Covid-20 math puzzle

A microbe, let’s call it Covid-20 can spawn a new microbe at any time. The probability of a particular microbe spawning at any given time is always the same, regardless of the time since the last spawning. The average time between spawnings from the same microbe is one day.

Once a microbe enters your lungs, what is the expected number of microbes will you have after seven days?

Does anyone not understand what's being asked?

Quote: Wizard

Does anyone not understand what's being asked?

The only thing I don't understand is, what is meant by "an average rate of one per day." Average in terms of what?

Even if you say, "If there are an infinite number of them created simultaneously, the mean time before they reproduce is 24 hours," what is the curve of time to probability of reproducing (or, alternatively, not reproducing) by that time?

The easy guess is 2^6=64 microbes -- but that sounds too obvious.

Phrases that cause some ambiguity for me: "average rate", "same rate", "memory-less property"

I should have taken more mathematics classes in college.

Quote: Wizard

Does anyone not understand what's being asked?

Yes!

If a microbe always creates another one 24 hours after it was born, then clearly the number of microbes just multiplies by 2 every day. After 7 days there are 2^7=128 (assuming we're talking Monday 0h01 to Sunday 25h59).

However suppose a microbe either creates another one 12 hours or 36 hours, then the ones born earlier have a chance of multiplying earlier, which has a much bigger effect on the growth rate than the ones born later (average about 1154). As a silly idea suppose a microbe can either replicate after 1 minute (P=1/9) or about 8 days (Pr=8/9), then the average is 1 day but on average a new one will be born every 9 minutes.

btw I don't know how to work it out (I'm too lazy to do integrals etc.) if, as I suspect, it's an exponential distribution.

Quote: Wizard

Does anyone not understand what's being asked?

Me.

I’m sure the answer isn’t 2^6=64.....

I said the AVERAGE time between spawninngs is one day. That implies is not is always exactly one day. Otherwise, the answer would indeed be 64.

I just changed the wording as follows:

Quote:

A microbe, let’s call it Covid-20 can spawn a new microbe at any time. The probability of a particular microbe spawning at any given time is always the same, regardless of the time since the last spawning. The average time between spawnings from the same microbe is one day.

Once a microbe enters your lungs, what is the expected number of microbes will you have after seven days?

I'm trying to ask this without using mathematical jargon.

The time between spawnings is similar to the time between royal flushes in video poker if you averaged getting one per day. On any given day you might get zero or you could get 10, or anything. The difference is that with royal flushes there are only so many opportunities to get them in a day, about 40,000, but with the microbe there is no limit and they can happen anytime, even a nano-second after the last spawn.

Quote: gordonm888

Show Spoiler

If N is the number of microbes at any time t, the instantaneous growth rate is given by:

dN/dt = + N/t_m

where t_m is the mean time to double/spawn

Integrating between any arbitrary zero time, N_o and a later time t gives us:


N = N_o e^λt

where λ = 1/t_m

So, starting with 1 microbe, after 7 days there should be e⁷ = 1096.6 microbes approximately.

I agree!

Here is my solution (PDF)

Gordon, I would be interested in your thoughts on how to word the problem in plain simple English. I am having a hard time getting across the point how the microbes grow at an exponential rate.

Quote: Wizard

Quote: gordonm888

Show Spoiler

If N is the number of microbes at any time t, the instantaneous growth rate is given by:

dN/dt = + N/t_m

where t_m is the mean time to double/spawn

Integrating between any arbitrary zero time, N_o and a later time t gives us:


N = N_o e^λt

where λ = 1/t_m

So, starting with 1 microbe, after 7 days there should be e⁷ = 1096.6 microbes approximately.

I agree!

Here is my solution (PDF)

Gordon, I would be interested in your thoughts on how to word the problem in plain simple English. I am having a hard time getting across the point how the microbes grow at an exponential rate.

I thought you worded it almost perfectly. I immediately recognized from your write-up that this problem was identical to radioactive decay of an unstable atom, except that it involves growth -hence exponential growth.