The Mathematics of Contagion

This is something I wrote up for people I know on facebook to understand the mathematics of contagion and why you should care.

Let’s talk about covid-19 in the US from a strictly math point of view. I’m going to keep the math as simple as I can. This isn’t going to get any more complicated than high school math. This is going to be long, but it’s important to understand.

Stick with me.

Many things in life are proportional relationships. For example, if I’m putting together a set of hand outs for a conference and each hand out is 5 pages, I can find out how many sheets of paper I need by multiplying the number of pages (5) by the number of attendees. If someone decides to show up, add 5 pages. 2 people cancel, subtract 10 pages. This is also called a linear relationship because if you draw a graph with “number of people” on the x axis and “number of pages” on the y axis and fill in the points, they make a lovely straight line.

Disease propagation does not work that way. It’d be great if it did, but it doesn’t. Instead it follows a pattern called exponential. Here’s an illustration of one exponential relationship.

Let’s say I have a checkerboard which is 32 squares. I ask you to put 1 penny on the first square, 2 on the second, 4 on the third, 8 on the fourth and keep doubling for each square. By square 26, you’d need more than $1,000,000 in pennies. On the last square, you’d need a stack of $21,474,836.48 in pennies.

Let’s compare that to a proportional relationship. On the first square you put 5 pennies, 10 on the second, 15 on the third, etc. Doing this, on the last square you would have $1.60.

That’s some difference, right?

How do we represent that in a way we can calculate? Let’s figure it out.
Here are your days and their relationship to pennies:
1 : 1
2 : 2 * 1 = 2
3 : 2 * 2 * 1 = 4
4 : 2 * 2 * 2 * 1 = 8
8 : 2 * 2 * 2 * 2 * 1 = 16
Since we know that 2 * 2 is 2 squared or 22 and that 2 * 2 * 2 is 2 cubed or 23, we can simplify the table:
1 : 1
2 : 21 * 1 = 2
3 = 22 * 1 = 4
4 = 23 * 1 = 8
8 = 24 * 1 = 16
In general, the number of pennies on a given square is 2n * 1, where n is the number of the square on the checkerboard.

You can see that the number of pennies needed grows really fast. The good news is that COVID-19 doesn’t grow this fast. The bad news is that it is still an exponential relationship. The problem is almost the same as the pennies, but instead pennies it’s confirmed cases of COVID-19 and instead of squares, it’s days. The number we don’t have is called the base. In the penny problem, the base was 2 and I gave it to you. Can we figure out the base for COVID-19? Yes. Take the number of cases on any given day and divide it by the number of cases on the previous day. Why does this work? Remember with the pennies, I said multiply the previous number by 2. So if I have the equation:

x * prevday = nextday

I can solve for x:

x = nextday / prevday

I did just that for COVID-19. CNN (and several other sources) have been posting the number of positive tests each day. I put that into a spread sheet and did the calculation you see here for each day since March 1st. The result is not always consistent, so I calculated the trend and it comes out to about 1.3. That means to predict the number of cases tomorrow, multiply today’s cases by 1.3. That means that the number of cases doubles about every 2.5 days. We are on a pace to have 181,000 cases by the end of the month.

There are estimated to be 160,000 ventilators in the US.

See the problem? And even though not all of those 181,000 cases will need ventilators, just wait 10 days and you’ll have 2.8 million cases.

Now let’s talk about the death rate. The estimates vary a lot. The lowest is around 1.4% and the highest is 4%. The reason why this is such a wide range is that it depends on treatment and it depends on patient age. This means that by the end of the month, you can expect to see between 2240 and 6400 deaths in the US.

There’s good news and bad news. The good news is that COVID-19 will not spread without limit. The limit is the number of people available to be infected. If we had unlimited people, we would infect 330,000,000 people in by May 1st. But we don’t have unlimited people, so the curve can’t grow without limit. Also not every case requires hospitalization.

Further good news – this model is incomplete. The world is a sticky place with many more complications in it. For example, not every person who is infected will need to go to the hospital or even need ventilation. Not only that, once someone is infected, there is a good chance they will recover and the number of cases goes down.

The bad news is that we have a very real limit on the number of doctors, the number of hospital beds, the number of masks, the number of test kits and so on. If we don’t reduce the base, you will see growth to the point where our medical system can’t handle the number of cases it gets. This will get worse as people who work in hospitals get sick. At some point (and it’s going to happen soon), doctors are going to have to make decisions like “which patients will last without ventilation?” or “which patient will recover faster?” or “which patients do we have to let die because we don’t have the staffing or equipment to properly care for them?”

You should ask yourself, do you want you or someone close to you to be one of the patients on the short end of one of those decisions?

Let’s finish this with a call to action. What can you do? Wash your goddamn hands. Seriously. Soap and water wreaks havoc on COVID-19 and is dirt cheap. Keep away from groups and group activities and limit contact (reduce shopping). Feel sick? See if you can use telemedecine. Call your government and demand more testing.

For the curious – here’s my math, which I will try to keep updated.

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