Testing Has Plateaued
I recently posted that I wouldn't have time to run any more models or projections until December, but I do still have time to occasionally plot the raw data.
So two plots for you today. The first is the number of tests that have been performed in the US. You can see that testing has been increasing slowly but steadily since mid-March. However, this stopped come late-July/early-August and has held steady since then.
Widespread, asymptomatic testing is the key to ending the pandemic pre-vaccine. For example, the few universities that are doing this are still open; those that are testing only the symptomatic are closing quickly (e.g., Duke vs. UNC -- I am embarrassed to point out...)
The question has always been would testing continue to scale? Apparently not. (However, new, quick, cheap, saliva-based tests are very encouraging -- we'll have to keep our eye on them over the next month or two.)
In the second figure, I plot (1) the doubling timescale -- how long for the number of reported cases to double -- which is now up to 106 days (blue curve), and (2) the testing timescale -- how long it would take to test everyone in the country -- which was decreasing but has stalled out at 445 days (orange curve).
The closer these two curves get to each other, the more the gray curve will drop. The gray curve is what the doubling time/blue curve would need to be to end the pandemic tomorrow (i.e., no new reported cases). It also happens to be, approximately, the number of reported cases that are currently infectious.
Unfortunately, all three of these curves appear to be stalling out now...which suggests a bumpy ride ahead.
So...keep social-distancing, mask-wearing, and hand-washing. And keep pushing for more testing and contact tracing!
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8/18/2020
Reason for Cautious Optimism
It’s been about two weeks since our last update. We’ve now fitted our — fully empirical — doubling-time model to the single-day doubling-time measurements through last Saturday, and find that the doubling time is continuing to rebound. As of last Saturday, it had reached 75.4 +/- 1.2 days — the highest value we’ve reported yet — which is great.
Even better though, we measure this number to be increasing at a rate of 45.6 +/- 1.0 hours each day. This is well above the “one day each day” benchmark, and consequently the number of new cases each day is falling again.
So we have been through two peaks. The first corresponding to the northeast and the second corresponding to the south and west.
Now, everyone has learned what is required to boost the doubling time and bring the number of new cases down.
If everyone stays on this trajectory, we still project a sobering 347 +/- 51 thousand deaths by September 1, 2021 (the date we’ve adopted for when a vaccine will be widely available).
Of course, a lot will happen between now and then, including the reopening of schools, wider-spread testing and tracing, the winter season, and hopefully an earlier release and widespread distribution (and widespread adoption!) of a vaccine. So don’t take this projection too seriously — it is more a measure of our current trajectory than it is of our destination.
The first of these challenges before us is the reopening of schools. Given that our team is based at UNC-Chapel Hill, we know better than most that this isn’t going to be easy. I believe that our classrooms were sufficiently safe. And I believe that most of our students made good decisions outside of class as well.
But this thing is so virulent, it only takes a few to ruin it for everyone. We didn’t last a week.
We’ll see if other schools can do better, and if K-12 schools can do better. It will likely be another couple of weeks before any of this is reflected in the doubling-time data.
Until then, keep social-distancing, mask-wearing, and hand-washing. And keep pushing for more testing and contact tracing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
8/5/2020
Still in the First Wave, but Past the Second Peak
We haven’t put out an update in over a month!
The reason why is that we have been going through all of the old data, day by day, and updating our fitted model and projections. Why do this? Having gone through this exercise in real time, we had to guess when we should add a new phase to our underlying doubling-time model, or more terms/flexibility to our weekly reporting cycle model. We have now introduced good statistical measures to make these decisions for us, so we’re systematically updating everything.
And learning some interesting things as we go. For example, did you know that in addition to the peaks in mid-April and late-July, there was a mini-peak in late May?
Anyway, this work is almost done, and we look forward to reporting on it soon.
However, a few days ago, we decided to jump ahead and do another present-day fit to the data, through August 1st. The results are presented in the attached figures.
As you can see, throughout most of June, the doubling time was plummeting, but by very late June this began to slow, and by July 9th we finally found the bottom, at a (weekly reporting cycle corrected) doubling time of about 36 days.
Since then, the doubling time has been increasing again, and as of August 1, we’re back up to 49.8 +/- 1.1 days, and increasing at a rate of 22.1 +/- 1.8 hours each day — just a little short of the critical “one day each day” benchmark that we need to surpass to really quash the pandemic.
By now, the pattern has become clear. (Enough) Americans aren’t willing to do what’s needed to end the pandemic quickly — we’ve spent much of the pandemic coasting just below the “one day each day” benchmark, and that’s where we find ourselves again.
But when we get lax, and the doubling time starts dropping and the case numbers start skyrocketing — we are willing to change our collective behavior and do what’s needed to prevent the health care system from getting overwhelmed. This is what happened in the northeast early in the pandemic, and in the south and west this past month. That’s not nothing.
The trajectory that we’re currently on isn’t great, but it isn’t exponential/super-exponential growth either. The question in our minds is what happens next, with K-12 and colleges and universities re-opening over the next few weeks? It is bound to have some negative effect, and possibly (probably?) a great negative effect.
Had the pre-second peak trend held, we would be taking this leap from a height of about 10,000 new cases per day. But because the second peak reset everything, we’ll instead be leaping from about 50,000 — 60,000 new cases per day.
So although we don’t know what the effect of re-opening the schools will be, we do know: (1) It will be about 5 — 6 times worse than it would have been, had we not gone through the second peak; and (2) It will be worse in the south and west, where the case numbers, although now declining, are still the most elevated.
Whatever happens, we’ll be ready to add another (#7!) phase to our underlying doubling-time model if/when the data request it.
Until then, keep social-distancing, mask-wearing, and hand-washing. And keep pushing for more testing and contact tracing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
7/1/2020
Super-Exponential Growth Continues, Unabated
Eight days ago we reported some bad news: The doubling time had stopped increasing and began decreasing, by 2.0 +/- 0.5 days each day. Negative growth of the doubling time corresponds to the number of new, reported cases increasing not exponentially, but super-exponentially.
Eight days later, we are still on this trend. The doubling time has plummeted from a high of approximately 66 days on June 11th to only a bit more than half of this yesterday: 36.4 +/- 1.3 days. And it continues to drop, by 1.6 +/- 0.1 days each day.
We have included the same two plots as last time, so you can compare and see how little the trend, and the projection, have changed.
The first plot shows the (weekly reporting cycle corrected) doubling time. The second plot shows the number of new, reported cases, and our projection based on the current doubling-time trend. The projection is virtually unchanged since last time, with a new peak expected around July 25th.
Keep in mind, this projection is a worst-case scenario, assuming that the doubling time continues to drop all the way back to its initial value of only 2 days. Since many people are continuing to practice social distancing, mask wearing, etc., we do not expect it to drop that far.
But our philosophy has always been to not try to model human behavior — just to model the doubling-time trend and show people where we’re heading if their behavior doesn’t change faster or slower than it already is.
That said…we’ve been on the decline now for 18 days and have yet to hit bottom — or even slow down. At the current rate of decline, a doubling time of 2 days is only about 3 weeks away. Given this, we expect to find the bottom over the course of the next week or two (sooner would of course be better than later).
But even once we do find the bottom, the projection won’t improve significantly until we reverse the trend and start digging our way out. This of course does not happen automatically, but requires people to change their behavior…again. (Given the projected case numbers, and the very real prospect of hospitals being overrun in the south and west, this is likely to happen.)
Alternatively, widespread testing and contact tracing could solve all of this for us. But this still needs to ramp up by another factor of 5 — 10, which won’t happen overnight.
Until then, keep social-distancing, mask-wearing, and hand-washing. And keep pushing for more testing and contact tracing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
6/23/2020
We Have Entered Super-Exponential Growth
In our last (6/18) update, we reported that the single-day doubling-time measurements may be taking a turn for the worse. Unfortunately, this appears to be correct — and much worse than we initially thought.
We now measure the doubling time to be 51.7 +/- 1.7 days — and decreasing — by 2.0 +/- 0.5 days each day.
So far, most of our posts have been about how the doubling time — the time it takes for the number of reported cases to double — hasn’t been increasing quickly enough to end the pandemic. Except for 1 — 2 weeks near the beginning of May, the doubling time has been increasing by less than the critical “one day each day” rate needed to get us over the peak.
But as you can see in our first figure (in which we have modeled out the effects of the weekly reporting cycle), the doubling time is not only not increasing quickly enough — it’s decreasing — and has been for the past week or so.
As long as the doubling time is increasing (below the critical rate), we have sub-exponential growth. If it holds steady, we have classical exponential growth (at least until most of the population is infected, and herd immunity effects kick in).
But if the doubling time is decreasing, we have super-exponential growth, which is (1) ugly and (2) unsustainable.
Let’s take “ugly” first. If the doubling time continues to decrease at this rate, there will be a new peak (let’s not get into debates about whether this is a second peak or still the first peak) in late July (July 20 — 26), reaching 1.3 — 7.0 million new cases per day. We plot this in our second figure. Note that the previous peak was around 30 thousand new cases per day, in mid-April. It is doubtful that our healthcare system could handle more than 100 thousand new cases per day, let alone millions.
However — and fortunately — a decreasing doubling time is not sustainable. At worst (which the above numbers assume), the doubling time stops decreasing once it’s back to 2 days, the natural rate at which the virus spreads.
Why is this happening? The doubling time reflects the measures that we, as a society, are taking to combat the virus’s spread. At the peak of stay-at-home, social distancing, etc., the doubling time was increasing by nearly 2 days each day. That it’s now decreasing at this rate reflects the rate at which we are rolling back these measures — but without replacing them with anything that is equally effective (such as sufficiently widespread mask wearing, testing and tracing, etc.)
The only question is at what doubling time does the current free-fall stop (and if, and how quickly, it rebounds after that). But none of these things happen on their own — they are direct reflections of the choices that we, as a society, are currently making.
Lastly, keep in mind that changes in behavior take about two weeks to show up in the data. Assuming that we have been on the same trajectory for the past two weeks, and that it will take another two weeks for changes in behavior to show up in the data, that gives us only 2(-ish) weeks to make sufficient changes to avoid, or at least flatten, the next peak.
I know that this seems like deja vu all over again — but exponential growth is unforgiving in this way.
Keep social-distancing, mask-wearing, and hand-washing. And keep pushing for more testing and tracing.
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
6/18/2020
After 5 Weeks, the Doubling-Time Trend May be Changing Again — and Not for the Better
Wednesday’s doubling-time measurement was unusually low — 58 days — when we were expecting 73 days (taking in to account the weekly reporting cycle, which we also, and simultaneously, model). Furthermore, Tuesday’s measurement was also low — 61 days.
This could be due to:
A dramatic increase in testing, artificially driving down the single-day doubling-time measurements. But this does not appear to be the case — Wednesday’s 466 thousand tests are typical of what we’ve been doing in the US for a couple weeks now.
Meaningless noise in the data. However, this would be a very rare noise fluctuation. Our combined doubling time + weekly reporting cycle model has been good to +/- 6.9%. But Wednesday’s doubling-time measurement came in 26% low, which should happen only once every 6000+ days.
A dramatic change in the weekly reporting cycle. Since the beginning of the pandemic, the under-reporting period has shifted from the weekend to the early week, and lengthened to many days. This leads to higher single-day doubling-time measurements almost all week, followed by a couple days us catch-up/backlog reporting (and low single-day doubling-time measurements) at the very end of the week. Perhaps this has changed. However, the weekly reporting cycle has changed only gradually over the course of the pandemic. A sudden change would be difficult to explain.
The doubling-time trend line could be changing again, and not for the better.
#4 would not be unexpected, given most states’ moves to reopen the economy, and given the continued politicization of mask wearing (which might be one of the most effective weapons we have against the virus).
Two graphs for you today. In our first graph, we plot the overall rate at which the doubling time has been increasing since about May 10th. Over the past 9 days, this has dropped from 20.4 +/- 2.4 hours each day to only 15.7 +/- 2.3 hours each day — well short of the “one-day-each-day” benchmark that we must surpass to end the pandemic.
In our second graph, we plot the number of new, reported cases each day. If #4 is correct, we may now be exiting the plateau phase that we’ve been stuck in since May 10th, and entering a period of resurgence.
Right now, this resurgence appears to be led by FL, TX, CA, and AZ, but many states are on the upswing (and some, primarily in the northeast, are still on a downswing).
Our plan is to wait until early next week and reassess.
Until then, keep social-distancing, mask-wearing, and hand-washing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
6/11/2020
The Recovery Has Stalled Out Again
Over the past week we’ve made a few final refinements to our code base, and should now be in a position to resume updating our new-and-improved model (which takes into account variations due to the weekly reporting cycle) daily, or at least near-daily.
As such, we’ve added another week’s worth of data and find that we are still stalled out: The doubling time is gaining 19.2 +/- 0.4 hours each day, which is short of the critical “one-day-each-day” level that we need to recover.
This is the second time that the recovery has stalled out. The first time was throughout the month of April. You can see this in both of the attached plots. In the first plot, you can see that after the number of new reported cases peaked in early April, then held steady throughout the rest of the month. This is because the rate at which the doubling time was increasing also stalled out, at a sub-critical 19.3 +/- 0.4 hours each day, which you can see in the second plot.
Then in early May the doubling time began to accelerate again, and the number of new reported cases dropped from about 30,000 per day to about 22,500 per day.
However, since then, the doubling time has been increasing sub-critically again, and consequently, we are seeing another plateau in the number of new reported cases.
If this doesn’t change, we’re in for a long, slow, and painful post-peak tail. Projecting forward, we’re finding 460 (+90, -80) thousand deaths by September 1, 2021 (which is when we have been assuming a vaccine will become widely available).
However, a very slow decline also leaves us at risk of a second peak, especially as people continue to become more relaxed about social distancing, mask wearing, etc. before testing and tracing ramps up enough to identify and isolate the asymptomatic or mildly symptomatic cases that are currently being missed.
Anyway, in our next post we’re going to shift gears and talk more about our weekly reporting cycle model, and interesting results that it has been uncovering.
Until then, keep social-distancing, mask-wearing, and hand-washing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
6/8/2020
Some Good News: The Gap between 2 Months and 2 Years is Closing (Relatively) Fast
We hit a pair of milestones this past week, around June 3rd.
First, the doubling time – the time that it takes for the number of infected to double (blue curve in today’s figure) – has finally clawed its way up to 2 months. This was increasing by as much as 2.5 days each day until the pushback against stay-at-home, social distancing, mask wearing, etc. kicked in. Since then, it’s been increasing by only about 3/4 of a day each day, which is not enough to save us from a long, post-peak tail, and puts us at risk of a second peak.
With stay-at-home, social distancing, mask wearing, etc. on the decline, and a vaccine still far on the horizon, the only thing that can turn the tide now is testing and tracing. But to be effective, we need to be able to test nearly everyone in the country on a doubling timescale. This way, even asymptomatic carriers can be identified and quarantined for the 5 – 14 days it takes to no longer be infectious to others. Technically, some people would still get it after being tested but before everyone else is tested, but the odds of this decrease rapidly when you can test everyone on the doubling timescale. One pass through the population and America would be very safely open for business again, with the remaining pockets eliminated on the next couple passes.
The time that it would take to test everyone is what we’re calling the testing timescale, and it’s been dropping fairly quickly (orange curve in today’s figure). On March 1st, it was about 10,000 years. By April 1st, it had plummeted to about 8 years, and by May 1st it had dropped again to about 3.5 years. As of last week, it’s now only 2 years.
2 years is still too long – but not for long: The testing timescale has been dropping at the same, consistent clip for the past two months now. If this continues – meaning if capacity continues to scale – the testing timescale should equal the doubling timescale by late summer/early fall. At this point, we would be only a few months of testing and tracing away from the pandemic being effectively over.
Of course, this also assumes that the doubling time continues to increase at its current rate, which is the big question right now. The decline in stay-at-home, social distancing, mask wearing, etc. very much works against this. But increasing testing and tracing boosts the doubling time. Everything really depends on which of these two forces wins out over the next couple months.
The gray curve in our figure is the “ending” timescale (and it also happens to equal the number of reported cases that are currently infectious). As the doubling timescale (blue curve) increases, the ending timescale (gray curve) decreases. We’re down to our last new case when these two curves cross, somewhere in the middle. But this won’t occur until a doubling timescale or two after the blue and orange curves cross.
Keep social-distancing, mask-wearing, and hand-washing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
6/4/2020
Stay-at-Home was Finally Beginning to Deliver, Right before Too Many People Gave Up on It
Yesterday, we presented the single-day doubling-time measurements in a new and powerful way. Much of what has been going on with the doubling time has been obscured by the weekly reporting cycle – weekend (and now early week) underreporting artificially boosts the measured doubling time, and later-week catch-up/backlog reporting artificially suppresses it.
But since this is something that my team is now able to model, we can divide this out of the data and see what’s really going on beneath all that additional scatter/noise. Check out yesterday’s post for the details, but the data show that stay-at-home and mask-wearing was finally beginning to pay off – right before enough people gave up on it.
Yesterday we presented the reporting cycle-corrected, underlying doubling-time trend. Today, we’ve simply plotted its slope – the rate at which the doubling time is increasing. You can see that after a month of virtually no progress in March, this began to increase in early April, but then leveled off again in late April.
It leveled off to almost – but not quite – one day of progress each day. This is a frustrating place to be. It means things aren’t getting worse, but they’re not getting better either. It also means that things could get worse at any point, including the possibility of a second peak.
But then in early May, it finally happened. Stay-at-home orders had been in place not only in the early states, but now in most states for weeks. And after being told that masks would not help us (probably to save them for health care workers who needed them more), they became more available, the government reversed its position, and people began wearing them. The doubling time accelerated like we hadn’t seen before, eventually increasing by about 2.5 days each day.
But it didn’t last. Opposition to stay-at-home and mask wearing coalesced almost immediately, and by the time this also worked its way into the doubling-time numbers, the rate dropped sharply, back to almost, but not quite, one day of progress each day.
And three weeks later, that’s where we still are. With stay-at-home, mask wearing, etc., now in decline, the question becomes can testing and tracing ramp up quickly enough to make up the difference.
We’ll return to this in a future post. Until then, keep social-distancing, mask-wearing, and hand-washing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
6/3/2020
Guess What Almost Happened this Past Month…
We took a month off. What did we miss?
In yesterday’s post, we introduced you to our new-and-improved doubling-time model. In particular, we can now model the weekly reporting cycle, which (up until now…) has been causing a lot of scatter in the data, obscuring the underlying trend.
This cycle – and how its changed from the beginning of the pandemic to now – is very interesting, and we intend to return to the topic in upcoming posts.
But in this post, we’ve simply taken our best-fit weekly reporting cycle model and used it to correct the data. (Note, we did this after fitting to the data – correcting data before fitting a model to it is a big no-no.)
This reduces the scatter about the underlying trend by a factor of about 2, and makes it a lot easier to see what’s been going on. In short, we have gone through four separate phases:
For most of March, the doubling-time increased at a pathetic rate of only 26 +/- 13 minutes each day. Almost indistinguishable from pure exponential growth.
But then the initial social distancing and handwashing effects kicked in, and this rate increased to 19.4 +/- 0.8 hours each day. Although significantly better than in Phase 1, it was shy of the critical “one-day-each-day” rate that we need to exceed to bring the number of new cases down quickly. This phase lasted a terribly long time, until early May.
Phase 3 began shortly after we began our month off, so this is the first we are reporting on it (we’ve marked its beginning and end in the attached graph with arrows). The doubling-time began to accelerate again(!) It is difficult to measure what it was ramping up to – our modeling suggests multiple days each day. This was almost certainly stay-at-home and mask wearing, on top of social distancing and handwashing, finally having the effect that we needed them to. Had this been sustained, the number of new cases would have plummeted, New Zealand style. The economy might even be fully re-opened by now. But alas…
Stage 3 was not sustainable. Too many people gave up on stay-at-home, etc., even before the orders were relaxed. Since mid-May, we’ve been back to only 18.4 +/- 4.8 hours each day: The doubling time is once again increasing too slowly to avoid a long, slow, painful post-peak tail, and leaves us vulnerable to a second peak later in the year.
So we got close, but failed to solve this New Zealand style.
Does this mean we’re doomed? No – testing and tracing are ramping up at a respectable speed, now approaching half a million new tests per day. We’re still a factor of (roughly) 10 shy of where we need to be, but if testing and tracing continue to scale as they have been, we could be there by end of summer: Once we have the capacity to test everyone (and quarantine the infected) faster than the doubling time – then we never double again. It would be the beginning of the end, and long before a vaccine is ready.
So there are two competing forces: (1) the relaxation of stay-at-home, social distancing, etc. pushing for a return to exponential growth, and (2) testing and tracing ramping up and pushing back. Which will win? It is difficult to say. Regardless, it’s a battle I would much rather we be waging above the critical “one-day-each-day” level – but you seldom get to pick your ground in war.
More on this in future posts. But until then, keep social-distancing, mask-wearing, and hand-washing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
6/2/2020
We’re Back! And with a New-and-Improved Doubling-Time Model
After a month-long hiatus, we’re back. Half of that delay was due to end-of-semester distractions. But then we decided to do some spring cleaning – of our codebase. The modeling codebase that my group has been developing (since long before COVID-19) has grown so much in the past year that we decided it was time to hit the pause button, reorganize it, and subject it to some stress tests. That’s done now.
But while we were doing this, we put together a new-and-improved doubling-time model, which we’re rolling out today.
Just to remind you, the goal of our COVID-19 effort is to model the underlying, doubling-time trend. However, there has always been a lot of scatter about this trend, in the single-day, doubling-time measurements. And much – even most – of this scatter was not random, but cyclic, due to the weekly reporting cycle: Sunday and Monday doubling-time measurements are almost always high, due to weekend underreporting, and Thursday and Friday doubling-time measurements are almost always low, due to catch-up, backlog reporting.
In addition to modeling the underlying, doubling-time trend, we are now modeling the weekly reporting cycle. By doing so:
Our projections should no longer oscillate from too optimistic to too pessimistic with this cycle, as each single-day measurement comes in: I.e., we now expect Sunday and Monday measurements to be higher, and Thursday and Friday measurements to be lower. As long as they are as high or as low as expected, our measurement of the underlying, doubling-time trend will no longer budge.
We have reduced the scatter about the best-fit model by a factor of about 2. This allows us (a) to measure the underlying trend that much more accurately, and (b) to be that much more sensitive to picking up changes in this trend. (Indeed, we have picked up two significant changes in the underlying, doubling-time trend over the past month – these will be the subject of our next update.)
Although less important, we can now also measure and study the weekly reporting cycle, and how it’s changed over the course of the pandemic.
In today’s figure, we have plotted the single-day doubling-time measurements, and our new, best-fit model to them (blue curve), on a logarithmic, or powers-of-ten, scale. This is so you can better see all of the oscillations (instead of just the most recent ones, which dominate when we plot on a regular, linear scale), and see how well we are able to model them.
We model these oscillations with three quantities: (1) an amplitude, (2) a phase – e.g., does the weekly cycle begin on a Sunday or a Monday? – and (3) a “concentration” – e.g., is the underreporting concentrated to the shorter weekend period, and the catch-up, backlog reporting more spread out, vice versa, or something in between?
Furthermore, we allow these three quantities to change (slowly) over the course of the pandemic. For example, the amplitude appears to have gone up, the phase appears to have shifted from a Sunday start to a Monday start, and underreporting used to be concentrated to only a couple days, but now appears to be much more spread out, with backlog reporting now being carried out in a fury at the end of each week.
We’ll cycle back (ha…) to these findings in later posts. But first, where do we stand today?
With the weekly reporting cycle factored out, as of yesterday, the doubling time was 58.7 +/- 2.4 days, and it is increasing at a rate of 18.4 +/- 4.8 hours each day. Sadly, this is still less than the critical “one-day-each-day” rate of increase that we need to avoid a long, post-peak tail, and the possibility of a second peak.
Sad…but even sadder once you learn the story of progress made, and lost, over the past month.
We’ll tackle that in our next update. Until then, keep social-distancing, mask-wearing, and hand-washing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
5/11/2020
End Game Update
It has been nine days since our last update. Apologies! A key team member had to take his final exams, graduate, and move into a new apartment — congratulations to Nick Konz for graduating from the University of North Carolina at Chapel Hill with a B.S. in Physics and Astronomy, a B.A in Mathematics, and for graduating with Highest Distinction and Highest Honors!
Despite all of this, we have been busy developing a new, more advanced doubling-time model — one that additionally accounts for the weekly reporting cycle. We are fine-tuning it now, and so far it looks like it will make us about 2.5x more sensitive to picking up on subtle changes in the doubling timescale, and in its trajectory. This should prove very helpful in the weeks ahead, now that stay-at-home orders are being relaxed around the country.
We hope to roll this new model out later in the week. But today, we have another plot for your consideration.
Many countries went all-in on stay-at-home and social distancing. They peaked, after which their numbers of new cases and new deaths dropped, and quickly. Now, they are essentially done with COVID-19 and are safely re-opening their economies.
However, in the US we did not get enough buy-in from enough of the population. We peaked well before nearly everyone got infected, and at a level that did not overwhelm the healthcare system — which is great! But we never came back down the other side of the peak — we are stuck in a long, post-peak tail. And now we are relaxing stay-at-home, so this should only get worse.
Short of more people getting wise and buying in, we next have to look to wider-spread testing and contact tracing. In the attached figure, we plot the doubling timescale in blue — again, this is how long it takes for twice as many people to become infected. We plot something that we are calling the “testing timescale” in orange — this is how long it takes to test everyone in the country — all 330-ish million of us — given current testing capacity. The pandemic ends as these curves draw closer together and cross: If we can test everyone, and quarantine the infected (and their contacts), faster than the number of infected can double — it won’t double anymore.
Sure, there would be some who get infected after they are tested but before everyone is tested, but this would be a small number, and most of them, and those they infect, would be caught on a second pass through the population.
Currently the doubling timescale is about 35 days. And the testing timescale is about 3 years. But a month ago, these numbers were 11 days and 6 years. They are getting closer together — if the current trajectories continue, they cross in perhaps 2 months. If…
The gray curve is how high the doubling time would need to be to end this tomorrow — essentially no new, reported cases. As the testing timescale decreases, and as the doubling timescale increases, this — the ending timescale — will decrease as well. (As we discussed previously, this ending timescale is approximately equal to the number of people in the US who are currently infectious — unfortunately, this number is stuck on a fairly flat trajectory right now).
Keep social distancing. But since everyone ins’t, keep pushing for more testing and tracing as well!
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5/2/2020
The Ups and Downs of the Weekly Reporting Cycle, and Benchmarks for the Week Ahead
As expected, Friday’s doubling-time measurement was the weekly low, at 22.0 days. The weekly high was last Monday, at 28.3 days (usually it’s Sunday). As we all understand now, the weekly high is driven by weekend (and Monday) under-reporting. The weekly low is driven by backlog over-reporting.
In today’s graph, we’ve plotted each week’s high and low single-day doubling-time measurements, for the past eight weeks. This week, the doubling time appears to have increased by about 21 hours each day. The previous week, the increase was a disappointing 11 hours each day — probably artificially depressed by another ramp-up in testing (which is a good thing). The week before that was about 20 hours each day.
If the doubling time continues to increase by 21 hours each day, we can expect the weekly high to be about 34.5 days, around Sunday, and the weekly low to be about 28.2 days, around next Friday. We can use these as benchmarks for the week ahead, to see if we are maintaining pace, or falling behind.
Note however that we really need to exceed these benchmarks: We are post-peak, but appear to be in the beginning of a long, slow, painful post-peak tail. For the tail to drop faster, the doubling time needs to increase by more than a day each day. Hovering just below this critical rate of increase is not where we want to max out.
Keep social distancing!
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4/29/2020
What is the End Game?
We have been tracking the doubling time for about a month and a half now. And it is increasing. In fact, the past three days have been record-setting, with single-day measurements of 24.9, 28.3, and 28.0 days.
But when does it end?
It ends when the doubling time is approximately equal to the number of cases that are still infectious.
In today’s graph, we’ve plotted the doubling-time data and our fitted model to it in blue. And we’ve plotted the approximate number of reported cases that are still infectious in orange. (Note, this is likely an underestimate by a factor of a few because of the asymptomatic population, but since this is a powers-of-ten plot, that doesn’t matter too much.)
For both curves, uncertainty ranges are indicated in green (optimistic) and red (pessimistic).
Once these curves meet, we’ll be down to only one(-ish) new, reported case per day. It’ll essentially be over.
In fact, if you count the number of powers of ten between the blue and orange curves, that tells you how many new, reported cases we should expect each day, at least to within a factor of a few. For example, there are currently roughly four powers of ten between the two curves, so we should expect roughly 10^4 = 10,000 new, reported cases each day, which is correct to within a factor of 2 — 3.
Once we’re down to three powers of ten, that’s roughly 1,000 new cases per day. Two powers of ten is roughly 100 new cases per day. And one power of ten is roughly 10 new cases per day. So the curves actually don’t have to cross for things to get a lot better. (But they will need to cross for us to get completely back to normal, without fear of exponential growth rearing its ugly head again.)
However, the problem, as you can see, is that the blue curve isn’t accelerating upward anymore. And the orange curve has stopped increasing, but doesn’t yet show any signs of decreasing. We’re making some progress, but it’s very…very…very incremental.
In short, this is probably not the best time to relax social distancing, at least not without widespread testing and sophisticated contact tracing ready to go.
So…keep social distancing!
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4/28/2020
We Can Now Measure the Infectious Period (It’s About 1 Week)
We have been modeling the “infectED” doubling time — the amount of time that it takes for as many people who have already been infectED to again become infectED.
But in our 4/22 update, we described how this is actually equal to the “infectIOUS” doubling time — the amount of time that it takes for as many people who are currently infectIOUS to again become infectIOUS — divided by “f” — the fraction of the the infectED who are still infectIOUS.
The fraction “f” is easy enough to measure from the daily case totals, for different guesses for the infectious period — the amount of time the infectED typically stay inctectIOUS (i.e., before they are quarantined, or if not quarantined, before they recover or die). (Nor do you really need to know what fraction of the infected are symptomatic vs. asymptomatic to calculate this, which is nice.) We calculated “f” in our 4/22 update for 2-week, 1-week, and half-week infectious periods.
For this update, we have multiplied “f” by our daily measurements for the “infectED” doubling time, to get daily measurements for the “infectIOUS” doubling time. We plot these in the attached figure for these same three guesses for the infectious period. (We have removed values that are contaminated by the primary testing catch-up phase between March 18 — 21.)
This plot is interesting for two reasons.
The first is this is actually the quantity that our social distancing efforts have been impacting. The “infectED” doubling time is a lesser measure in that it also depends on “f”. The only reason that we model it instead of this “intectIOUS” doubling time is because we can do so without having to guess at the infectious period.
But it is interesting to see it. Although there is a lot of scatter, social distancing has been driving up the “infectIOUS” doubling time in a slow and steady fashion (how much depends on one’s guess for the infectious period). It will be interesting to see if this continues, or if it levels off, now that states are beginning to relax social distancing.
The second reason that this plot is interesting is that it can be used to, at least approximately, measure the infectious period.
Consider the number of people who are currently infectious. Some are nearly done. Some have the whole infectious period to go. On average, this group has half an infectious period to go. These timescales are represented by horizontal lines in the plot.
New cases should peak when the “infectIOUS” doubling time equals this timescale. We’ve fitted lines to these “infectIOUS” doubling-time measurements and find that if the infectious period is 2 weeks, new cases should peak around May 19. If the infectious period is 1 week, new cases should have peaked around April 18. And if the infectious period if half of a week, new cases should have peaked around March 17.
Qualitatively, this makes sense. If the infectious period is 0 days, the virus would die out immediately and the peak would be on Day 0. If the infectious period is huge, more people would still be spreading the virus, and the peak would be much later.
Although the peak appears to be broad (e.g., see our 4/24 update), we do appear to be on a slow decline, and our projections put the peak around April 18th. Consequently, we find the infectious period — the typical amount of time between becoming infectious and being removed from the infectious pool, either through quarantine, or if not quarantined, through recovery or death — to be about 7 days. Of course, some people will take longer, and some will take less time.
If this number feels a bit low, remember that (1) when the peak occurs is almost certainly driven by the almost certainly larger asymptomatic population, so this number probably most reflects their average infectious period. But (2) even for the symptomatic population, this number would suggest that most of them are being identified and quarantined soon after they become symptomatic — although they may remain infectious for up to two weeks, they no longer count as infectious in this analysis once they’re quarantined.
Measuring this average timescale eliminates one of our model’s free parameters. However, the post-peak tail was never very sensitive to this parameter anyway, so measuring it doesn’t really affect our projections, up or down. (But it is interesting to know.)
In other news, yesterday’s doubling-time measurement was very good: 28.2 days. But this has already been a long post, so we’ll redirect you to our main page (linked above) for the updated projection.
Keep social distancing!
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4/27/2020
We Failed to Meet Two Benchmarks Over the Weekend
As we pointed out in our 4/21 and 4/24 updates, the doubling time is increasing, but it is no longer accelerating. And acceleration is what we need: As long as the doubling time increases by less than one day each day, it will take a very long time to get past the peak. And then, if it is still increasing at this, constant rate, we will be faced with a very long post-peak tail, with many new cases, and many new deaths.
How high does the doubling time need to get? To get down to only one new, reported case per day, the doubling time has to get as high as 0.7 times the number of people who are still infectious. Currently, there are probably 300,000 — 700,000 people still infectious, and this number is dropping, but clearly we have a ways to go before these two numbers meet, somewhere in the middle.
Last Thursday presented us with a disappointingly low doubling-time measurement of 17.5 days. However, Thursday measurements tend to be low because of backlog reporting, so in our last (4/24) update, we set two benchmarks for the weekend, to see if we’re still on the same (albeit non-accelerating) trajectory, or if we might be losing ground.
The first was if Friday’s measurement was equally disappointing, since Friday measurements tend to be low by the same amount as Thursday measurements. And it was worse: Only 16.9 days.
The second was Sunday’s measurement, since Sunday measurements tend to be the biggest bump that we get all week (due to weekend under-reporting), and they kind of set the scale for the week ahead, until next Sunday’s bump. The trend of Sunday-to-Sunday increases over the past three weeks suggested that we should expect something around 27 — 28 days for this Sunday’s measurement. And it also came in low: 24.7 days (a record, but not as big of one as we should have had).
So our tentative conclusion is that something is changing. Here are four possibilities:
The national peak is being stretched out because not all of the states’ peaks are occurring at the same time.
Social distancing is not being taken as seriously as it once was.
Perhaps attitudes about social distancing aren’t changing, but it is being taken more seriously in states that have already peaked than in states that are peaking now and next.
Testing is increasing again (which is great!), though it deflates the doubling-time measurements a bit until testing levels off again.
Personally, I think all four of these are playing some role, though it is difficult to say which, if any, are playing a dominant role at this point.
Keep social distancing!
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4/24/2020
It’s Not a Peak — It’s a Plateau
The number of new, reported cases technically peaked on April 10th. Due to weekend-weekday reporting variations, the “true peak” could have been somewhat before or somewhat after then. Since then, we have been experiencing a slow, if undulating, decline.
At least until yesterday, when 34,856 new cases were reported (at least as of this morning — the number will likely change by a few hundred, in either direction, as it is updated over the next few days). This is only 59 below April 10th’s value. We don’t appear to be experiencing a peak, but a plateau (see attached plot).
As we speculated in our 4/21 update, this could be because the whole country isn’t peaking at the same time. Some states, such as NY, are now on the decline, but others, including more rural areas, are still ramping up. In this case, the plateau could go on for weeks more, and possibly even rise to form a new, later peak. But then the numbers should decline sharply — at least if social distancing is still being practiced to the degree that it has been.
However, the plateau could also be, at least partially, due to people growing tired of social distancing and not taking it as seriously as they had a week or two before. In this case (or if this becomes the case), there won’t be a sharp decline, but a long tail, of new cases, and new deaths.
Thursday’s near-record number of new, reported cases can also be seen in the doubling-time measurements: After reaching an all-time high of 21.4 days (after updates) on Tuesday, the single-day doubling-time measurement dropped to 19.5 days on Wednesday and to a disappointing 17.4 days Thursday.
Note that Thursday and Friday values tend to be low by 8% — 9%, due to backlog reporting. But even so, this Thursday’s value comes in below expectations, and pulls our trend line down: We measure the doubling time to currently be 21.0 +/- 0.7 days, increasing by 19.3 +/- 1.3 hours each day. This is still below the “one day each day” benchmark that we need to achieve to get past the peak.
It will be interesting to see if Friday’s value is similarly bad, and if we get our usual weekend bump: Over the past three weeks, we’ve experienced Sunday-to-Sunday increases in the doubling time of 70%, 58%, and 47%. This trend would suggest a 35% Sunday-to-Sunday bump this weekend, corresponding to a doubling time of 27 — 28 days this Sunday. I think that we can view this as a benchmark as to whether we’re still on the same trajectory, or whether something fundamental has changed.
Keep social distancing!
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4/22/2020
Let’s Break It Down: InfectED vs. InfectIOUS Doubling Time
After a small set-back on Monday, Tuesday’s doubling-time measurement was another record: Monday’s was 19.2 days and Tuesday’s was 21.0 days. Overall, we measure the doubling time to be 20.5 +/- 0.8 days, increasing by 22.9 +/- 1.8 hours each day.
We’ve come a long way: It took nearly 20 days for the doubling time to climb from 2 days to 3 days. Now it’s increasing by about a day each day. Is this all due to social distancing?
No. And yes. But to explain, I have to get a little mathy. Buckle up!
Let’s start by reminding ourselves what the doubling time actually is: It is the time that it takes for as many people as have already been infectED to also become infected. I.e., it is the time that it takes for the number of infected to double. For the remainder of this post, let’s instead call this the “infectED” doubling time.
For epidemics that peak before spreading to most of the population (such as this one, fortunately), the mathematics of the “infectED” doubling time simplifies a great deal, to:
“infectED” doubling time = “infectIOUS” doubling time / “f”
where the “infectIOUS” doubling time is how long it takes for as many people as are currently infectIOUS to also become infected, and “f” is the fraction of the infectED who are still infectIOUS.
The “infectIOUS” doubling time is actually what all of our social distancing efforts have been increasing. And this increase has been slow and steady (i.e., not accelerating). We’ll come back to what its current value is in a later post.
In this post, let’s instead focus on “f”. If you know the infectious period — how long someone who is infected typically stays infectious (i.e., until they recover, are quarantined, or die) — this fraction is actually super easy to calculate from the daily case totals. Nor does it really depend on how big the asymptomatic population is. We plot this — the fraction of the infectED who are still infectIOUS — in the attached figure, for three guesses for the infectious period: 2 weeks, 1 week, and 1/2 of a week. (We have eliminated values that are contaminated by the primary testing catch-up phase between March 18 — 21.)
Whatever the the infectious period happens to be, you can see that this fraction started out fairly constant. This is because of exponential growth — creating new, infectious cases much faster than they could be retired.
However, as social distancing, hand washing, face-mask wearing, etc. increased the “infectIOUS” doubling time, we began to make new, infectious cases more slowly, and the fraction of retired, no-longer-infectious cases began to dominate, driving “f” down — slowly at first, but then faster and faster.
And since “f” is in the denominator of the above equation, the “infectED” doubling time — the quantity that we’ve been tracking for the past month — increased, slowly at first, but then faster and faster.
Or to put it another way, the “infectED” doubling time is accelerating now because, although we have had a lot of infectED people, only a diminishing fraction of them are still infectIOUS.
However, this does not mean that we can stop social distancing yet. If we stop too soon, the “infectIOUS” doubling time will decrease, we will start creating new, infectious cases more quickly, and “f” will stop dropping. If “f” becomes constant again — even a low constant — exponential growth resumes.
So…keep social distancing!
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4/21/2020
Are We Stalling Out?
Monday’s doubling-time measurement was 18.9 days, and mostly in line with expectations: As of today, the doubling time is 19.2 +/- 0.8 days, increasing at a rate of 21.4 +/- 1.7 hours each day. You can check out the remainder of our projections (which haven’t changed much from yesterday) at the link above.
However, what is beginning to concern us is that the rate at which the doubling time is increasing is beginning to level off (see attached plot — it is the slope of today’s fitted doubling-time model). Now that we are near or possibly a bit past peak, the doubling time should be increasing faster and faster, because the fraction of the infected population that is still infectious is getting smaller and smaller.
But it’s not. Two possible explanations:
We could be becoming prematurely relaxed in our social-distancing efforts. In my opinion, this would have to be broader than the isolated protests that we are seeing. (However, I do expect infections to jump — and maybe even skyrocket — among the protesters and their immediate families in the weeks to come.) However, these protesters might just be the tip of a much larger, and much more dangerous, iceberg of people who are growing weary, and now that things are beginning to look a bit better, are slacking off — unfortunately too soon: This really can’t occur until we are well past “one day each day” of gain, else there will be a second peak.
Another, more benign, explanation is that different regions, states, and cities are peaking at different times. This artificially broadens the national peak, which could also make it look like we’ve stalled out, at least until we are post peak for not some, but most of these areas.
It could also be a combination of both of these effects.
Regardless, remain vigilant, and keep social distancing!
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4/20/2020
Three Days, Three Record-Long Doubling Times — But Expected and Nothing to Get Excited About
Friday’s doubling-time measurement was 15.4 days, Saturday’s was 17.8 days, and Sunday’s was 20.4 days — all records. However, Friday’s was probably low, due to backlog reporting, and Sunday’s was probably high, due to lower weekend reporting. We measured this weekend-weekday effect in our 4/17 update. Taking it into account, the doubling time probably increased by only 1 — 2 days during this 3-day period, not by 5 days.
We are developing an empirical model for this weekly cycle that we hope to fold into our regular fits later in the week. Once we do this, we won’t have to do these day-of-week qualifications anymore — the model will expect higher values on Sundays and Mondays in particular, and lower values on Wednesdays — Fridays in particular. The curves in the attached plot will instead oscillate as they go, taking this effect into account.
This should reduce the unaccounted-for scatter about the fitted curve, allowing us to model the underlying trajectory with greater precision. Also, our projections should become steadier — no longer caught in a cyclic tug-of-war between these high weekend and low weekday measurements.
But until then, here’s today’s update: We measure the doubling time to be 18.4 +/- 0.8 days, improving by 21.7 +/- 1.9 hours each day. Projecting this forward, we find that approximately 6.3% of Americans will become infected before we get a vaccine, 2.1% (+0.8%, -0.5%) symptomatically. Of the latter, 6.6% (+0.7%, -0.6%) are currently dying, corresponding to 450 (+240, -140) thousand dead before we get a vaccine — but only if we can’t increase the doubling time faster than we have been.
Keep social distancing — until we have widespread testing and contact tracing, it’s the only thing keeping this under control.
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4/18/2020
Report from the Low Point in the Weekly Cycle
As we discussed in some detail yesterday, because of the weekend-weekday reporting cycle, we’ll usually put out slightly too-rosy reports on Monday/Tuesday, and slightly too-grim reports on Saturday. And today is Saturday — so it should only get better from here!
The last six single-day doubling-time measurements were all between 13.8 days and 15.3 days. 15.3 days is our newest value, from yesterday, and it is also our highest value so far (after revisions brought down some of the Easter weekend measurements). Still it is below what one might have expected, given the overall trend line (but not below what one might have expected given the overall trend line and the weekend-weekday reporting cycle).
Overall, we find the doubling time to currently be 15.9 +/- 0.8 days, increasing by 18.2 +/- 1.6 hours each day. Notice that this has slipped below the “one day each day” benchmark that we want to see.
Still, the projections find that the peak is behind us, though just barely — and only tenuously in our pessimistic scenario (in which we assume a smaller asymptomatic population, corresponding to less herd immunity, and a longer infectious period). If the rate of progress drops much further below the “one day each day” benchmark, our pessimistic model will likely change its mind and push the peak out in front of us again. But our baseline and optimistic models will likely hold firm.
Assuming a 5 +/- 2 day delay between a case being reported and death occurring (for those who do die of the virus), 6.6% +/- 0.7% of reported cases are currently resulting in death. And since we are still projecting a long, post-peak tail, this implies 570 (+490, -200) thousand deaths before September 1, 2021.
However, if we can push the doubling time a bit higher, and do so a bit faster, this tail should begin to fold, bringing these more dire numbers down considerably. (They are down 36% from where they were a week ago.)
Keep social distancing!
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4/17/2020
Weekly Variation in Single-Day Doubling-Time Measurements
We’ve spent a lot of time talking about how single-day doubling-time measurements are artificially high on weekends due to under-reporting, and artificially low on weekdays, due to backlog reporting. This is something that we should be able to measure. In the attached plot, we’ve calculated by what percentage these measurements have been high or low compared to our most-recent, best-fit model, averaging over the past three weeks.
The effect is certainly real. As one might expect, the overestimate is greatest on Sundays, resulting in values that are typically 17% too high. This then drops throughout the week, bottoming out on Thursdays and Fridays, with values that are typically 8% — 9% too low.
(This long, sustained drop usually cancels out genuine progress that is being made in the doubling time over the course of the week, resulting in projections that don’t change much until Monday and Tuesday, when we factor in Sunday and Monday’s measurements.)
In other news, we probably won’t have a model update today — my student who runs the fits (the amazing Nick Konz!) has actually chosen to prioritize his homework and exams today (go figure!).
However, next time we update the model and projections, expect to see somewhat larger uncertainty ranges. As we’ve been adding data, these ranges have been shrinking (as one might expect them too), but recently we’ve felt that they’ve been shrinking too quickly. We’ve been poking at this behind the scenes, and after updating part of our code base yesterday (technically, the adaptive rejection sampler (ARS) of our Metropolis-Hastings Markov chain Monte Carlo (MCMC) algorithm…) we think we’ve resolved the issue.
We’ll update the past few models and projections at the same time, but we already know that this doesn’t change yesterday’s conclusion — that we are at peak, plus or minus a few days.
Note, people will likely squabble about which day actually was the peak, but (1) the true peak is lost in the above weekend-weekday reporting effect, and (2) we cannot forget that the peak is actually occurring at different times in different states and cities.
So remain vigilant, and keep social distancing!
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4/16/2020
Today’s Model Puts the Peak…Yesterday
The last three doubling-time measurements have been between 14.3 days and 15.8 days, and we are now well in to the part of the week where healthcare workers would be catching up on any backlog, so these values, if anything, are low. Overall, our projections remain steady, which is typical for us — suppressed end-of-week measurements hide progress, which is then revealed all of a sudden with inflated values come the weekend…our weekly roller-coaster ride.
Today, we measure the doubling time to be 15.5 +/- 0.3 days, increasing by 23.6 +/- 0.5 hours each day. And for the first time, our model puts the peak not in the future, but in the past: April 15th. Today’s attached plot shows how our projections for how strong the peak would be decreased with time.
This plot shows your progress — all of your progress. Remember, the point of our page isn’t to predict what will happen, only what will happen if we don’t increase the doubling time faster than we have been. Initially things looked quite grim — progress was incremental, implying many million cases per day at peak. The healthcare system simply and utterly would have collapsed under such strain.
But then we began to see the effect of social distancing, and eventually of the stay-at-home orders. The projected peak dropped to hundreds of thousands per day, and then to tens of thousands per day. The heathcare system was stretched, especially in places (e.g., NYC), but not overwhelmed.
Enough people took social distancing and stay-at-home seriously enough to make the difference that had to be made. Of course, some (fools!) would say that the whole thing was overblown. But of course it wasn’t — had we not made the efforts that we did, this plot shows you exactly (or at least approximately) what the consequences would have been.
But it ain’t over yet!
Our future posts will focus on the post-peak tail. Our conservative, empirical approach (https://www.danreichart.com/covid19-rationale) currently projects a long, drawn-out tail, corresponding to 360 (+150, -100) thousand dead by 9/1/2021.
We can (and should) take a measure of pride in our collective accomplishment of avoiding the imminent threat of a peak that would have broken our healthcare system, resulting in so many more deaths. But, it is too soon to let up: Unfortunately, we still need to push the doubling time higher, and higher faster, if we’re to also squash the post-peak tail.
Zeroing out, or nearly zeroing out, the number of new, reported cases is also a necessary (but not sufficient!) condition to begin to reopen the economy. Because without periodic testing of nearly everyone, and a sophisticated, national system of contact tracing, once social distancing ends, exponential growth will begin all over again. (We did so well squashing the peak that there isn’t enough herd immunity to prevent a resurgence, even if there is a reasonably large asymptomatic population out there.)
Our best bet it to run this thing into the ground, and hope that the government has figured out blanket testing and contact tracing by then. Else, we’ll be able to open things back up, but probably only for a month or two before we’ll have to shut it all down again, to deal with a new peak (see Figures 3 and 4 of https://www.imperial.ac.uk/media/imperial-college/medicine/sph/ide/gida-fellowships/Imperial-College-COVID19-NPI-modelling-16-03-2020.pdf?fbclid=IwAR12Ca7H2kIG-yd9MUc5gTzi6eiwfHT-xInmKg0hTHLjhLA-RgXPAmPqu8I).
Anyway, more on the post-peak tail to come. Until then…keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/14/2020
Doubling Time Now Increasing by More than One Day Each Day: We are Likely at Peak!
Over the holiday weekend, the doubling time increased from 9.1 days on Thursday to 15.0 days on Monday. If genuine, this implies that another, significant improvement to the doubling-time trajectory has taken place – enough to suggest that we are now at peak!
If genuine. This has not only been a weekend, but a long, holiday weekend, stretching from Good Friday through Easter Monday – also a day off in many places. And as we all know very well now, weekends and holidays result in reduced testing, fewer reported cases (and reported deaths), and consequently artificially longer doubling-time measurements.
So we have waited for Tuesday’s measurement to come in before posting this update. This is when healthcare workers would have begun to catch up on any long weekend/holiday backlog – resulting in an artificially shorter doubling-time measurement. But despite this, Tuesday’s (preliminary) measurement is also record-breaking: 15.6 days.
Our philosophy all along has not been to try to predict what will happen – only what will happen if the doubling-time trajectory (and the testing-efficiency trajectory – see yesterday’s update) do not change. But we have once again had a genuine change, and a significant one, for the better.
As such, our projections have improved. Not yet including Tuesday’s measurement, we find the doubling time to now be 13.8 +/- 0.3 days, and we find it to be increasing by 25.9 +/- 0.6 hours per day – this is the first time that we have found the doubling time to be increasing by more than one day each day. Projecting forward, this means that we are now at peak!
And if this trajectory continues: (1) 4.8% (+0.5%, -1.0%) of Americans will become infected, (2) 1.6% +/- 0.3% will become symptomatic and reported, and (3) 290 (+110, -80) thousand will die (or at least will be reported dead, due to COVID-19). Although these numbers are 2 – 3 times better than they were even yesterday, they could still be improved upon significantly, if the doubling-time trajectory continues to accelerate.
So…keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/13/2020
No Need to "Throw the Baby Out with the Bathwater"
In three days, the doubling time has increased from 9.1 days to 9.9 days to 11.9 days to 13.7 days. Are these gains real? Partially, but probably not totally -- it's Easter weekend, and even healthcare workers need some time off: This results in fewer cases being identified and/or reported, and consequently artificially longer single-day doubling-time measurements.
Some healthcare workers might have taken off a day early, on Good Friday (9.1 to 9.9), more probably took Saturday off (9.9 to 11.9), and even more almost certainly took Easter Sunday off (11.9 to 13.7). It will probably take a few days to catch up on this backlog, so expect a rougher end of week. But it should still be better than last week's end of week, since the doubling time is gaining over half a day per day now. (This will need to ramp up to over 1 day per day before our projections lose their long, post-peak tail.)
Yesterday, we posted about how we deal with these day-to-day variations in testing and reporting efficiency (it was kind of technical -- sorry about that). Today, we promised to post about how we deal with longer-term variations in testing efficiency. Let's break these into long-term and intermediate-term changes:
Long-term changes in testing efficiency: Overall, we have moved from not enough testing to probably enough testing to catch most of the symptomatic cases (but not the asymptomatic cases, which will require periodic testing of essentially everyone). And as you might expect, this also gets folded into the daily doubling-time measurements, lowering each by a little bit.
Is this a problem? The answer is no, because of the approach that we have been taking. The goal of our modeling effort is not to predict what will happen, but what will happen if we stay on the same doubling-time trajectory. And since long-term changes in testing efficiency have been folded in to these measurements, we are projecting what will happen (1) if the doubling time continues to increase as it has been, and (2) if testing efficiency continues to increase as it has been.
Many experts (including my favorite data journalist, Nate Silver) have been guilty of saying that without understanding testing efficiency, the reported number of cases is meaningless. And in an absolute sense, this is true. But relative to where we currently are, and given the empirically determined trajectory that we are already on, this fits in nicely with our approach (i.e., one doesn't have to "throw the baby out with the bathwater").
Of course, like everyone, we still have to make assumptions about what fraction of the infected population is asymptomatic, but as we get closer to peak, this might be measurable from the reported cases data (at least approximately). More on that in a later post.
Intermediate-term changes in testing efficiency: A good example of this is the rapid ramp-up in testing that took place in mid- to late May, causing a brief period of very low doubling-time measurements. Another example (which has not happened) is if demand were to briefly outstrip supply again (which would cause a brief period of very high doubling-time measurements).
Our empirical doubling-time model has only so many parameters -- enough to capture long-term changes in the doubling time, and long-term changes in testing efficiency -- but not enough to capture such intermediate-term changes.
Consequently, we excise these periods from our data set -- but not willy-nilly. There is a simple statistical test that you can perform, called Chauvenet's criterion, which can tell you if "one of these things is not like the others" (to quote Sesame Street).
We explained it in some detail in our 3/22 update, so won't repeat it here.
But so far, only four measurements meet Chauvenet's criterion, and these (conveniently!) correspond to the beginning of the testing ramp-up period. The remainder of the testing ramp-up period appears to fall under the category of "long-term changes" and was absorbed into our, empirical, doubling-time model (as described above).
To really see this, today we have plotted the doubling-time data and fitted model in "logarithmic space", which is where the fit is actually done (see attached plot). The dashed curves capture 68% of the day-to-day variations (see yesterday's update).
32% of the measurements are allowed to be outside of this envelope...but not too far outside. The open circles are too far outside, according to Chauvenet.
Well, that's it for our modeling infrastructure (at least so far) -- back to regular updates tomorrow. If you want to see what effect the Easter weekend measurements have been having on the projections, check out our main page (link above). But take them with a (small) grain of salt until we get some post-Easter measurements to balance them out.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/12/2020
How Do We Deal with Day-to-Day Variations in Testing/Reporting Efficiency?
Yesterday, the doubling time jumped again, from 9.8 days on Friday to 12.0 days on Saturday.
But wait…we’ve seen this one before. On weekends, the number of new, reported cases tends to drop, presumably due to shorter staffing at healthcare facilities. This results in an artificially high doubling time (which will be followed by artificially low doubling times later in the week, as the backlog is caught up on). (Check out our 4/6 update, for a lengthier discussion of this effect.)
But what we haven’t discussed yet is how our model deals with (1) day-to-day variations in testing efficiency. Nor have we discussed how we deal with (2) longer-term variations in testing efficiency. Let’s tackle (1) today and save (2) for tomorrow.
Day-to-day variations: You’ve probably noticed that the daily doubling-time measurements vary about the best-fit curve by more than can be accounted for by their statistical error bars alone. Indeed, the statistical error bars are so small now, we can no longer see them under the data points themselves. This, non-statistical, variation is primarily due to day-to-day variations not in cases, but in testing and reporting efficiency (such as the weekend-weekday effect mentioned above).
We treat this by fitting not a curve, but a distribution, to the data. Think of this as the curve that we normally plot, plus or minus a value that could have been drawn randomly from a scatter distribution. The shape of this distribution doesn’t actually matter all that much — we model it as Gaussian. But what does matter is it’s width. We model this as constant in log-space…or in English: This corresponds to less scatter for low doubling times and more scatter for high doubling times. This makes sense in that there is a pretty big difference between doubling times of, say, 2 and 3 days, but not nearly as much difference between doubling times of, say, 12 and 13 days.
We normally do not plot this scatter envelope with our best fit, since day-to-day variations should not affect long-term projections. (Instead, we normally plot an envelope corresponding to the uncertainty in our best fit, which is much smaller, and which does affect long-term projections). But we have plotted the scatter envelope in the attached plot today, so you can see it. If we are doing this correctly, about 68% of the data points should fall between the two dashed curves. Indeed, 23 out of 34 (non-rejected, see tomorrow’s post) data points do, corresponding to 68% (+/- 6%, given the small numbers).
Nor do we assume how big this log-constant should be, but we extract it from the data as part of our fit (using our TRK statistic). And even though this day-to-day scatter does not impact long-term projections, failure to account for it would lead to a greatly underestimated uncertainty in our best fit, which would impact long-term projections. (This is a standard modeling failure that occurs again and again in most scientific fields…but not here!)
Anyway, all this was perhaps a bit technical, but transparency is important to us. We will tackle the arguably trickier subject of long-term variations in testing efficiency tomorrow.
And our updated projections can be found on our main page (link above).
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/10/2020
Doubling Time Still on the Same Trajectory, and a Few Model Notes
Just a quick update tonight. Yesterday’s doubling-time measurement was 9.2 days, and was almost exactly on the previous model’s trajectory. As was the previous doubling-time measurement. As was the one before that. (See the attached plot.)
In summary, the doubling-time is currently 9.2 +/- 0.2 days, and increasing by 11.8 +/- 0.5 hours each day. Extrapolating this (linear) trajectory forward yields a peak of 49 (+19, -9) thousand new reported cases per day, on June 4th (+60, -28 days).
Model Note #1: We have not implemented the full SIR model (https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology), but an approximation that holds when only a small fraction of the population becomes infected. We improved this approximation a bit today, which raised our projected peak by about 3%. (Should the full SIR model be needed, we will switch to it then.)
Overall, we project that 15% +/- 2% of Americans will become infected, and that 6% +/- 3% of Americans will become symptomatic/reported, prior to September 1, 2021 (which is when we are assuming that a vaccine will become widely available).
Model Note #2: This is quite a few Americans, most of whom become infected and/or symptomatic (and/or die, see below) in a long, post-peak tail. Many of the epidemiological models don’t have such a tail, so we spent some time today investigating what will be required of our empirical (see https://www.danreichart.com/covid19-rationale) doubling-time model before our projections can also lose their tails:
The post-peak tail becomes significantly suppressed once the doubling time is increasing by more than 24 hours each day (which makes sense if you think about it).
But even in this case, a tail persists, albeit at a lower level. To lose the tail completely — which would have to happen before social distancing could end, else a second peak would begin to ramp up immediately — the doubling-time measurements really need to increase faster than linearly (e.g., as a power-law or exponential). We of course hope that this happens, and soon, but given our page’s conservative philosophy, we will wait for the data to request this before making the change, instead of just assuming that it will happen.
Lastly, of the 6% +/- 3% of Americans that we we project to become symptomatic, we project that 5.6% +/- 1.1% will die.
Model Note #3: As we discussed in yesterday’s update, we calculate this by assuming a 5-day lag between becoming a reported case and dying (for those who do die of the virus). But lags as low as 3 days and as high as 7 days also appear to be reasonable. Based upon a recommendation from Susan C. Eaton through Facebook, we now assume 3 days for our optimistic projection, 5 days for our baseline projection, and 7 days for our pessimistic projection.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/9/2020
The Death Rate is STILL Increasing
Yesterday’s doubling-time measurement was the highest yet (8.8 days), but completely in line with our previous model, so our projections did not change much today (you can find them on our main page, which is linked above). Today’s measurement is shaping up to be even higher, but also in line with our previous model.
So today, we revisited the death rate. Not the overall death rate, because no one knows how many asymptomatic cases are out there in America. But the percentage of reported cases that die.
Our expectation is that this number would start out high: Due to the initial shortage of testing kits, only the most severe cases were being tested, and these are more likely to result in death.
But then testing ramped up, and by late March most symptomatic cases were probably being identified. This should then be followed by a relatively constant death rate.
At least until/if our healthcare facilities get overwhelmed. This would be accompanied by an increase in the death rate.
Since there is a delay between becoming a reported case and dying (for those who ultimately do die from the virus), the proper way to calculate this is the total number of deaths reported so far divided by the total number of cases reported — not so far, but this delay ago (times 100%).
The problem is that there isn’t great information available on what this delay should be. Consequently, we have been calculating the death rate for a variety of delays, and looking for the above pattern: A decline followed by a constant phase, and possibly followed by an increase.
In our 4/1 update, we noticed a decline followed by a constant phase if one used a 3-day delay (see the orange curve in the attached plot). The constant phase began around March 22nd, which was perhaps halfway through the primary testing catch-up phase in the US. (We adopted this value.)
However, beginning around March 30, this measure of the death rate began climbing again, from about 3% to 4.4% today. This could be due to our hardest-hit healthcare facilities beginning to become overwhelmed. But it could also be because we are still underestimating the delay (though 3 days is certainly better than assuming no delay, as we had been prior to our April 1 update).
We now see a similar constant phase with a 7-day delay (see the gray curve in the attached plot), beginning around April 1 and still ongoing. April 1 is probably a bit after testing caught up with (symptomatic) demand, but not by much.
This scenario would correspond to our healthcare facilities not/not yet being compromised by still increasing demand. But it does also correspond to a higher death rate, 6.8% today.
It is difficult to decide between these two scenarios — both are plausible. Consequently, we have decided to switch to using a delay of 5 days, their average. This corresponds to a slowly increasing death rate, with a current value of 5.3%.
Keep social distancing.
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/8/2020
We Now Have Enough Data to Make a Further Improvement to Our Doubling-Time Model
The last four single-day doubling-time measurements, including today’s (since it is now after midnight), have all been between 7.9 and 8.8 days. This has been pulling our doubling-time model upward.
We made another improvement to our model today. Previously, we had been modeling the single-day doubling-time measurements with a “sharply broken” linear model — one rate of increase at early times and a different, faster rate of increase at later times. However, the rate at which the doubling time has been increasing probably didn’t jump overnight, on March 26. Consequently, today we switched over to a “smoothly broken” linear model (see attached plot).
This model asymptotes to straight lines both at early times and at late times. We like this because ending up with a linear rate of increase is conservative. If we ended up with a power-law or an exponential rate of increase, our projections would truncate much earlier — with fewer infected, fewer symptomatic, and fewer dead.
This would of course make us feel better, but it wouldn’t be for the right reasons. Some epidemiologists are projecting an early peak and few deaths — I hope they are right. Some are projecting the worst case scenario. If you’ve been following us for a while, you know that we are taking a very different approach: We will apply conservative (and fully empirical — see April 2nd’s update) models until the data requests something different.
Why? Because we are not trying to project what will happen, but what will happen if we stay on the current trajectory. And the only thing that can change that trajectory is you, through social distancing, hand washing, mask wearing, etc.
We are all staying at home as much as possible, doing all of these things, and wondering what, if any difference, we are making. That’s where our projections come in — they are meant as tangible feedback for a very intangible problem.
That said, here’s today’s feedback. We now measure the doubling time to be 8.2 +/- 0.2 days, and we measure it to be increasing by 11.8 +/- 0.5 hours each day. If the doubling time continues on this trajectory, we project a peak of 48 (+19, -10) thousand new, reported cases per day — which suggests that we might be getting close to the peak, since today’s number of new, reported cases was 33 thousand.
However, we are very uncertain as to when the peak will occur. We are currently projecting June 3rd (+59, -28) days. This uncertainty is almost completely due to us not knowing what fraction of the infected are asymptomatic, and not reported. If it’s a large fraction (e.g., 3/4), herd immunity is setting in and the peak will be soon. If it’s a smaller fraction (e.g., 1/2), we still have a ways to go.
Overall, we project that 5% (+4%, -2%) of Americans will become symptomatic and reported, and of these 640 (+490, -230) thousand die. But again, these last two number are very sensitive to whether the doubling time continues to increase at its current rate, and whether it continue to increase linearly. So take this as a conservative (but certainly not worst-case) scenario, that we should continue to endeavor to improve through our collective efforts.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/7/2020
Doubling Time on Rapidly Increasing Trajectory!
For four days, the single-day doubling-time measurement inched up from 5 days to 6 days. And then Sunday’s value clocked in at 8.8 days. Was it real? Was it a fluke? We presented four different possible explanations in yesterday’s update, but in the end we decided that more data was needed to decide between them.
Well, that data is now in…and it’s very good news: Monday’s doubling-time measurement was 8.0 days. And Tuesday’s preliminary value (just in) is 8.5 days. This appears to be for real.
Social distancing is making a real difference. And given that there are likely more asymptomatic (but still infectious) cases out there than reported, symptomatic ones, it is probably making a bigger difference than wider-spread testing has. Well done everyone!
Our new baseline model can be seen in the attached plot. We conclude that the doubling time is currently 7.5 +/- 0.1 days, and that it is increasing by 9.3 +/- 0.5 hours each day. This is so much better than when we started this only 20 days ago, with a doubling time of about 2 — 3 days, increasing by only 1 — 2 hours each day.
And if we project this new trajectory forward, accounting for reasonable uncertainties in the size of the asymptomatic population, and in the infectious period, we find:
At peak, there will be approximately 220 thousand new infections per day, corresponding to 72 (+44, -20) thousand new symptomatic/reported cases per day. Although still not great, this is much better than where we were, even yesterday. In particular, this is a number that should stretch — but not break — the US health-care system.
The peak has been pushed back to June 14th (+45, -25) days. This is a big range, and depends primarily on how big the asymptomatic population is. In fact, as we approach the peak, we hope to measure this directly from the data.
By September 1, 2021 (which is when we are assuming that a vaccine will become widely available), only 19% (+6%, -4%) of Americans will have been infected, and only 6% (+6%, -2%) symptomatically so. Again, this is much better than even yesterday’s projection.
Finally, given the current death rate of reported cases (about 4%), this implies 800 (+780, -320) thousand Americans will die of COVID-19. 4% of 6% of all Americans is unfortunately still a large number.
So, that is where we stand on April 7th. Keep up the good work, and if we can steepen the trajectory again, the outlook will get even better.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/6/2020
What the Heck! (And Model Update…)
I am done with my grant proposal, so back to the daily updates. (Actually, I made one late last night, but it was erroneous, so I took it down in the morning — my apologies to any 2 AM — 8 AM readers who may have been confused by it.)
So what the heck is going on! (Yes, that’s a statement, not a question.)
Friday’s doubling-time measurement was 5.6 days. Saturday’s was 6.0 days. And Sunday’s was 8.8 days. Wait, what?!? Yes, 8.8 days.
But before everyone gets too excited, let’s consider the possibilities:
First of all, Sundays have become notorious for higher-than-expected doubling-time measurements that later get revised downward. Why? Because everyone needs a break and Sundays are generally when people take them. Perhaps the newly symptomatic are less likely to act on it if it’s a Sunday. Or perhaps they act on it, but fewer health care workers are available on Sunday, so they either get put off until Monday, or at least their reports don’t get filed until a day or two later — even the reported number of deaths go down on Sundays.
That said, we’ve never seen a jump in the doubling-time data this dramatic. Perhaps the case load has gotten so high that we’ve hit a wall as to what can be processed on shorter-staffed Sundays. Or perhaps this is due to COVID-19 now putting a strain on smaller, suburban and rural health care facilities, which might have even more stringent Sunday limitations.
Still, today is Monday, and with an hour left to midnight, today’s doubling-time is shaping up to be similarly high — the opposite of what one might expect if Mondays are spent catching up on a backlog.
Another possibility is that although testing has ramped up in the US, perhaps new cases are once again outpacing it. If tests have to be administered more sparingly one day than on the previous day, this results in fewer new cases being identified, and artificially increases the doubling-time measurements for a while (just as they were artificially decreased in the testing ramp-up phase in the US, between March 18 — 28). This is a tricky subject, and we hope to dedicate a full update to it later in the week. But the short version is there are ways to deal with this, and one shouldn’t throw the baby out with the bathwater.
All that said, testing appears to be on the rise again, which suggests that this probably isn’t the explanation either.
We should probably get used to more scatter in the doubling-time measurements — in both directions. This is simply because there’s a big difference between doubling times of 1 and 2 days, but not really much of a difference between doubling times of 10 and 11 days: Instead of fitting to doubling-time data assuming a constant amount of the scatter about the model, we should probably be fitting to doubling-rate data (equal to 1 divided by the doubling time) assuming a constant amount of scatter about this model. We’ll probably switch to this tomorrow (behind the scenes — we’ll continue to report everything in terms of the doubling time, not the doubling rate.)
And last but certainly not least, perhaps the doubling time is actually increasing faster than before. It appears to have increased at one rate until around March 16, and then at a different, faster rate until around now. Perhaps it has taken another change for the better — we can hope.
Regardless, we will wait a day or two before including this new, 8.8 day doubling-time measurement in our fits. As things stand, it can be formally, statistically rejected as outlying, using Chauvenet’s criterion (which we’ve discussed in previous updates).
However, we do have one more thing to report. Since the end of the testing ramp-up phase in the US, we have been modeling the doubling-time data with a sharply-broken line (see attached plot), with the break-time fixed to the beginning of the testing ramp-up phase. There is now enough new data that we no longer need to fix this break time, and have switched over to treating it as a free parameter. This doesn’t really change our baseline projection, but it does increase its uncertainty. For example, Saturday’s projected peak was 270 (+300, -110) thousand new reported cases per day. Sunday’s, with the more flexible model, is 260 (+660, -150) thousand. This uncertainty should decrease again as more doubling-time data becomes available.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/3/2020
Two Days, Two Record-Long Doubling Times…But Still on the Same Trajectory
The last two updates have been fairly lengthy, so I’m going to keep this one short(er). Wednesday’s doubling-time measurement was a record-long 5.3 days (after revisions). Thursday’s was 5.4 days. And with only an hour to go, today’s is looking similarly strong.
However, this is all consistent with the current trajectory.
In one sense, this is good. After the testing catch-up phase ended, and we started getting usable doubling-time measurements again, it was not immediately clear how we should connect the new data to the old. We opted to model this transition conservatively, to not overestimate the rate at which the doubling time is increasing — better to be too conservative and pleasantly surprised than to be overly optimistic and then disappointed. And now that we have 5 new measurements under our belt, it appears that we made the correct choice — these measurements are consistent with the new trend line, within the expected errors.
But the flip side is that as long as the doubling-time continues to increase at the same rate (we currently measure 3.7 +/- 0.1 hours each day), our projections will not improve. Now, this rate of increase is roughly double what it was before the testing catch-up phase began, which is positive. But it will need to be triple, quadruple, or more (and soon) if we are to avoid the worst of what’s to come.
We’ll give it a few more days and see where we stand.
And fair warning: We’ll continue to update the model and projections, but I need to pause the daily updates until Monday night — I still have my day job (for which I am grateful), and I have a grant to write this weekend!
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/2/2020
Why We Do What We Do…and Today’s (Positive!) Update
Before getting to today’s update, let’s spend a moment reviewing our general approach, and why we chose it.
Our page features two separate models. The first is a purely empirical model of the daily doubling-time measurements. We take a purely empirical approach — i.e., what is the doubling-time trend line and what is its uncertainties? — because (1) this doesn’t require any specific knowledge of epidemiology (we are astronomers after all!), and (2) the alternative is nearly impossible, even for professional epidemiologists.
What is the alternative? A first-principles approach: What is the specific effect of more widespread, faster case identification and isolation? What is the effect of social distancing? What is the effect of stay-at-home orders? What is the effect if these (and many, many, many other factors!) are not applied uniformly? What is the effect of wearing vs. not wearing masks? Etc., etc., etc. Many, very smart, very experienced, professional epidemiologists are taking such, first-principles approaches — and their projections are all over the place (for example).
Why are they all over the place? There are simply too many variables, and too many unknowns. The result is a model for everyone: Optimists can find an optimistic model. Pessimists can find a pessimistic model. Politicians can find either, depending on what they hope to achieve, or on how they hope to appear.
This is why we chose to take the opposite approach. We do not know how all of these efforts that everyone is making translates into changes in the doubling time…but we can measure these changes. And we can measure them well — in near-real time, and without underestimating the uncertainties in the implied parameters. (The latter in particular is actually our little group’s area of expertise.)
The second model that our page features is the projection model: Given this doubling-time trend line, and its uncertainties, how many people will become infected and when? This (unlike the challenging epidemiological questions above) is actually a fairly easy problem to solve. Similar questions are assigned to students in introductory calculus classes.
And with not much more information, we can also project how many of the infected will be symptomatic, and how many of these will die. Each of these additional steps comes with additional uncertainties, but we are careful to propagate these uncertainties through to our final projections.
The only assumption that this approach makes, that is not reflected in our uncertainties, is that the doubling-time trend line continues to hold into the future. That is why we are always careful to emphasize that we are not projecting what will be, but what will be if we cannot make the doubling trend increase faster than it already is.
In this sense, the true goal of our little project is to give us all near-real-time feedback as to whether or not our collective actions are making a difference, and if so to what degree. When fighting an invisible enemy, it is hard to know if we are winning or losing, or at least turning the tide. But each time our projections improve, we know that at least some of the things that we are doing are having an effect. In that sense, this page is meant to be encouraging, and even empowering.
And on that note…today’s update: Yesterday’s doubling-time measurement was the longest yet: 5.24 days. And with only minutes to go until midnight, today’s looks like it will be even longer. We currently measure the doubling time to be 5.2 +/- 0.1 days, increasing at a rate of 3.7 +/- 0.2 hours each day.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
4/1/2020
Why is the Death Rate Increasing?
But before we get to that, a quick update. Yesterday’s doubling-time measurement (which we added today) was 4.8 days, similar to 4.9 days the day before, and 5.2 days the day before that. And today’s measurement is shaping up to be around 5 days as well. As we concluded yesterday, we are through the initial testing ramp-up phase, and the doubling time has been increasing faster than it was before the testing ramp-up began. And since our newest measurement is similar to our previous measurement, our projections haven’t changed much either (with one exception, which I’ll describe below). I encourage you to go to our model and projections page (linked above) to see how our collective social distancing and other efforts are working out so far.
But what about the death rate? So far, I have been reluctant to talk about projected deaths, because it is so serious, and so scary, of a topic. And also because we are not trying to project how many *will* die, but how many *will die if the doubling time continues to increase at its current rate*. This gives us an idea if our collective efforts are working, to what degree that they are working, and if we need to do more.
Currently, most people calculate the death rate by taking the total number of deaths, dividing by the total number of reported (i.e., symptomatic, for the most part) cases, and multiplying by 100%. If you do this, you currently get 2.2% of reported cases result in death.
But this isn’t actually the correct way to do this (hat tip to Charles Fischer, who brought this to my attention yesterday): It takes time to progress from becoming a reported case to potentially dying of COVID-19. Consequently, we should instead be dividing by the total number of reported cases *not now*, but *this amount of time ago*.
The problem is that since the total number of reported cases is growing exponentially, most cases have occurred in just the past 5.1 days (the current doubling time). So if the typical time to progress from becoming a reported case to potentially dying is long, we should instead be dividing by a much smaller number of reported cases, resulting in a much higher death rate.
Of course, both techniques yield the same result once all of this is over. Or to put it another way, our (incorrect) technique will begin to yield larger death rates, closer to that of this better technique, as we approach the peak. And indeed, the death rate, as we have been measuring it, has been inching up, from 1.3% 11 days ago to 2.2% now (see the blue curve in the attached plot).
So how long does it take to progress from becoming a reported case to dying? Unfortunately, I can’t find any good data on this. Plus it will likely vary from country to country, depending on demographics, and on quality of health care.
However, we decided to take the following approach to get a “best guess” for the American data set.
Initially, the death rate should be high, because only the most severe cases are being identified. But eventually, testing ramps up such that most of the symptomatic population is being identified. After this point, the death rate should be fairly constant, at least as long as the country’s health-care capacity is not exceeded. If/when that happens, the death rate should go up again.
So we took the publicly available data and calculated how the death rate has been changing, assuming delays of 0, 1, 2, 3, etc. days. As you can see from the attached plot, a 3-day delay yields a constant death rate after the initial, expected decrease. Here we are of course assuming that our health-care capacity has not yet been exceeded. Although it is being stretched, especially in New York, this is probably a reasonable assumption.
Adopting a 3-day delay implies that the percentage of reported cases that result in death is not 2.2% but 3.3%. Given the current doubling-rate trend, this increases our projected number of deaths by 50%, to 1.0 (+1.3, -0.5) million Americans before September 1, 2021 (when we are assuming a vaccine will be widely available).
This is a sobering number, and should encourage us to do more. Because — and I wish to emphasize — this is only our projection *if the doubling-time trend continues to increase at the rate that it has been*.
Although adopting a 3-day delay is probably not perfect, it is more realistic than assuming a 0-day delay, as we had been (corresponding to all deaths occurring immediately after testing), so we have added this to our model permanently (or at least until a better measurement becomes available).
Keep social distancing.
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
3/31/2020
New Measurement Confirms that the Doubling Time Has Indeed Accelerated
Yesterday’s update was lengthy, so we’ll keep today’s brief. Sunday’s doubling-time measurement was revised upward, from 4.8 days to 5.1 days — an all-time high. Yesterday’s measurement, which we added today, came in at a slightly lower but still confirming 4.8 days. And with two hours to go, today’s measurement is shaping up to also be similar. As we described in yesterday’s update, the doubling time is indeed increasing at a faster rate than before.
Today we find the modeled doubling time to be 5.1 +/- 0.2 days, increasing by 4.0 +/- 0.3 hours each day. This implies a peak of 780 (+380, -190) thousand new infections per day, corresponding to 260 (+320, -110) thousand new reported cases per day, on May 27th (+15, -9) days. The optimistic side of this range is on the cusp of what our health care system might be able to handle, which does not sound great, but is significantly better than the projections that we were facing even only a couple days ago.
This rate of increase in the doubling time also implies that 28% (+15%, -7%) of Americans will become infected prior to September 1, 2021 — our current best guess as to when a vaccine will be readily available. This corresponds to only 9% (+12%, -4%) of Americans becoming symptomatic.
However, currently 1.9% of such cases in the United States are resulting in death. If this percentage holds, this would correspond to a total of 600 (+770, -270) thousand deaths.
But again, all of this is only what happens if the doubling rate does not continue to accelerate. It will take probably another week’s worth of data to know if we are on a linear trajectory again (albeit one with a steeper slope), or if this trajectory is continuing to accelerate with time.
Keep social distancing.
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
3/30/2020
The Moment We've Been Waiting For...Sort Of
Yesterday saw another, significant rise in the single-day doubling time, from 4.0 days to 4.8 days. Furthermore, today's measurement (which we will add tomorrow) also appears to be 4.8 days.
This means that we can no longer model the doubling-time measurements with a linear model -- the doubling time is not only increasing, it's accelerating!
Wider-spread testing, resulting in earlier case identification and isolation, as well as the social distancing, hand washing, etc. measures that we have all been practicing is finally beginning to work.
Now, we do not yet have enough data to model this exquisitely. As such, we have chosen a simple, conservative, broken-line model, with the time of the break fixed to when the testing ramp-up began in earnest. ***This model is conservative in that it will not overestimate the doubling time.*** (We will be able to relax some of its constraints in the days ahead, as more data come in.)
So...we now measure the doubling time to be 4.5 +/- 0.2 days, increasing at a rate of 3.2 +/- 0.3 hours each day.
And as you should expect, this corresponds to fewer cases at peak, fewer cases overall, and fewer deaths overall.
But...although these numbers are significantly improved, they are not yet improved enough:
We now project 360 (+520, -170) thousand cases per day at peak, which we project to occur on May 18th (+10, -6) days;
We project 10% (+13%, -4%) of Americans to become symptomatic and identified; and
We project 580 (+750, -260) thousand American deaths, assuming the current death-to-reported cases percentage (1.8%).
These numbers are significantly improved, but still nowhere near good enough. However, if we can continue to increase the doubling time even faster, these numbers will continue go down.
Also:
We have incorporated new knowledge about what is known about asymptomatic carriers into our projections (this is a topic that we explored in our update only two days ago); and
We have updated our main page's text, plots, and numbers. If you haven't read through it in a while, now might be a good time for a refresher.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
3/29/2020
Yesterday’s Doubling-Time Measurement was Tantalizing…But Let’s Not Get Too Excited Just Yet
Thursday’s doubling-time measurement was 3.19 days. Friday’s was 3.49 days. And Saturday’s was 4.01 days — the first measurement in our data set to exceed 4 days.
As you can see in the attached plot, this most recent measurement exceeds our model by a fair amount. Not by so much to qualify as an outlier, given Chauvenet’s criterion (which we described in some detail in our 3/22 update). But it is tantalizing.
So what may be going on? Three possibilities:
Saturday’s number of new cases might have been under-reported. This can happen on weekends, resulting in an artificially high doubling-time measurement as you transition from weekday Friday to weekend Saturday. In this event, Saturday’s number of new cases will be revised upward in a day or two, and Saturday’s doubling-time measurement will move downward.
Saturday’s numbers might require no serious revision, but at the same time may indicate no change in the doubling-time trajectory: There is a lot of scatter in these measurements, and we should take care not to get too excited every time a new measurement lands on the model’s high-side, just as we shouldn’t get depressed every time one lands on its low side.
And finally…this might be the beginning of an honest-to-goodness change in the doubling-time trajectory. But we’ll need at least another day or two of measurements to know for sure.
Of course, earlier case identification and isolation, made possible by wider-spread testing, and the social distancing that we have all been practicing, *should* result in a positive change in the doubling-time trajectory. Not right away of course, but after a week or two — which would be around now.
And with only two hours to go, today’s doubling-time measurement is looking even more tantalizing than Saturday’s.
Given all of this, we have dusted off our “smoothly-broken linear” model and have begun trying it out with these data. If #3 is correct, we’ll switch to it in a day or two, which should significantly change our projections — simultaneously making them better, but also making them more uncertain, at least until more data can be collected along this new trajectory.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
3/28/2020
How Would a Population of Asymptomatic Carriers Affect Our Projections?
This is a good question, and we spent a lot of time building this into our model today. So much that we didn’t get around to updating our projections — we’ll do this first thing tomorrow.
Currently, our model assumes that all or nearly all cases are eventually identified. However, this is probably oversimplistic — there are reports of people who have tested positive for the virus but, at least so far, have shown no, or only very mild, symptoms. Not many reports, but this could be a selection effect: As long as we test only the symptomatic, such a population would be almost completely missed. But they would still be vectors for spreading the virus.
So what would the effect of such a population be? It appears to be twofold:
If there is an asymptomatic population out there, unknowingly spreading the virus, this results in more cases in the short term, and an earlier peak.
But it also results in a milder peak — basically, by the time we get to peak, there will be fewer people out there left to infect, because a larger fraction will have already had it (again, many of them without even knowing it). In other words, herd immunity begins to set in.
So, we built this into our model and can present you with a few example scenarios. Beginning with yesterday’s baseline model, which assumes that all or nearly all cases are eventually identified, our projection is for 6.6 (+3.6, -2.3) million new cases per day on May 19 +/- 5 days. But if we instead assume that there is one asymptomatic case for every two identified cases, the peak shifts to May 12 +/- 3 days, but with a smaller, 4.7 (+2.5, -1.6) million new cases per day at peak. If we assume that the relationship is one to one, the peak shifts to May 7 +/- 3 days with a peak of 3.7 (+1.9, -1.2) million new cases per day.
Of course, all three of these scenarios would be devastating, given that our health case system can probably support no more than a few hundred thousand new cases per day (if that). We must continue to hope that more rapid case identification and isolation, and the social distancing measures that we have been taking, will ramp up the doubling time so we can avoid this outcome.
We have also begun to look into whether the size of this population can be identified from how the reported case numbers are changing. The answer is yes, but not yet. At least a few percent of the population, if not more, would need to become infected before we could measure if there is a second, asymptomatic population out there, changing the numbers.
Until then, we will continue to assume that this population is small when making our projections.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
3/27/2020
As Doubling-Time Trend Continues to Hold, Date of Projected Peak Becomes Better Defined
Another day, another doubling-time measurement…on the same trajectory.
Wednesday’s doubling-time measurement was 3.1 days. Thursday’s doubling-time measurement, which we added today, was 3.2 days. So overall, not much has changed with our projections: Currently, we model the doubling time to be 3.40 +/- 0.15 days, and find that it is increasing by 1.34 +/- 0.26 hours each day.
However, what is changing in our forecast is the degree of uncertainty. As more and more measurements are added — and as long as they continue to fall along the pre-existing trend — the better defined this trend becomes. And consequently, the better defined our projections become.
You can see this in the attached plot, which shows our daily projections for when the peak will occur. They started out fairly uncertain, but now the uncertainty is down to only +/- 4 — 5 days, currently centered on May 20th.
Of course, if new measurements show the doubling time to be increasing faster than it has been, our projected peak will be pushed back to a later date, and at the same time will become more uncertain, at least until we collect enough data to establish the new trend.
But if the current trend instead continues to be reaffirmed, we are looking at a peak about 7 +/- 1 weeks away.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
3/26/2020
Doubling Time Still Consistent with Previous Trend
Just a short update tonight. After a huge jump in the doubling time two days ago (from 2.5 days to 3.8 days, given this morning’s updates to the numbers), yesterday’s doubling time was a more modest 3.2 days. And with only an hour and a half to go, today’s doubling time is shaping up to be similar.
These measurements are all consistent with the doubling-time trend that we measured before the testing ramp-up began, in which case, we measure the doubling time to be increasing, but only by 1.5 +/- 0.3 hours each day.
This rate of increase buys us a little time, with a projected peak infection rate on May 24 +/- 6 days…but not enough time.
We’ll know for sure if we are still on this trend, or if the trend is (hopefully) bending upward, within the next couple days.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
3/25/2020
Longest Single-Day Doubling Time Yet: Back in Business!
Good news! Yesterday, approximately 2,000 fewer cases were reported than the day before, resulting in the largest single-day doubling time measured yet: 3.9 days.
This means that we are probably past the most dramatic part of the testing ramp-up (during which single-day doubling-time measurements will be artificially suppressed). In fact, yesterday’s measurement, which we added this morning, is in line with the last model that we made before the testing ramp-up began.
So we have resumed updating our model. We find that the doubling time is currently 3.6 +/- 0.2 days, and increasing by 1.9 +/- 0.4 hours each day.
However, this won’t be enough to avoid the worst of it. If the doubling time continues to increase at only this rate, we are looking at 3 — 7 million new cases per day at peak, around June 6th +/- 10 days. This is an order of magnitude more than our health care system can handle (at least).
And with 2 hours left until midnight, today’s doubling-time measurement is looking to be closer to 3 days than to 4 days.
But we should not worry yet. This is a noisy data set: It will probably be a couple more days before we know if the old trend is continuing to hold, or if social distancing, etc. has put us on a new, steeper progression.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. And like them when they appear, to train FB's algorithm. Otherwise, they are also posted here.
3/24/2020
Almost There…
The doubling time climbed again yesterday, from 2.56 on Sunday to 2.83 on Monday (which we added today).
In fact, neither of these measurements are sufficiently below the last model that we fitted to the data, before the testing ramp-up began, to formally reject.
This means that:
(1) The testing ramp-up is leveling off, and we can begin to trust the doubling-time measurements again; or
(2) The testing ramp-up is continuing to reveal a significant number of cases that otherwise would have been missed, even the day before, but social distancing, etc., is beginning to have an even greater effect.
Although these last two measurements cannot be formally rejected (using Chauvenet’s criterion), like the previous four measurements are, we have decided to hold off on including them, and on significantly updating the model, for another day or two.
The reason for this is that we are no longer confident that the previous trend still holds: A lot has happened over the past week, and if social distancing is going to have an effect, it should be beginning to manifest itself now.
However, do know that we have prepared a new model — one that asymptotes to a new (but still linear) trajectory. And we are eager to have the opportunity to try it out :)
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. Otherwise, they are also posted here.
3/23/2020
Doubling Time Continues to Bounce Back
Just a brief update today. The doubling time continues to climb: After bottoming out at only 1.4 days on Thursday (after today’s revisions), Sunday’s value was back up to 2.6 days, and today’s value (although we still have 2 hours to go) *might* be above 3 days. If so, we will resume our projections either tomorrow or the next day.
Today’s doubling time, extrapolated from before the testing ramp-up began, is 3.2 +/- 0.2 days, increasing 1.4 +/- 0.5 hours each day. If this trend resumes, we project the peak to be May 23 +/- 11 days. But in this scenario, nearly everyone would get it. Hopefully the post-ramp-up phase has a significantly steeper trajectory.
Keep social distancing!
P.S. If you want to receive these updates in your FB news feed, just send me a friend request. I add everyone automatically. Otherwise, they are also posted here.
3/22/2020
Revised Numbers Worsen Projections
Each day not only are new cases reported, but old numbers are revised. And today (Sunday), we saw significant revisions to Monday through Wednesday’s numbers, causing Monday and Tuesday’s doubling times to move upward and Wednesday’s doubling time to plummet.
Previously, we had excluded all three from our model because they fell so far below the previous model that we deemed them to be outlying. If a measurement is outlying it does not necessarily mean that it is wrong, just that it is attributable to something else. We attributed these measurements’ very low values to the recent ramp-up in testing, which can artificially suppress the measured doubling time (as we discussed in our last three updates).
Note, we do not take the decision to exclude data lightly — nor do we make the decision ourselves. There is a simple test called Chauvenet’s criterion that we use. First you calculate how many standard deviations the measurement is outlying. Let P be the probability of a measurement being at least that many standard deviations away, and let N be the number of measurements in the data set. A measurement can be rejected if P x N < 0.5.
But now that Monday and Tuesday’s measured doubling times have moved upward, they no longer satisfy Chauvenet’s criterion, and consequently we have no choice but to include them — although they may still be somewhat low from the recent ramp-up in testing.
This has lowered both our estimate of the current doubling time (now 3.0 +/- 0.2 days) and our estimate for how quickly it is increasing (now 1.3 +/- 0.4 hours each day). This in turn has increased our projected peak number of new cases per day (now 4 — 12 million) and decreased when this peak is projected to occur (now May 19th +/- 9 days).
Thursday and Friday’s measurements of the doubling time still do meet Chauvenet’s criterion, as does Saturday’s (which we added today). The silver lining in this cloudy forecast is that these are still increasing, as we reported yesterday, now from 1.2 days on Thursday to 2.2 days on Saturday.
Once the reported doubling time rises to the point of no longer meeting Chauvenet’s criterion, we will be able to resume making new, daily projections.
3/21/2020
Doubling Time Has Stopped Falling
After decreasing for four days in a row, the doubling time is finally increasing again.
And as we discussed in yesterday’s update, those four days are probably an anomaly — driven by a greater availability of testing kits, and consequently fewer cases being missed.
But after bottoming out at 1.6 days, yesterday’s doubling time ticked back up to 2.1 days, and today’s (although the day isn’t over yet) looks to be similarly promising.
So either the testing ramp-up is beginning to level off, or social distancing, hand washing, etc. is finally beginning to pay off.
In either case, we may be only a few days away from being able to resume regular updates to our projections.
3/20/2020
Increased Testing Continues to Suppress the Doubling Time
Only a short update tonight. As we described in yesterday’s update, if any day’s testing is more widespread than the previous day’s, this results in an artificially suppressed doubling-time measurement from that day’s data. This began to manifest itself on 3/16 or 3/17, and has continued through yesterday’s measurement (3/19), which we added today. Yesterday’s value was only 1.6.
In fact, this downturn in the doubling-time data appears to be following a parabolic trajectory (dotted curve in the attached plot).
And although it is still an hour to midnight, it looks like this trend will continue tomorrow, but with an upturn. If so, this suggests that the testing ramp-up might be beginning to level off, and/or that society’s collective efforts to increase the doubling time might be beginning to have an even greater effect.
3/19/2020
Bad News But Good News
Each day we get a new measurement of COVID-19's doubling time in the United States. Over a week and a half, we had inched up from taking only 2 days for the number of infected to double, to this instead taking nearly 3 days. But Tuesday's value, which we added yesterday, was disappointing -- back to 2.1 days. And Wednesday's value, which we added today, is even worse -- 1.85 days. (The open circles in the attached plot.)
Are we backsliding, despite all of the lifestyle changes that we have been making? Or is something else going on?
Something else appears to be going on: Testing is finally becoming more widespread in the United States. This is resulting in many more cases being identified, and being identified earlier. This is of course good news.
But it does affect our modeling effort: As long as fewer cases are being missed today than yesterday, today's doubling-time measurement will be underestimated. And if we include this underestimated measurement in our modeling, this will cause us to underestimate how quickly the doubling time is increasing, and to overestimate how many will ultimately become infected and die, should this trend continue.
To deal with this, we performed an outlier rejection test (Chauvenet's criterion, which my team recently published a large paper on) on these two new measurements. And indeed, they are statistically inconsistent with the overall trend.
Taking the view that this is not backsliding (which we would instead have to model), but a side effect of widespread testing finally ramping up, we have decided to eliminate them from the data set.
Furthermore, it is looking like tomorrow's doubling-time measurement will be even lower, and consequently will also be eliminated.
As such, our model, and our model-based projections, are effectively paused until new, usable data are available: We will have to wait for this ramping-up phase to level off, or for the true, underlying doubling time to start increasing rapidly enough to overtake it.
We will continue to update the plots each day, but don’t expect the projections to change much in the short term.
In other updates, we have split our page into two. One for the model and projections:
https://www.danreichart.com/covid19
and one for these update reports:
https://www.danreichart.com/covid19-reports
Stay socially distant,
DR
3/18/2020
Encouraging Model Update…But Discouraging New Data
The response to our COVID-19 modeling effort has been overwhelming. I didn't mention them yesterday, but extra thanks to Nick Konz and Adam Trotter for helping to pull this together so quickly.
A couple of updates for you:
1. Yesterday's model assumed that everyone who is infected stays infectious. Since we are modeling the doubling time empirically, this does not affect the projections in the near term, but it does affect them once an appreciable fraction of the population becomes infected. In particular, this would cause us to overestimate what the peak number of new cases will be.
Consequently, we have now updated the model to have an infectious-period parameter. Currently, we are assuming a very short period for our optimistic projection (green curves) and a very long period for our pessimistic projection (red curves). If an appreciable fraction of the population does become infected, this is something that we should be able to measure directly from the data.
This update did lower our projected peak number of new cases by about 20%. But...
2. We added yesterday's doubling time to the data set and it was only 2.2 days. This lowered the rate at which the doubling time appears to be increasing to only 1.1 +/- 0.5 hours each day. Consequently, our projected peak is now stronger (4 -- 15 million cases per day), and sooner (May 18th +/- 14 days), despite our above update to the model.
Furthermore, today's doubling time looks like it is going to be even lower, <2 days. This will likely make tomorrow's projection even worse.
However, we should not despair yet. We might simply be getting better at identifying cases -- which would be a (very) good thing. If so, the doubling time will be underestimated until our efficiency at identifying cases levels off -- after which I am hopeful that we'll see a sharp rise in the doubling time, due to everyone taking social distancing, hand washing, etc. seriously.
Otherwise, I cleaned up some of the explanatory text, and added a log of these posts at the bottom of the page.
3/17/2020
Open For Business: First Take
Alright, this took all day, but we now have projections for:
1. COVID-19's doubling time in the United States;
2. The number of new cases that we can expect each day if we cannot increase the doubling time faster than we already are; and
3. The ultimate number of deaths that will result if we cannot increase the doubling time faster than we already are.
Everything is explained in detail here:
https://www.danreichart.com/covid19
But in short, the good news is that the doubling time does appear to be increasing. The bad news is that it is not yet increasing nearly fast enough to matter (note -- I got this wrong in my first attempt yesterday).
However, it is still early, and social distancing is just now beginning. We should know more in the days ahead.
We will update these projections daily, so bookmark the page and visit it as frequently as you like.
Although the forecast is currently pretty scary, I am hoping that many will find the page empowering. It should give us near real-time feedback as to whether the extreme lifestyle changes we are making are working.