3 ms·
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
by 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.
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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:
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.
- Certhas 7y agoOkay 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.
- hef19898 7y agoSee my comment above. I don't think we really disagree.