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PROFESSOR: So in order to use
our results for the analysis

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of the well-mixed room
into the safety guideline,

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let me just
summarize the results

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for the general case of
transient build-up of aerosols

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in the room and the
associated transmission.

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So here is the result
that we derived

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earlier, which is
that the transmission

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rate as a function of time is
an integral over all the droplet

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sizes.

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And then you have here the
mask filtration factor,

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which depends on size P_m.

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You have the
breathing rate Q_b that

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comes in squared because there's
one person breathing out,

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another person breathing in.

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You got the volume of the room.

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You have here the
relaxation rate

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for the concentration
of aerosol in the room,

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which here is given by four
factors for ventilation,

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sedimentation, filtration,
and deactivation.

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And then for the
production of aerosols,

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there's this n_q(r), which is
the number of exhaled infection

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quanta per volume,
per air volume leaving

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the breath per radius
because it's still resolved

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by the different droplet radii.

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And this has several
contributions.

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It has n_d(r), which is the
distribution of droplet sizes.

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V_d(r) is the volume
of each droplet.

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Depends on the
respiratory activity.

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C_v is the viral load,
which we are typically

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assuming is near the
maximum when we're concerned

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about controlling spreading.

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And C_i(r) is the
infectivity per virion,

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which we have discussed before
also may have a size dependence

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and is most likely higher
in the aerosol droplets.

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So that's the general solution.

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And the safety guideline
we just discussed

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has in it the time average beta,
so beta with the two brackets,

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which is the integral in time
of beta divided by the time tau.

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So you break that
into two parts.

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So we integrate this here.

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The one here is--

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it gives you a
steady state term,

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and the interval is
shown right here.

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And that's basically the--

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ultimately the average
that remains, but initially

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when the infected person
first walks in the room,

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there's a time to build up
the concentration, which only

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lowers the transmission rate.

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So the average
transmission rate is always

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less than the steady state.

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You're approaching
the steady state

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from below because you
need the time to build up

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those droplets.

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And so the DELTA beta here,
which is that correction,

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takes the following form.

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What you can do is bring the
interval over time and switch

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places with the
integration over r

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and do the time integral
inside the integral.

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And so that allows you to
get-- instead of lambda_c here,

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you get lambda_c
squared, and you

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get the following expression
for the DELTA beta.

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It may not be obvious
looking at it,

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but if you take a look
at tau going to 0,

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this expression leads
to just beta bar.

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So DELTA beta of 0 is
beta bar, and that's

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because if you take
this exponential here

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and you go to small times, you
can linearize that and find

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its lambda_c t.

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So it factors--
cancels one factor

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of lambda_c, one factor
of tau, and you end up

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with just a single
lambda_c as above.

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So what that means is that
the average beta, which

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we're plotting here, as a
function of the time tau

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starts out at 0.

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It ramps up and then eventually
approaches a steady state,

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and here's the full solution.

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So all the
information that we've

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talked about before in terms
of filtration, sedimentation,

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other phenomena in
the well-mixed room

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are all included
in this framework

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and can then be put into
the safety guideline

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to derive a general
safety guideline that

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has all of the physics
that we want in there

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and allows you to define a
safe occupancy for a room.