Gompertz, no Farrs, yeah check me out…
If I hear another armchair epidemiologist throw either of these at me again I’ll scream. Apart from anything else, I doubt most people even understand that a Gompertz curve, or a sigmoid, is not the death decay curve. The Gompertz curve’s part in all this, is that it approximates the size of the susceptible, or infectible, group over the course of the spread of a virus.
Specifically in this mock up of what IgA/Tcell believers, like Gupta, think happened in the first wave, it is the blue line of people still in the market to get the virus. In this example defined around the UK population, I set it to 18%, or about 12m people.
Now, those currently infected, and as a direct result dying, which is what most people are usually referring to when they incorrectly refer to the Gompertz curve, is the red curve. Which is derived from the Gompertz curve and looks like a bell curve.
Now, you might be wondering why I’m writing this, well it is response to a complaint from the @fatemperor, who didn’t like my brief twitter attempt to express a complex point related to this. Which is, that one of the reasons we can tell that the burn out that happened after the first peak was not caused by us hitting an IgA/Tcell driven, natural herd immunity, is because of the shape of the death curve, and the burn out we had.
The first confusion to dispell is that both theories involve the population hitting a herd immunity ceiling, and then burning out. The reason for this is, the ceiling is a function of the transmissibility. If you have a higher spread rate, you’ll need to infect more to burn out, lower fewer. So if you are able to change the dynamic, you can change the ceiling.
With that in mind, let’s consider the scenario in this plot where we are tundling along an infection curve that is merrily trying to infect 80% of the population, but we suddenly dramatically reduce the remaining number of susceptible people, as per the blue line suddenly schisming.
What will happen to the infections is they will suddenly curtail, and you won’t get the nice round topped infection peak. Now this very much matches deaths curves that we have for the 1st peak, but it matches those in the 2nd and 3rd less, and this is because we were already so much more locked down, that the change in tramission dynamics was far smaller.
To make this clear, here are the two curves in the same plot.
With all this said we need to be bear in mind that this is just a model for what happens in the real world. So we do need to make sure that what we’ve done with the model matches what happened in reality.
The best way to think about it is to review what the model is actually trying to approximate. Which is the idea that you start with a certain R0, or level of transmissibility. In C19’s case about 5, but as it spreads, newly immune people get in the way of trasmission more and more, lowering the degree to which each newly infected person passes on the infection. This is know as the RE, or decayed version of this tramissibility. Eventually, the RE will get to 1, mapped onto the Gompertz model, this happens at the inflection point, which corresponds to the peak of cases, after this, each new infection infects fewer, on average, than one other person, and the virus slowly burns out, with the RE eventually hitting zero.
Now, in the natural burn out scenario, the RE decays smoothly, and we get the nice round topped infection curve. However, a world where what we did was the rapidly change the transmission dynamics, what happens is you’re trundling along with RE having decayed from 5 to say 4, still on an steep upward trajectory, and the world gets rebooted under it. People stopped writhing in nightclubs, and hide under their duvets, and suddenly the RE drops to about 0.6. So instead of the rounded peak caused by the RE spending time just above, then at, and then just below 1, generating lots of days with similar infection levels of new infections, they just drop off a cliff.
So this is the reason, why we can tell it was measures. If it had been a natural ceiling deaths would have stayed at a consistently higher level for far longer.
