6.3.
LINEAR MODELS IN BIOLOGY
767
6.3
Linear Models in Biology
An important class of models are described by the
linear differential equation
dy
dt
=
c
0
+
c
1
y
where the constants
c
0
and
c
1
are model parameters that have specific physical or biological interpretations.
For
example, in Section 6.1 we saw how models with
c
0
= 0 and
c
1
=
r
were used to describe exponential population
growth (
c
1
=
R
) and radioactive decay (
c
1
=
−
λ
). In this section, we discuss further applications where the constant
coefficient
c
0
is nonzero.
Mixing models
Mixing models are formulated on the premise that the density of individuals or concentration of molecules, which are
generically characterized in terms of a number of
objects
per unit area or volume, form a homogeneous
pool
such that
the flow of objects into the pool is controlled by an external constant rate while the flow of objects out of the pool
is in proportion to the density of objects in the pool. This latter assumption implies that the greater the density of
objects in the pool, the faster the total flow of objects out of the pool.
Mixing Model
Let
y
(
t
) represent the density of objects in a pool at time
t
. If objects flow into this
pool at a constant
total rate
a >
0 and out of this pool at a rate
by
(
t
)
>
0, i.e at a
constant
percapita rate
b >
0, then the density of objects in the pool over time is
governed by the equation
dy
dt
=
Rate In
−
Rate Out
=
a
−
b y.
Example
1.
Modeling HIV
Human immunodeficiency virustype 1 (HIV1) has many puzzling quantitative features. For instance, most HIV
patients undergo a 10 year period during which the concentration of virus in the plasma is very low. It is only after
this quiescent period that a patient experiences the onset of AIDS. The reason for this quiescent period is unknown,
and it was presumed that during this period the virus was relatively inactive. Using models, Perelson and colleagues
©
2010 Schreiber, Smith & Getz
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
768
6.3.
LINEAR MODELS IN BIOLOGY
quantified viral levels in the blood of infected individuals during this quiescent period.
∗
Specifically, Perelson and colleagues let the concentration of viral particles in the blood plasma be represented by
the variable
V
(
t
). They assumed that HIV viral particles infused into the blood, from production sites in lymphatic
tissue, at a constant rate
P >
0 and were eliminated from the blood at a rate
cV
(
t
), where
c >
0 is referred to as
the
elimination rate constant
. From these assumptions they obtained the mixing model
dV
dt
=
P
−
c V,
where
t
is measured in days. Both
P
and
c
are unknown constants.
a.
Data showed that after being put on a potent antiviral drug, the viral concentration fell exponentially
in the blood. Assuming that the drug killed the production of new virus completely in lymphatic tissue,
Perelson
et al.
estimated the halflife of the viral particles to be 0
.
2 days. Use this information to estimate
the elimination rate constant,
c
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '10
 WayneM.Getz
 Orders of magnitude, Constant of integration, ©2010 Schreiber

Click to edit the document details