1550078481-Ordinary_Differential_Equations__Roberts_

472 Ordinary Differential Equations

S > r / (3), we have dI / dt > 0 and the number of infectives increases. When

(JS - r < 0 (that is, when S < r / (3), we have dI / dt < 0 and the number of

· infectives decreases. The quantity p = r / (3 is call ed the relative removal
rate. Since S(t) is a strictly decreasing function, 0 :::; S(t) :::; So where So
is the initial number of susceptibles. If So is less than p = r / (3, no epidemic
occurs since dI/dt = (f3S-r)I:::; (f3So -r)I < 0 which implies I(t) is a strictly

decreasing function. That is, if So < r / (3, the number of infectives decreases

monotonically to zero from the initial value of I 0. On the other hand, if

S 0 > r / (3, the number of infectives increases from the initial value of Io to a

maximum value which occurs when the number of susceptibles has decreased

to the value r / (3 at some time t > 0. For t > t, it follows that S(t) < r / f3

and the number of infectives decreases. This result is what epidemiologists

call the threshold phenomenon. That is, there is a critical value which the

number of initial susceptibles must exceed before an epidemic can occur. The

threshold theorem for system ( 4) stated below was proven by W. 0. Kermak

and A. G. McKendrick in 1927:

THRESHOLD THEOREM FOR EPIDEMICS, SYSTEM ( 4)

If So < r/ (3, then I(t) decreases monotonically to zero.

If So > r / (3, then I(t) increases monotonically to a maximum value and

then decreases monotonically to zero. The limit, limt_, 00 S(t), exists and is

the unique solution, x, of

(5) Soe-(3(N-x)/r = x.

In order to prevent epidemics, medical personnel try to decrease the number

of susceptibles S or to increase the critical value r / (3. This is sometimes

accomplished by inoculation and by early detection of the disease followed by

quarantine procedures.

EXAMPLE 12 Computer Solution of an Epidemic Model

Solve the epidemic model

(6)

dS

• = -.005SI
  dt
  dI

dt = .005SI - .7I

~~ = .7I