So, how far & fast does a virus spread?

Here’s a crash course on the subject.

In a press conference last week, Gov. Cuomo started talking about the “virus reproduction rate” and, channeling Germany’s Chancellor Merkel, declared that NY can’t be reopened until it is under control. WSJ

Sounds reasonable, right?

Yeah, but what the heck is he talking about?


Fasten your seat belts and let’s do some fun math today…


Let’s start with the basics…

The “virus reproduction rate” is a number that indicates how many new cases (“infectees”) an infected person generates.

For example, does an infected person infect 2 other people (e.g. as was evidenced with Ebola)? 3 other people (e.g. SARS)? 15 other people (e.g. measles)?

In math-speak, the “number” is symbolically coded as “R0” … which is pronounced “R naught” or “R zero”.

The viral reproduction process starts with one infected person … called a “viral seed” or “patient zero”.

Think: the first infected traveler from Wuhan or Europe who landed in the U.S.

That person infects others … who infect others … who infect others.


Again, R0 is the number of other people that an infected person infects.

So, for example, if R0 =3 … then the viral seed person infects 3 other people in the first cycle (called a “phase”) of the virus transmission.

Stat note: It’s estimated that each coronavirus phase is about 1 week long. Source

In the 2nd phase, the 3 new infectees each infect 3 more people.

In phase 3, each of those infectees infects 3 more … and on and on and on … until all susceptible people have been infected or an intervention terminates the virus (e.g. a vaccine).

Got it?

Here’s how the process looks visually for 3 phases with an R0 =3:

Image source


OK, let’s talk coronavirus…

The R0 for the coronavirus is estimated to be between R0=2 and R0=3.

Technical note: Experts concede that calculating R0 after-the-fact is a challenge … and its nearly impossible to calculate in real time when a virus is mid-stream in the transmission process. So, estimates are really guesstimates derived from other incidence data.

Now, let’s work the numbers…

If we split the difference and assume that R0=2.5 … then in 3 phases, there would be a cumulative total of about 25 infectees.


That’s simple enough, right?

But, in mid-stream (as we are now), there isn’t just 1 “seed” infectee … there are many infected people in circulation … each of whom is a spreader.

Say that we did random diagnostic surveillance testing in our local area and ID’ed 50 or 100 infected people.

Here’s how the numbers move…


Note that 100 seed infectees would give us 2,538 cumulative infectees over 3 phases.

Math note: You just have to multiply the number of infectees in the single seed scenario (the left-most column) by the number of seeds in the higher number of seed scenarios.


What happens if the process runs more than 3 phases?

Technical note: The process would continue until the entire susceptible population has been exposed to the virus … or until there is an intervening terminator (e.g. a vaccine).

For example, over 8 phases (about 8 weeks for the coronavirus), with 100 starting “patients zero”, there would be over 250,000 infectees.

That’s what’s called exponential growth … or what finance people call the “miracle of compounding”.



What if there are more than 100 people currently infected?

Simple: Just multiply the numbers in the single seed cumulative (“cume”) column by the estimated number people who are currently infected.

For example, if there are 1,000 people currently infected, multiply 2,542 times 1,000 to get 2,542,000.


What if R0 is something other than 2.5?

Simple: Just use the below table … pick your R0 and number of phases, then multiply times the estimated number people who are currently infected (in phase 0).


Note that when R0=1, the virus is still spreading, but is relatively under control.

Technical note: When R0 is less than 1, the virus eventually stops spreading.

Also note the HUGE difference between the cases where R0=2.0, 2.5 (prior chart), and 3.0.

The 8th phase numbers increase exponentially from 511 to 2,542 to 9,841.

Takeaway: Relative small differences in R0 values have a big impact on viral spread.


That’s the math, but what actually drives the R0 virus reproduction rate?

Answer: Several factors, including …

· The inherent infectiousness of the virus … which, for the coronavirus is very high

· The length of time that an infected person is contagious … for the coronavirus, generally assumed to be about 2 weeks, starting about 2 days before symptoms are present

· The size of the susceptible population … for the coronavirus, it is assumed that practically everybody is susceptible … children may be less susceptible than adults

· The density of the susceptible population … which increases the odds that an infected person bumps into a susceptible person … and which is vastly different between locales (e.g. Montana versus NYC)

· The number of contacts that an infected person has with susceptible people … or vice versa, the number of contacts that a susceptible person has with infected people.

· The nature of the contacts that are made … are they distant and incidental or close and prolonged?

· The mitigation actions that are taken … most notably: wearing masks or safe distancing

Note that the first 3 factors are largely “givens” that we can’t control … and, the 4th factor — local density – can only be controlled by quarantine (i.e. in effect, redefining your local area) or moving to a less dense area.

But, the last 3 factors are largely under our individual control (unless government mandated).

Bottom line: Stay safe, take control !


P.S. To quant jocks…

For great dynamic graphics and a nifty interactive estimation tool, see:
“How coronavirus spreads through a population and how we can beat it”

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