Is R0 larger than people think?

EDIT: as a quick update to this, I'd just like to point to the new IC report out here, which vindicates what I've written below. Yes, R0 is higher than they thought, and this does mean that control is far harder to achieve (and quite possibly will not be achieved with current policies).

You read it here first...

--------


Is R0 really big?  R0 here being the average number of people that each infected person passes the disease on to, assuming a population with no immunity. The reason I wonder, is that the doubling rate in the death data is well under 3 days - under 2.5 in fact over the full time series from the first death to yesterday's data. There hasn't been a single period of 6 days in which the total number of deaths in the UK has not at least quadrupled, apart from when the govt sneaked in a short 8 hour “day” into the official time series recently. I see this has been adjusted for in some data sets, but not the one I have to hand - it won't matter in the long run of course. 

Incidentally, it should not be forgotten that the official death toll now only counts hospitalised cases. Those who don't get that far, and aren't tested, aren't counted. I'll assume this is a minor factor for now and ignore it completely in this analysis.

Doubling time depends on 3 parameters in the SEIR model - R0, and the latent and infectious periods which I'll abbreviate to L_p and I_p. I_p is the time interval during which ill people are passing on the disease, which can hardly be less than a day. I'll fix it at 2, which is also much shorter than most estimates I've seen. It's important to realise that L_p and I_p are basically biological parameters of the disease. They may vary a bit from place to place, but it's hard to imagine wild variation. R0, on the other hand, is substantially a social construct. It varies with how many people we meet, and how we behave with hygiene and cleanliness. It is R0 we aim to change through social distancing and disease control measures such as quarantining etc. A value of R0 calculated from another country and culture may be rather unsuitable for our situation. The value of R0=2.4 that Ferguson et al preferred for their main calculations (with a range of 2.0-2.6) was from the first few people infected in China, and may be unrepresentative for a number of reasons.

Even with I_p fixed at a very small value, in order to get a short doubling time, you still need either to have a small value for L_p, or a large R0. Most studies put L_p around 4-5 days which would mean R0 has to be pretty huge. You do have to be a little careful in interpreting the epidemiological analyses, as people often define L_p as the time taken to develop symptoms, whereas the model actually only cares about when you start to infect others - and there is evidence that this is a slightly quicker process. But still, a latent period of 3 days would be very short compared to what most people are saying.

If the latent period is 3.5, and the infectious period is 2 - both value which seem to be below the range of widely accepted values - then R0 has to be 2.7 even to get a doubling rate of 3 days, which is a bit slower than the data suggest. R0 = 2.7 is outside the range of values that were even tested by Ferguson et al. If you take a more accurate but pessimistic view that the doubling rate is actually 2.5 (it's actually comfortably quicker even than this in the data!) then R0 has to be 3.2.

Why does this matter? The control, mostly. Ferguson et al assume we can reduce R0 substantially by the extreme control measures that are put in place. Their “suppression” strategy takes a basic value of R0=2.2 and assumes it will be reduced by about 70% though social distancing and all the other measures which are put in place.

Reducing 2.2 by 70% gets it well below 1 and control is rapidly achieved. Reducing 3.2 by 70% just gets to below 1, barely. If R0 is any bigger, then a 70% reduction still leaves it greater than 1 meaning we progress through a full epidemic, albeit a smaller one than we'd have had without the controls.

In the graphs below I've tried a range of combinations with different pairs of R0 and L_p, all of which have exactly the same doubling time of 3 days - at the slow end of what might be compatible with the data, though the quick end of what the Govt is currently planning for. These following simulations are also all tuned to have 1000 deaths at the present date.

With no intervention, the epidemics are identical at first and broadly similar in their entirety, only differing in the point at which herd immunity is reached. We see about 600,000 deaths at a 1% death rate.

If we introduce controls in the middle of March to reduce R0 by 70%, then the second plot results. What a difference! For the low R0 values, the epidemic is rapidly controlled, though the deaths lag this a bit. If R0 is large, even though the growth is completely indistinguishable up to this point, the futures are very different. The growth remains positive for a while, before the epidemic levels out and decays. Note this assumes we keep the control measures in place indefinitely.

Here's a blow-up of the last plot just focussing on the period around the present so you can see the detail. And I will also show daily deaths as this gives a clearer indication of changes.

I suspect this model reacts too sharply to changes in parameters, compared to reality where there are numerous sources of variability that this model does not capture. That tends to smooth things out rather. Nevertheless, it suggests we'll be a few days into April that we could possibly expect to recognise a levelling off in the death rate. A key question will be whether we keep below 1000 deaths per day, which I think would be an optimistic hope at this point. I emphasise that the lower lines depend on really rather extreme values for the time scale parameters of the virus, which in principle I'd think should be reasonably well understood by virologists and the like. The lower two lines have latent periods of just under 2 and 3 days respectively!

The Govt's talk of 20,000 deaths looks to be a bit hopeful to me.