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Why should infections follow a bell curve? Shouldn't it be the derivative of a the logistic curve, which should be approximately exponential on both ends. I don

Why should infections follow a bell curve? Shouldn't it be the derivative of a the logistic curve, which should be approximately exponential on both ends. I don't see how a quadratic should show up in the exponent; either mathematically or epidemiologically.

mikekij 6y ago
You're pointing out a great ambiguity in the state's graph. Is the hospitalization line a cumulative count of all hospitalizations? Or hospitalizations at any given time? Since the state's hospital capacity is related to hospitalizations at a given time, and not cumulative, I'm assuming the fit of that graph would be a bell curve with a max, followed by a decrease. People either die or leave the hospital.
wtvanhest 6y ago
Does anyone have the ability to walk through these explanations in a little more detail. I'm having trouble understanding why infections would be a bell curve, or derivative of a log curve.
- mikekij 6y ago
  Cumulative infections would not be a bell curve. They'd be a sigmoid curve, approaching some upper limit (e.g. 100,000). Current Hospitalizations, which is the number we should be most concerned about, and is reflected by the orange dots on the state's graph, should start low, max out at a number, and eventually return to 0. That should follow something like a bell curve. I think the confusion is between the "total infections" curve and the "current hospitalizations" curve. The difference between them is huge, and really important. The "Flatten the Curve" idea refers to keeping the number of hospitalizations at any one moment below some upper limit, ideally the state's number of available ICU beds. That's a bell curve.
  - SquishyPanda23 6y ago
    > think the confusion is between the "total infections" curve and the "current hospitalizations" curve. I think the confusion is that if the CDF is the logistic function, then the PDF is the logistic distribution. This is what archgoon is saying. Bell curve refers to a Gaussian distribution, which is different but looks similar. https://en.m.wikipedia.org/wiki/Logistic_distribution https://en.m.wikipedia.org/wiki/Logistic_distribution http://visionlab.harvard.edu/Members/Anne/Math/Logistic_vs_Gaussian.html http://visionlab.harvard.edu/Members/Anne/Math/Logistic_vs_G...
- archgoon 6y ago
  So the basic assumption of an exponential growth curve is: "rate of change is proportional to a population." However, in the case of a virus, you eventually run out of population to consume; so you tweak the model to be "Let the rate of infection be the probability that a infected person encounters a non-infected person" (nicely, for a large population, this will give you the same results initially). This is, approximately, proportional to the product of the number of people who are infected times the number of people who are not infected (think back to chemical reagents and reaction rates). That is, (1-infected(t)) * infected(t). A function who's rate of change is that is the logistic function (and you can verify by taking the derivative of 1/(1+e^-t)) This is a simple model, ignores geography, ignores population change, and ignores changing behavior. It basically pretends everyone is an ideal gas molecule in a volume. But if you want to model something that "grows exponentially with a limit" it works alright. I don't know under what assumptions the rate of change would yield a bell (normal) curve. (e^-x^2).
- nomel 6y ago
  Here's a neat video by 3Blue1Brown that shows some simple simulations of an epidemic, and give a good visualization of the bell curve.
Animats 6y ago
That's a good question. If control measures are undertaken to "flatten the curve" down to the capacity of the medical system, we'd expect to see new infections per day and deaths per day level out, maybe decline a little, but not decline a lot. Which is what the actuals show for the areas worst off.[1] At some point, there's saturation, and "herd immunity". For this virus, that's somewhere in the 60%-80% range of the population. US population is around 333,000,000, so saturation is somewhere above 200 million. We're nowhere near that. Current known US total cases are nearing 2 million. Current estimates of undetected cases are in the 80% range, so current total cases is probably around 10 million. So about 20x more people have to get sick before herd immunity. That's well over a year away. [1] https://www.ft.com/coronavirus-latest https://www.ft.com/coronavirus-latest
- apatters 6y ago
  It seems like both the perception of and response to this pandemic are not reflective of reality. What am I getting wrong here? - Around 200 million Americans will eventually get it, barring extraordinary events (like a vaccine arriving in 6 months). We cannot change this. - Around 0.5% will die, overwhelmingly those who had a low remaining life expectancy to begin with. This is a much larger body count than most people seem able to accept, but also much less dangerous to the average American than many believe. - All we can do is alter the timetable to keep the hospitals functioning, which has been done successfully in many places throughout the world albeit at great cost. There's both good news and terrible news in those facts but none of it really seems to be in the consciousness of the public or of leaders, whereas "oh no you can maybe get it 8 hours after a guy sneezed in a lobby" is everywhere. Along with "everyone who stays at home is a hero" as we set ourselves up for a second Depression.
  - digitaltrees 6y ago
    It’s not 0.5% it’s closer to 2%. You don’t have a normal economy if people don’t want to go out in public or the supply chain is interrupted as workers get sick. A pandemic will harm the economy no matter what we do, these efforts, it is hoped, will reduce the impact while also saving lives and the hospital system.
jariel 6y ago
It won't follow a 'bell curve' but the curve will 'look like a bell curve'.
squillesque 6y ago
You can approximate one with the other; it used to be common when normals were harder to work with early computers. Eg, https://www.johndcook.com/blog/2010/05/18/normal-approximation-to-logistic/ https://www.johndcook.com/blog/2010/05/18/normal-approximati...

mikekij 6y ago

You're pointing out a great ambiguity in the state's graph. Is the hospitalization line a cumulative count of all hospitalizations? Or hospitalizations at any given time? Since the state's hospital capacity is related to hospitalizations at a given time, and not cumulative, I'm assuming the fit of that graph would be a bell curve with a max, followed by a decrease. People either die or leave the hospital.

wtvanhest 6y ago

Does anyone have the ability to walk through these explanations in a little more detail. I'm having trouble understanding why infections would be a bell curve, or derivative of a log curve.

mikekij 6y ago
Cumulative infections would not be a bell curve. They'd be a sigmoid curve, approaching some upper limit (e.g. 100,000). Current Hospitalizations, which is the number we should be most concerned about, and is reflected by the orange dots on the state's graph, should start low, max out at a number, and eventually return to 0. That should follow something like a bell curve. I think the confusion is between the "total infections" curve and the "current hospitalizations" curve. The difference between them is huge, and really important. The "Flatten the Curve" idea refers to keeping the number of hospitalizations at any one moment below some upper limit, ideally the state's number of available ICU beds. That's a bell curve.
- SquishyPanda23 6y ago
  > think the confusion is between the "total infections" curve and the "current hospitalizations" curve. I think the confusion is that if the CDF is the logistic function, then the PDF is the logistic distribution. This is what archgoon is saying. Bell curve refers to a Gaussian distribution, which is different but looks similar. https://en.m.wikipedia.org/wiki/Logistic_distribution https://en.m.wikipedia.org/wiki/Logistic_distribution http://visionlab.harvard.edu/Members/Anne/Math/Logistic_vs_Gaussian.html http://visionlab.harvard.edu/Members/Anne/Math/Logistic_vs_G...
archgoon 6y ago
So the basic assumption of an exponential growth curve is: "rate of change is proportional to a population." However, in the case of a virus, you eventually run out of population to consume; so you tweak the model to be "Let the rate of infection be the probability that a infected person encounters a non-infected person" (nicely, for a large population, this will give you the same results initially). This is, approximately, proportional to the product of the number of people who are infected times the number of people who are not infected (think back to chemical reagents and reaction rates). That is, (1-infected(t)) * infected(t). A function who's rate of change is that is the logistic function (and you can verify by taking the derivative of 1/(1+e^-t)) This is a simple model, ignores geography, ignores population change, and ignores changing behavior. It basically pretends everyone is an ideal gas molecule in a volume. But if you want to model something that "grows exponentially with a limit" it works alright. I don't know under what assumptions the rate of change would yield a bell (normal) curve. (e^-x^2).
nomel 6y ago
Here's a neat video by 3Blue1Brown that shows some simple simulations of an epidemic, and give a good visualization of the bell curve.

Animats 6y ago

That's a good question. If control measures are undertaken to "flatten the curve" down to the capacity of the medical system, we'd expect to see new infections per day and deaths per day level out, maybe decline a little, but not decline a lot. Which is what the actuals show for the areas worst off.[1] At some point, there's saturation, and "herd immunity". For this virus, that's somewhere in the 60%-80% range of the population. US population is around 333,000,000, so saturation is somewhere above 200 million. We're nowhere near that. Current known US total cases are nearing 2 million. Current estimates of undetected cases are in the 80% range, so current total cases is probably around 10 million. So about 20x more people have to get sick before herd immunity. That's well over a year away. [1] https://www.ft.com/coronavirus-latest https://www.ft.com/coronavirus-latest

apatters 6y ago
It seems like both the perception of and response to this pandemic are not reflective of reality. What am I getting wrong here? - Around 200 million Americans will eventually get it, barring extraordinary events (like a vaccine arriving in 6 months). We cannot change this. - Around 0.5% will die, overwhelmingly those who had a low remaining life expectancy to begin with. This is a much larger body count than most people seem able to accept, but also much less dangerous to the average American than many believe. - All we can do is alter the timetable to keep the hospitals functioning, which has been done successfully in many places throughout the world albeit at great cost. There's both good news and terrible news in those facts but none of it really seems to be in the consciousness of the public or of leaders, whereas "oh no you can maybe get it 8 hours after a guy sneezed in a lobby" is everywhere. Along with "everyone who stays at home is a hero" as we set ourselves up for a second Depression.
- digitaltrees 6y ago
  It’s not 0.5% it’s closer to 2%. You don’t have a normal economy if people don’t want to go out in public or the supply chain is interrupted as workers get sick. A pandemic will harm the economy no matter what we do, these efforts, it is hoped, will reduce the impact while also saving lives and the hospital system.

jariel 6y ago

It won't follow a 'bell curve' but the curve will 'look like a bell curve'.

squillesque 6y ago

You can approximate one with the other; it used to be common when normals were harder to work with early computers. Eg, https://www.johndcook.com/blog/2010/05/18/normal-approximation-to-logistic/ https://www.johndcook.com/blog/2010/05/18/normal-approximati...