MathBench > Population Dynamics

Stage-based Models

Getting Better at Getting Sick

We intuitively expect an outbreak to start slow and gain momentum, so it didn’t make much sense to expect that the same number of kids would get sick each day. Likewise, we expect the outbreak to fade away slowly, whereas D’s original equations would have meant the outbreak kept going and going… even to the point where there are more kids sick than there are students at Hogwarts.

So what kind of assumptions would be reasonable? How does the process of getting sick work? Let’s see:

There is a common theme here – kids are infected by other kids.

When there are only a few infected kids, even if they sneeze and cough for all their worth, the total population of infected kids doesn’t increase very quickly. On the other hand, when there are a decent number of infected kids, AND a decent number of uninfected kids, then the process of infect will speed up. On the third hand, once most of the kids have already been infected, there aren’t many left to find, so the process slows down again.

The applet below demonstrates this process. We assume that the kids are coming into contact randomly (for example, there are no quarantines that would keep uninfected kids away from the general population, and there are no organized gangs of sick kids seeking out uninfected kids). Notice that the number of sick kids increases slowly at the beginning when there are few infected kids, then speeds up, and finally slows down again when there are few uninfected kids:

(applet – color code – graph)