1550078481-Ordinary_Differential_Equations__Roberts_

476 Ordinary Differential Equations

EXERCISES 10.6

1. Consider the epidemic model

(7)

dS = -.OlSI

dt

dI

- = .OlSI -3I

dt

dR = 3I

dt

a. Calculate the relative removal rate for this model.
b. Use SOLVESYS or your computer software to solve system (7) on the
interval [O, 5] where t is in days for the initial conditions:

(i) So = 300 , Io = 10 , Ro = 0 (ii) So = 500 , Io = 10, Ro = 0

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

graph of I versus S.

c. From the solution or phase-plane graph of I versus S for system (7) for
the initial conditions (i) and (ii), estimate the remaining number of suscepti-

bles at the end of the epidemic. That is, estimate S= = limt-+= S(t) which

is approximately S(5) and verify that S= satisfies equation (5).

2. Suppose when an epidemic begins medical personnel inoculate members
   of the susceptible group at a rate a which is proportional to the number of

susceptibles. Then at each instant in time aS individuals are subtracted from

the susceptible group and added to the removed group. So one model for an
epidemic with inoculation is

dS

dt = - {3SI - aS

dI

dt = {3SI-rI

dR

di= rI + aS

where a, {3, and r are positive constants.