MathBench > Population Dynamics

Stage-based Models

Getting Better at Recovering

Now we have rates of change for the S and I compartments, but still nothing for the R compartment. This has the unfortunate side effect that while kids get sick, no one ever gets better (and that’s really unfortunate when you’re talking about pigeon pox!)

Remember that Nurse () says that the disease takes 1 week to recover from? Let’s assume that we have a bunch of sick kids. Presumably some of them got sick a week ago, some 6 days ago, some 5 days ago, and so on. If the kids are evenly spread in terms of when they got sick, then they should also be evenly spread in terms of when they get better. In other words, one-seventh of the sick kids should get better each day, or

Rt = Rt-1– 1/7 * It-1

or

delta R = 1/7 * I

and, since we know what gets added must also get subtracted from somewhere else, we’ll subtract the recovering students from the infected group:

delta I = 2 – 1/7 *I

At this point, you should be able to write down all three equations. Remember, two new students get sick every day, and 1/7 of the students that are sick get better each day. Try to jot them down quickly before you click below to see them:

(applet Delta S = -2 Delta I = +2 – 1/7 * I Delta R = + 1/7 * I Graph)

Notice that the rates add up to zero. This is what we assumed earlier – the total population size is not changing.

However, our graph still does not look like we expect an outbreak to look. Pretty soon, we’ll focus on the kids getting sick and fix that problem. But first, a few more words about iteration.