Homework in Preparation for the Midterm

In preparation for your exam, please do the following (by Monday, 3/12):
  1. We discussed how we cast non-linear problems into the linear domain, and solve an easier linear problem; then iterate (i.e., do it again, and again, and again....). My first example of this was Newton's method.

    Use Newton's method to find a root of the non-linear function \[ f(x)=3-2e^{x/2} \] (or rather, a good approximation of one), starting from an initial guess. For that, take the number of letters in your full first, middle, and last names: this creates a four digit number $-1.fml$ (perhaps $mod_10$ -- that is, if it's 10 characters or longer, subtract multiples of 10 until you get a one-digit number). So for me, the number is -1.664 (Andrew Edmund Long). \[ x_0=-1.664 \] But if your name were Engelbert Fitzgerald Humperdinck, your starting value would be \[ x_0=-1.901 \]

    1. Find the true value of the root: that is, $r$ such that $f(r)=0$.
    2. Draw the first tangent line (linear function) to the graph of $f$, through the point $(x_0,f(x_0))$.
    3. Find the root $x_1$ of the tangent line function, $T_1(x_1)=0$, and use this as ``improved'' guess.
    4. Iterate this process (that is, do it again -- though you don't need to graph the tangent line) four more times, producing a table of estimates $x_0, x_1, x_2, x_3, x_4, x_5$. Remember, these are solutions of a linear root finding problem, which is easy. What do you notice about the estimates from $x_1$ on?
    5. Discuss: Are you there yet? Or do you need to iterate more? How does your current best estimate for the root compare to its true value?

  2. Use non-linear regression to fit a
    1. Hollings Type III model
    2. Scaled logistic model
    3. von Bertalanffy model
    to the cadaver cumulative data (see the Bestiary of functions, from Ben Bolker's Ecological Models and Data in R for the definition of each model).

    I would suggest simply making the necessary modifications to my cadaver modeling code in Mathematica, with the Manipulate command perhaps helping to find good starting values. Then

    1. Compare all models with the one I chose.

  3. A reading: Chapter 1 from Mathematical Epidemiology. It gets a little technical at the end, but does a good job of summarizing some aspects of modeling epidemics using SIR models. And there are lots of cool examples! Don't worry too much when they go a little far afield mathematically....

    Write a paragraph on each of the following (two pages max)

    1. Our particular SIR is not discussed. What interesting twists (which we do not include in our SIR) do these authors discuss?
    2. What do you learn about the stability of the asymptotic solutions of SIR models? Can we conclude what happens in the long run to our sleek SIR?
    3. What are the authors' suggestions for introducing more realistic dynamics into an SIR model?
    4. What's your favorite story about a disease?:)

Website maintained by Andy Long. Comments appreciated.