Day 18 in math modeling

Last time

Next time

Announcements
- You have a preliminary part of mini-project 3 due this Friday (your evaluation of mini-projects 2), and
- the third report itself, concerning rainfall, will be due on the Friday just before break.
- I've given you a homework assignment which will be due the Monday after break (boo!), just to get you ready for your exam (which is the Friday after break):
  "Exam prep" homework due the Monday after break, 3/12. Start early, and ask questions as needed.
Last time:
- I went over the derivation of the related difference equation solution for the SIR with Infecteds only (neither Susceptible or Recovered were present).
  The reason difference equations are important is that we solve differential equations numerically by converting them to difference equations, so it was heartening when the solution of the difference equation converged to the solution of the differential equation.
- We continued our examination of an InsightMaker SIR model for infectious disease. We got through adding in the infection flow, which was "bi-linear" in S and I:
  The idea is that the number of infected susceptibles would double if we double either of S or I. Hence, the flow looked like $r_I S I$.
  At the end of the hour, we were looking at the Mathematica file, and the differential equations that corresponded to the compartment model created in InsightMaker.
Today:
- I want make some final fixes on the SIR model. It should be as sleek and insightful as possible, so we'll be doing three things mathematically:
  - Modifying our infection term a little, so that the system scales properly with population size;
  - Scaling all our sub-populations, to make them proportions;
  - Scaling time (that sounds a little weird), so that it relates directly to the infectivity of the disease.
  What kinds of behaviors can an infection event undergo according to this model?
  It turns out that the results are fairly limited....
  I'm going to lead you through my SIR Model and DiffEQs page, which I created yesterday. Time to battle test it!:)
- Then we'll move back to InsightMaker, to finish with the sleekest, most beautiful "compartmental model" for the SIR.
  Clone the "sleek" one. And I'll tell you a little bit about it before I let you play.
- Finally we'll have a look at the same thing within Mathematica, where we can more easily see the results of changing the two parameter values that determine the dynamics.
Links:
- Chapter 1 from Mathematical Epidemiology.
- InsightMaker has a demo (here) of the SIR building process, in case you struggled last time -- or would just like to go over it again. They take you through it step-by-step.
- I have a Newton's method sidenote to show you, to illustrate how or why it works for non-linear regression: fitting a single parameter model to a single data point. I borrowed the strategy from the Mathematica demo file for Newton's Method.

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