Coronavirus: a mathematical prediction and what you can do about it

Caveat: I’m not the CDC. I’m just a former math professor who had a long flight last week and wanted to figure out the odds that someone on my plane had coronavirus. That said, the model I describe below is a reasonable approximation of the situation we’re in for the time being; that is, unless the situation changes and we seriously start to slow the spread of the virus, this model will remain at least mostly applicable (and that is a Very Bad Thing). However, it is a crude tool, and is used to get some ballpark numbers. It will not be exactly right by any stretch of the imagination.

TLDR: Unless we do something to slow down coronavirus’ spread (tips for doing so at the end of the article), by mid May we could have a full on crisis on our hands with the potential to cascade into a near total collapse of society as we know it.

One way to describe the spread of a disease is using a logistic growth model. Basically, it’s a mathematical model that’s a twist on exponential growth. Specifically, we use it when otherwise exponential growth is limited by a ‘carrying capacity,’ or a hard limit to the growth.

This is great for modeling uncontained spread of a transmissible disease within a population, because growth would be exponential (the more people who have it, the more people it can spread to) except that there are only so many people in the country/world/universe. Here’s the actual formula:

Screen Shot 2020-03-07 at 10.05.54 PM

and here’s what it looks like when you graph it:

Screen Shot 2020-03-07 at 10.03.44 PM


Where f is the number of cases at a given time, L is the carrying capacity, t is time, k is some number we’ll calculate, and t_0 is the time at which half the population has been infected.

Now for some assumptions. These are all very back-of-the-napkin, so feel free to replace them with your own. Let’s say that 10% of coronavirus cases require hospitalization. As of today (Saturday, March 7th), there are 352 confirmed cases in the US. Assuming that confirmed cases = total cases requiring hospitalization (that is, there’s no one in the hospital with coronavirus who hasn’t been confirmed as having it – a big assumption), then there are approximately 3500 people infected with coronavirus in the US.

Multiple epidemiological studies indicate that the doubling rate is around 6 days. (Math note: the doubling rate means that every 6 days the number of cases doubles; this is a common way of quantifying exponential growth. In the beginning of an outbreak that’s fine, and logistic growth looks almost identical to exponential growth during that time – it starts to change after the halfway mark, however, and the exponential growth will stop being as useful a tool. That’s why I chose to use this model.) So that means that on March 2nd, there were about 1750. Using these two times, we can work out what k and t_0: about 0.1155 and 105.148 respectively.

Now we can use our logistic growth formula to make some predictions.

A week from today, we can expect there to be around 7000 infections, and around 700 confirmed cases.

By the end of March, we would expect ~50k infections nationwide and 5000 confirmed cases. With the current estimated fatality rate, about 1700 of those people will die, or already be dead.

By May, we’re looking at upwards of 1.8 million infections, 180k hospitalizations, and 60k dead.

Around May 8th, with 3.9 million infections (344k of which actively requiring hospitalization), we’ll run out of hospital beds. That means people will start being turned away from the hospital and told to hole up in their homes.

Jared Diamond’s Collapse, a popular, well-researched book on what causes civilizations to implode, suggests that, in our modern society, the deaths of as little as 2% of the workforce could sufficiently disrupt society to cause cascade of failures resulting in a complete collapse. He further suggests even the absence of 10% of the work force for two weeks could have the same result. It is worth noting that, as of this writing, coronavirus’ case fatality rate is estimated to be 3.4%, and we’ve been assuming that around 10% of people will need hospitalization. If Diamond’s numbers are correct, we may be teetering on the edge of a total breakdown of civilization.

Ultimately, the problem with this disease is how quickly it’s spreading. If many of the assumptions I make in this article are wrong, little is changed; for instance, if 5% of people who catch it require hospitalization instead of 10%, that gets us a week longer before we run out of beds. The only thing that significantly affects the results is the doubling time. If we can prolong that time, the epidemic becomes much more manageable. If we could bring the doubling time from 6 days to 10 days, we’d have until June before running out of hospital beds. At 12 days, we’d have until the middle of July. Those precious months would allow us more time to ramp up production of necessary supplies and take measures to protect those most at risk.

So how can you help prolong the doubling time?

First, avoid other people as much as you can. Obviously, few people can afford to skip work – but every little bit helps. If you can afford to order groceries online, do it. Maybe postpone any flights you’ve been planning on, if you can. Perhaps skip that party/potluck/bar crawl.

When you do have to interact with others, try to minimize physical contact. Don’t shake hands, don’t hug, maybe stand a little further away. Subsequently, don’t touch your face until you’ve been able to wash your hands, with soap, for more than 20 seconds. Maybe wear gloves, and learn to take them off without contaminating yourself. Definitely cough/sneeze into your elbow.

This situation has the potential to get out of control. However, we’re not there yet. As cases continue to develop, governments around the world will take action to slow its spread. But all of us should be prepared, and we need to do our part to protect those in the most danger.

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