Guest post: doubling times redux (image issues fixed!)

27 March 2020; updated 28 March to fix broken image links in some clients

Much of what I share in these notes is the result of conversations with many well-informed, intelligent, and wise people. The constant stream of suggestions, corrections, and criticism is invaluable.

One regular correspondent, who has asked to remain anonymous as he has a high-profile role in Silicon Valley, shares my interest in doubling times and second derivatives, but has taken it to another level. He’s gone so far as to assemble a small team to model this at the country level, and has given me permission to share the following.

Totally agree with your post this morning. Many countries are making progress! I am now confident that social distancing is working. Italy has seen its seventh day of ~5-6k daily new cases. That means that the doubling number is sure to keep falling. Even at our conservative Scenario B (copied below; not sure if you saw my email yesterday), I think we are within days of peak active cases for Italy (more on why that matters in a minute). Of course, with the fire now smoldering, the key question will be how it responds when social interaction resumes.

Loved your look at doubling times. When you look at Italy zoomed-in, the slope of progress becomes even more apparent. I was a bit worried about daily volatility, so I used a 2-day, 3-day, and 4-day doubling time measurement. Was remarkably smooth. Also, what do you make of the blips every seven days? I have a suspicion it’s related to slower weekend testing or a lag in reporting over the weekends.

To the point you made in you post, it’s all about the rate of change in new cases. For the graph below, I calculated the day-over-day change and then calculated the average of that change for the trailing five days.  The slope of the line looks to be statistically significant … and Italy just crossed over to negative 5-day trailing growth. 

So, Italy may have seen its peak reported daily new cases 14 days after lockdown. Remind you of another curve? Wuhan/Hubei! Lockdown works. It just takes 14 days before the bad news peaks. 

Another idea on which I’ve been reflecting: We have been focused on Total Cases for the first few weeks, because that is roughly equal to Active Cases in the early days. However, what really matters to the stability of our healthcare systems is Active Cases, i.e “flattening the curve”. If we can keep the healthcare systems from being overrun, we can tolerate a certain number of Active Cases at a time, especially once we have effective therapeutics. Now … given just two variables, a doubling rate (in days) and an average case duration rate (also in days), isn’t there a theoretical doubling rate where Active Cases plateau? If the doubling rate (in days) is below that theoretical threshold, Active Cases will grow forever. However, once the doubling rate goes above that threshold, Active Cases will eventually peak and then start to fall. Going to see if we can model this theoretical number for COVID-19 today. That’s the “bogey” that every country should be shooting for. 

To your point, on the US, it appears that there is still a long way to go. Hard to even know the trajectory until we have more days under broader testing availability. When I smooth out the doubling calculations over 4-days, the US is at a 4.6 day doubling rate. However, that was a terrifying 1.5-2.0x just a few days ago. Suspect most of this volatility is related to changes in testing availability. The next few days will be really important. FWIW, NY and CA are both at a doubling rate of ~3x, so the higher US rate is related to a bunch of other regions that are earlier on the exponential curve. Does not bode well for news coming out of the US over the next couple of weeks.  


(Chris here again.). If you’re interested in going deeper into the approach this team is using, I’ll also share their description of their methodology:


Our first, rough approach (will be refined later) was to fit curves to China’s and Korea’s daily new cases over time, after they hit peak daily new cases. We then created two decay curves we could apply to Italy that were more conservative, adjusting for the exceptional containment capabilities of China and Korea. Our Scenario C, followed China on the decay, but bottomed at 20% of peak daily new cases, recognizing that many countries won’t be able to entirely eliminate transmission. Our Scenario D assumed that the decay was twice as slow as China and bottomed out at 40%. 

We then applied Scenario C and Scenario D, as well as two other scenarios, to Italy’s current new case count. Scenario A took the 3/23 # of daily new cases and assumed that continued in perpetuity, just to conservatively see what it would do to Peak Active Cases. Scenario A went even more conservative, and took the average of the prior five days (including more of the peak) out in perpetuity. 

The key output was then Active Cases using  our four daily new case scenarios. We assumed 14 days on average from case reporting to case resolution. We thought Active Cases were far more important as a measure of burden on the health system than were Total Cases. As you can see below, our models suggest that Italy may be somewhere between 3 and 10 days from peak healthcare system utilization related to COIVD-19. More important, the current 55k Active Cases may only top out at 60-80k. Suggests that the terror of exponential growth may be behind us. 

We also looked at log curves of Total Cases, which our modeling suggests are about to flatten quickly and significantly. 

This is just a first draft of half a curve for one country, but it suggests to me that we may be surprised at how quickly the curves start to flatten after we implement social controls. Of course, the bigger question is what happens as we start to release those controls. I think if the public health regulators around the world can learn from China and Korea (aggressive testing, isolating, and tracing), we might be able to avoid lighting the uncontrollable community spread again. 

The exponential total case curves still look scary for most countries, but if we focus on daily new case, I think we’re going to see a similar effect as Italy once we get another week into social controls for the rest of the world. My first moments of optimism in about four weeks!

One thought on “Guest post: doubling times redux (image issues fixed!)

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google photo

You are commenting using your Google account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s

%d bloggers like this: