Covid-19 and exponential growth: Actual cases = 22 × Reported cases

Covid-19 is growing exponentially. Exponential growth is funny (counterintuitive). For a while, nothing seems to be happening. Then very quickly “everything explodes.” (Also strange: this visual behavior repeats at different scales.) For example, here is a chart of reported cases in Italy. (Source: dev_thetromboneguy )

Exponentially growing data looks small at first; then it appears to zoom upwards

As an applied mathematician (among other things), I’ve worked with lots of exponential growth. I fear that reporters and others are not understanding the implications.)

  • Covid-19 infections are growing exponentially, with a doubling time of 5 days. This is based on Italy, and on China before they took drastic measures to limit spread. (On the Italian chart above, it’s even faster.)
  • The incubation period between getting sick and knowing you are sick averages 14 days.
  • In the US, it is taking multiple days to get tested when you ask for it. Optimistically, a 4 day delay on average.
  • The test results are apparently taking days to come back. Approximately 3 days
  • So total lag between being infected, and being reported in the statistics as a “confirmed new case” is 14 + 4 + 3 = 21 days.
  • This is 4.2 doubling intervals = 21/5
  • 2^4.2 = 22.6 ;2 raised to the 4.2 power is 22
  • So for every confirmed case, there are about 21 more that are sick but not yet reported.
  • This assumes everyone with the disease is eventually tested. If half of people who are sick are never tested, double this number.

This is why we each need to take personal defensive action before there are many reported cases in our city. By the time we know there are many cases near us, it is 3 weeks later and the real number is 22 times higher.

Of course there are many approximations in this model. If you know anywhere that has done similar analysis more rigorously, please post it in the comments or tweet it to me, @RogerBohn

Update March 17: The 20:1 calculation is driven by the 21 days assumed between being infected, and getting positive test results. The average may be lower, and it will fall as more tests become available. On the other hand Italy reports a case growth rate of 35% per day! So the ratio could be as low as 8.

John IOANNIDIS calls for random testing of entire population, which would give an unbiased estimate. That’s not yet feasible for the U.S. due to CDC’s inexplicable foot dragging on test development and procurement. But Korea could do it. The required sample size is large due to uneven incidence in different parts of a country. But a running sample e.g. 1000 per day would give very useful numbers.
If the test is sensitive enough, random sampling would also show how many people are sick with few or no symptoms.

Don’t expect Level 5 Autonomous cars for decades

Why I don’t expect fully autonomous city driving in my lifetime (approx 25 years).

Paraphrase: The strange and crazy things that people do. .. a ball bouncing in front of your car, a child falling down, a car running a red light, head-down pedestrian. A level-5 car has to handle all of these cases, reliably.

These situations require 1) a giant set of learning data 2) Very rapid computing 3) Severe braking. Autonomous cars today are very slow + very cautious in order to allow more time for decisions and for braking.

My view:

There is no magic bullet that can solve these 3 problems, except keeping autonomous cars off of city streets. And all 3 get worse in bad weather, including fog much less in snow.

Also, there are lots of behavioral issues, such as “knowing” the behavior of pedestrians in different cities. Uber discovered that frequent braking/accelerating makes riders carsick – so they re-tuned their safety margins, and their car killed a pedestrian.

A counter-argument (partly from Don Norman, jnd1er): Human drivers are not good at these situations either, and occasionally hit people. Therefore, we should not wait for perfection, but instead systems that on balance are better than humans.  As distracted driving gets worse, the tradeoff in favor of autonomous cars will shift.

But there is another approach to distracted driving. Treat it like drunk driving. Make it socially and legally unacceptable. Drunk driving used to be treated like an accident, with very light penalties even in fatal accidents.

Finally, I’m not sure if any amount of real-life driving will be good enough to develop  training datasets for the rarest edge cases. Developers will need supplemental methods to handle them, including simulated accidents and some causal modeling. For example, the probabilities of different events change by location and time of day. Good drivers know this, and adjust. Perhaps cars will need adjustable parameters that shift their algorithm tuning in different circumstances.

Source of the quotation: Experts at the Table: The challenges to build a single chip to handle future autonomous functions of a vehicle span many areas across the design process.

Source: Semiconductor Engineering – Challenges To Building Level 5 Automotive Chips

Semiconductors get old, and eventually die. It’s getting worse.

I once assumed that semiconductors lasted effectively forever. But even electronic devices wear out. How do semiconductor companies plan for aging?

There has never been a really good solution, and according to this article, problems are getting worse. For example, electronics in cars continue to get more complex (and more safety critical). But cars are used in very different ways after being sold, and in very different climates.This makes it impossible to predict how fast a particular car will age.

Electromigration
Electromigration is one form of aging. Credit:  JoupYoup – Own work, CC BY-SA 4.0, 

When a device is used constantly in a heavy load model for aging, particular stress patterns exaggerate things. An Uber-like vehicle, whether fully automated or not, has a completely different use model than the standard family car that actually stays parked in a particular state a lot of the time, even though the electronics are always somewhat alive. There’s a completely different aging model and you can’t guard-band both cases correctly.

The First Smart Air Quality Tracker?

The first microprocessor is almost 50 years old, but microprocessors (MPUs) continue to revolutionize new areas. (First MPU = Intel 4004, in 1971, which Intel designed for a calculator company!) In concert with Moore’s Law and now ubiquitous wireless two-way wireless data transmission (thanks, Qualcomm!). smartphones have become a basic building block of many products.

A companion to explain what’s in your air, anywhere. Flow is the intelligent device that fits into your daily life and helps you make the best air quality choices for yourself, your family, your community.

Source: Flow, by Plume Labs | The First Smart Air Quality Tracker

Here is a quick review I wrote of the “Flow” pollution meter, after using it for a few months.  I wrote it as a comment on a blog post by Meredith Fowlie about monitoring the effects of fires in N. California.

Memorial Sloan Kettering’s Season of Turmoil – The New York Times

America’s health care research system has many problems. The overall result is poor return on the money spent. The lure of big $ is a factor in many of them. Two specific problems:

  • What gets research $ (including from Federal $) is heavily driven by profit potential, not medical potential. Ideas that can’t be patented get little research.
  • Academic career incentives distort both topics of research (what will corporate sponsors pay for?) and publication. The “replicability crisis” is not just in social sciences.

This NYT article illustrates one way that drug companies indirectly manipulate research agendas: huge payments to influential researchers. In this article, Board of Directors fees. Large speaking fees for nominal work are another common mechanism. Here are some others:

Flacking for Big Pharma

Drugmakers don’t just compromise doctors; they also undermine top medical journals and skew medical research. By Harriet A. Washington | June 3, 2011

I could go on and on about this problem, partly because I live in a biotech town and work at a biotech university. I have posted about this elsewhere in this blog. But since it’s not an area where I am doing research, I will restrain myself.

Scrivener 3 for Academic Writing: An In-Depth Review

I use a variety of specialized software for note taking, managing academic papers, etc. Rather than write my own review of Scrivener, I link to someone else’s here. I added a comment to her post, about using it with bibliography software.

Feel free to add links to your own favorite Scrivener reviews, in the comments. There is a fair amount of overhead in learning Scrivener, but for longer projects (eg > 10,000 words) it saves writers from “multiple version hell.”

In this in-depth Scrivener 3 review, I show you why Scrivener is the best word processor for academic writing. Unlike Word, Scrivener 3 keeps all your research and writing in one place. Its best features? The ability to drag and drop to reorganize your draft, split screen mode, word targets, and linguistic focus.

Source: Scrivener 3 for Academic Writing: An In-Depth Review

Rescuing a medical treatment from failure in a clinical trial by using  Post Hoc Bayesian Analysis 

How can researchers maximize learning from experiments, especially from very expensive experiments such as clinical trials? This article shows how a Bayesian analysis of the data would have been much more informative, and likely would have saved a useful new technique for dealing with ARDS.

I am a big supporter of Bayesian methods, which will become even more important/useful with machine learning. But a colleague, Dr. Nick Eubank, pointed out that the data could also have been re-analyzed using frequentist statistics. The problem with the original analysis was not primarily that they used frequentist statistics. Rather, it was that they set a fixed (and rather large) threshold for defining success. This threshold was probably unattainable. But the clinical trial could still have been “saved,” even by conventional statistics.

Source: Extracorporeal Membrane Oxygenation for Severe Acute Respiratory Distress Syndrome and Posterior Probability of Mortality Benefit in a Post Hoc Bayesian Analysis of a Randomized Clinical Trial. | Critical Care Medicine | JAMA | JAMA Network

Here is a draft of a letter to the editor on this subject. Apologies for the very academic tone – that’s what we do for academic journals!

The study analyzed in their article was shut down prematurely due to the unlikelihood that it would attain the target level of performance. Their paper shows that this might have been avoided, and the technique shown to have benefit, if their analysis had been performed before terminating the trial. A related analysis could usefully have been done within the frequentist statistical framework. According to their Table 2, a frequentist analysis (equivalent to an uninformative prior) would have suggested a 96% chance that the treatment was beneficial, and an 85% chance that it had RR < .9 .

The reason the original study appeared to be failing was not solely that it was analyzed with frequentist methods. It also failed because the target threshold for “success” was set at a high threshold, namely RR < .67. Thus, although the full Bayesian analysis of the article was more informative, even frequentist statistics can be useful to investigate the implications of different definitions of success.

Credit for this observation goes to Nick. I will ask him for permission to include one of his emails to me on this subject.