Table of Contents

• Did Harry Potter ever get the measles?
• Week 1 and the Search for a Crystal Ball
• 1: Week 0: The crystal ball
• 2: Compartmentalizing the students
• 3: A Muggle Interlude: Models
• Week 2: The plot thickens...
• 4: ...for a week
• 5: Muggle Interlude: Compartments
• 6: A really simple model
• 7: Iterating the Simple Model
• 8: Getting Better at Recovering
• 9: Iteration vs. Closed Forms
• 10: Don’t get them confused!
• Week 2.142057149: Crisis at Hogwarts
• 11: Students dropping like flies
• 12: Getting Better at Getting Sick
• 13: Writing Equations for Contact
• 14: Week 2.142057149…: Crisis at Hogwarts
• 15: The Whole Enchilada
• 16: Comparing Rate-of-Change to Closed Form Equations (again)
• 17: When will the pox peak? And will the tournament go on?
• Week 3: Exploring the model
• 18: Week 3: The mathemagician gets busy
• 19: How many people got sick in the end?
• 20: Do initial conditions matter?
• 21: Does recovery time matter?
• 22: Does transmission rate matter?
• 23: Does quarantine matter?
• 24: Review

Don’t get them confused!

OK, one last thing before we head on. Iterative (or rate-of-change) equations are QUITE DIFFERENT from closed-form equations. Consider the following:

Delta I = 2 – 1/7* I

I(t) = 2 – 1/7 * t

These two look kinda similar. The one on the right you should be pretty comfortable with. It’s just a line with a shallow negative slope. You’ve been doing those since highschool at least. But the one on the left you may not have encountered much, unless you’ve studied differential equations. You might expect its graph to look like a straight line, BUT IT WON’T.

Let’s think about this a little. When you move forward a timestep, you’re always going to add 2, but you’re ALSO going to remove 1/7th of what was there before. So on balance, does the compartment increase or decrease? Or better yet, when does it increase or decrease?

• What if I has only a few kids in it? For example, I=7, then one kid recovers but 2 get sick, so I grows. In fact, if 1/7 of I is less than 2, then the compartment will increase – in other words, if I < 14.
• Likewise, if I > 14, then the compartment will shrink because more kids will get well compared to the 2 that get sick.
• Right at I = 14, the getting sick and getting well balance out and the compartment doesn’t change size. So 14 is what we call an equilibrium – the forces balance out.

So, since we start with a very small compartment (1 kid), we know it has to grow, until it reaches 14. At that point, the curve will level out – or, if the compartment overshoots its equilibrium, it will get drawn back towards it (remember, the compartment will shrink if it gets bigger than the equilibrium size).

Below is an applet to play with. See if you can predict the equilibrium before clicking on the graph button.

(applet Slider input Slider recovery rate Free play button Graph button Predict equilibrium textbox )

So to conclude, finish up, end, and wrap up this section, just remember that the methods for dealing with closed form equations are VERY DIFFERENT from those that we use with iterative (or rate-of-change) equations. Rate-of-change equations are more flexible, and fairly simple formulations lead to surprisingly complex, curvy, biologically interesting graphs.