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Comment by chipperyman573 | Hacker News Reader

It's been a long time since I took calc. What does the second derivative show us?

Acceleration.

The rate of change of the rate of change of deaths.

The word exponential only occurs once in the paper, and the rate of change of an exponential is an exponential, so take as many derivatives as you want and it's still going up. Why they don't take the log of deaths over time isn't explained.

I've seen very little coverage that really uses the term exponential in a way that suggests those reporting know the implications of that. I've seen one plot with a logarithmic y-axis, and that was in the Guardian some time ago. From it, it appeared a crude heuristic could be drawn that from the point that lockdown is implemented, the cumulative deaths appear to grow between two and three orders of magnitude.

I've been grabbing the data put out by Johns Hopkins [0] for the last few days to update some plots in a very hacky Jupyter notebook.

Crucially, the plots are normalized to (estimated) populations and the y-scale is logarithmic. I've put them on github in case anyone else is interested [1].

[0] https://github.com/CSSEGISandData/COVID-19

[1] https://github.com/dpwm/covid19-analysis/blob/master/Coronav...

edit: Clarified that though exponential growth is widely used, the implications of that are not followed up. It's past my bedtime!

Even fewer people understand logarithms than exponential growth. If you publish graphs with a log scale you will certainly confuse some people into thinking that the illness is tapering off.

Hmm...I see the term exponential everywhere, but based on comments I read on public Facebook posts, almost nobody in the general public has any clue what it means or implies. But this site is widely referenced. It has a log scale option: https://www.worldometers.info/coronavirus/country/us/

> I've seen one plot with a logarithmic y-axis, and that was in the Guardian some time ago.

You mean like the widely cited NYT Coronavirus death tracker[1]?

[1] https://www.nytimes.com/interactive/2020/03/21/upshot/corona...

yep. My first thought was why the hell they didn't they use the word acceleration for better readability.

It's not an exponential. It's a shifted sigmoid.

The logistic function specifically, which is a blend of an exponential growth and an exponential asymptote. So initially it behaves indistinguishably from an exponential function, until you get close to the inflection point.

No it's not the logistic function. Sigmoid is a general term of the shape, logistic specifically refers to an equation that's almost certainly not correct. It's probably close to stretched exponential, which is a reasonable approximation of the curve of an autocatalytic process that is controlled by stochastic collisions that can self-exhaust, but even that's not exactly correct.

As others have mentioned, acceleration, or the rate at which the rate is changing. Now... I admit to being an armchair epidemiologist. I've been graphing the data from the JHU website myself, but I've just been using semi-log plots. This is an option at some of the "dashboard" sites.

In a semi-log plot (vertical axis is logarithmic), a constant rate of exponential growth is seen as a straight line, the slope of the line is proportional to the doubling rate, and changes in that slope show that the exponential growth rate is changing. This is a way to "eyeball" the graphs without trying to read anything too profound into them. But just comparing the graphs of the US, Italy, and South Korea is interesting.

What I'm not doing is publishing conclusions from this armchair analysis.

Rate of change of rate of change.

They explain this in the paper: a constant (flat line across the graph) would correspond to exponential growth. The actual graphs drop (mostly), indicating sub-exponential growth, which is a good thing.

A constant 2nd-derivative means quadratic growth, not exponential.

I'm bit confused by the graphs, because they show the 2nd-derivative going to 0. But to stop this thing, we actually need to decelerate, eg going negative on the 2nd-derivative.

I think they're talking about the 2nd-derivative of the cumulative deaths. As they can (presumably) only increase, the first derivative is >= 0. The second derivative tends to zero as the cumulative deaths flattens off.

Edit: Now that I think about this some more, you're right. There has to be a decrease in the daily cases, which implies a negative 2nd-derivative. They mention "relative 2nd-derivative." It seems to be defined in the paper in a way that leaves me more confused:

> Daily fatality rates from the included countries were then used to calculate estimates of the relative second derivative of total deaths, N, 1/N d²N/dt², for a period of at least ten days.

Does this mean they are taking the second derivative of the reciprocal of the cumulative deaths?

They are taking the 2nd derivative of total deaths and dividing that by the number of total deaths and calling it the relative second derivative (rate of increase of rate of increase of total deaths, relative to total deaths).

The only way the velocity (number of cases) would go negative is if they discovered a disproportionate number of false positives or (number of deaths) if deaths were discovered to be due to something else [or people came back to life]. As time goes on, the velocity will reach a constant of 0, presumably. The acceleration at a constant velocity is 0.

Eventually acceleration would go to 0, but it needs to go negative first. Right now acceleration is positive and velocity is positive, hopefully soon acceleration will become negative, then eventually velocity will approach 0 from above and acceleration will approach 0 from below.

I believe that you are correct in your calculation, but incorrect in your assumption.

You are treating the curve of daily deaths as the thing to be differentiated. They are treating the curve of total deaths as the thing to be differentiated.

While the curve of daily deaths certainly would have a negative acceleration, the curve of total deaths is monotonically non-decreasing and therefore the second derivative is always positive but trending towards zero.

Like I mentioned above, the only way for total deaths to decrease (making the function not monotonically increasing and therefore capable of having a negative acceleration) would be through errors of accounting.

The goal was never to "stop" Coronavirus. Just slow it down enough that it doesn't crush the medical system.

> A constant 2nd-derivative means quadratic growth, not exponential.

A constant relative second derivative of total deaths is what they are talking about, meaning that they are flattening the 2nd derivative of the exponential by dividing by the exponential, making it constant.

Of course, this is all numerical methods, not analytical methods, so it doesn’t necessarily make pure analytical sense that they are even talking about an exponential.

1st: Velocity

2nd: Acceleration

3rd: Jerk

4th: Jounce

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