MathBench > Microbiology

Frank Gets Messy: in search of Exact Doubling Time

In search of ... the exact doubling time

Poor Frank! He still has meningitis, and we still can't tell him anything very accurate about how fast the disease will progress.

So far, we've been guesstimating doubling time from a graph of population over time. But there is a problem...

Guesstimating a parameter from a graph is not really approved scientific procedure. It would be hard to imagine publishing a report in the Journal of the American Medical Association that started out "we looked at the graph, squinted a little, and decided that the doubing time was 23 minutes...".

It can be possible to calculate the doubling time from a table of data, but only if the table happens to include an exactly doubled population. If the table does NOT contain a doubled population entry, then you're in trouble. In other words, we can figure out the exact doubling time for nice timeseries below, but not for messy one:

time nice messy
10:00 am 10 million 10 million
10:20 am 20 million 15 million

Clearly, nice doubled in 20 minutes. And at first glance, it appears that messy got "halfway" to doubling, so doubling time should be 40 minutes. Let's see if that's true -- if the population keeps growing at the same rate, will it double in 40 minutes?

time messy
10:00 10 million
  ... multiply by 1.5 to get
10:20 15 million
  ... multiply by 1.5 to get
10:40 22.5 million


Oops, we got to 22.5, not 20 million. Granted, this is a small difference, but it's still not the right answer. Like the old joke goes, a million here, a million there, pretty soon you're talking about a real epidemic. What we need is a foolproof way to determine exactly what the doubling time is.