The logistic equation is included in most every calculus text covering simple differential equations. Generally it is used for population growth with some restriction of resources required for growth. The resource restriction is assumed to be proportional to the population at a given time, The modification here is that the growth is really of the virus population measured by numbers of individuals infected at each time point. The limitation to growth is the quarantine method, which must be nearly totally effective for the logistic model to work. A partial quarantine just slows the rate of spread until everybody is eventually infected. An effective quarantine must isolate infected persons and their contacts faster than the virus can spread to new populations.
Unfortunately the model doesn't work with the COVID-19 epidemic very well because the voluntary quarantine methods are not very effective. Initially it seemed to be working with the Chinese data, but most people now think that data was probably not very accurate.
