The Corona Virus pandemic appears to be a relevant example of logistic growth. It grows exponentially at first but then tends to level out, [as in China](https://publichealth.gsu.edu/coronavirus/).
As mentioned in comment #2 above, I have a novel mathematical derivation of this logistic sigmoid which has absolutely nothing to do with the logistic equation, but instead uses stochastic principles of the competing processes of a dispersive exponential growth and a range of limiting populations in which to draw from -- this is on [p.85 of our book Mathematical Geoenergy](https://www.google.com/books/edition/Mathematical_Geoenergy/xb17DwAAQBAJ?hl=en&gbpv=1&bsq=%22LOGISTIC%E2%80%90SHAPED%22).
Just because a sigmoid-shaped curve follows a shape such as 1/(1+A exp(-t)) doesn't mean that it comes solely from the logistic equation. As noted in #2, consider that just as the logistic sigmoid also maps to the [Fermi-Dirac distribution](https://en.wikipedia.org/wiki/Fermi%E2%80%93Dirac_statistics#Fermi%E2%80%93Dirac_distribution), the heuristic logistic equation derivation also appears to be just a quirky coincidence.
As an exercise amongst the mathematicians, can anyone else derive the logistic sigmoid function without relying on the logistic equation?
**EDIT**: This YouTube was recently posted and goes through the conventional derivation
John has a Twitter thread on the virus here : https://twitter.com/WHUT/status/1238148317739089920