Friday, March 27, 2020

More on parameters and rates

So it seems the Govt got the memo (I'll add a link when I see a write-up of Gove's comments). Though their estimate of 3-4 days for the doubling period is still longer than the data indicate - it was probably under 3 in my estimation, and that was way back in the mists of time when the latest figure for deaths was only 422. Two days on (plus a bit, shifting the reporting deadline from 9am to 5pm so really 56 hours) and it's 759. That's a doubling per 2.75 days unless my maths skillz are all gone to pot.

I hope that they have also realised the consequences for the timing and magnitude of the peak, because it really makes a bigly huge difference in bringing it all upward and forward as I showed yesterday. An awful lot hangs on whether this social distancing thing really works, because if not, it's going to be pretty dreadful. It's already pretty bad, and there are several doublings to go.

One important detail to remember in the parameter estimation/model fitting process is that with 3 parameters all playing a role in determining the net growth rate, we have a highly underdetermined problem when we look at population-level time series data alone, as this is just a simple exponential curve with a particular magnitude and growth rate. If I tell you that AxBxC = 35.4, you are still none the wiser as to what A, B, and C are. The converse of this is that this lack of determinacy isn't a huge problem in the simple simulation of the initial exponential growth curve - all that matters is the growth rate, and the data do constrain that. It does start to matter when you consider the later stages where herd immunity comes into play, or the response to a changed social situation such as controls and shutdowns. These are primarily linked only to R0.

So anyway, my first assumption when wondering what had lead Ferguson et al astray was that they might have used R0 from one epidemiological study and the latent period from another, picking values that were separately plausible but jointly wrong. I was a bit surprised to see the same ref listed for both. It reports a doubling time of 7.4 days for the early stages of the Wuhan outbreak. So, shrug. Either the data there are unreliable or the social situation is sufficiently different as to be inapplicable. In any case, it was a clear mistake to rely so strongly on one small data set. We have an abundance of empirical evidence that the intrinsic doubling time in multiple western societies (prior to any large-scale behavioural changes) is about 3 days, quite possibly even shorter.

A consequence of the parametric indeterminacy is that we don't know if the latent period is short and R0 smallish, or whether both are rather larger. This doesn't matter for the historical curve (which is of course precisely why they are undetermined) but it does when you start to consider interventions. If we assume (ok, it's an assumption, but we have little choice here with this sort of model) that a given social intervention acts so as to halve the amount of contact people have and consequently halve the value of R0, then a change of R0 from 8 to 4 (assuming reproductive time scale of 9 days) is much less effective than a change from 4 to 2 (and a 6 day time scale) or 2 to 1 (with a 3 day generation). The doubling time was initially 3 days in each case, and changes to 4.5 days in the first case, 6 days in the second and the epidemic stops in its tracks for the third. I should probably make a pic of this but I can't be bothered. Use your imagination.

It seems to me that the only way these parameters can be reasonably estimated is from detailed case studies, as in the cited Wuhan study which actually looked at time from infection to symptoms. Though, if the disease is infectious sooner than that (as appears to be the case) this isn't really the right time scale to use anyway. Regardless of the reasons, it's doubling much more rapidly than every 7.4 days.


Kaivey said...

Are you on Twitter, James?

James Annan said...

Most certainly....

Everett F Sargent said...

I think that I'm starting to believe those high numbers, particularly for the USA and Europe.