On: Log-log plot of new vs. total Covid-19 cases by country

[fxj]

This kind of analysis is also called a phase space plot: the function is plotted against its derivative. And when the function is an exponential growth then the derivative is the same which gives a similar plot for all the different countries. When the function deviates from the exponential like in China you can spot the difference very early in these kind of plots.

https://en.wikipedia.org/wiki/Phase_space

or for an example:

https://en.wikipedia.org/wiki/Duffing_equation

or in math notation: the governing equation for exponential growth is :

y' = ay

which is a linear function where the slope is the growth rate. The plot shows y' vs y. This is the straight line in the plot. Any deviations from exponential growth can be easily spotted now.

[jdc → fxj]

https://en.wikipedia.org/wiki/Logistic_function

[fxj → jdc]

For the logistic function the relation is

y' = y - y^2

This is why the linear relation later on bends back to lower values until y' becomes 0 for the end of the infection.