Biology Reference
In-Depth Information
I
ð1Þ ¼
lim
t
I
ð
t
Þ¼
N
S
ð1Þ
R
ð1Þ
!1
exists as well. There are two possibilities for I(
1
): either I(
1
)
>
0or
I(
1
)
¼
0. We shall show that I(
1
)
¼
0.
Assume I(
1
)
>
0.
Because
dR
dt
¼ b
I,ifI(
1
)
>
0,
then we would have
dR
dt
¼ b
lim
t
I
ð1Þ >
0
;
!1
and then R(t) would have to go to infinity (see Exercise 2-8, below). This
is impossible, because R(t)
N for all values of t. Thus, it is impossible
for I(
1
)
>
0, and this implies that the alternative I(
1
)
¼
0 holds. That is,
the disease dies out.
Combined with our earlier result that S(
0, this means that the
disease dies out because all infectives have been removed from the
population, and not because all susceptibles have been infected.
1
)
>
E
XERCISE
2-8
dR
dt
>
If lim
t
!1
0, why does this mean R(t) would have to go to infinity?
It is unusual to find actual populations as isolated as hypothesized in
the SIR model. One famous example occurred at an English boarding
school in 1978, as described in The Communicable Disease Surveillance
Centre Report (1978). We present this example in Figure 2-6.
Summary. To summarize the results of the SIR model:
Þ >
b
1. We have an epidemic if, and only if, S
ð
0
a
;
2. The average lifetime of an infection is
1
b
;
3. Under optimal conditions, the average number of secondary infec-
tions one infective can produce in a fully susceptible population is
a
b
ð
Þ;
S
0
4. The disease dies out, and not all susceptibles will catch the disease.
In the SIR model, it is impossible to solve explicitly for S(t),I(t), and R(t).
This is typical for coupled systems of nonlinear differential equations.
Furthermore, what is often most important to know is how one group
responds to a change in the other groups. When only two groups are
