Lotka's Logistic Equation

 

  An NIH webpage discusses the use of Lotka's Logistic Equation for Covid-19 forecasting. This equation is found in Lotka, Elements of Physical Biology, p. 65. Lotka's assumes the general growth equation can be expanded as the sum of individual factors and the product of their powers. For a simple quadratic growth function the solution is a geometric function which he sums with using the usual formula for the sum of a geometric series. The resulting formula is applied to the growth in the US population between 1790 and 1910. I tried to see if I could improve on his fit using correction equaton method. My first attempt failed but the fit indicated that the value for the year 1860 was an outlier so that data point was eliminated from the fit. Here are some images showing my fit.

So the correction equations succeeded in finding a better fit to the US population growth. Losses during the American Civil War may have affected the data after 1860. δ is the difference between calculated and observed values.

Supplemental (Sep 5): Lotka's fit is slightly better than the fit using the nonlinear least squares correction equatons. Lotka's standard deviation is 325781 while my fit had a stdev of 384743 and that was without the 1860 data point.

Supplemental (Sep 5): Lotka appears to have done a three point fit of the derived equaton. A similar fit produced these results.