MathBench > Population Dynamics

Stage-based Models

Iteration vs. Closed Forms

With our first set of equations

Delta S = -2

Delta I = +2

You might have thought this was just an idiosyncratic and frustrating way of expressing two straight lines, and in a sense you would have been right. With a system that simple, there’s not much point in using rates of change – it would be just as easy to write

S = 600 – 2t

I = 2t

You would get the same graph, and as an added bonus, it would be a lot easier to evaluate the system on, say, day 300. These kind of equations are called “closed-form” expressions. Closed-form expressions are generally easier to work with than iterative equations, but its not always (or even often) possible to come up with a closed-form way of expressing a given process.

So, iterative expressions have the great advantage of being extremely flexible. Take the rate of change of infected kids: I can make it depend on how many kids are already infected, or how many susceptible kids there are, or what the temperature is today (assuming I have a time series of temperature), or even something totally extraneous like the length of Dumbledore’s beard:

Delta I = + 0.5 S – 1/7*I + 2*(centimeter of rain) + 5 (length of Dumbledore’s beard in meters)

This says that half of the still-susceptible students will get sick each day, 1/7th of the sick ones will get well each day, AND two extra students will get sick for cm of rain, AND 5 extra for each meter in length of D’s beard. It would be impossible to come up with a closed-form expression for this, but its easy to write a rate-of-change, or iterative, equation.

Very important caveat!!!!