NOTE FROM TIM: The following was written by a close friend who has decades of experience in healthcare systems and advanced training in computer science and healthcare data science. He preferred to publish anonymously to avoid political headaches, so I’ll call him “Felix.”

We are publishing this quickly for reasons that will soon be obvious, so please excuse any typos. This is a work in progress, and we encourage leaving comments if you have any suggestions for improvement.

Enter Felix…

During this coronavirus crisis, the paramount objective has become avoiding hospital overflow, which would make the mortality rate skyrocket. We’re trying to accomplish this with voluntary social distancing, but it’s evident that that’s not going to be enough. So how early should governments start introducing mandatory quarantines? The answer is much, much earlier than you might think. 

In the following example, we can show mathematically that a city that institutes a city-wide quarantine when its hospitals are anywhere from 4% to 20% full can still easily exceed its total hospital capacity. The assumptions underlying this model are at the high end of current estimates in order to make a point, but they are well within the realm of possibility.

There are two important concepts here that humans are not wired to understand, and it takes some time to wrap your head around them.  Exponential growth, when combined with lag time, can create some wildly counterintuitive effects. Say you have a city (let’s call it Springfield) where new COVID infections per day are growing. The numbers are doubling every 5 days:

10–20% of infected people will need to be hospitalized. From the day a person gets infected, it takes an average of 10 days for them to seek hospital care. So if we had 1000 new cases of coronavirus on Day 1, using the high end of the estimate, we can expect 200 cases to arrive at the hospital on Day 11. We can chart the new daily hospital cases for our beloved city of Springfield.

Let’s also say that the average hospital stay is 12 days. So coronavirus cases start to accumulate in the hospital:

Now, let’s say there are 5000 total hospital beds in all of Springfield. The mayor is watching dutifully, but his team, overwhelmed with other duties, aren’t carefully considering exponential growth in combination with lag time. He sees that about 80% (4000) of his beds are occupied on Day 20, and realizes the city needs to do a full quarantine. He orders it promptly on Day 20, so that he can avoid his hospitals overflowing. New infections reliably start to fall on Day 21 and continue to fall forevermore:

What happens to new daily hospital cases? Remember there’s a 10-day lag. So on Day 21 you’ll get 20% of Day 11’s new infections:

Hospital cases continue to increase after Day 20 even though we instituted a quarantine! On Day 20 we were only getting Day 10’s victims into the hospital.  The people infected on Day 20 won’t show up at the hospital until Day 30. Will we overflow our hospitals? How many total beds will we need? Remember: the patients will accumulate since it takes 12 days to be discharged. Let’s extend our Total Beds Occupied graph:

Total hospital beds needed doesn’t peak until Day 35, even though we quarantined on Day 20. It peaks at around 22,000—more than 4 times as many beds as there are in Springfield. Given the numerical assumptions above, if you quarantine when your hospitals are 80% full, you can expect to exceed your total number of hospital beds by more than 300%. You can see this playing out in Italy right now. They’ve quarantined, but the hospital cases will still rise for many days after the quarantine is instituted.

So how early should we start the quarantine in order to avoid our hospitals overflowing? Let’s see what the max hospital bed need is for different points of quarantine:

If you quarantine at 50% full on Day 17, you’ll still have a peak hospital need of 16,812 beds. (Remember: there are only 5000 total hospital beds in all of Springfield)

If you quarantine at 25% full on Day 14, you’ll still have a peak hospital need of 11,092 beds.

If you quarantine at 4% full on Day 10, you’ll still have a peak hospital need of 6,370 beds.

In this example, if you want to avoid hospital overflow, you have to start quarantining when your hospitals are 3% full. Of course, this extreme example starts at 1,000 cases on Day 1 with only 5000 total beds. The effects of exponential growth are less extreme if you start with a smaller number of initial cases. Nevertheless, it’s valuable to have this example drive the point home that in order to prevent hospital overflow, you have to quarantine surprisingly early.

There are some fundamental assumptions I’ve made up here that you may disagree with, such as the initial case count, growth rate, the hospitalization rate, etc. If you’d like a rough estimate of when would be the appropriate time to quarantine for your particular geographic region, you can modify all the numerical assumptions and run different scenarios by copying the template here:

https://docs.google.com/spreadsheets/d/1hUIFj98V53xV1lIwRy_83WAPZwtEQBHL5H1gmSVUjPs/edit#gid=571616345

[Note from Tim: This spreadsheet is read-only. Click on File –> Make a Copy to duplicate it for your own scenarios.]

ADDITIONAL COMMENTS:

The author is a data scientist but not an epidemiologist. Any feedback from epidemiology experts about underlying factors is encouraged in the comments.

The underlying assumptions have been pulled from papers about COVID-19 published in recent months. They are at the high end of estimated ranges but not outside of what experts believe to be the possible ranges.

There are far more sophisticated ways to model epidemics, such as SIR and stochastic input modeling. The point of this article is to spread understanding so we chose to use the simplest possible model that still shows the dangers of exponential growth.

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