7 ms·

An international comparison of the second derivative of Covid-19 deaths [pdf]

58 points by hdivider 7y ago

hdivider 7y ago
Full title (not possible to put as title because of HN limits): An international comparison of the second derivative of COVID deaths after implementation of social distancing measures
chipperyman573 7y ago
It's been a long time since I took calc. What does the second derivative show us?
- deleted 7y ago
  [deleted]
- deleted 7y ago
  [deleted]
- lsb 7y ago
  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.
  - deleted 7y ago
    [deleted]
  - dpwm 7y ago
    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 https://github.com/CSSEGISandData/COVID-19 [1] https://github.com/dpwm/covid19-analysis/blob/master/Coronavirus%20Deaths.ipynb 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!
    - mr_toad 7y ago
      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.
    - dboreham 7y ago
      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/ https://www.worldometers.info/coronavirus/country/us/
    - grandmczeb 7y ago
      > 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/coronavirus-deaths-by-country.html https://www.nytimes.com/interactive/2020/03/21/upshot/corona...
  - 3solarmasses 7y ago
    yep. My first thought was why the hell they didn't they use the word acceleration for better readability.
  - babesh 7y ago
    It's not an exponential. It's a shifted sigmoid.
    - the8472 7y ago
      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.
      - dnautics 7y ago
        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.
- dboreham 7y ago
  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.
  - zeroimpl 7y ago
    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.
    - dpwm 7y ago
      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?
      - 9wzYQbTYsAIc 7y ago
        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).
    - 9wzYQbTYsAIc 7y ago
      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.
- fortran77 7y ago
  1st: Velocity 2nd: Acceleration 3rd: Jerk 4th: Jounce
- analog31 7y ago
  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.
stefan_ 7y ago
This looks like one of those P versus NP papers.
bernardv 7y ago
Very little details on the methodology
- dboreham 7y ago
  How so? They said they computed the second derivative. What more is there to say?
  - willis936 7y ago
    You can’t compute the second derivative of data from May 2020 when it is currently March 2020. Where did the data come from?
    - 9wzYQbTYsAIc 7y ago
      China data time shifted to match the country of interest.
      - softwaredoug 7y ago
        That seems to make a pretty big assumption western countries can achieve that? My understanding even a small amount of non compliance has big consequences...
    - Someone 7y ago
      FTA: The cumulative deaths for each country, N(t), were estimated by deriving a multiplier Nfinal/N(tconv) at the time of convergence, tconv, to the Chinese trajectory, Nfinal being the total number of deaths in China and applying this multiplier to Chinese data for times beyond convergence.
lonelappde 7y ago
Better comparison: http://91-divoc.com/pages/covid-visualization/ http://91-divoc.com/pages/covid-visualization/
- xiphias2 7y ago
  This visualization doesn't show the lockdown effects that the paper estimates
- internet_user 7y ago
  No Taiwan? They had the most proactive response in the world.
bigpumpkin 7y ago
Can't expect China-like second derivative when countries are not doing these counter measures: Isolation of suspected cases and close contacts Universal masking Restrictions on travel inside country Sending doctors from the rest of the country to epidemic centers.
- kspacewalk2 7y ago
  This presumes all of these measures are highly effective, which is far from certain or obvious.
  - justicezyx 7y ago
    "No decision is the worst decision"
  - daxfohl 7y ago
    No, it only requires that one of the methods is slightly effective.
  - vkou 7y ago
    Not taking any of these measures is highly effective at failing to control the virus. We're currently seeing it play out across Europe and the US.
  - FuckButtons 7y ago
    Or that they have significant additional effect over social distancing alone, which we won’t know for some time yet.
- avs733 7y ago
  >Can't expect China-like second derivative when countries are not doing these counter measures * not reporting cases accurately.
squidproquo 7y ago
If anyone has a forecast for the United States, this website is aggregating forecasts: https://www.unitarity.com/app/challenges/us-coronavirus-outbreak/events/mar-20 https://www.unitarity.com/app/challenges/us-coronavirus-outb...
daxfohl 7y ago
I have a hard time believing Spain will have a higher peak than USA.
- fermienrico 7y ago
  You can't look at Spain and USA without considering the total population. As someone in this thread pointed out: http://91-divoc.com/pages/covid-visualization/ http://91-divoc.com/pages/covid-visualization/ Take a look at the charts of cases/1M people. It is obvious that US will have more cases simply because of the population.
avs733 7y ago
it is notable that this work is done by an electrical engineer and a cardiologist, not epidemiologists. More than anything, the arm chair epidemiology is the our current second biggest danger. Epidemiology is hard. incredibly hard. It isn't viral marketing. It isn't electrical engineering. Data isn't pure or assumed to be correct. There data source was basically websites.
- anacrolix 7y ago
  Their
  - avs733 7y ago
    thanks. there are a couple other typos I noticed this morning as well. shouldn't HN with Bourbon.
- thu2111 7y ago
  Based on what we've seen so far, electrical engineering and cardiology are significantly more rigorous than epidemiology, which appears to be more like economics or social psychology in terms of the robustness of its methods and quality of its work.
  - avs733 7y ago
    Define rigor. In context I suspect you mean something like "treat data as objective." I can go to France and pickup and hold the literal kilogram. U can't do that, yet, with people's brains to the level needed for psych measurement. Personally, I prefer social sciences and epi methods (get economics out of here...) Because they are more transparent about the role of the researcher and the limitations of their data. They don't bluntly trust it...the engineers I work with largely do. They assume data represents truth and is largely without meaningful error.
    - thu2111 7y ago
      I've elaborated here on what I mean here: https://news.ycombinator.com/item?id=22737948 https://news.ycombinator.com/item?id=22737948 It's not just how much data is trusted (though note: Professor Ferguson at Imperial appears to trust the data coming out of Italy almost completely). It's the whole set of problems.
X6S1x6Okd1st 7y ago
The methodology don't really explain how they aligned the second derivative curve to China or why that's reasonable. I wouldn't put much faith in their predictions.
twoslide 7y ago
I tried fitting a third order polynomial for JHU Covid data, just as a way to kill time. For all countries, the 95% confidence interval of the second derivatice overlapped zero (i.e. it can't be estimated very well). We only have about three weeks of data from the 10th death, for most countries, you can't fit a very good curve with that.

58 points by hdivider 7y ago

Full title (not possible to put as title because of HN limits): An international comparison of the second derivative of COVID deaths after implementation of social distancing measures

chipperyman573 7y ago

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

deleted 7y ago
[deleted]
deleted 7y ago
[deleted]
lsb 7y ago
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.
- deleted 7y ago
  [deleted]
- dpwm 7y ago
  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 https://github.com/CSSEGISandData/COVID-19 [1] https://github.com/dpwm/covid19-analysis/blob/master/Coronavirus%20Deaths.ipynb 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!
  - mr_toad 7y ago
    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.
  - dboreham 7y ago
    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/ https://www.worldometers.info/coronavirus/country/us/
  - grandmczeb 7y ago
    > 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/coronavirus-deaths-by-country.html https://www.nytimes.com/interactive/2020/03/21/upshot/corona...
- 3solarmasses 7y ago
  yep. My first thought was why the hell they didn't they use the word acceleration for better readability.
- babesh 7y ago
  It's not an exponential. It's a shifted sigmoid.
  - the8472 7y ago
    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.
    - dnautics 7y ago
      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.
dboreham 7y ago
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.
- zeroimpl 7y ago
  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.
  - dpwm 7y ago
    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?
    - 9wzYQbTYsAIc 7y ago
      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).
  - 9wzYQbTYsAIc 7y ago
    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.
fortran77 7y ago
1st: Velocity 2nd: Acceleration 3rd: Jerk 4th: Jounce
analog31 7y ago
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.

stefan_ 7y ago

This looks like one of those P versus NP papers.

bernardv 7y ago

Very little details on the methodology

dboreham 7y ago
How so? They said they computed the second derivative. What more is there to say?
- willis936 7y ago
  You can’t compute the second derivative of data from May 2020 when it is currently March 2020. Where did the data come from?
  - 9wzYQbTYsAIc 7y ago
    China data time shifted to match the country of interest.
    - softwaredoug 7y ago
      That seems to make a pretty big assumption western countries can achieve that? My understanding even a small amount of non compliance has big consequences...
  - Someone 7y ago
    FTA: The cumulative deaths for each country, N(t), were estimated by deriving a multiplier Nfinal/N(tconv) at the time of convergence, tconv, to the Chinese trajectory, Nfinal being the total number of deaths in China and applying this multiplier to Chinese data for times beyond convergence.

lonelappde 7y ago

Better comparison: http://91-divoc.com/pages/covid-visualization/ http://91-divoc.com/pages/covid-visualization/

xiphias2 7y ago
This visualization doesn't show the lockdown effects that the paper estimates
internet_user 7y ago
No Taiwan? They had the most proactive response in the world.

bigpumpkin 7y ago

Can't expect China-like second derivative when countries are not doing these counter measures: Isolation of suspected cases and close contacts Universal masking Restrictions on travel inside country Sending doctors from the rest of the country to epidemic centers.

kspacewalk2 7y ago
This presumes all of these measures are highly effective, which is far from certain or obvious.
- justicezyx 7y ago
  "No decision is the worst decision"
- daxfohl 7y ago
  No, it only requires that one of the methods is slightly effective.
- vkou 7y ago
  Not taking any of these measures is highly effective at failing to control the virus. We're currently seeing it play out across Europe and the US.
- FuckButtons 7y ago
  Or that they have significant additional effect over social distancing alone, which we won’t know for some time yet.
avs733 7y ago
>Can't expect China-like second derivative when countries are not doing these counter measures * not reporting cases accurately.

squidproquo 7y ago

If anyone has a forecast for the United States, this website is aggregating forecasts: https://www.unitarity.com/app/challenges/us-coronavirus-outbreak/events/mar-20 https://www.unitarity.com/app/challenges/us-coronavirus-outb...

daxfohl 7y ago

I have a hard time believing Spain will have a higher peak than USA.

fermienrico 7y ago
You can't look at Spain and USA without considering the total population. As someone in this thread pointed out: http://91-divoc.com/pages/covid-visualization/ http://91-divoc.com/pages/covid-visualization/ Take a look at the charts of cases/1M people. It is obvious that US will have more cases simply because of the population.

avs733 7y ago

it is notable that this work is done by an electrical engineer and a cardiologist, not epidemiologists. More than anything, the arm chair epidemiology is the our current second biggest danger. Epidemiology is hard. incredibly hard. It isn't viral marketing. It isn't electrical engineering. Data isn't pure or assumed to be correct. There data source was basically websites.

anacrolix 7y ago
Their
- avs733 7y ago
  thanks. there are a couple other typos I noticed this morning as well. shouldn't HN with Bourbon.
thu2111 7y ago
Based on what we've seen so far, electrical engineering and cardiology are significantly more rigorous than epidemiology, which appears to be more like economics or social psychology in terms of the robustness of its methods and quality of its work.
- avs733 7y ago
  Define rigor. In context I suspect you mean something like "treat data as objective." I can go to France and pickup and hold the literal kilogram. U can't do that, yet, with people's brains to the level needed for psych measurement. Personally, I prefer social sciences and epi methods (get economics out of here...) Because they are more transparent about the role of the researcher and the limitations of their data. They don't bluntly trust it...the engineers I work with largely do. They assume data represents truth and is largely without meaningful error.
  - thu2111 7y ago
    I've elaborated here on what I mean here: https://news.ycombinator.com/item?id=22737948 https://news.ycombinator.com/item?id=22737948 It's not just how much data is trusted (though note: Professor Ferguson at Imperial appears to trust the data coming out of Italy almost completely). It's the whole set of problems.

X6S1x6Okd1st 7y ago

The methodology don't really explain how they aligned the second derivative curve to China or why that's reasonable. I wouldn't put much faith in their predictions.

twoslide 7y ago

I tried fitting a third order polynomial for JHU Covid data, just as a way to kill time. For all countries, the 95% confidence interval of the second derivatice overlapped zero (i.e. it can't be estimated very well). We only have about three weeks of data from the 10th death, for most countries, you can't fit a very good curve with that.