1550078481-Ordinary_Differential_Equations__Roberts_

140 Ordinary Differential Equations

of these differential equations can b e solved and the remaining function (I(t)

or S(t)) can be determined from equation (1). Observe that equation (4),

I' = f3N I - (3!^2 , has the same form as the logistic law model, P' = kP - EP^2 ,

which we studied in the previous section.

EXAMPLE 1 Solution of a Simple Epidemic Model with

with One Initial Infective

Solve numerically and graph the solution of the differential equation

(4) I'(t) = f3(N - I(t))I(t)

for the number of infectives, I(t), on the interval [O, 10] if f3 = .002, N = 1000 ,

and I(O) = 1. What is limt-+ooI(t)? What is limt-+ooS(t)?

SOLUTION

We associated I(t) with y(x) and ran SOLVEIVP by setting f(x, y) =

.002(1000 - y)y, by inputing the interval of integration as [O, 10], and by

inputing the initial condition y(O) = 1. A graph of the solution is shown in

Figure 3.11. Notice that as t approaches 6, I(t) approaches 1000. And since

I(t) + S(t) = N = 1000 , as t approaches 6, S(t) approaches 0.

(^1000) I(t)
800
600
400
200
(^0) t
(^0 2 4 6 8 10)
Figure 3.11 Numerical Approximation to the IVP:
y' = .002(1000 - y)y; y(O) = 1