But the early stages of disease spread _are_ exponential. And I find that people really fail to understand the meaning of that, and how scary numbers like "new confirmed cases doubling every two days, 500 new cases in France yesterday" truly are.
The problem is, exponential for how long?
For instance, all seems to indicate measures in Italy are taken too late: and the exponential growth is still happening there (almost 3 weeks after the problem started)
import pandas as pd
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
it = pd.read_json('https://raw.githubusercontent.com/pcm-dpc/COVID-19/master/dati-json/dpc-covid19-ita-andamento-nazionale.json',
convert_dates=True).set_index('data')
plt.plot(it[['deceduti', 'totale_casi']])
plt.xticks(it.index.values, rotation=90);
It's to early to say. The people who are popping up with symptoms now might have been infected up to two weeks ago.
If you look at the Wuhan data, you see that the lockdown worked, but the number of confirmed cases at the time continued to rise dramatically until 12 days after the lockdown went into effect. We are only three days in on the Italy lockdown.
Yes, but extrapolating the early exponential phase is, plain and simple, mathematically wrong. No idea why data scientists aren't going the first mile and look up the relevant functions before spitting data out. Total fail on their part, IMHO.
It's not wrong: it has produced extremely good predictive results day after day for total cases outside of China for months now. Using a simpler approximate function in place of a more complicated exact function is far from wrong, it is bread and butter for getting practical results in all areas of science.
Countries that implemented extreme measures to slow the spread of the virus. It will be different for countries which haven't. The WHO and other health officials around the world are saying the same thing. This will spread to a large number of people. It's too late for containment, you can only hope to flatten the curve. That's why it's now a pandemic.
Unless you simultaneously ramp up your healthcare resources (what China did), even as "low" as 100 cases/day still has very real potential to go exponential (R0=2.5), overwhelm your healthcare, and kill 10% of our population if not more (especially if your population is aged).
I hear that, what I'm saying is, follow that logic through... If it's crucial to maintain strict protocols even during a rate of only 100 new cases/day, that means these protocols stay in place roughly... indefinitely.
Counting individual cases and deaths in a country of 300 million is nonsensical. You could do the same thing with the flu and you'd hit 20,000 deaths a year and you'd destroy the economy in the process.
IMO, the solution is best preparing to treat the number of people who do become severely ill from it, and getting the rest of the country fully back to work as quickly as possible.
Although I disagree strongly on the "kill 10% of your population" bit, which is just fear-mongering, it's kind of besides the point.
>Although I disagree strongly on the "kill 10% of your population" bit, which is just fear-mongering, it's kind of besides the point.
If every case went untreated, 10% is a reasonable guess about what the death rate would be. Unchecked growth would lead to almost every case going untreated. That's the penalty for allowing the number of active cases to rise above five times the number of open beds/ventilators.
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...
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.
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.
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.
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.
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.
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.
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Here is just one paper I found with two seconds of google that looks at this, very recent, a bit basic:
"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"
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.
Okay you're just trolling now. We are very much in the initial exponential phase of Covid-19. As I derived above, once you get to 10% you might start seeing deviations from exponential in the SIR model. The bump that you saw in China is not due to this deviation from exponential growth behaviour. We luckily never got close to 10% infection. It's due to getting to negative exponential growth. The beta(t) part of my answer above.
You're free to use whatever sensible model you want in your analysis, but you are criticising people for using exponential models. You have no arguments for that. Then you throw out Wikipedia, falsely claiming that exponential models are nowhere to be found there. Then you demand peer reviewed studies that use an exponential model, which I provide. But somehow that is not enough either.
In fact there are numerous assumptions in the SIR/exponential growth model. These might not hold. We might get slightly sub-exponential growth in some diseases. There is vast literature on that. But the base line in all the vast literature on this is that the most natural, obvious and common behaviour is exponential.
