"Thoughts on a logplot." My latest thoughts on logplot I made regarding the Corona virus.
Usual disclaimer, this all initial theorizing regarding numbers I sloppily collected from the Internet, which were hastily gathered themselves, and may suffer from a myriad of other distorting effects. Hardly affected people may have gone unnoticed, doctors may simply have become better at collecting data leading to initial exaggerated growth, people might have run out of test kits in the field, etc.. Use a lot of caution regarding the data.
Second, even if high-school math, it may very well be plain wrong. It's unconfirmed theorizing, and -honestly- I am a bit embarrassed how this simple math still takes considerable effort from my part.
Now we have that out of the way, the initial thoughts:
We have a logplot with two lines: The confirmed and the dead cases. I am going to treat them as is. [BIG IF]
Hypothesis: The illness follows an exponential growth modeled as a function f(t) = c_0^t. The deaths can be modeled as percentage time-lagged dependent function g(t) = m*f(t-t_0) where m is mortality rate and t_0 is average time to death or lag [SECOND BIG IF]. c_0 (growth), m (mortality rate), t_0 (lag) are constants.
In the logplot, c_0 can just be deduced from the slope. The fact that both f and g have the same slope seems to confirm stable exponential growth and the dependence of g on f.
Because we know c_0, we actually know the start of the disease, just extrapolate f to where it meets the horizontal axis. I am going to assume 1/1/2020, twenty days prior.
Now, we have _two_ unknowns: m (mortality rate) and t_0 (lag). Hypothesis: I can fiddle with either to make f map on g corresponding to I can move f either horizontally or vertically to place it on g. [THIRD BIG IF]
Support for that: Is an academic case of a mortality rate of 100% possible? Sure, just assume m is 1 and t_0 is twenty days. This corresponds to shifting f twenty days to the right; i.e., everybody dies, f is exactly reproduced twenty days later. Orthogonal: Is an academic case of 2% possible? just shift directly f downward, this corresponds to 2% dying within 0 days.
Do I know the mortality rate? No. But I could tell you if the above is correct and I would know the average time it takes to die. Only thing I know now it's between 2% (unlikely) and 100% (unlikely).
This seems to confirm the old adagium that you cannot know the details of a disease in the initial phase because you have two unknowns you're trying to fit. Visually you're trying to fit a line on another one by moving either horizontally or vertically and both works.
Again, no idea whether the above is bullshit. The only thing is that I can tell you that I think this model seems to suggest that we're still firmly in the unknown regarding anything.
