However, in the case of a virus, you eventually run out of population to consume; so you tweak the model to be "Let the rate of infection be the probability that a infected person encounters a non-infected person" (nicely, for a large population, this will give you the same results initially). This is, approximately, proportional to the product of the number of people who are infected times the number of people who are not infected (think back to chemical reagents and reaction rates). That is, (1-infected(t)) * infected(t). A function who's rate of change is that is the logistic function (and you can verify by taking the derivative of 1/(1+e^-t))

This is a simple model, ignores geography, ignores population change, and ignores changing behavior. It basically pretends everyone is an ideal gas molecule in a volume. But if you want to model something that "grows exponentially with a limit" it works alright.

I don't know under what assumptions the rate of change would yield a bell (normal) curve. (e^-x^2).

Current Hospitalizations, which is the number we should be most concerned about, and is reflected by the orange dots on the state's graph, should start low, max out at a number, and eventually return to 0. That should follow something like a bell curve.

I think the confusion is between the "total infections" curve and the "current hospitalizations" curve. The difference between them is huge, and really important. The "Flatten the Curve" idea refers to keeping the number of hospitalizations at any one moment below some upper limit, ideally the state's number of available ICU beds. That's a bell curve.

I think the confusion is that if the CDF is the logistic function, then the PDF is the logistic distribution. This is what archgoon is saying.

Bell curve refers to a Gaussian distribution, which is different but looks similar.

https://en.m.wikipedia.org/wiki/Logistic_distribution

http://visionlab.harvard.edu/Members/Anne/Math/Logistic_vs_G...