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A logistic curve describes the European experience where quarantines etc. WERE enforced and effective.. In the US by contrast, some .people see voluntary quarantine as an infringement of their rights, and lacking firm leadership and with police forces declining to enforcing such local regulations in some cases, on a similar basis, the results are an uneasy leveling then resumption of total case increase. Now as the World leading sore spot of infections, an effective vaccine is the best hope for the yet uninfected population. Sad.
Thanks for your post!. Could you recommend additional references?
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.
I received some helpful ideas from David Bowman at Georgetown. He understood my need for something better than the logistic function which uses thesame exponential for growth and decay to a high limit. At the cost of a fourth parameter, he provided a function which can fit a curve that grows and levels on different time constants. Sadly, the only curve where this is presently finding a use is US Deaths for whatever reason. US, Cases and Oklahoma Cases and Deaths are best fitted with a linear slope presently. This is the function: deaths = k1 / (1+ aexp(ab*(Start day number - present day number)))^(1/c) Four parameters. Here is what a plot looks like:
By contrast here is a linear US CASES plot:
I find a logistic function helpful in conjunction with a non linear curve fitter. It does have a short-coming. After locating the probable peak rate from a cumulative time series, it is likely to underestimate the decay portion of the rate plot. This is true where the exponential for the upper half is in fact on a longer time constant, so the rate has a longer decay time. All in all, it is a useful approach - and this was true when it was used to model the AIDS epidemic last century.
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