Can anyone explain the math to me?

I’m talking to people who are telling me “experts project that the virus will reach its peak at X date,” and I keep saying “How do experts know that?”

And basically, that’s my question for you guys. To give one example, in my home state of Virginia, I’m looking at this graph showing that in about two months, the number of cases will be going down each day rather than up: COVID-19. How the hell do they calculate something like that? I mean, I can understand seeing a line on a graph and extending that line out, like if something was increasing exponentially for two weeks, to increase it at that same rate for another two weeks.

What I don’t understand is what you can use to project a turning point. Only thing I can think of is if mathematicians took data from elsewhere in the world and tried to fit current data to the same curve, but my understanding is that most countries are still on the uptick.

One thought I had was that maybe they were projecting the point at which so many people were exposed to the disease that it could no longer grow exponentially, but when I read their FAQ page, it said that “By the end of the first wave of the epidemic, an estimated 97% of the population of the United States will still be susceptible to the disease and thus measures to avoid a second wave of the pandemic prior to vaccine availability will be necessary,” so that’s not it.

I think they are estimating the average number of people that one person can infect (based on observation of the increase in infections) - and from that number, you can extrapolate the future number of infections, and when the rate will start to decrease. This video is pretty helpful and shows how changes in the number of people infected by each person affects the curves Simulating an epidemic - YouTube

In order to spread the virus needs hosts and if there are lots of other hosts around it can spread easily and continue to infect more and more people. After being infected, the host will either die or recover, but either way, they are no longer a viable host. The virus, as it spreads, is essentially using up all the available hosts. Eventually, the number of available hosts begins to become sparse and the spread slows and, hopefully, stops.

Understanding the transmission rate of the virus dictates how long it will take before the peak spread is reached and eventual decline.

Imagine 1 person in a room with 99 viable hosts. It will spread pretty fast! Now imagine what happens sometime later. Most people in the room have had it, so the the virus has a harder time finding new hosts.

Start with this Wikipedia article:

Okay, but did you guys see the last paragraph of the OP? It seems like what you’re saying is, essentially, the disease will die down when there aren’t as many people to spread the disease to.

If 97% of the population is still susceptible to the disease, then what you’re describing doesn’t seem to fit the model that is providing. If they were projecting that the number of new cases would start falling in correlation to when new isolation measures were put into place and the disease incubation period had passed, I would understand that.

But to say that they’re projecting the curve to start declining on May 24 (for the state of Virginia) doesn’t seem to hinge on anything. It’s too early for the disease to have reached a critical mass of people, and it’s too late for the decline to be a result of any additional isolation measures.

Yes, to the first part (too early to get to everybody) - but not too late to be the result of isolation. Isolation has a built-in lag - there will be people infected before the isolation started who still turn up sick, and who infect other people who are in isolation and who can still infect the few people they still encounter (family members, etc.), but isolation will gradually start to break the growing chains of infection that would otherwise go on until everyone got sick.

The simulations I mentioned above may help demonstrate that.

Okay, thanks, that explanation helps. I do intend to watch the video you linked to, but my dog was pestering me to go outside so I haven’t yet.

Various numbers have been proposed for when we get to “herd immunity”, which means basically the density of viable hosts falls to a point where the disease starts to die out. Those numbers are anywhere from 20% to 80%. You don’t need 100% of the population to have been infected to have the disease die out. If, for example, you get to 50% that have been infected, it will start to die out.

Looking at the data from that website for Wes Virginia, peak happens on May 4. West Virginia has a population of 1.86Million. I don’t know the current hospitalization rate, but I’m going to guess it is below 20%. In China I saw a report it was 15%.

West Virginia population: 1.8 million
Number needed to pass the peak: 900,000
Corresponding hospitalization: 900,000 * 0.2 = 180,000
If you add up the total hospitalizations from that website up to May 4, it is 261,000.
So maybe the authors of that site think the hospitalization rate is higher or they feel a higher percentage is needed for herd immunity.

Either way, May 4 is about right for when we will begin to see herd immunity kick in for West Virginia.

Sorry, ignore this. Screwed up some numbers.

At this stage of the game, are we sure that being infected with the virus and recovering results in 100% immunity? I seem to recall not long ago that it was uncertain. And what if you are infected just a little (there seem to be gradations), do you get just a little or full immunity?

Outside of some very dubious reports of reinfection, I know of no instances of reinfection happening. If you are infected “just a little”, I’ve never heard of the giving you lesser immunity than someone who received a larger viral load, but that is a better question for the experts. I just have never heard of it. The duration of the immunity is not known, but based on the reports I’ve read it is not expected to be less than a year or two.

I know you said to ignore this because you screwed up some numbers, but regardless of numbers, doesn’t this also assume that West Virginia has an alligator filled moat around it?

Another way to think of the math is to think of R0, a concept that most by now are well aware of.

R0 is that concept of how many each average infected person will infect. More above 1 the faster it spreads. At 1 it stays stable Below one the transmission slows down. As SARS-CoV-2 first hit a population not practicing any social distancing R0 was thought to be between 2 and 3.

Crowded conditions and lots of mixing of people makes for a higher R0. Having almost everyone around being susceptible to the infection makes for a higher R0. Herd immunity in a particular circumstance is when R0 is effectively less than 1.

Social distancing lowers the effective R0. Having more as not susceptible lowers the effective R0. Between the two, and the nature of the bug, at some point it is effectively under 1.

The various models make assumptions about the bug, some very likely, some that are the best guesses they can pull out of the air. But the basic approach is considering three buckets: Susceptible; Infectious; and Recovered. It gets called the SIR model and Wiki goes over the math of it here. The problem is that the models can only be as sure as we are of the assumptions that go into it.

Post-infection immunity probably isn’t 100%. I mean, everyone talks about chickenpox working that way, but my sister and I both managed to catch it twice, as kids. But even if post-infection immunity is only 99%, the net effect is nearly the same.

It should also be pointed out that there’s now loads and loads of data available for this disease, more than enough to make a fit to even a very complicated model. Everyone keeps saying “But we just don’t know”. That was true in January, maybe, but it isn’t any more.

Regarding reinfection/immunity, here’s one doctor’s opinion, posted March 30 (Jen Gunter, obstetrician and gynecologist):

Source: (NYT paywall)

Mathematicians and epidemiologists have been collecting data on pandemics and plagues for centuries. Through exquisite analysis of these data, they have constructed models on how a disease moves through a population. People with big brains have earned doctorate degrees and gone on to extensive post-doctoral studies on this critical topic.

I believe Dr Anthony Fauci is one of these experts. You can see the man’s frustration as he is forced to listen to Trump’s off the wall projections and assumptions about medication. Fauci works hard to break down the progress of the disease as it spreads through our population, and he has been quietly sounding the alarm for some time: it will get worse before it gets better.

There are no cut-and-dried predictions. Nobody can say, “In two weeks, we’ll see the decline in new cases.” This is a numbers game, and it works on probability. The slope of the curve created by new cases is the crucial focus now. There will not be a peak, or a hinge point. The curve will begin to level off. Eventually. No one knows when.

The changing shape of this curve right now depends upon the behaviors of the uninfected, and if they can STAY uninfected. Not just for today and tomorrow. We’re talking weeks or even months.

And always remember, that area under the curve represents all the sick/dying/dead from this disease.

Sit back. Be patient. Stay home. Wash your hands. Don’t be stupid. And continue doing this for weeks, perhaps even months.

The loads and loads of data on this disease are NOT ENOUGH. There needs to be extensive testing for the disease so it can be accurately tracked and asymptomatic carriers identified.

Until then, what we have now is “best guess.”

In the hands of an expert like Dr Fauci, I’ll take his best guess as an extremely informed best guess. And he has said it will get a lot worse before it gets better.

Loads of data still is not the key data the models need. “We just don’t know” is in fact the true current answer.

Yes, that’s what a model produces - a prediction based on past data.

One problem I see often is, when scientists/mathematicians produce predictions, there’s always a confidence interval (error bar) associated with it. Journalists tend to throw that out and just report the prediction, giving the impression that it’s a highly precise prediction.

Yes, but I’m afraid that, even then, most people aren’t going to know what a confidence interval is, and are going to assume that the actual value is guaranteed to be within that interval.