Biology Reference
In-Depth Information
S
ð
t
Þ¼
the number of susceptibles at time t
;
I
ð
t
Þ¼
the number of infectives at time t
:
With this notation, the assumption that the size of the population N
N, so that
dS
dI
dt
¼
remains fixed translates into S(t)
þ
I(t)
¼
dt
þ
0
:
We need
equations that describe how each of the groups changes. Earlier, we
noted that the susceptibles become infected at the rate
a
IS. The number
a >
0 is called the infection rate. It can be interpreted as the probability
that a particular susceptible is infected by a particular infective within a
unit of time. We justify this interpretation in Section VIII.
Individuals who recover from the disease rejoin the group of the
susceptibles immediately. If we assume that the infectives recover at a
constant per capita rate
b
, then the recovery rate is
b
I. That is, the flow
from I to S occurs at a rate
I. Accounting for outflow and inflow, the
rate of change for the size of S is therefore given by:
b
dS
dt
¼a
SI
þ b
I
:
(2-1)
Now, because
dS
dI
dt
¼
dt
þ
0, the rate of change for I will be:
dI
dt
¼ a
SI
b
I
:
(2-2)
b
I
The mathematical model composed of Eqs. (2-1) and (2-2) is often called
the SIS model. A pictorial representation of the model is given in
Figure 2-3. The rectangles represent the different groups, and the arrows
represent the flows between the groups. Each arrow is labeled with the
rate of flow between the groups. Such diagrams are often helpful in
formulating mathematical models.
S
I
a
SI
FIGURE 2-3.
Schematic representation of the SIS
model. S and I represent the susceptible
and infected populations, a is the
infection rate, and
b
is the per capita
We next focus on the meaning of parameter
b
and the long-term
recovery rate.
behavior described by the SIS model.
2. Interpreting the Parameter
b
The per capita recovery rate
in the SIS model is related to the average
length of the infection, d, in the following way:
b
1
b
:
d
¼
Thus, the smaller the value of
b
, the longer lasting the disease would be
on average.
As an optional reading, we next present a mathematical justification for
this relation. It requires a certain higher level of calculus proficiency, and
its omission will not affect the subsequent sections.
