The Brutal Math Behind Flattening the Curve
So, I currently live in Toronto, where some people think the shut down of the city is an overreaction, or—to a local community member I just chatted with—a conspiracy by the powers that be. Toronto has arguably done a better job at controlling transmission than many other cities, including many in the United States (though not anywhere near as well as say, Singapore and Taiwan). Unfortunately, because it’s done a reasonable job, many people don’t personally know anyone with coronavirus, or they think of it as just a bad flu and nothing to overreact about.
You’ve probably seen experts talking about how to #flattenthecurve — how we have to enact “social distancing” now or else risk the possibility of 2.2 million deaths in the USA¹. I’m no epidemiologist, and I defer to them on actual recommendations. But I am a math professor, and I wanted to try and make a bit more accessible some of the math behind it all. In short, social distancing now is important because by slowing down disease spread (the “flattening the curve” you hear about), medical professionals will be less likely to be overwhelmed, and scientists will have more time to try and come up with ways to combat coronavirus.
There are three reasons why we should overreact while we still can:
- Delayed onset
- High rate of hospitalizations
- Exponential growth
The most insidious thing about coronavirus is that you are infectious even before you show any symptoms. You may not know you are infected yet, but you can still transmit it to your friends and family (or strangers, of course). This means that we have to be super proactive in preventing transmission before people show any symptoms. If we wait too long, until many cases start showing up in hospitals, like Italy did, it’ll be too late. Notably, Taiwan is right next to China, but they immediately went into action before the WHO even recognized coronavirus as a threat²; they were able to keep it contained. Italy dragged their feet until it was obvious, and by then it was too late³. Fingers crossed that the USA didn’t wait too long, but because symptoms don’t show up until later, we won’t know for a couple weeks.
This is why we have to act (or “overreact”) when it seems like not many people actually have coronavirus.
High rate of hospitalizations
But then the question becomes: is it actually so bad? We already deal with things like seasonal flu every year. Well, roughly 84%⁴ of coronavirus cases are not that bad. You might not even know you have it, but for many people it’ll just be between a bad case of flu and a mild cold.
It’s the other 16% that matters though. Especially for older people, or people who already have heart or lung disease, coronavirus is a serious illness that often requires hospitalization and being hooked up to a ventilator. The death rate of coronavirus is still not completely clear, but it seems to be around 2.3% total⁵ (compared to less than 0.1% for seasonal flu).
But this number is drastically different when comparing different places. When we have a hospital infrastructure that can treat all the patients who are sick, the death rate is estimated at around 1%. But when the medical system is overwhelmed, the death rate spikes up to over 5%⁶, such as in Wuhan, which was very hard hit.
Now here’s the kicker: we probably have fewer than 100,000 ventilators⁷ in the USA. Remember how just a few days ago, people made a run on the grocery stores, leaving stores empty of toilet paper and hand sanitizer? Imagine that happening with ventilators or hospital beds.
Here’s where the math gets scary: if we do nothing, Dr. Anthony Fauci, directory of the National Institute of Allergy and Infectious Diseases, thinks we could have 70 million to 150 million cases⁸ ⁹. But if even 2% of patients need ventilators (and keeping in mind that with an estimated death rate of 2%, this is a drastic underestimate), that’s between 14–30x more ventilators than we have.
Luckily, if people don’t get sick at the same time, we can reuse equipment. But this is where the final whammy hits us. Coronavirus is a new disease, which means that it has the power of exponential growth. As a math professor, I make no apologies now for giving you a word problem and taking you back to high school math class. Imagine a large pond that is completely empty except for 1 lily pad. Each month, the lily pad doubles, so there are 2 lily pads in the second month, 4 in the third month, etc. Let’s say that the pond has about 1 million lily pads after 20 months. When did the pond have half a million lily pads? The answer is after 19 months. Because of the doubling property, the newest half a million lily pads all were added in the last month.
The real answer with coronavirus is more nuanced of course, and I’m simplifying a lot of the math, but the same principle applies. If the number of new cases doubles each week, then half of all cases ever in the country will be from the latest doubling. This is real bad, because it means that a whole lot of people will be sick at the same time, and we’ll run out of hospital beds and ventilators, not even counting on the fact that if our doctors and nurses get sick, we might not have people to run the hospitals.
So why doesn’t this exponential growth happen with other diseases, like flu or measles. The reason is that the effective reproduction number is much lower. In the doubling example, the effective reproduction number is 2, because every person infects 2 new people before they recover. For something like the flu though, if enough of us get vaccinated, the reproduction number lowers down. In fact, if we can get the reproduction number down below 1, then the disease dies out, because the number of new patients is decreasing every week.
This is related to the concept of “herd immunity”, because if enough of us are immune (e.g. through vaccinations or prior exposure) to a disease, then the disease reproduction number can be lowered enough that it can’t ever spread. Remember the curve that we’re trying to flatten? The reason it peaks and goes back down is that eventually, most people have already been sick and developed immunity the hard way. Unfortunately, that kind of immunity happens too late to stop coronavirus from overwhelming hospitals.
A fighting chance
Even if we can’t get the effective reproduction number to below 1, lowering it still helps a lot. A reproduction number of 2 is terrible, because it means that half of all cases ever are happening right now, and twice as many as the previous week. A reproduction number of 1 would mean that the number of cases is steady every week, which is all right, because then our health-care system can easily keep up. But even lowering that number a bit helps a lot because it means fewer simultaneous cases that hospitals have to deal with, and with the various states of emergency that have been declared, we can put in more resources to the medical system to help with the increased load. This is the math behind the drive to #flattenthecurve.
So, how do we lower the reproduction number? As mentioned above, it’ll naturally lower over time because as people recover, they’ll have immunity to coronavirus. But this is too slow, and we’ve seen in Wuhan, China that the entire hospital system gets overwhelmed (even though they constructed two new hospitals in just over a week!¹⁰) and people die at a very high rate. Unfortunately, we don’t have a vaccine yet, though scientists are working on it. This leaves us with only the option of changing our own behavior. Things like washing your hands and not touching your face will help. But in order to really fight coronavirus, we need “social distancing”. If people don’t hang around each other as much, the virus doesn’t have a chance to spread, lowering the reproduction number.
And that is why we’ve had to shut down so many public gatherings. It’s not some crazy conspiracy theory. It’s just math. And while sometimes math is brutal, our collective efforts can yet stem the tide of disease.