Biology Reference
In-Depth Information
leading to the difference equation:
S
n
þ
1
S
n
¼
pS
n
I
n
:
This converts to
dS
pSI in the continuous model. The main idea of
this conversion is outlined next.
dt
¼
For a very small number h, choose a value of t and pick the integer n for
which nh is as close to t as possible (see Figure 2-25). Let S
n
¼
S
ð
nh
Þ
and I
n
¼
I
ð
nh
Þ
. Then S
n
þ
1
S
n
¼
S
ðð
n
þ
1
Þ
h
Þ
S
ð
nh
Þ
and
S
n
I
n
¼
S
ð
nh
Þ
I
ð
nh
Þ
.
t
Note that under the model assumptions, the probability of a susceptible
being infected by an infective in an interval of time h is now
approximated by ph. Thus:
h
h
time
(
n
+
1)
h
h2h
3
h
nh
0
FIGURE 2-25.
Locating t in a subinterval. Fix a number h >
0 and consider the intervals [0,h], [h,2h], [2h,3h],
and so on. Next, choose a value t > 0, and find
the integer n for which nh is closest to t.
Þ
~
S
ð
nh
þ
h
Þ
S
ð
nh
hpS
ð
nh
Þ
I
ð
nh
Þ:
Because nh is very close to t, this is approximately:
S
ð
t
þ
h
Þ
S
ð
t
Þ
hpS
ð
t
Þ
I
ð
t
Þ:
So:
S
ð
t
þ
h
Þ
S
ð
t
Þ
pS
ð
t
Þ
I
ð
t
Þ:
h
Taking the limit as h
!
0 gives
dS
dt
¼
S
ð
t
þ
h
Þ
S
ð
t
Þ
lim
h
¼
pS
ð
t
Þ
I
ð
t
Þ:
h
!
0
We thus obtained that the rate of change from S to I is given by the
differential equation
dS
dt
¼
pS
ð
t
Þ
I
ð
t
Þ
exactly as assumed in the SIS model
from Eqs. (2-1) and (2-2) and the SIR model defined by Eqs. (2-4) (with
p in place of
). As p is the probability that a particular susceptible is
infected by a particular infective in a unit length of time, this also
confirms the interpretation of the parameter
a
a
given in Section II.B.1.
REFERENCES
Beltrami, E. (1987). Mathematics for dynamic modeling. New York: Academic Press.
Crosby, A. W. (1989). America's forgotten pandemic: The influenza of 1918.
Cambridge, UK: Cambridge University Press.
Gause, G. F. (1927). Experimental studies on the struggle for existence. I. Mixed
population of two species of yeast. Journal of Experimental Biology, 9, 389-402.
