Infection delays 4

I have grabbed some data from a UK government website (https://coronavirus.data.gov.uk/details/cases) and amalgamated the daily data into weekly bins, since I think people react on a longer scale than daily, but monthly would give too few data points. The orange curve above is the data up to 21 December 2020, and the blue is an attempted fit, based on an exponential growth term and two complex terms that combine to give a real response. I have not as yet (this may not be true in the future) described how complex powers combine oscillatory and exponential behaviour, but they do. Anyway, I looked at the data and saw two peaks, the first around week 15, and the second around week 45. So we are looking for an oscilliatory component with about a thirty week period.

So what I am proposing is a solution of the form:

\[I_{w} = A r^w + B \rho^w e^{i w \theta} + B^* \rho^w e^{-i w \theta} \]

With this template, we can work back to the recurrence relation. First we construct the characteristic polynomial:

\[\left( a - r \right) \left( a - \rho e^{i \theta} \right) \left( a - \rho e^{-i \theta} \right) = 0\]

Multiplying out with the information \(e^{i \theta} + e^{-i \theta} = 2 \cos \theta \) (this is where the oscillation comes in) and \(e^{i \theta} e^{-i \theta} = 1\):

\[a^3 - \left( r + 2 \rho \cos \theta \right) a^2 + \left( 2 r \rho \cos \theta + \rho^2 \right) x - r \rho^2 = 0\]

The corresponding recurrence relation is:

\[I_w = \left( r + 2 \rho \cos \theta \right) I_{w - 1} - \left( 2 r \rho \cos \theta + \rho^2 \right)I_{w - 2} + r \rho^2 I_{w - 3}\]

To give a thirty week period, one needs \(\theta\) to be about \(2 \pi / 30 \) in radians. This comes out at about 0.21 radians, or about 12°. I have started the recurrence at a fairly high value in order to get something of the size of the peaks. When I looked at the UK data I was surprised to see one case in week 2 of 2020 — that would have been in January. Since the disease was barely a blip on the radar, possibly there was far more infection around in the UK than suggested by that single case. The ability to recognize and test for the disease must have improved with time, so perhaps we are also not comparing like with like in our weekly series. One could add a phase factor to give a bit more nudge space on the fit, but the more parameters, the less value: it is said that von Neumann, a mathematician of the early-to-mid 20th century, would claim to be able to fit an elephant with four parameters, and even make its trunk wiggle with a fifth.

What does this all tell us about anything? I’m not sure much. The real behaviour is sharper, particularly on the upside (for the SARS-CoV-2 virus, at least). This is the result of inertia on the part of governments and people — “Perhaps it will just go away. It’s not real. Why should I wear a mask?” So society’s behaviour is more like a switch (a bit like nerve signals that only fire at a certain threshold, let alone the analogy of neural adaptation where a repeated stimulus evokes less of a response as time goes on). Change only happens when a crisis point is reached, otherwise it is the line of least resistance. For my future readers, if any, the upswing in the real data around week 47 seems to be due to a new kid on the block, a variant virus that seems to have been developed in the UK evolutionary laboratory of schools and a government programme, “Eat out to help out” during the summer, designed to help the ailing catering industries and small businesses stay afloat. There also is apparently another variant that was brewed in southern Africa. Whether these new variants will respond to the vaccine developed for the original virus worries me, and I suspect others. This doesn’t seem to be ending soon.