1

I was having trouble trying to understand the parameters of the simplest SIR model.

enter image description here

If beta is the effective contact rate and s is the percentage of people who are susceptible, then how do the units cancel out such that ds/dt is measured in percentage per unit time? When I multiply -ßsi out, the units cancel out to people/time, which isn't how ds/dt is measured. Am I missing something here?

Krishna Kalakkad
  • 11
  • 1

1 Answers1

1

The variables $s, i, n$ are dimensionless. (You consider the dimension 'percentage' but that is not a dimension. A percentage is dimensionless.)

So the differential on the left $\frac{di}{dt}$ has units 'per unit time' and not 'percentage per unit time'.

On the right the term $\beta si$ has also units 'per unit time'. The $\beta$, a rate, has units 'per unit time', and the $s$ and $i$ are dimensionless.

Sextus Empiricus
  • 43,080
  • 1
  • 72
  • 161
  • thanks for keep answering SIR model related questions. I have been thinking an interesting problem: if I do not know SIR model, just use the old machine learning and curve fitting thing for number of infections over time, I may get some polynomial fit on I(t). What is pros and cons for polynomial fit and SIR fit? I may want to ask such question later in CV. – Haitao Du Apr 21 '20 at 09:46
  • @HaitaoDu That may certainly be an interesting question on its own. I do not have much knowledge of machine learning, and despite some understanding of fitting with regularization, I am still skeptical about the fits with neural networks (machine learning is a bit ambiguous and I am throwing in NN, I am not sure what you refer to). I believe that these sort of technologies only work in the proper invironment.... – Sextus Empiricus Apr 21 '20 at 10:03
  • ... There is apparently [some machine learning type model](https://www.statnews.com/2020/04/17/influential-covid-19-model-uses-flawed-methods-shouldnt-guide-policies-critics-say/) and it is being criticized by the theoretical, ab-initio-oriented people, for not providing any insight. Machine learning may certainly provide predictions (as good as they can get which, I believe, is not much), but it will not provide much insight. I believe that we are operating with limited knowledge/data, but [still can *get* some of the principles](https://stats.stackexchange.com/a/461455/164061). – Sextus Empiricus Apr 21 '20 at 10:13
  • .... normally I would be less skeptical about machine learning, but with the covid-19 data, I suspect a lot of bias in the data and also I am very uncertain about the pattern that a machine learning method would learn. There is not a lot clear and unbiased data and the learning will be likely a lot biased. – Sextus Empiricus Apr 21 '20 at 10:18
  • I think that problem can be viewed as parametric or non-parametric model, or similar to [generative model vs. discriminative model](http://papers.nips.cc/paper/2020-on-discriminative-vs-generative-classifiers-a-comparison-of-logistic-regression-and-naive-bayes.pdf), the key idea is that, if the model assumption is right, then parametric (SIR) model will be better. On the other hand, if SIR is way off, then non-parametric will be better. – Haitao Du Apr 21 '20 at 10:22
  • welcome to comment or answer. I will answer it by myself later https://stats.stackexchange.com/questions/461816/what-are-the-pros-and-cons-to-fit-data-with-simple-polynomial-regression-vs-com – Haitao Du Apr 21 '20 at 11:05