1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations 473

on the interval 0 :::; t :::; 10 days for the initial conditions

(i) So = 500, Io = 5, Ro = 0 (ii) So = 100, Io = 100, Ro = 0

In each case, display S, I, and Ron a single graph and display a phase-plane

graph of I versus S.

SOLUTION

(i) Identifying S with Yi, I with Y2, and R with y3, we ran SOLVESYS by

setting n = 3, fi(t, Yi, Y2, y3) = -.005yiy2, h(t, Yi, Y2, y3) = .005YiY2 - .7y2,

and f3(t, Yi, y2, y3) = .7y2. We input the interval of integration as [O, 10]

and input the initial conditions: Yi (0) = 500, Y2 (0) = 5, and y3 (0) = 0.

After integration was completed, the graph of S , I, and Ron the rectangle

0 :::; t :::; 10 and 0 :::; S, I, R :::; 500 shown in Figure 10 .17 was displayed on the
monitor. Notice that S decreases monotonically from 500 to approximately 15
and R increases monotonically from 0 to 485. Also observe that I increases
monotonically from 5 to a maximum value of approximately 190 when t is
approximately 3 days and then I decreases monotonically to 5 as t approaches
10 days. Since I increases before decreasing, an epidemic occurs.

400

300

200

100

o~~::__~~~~==:=:::=::'.::::::::=:;~~

0 2 4 6 8

Figure 10. 17 A Graph of S(t), I(t), and R(t) for System (6)

for Initial Conditions (i)

10

Since we still wanted to display a phase-p lane graph of I versus S, we
indicated to SOLVESYS that we wanted S(t) assigned to the horizontal axis
and I(t) assigned to the vertical axis and that we wanted the graph displayed