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Exponentially Increasing

Humans are woefully terrible at understanding the impact of exponential growth. Here’s an example.

An article in the NY Times looked at CDC data for COVID-19 and found something interesting: “By the time 50 cases were officially confirmed, at least 1,200 people had already started showing symptoms of COVID-19, the disease caused by the virus.”

If you do the math, this means that there were 1200 / 50 = 24 times as many people infected compared to the number of confirmed cases. As I’m writing this Thursday evening, there are 245,080 Coronavirus cases known, which means that there might be 5.9 million people actually infected right now.

Take that number with a huge grain of salt (or a barrel of regular sized grains of salt). Because the growth rate isn’t constant, and external factors — like how well we are behaving at social distancing and washing our hands — can reduce growth considerably.

And who knows, maybe someday we will even have enough test kits and doctors so that when Donald Trump says anyone who wants to get tested, can be tested, he won’t be a pathological, pants-on-fire, lying asshole. That would help, too.

However, there are other potential external factors — like hospitals becoming overwhelmed, people running out of food or losing their jobs or homes — that could make the growth worse. Much worse.

But let’s say that the growth stays the same (even though it won’t). What is the interval of time that it takes between the time that someone gets infected and when they are a confirmed case? This includes how long it takes after someone is infected before symptoms appear (estimated to be 5 to 14 days), how long it takes before they can take the test (highly variable), and how long to get the results back (around 4 to 8 days now, but will get longer if hospitals get overloaded).

We don’t actually need to know precisely how long it takes. Whether it takes a week or a month (much more likely to be somewhere between that) the result of exponential growth is the same, it just takes longer.

In that amount of time, the number of people who are likely to be currently infected right now (roughly 5.6 million people) will grow around 24 times and turn into 141 million people. It just doesn’t matter if that takes a week or a month.

The only solution is to stop the rate from growing at all, by any means at our disposal. The magic number is to make it so that on average each infected person infects less than one other person. We know how to do it, now it is up to us to just do it.

(While I was writing this, the number of cases rose again, and the number deaths also increased a bit.)



  1. Patriotsgt wrote:

    Thanks for the article IK. I had early on surmised similarly, based on reporting that folks were told to wait for more symptoms. Given a good size percentage of people who contract the disease only have mild to moderate symptoms the numbers seem rational.
    With that information, there may be a silver lining in that data. By that I mean, they seem to be much more accurate in tracking the fatalities form the disease, and if the actual cases are really exponentially higher. Then it follows that the mortality rate is actually much smaller.

    Just trying to have a better perspective.

    Friday, April 3, 2020 at 12:00 pm | Permalink
  2. ebdoug wrote:

    Deaths are going up at 1000 a day now. All the democrats fault says my neighbor who helps me the minute I need help. Like to fish dead cats out of the pond when they go through the ice, like to take me to the ER twice, like to bring me home from surgery:Just spreading so fast and so deadly. “So sick of the Dems bashing Trump for EVERYTHING. Now Peleosi and Schiff up to their games AGAIN”

    Friday, April 3, 2020 at 3:56 pm | Permalink
  3. Iron Knee wrote:

    PSgt, I hope that you’re right, but fear that you are not.

    Friday, April 3, 2020 at 7:22 pm | Permalink
  4. William wrote:

    IK, the testing regime we had early on was so VERY limited that basing anything on it seems very fraught. The factor of 24 seems, to me at least, more an artifact of the low testing rate than an indication of a massive undercount of cases now. At least, I can HOPE that your numbers prove incorrect. If Corvis-19 has even a 1% fatality rate, your estimate of 5.9 million PRESENTLY infected would imply 59000 deaths. If each infected person from here out only managed to infect on average 0.5 new person (and each of those infects 0.5..), we’d end up with double that total case count, and nearly 120000 deaths. If R were 0.75, we’d end up with 4 times instead of double, and 240000 deaths. We really want R to be well under 1.0, otherwise we’d have a continuing crisis where the newly infected just keep infecting more.

    The hope for social isolation is not to just knock R down some, to reduce and delay the peak somewhat, but really to drive R down WELL BELOW 1, so that we get to the other side of the peak count of active cases (without overwhelming our health care system in the mean time), but also so that we have fewer active cases, and can actually trace back their contacts, and get the situation under control, while waiting for both a vaccine and a treatment regimen.

    Of course, we don’t know the real fatality rate either. I used 1% as a figure between the 0.5 from South Korea, with lots of testing, masks, sanitizer, and temperature checks, and higher figures of 2-5% that I’ve seen.

    A point late in the article says:

    “The C.D.C also estimates that as many as 25 percent of people infected may not show symptoms at all.”

    If these folks don’t get tested, they don’t show up in the count of confirmed cases. Good news: These would support PSgt’s point that the mortality rate is lower (perhaps by 1/5), because we may only know about 4/5 of the overall cases.

    The very bad side of it would be if those additional people are infected, contageous, and unaware of both.

    Another more point: “The data also does not reflect the real timeline of infections[…]. The virus’s incubation period is estimated to be five days on average following exposure, but can last up to 14 days. That means people reporting symptoms today may have been infected up to two weeks ago.”

    So if the only people tested are those showing symptoms, and for those who even show symptoms, they may have been infected and potentially contageous 5-14 days earlier… We may really be much further behind the curve, and don’t know it yet.

    Scary times!

    Friday, April 3, 2020 at 9:28 pm | Permalink
  5. Iron Knee wrote:

    William, I totally agree with you. However, you seem to think my factor of 24 is high, and calculate that if it is correct, that would lead to between 120,000 and 240,000 deaths. So you hope my number is incorrect. However, around 120,000 to 240,000 deaths is what Trump and others say is expected.

    A nice source of numbers for all of this is

    If you scroll down a little, there are graphs showing both the cases and the deaths, and for each of those both the total numbers, and the “total new” numbers. ALL of these are increasing. In order to reach an R of 1, the “Daily new cases has to go flat”. And as you say, we really want it to go down steeply.

    If you read the numbers, the time in days that it takes the number of total deaths to double has been running between 3 and 4 days for the last two weeks. The total deaths as of April 3 is 7392, so if we round up to 4 days, that means that if we don’t bring this number down soon, in around 16 days we will have passed 120,000 deaths.

    Now the good news. The slope of the “total deaths” graph is straightening out just a little bit. I’m assuming that now that most states have a “shelter in place” order and people are taking social distancing seriously, things will start getting better (as it has in some countries). I just wish it had happened much sooner.

    Saturday, April 4, 2020 at 2:40 am | Permalink
  6. Jonah wrote:

    The fatality rate is likely a range right now. Probably somewhere between 1-3%. I think people are underestimating a couple of things. Even 1% is too high since its 10x more than flu. 1% of several 100 million people is a large number. The other point to make note of is a large number of people die of other causes but attributable to Covid-19 since hospitals are unable to cope.

    All in all, even if people argue that the fatality ratio of deaths directly attributable to Covid-19 is as low as 1% its still too high for the reasons above.

    Saturday, April 4, 2020 at 3:25 am | Permalink