Calculus: The Language of Change

Exercise Set 2.3
The Continuous Variable Model

by Kevra Tanke

Exercise 1

What is the meaning of fractional values of S, I, and R in Excercise 2.1.1 of Section 2.1?  Are the fractional variables s, i, and r meaningful  in a small population?

Solution

Looking back at 2.1.1 we find that 

	S[0] = 10000 = number of suseptable individuals
 	I[0] = 1000 = number of infectious individuals
	R[0] = 19000 = number of removed individuals
	n = 30000 = total population
	
	s = (S/n)= (10000/30000) = (1/3)
	
	i = (I/n)= (1000/30000) = (1/30)
	
	r = (R/n)= (19000/30000) = (19/30)

Fractional values of S, I and R represent less than a complete person and we should therefore round off the values to the nearest "whole person."

s, r, and i are the fractions of the population that  are susceptable, removed, and infectious, respectively.  Representing a fraction of the total population, each of these ratios is then a smaller population.  However, if the total population is too small, say less than 30, the ratios may not be accurate because you cannot have a fraction of a person removed.  Another thing to consider is whether the population is so small that everyone comes into contact with everyone everyday.  That would also affect the ratios.  As in most experiments, larger samples generally yield more accurate findings.

Exercise 2

The total population is part of a model with S, I, and R, but not with s, i, and r.  Explain how we might use the history of a rubella epidemic with one size population to make predictions about a new and different size one.

Solution

To make predictions with a new size population use the fractional values to find the ratios of the population that  are infected, susceptable, and removed.  Use these ratios with the new population.

For example: to find the new number of suseptable people, (Subscript[S, n]), for a new population of 20000:
		    
		(10000/30000)= ((Subscript[S, n])/20000)        =>     ((10000 * 20000)/30000) = (Subscript[S, n]) = 6667

Exercise 3

Computer Program.

Exercise 4 : The Geometry of Euler's Method

Let x and y be real variables.  You are  given that  when x=0, then y=1, 

	y[0] = 1  and that  y changes with respect to x by the rule (dx/dy) = -y
	
Sketch the graph of y = y[x] by graphically using Euler's Method.

Solution

Sketch the graph for dx = 0.5.

\(Clear[f, G, x, y]; 
f[x_, y_] := \(-y\); 

    G[{x_, y_}] := {x + dx, y + f[x, y]\ dx}; 
x0 = 0; 
y0 = 1; 

    finalTime = 1; 
dx =  .5; 
steps = Floor[finalTime\/dx]; 

    Print["Time step = ", dx]\)\(solnList = NestList[G, {x0, y0}, steps]; 

    ListPlot[solnList, PlotJoined -> True]; \)

Notice how the graph is not smooth.  That  is because Euler's method approximates  the slope at a point, using the slope that it calculated it draws a line to the next point and calculates the slope again.  To get the graph looking more 'smooth' we must decrease our step size, say to 0.1.

\(Clear[f, G, x, y]; 
f[x_, y_] := \(-y\); 

    G[{x_, y_}] := {x + dx, y + f[x, y]\ dx}; 
x0 = 0; 
y0 = 1; 

    finalTime = 1; 
dx =  .1; 
steps = Floor[finalTime\/dx]; 

    Print["Time step = ", dx]\)\(solnList = NestList[G, {x0, y0}, steps]; 

    ListPlot[solnList, PlotJoined -> True]; \)