Biology Reference
In-Depth Information
V. EPIDEMIC MODELS WITH DELAY AND MODELS
WITH INTERMEDIATE GROUPS
A. Models with Delay
As already noted, one of the advantages of the SIR model is that it allows
for immunity. However, the model is built on the assumption of
permanent immunity upon recovery, and this assumption, although
reasonable for certain diseases, is not accurate for others. For certain
diseases, such as brucellosis or the sexually transmitted diseases
gonorrhea, chlamydia, and syphilis, only limited immunity is conferred
by having had the disease. This immunity is lost after some time interval
has elapsed, and the length of this interval varies, depending on the
disease. We shall model this situation using an SIR model with delay. As
a first approximation, we assume that the length of this temporary
immunity is constant—those who have recovered from the disease
lose immunity after a fixed time D
>
0.
Suppose, for example, that when an individual recovers he or she
maintains immunity for 60 days. Consider how S is changing at a time t:
as before, S is decreasing in size because of individuals who are
becoming infected, and the rate at which this happens is proportional to
S(t)I(t). Now, however, S is also increasing because of the individuals
who have recovered and have subsequently lost their immunity. The
rate at which this happens is exactly the rate at which individuals
moved from the infected to the recovered group 60 days before t, and
this is
I(t - 60). The block diagram representing this model is depicted
in Figure 2-13. The delay of length D is represented by a triangle on
the arrow from R to S.
b
The following equations mathematically describe these modifications to
the SIR model:
dS
dt
¼a
S
ð
t
Þ
I
ð
t
Þþb
I
ð
t
D
Þ
dI
dt
¼ a
S
ð
t
Þ
I
ð
t
Þb
I
ð
t
Þ
(2-10)
dR
dt
¼ b
I
ð
t
Þb
I
ð
t
D
Þ:
D
a
SI
b
I
S
I
R
FIGURE 2-13.
Schematic representation of an SIR model with temporary immunity. The recovered individuals
remain in R for a fixed time D that corresponds to temporary immunity. After time D, the recovered
individuals are again susceptible to the infection.
