Biology Reference
In-Depth Information
Notice that for the SIR model, the parameters
a
and
b
have the same
meaning as in the SIS model.
2. Does an Epidemic Occur?
Recall that an epidemic occurs if the number of infectives in the
population increases. We now examine this question for the SIR model.
For the SIR model described by Eqs. (2-4),
dI
dt
¼ a
SI
b
I
¼
I
ða
S
bÞ:
Thus, the number of infectives will increase when
a
S(t) -
b >
0; that is,
when
a
S(t)
> b
, and decrease when
a
S(t)
< b
. Because S(t) is largest
when t
¼
0, an epidemic will not take place if
Þ <
b
a
S
ð
0
Þb <
0or
;
equivalently
;
S
ð
0
a
:
We shall see that this is related to the average number of new infectives
that each infective causes (i.e., the number of secondary infections).
We would like to know how many susceptibles are infected by a typical
infective. Upon reflection, we might decide that this depends on
how many susceptibles are available and how long the infective is
available.
E
XERCISE
2-5
Are there any other assumptions or factors that should be considered in
determining how many susceptibles an infective can infect?
To estimate the average number of secondary infections, consider the
following example. Assume that the average number of infections
caused by 1 infected individual per unit time is 3 per hour. Then, if the
infective remains sick, on average, for 5 hours, he or she would infect
ð
5 hours
Þð
3 susceptibles per hour
Þ¼
15 susceptibles
:
In the SIR model, the rate of new infections is given by
dS
dt
¼a
SI.
Recall that the outflow from S equals to the inflow to I. Thus, the rate of
new infections is given by
a
SI
¼
(
a
S)I, meaning that the (average) per
capita infection rate at time t is
a
S.
0,
dS
If I(0)
. Since S(0) is
the largest value of S(t), one infected individual can infect, on average,
no more than
¼
1, we obtain that at t
¼
dt
¼a
S
ð
0
Þ
I
ð
0
Þ¼a
S
ð
0
Þ
a
S(0) susceptibles per unit time. Recall now that the
