More math of infectious diseases

2020 March 2
by Daniel Lakeland

Thanks to Coronavirus, let’s talk about the mathematics of infectious diseases… This is the mathematics of exponential growth. It works like this… One person has a virus in a transmissible state without severe symptoms. They walk around throughout their day contacting people. Some fraction of the people they contact contract the virus. Several days from now, those people are in the transmissible state. They go about their day, spreading the virus… Etc The number of new cases tomorrow is some small multiple of the number of people who got the virus in the last 4 or 5 days (once you’re sick you don’t spread the virus further necessarily, once you recover or die you don’t spread the virus at all). Of course, there are some people who never get severely ill… and they may continue to spread the virus for tens of days.
The result is that if you plot N vs time it is an exponential curve, or log(N) vs time is a line… which is exactly what you see when you click the “Logarithmic” graph of cases vs time in the Johns Hopkins data display. (actually it’s more complicated, you first see an alarmingly fast line, then it turns over and continues at a different slope, this probably reflects our response, which is delaying and slowing the spread). Eventually enough of the population is sick that there aren’t new people to infect, or everyone becomes extremely wary of being around sick people, and the curve stops growing exponentially. So the result is only exponential in the initial stages (described by mathematicians as “asymptotically for small times”).

Now, exponential growth is *really fast*. Most people can’t really “get” it. Think of it this way, if you start with a ruler that’s 1 foot long, and you double it every day. It takes you 30 days to get the moon, 39 days to get to the sun, and 57 days to get to Alpha Centauri. Since it’s 4 light years to Alpha Centauri, evidently somewhere in the first 40 days or so you substantially exceeded the speed of light extending your ruler…
Next let’s talk about bias in measurement. In general there are two ways to get a biased estimate of how many people have a disease. One is that a bunch of people get the disease, it’s not very bad, and they don’t get tested or counted. This biases you LOW. Obviously the very sick do get tested, and so they are the only ones counted. If there is variability in symptoms, and there always is, you are essentially *always* biased low looking at the “confirmed cases”. The other bias is when people stop testing for the disease and report that everyone with some broad set of symptoms probably had the disease. This biases you higher… It’s primarily an issue with rare diseases rather than epidemics.

In general how big this bias is is unknown for any given disease, but for coronavirus which is known to only cause “a bad cold” in many people, and often very mild symptoms in the young, it’ll be significantly biased low.

As of this morning (Mar 2 2020), there are 86 confirmed cases reported on Johns Hopkins data display, and 2 deaths mostly involving people with existing health complications. The largest outbreak in CA is in Santa Clara County (Silicon Valley) with 6 confirmed cases. How many actual cases are there in the US? We don’t know, but given the known bias that people with mild symptoms will never be tested, and the fact that there are circulating cases (people who got it from an unknown source) it would be silly to estimate less than say 2x the number of confirmed cases, and it would be reasonable to estimate perhaps up to 10x. So that means somewhere between around maybe 160 and 860 real cases in the US today. If you look at the “other locations” graph, you’ll see that in the month of February the reported cases doubled about 6 times, or doubling every 5 days or so. If there are say 200 actual cases in the US, how many will there be by saturday when the Southern California Linux Expo is supposed to take place? The answer is perhaps 400. How many reported cases would there be? Perhaps 400 * 86/200 ~ 170. How many cases will there be 1 week from then? Around 800, how many by Mar 21… about 1600. With reported cases around 700.

The good news is this reflects a substantially reduced growth rate compared to the early days in China, when cases went from 200 to 10,000 in about 10 days (Jan 20 to Jan 30). If we had that kind of spread rate here in the US, we could expect 10000 cases *reported* by Mar 12. That’s a lot faster than the numbers above, and even more scary. In general it’s good to slow the spread, because slowing the spread prevents the health care system from being overwhelmed and unable to care for people. That leads to much higher death rates than would occur at a slower rate of spread.

So, what can we do about all this? The number one thing for a virus that sometimes has mild symptoms but occasionally very bad, is to start *early* infection control procedures. Social distancing is the term used for things like closing schools, working from home, canceling conferences, canceling sporting events, etc. When should we start social distancing? The answer is basically right now or very soon at least. On the order of 7 to 10 days from now. Remember, exponential growth? In china on Jan 20 the epidemic would have seemed not that big a deal, 278 people in China were reported having the disease. By Jan 30 it was 10,000! Anything we can do to slow the spread of the virus thereby reducing the number of cases at any given time that need severe treatment will save lives.

I won’t be going to the SCALE linux conference, even though it’s right here in Pasadena, and even though my kids get free entrance through their schools. It simply doesn’t make sense for our family, as the value we’d get isn’t high enough to overcome the general risk of being around maybe ten thousand people or more milling around sneezing. And they will be sneezing… it’s allergy season in LA.

4 Responses leave one →
  1. Phil permalink
    March 4, 2020

    I’ve been a bit disappointed to not see clearer statements or estimates in the popular press about what is actually going on. I guess by ‘popular press’ I mean the New York Times. They report on known cases and deaths, and on measures taken to slow the spread, but I haven’t seen, for instance, projections of future cases and deaths under various scenarios. I realize the uncertainties are huge but that’s fine, we’ll see a big range of numbers. And then as we learn more about transmission rates and the effectiveness of public health interventions the uncertainties will come down some.

    I think a really big uncertainty is temperature response. Flu is much more transmissible in winter than summer, but that may or may not be true of this coronavirus. But that’s fine, they can show us different trajectories for high, medium, or low temperature-sensitivity.

    Why do we care? Well, for one thing it affects both personal and public health decision-making. Consider travel restrictions: there’s no point restricting travel from Place A to Place B if lots of people in both places are already infected, so if I expect that there will be cases pretty much everywhere by time T, I can go ahead and plan travel for time T without much worry that it’s not going to work out.

    I also wish there were a clear statement to the effect that we do not expect to be able to stop this from spreading. Authorities are still doing the ‘race to trace’ for each infected person and to test and isolate people, but I’m almost positive they don’t expect to be able to stop transmission this way, they’re just trying to make it a slow-moving epidemic than a fast-moving one, to give the medical system more time to prepare and to avoid having a huge crush of cases all at once. If that’s the case then they should say so. Maybe I have, but if so I haven’t seen it.

    • Daniel Lakeland
      March 4, 2020

      I agree with all that Phil. But I think there are some obvious issues that make the reporting not so good as you mention.

      1) The bottom dropped out of the market for in-depth reporting. Most reporting is writing up press releases these days, though there are still a few top tier organizations, maybe NYT and LATimes and Washington Post and such. Still, for the most part the skill isn’t there.

      2) The organizations with the data aren’t sharing raw forms of it. So WHO and Johns Hopkins and whatnot have some data based displays but there’s not a database of every known case for example. So, if you want to do some good projections and things, then how do you do it?

      Also, I disagree with the idea that “if there are cases everywhere” you can “plan travel for time T and it will work out”. Travel exposes you to a lot of people, airports and train stations and such. Staying in and around your home doesn’t. The math of large numbers of people is that as the crowd size grows the chance that there’s one or more people with the disease climbs and eventually approaches 1. You can go hundreds of days without exposure if you are hanging out at home, doing a couple things with a few of your friends, working via teleconference etc… You can go 1 day without exposure at a soccer stadium.

      • Phil permalink
        March 5, 2020

        Oh, sure, I didn’t mean to imply that it would be _wise_ to travel, just that there are unlikely to be travel restrictions once there are cases everywhere. I assume people will still avoid travel, and that I’ll try to do so myself. On the other hand, though, I expect to catch this someday. I’m willing to curtail activities to try to slow the spread of the virus right now, and I definitely would like to avoid being one of the many many people who will have it at the peak (because that’s when care will be the worst), but I’m not going to curtail my activities forever to avoid catching this. Eh, actually I guess things may depend on the outlook for a vaccine: if it looks like one is in the works in the next year, OK, I could imagine changing my life for a year…maybe. But…well, for example my wife and I have travel plans for May. Unless it’s really untenable to do so, I think we will go ahead and take that trip.

        • Daniel Lakeland
          March 5, 2020

          I think that it’s still possible to have a single wave of this disease, lasting as a pulse around 3-4 months, and then a reprieve, possibly with a rebound several months out, but all of that lasting eventually long enough that a vaccine is available say 1 year out. I think it’s possible to keep the cases down in the 100k range in US that china has had over maybe a couple waves, before the vaccine is ready and it’s eradicated.

          So, long term strategy for me is to avoid getting it.

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