Coronavirus, Cons and the Tallyman of Nurgle

We’re going to take a bit of a break from MathHammer to consider what I do in my real life – the study of infectious diseases, including COVID-19. We’ve used similar math before in other posts – looking at Lanchester’s Laws, and considering “the meta” as an ecological model.

But let’s talk about my boy Nurgle.

And cons.

The Math of Epidemics

The basic building block of the math behind epidemics is what is known as the “SI model” – where the population is divided up into two states – “Susceptible” and “Infected”, with equations describing the transition between them.

$$\frac{dS}{dt}=-\beta*S*I$$

and

$$\frac{dI}{dt}=\beta*S*I$$

This is a very simple way to describe epidemics, where essentially both population groups mix randomly together, and a term $\beta$ describes the rate at which they contact each other, and the probability that someone in S and someone in I come into contact results in the susceptible person getting infected. Much of my work is, in essence, gluing things on to these basic equations.

We can use these models to ask hypothetical questions, study interventions that don’t exist yet (like in-development vaccines), and forecast the path of epidemics. One of the most frightening and important insights of these models is that – for sufficiently transmissible diseases – you can get exponential growth in a population.

For COVID-19, we can use a slightly more complicated model, which adds two additional states – E and R. E stands for “Exposed”, which means individuals who have been exposed to an infectious diseases, but are not yet infectious. R stands for, depending on who you ask, “Recovered” or “Removed” – people who have permanent immunity, or people who are dead. From a somewhat macabre mathematical standpoint, the two are somewhat similar, in that they have little impact on transmission in the future, with the exception of a very few diseases (like Ebola, where transmission at funerals is a concern).

This is a simplified model (the one I’m currently working on for COVID-19 involves several dozen equations), but it’s a good illustrative one.

$$\frac{dS}{dt}=-\beta*S*I$$

$$\frac{dE}{dt} = \beta*S*I – \delta*E$$

$$\frac{dI}{dt} = \delta*E – \gamma*I$$

$$\frac{dR}{dt} = \gamma*I$$

In essence, you have the same process as before – Susceptible and Infected mix with each other, and sometimes a Susceptible person gets infected. That person becomes “Exposed”, where there’s a time called the latent period, where the virus is multiplying, but symptoms haven’t developed yet – denoted by $\delta$. Once those symptoms develop, they become Infected, and can start spreading the disease, and then recover at a rate $\gamma$.

For COVID-19, the latent period appears to be about five days, while the recovery rate is less obvious, but is likely fairly long. I’ve seen estimates as high as 22 days, though this will vary wildly as testing becomes more common, how alert people are to the infection, if we develop treatments, etc. Because they’re rates, $\delta$ and $\gamma$ are actually 1/5 and 1/22 respectively if you’re working in days.

We can get $\beta$ by reverse engineering from estimates of what is called the Basic Reproductive Number, or R0. If you’ve seen Contagion there’s actually a really solid explanation of R0.

In essence, it’s the average number of infections caused by a single infected individual in an entirely susceptible population. To get an epidemic, you need an R0 of greater than 1. For COVID-19, this is about 2.5. R0 = $\frac{\beta}{\gamma}$, so with a bit of math, you can solve for $\beta$.

So if you dropped a single infected person in a teeming mass of unsuspecting gamers, on average, they’d infect 2.5 people. Who would each infect 2.5 people, etc., etc.

Funny you should mention that…

Travel and Conventions

You’ve likely noticed a lot of conventions are canceling – Emerald City Comic Con in Seattle is delayed, SXSW is canceled, GDC has been canceled, etc.

Why?

Large group gatherings boost how often people come into contact with each other, which in turn boosts R0, which means a bigger, sharper, nastier epidemic. And they take people from a lot of places, mix them all together, and send them back to where they came from. But that can’t be that big of a deal…can it?

Let’s consider a slightly more complicated model. We’ll start with a population of 6,000 gamers, one of whom is Exposed – that is, they’ve gotten COVID-19 in the airport. For four days, they mix twice as much as they do normally – shaking hands, admiring models from vendors, chatting in the Forge World line, etc. After that, they stop mixing with each other, and instead move back to a much larger population of susceptible people and mix at the normal rate.

How many people get sick at the con? And more importantly, how many people do they get sick, just from that one person? And because I love variability (obviously), lets simulate this 1,000 times.

On average? No one else gets sick at the con.

The variability isn’t even all that bad. There’s an outside chance of three or more additional cases just from the convention, but this is pretty rare.

But it’s not really just the con-goers we’re worried about. It’s the con-goers and the people they come home to. And there, things get a little worse – a median of 7.5 cases. And the variability there is a little less sanguine.

There’s a 25% chance of 21 or more cases, and a 10% chance 38 or more cases. And remember, this is just cases originating from people infected at the convention – this model doesn’t count the people they infect go on to infect. Those numbers get big fast. But we’re still below a magic number in my mind – 50. We’ll get to why that would be in a bit.

But we’re also being pretty optimistic. The contact rate is only double, and “Con Patient Zero” starts in their latent period, so they’re unlikely to be transmitting for the first few days. Let’s kick it up a notch, and look at something slightly worse – three times the mixing, and that first individual is sick when they arrive.

The actual con is, again, not too bad. The median number of new cases is one, there’s a 25% chance of three or more, and a 10% chance of four or more. The total “con induced epidemic” on the other hand, is a little worse – the median number of new cases is 20, there’s a 25% chance of 41 or more cases, and a 10% chance of 68 or more cases.

Remember that magic number 50 I mentioned? That’s where, on average, with a 2% case fatality rate, the event itself results in someone dying.

That analysis is something I learned from a senior colleague in my field – looking at things that are largely optional, or controversial, and translating into an expected number of deaths. In this case, it’s still less than one death on average, but remember – we’re not counting any infections because the people the con-goers got sick in turn got sick. We’re not counting exposure during air travel, etc.

And just for kicks, let’s take that pessimistic scenario, and have five people who don’t know they have coronavirus at the event, instead of just one.

Now the respective median number of new infections is 9 and 134 respectively.

So I Should Panic, Right?

No. This is not a rigorous and robust forecast. This is a night’s thought exercise while I watch YouTube videos.

One of the points of going through this is, beyond just illustrating what I do for a living with a somewhat tenuous tie to this blog, is to note how much a few assumptions can change things. Do you interact with people twice or three times as often at a con as normal (or do you interact in a way that’s more likely to result in transmission)? Are we talking about 0.02% or 0.08% congoers having COVID-19?

Things that grow exponentially change radically if you change those kinds of things.

So if you’ve noticed how much the media is struggling to be consistent with the predictions in question? That’s because fundamental questions about the disease are still emerging. Imagine having to predict this not for a single convention, but for everything. It gets tricky, and things evolve rapidly.

So What Should I Do?

First, stop getting your advice on infectious diseases from a wargaming blog.

Even this one.

If you want a reliable, approachable infectious disease scientist, I highly suggest starting with following @aetiology on Twitter.

But since you’re here, and there are cons coming up:

• Think seriously about whether or not you want to go. Do you have underlying respiratory or health problems? Do you care for elderly parents? Do you work in a healthcare setting? A few days of wargaming may, to be blunt, not worth the risk.
• Wash Your Hands. All the damned time. For twenty seconds. Enough time to recite the Primarchs and their Legions, or the Litany Against Fear. With soap.
• Try not to touch your face. A common mode of introduction is to pick the virus up from a surface, and introducing it to the nice, inviting tissues of your nose, mouth and eyes.
• Try not to touch other people. Now would be a good time to pass on the friendly pre-game handshake. Don’t share tape measures. Just…try not to interact with people as heavily as you otherwise might. I know that’s frustrating, because that’s the point of a con, but…yeah.
• Elmo Sneeze.

• Wash Your Hands.
• Do your body a favor. Maybe don’t drink quite as much as you otherwise might. Stay hydrated. Do your best to eat something better than con food.

Basically, if you’re familiar with “con crud”, think about how that’s indicative of just how easy it is to get infected at a con. Except this is potentially considerably more serious. So do your part to keep $\beta$ low and keep Grandfather Nurgle at bay.

And wash your hands. Again.

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