This great preprint recently came out from a team of Katelyn Gostic and others. It uses simulations to test various methods of estimating the effective reproduction number \(R_t\). If you are following the Covid-19 pandemic from a data angle at all, you will no doubt have come across the effective reproduction number and will know that it is an estimate, at a point in time, of the average number of people an infected person infects in turn. It drives the exponential growth of an epidemic, and the way it varies over time is a clear, interpretable way of understanding how that growth rate gets under control, or doesn’t.
The paper is thorough, timely, and transparent - all the code is available, and the criteria on which the authors test different methods are clear and sensible. It’s certainly a must-read for anyone thinking of trying to estimate the reproduction number in real time. Some of the key issues that it considers include (in my sequencing, not their’s):
- the confirmed cases on any day will be a shifting underestimate of the number of actual cases. Most notably, when testing is constrained, high test positivity is very likely an indicator of more un-tested cases in the community the rate of which changes dramatically over time, but there is no agreed way of adjusting for this (in a previous blog I have suggested a pragmatic method of doing this which puts the role of tests in one day somewhere between a census which implies one should use just counts and a random sample which implies one should use just positivity, but it leaves a lot of space for judgement).
- the impact of the delays between infection, symptoms, testing and confirmation of a case. Some method of allocating today’s cases to various past dates is needed. One common method of doing this - subtracting days according to the distribution of the delay from infection to confirmation - makes things worse.
- the impact that correcting for that delay will have on our estimates of today’s cases, which will be right-truncated. That is, the infections in recent days will only be reflected fully in future data, so recent days’ estimates of infection numbers will be biased downwards
There’s also a complication about the difference between instantaneous or case/cohort reproduction number. As I understand it, for any given “today” the former is an estimate of the number of people being infected today per active cases today; but the latter is the average number of people who will be infected (today and later) by people who are infected today. The instantaneous measure is more appropriate for real-time surveillance.
Cases and test positivity in Victoria
I wanted to understand these issues better, particularly in the context of the recent marked increase in Covid-19 cases in Melbourne, Victoria, Australia where I live. Let’s start with looking at confirmed case numbers and test positivity rates. The Guardian compiles various announcements, media releases, dashboards and the like from the different state authorities around Australia into a tidy Google sheet. Thank you The Guardian. The data are presented as cumulative totals of tests and of cases whenever data comes in - so they are event data, with between zero and four observation events per day per state. This needs a bit of tidying to turn into daily increases in cases and tests.
Here’s the number of cases in Victoria, with and without a modest adjustment for test positivity as set out in my earlier post:
The history here will be familiar for Australians, but for other newcomers, the first burst of infections was driven nearly entirely by international travellers. There was a very low level of community transmission at that point, and the social distancing, test and trace, and lockdown measures pursued by governments at both Federal and State level were effective in suppressing the cases down to single digits. Then, since the second half of June, there has been a big increase in cases, this time locally contracted.
The adjustment I’m making for positivity (the aqua-blue line) is very mild - I’m multiplying the confirmed cases by positivity to the power of 0.1 and then scaling the result so that the cases at the lowest test positivity are left as the originally were. I’m not using the power of 0.5 (ie square root) that I used in my previous post in the US context and that has been taken up by some other analysts. I don’t believe testing bottlenecks are as much of an issue here as in the US; I think our testing regime comes much closer to the ideal of a census of cases, or at least a relatively constant proportion of cases allowing for a big but relatively stable proportion of asymptomatic cases below the radar. To see why, here’s the time series of test positivity in Victoria:
That uptick is an issue, but it’s clearly a different order of magnitude of challenge to that faced in the US, where several locations have seen 20% of tests returning positive or higher (even 50% - one in two tests returning positive!), a major flashing red light that many many cases remain undiagnosed. The bottom line - there’s a lot of testing going on here in Victoria at the moment, I’m not sure it’s a major bottleneck for counting cases, and I only want to make a small adjustment for the positivity rate which is so close to zero for a lot of the period of interest.
Here’s the code to download that data from The Guardian, estimate daily changes, model and smooth the test positivity rate and estimate an adjusted rate. It also sets us up for our next look at the issue of dealing with the delay, and with the downwards bias in recent days’ counts resulting from how we deal with the delay.
Post continues after R code
Convolution and Deconvolution
One of the obvious challenges for estimating effective reproduction number (\(R_t\)) is the delay between infection and becoming a confirmed case. The Gostic et al paper looked at different ways of doing this. They found that with poor information, an imperfect but not-too-bad method is just to left-shift the confirmed cases by subtracting the estimated average delay from infection to reporting. A better method takes into account the distribution of that delay - typically a Poisson or negative binomial random variable of counts of days. However, a common approach to do this by subtracting delays drawn from that distribution is noticeably worse than simply subtracting the mean. In the words of Gostic et al:
“One method infers each individual’s time of infection by subtracting a sample from the delay distribution from each observation time. This is mathematically equivalent to convolving the observation time series with the reversed delay distribution. However, convolution is not the correct inverse operation and adds spurious variance to the imputed incidence curve. The delay distribution has the effect of spreading out infections incident on a particular day across many days of observation; subtracting the delay distribution from the already blurred observations spreads them out further. Instead, deconvolution is needed. In direct analogy with image processing, the subtraction operation blurs, whereas the proper deconvolution sharpens”
To be clear, it’s not just armchair epidemiologists who are wrongly using convolution backwards in time here, but specialists performing Covid-19 surveillance or research for their professional roles. So this paper does a great service in pointing out how it makes things worse. I think that at least one high profile dashboard of \(R_t\) estimates has modified its method based on the findings in this paper already. Self-correcting science at work!
Recovering unobserved past infections with toy data
To see this issue in action I made myself two functions
sharpen() which respectively do the simple convolution and deconvolution tasks described above in a deterministic way. The job of
blur() is to delay a vector of original cases in accordance with a given distribution. In effect, this spreads out the original vector over a greater (and delayed) time period. The job of
sharpen() is to reverse this process - to recover an original vector of observations that, if blurred, would create the original (unobserved) incidence counts.
I tested these two functions with a super-simple set of six original incidence counts, the vector
4, 6, 8, 9, 7, 5. I blurred these into the future in accordance with expected values of a Poisson distribution with a mean of 3 days. Then I used
sharpen() to recover the original values, which it does with near perfect success:
My blur() and sharpen() functions (which are completely deterministic and not fit in my opinion for dealing with real-life random data) are in the
frs R package where I store miscellaneous stuff for this blog.
Here’s he code that makes those toy example original, delayed and recovered time series:
… and for completeness, here are the definitions of my
sharpen() functions. Most of the versions of these I’ve seen in R, Python and Stan use loops, but the particular nature of the “multiply every value of a vector by the probabilities for various lags” operation suggested to me it would be simpler to write as a join of two tables. Maybe I spend too much time with databases. I did think at one point in writing these that I seemed to be re-inventing matrix multiplication, and I’m sure there’s a better way than what I’ve got; but the complications of forcing the recovered vector of original infections to match the observed vector were too much for me to come up with a more elegant approach in the hour or so time budget I had for this bit of the exercise.
A big limitation on these toy
sharpen() functions is that they assume not only that we know the probability mass for each possible day of lagging, but that the process is deterministic. Of course, this isn’t the case. The
backprojNP() function in the
surveillance R package estimates those unobserved latent series on the assumption that the observations come from a Poisson process. Here’s how that looks when we apply it to my toy data:
This uses a method first developed in the context of understand the spread of HIV, which has a particularly long and uncertain delay between infection and confirmation of a case.
It doesn’t do as well at recovering the original as my
sharpen() did, because
sharpen() has the luxury of knowing the blurring took place deterministically. With these small numbers that makes as big difference. But
backprojNP() does an ok job at recovering at least some of the original structure. It’s a bit sharper than the blurred observations. Here’s the code applying
backprojNP() to my toy data (mostly this is plotting data; I stuck to base graphics to give me a bit more control over the legend):
Not only does
backprojNP() do a better job at coping with real-life random data, it’s much faster than my
sharpen() function. So I should emphasise again that
frs::sharpen() is here purely for illustrative purposes, to help me get my head around how this whole single dimensional deconvolution thing works.
Recovering unobserved past infections with toy data
OK, so how does this go when we apply it to real Covid-19 data? A quick google suggests that the delay from infection to symptoms is approximately a Poisson distribution with a mean of 6 days. The time from symptoms to becoming a confirmed case is unfortunately very much unknown, and also is likely to change over time (for example, as queues for testing lengthen or shorten, and testing regimes become more or less aggressive). I took a guess for illustrative purposes that it is going to be a Poisson distribution with a mean of about 3 days, shifted one day to the right (because it takes at least one day to get one’s test results). As the sum of two Poisson distributions is another Poisson distribution, this means we can model the overall delay from (unobserved) infection to confirmation as a Poisson distribution with mean of 9 days, plus one day for good luck. In the code below the probability of delay by various lags is defined on this basis in the
Here’s what this looks like when I apply it to the Victorian data:
Overall the curve looks like how we’d expect - earlier than the observed cases, and a little steeper (not as much steeper as I expected though - in the course of writing this blog over a few days I noted this curve can change noticeably as individual data points come in, so it should be treated with caution - see later for a better chart).
Note how this chart really draws attention to the final challenge we’ll be looking at - the right truncation of the data. That massive drop in the aqua-blue line is very misleading, because it’s based on there being zero confirmed cases from tomorrow onwards! We’ve got a way to go yet before we want to turn this into estimates of effective reproduction number, but we’re on track.
Here’s the code for that bit of analysis:
Adjust for testing, delay and right truncation all at once
Clearly the problem of adjusting for the delay is going to need a more careful evidence base than my casual guess at a Poisson variable with mean of 9 days displaced one day to the right; we’ll need an effective way to manage the smoothing of our inferred original infection estimates; we need a way of estimating the unobserved cases; and a method for dealing with the “right truncation” ie the absence of data that can be mapped to infections today, yesterday, and other recent days.
Following one of the footnotes in Gostic et al (referring to “Many statistical methods are available to adjust for right truncation in epidemiological data”) led me to the very excellent
EpiNow2 R package, which tries to do most of this simultaneously. It wraps a Bayesian Gaussian process model implemented in Stan, which is exactly the way I think this problem should be approached. Their documentation helpfully includes instructions on using their package for real-time estimation of Covid-19, exactly my problem. It even includes literature-based estimates of the key characteristics of the appropriate distributions to use for estimating and combining different types of lags (generation, incubation and reporting).
EpiNow2 is one of the tools developed and used by the team behind epiforecasts.io - Sebastian Funk and others at the London School of Hygiene and Tropical Medicine. The guts of the software is this Stan program.
While it’s still under development, it’s very impressive. Hats off to the team. Here’s the full citation, noting a fair bit of cross-over with both the epiforecasts.io team and the Gostic et al paper I started this blog with a link to:
Sam Abbott, Joel Hellewell, Robin Thompson, Katelyn Gostic, Katharine Sherratt, Sophie Meakin, James Munday, Nikos Bosse and Sebastian Funk (2020). EpiNow2: Estimate Realtime Case Counts and Time-varying Epidemiological Parameters. R package version 0.3.0.
Here’s what I get when I apply their approach out-of-the-box to Victoria’s case counts:
As can be seen, several sets of estimates are returned. Panel B is important - it shows the estimated actual infection counts (the curving ribbon, which reflects a credibility interval) in contrast to the reported confirmed cases (the grey columns). It makes it easy to see the broadening uncertainty as we get to today and even a bit into the future; and also how the estimated infections precede the confirmed cases.
From panel B, the estimates of instantaneous reproduction number follow in straightforward fashion, as seen in the bottom panel. I’ve opted to only show the estimates of \(R_t\) from late April onwards because prior to that the cases were dominated by international arrivals. While methods exist to estimate reproduction number appropriately in this circumstance, I’m not sufficiently interested in that bit of history (months ago now…) to go to the effort to do it.
Of course, this chart is cautiously good news for Victorians. The brown ‘nowcasting’ segment of the plot shows a decided downwards trend in estimated infections, timing closely with the extra shutdown and distancing measures brought in just under two weeks ago. And it shows the best estimate of today’s effective reproduction number to be (just) less than 1. Let’s hope that stays the case.
That first fit was with the confirmed case numbers directly from The Guardian. If I instead apply them to the numbers after my mild correction for test positivity, we see a similar picture. Recent estimated case numbers are higher because of the higher test positivity, but the estimate of \(R_t\) today is similar - still just below 1.0, and on its way down (albeit with lots of uncertainty).
That uncertainty issue is actual one of my main motivators for writing this blog. The fact is, we don’t know what’s going to happen here. This is definitely one of those cases when one of the most useful things a forecast or nowcast can do is highlight the range of possibilities that are consistent with the data we have. And critically, what happens in the future - that big blue credibility interval in the last couple of charts - depends on actual people’s actual decisions and actions.
“The fault, dear Brutus, is not in our stars, but in ourselves”
That reminds me, a Shakespearean blog post really is on the way.
So the situation in Melbourne is clearly on a knife edge. Today’s numbers (Saturday 18 July) are good (in fact exactly what and when would be hoped if the suppression strategy is doing its job), but we know not to pay too much attention to one point in these noisy processes. In a week’s time, any number of cases per day from zero to more than 1,000 (the chart is cut off well below its top point) is possible. Let’s remember those big, uncertain prediction intervals; not be too confident about which way things are headed; and do our best to point them in the right direction by our own behaviour. Stay safe out there! Practice social distancing, stay at home as much as possible, wear a mask when going out, and pay attention to your local public health experts. New Zealanders are excluded the first three of those four things, because they did the fourth one reasonably well.
Anyway, here’s the code for those final bits of analysis. Running the estimation processes took several hours each on my laptop: