Sunday, March 29, 2020

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.


jules said...

The doubling time in Italy now seems to have gone up to about 6 days. Lombardy may be getting towards herd immunity, so the curve there might flatten ‘naturally’, but not the rest of Italy yet, surely... maybe it is too soon to tell and the doubling it will rise much further but the blue one looks truthy to me for lockdown European stylee. R0=3.2 ?!

steven said...

R0 is a bitch

Phil said...

R0 of people sitting alone in rooms is almost exactly 0. Italy isn't quite to that yet. The doubling time should no long exist in Italy as there is a tipping point at R0 = 1.0

What you would see after R0 goes below 1.0 is rising deaths for something like 3 weeks to a month. The time between infection and death. Roughly a month after lock down, death rate should start to fall. See Wuhan or South Korea. I'm not sure Italy can remain a stable state for another week... Italian government needs to make sure people have food.

Confirmed cases most likely are a function of the number of tests that can be run.

Herd immunity isn't likely in Italy yet. See San Marino.
(then sort by deaths/1M)

San Marino is a tiny country with a long history located in Northern Italy. This is probably a reasonable sample of Northern Italy. If the number of cases was even close, the infection rate is 0.7%. The recovered rate is 0.2%. Estimating from the number deaths (1% death rate guess) might be as much as a 7% infection rate. Herd immunity happens at much higher rates, usually 50% or higher. This is very roughly about 10% of a full scale epidemic.

Unknown said...

To the virus, an immune host and a perfectly isolated host are the same thing: the virus is going to die. SARS-CoV in 2003 infect ~8,000 and killed ~800. Despite difficult transmission characteristics, it managed to leap oceans on airplanes. SARS was a limited spread virus with 10% lethality. Be extremely afraid. SARS-CoV-2 is a better spreader, but it can be headed off: either by government initiatives at tract ID, isolate, or it can simply dead end: fail to find a host, and the chain dies. To overcome its spreading deficiencies takes time: a slow jerky build. It hits a moment, and it suddenly becomes a spreading machine: Hubei, but not to neighboring provinces (because the Chinese countermeasures were extreme to an extreme): Italy (where countermeasures were a sieve): to the rest of Europe (sieve): to UK (sieve). In Italy the virus is most likely hitting the low effectiveness of their sieve countermeasures, which indicative of what this virus is: ultimately easy to stop, even if you're really bad it. Because an isolated person, even if they're bad at isolation, remains less accessible to the virus as a host. In Italy, hopefully, the virus is hitting the slightly better isolators and failing to thrive as well as a result.

David Young said...

I just don't see how 600K deaths in the UK is possible based on the experience in places like South Korea and Japan.

I do wonder how biased the death numbers are. If you die and test positive for covid19 you are included in the statistics. If you die from respiratory syndrome or any other condition, you are not tested for flu generally and you are not counted in flu statistics. Flu mortality numbers are guesses anyway. I could see covid19 death numbers being exaggerated by a factor of 2-5 because most of the people who die are already pretty seriously ill with other conditions. Covid19 might have been only a small contributory factor to their deaths.

Even in the wildly biased WHO fatality statistics for those under 40 the death rate is 0.2% meaning in reality its probably less than 0.05%. These people just don't have a lot to worry about. They just need to worry about exposing their elderly and already ill relatives and friends.

James Annan said...

Japan hasn't had its epidemic yet - it is growing more slowly there, for a number of possible reasons, but it's still growing! And as for SK, it managed to achieve control - for now - with methods that we haven't managed to implement.

But I'm sure you are right, these doctors working up to shortly before their own deaths on trying to save CV patients were probably at death's door already....

David Young said...

James, Your sarcastic comment about doctors dying is of course anecdotal and meaningless as I'm sure you know. The media has misreported some of the deaths amoung the young either misstating whether they were "healthy" and also the cause of death in some cases. Fear mongering is not helpful and actually does harm. It's just a fact that all the statistics show very low fatality rates amoung those under 40 who are otherwise healthy. Those under 20 apparently are essentially not getting sick at all.

Everett F Sargent said...


Really nice post!

I posted a doubling time plot (deaths) over at ATTP's ...
I don't have a model, but it does need a model, badly. No amount of curve fitting or extrapolations yields reasonable long term numbers (e. g. extinction events unless you get doubling times into the range of double figures rather quickly).

For Europe and especially the USA doubling times must increase significantly over the next 2-3 weeks.

I don't know if your model can use local regressions or whatever, but a form of doubling time plot, similar to what I've shown above, is that possible from your modelling efforts?

Everett F Sargent said...


Oops, if possible, call it effective doubling time based on the output data from your model.

steven said...

"And as for SK, it managed to achieve control - for now - with methods that we haven't managed to implement."

we are playing wack a mole in Korea. cases stable at about 100 per day, large percentage imports.
the wack a mole approach will work until it doesn't. them boom. tick tock......
Here is what we are up against. every few days a hospital comes down with it.
ALL patients are tested, ALL staff, All family members, all contacts,, etc
typical profile will be... 1 staff has it, 30% of patients, 10% of family, 15%
of visitors... numbers "like" that. which looks like the staff started it.
we have the luxury of going crazy with contact tracing and mass testing.
patient in building X? boom, test the whole building. positive rates with this
approach is <5%. Asymptomatic rates are 20% So, person X gets it. Test his whole
company.. find 40 more cases, 20% symptomatic.

Same thing for larges churches, out current and recurrent problem.
God will save them, so they go to church, packed like sardines.. 1 has it, then
BOOM, we get another 50. News covers it, subways they rode on are all shown on the
news, phone apps go off alerting you.. news shows their route through the city
Bus they took, what times, what stops, etc.

My sense? not sustainable. Its just a matter of time before we get another patient 31
who gave it to something like 1000 people.

Some stupid shit they do in Korea. quarantines with your family. Yikes.

James Annan said...

Everett, yes I can do an instantaneous doubling time and how it varies over time. Basically a constant until a significant proportion affected, at which point it lengthens I dunno exactly how the shape would go. Maybe have a look next time I fire up the code.

steven, it will be a challenge for sure. Will be interesting to see how it goes long-term. In the long run of course we are all dead, but maybe we get a vaccine first :-)

Unknown said...

Yes, South Korea, Japan, etc. they are all playing with a lit fuse. They think they've snuffed it out, and it starts burning again. They could succeed at it forever, or: boom.

After reading about Japan, good gawd. I think they're going to get kneecapped; baby boom in a couple of years.

It's China. Only China knows what they are doing. They did it like a veterinarian would do it. Kill the gawd damned thing. I watched my Dad and his colleagues kill every hog on a farm, and inform the farmer he could not have hogs on his land for years. Can't remember the exact, but I think it was 6 or 7. -JCH

Phil said...

China is likely not reporting the whole truth. Likely in the same place as South Korea, and that is not a bad place. Playing wack-a-virus.

I see three ways we go:
The killing field, where the virus does its worst. Millions dead. India and Africa may go this way. Much of the Middle East. Much of South and Central America. Survivors will not need a vaccine, as most will become immune.
Wack-a-virus for a year or so, then vaccine.
Shutdown everything until you can play wack-a-virus, then a vaccine in a year or so.
Korea might need to have a shutdown if it gets away from them. Hopefully just a local one, long enough to find enough cases to snuff the sparks out.

James Annan said...

I'd just like to point out that the new report from IC vindicates everything I've said in this post. Link added at the top of the post.

David B. Benson said...

For the USA, daily new cases continue to rise, but the slope is much less.
The estimate for Washington state is that R0 has been reduced from 2.7 to 1.4. Already.

steven said...

have a look

lucia said...

Willis E has been posting simple plots of death vs. days on facebook and at WUWT. Washington state appears to be an outlier as far as slope. Most states are zooming up-- I think I recall he said doubling is on the order or 2-3 days using simple fits.

Eyeballing, I'd say the US slope isn't much different from UK. There was a bit of a different trajectory early on, but once it set in, the two appear to have about the same slop on his graph.

David B. Benson said...

Yes, Washington state is an outlier, especially since New York City and vicinity dominates.

David B. Benson said...

Tommy the robot nurse: