I tried to do a more accurate version of Lotka's fit of US population growth. The data used came from Appendex A of the US Census publication Measuring America. Since the day of the year that the census was taken on has not been constant I computed the number of days from the beginning of 1790 and divided by 365.25, the number of days in a Julian year and used that as the time scale. The growth appears to be approximately geometric until about 1930 and may have been affected by the Depression, WWII and the Cold War as well as government domestic and immigration policy.
The second plot shows the deviation from geometric growth. Lotka's measured time with t'=t-t0 with t0 being the time of peak growth which he gave as Apr 1, 1914. The values I got were the following.
The formula for the time of peak rate of change is determined as follows,
The population growth of the US appears to have been slowing down. The last plot of the rate of change in population growth appears to be symmetrical. So if one uses this model to predict the number of future Covid-19 cases of infections one is likely to be off since the coronavirus tends to linger. The SIR model is a better predictor since it can match an asymmetric curve. For an accurate prediction we need both an accurate model and an accurate fit.
Supplemental (Sep 8): If the data points are not evenly spaced the formula that has to be solved to determine A for the 3 point fit is f=α1(A)-α2(A)=0 or,
Supplemental (Sep 10): Lotka's solution for the integral of the logistic equation is exact. dx/x(a+bx) is a standard integral and gives the same solution.
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