Biology Reference
In-Depth Information
elements in each group. To complement the heuristic justification for this
assumption presented earlier, we present its mathematical explanation.
We begin by outlining the basic ideas; the first two come from
probability, while the third uses calculus.
A. Idea 1
If we roll a standard 6-sided die, the probability that a particular
number, say 4, comes up is 1/6. The probability that a 4 does not come
up is 1
1/6. If we roll 10 dice, the probability that none of the
10
1
6
10 dice is a 4 is
1
:
The spread of an epidemic model is a good example of a problem where
we need to measure the contact between groups. In what follows,
when a susceptible and an infective interact we mean that the two
contact one another and the disease is passed to the susceptible. We
hypothesize that the probability that a particular susceptible and a
particular infective interact in a unit of time is known, and we denote it
by p. (Note that p is a number between 0 and 1. In most cases, it will be
close to 0.) So the probability that two random members do not interact is
1
p.
Now keep the susceptible element fixed—call it s
*
. Suppose we denote
the number of infectives by I. Then the probability that s
*
does not
interact with any of the infectives is (1
p)
I
. In this description, we are
assuming that both populations are uniformly mixed. Is this assumption
reasonable?
B. Idea 2
Suppose that a basketball player makes 70% of her free throws over the
course of a season. If she attempts 20 free throws in a game, we
would expect her to make about:
20
0.70
¼
14
of these. This does not mean that she will make exactly 14 free throws in
every game where she has 20 attempts, but it does mean that if she
had many games in which she shoots 20 free throws, we would expect
that the average number of successes would be close to 14.
This is an example of the following principle: If we have an experiment
such that on each trial the probability of success is p, and we do k trials
of the experiment, then the expected number of successes is kp.
Back to our example: Suppose we have two populations S and I, and
we want to know the expected number of susceptibles that do not
interact with an infective.
