching
the steady state, which is P/Q. So there's always that kind
of balance which is reached.
Now let's also ask ourselves, briefly at
this point, how reasonable is the well-mixed approximation?
We will come back to this and analyze it much more carefully,
taking into account all the different processes I described
at the beginning, including breathing, motion of people.
But let's just think about the motion caused by the airflow
itself.
So in the case of mechanical ventilation
where that flow rate can be sometimes rather high,
that can be a significant source of mixing in the system.
And the way we'll think about that
is by writing down the typical velocity of the air
due to the outdoor air flow.
That is Q. We can write it as either Q divided by the area,
so some kind of representative area of the room.
It could be, let's say, the floor area,
but depending on the shape of the room,
it might be a little bit different value than that.
We can also write this as Q times H over V, where
V is the volume of the room and H
is some characteristic height, like, for example, a ceiling
height of the room.
OK.
So this is the mean airspeed due to ventilation.
And we'll come back to a more deeper investigation
of these fluid mechanics of the room.
But as a first example of what we'll be interested in is we'd
like to calculate the Reynolds number due to the airflow.
So I'll put a subscript a there.
And that is the typical velocity times a length scale, which
could be, let's say, the height of the room
or some linear length scale of the room,
depending on the direction of the airflow.
And then the kinematic viscosity of air,
so I'll write that as new a.
So this here is the Reynolds number.
This is basically telling us how important inertia of the fluid
is compared to viscous stresses that slow the fluid down.
So, basically, it tells us how quickly--
how much of a tendency there is for momentum
to be carried in the fluid, which then
leads to sort of swirling motions and complex flows,
as I've sketched here.
And this new a is the viscosity of the air
that we've already talked about when looking at Stokes flow
but divided by the density of the air.
OK.
So that's the kinematic viscosity.
And for air, the kinematic viscosity
is 1.5 times 10 to the minus 5 meters squared per second.
And so if I plug these numbers in and I pick,
let's say, a typical ceiling height of maybe a few--
I think this-- if the H is of order 3 meters--
or 2 meters might be a typical scale.
Just to get an approximate sense of the scale here,
the Reynolds number will be varying from around 50 or tens,
up to 5,000, which would be in the case of very fast
ventilation like the 30 ACH.
So this would be if we have 0.3 ACH up to around 30 ACH.
That's just a rough number.
And from fluid mechanics, we know
the significance of these large Reynolds numbers
is that the flows really do look a bit like I've shown here.
So, basically, when the air is sitting in the room--
you know this from looking at the smoke
from a candle or other flows that you can visualize--
it's not just a sort of uniform flow.
But instead there are all these sort of plumes and swirls
and vortices.
And when the Reynolds number is on the order of tens,
when there is a motion, it tends to lead
to a shedding of a vortex and to some kind of swirling flows.
And when the Reynolds number gets up as high
as several thousand, then in most geometries
that starts to lead to a transition to turbulent flow.
And that's when the flow is getting
so complicated there are eddies of different sizes
and very rapid mixing.
So I just wanted to show this right
at the beginning of the discussion
to point out that for typical flows
we would expect that there is going
to be a decent amount of mixing occurring just
because of the ventilation.
And now if you add to the fact that people are breathing,
imparting momentum to the fluid, we
have people moving and other kinds of activities in a room,
that all of those processes lead to giving us
a reasonable assumption of a well-mixed room.