3 ms·

Mathematically wrong is not a relevant category here. There is no "ground truth" that is approximated by an exponential growth. Every disease model is counterfa

Mathematically wrong is not a relevant category here. There is no "ground truth" that is approximated by an exponential growth. Every disease model is counterfactual "spread of the disease if no countermeasures were taken", etc... Modelling the beginning of an epidemic as exponential growth is useful, sensible and empirically validated. As, countermeasures are taken during the beginning of the epidemic, for discussions of the effect of countermeasures, exponential is the correct model. In fact, hopefully the countermeasures taken will be sufficient that the intrinsic disease dynamic will never get past the exponential stage...

hef19898 7y ago
I don't find any differentiation between early stage and later stage nor references to exponential functions: https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease https://en.wikipedia.org/wiki/Mathematical_modelling_of_infe... If you find any source, idealy scientific papers on the subject, that shows that contagious diseases can be modelled with simple exponential functions, please share them. I am not an expert on that matter, so I would be really interessted to see some reliable sources confirming the use of exponential functions.
- cvwright 7y ago
  Ok, look at the differential equations on this page about the Kermack-McKendrick model: https://mathworld.wolfram.com/Kermack-McKendrickModel.html https://mathworld.wolfram.com/Kermack-McKendrickModel.html Notice that the rate of change in the infected population (dI/dt) is proportional to the current number of infected (I) and the current number of susceptible (S). In the early stages, when S is large and I is still small, this acts just like an exponential function in I. For example, think about what happens when you double I the first several times. S stays relatively unchanged, and so dI/dt roughly doubles each time you double I. Does that make sense?
  - hef19898 7y ago
    Yeah, exactly. In the beginning exponential, later on not anymore. Without changing the equation. So, no, not exponential. Because, you know, very early on an exponential graph can approximated by something non-exponential. Would be utter BS, sure, so nobody does it.
    - cvwright 7y ago
      Exponential growth. It's the growth phase that matters. People keep mentioning the exponential because both of these curves, at an important point in time, grow very fast. Much faster than anything you're likely to encounter in daily life, so the public really doesn't have the requisite intuition. But I can see you're not interested in understanding the point. You asked for equations, I gave you equations. Good day sir.
      - hef19898 7y ago
        We are, I think, not disagreeing at all. May sound crazy, but bear with me a second here. I know that the initial phase of any epidemic is exponential. Key are these three small words, initial growth phase. Because they make all the difference, as every epidemic will ultimately peak. These three words are simply to often drowned out in discussions about COVID-19. That's we you see extrapolations of this growth against, for example, the population of Italy. And this is dangerous. And I don't think we disagree here, I never disputed the exponential nature of the early phase. We just shouldn't make the mistake in assuming everybody catches these three little words, especially not online. The most critical question everybody is trying to answer right now is, when the peak will be. And whether this peak will to high for our medical infrastructure. With all the measures being taken, the conclusion seems to be the point will be too high. Hence the counter measures. So we should all do what we can to flatten the curve as much as possible. I never said anything else, and honestly, I don't see where are disagreeing here.
- Certhas 7y ago
  In your link, look at this: https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease#The_SIR_model https://en.wikipedia.org/wiki/Mathematical_modelling_of_infe... S = N - I - R, and lets used normalized ratios, so i = I/N, r = R/N, etc... so the dynamics of i are given by: di/dt = beta (1 - i - r) i - gamma i di/dt = (beta - gamma) i - beta (i^2 + ri) At the beginning of the disease, the ratio of infected and recovered is very small, say 1e-5 for 100 cases in a population of 10 million. So initially the second term is 1e-10 whereas the first is 1e-5. The initial dynamics of a new disease are given by: di/dt ~ (beta - gamma) i exponential growth. It will start deviating from exponential growth once i becomes large enough. If 10% of the population have had the disease it will deviate from exponential growth by 10%. Once it gets to half the population being infected or recovered you start seeing a real deviation. In reality this might never happen, because we are taking a lot of measures to get beta down. So really you have something like di/dt ~ (beta(t) - gamma) i where beta(t) will capture all the countermeasures people take. If the countermeasures are effective and manage to push beta(t) below gamma before a large chunk of the population is infected, we might never see non-exponential behaviour from the disease. It will have an exponentially growing phase, and then an exponentially shrinking phase. --- Here is just one paper I found with two seconds of google that looks at this, very recent, a bit basic: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6962332/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6962332/ But here is what it says in the abstract: "The initial exponential growth rate of an epidemic is an important measure of the severeness of the epidemic, and is also closely related to the basic reproduction number." "Classical compartmental transmission models assume exponential growth during the early phase of a well-mixed population"
  - hef19898 7y ago
    "The initial exponential growth rate" Key word, initial. Later on, not so much. Which is the whole point, pandemics aren't exponential over the complete run time. So, why not using these formulas at least in the models instead of using exponential equation limited by world population? Edit: "It will have an exponentially growing phase, and then an exponentially shrinking phase" Confirming the point above. Math is a precise science, it's a while since I did stuff like that, but getting equations right or not used to be a big deal back than. And now we talking more serious things than just a grade in an exam.

hef19898 7y ago

I don't find any differentiation between early stage and later stage nor references to exponential functions: https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease https://en.wikipedia.org/wiki/Mathematical_modelling_of_infe... If you find any source, idealy scientific papers on the subject, that shows that contagious diseases can be modelled with simple exponential functions, please share them. I am not an expert on that matter, so I would be really interessted to see some reliable sources confirming the use of exponential functions.

cvwright 7y ago
Ok, look at the differential equations on this page about the Kermack-McKendrick model: https://mathworld.wolfram.com/Kermack-McKendrickModel.html https://mathworld.wolfram.com/Kermack-McKendrickModel.html Notice that the rate of change in the infected population (dI/dt) is proportional to the current number of infected (I) and the current number of susceptible (S). In the early stages, when S is large and I is still small, this acts just like an exponential function in I. For example, think about what happens when you double I the first several times. S stays relatively unchanged, and so dI/dt roughly doubles each time you double I. Does that make sense?
- hef19898 7y ago
  Yeah, exactly. In the beginning exponential, later on not anymore. Without changing the equation. So, no, not exponential. Because, you know, very early on an exponential graph can approximated by something non-exponential. Would be utter BS, sure, so nobody does it.
  - cvwright 7y ago
    Exponential growth. It's the growth phase that matters. People keep mentioning the exponential because both of these curves, at an important point in time, grow very fast. Much faster than anything you're likely to encounter in daily life, so the public really doesn't have the requisite intuition. But I can see you're not interested in understanding the point. You asked for equations, I gave you equations. Good day sir.
    - hef19898 7y ago
      We are, I think, not disagreeing at all. May sound crazy, but bear with me a second here. I know that the initial phase of any epidemic is exponential. Key are these three small words, initial growth phase. Because they make all the difference, as every epidemic will ultimately peak. These three words are simply to often drowned out in discussions about COVID-19. That's we you see extrapolations of this growth against, for example, the population of Italy. And this is dangerous. And I don't think we disagree here, I never disputed the exponential nature of the early phase. We just shouldn't make the mistake in assuming everybody catches these three little words, especially not online. The most critical question everybody is trying to answer right now is, when the peak will be. And whether this peak will to high for our medical infrastructure. With all the measures being taken, the conclusion seems to be the point will be too high. Hence the counter measures. So we should all do what we can to flatten the curve as much as possible. I never said anything else, and honestly, I don't see where are disagreeing here.
Certhas 7y ago
In your link, look at this: https://en.wikipedia.org/wiki/Mathematical_modelling_of_infectious_disease#The_SIR_model https://en.wikipedia.org/wiki/Mathematical_modelling_of_infe... S = N - I - R, and lets used normalized ratios, so i = I/N, r = R/N, etc... so the dynamics of i are given by: di/dt = beta (1 - i - r) i - gamma i di/dt = (beta - gamma) i - beta (i^2 + ri) At the beginning of the disease, the ratio of infected and recovered is very small, say 1e-5 for 100 cases in a population of 10 million. So initially the second term is 1e-10 whereas the first is 1e-5. The initial dynamics of a new disease are given by: di/dt ~ (beta - gamma) i exponential growth. It will start deviating from exponential growth once i becomes large enough. If 10% of the population have had the disease it will deviate from exponential growth by 10%. Once it gets to half the population being infected or recovered you start seeing a real deviation. In reality this might never happen, because we are taking a lot of measures to get beta down. So really you have something like di/dt ~ (beta(t) - gamma) i where beta(t) will capture all the countermeasures people take. If the countermeasures are effective and manage to push beta(t) below gamma before a large chunk of the population is infected, we might never see non-exponential behaviour from the disease. It will have an exponentially growing phase, and then an exponentially shrinking phase. --- Here is just one paper I found with two seconds of google that looks at this, very recent, a bit basic: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6962332/ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6962332/ But here is what it says in the abstract: "The initial exponential growth rate of an epidemic is an important measure of the severeness of the epidemic, and is also closely related to the basic reproduction number." "Classical compartmental transmission models assume exponential growth during the early phase of a well-mixed population"
- hef19898 7y ago
  "The initial exponential growth rate" Key word, initial. Later on, not so much. Which is the whole point, pandemics aren't exponential over the complete run time. So, why not using these formulas at least in the models instead of using exponential equation limited by world population? Edit: "It will have an exponentially growing phase, and then an exponentially shrinking phase" Confirming the point above. Math is a precise science, it's a while since I did stuff like that, but getting equations right or not used to be a big deal back than. And now we talking more serious things than just a grade in an exam.