Applications of the Initial Value Problem y' = f(x, y); y(c) = d 141This example illustrates a common characteristic of all epidemic models
based solely on equations (1) and (2)- or equivalently on equation (3) or
( 4) - namely, regardless of how few infectives there are initially, all members
of the population acquire the disease before the epidemic ends. This is a
shortcoming of this particular model. By definition an epidemic ends when
there are no new infectives over a certain period of time. When a real-life
epidemic ends there are still many susceptibles in the population. Therefore,
the model we have just studied must be modified in order to improve the
results so they more closely reflect the results observed in real life. We will
study such models in section 10 .6.
EXERCISES 3.4
l. When an epidemic is discovered, steps are normally taken to prevent
its spread. Suppose health officials begin inoculations at a rate of i'(t)
per unit of time and this procedure is continued until the epidemic ends.
As before, we let I(t) denote the number of infectives at time t and S(t)
denote the number of susceptibles at time t. We denote the number
of people who h ave been inoculated at time t by i(t). Under these
assumptions,(5) I(t) + S(t) + i(t) = N for all t.
In this case, it is assumed that(6) dS = S'(t) = - {JS(t)I(t) - i'(t) for all t
dt
where f3 > 0 is the infection rate. The first term on the right-hand
side of equation (6) represents the rate of decrease in susceptibles due
to contact between susceptibles and infectives while the second term
represents the rate of decrease in susceptibles due to the inoculation
of susceptibles. Differentiating equation (5) and solving the resulting
equation for J'(t), we find(7) I'(t) = -S'(t) - i'(t).
Substituting for S'(t) from equation (6), we obtain(8) I'(t) = {JS(t)I(t).
Solving equation (5) for S(t) and substituting the result into equ a-
tion (8), yields the following differential equation for the number of
infectives