If one wants a fit that is useful for predictions the fit function for dK/dt should tend to zero as K becomes large. Since polynomials tend to infinity as K goes to infinity we can try fitting dK/dt=1/P(K). An even power for P(K) will guarantee that the function will go to zero as K goes to ±∞. Fitting a fourth power in this manner gave the following results for a dt/dK fit.
Another advantage is that extrapolation of the data gives more realistic prediction results.
Finding a mathematical model for the data was challenging. To do so I tried modifying the SIR pandemic model to give something closer to the data. If one replaces rS-a with Q in the SIR formula for dI/dt one can derive a formula for Q in terms of the polynomial P(K).
This can be used to define a function S* which is related to S.
Comparing S* with I for the given data shows A is small but not quite constant.
This is as far as I've been able to get so far in analyzing the given date. More data appears to be needed to determine the parameters r and a.
Supplemental (Nov 2): The coefficients given above are those for a polynomial of X=K-K0 with K0=356834.7 being the value of K for the peak of dK/dt for the fit. X was used to get a more symmetrical fit about the peak with a positive 4th power coefficient of X so dK/dt would approach zero as the magnitude of X grew large.
Supplemental (Nov2): I probably made a mistake using a SIR model formula to calculate Q which assumes the logistic function for dI/dt. It is probably more general and the mathematical model associated with the fit may be the best one.
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