With Michigan and most other places in the world sheltering from COVID-19, I thought it might be interesting to talk about exponential growth, and how it relates to disease spread. Disease is transferred from one person to others, so the number of new people who get a disease is related to the number who have it now. We can write that as a differential equation:
N is the number of people with the virus, and the dot indicates the change in time.
τ is a time constant that tells how quickly the disease spreads. Solving this equation gives
At
t = 0, we have one case, and after every
τ, the number increases by a factor of
e (= 2.71828).
A couple days ago, the blog Hack-a-Day (which I've
mentioned before) had a
post about getting realtime data about COVID-19 statistics. I wrote up a Python
script to pull the data we're interested in, and make some plots. First, we can look at the total number of US cases over time:
According to the equations above, if we look at the increase from day to day, it should be linearly related to the number of cases on that day. Plotting gives
There's some variation, but overall it's fairly close to the fit line. According to that fit, we have
τ = 4, meaning the number of cases increases by
e every 4 days. To put that in more relatable terms, that's doubling the cases every
2.8 days!
Looking at the US cases compared to the global total,
we can see that the global rate is beginning to taper off compared to the US, suggesting we should do more to limit the spread!
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