Table of Contents

• Section 1. Frank's Story
• 1: The story
• 2: Biological interlude
• Section 2. How the meningitis population grows
• 3: How many meningococci can kill you?
• 4: Build the basic growth equation
• 5: Draw exponential growth
• Section 3. So why isn't Frank dead already?
• 6: How long has poor Frank got?
• 7: All we need now is data
• 8: Looking at data
• 9: Log transform straightens out exponential growth
• 10: Do ALL curves get "straightened" by taking the log?
• 11: Phases of bacterial growth
• Section 4. How can we tell when the population is growing exponentially?
• 12: A test for exponential growth
• 13: And the answer is...
• 14: No way its growing that fast!
• 15: Some more practice
• 16: In search of ... the exact doubling time
• 17: Finding doubling time with messy numbers
• 18: Logs = Exponent-busters
• 19: Let's bust some exponents
• 20: Let's bust some exponents
• 21: Extended problem #1: Exponential growth ends (with a whimper)
• Section 5. Back to Frank
• 22: Extended Problem #2: How long DOES poor Frank have?

In search of ... the exact doubling time

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, that 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.