Calculus: The Language of Change

Exercise Set 2.6
Calculus and the S-I-R Invariant

by Kevra Tanke

Exercise 1 : Math Nerd Fever

At the beginning of the semester last year, 96% of the students were not interested in calculus, but word of this chapter spread around campus infecting the population with a mythical disease Math Nerd Fever.  Victims of the disease find that they become interested in calculus - and stay that way, though they are only successful in spreading the infection for a brief period when they are first excited about the chapter.  At the end of the last academic year, 44% of the population was interested in calculus.  What is the contact number for Math Nerd Fever?  On the average, how many new math nerds does each infected student create?

Solution

From the text we know that c (the number of contacts) is approximately:

	c ≈ \(TraditionalForm
      \`\((Log\ [s[∞]] - Log[s[0]])\)/
        \((s[∞] - s[0])\)\)

We are given that 96 percent are initially susceptable so s[0] = .96 and that the final value is 56 percent (100-44) so s[∞] is .56.

c = ((Log [ .56] - Log[ .96])/( .56 -  .96))

So the average total number of contacts per infectious person is about 1.3.

Exercise 2

Suppose we measure a and b in a college population for a new strain of flu.  What limitations would there be in using these parameters  in predicting an epidemic in a nursing home?

Solution

The daily rate of contacts per infective, a, for a college population would most likely be larger than that for a nursing home since the student population would probably be larger and more mobile.

The probable difference in age may or may not affect the number of days infective, so b may not be the same value for both groups.

Also to be considered: the initial values of s, i and r may be different for the two populations.

Exercise 3

Computer Program