Biology Reference
In-Depth Information
E
XERCISE
2-2
b
Let N
:
(a) Show that if I
a
>
0
b
then the number of infectives is
increasing with time, and the number of susceptibles is decreasing
with time. Show that lim
t
!1
ð
0
Þ <
N
a
;
b
a
Þ¼
b
I
ð
t
Þ¼
N
and thus lim
t
!1
S
ð
t
a
:
b
(b) Show that if N
then the number of infectives is
decreasing; the number of susceptibles is increasing; and, once
>
I
ð
0
Þ >
N
a
;
Þ¼
b
again, lim
t
!1
S
ð
t
a
:
b
(c) What happens for I
ð
0
Þ¼
N
a
?
Figure 2-4 shows typical trajectories for the SIS model. For certain
values of the parameters, the trajectories of both S(t) and I(t) stabilize
at nonzero levels, meaning that the disease does not die out but
remains endemic in the population. Thus, for situations where
accounting for the continuing presence of disease is important,
the SIS model may offer a good starting point for describing the
dynamics of the epidemic.
E
XERCISE
2-3
Criticize the SIS model. What assumptions were made to create the
model that may not be quite realistic? Suggest improvements and
refinements for the model.
1200
1000
800
600
400
200
0
0
2
4
6
8
10
12
14
16
18
20
Time
FIGURE 2-4.
Progression of disease in SIS model. Solution trajectories for S(t) (solid line) and I(t) (dashed line) for
the SIS model with initial conditions S(0) ¼ 1000 and I(0) ¼ 10 and parameter values a ¼ 0.001,
b ¼ 0.22.
