Biology Reference
In-Depth Information
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FIGURE 2-2.
Measles in England and Wales. Weekly case notifications of measles in England and Wales before
general vaccination was initiated. (Data from
http://www.zoo.ufl.edu/bolker/measdata.html.)
least some of the differential equations involve more than one variable.
For example, we might have:
dA
dt
¼
2 A
3B
dB
dt
¼
þ
:
A
B
This means that the behaviors of the groups A and B are linked.
To provide concrete examples, we shall consider two classical models
from mathematical biology—the spread of an epidemic, and predator-
prey models. In these examples (and many other problems in biology
and chemistry), the dynamics depend on contact between groups. This
dependency is manifested by the product of two groups in a
differential equation. For example, in the spread of an epidemic, the
two groups considered will be the susceptibles (S) and the infectives (I).
We shall hypothesize that the susceptibles' rate of infection is
proportional to the amount of contact they have with the infectives,
and that the flow from S to I occurs at a rate
0isa
constant; that is, at a rate proportional to the product of the group
sizes.
a
SI, where
a >
Why SI instead of something else? An intuitive way to understand this
is to begin with the groups uniformly mixed, then double the number
of infectives while keeping the number of susceptibles fixed. We should
expect the number of contacts between the groups to then double.
