Biology Reference
In-Depth Information
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Date of 1978
FIGURE 2-6.
Data from an influenza epidemic in a boarding school in England. The solid line represents the
solution for the infectives (I) in a SIR model with S
0
¼ 762, I
0
¼ 1, N ¼ 763, b ¼ 0.0022, and
a ¼ 0.455. (From a report by the Communicable Disease Surveillance Centre and the
Communicable Disease [Scotland] Unit in the March 4 issue of the British Medical Journal, 1978, p.
587, reproduced with permission from the BMJ Publishing Group.)
involved, a phase diagram can be very helpful, and we describe this
concept next.
III. PHASE PLANE ANALYSIS
b
S
a
SI
g
I
A. The Phase Plane
S
I
Births
Deaths
To demonstrate the phase plane technique, we consider a slightly
modified SIS model with specific values for the constants. In this
version, we allow for births to the susceptibles and consider the
infectives to be removed by dying, meaning the population is no longer
fixed. The schematic representation of the model is presented in
Figure 2-7, and the differential equations are:
FIGURE 2-7.
Schematic representation of the model
represented by Eq. (2-5). S, I, and
a
are as above,
and
represent the per capita rates of birth
and death, respectively.
b
and
g
dS
dt
¼a
SI
þ b
S
(2-5)
dI
dt
¼ a
SI
g
I
;
where
0 is the per capita death
rate (corresponding to the recovery rate of the earlier model).
b >
0 is the per capita birth rate and
g >
Before proceeding further, we note an important mathematical fact and
establish some vocabulary. The mathematical fact is this:
