Biology Reference
In-Depth Information
Theorem. Suppose x and y are functions of t and dx
dy
dt ¼
dt ¼
f
ð
x
;
y
Þ;
g
ð
x
;
y
Þ
,
where f (x,y) and g(x,y) have continuous partial derivatives. Then for
any point (x 0 ,y 0 ) there is a unique solution (x(t), y(t)) to the above system of
equations with x(0)
¼
x 0 and y(0)
¼
y 0 .
This theorem is not nearly as complicated as the mathematical
symbolism might make it appear. It simply says the rate of change of
each variable depends only on the variables and not time, while the
conclusion states that if the initial values of each variable are given, then
there is exactly one way the process can evolve. All of the models that
we consider will fulfill the hypotheses of this theorem. Such theorems in
mathematics are referred to as existence and uniqueness theorems.
To gain some perspective on what the theorem means, we go back to the
example and let S and I play the roles of x and y, respectively. The values
of dS
dt and dI
dt are determined by S and I, rather than having t appear
explicitly in the equations. This means that the model from Eq. (2-5) fits
the hypotheses of the theorem.
We construct a Cartesian coordinate system, where the axes are the
variables of interest—in our case, S and I.
Suppose at some time, usually t
¼
0, we know the values of S and I in the
I
equations for dS
dt
and dI
dt given above. Then we have a point (S(0), I(0)) in
the (S,I ) plane (also called the phase plane). We can also compute what
dS
dt
( S (0), I (0))
and dI
dt are at that time, so we know the direction of travel. For
example, if the coordinates are as shown in Figure 2-8, and if dS
dt >
0 and
0
S
dI
dt <
then S would be increasing and I would be decreasing. In this
case, at this instant of time, we would associate an arrow that points
down and to the right. Thus, for each time t, we have a point, (S(t),I(t))
in the (S,I ) plane and a direction of travel. As t continues to change, we
have a curve called a trajectory traced out in the (S,I ) plane. A trajectory
constructed as we have just described defines a solution to the system of
equations. The theorem says:
0
;
FIGURE 2-8.
Using the phase plane to determine direction of a
trajectory. The arrow indicates direction of travel
at a point where and at t ¼ 0.
1. Given any initial point in the (S,I) plane, there is a trajectory
(solution) that starts at that point; and
2. Different trajectories never intersect.
B. Constructing the Phase Plane
Consider now the following specific example for the model defined by
Eqs. (2-5):
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