So the basic assumption of an exponential growth curve is: "rate of change is proportional to a population." However, in the case of a virus, you eventually ru

So the basic assumption of an exponential growth curve is: "rate of change is proportional to a population." 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).