Category: health
Coronavirus Roundup II
Some things I’ve found interesting and useful lately:
R0
What I think is a pretty important article from LANL: High Contagiousness and Rapid Spread of Severe Acute Respiratory Syndrome Coronavirus 2. This tackles the questions I wondered about in my steady state growth post, i.e. that high observed growth rates imply high R0 if duration of infectiousness is long.
Earlier in the epidemic, this was already a known problem:
The reproductive number of COVID-19 is higher compared to SARS coronavirus
Data
Epiforecasts’ time varying R0 estimates
CMMID’s time varying reporting coverage estimates
NECSI’s daily update for the US
The nifty database of US state policies from Raifman et al. at BU
A similar policy tracker for the world
The covidtracking database. Very useful, if you don’t mind a little mysterious turbulence in variable naming.
The Kinsa thermometer US health weather map
Miscellaneous
Nature’s Special report: The simulations driving the world’s response to COVID-19
Pandemics Depress the Economy, Public Health Interventions Do Not: Evidence from the 1918 Flu
Projecting the transmission dynamics of SARS-CoV-2 through the postpandemic period has some interesting dynamics, including seasonality.
Quantifying SARS-CoV-2 transmission suggests epidemic control with digital contact tracing looks at requirements for contact tracing and isolation
Models for Count Data With Overdispersion has important considerations for calibration
Variolation: hmm. Filed under “interesting but possibly crazy.”
Creative, and less obviously crazy: An alternating lock-down strategy for sustainable mitigation of COVID-19
How useful are antibody tests?
I just ran across this meta-analysis of antibody test performance on medrxiv:
Antibody tests in detecting SARS-CoV-2 infection: a meta-analysis
In total, we identified 38 eligible studies that include data from 7,848 individuals. The analyses showed that tests using the S antigen are more sensitive than N antigen-based tests. IgG tests perform better compared to IgM ones, andshow better sensitivity when the samples were taken longer after the onset of symptoms. Moreover, irrespective of the method, a combined IgG/IgM test seems to be a better choice in terms of sensitivity than measuring either antibody type alone. All methods yielded high specificity with some of them (ELISA and LFIA) reaching levels around 99%. ELISA-and CLIA-based methods performed better in terms of sensitivity (90-94%) followed by LFIA and FIA with sensitivities ranging from 80% to 86%.
The sensitivity results are interesting, but I’m more interested in timing:
Sample quality, low antibody concentrations and especially timing of the test -too soon after a person is infected when antibodies have not been developed yet or toolate when IgM antibodies have decreased or disappeared -could potentially explain the low ability of the antibody tests to identify people with COVID-19. According to kinetic measurements of some of the included studies 22, 49, 54 IgM peaks between days 5 and 12 and then drops slowly. IgGs reach peak concentrations after day 20 or so as IgM antibodies disappear. This meta-analysis showed, through meta-regression, that IgG tests did have better sensitivity when the samples were taken longer after the onset of symptoms. This is further corroborated by the lower specificity of IgM antibodies compared to IgG 15. Only few of the included studies provided data stratified by the time of onset of symptoms, so a separate stratified analysis was not feasible, but this should be a goal for future studies.
This is an important knowledge gap. Timing really matters, because tests that aren’t sensitive to early asymptomatic transmission have limited utility for preventing spread. Consider the distribution of serial infection times (Ferretti et al., Science):
Testing by itself doesn’t do anything to reduce the spread of infection. It’s an enabler: transmission goes down only if coronavirus-positive individuals identified through testing change their behavior. That implies a chain of delays:
- Conduct the test and get the results
- Inform the positive person
- Get them into a situation where they won’t infect their coworkers, family, etc.
- Trace their contacts, test them, and repeat
A test that only achieves peak sensitivity at >5 days may not leave much time for these delays to play out, limiting the effectiveness of contact tracing and isolation. A test that peaks at day 20 would be pretty useless (though interesting for surveillance and other purposes).
Seroconversion rates of 30% at onset of symptoms seem problematic, given the significant pre-symptomatic transmission implied by the Ferretti, Liu & Nishiura results on serial infection times. I hope the US testing strategy relies on lots of fast tests, not just lots of tests.
See also:
Antibody surveys suggesting vast undercount of coronavirus infections may be unreliable in Science
h/t Yioryos Stamboulis
A coronavirus prediction you can bank on
How many cases will there be on June 1? Beats me. But there’s one thing I’m sure of.
My confidence bounds on future behavior of the epidemic are still pretty wide. While there’s good reason to be optimistic about a lot of locations, there are also big uncertainties looming. No matter how things shake out, I’m confident in this:
The antiscience crowd will be out in force. They’ll cherry-pick the early model projections of an uncontrolled epidemic, and use that to claim that modelers predicted a catastrophe that didn’t happen, and conclude that there was never a problem. This is the Cassandra’s curse of all successful modeling interventions. (See Nobody Ever Gets Credit for Fixing Problems that Never Happened for a similar situation.)
But it won’t stop there. A lot of people don’t really care what the modelers actually said. They’ll just make stuff up. Just today I saw a comment at the Bozeman Chronicle to the effect of, “if this was as bad as they said, we’d all be dead.” Of course that was never in the cards, or the models, but that doesn’t matter in Dunning Krugerland.
Modelers, be prepared for a lot more of this. I think we need to be thinking more about defensive measures, like forecast archiving and presentation of results only with confidence bounds attached. However, it’s hard to do that and to produce model results at a pace that keeps up with the evolution of the epidemic. That’s something we need more infrastructure for.
Coronavirus Curve-fitting OverConfidence
This is a follow-on to The Normal distribution is a bad COVID19 model.
I understand that the IHME model is now more or less the official tool of the Federal Government. Normally I’m happy to see models guiding policy. It’s better than the alternative: would you fly in a plane designed by lawyers? (Apparently we have been.)
However, there’s nothing magic about a model. Using flawed methods, bad data, the wrong boundary, etc. can make the results GIGO. When a bad model blows up, the consequences can be just as harmful as any other bad reasoning. In addition, the metaphorical shrapnel hits the rest of us modelers. Currently, I’m hiding in my foxhole.
On top of the issues I mentioned previously, I think there are two more problems with the IHME model:
First, they fit the Normal distribution to cumulative cases, rather than incremental cases. Even in a parallel universe where the nonphysical curve fit was optimal, this would lead to understatement of the uncertainty in the projections.
Second, because the model has no operational mapping of real-world concepts to equation structure, you have no hooks to use to inject policy changes and the uncertainty associated with them. You have to construct some kind of arbitrary index and translate that to changes in the size and timing of the peak in an unprincipled way. This defeats the purpose of having a model.
For example, from the methods paper:
A covariate of days with expected exponential growth in the cumulative death rate was created using information on the number of days after the death rate exceeded 0.31 per million to the day when different social distancing measures were mandated by local and national government: school closures, non-essential business closures including bars and restaurants, stay-at-home recommendations, and travel restrictions including public transport closures. Days with 1 measure were counted as 0.67 equivalents, days with 2 measures as 0.334 equivalents and with 3 or 4 measures as 0.
This postulates a relationship that has only the most notional grounding. There’s no concept of compliance, nor any sense of the effect of stringency and exceptions.
In the real world, there’s also no linear relationship between “# policies implemented” and “days of exponential growth.” In fact, I would expect this to be extremely nonlinear, with a threshold effect. Either your policies reduce R0 below 1 and the epidemic peaks and shrinks, or they don’t, and it continues to grow at some positive rate until a large part of the population is infected. I don’t think this structure captures that reality at all.
That’s why, in the IHME figure above (retrieved yesterday), you don’t see any scenarios in which the epidemic fizzles, because we get lucky and warm weather slows the virus, or there are many more mild cases than we thought. You also don’t see any runaway scenarios in which measures fail to bring R0 below 1, resulting in sustained growth. Nor is there any possibility of ending measures too soon, resulting in an echo.
For comparison, I ran some sensitivity runs my model for North Dakota last night. I included uncertainty from fit to data (for example, R0 constrained to fit observations via MCMC) and some a priori uncertainty about effectiveness and duration of measures, and from the literature about fatality rates, seasonality, and unobserved asymptomatics.
I found that I couldn’t exclude the IHME projections from my confidence bounds, so they’re not completely crazy. However, they understate the uncertainty in the situation by a huge margin. They forecast the peak at a fairly definite time, plus or minus a factor of two. With my hybrid-SEIR model, the 95% bounds include variation by a factor of 10. The difference is that their bounds are derived only from curve fitting, and therefore omit a vast amount of structural uncertainty that is represented in my model.
Who is right? We could argue, but since the IHME model is statistically flawed and doesn’t include any direct effect of uncertainty in R0, prevalence of unobserved mild cases, temperature sensitivity of the virus, effectiveness of measures, compliance, travel, etc., I would not put any money on the future remaining within their confidence bounds.
The Normal distribution is a bad COVID19 model
Forecasting diffusion processes by fitting sigmoid curves has a long history of failure. Let’s not repeat those mistakes in the COVID19 epidemic.
I’ve seen several models explaining “flattening the curve” that use the Normal distribution as a model of the coronavirus epidemic. Now this model uses it to forecast peak hospital load:
We developed a curve-fitting tool to fit a nonlinear mixed effects model to the available admin 1 cumulative death data. The cumulative death rate for each location is assumed to follow a parametrized Gaussian error function … where the function is the Gaussian error function(written explicitly above), p controls the maximum death rate at each location, t is the time since death rate exceeded 1e-15, ß(beta)is a location-specific inflection point(time at which rate of increase of the death rate is maximum), and α(alpha)is a location-specific growth parameter. Other sigmoidal functional forms … were considered but did not fit the data as well. Data were fit to the log of the death rate in the available data, using an optimization framework described in the appendix.
One bell-shaped curve is as good as another, right? No!
Like Young Frankenstein, epidemic curves are not Normal.
1. Fit to data is a weak test.
The graph below compares 3 possible models: the Normal distribution, the Logistic distribution (which has an equivalent differential equation interpretation), and the SEIR model. Consider what’s happening when you fit a sigmoid to the epidemic data so far (red box). The curves below are normalized to yield similar peaks, but imagine what would happen to the peaks if you fit all 3 to the same data series.
The problem is that this curve-fitting exercise expects data from a small portion of the behavior to tell you about the peak. But over that interval, there’s little behavior variation. Any exponential is going to fit reasonably well. Even worse, if there are any biases in the data, such as dramatic shifts in test coverage, the fit is likely to reflect those biases as much as it does the physics of the system. That’s largely why the history of fitting diffusion models to emerging trends in the forecasting literature is so dreadful.
After the peak, the right tail of the SEIR model is also quite different, because the time constant of recovery is different from the time constant for the growth phase. This asymmetry may also have implications for planning.
2. The properties of the Normal distribution don’t match the observed behavior of coronavirus.
It’s easier to see what’s going on if you plot the curves above on a log-y scale:
The logistic and SEIR models have a linear left tail. That is to say that they have a constant growth rate in the early epidemic, until controls are imposed or you run out of susceptible people.
The Normal distribution (red) is a parabola, which means that the growth rate is steadily decreasing, long before you get near the peak. Similarly, if you go backwards in time, the Normal distribution predicts that the growth rate would have been higher back in November, when patient 0 emerged.
There is some reason to think that epidemics start faster due to social network topology, but also some reasons for slower emergence. In any case, that’s not what is observed for COVID19 – uncontrolled growth rates are pretty constant:
https://aatishb.com/covidtrends/
3. With weak data, you MUST have other quality checks
Mining data to extract relationships works great in many settings. But when you have sparse data with lots of known measurement problems, it’s treacherous. In that case, you need a model of the physics of the system and the lags and biases in the data generating process. Then you test that model against all available information, including
- conservation laws,
- operational correspondence with physical processes,
- opinions from subject matter experts and measurements from other levels of aggregation,
- dimensional consistency,
- robustness in extreme conditions, and finally
- fit to data.
Fortunately, a good starting point has existed for almost a century: the SEIR model. It’s not without pitfalls, and needs some disaggregation and a complementary model of policies and the case reporting process, but if you want simple projections, it’s a good place to start.
Once you have triangulation from all of these sources, you have some hope of getting the peak right. But your confidence bounds should still be derived not only from the fit itself, but also priors on parameters that were not part of the estimation process.
Steady State Growth in SIR & SEIR Models
Beware of the interpretation of R0, and models that plug an R0 estimated in one context into a delay structure from another.
This started out as a techy post about infection models for SD practitioners interested in epidemiology. However, it has turned into something more important for coronavirus policy.
It began with a puzzle: I re-implemented my conceptual coronavirus model for multiple regions, tuning it for Italy and Switzerland. The goal was to use it to explore border closure policies. But calibration revealed a problem: using mainstream parameters for the incubation time, recovery time, and R0 yielded lukewarm growth in infections. Retuning to fit the data yields R0=5, which is outside the range of most estimates. It also makes control extremely difficult, because you have to reduce transmission by 1-1/R0 = 80% to stop the spread.
To understand why, I decided to solve the model analytically for the steady-state growth rate in the early infection period, when there are plenty of susceptible people, so the infection rate is unconstrained by availability of victims. That analysis is reproduced in the subsequent sections. It’s of general interest as a way of thinking about growth in SD models, not only for epidemics, but also in marketing (the Bass Diffusion model is essentially an epidemic model) and in growing economies and supply chains.
First, though, I’ll skip to the punch line.
The puzzle exists because R0 is not a complete description of the structure of an epidemic. It tells you some important things about how it will unfold, like how much you have to reduce transmission to stop it, but critically, not how fast it will go. That’s because the growth rate is entangled with the incubation and recovery times, or more generally the distribution of the generation time (the time between primary and secondary infections).
This means that an R0 value estimated with one set of assumptions about generation times (e.g., using the R package R0) can’t be plugged into an SEIR model with different delay structure assumptions, without changing the trajectory of the epidemic. Specifically, the growth rate is likely to be different. The growth rate is, unfortunately, pretty important, because it influences the time at which critical thresholds like ventilator capacity will be breached.
The mathematics of this are laid out clearly by Wallinga & Lipsitch. They approach the problem from generating functions, which give up simple closed-form solutions a little more readily than my steady-state growth calculations below. For example, for the SEIR model,
R0 = (1 + r/b1)(1 + r/b2) (Eqn. 3.2)
Where r is the growth rate, b1 is the inverse of the incubation time, and b2 is the inverse of the recovery time. If you plug in r = 0.3/day, b1 = 1/(5 days), b2 = 1/(10 days), R0 = 10, which is not plausible for COVID-19. Similarly, if you plug in the frequently-seen R0=2.4 with the time constants above, you get growth at 8%/day, not the observed 30%/day.
There are actually more ways to get into trouble by using R0 as a shorthand for rich assumptions in models. Stochastic dynamics and network topology matter, for example. In The Failure of R0, Li Blakely & Smith write,
However, in almost every aspect that matters, R 0 is flawed. Diseases can persist with R 0 < 1, while diseases with R 0 > 1 can die out. We show that the same model of malaria gives many different values of R 0, depending on the method used, with the sole common property that they have a threshold at 1. We also survey estimated values of R 0 for a variety of diseases, and examine some of the alternatives that have been proposed. If R 0 is to be used, it must be accompanied by caveats about the method of calculation, underlying model assumptions and evidence that it is actually a threshold. Otherwise, the concept is meaningless.
Is this merely a theoretical problem? I don’t think so. Here’s how things stand in some online SEIR-type simulators:
| Model | R0 (dmnl) | Incubation (days) | Infectious (days) | Growth Rate (%/day) |
| My original | 3.3 | 5 | 7 | 17 |
| Homer US | 3.5 | 5.4 | 11 | 18 |
| covidsim.eu | 4 | 4 & 1 | 10 | 17 |
| Epidemic Calculator | 2.2 | 5.2 | 2.9 | 30* |
| Imperial College | 2.4 | 5.1 | ~3** | 20*** |
*Observed in simulator; doesn’t match steady state calculation, so some feature is unknown.
**Est. from 6.5 day mean generation time, distributed around incubation time.
***Not reported; doubling time appears to be about 6 days.
I think this is certainly a Tower of Babel situation. It seems likely that the low-order age structure in the SEIR model is problematic for accurate representation of the dynamics. But it also seems like piecemeal parameter selection understates the true uncertainty in these values. We need to know the joint distribution of R0 and the generation time distribution in order to properly represent what is going on.
Steady State Growth – SIR
The Chartjunk Pandemic
So much junk, so little time.
The ‘net is awash with questionable coronavirus memes. The most egregiously flawed offender I’ve seen is this one from visualcapitalist:
It’s interesting data, but the visualization really fails to put COVID19 in a proper perspective.
Exponential Growth
The biggest problem is obvious: the bottom of the curve is nothing like the peak for a quantity that grows exponentially.
Comparing the current death toll from COVID19, a few months old, to the final values from other epidemics over years to decades, is just spectacularly misleading. It beggars belief that anyone could produce such a comparison.
Perspective
Speaking of perspective, charts like this are rarely a good idea. This one gives the impression that 5M < 3M:
Reliance on our brains to map 2D to 3D is even more problematic when you consider the next problem.
2D or 3D?
Measuring the fur-blob sizes shows that the mapping of the data to the blobs is two-dimensional: the area of the blob on the page represents the magnitude. But the blobs are clearly rendered in 3D. That means the visual impression of the relationship between the Black Death (200M) and Japanese Smallpox (1M) is off by a factor of 15. The distortion is even more spectacular for COVID19.
You either have to go all the way with 3D, in which case COVID19 looks bigger, even with the other distortions unaddressed, or you need to make a less-sexy but more-informative flat 2D chart.
Stocks vs. Flows
The fourth problem here is that the chart neglects time. The disruption from an epidemic is not simply a matter of its cumulative death toll. The time distribution also matters: a large impact concentrated in a brief time frame has much greater ripple effects, as we are now experiencing.
Open Letter on Coronavirus
For all friends in the Bozone … this is letter I sent to a community list earlier today:
I appreciate the timely information from Chris Mehl. I’d like to emphasize a few points. I’m largely drawing on my thinking from an epidemic model that I posted here a few days ago, https://youtu.be/mxUsBI0Wr2M
It’s unfortunate that we now have our first confirmed COVID19 case, but we knew it was coming. One case likely means several more exposed asymptomatic are already here. Things could develop quickly: Italy went from 3 cases to 3858 in two weeks, although some of that is illusory due to the expansion of testing. However, that is not necessarily our fate – we may be luckier, and we definitely can change some things.
Two key points about this disease:
– The Covid-19 virus is transmissible before people become symptomatic. We therefore cannot wait to take action until there are confirmed or presumptive cases, or even people who have potentially been exposed, in our social networks, faith groups, communities, workplace, towns, or cities.
– This is not just about us individually. The disease disproportionately attacks older people, and people with compromised immune systems. The life you save through personal behavior may be your grandmother’s.
The response has a lot of moving parts.
1. The end of the line is treatment for the severely ill. The flu peak is typically 5-6% of ER admissions in Montana, so something just as widespread and 10x more serious is an obvious threat. It’s critical to avoid overwhelming (and infecting) our precious and limited health care workers, because if we don’t, fatality rates go up, as happened in Italy and Wuhan.
2. We can reduce contacts with the infected through monitoring, quarantine and isolation. This is the public health bailiwick, but I would expect that a large caseload could overwhelm them as well. I hope they’ll reach out to the community for help if needed. Independent of that, perhaps we can make it easy on people who are self-isolating, by organizing delivery of essential services? For employers, I hope that generous sick leave is a matter of enlightened self-interest: do you want 1 employee out for a week, or all of them out two weeks later?
We have two options that scale up well:
3. We can reduce the risk of each social contact by avoiding proximity and touching (elbow bumps instead of handshakes) and cleaning everything – handwashing, hard surfaces, etc. Lots of good info out there on this. (Sadly, hoarding sanitizers doesn’t contribute.)
4. Finally, we can reduce the number of contacts per person, aka social distancing. Cancelling nonessential group events, or moving them online, is very influential. One Biogen company meeting spawned 100 infections in Boston. The Atlantic has a nice discussion of the nuances: https://www.theatlantic.com/…/coronavirus-what-does…/607927/
If half the infected are isolated, and we have half as many contacts, and those contacts are made 50% safer, then we’ve reduced the transmission of infection by 87.5%. That’s enough to slow the infection rate to less than the recovery rate, so the disease will die out.
If we do that, we’re not quite out of the woods. Social distancing is going to be hard to sustain. But if you stop to soon, the disease comes back. So we’ll need a plan for transitioning from more disruptive social distancing measures to things we can sustain.
When we have to close schools, which I think is likely, we will need to find ways to provide good nutrition and safe spaces for kids, without risk of infection. We can help.
Social distancing is also disruptive to the economy. Our tourism industry and performing arts, and surely other sectors I haven’t thought of, are going to have a rough time. We need to mitigate that.
It’s hard on our social fabric, so things like the Renewal Network’s recent links are important. We need to figure out how to support, comfort and play interact with each other … six feet apart, like the Italians singing from their balconies in the darkness of locked-down Siena.
Fortunately, some bad outcomes are very unlikely. There’s no reason for the food system to break down, for example. Inventories are large and the working-age population won’t have high mortality. So keeping a decent stock of food in case you’re sick is prudent, but panic buying of massive quantities is unproductive. This is not the zombie apocalypse.
There is a Cassandra’s curse here. If we succeed, the worst-case scenarios won’t come true, and some will accuse us of overreacting. That takes courage.
Finally, a personal note. David Brooks had a gloomy piece in the New York Times a day or two back, detailing social breakdown during historic plagues. I think that is not our fate. We have new tools at our disposal, i.e. the internet, that did not exist in previous pandemics. Incredibly, I know – by just one hop on social media – people involved in several of the defining events of this epidemic, including the Siena singers. W now have a powerful and non-infectious way for us to stay coordinated; we just have to be sure that we find ways to reach out to people who are not so digitally connected.
There’s a huge trove of digital resources here:
https://coronavirustechhandbook.com/
Vensim SIR modeling primer
I’ve added an SIR modeling primer video to the Vensim coronavirus page, where you can download the models and the software.
This illustrates most of the foundations of the community coronavirus model. Feel free to adapt any of these tools for education or other purposes (but please respect the free Vensim PLE educational license and buy a paid copy if you’re doing commercial work).