Estimating the Infection Fraction

So, how did someone come up with the value 0.0151 for the Infection Fraction? Basically by mixing observation and some elementary calculus.

First, let's agree on some abbreviations for our variables. Let S be the number of susceptible people and I get the number of infected people. Call the infection fraction k. Let t represent time. We will also abbreviate the rate Get Sick as GS. Recall that the formula for Get Sick is:

GS = S * I * k

Since GS is a rate, it represents a change over time. Here, the change is the number of infected people per unit of time is equal to the number of susceptible people times the number of infected people times the infection fraction. Using notation from Calculus, this can be written:

dI/dt = S * I * k

dI means "the change in I" and dt means "the change in t."

Finding values for S and I is not too hard, but what k should be is less obvious. One way to calculate a reasonable value for k is to estimate it based on personal experience. If one sick person comes to school, how many students would you expect to get sick each day?

Suppose you decided that in two days, three new students would be sick. We can now substitue 3 for dI and 2 for dt in the equation above to get

3/2 = S * I * k

We also know that at S=99 and I=1 at time zero. So,

3/2 = 99 * 1 * k

Now, solving for k indicates what the infection fraction should be so that in two days there will be three new sick students.

k = (3/2) / (99 * 1) = 1.5 / 99 = 0.01515...

This is a repeating decimal, but rounding off to 0.0151 is more than generous if you consider significant figures.

If you plan to go through this calculatioon in a classroom setting, consider polling students about how long it would take for people in the model to get sick, averaging their responses, and using that estimate to calculate k.