PROFESSOR: So as an aside, for those of you
who have an understanding of probability
theory and stochastic processes, let
me just explain why it is valid to bound the main transmission
rate in defining the indoor reproductive number when
we might be more concerned about bounding
the probability of a transmission.
So let's let a capital T be a random variable which
describes the random number of transmissions
that occur in the room.
And this is specifically for the case for one infector
or infected person and n minus 1 susceptible.
So again, the situation of the reproductive number where
for every person that comes in we want to know
is there going to be transmission,
and typically, only one infected person would be seen.
So this is a random variable, and let's let ft of n
be the probability density function that
gives the probability of little n transmissions.
And then we can define the risk of a transmission, the risk
of at least one transmission, as the probability
that this random variable takes on a value, which
is greater than or equal to 1.
OK.
Well, in terms of the probability density function
then, that would be a sum from 1 to infinity of the ft event,
basically.
So we're just adding the probabilities
that we're not seeing a transmission to those--
or possible transition to those different numbers of people.
Now I'd like to do a little calculation
to get an upper bound on this quantity.
So we can say that this is less than
or equal to the sum from n equals 1
to infinity of n ft of n.
Now, this is just a mathematical trick here.
So n refers to the natural numbers 1, 2, 3, 4, et cetera.
Those are all positive numbers and they're
all greater than or equal to 1.
So if I take 1 in this expression over here
and replace it with little n, I'm
only increasing the value of that sum, because also,
this ft is a probability density that has to be positive.
And then now I can also say that this is actually
equal to throwing in n equals 0, because that is a term that
is actually identically 0.
So I can change the summation.
And then by definition here, this thing
is the expected value of the number of transmissions,
because I'm summing the number of transmissions
little n times the probability of that event occurring.
So that is the definition of the average.
So what we're seeing here is that the risk of a transmission
is rigorously bounded above by the expected
number of transmissions.
And so therefore, if we require that this is less than epsilon,
our risk tolerance that we've just introduced,
this is a conservative bound on the true risk, which let's
say here is defined by rt.
So if your goal is to control the probability of having
at least one transmission, so basically
to ensure that no transmissions occur,
then you would do well to bound the expected number
of transmissions, because that's an upper bound.
It can also be shown that as epsilon
goes to 0, so we're talking about very low probabilities
of transmission, and oftentimes that is the case,
then rt is asymptotically the same as the expected
number of transmissions.
So this overall risk of a transmission and the expected
number are the same.
And in fact, that's one way to understand
sort of even the definition of a probability
in terms of an expected number of events.
And we are typically thinking of cases where
epsilon is much less than 1.
So for those of you that have some background in probability,
you may recognize that what I've just done here
is an example of a much more general result, which
is called Markov's Inequality.
So now we can safely proceed by continuing
to work with average values of all the quantities of interest.