1550078481-Ordinary_Differential_Equations__Roberts_

474 Ordinary Differential Equations

on the rectangle 0 ~ S ~ 500 and 0 ~ I ~ 200. This phase-plane graph is
shown in Figure 10.18. Observe from the graph that S approaches 15 as I
approaches 0. Substituting S 0 = 500, /3 = .005, N = So + Io + Ro = 505,

x = 15 , and r = .7 into the left-hand-side of equation (5), we find

Soe-f3(N-x)/r = 500e-^3 ·^5 = 15.099.

Since this value is approximately x, we have verified that the limiting value
of the number of susceptibles at the end of the epidemic is approximately 15.

That is , limt___,= S(t) = 15. Thus, for this epidemic model, system (6) with

initial conditions (i), we have verified equation (5) of the threshold theorem.

200~~~~~~

150

I 100

50

O-+-~~~~~~~~~~~~~~~~~~~~~~~~ ...........

0 100 200 300 400

s

Figure 10. 18 A Phase-Plane Graph of I versus S for System (6)

for Initial Conditions (i)

500

(ii) Since we now wanted to solve the same system of differential equations

with different initial conditions, we entered the new initial conditions: S(O) =

100 , I(O) = 100 , and R(O) = 0. A graph of S , I , and R is displayed in
Figure 10.19. Notice that S decreases monotonically from the value of 100 to
approximately 30 , I decreases monotonically from the value of 100 to nearly 0,
and R increases monotonically from 0 to approximately 170. Since I decreases

monotonically to 0, no epidemic occurs in this instance. Verify that x = 30 is

the approximate solution of equation (5) for the given initial values. Hence, in
this case, as I , 0, S , 30. Thus, there are 30 susceptibles remaining when

the epidemic ends. A phase-plane graph of I versus Sis shown in Figure 10.20.