1. A short thread about best and worst case models of coronavirus.

I've talked about how the @UW_IHME model strikes me as a best case model (at least given current data inputs) and its confidence ranges represent uncertainty around that best case, not best-to-worst cases.
2. (NB: the IHME team might not accept that characterization.)

I say this because the IHME models projects a first wave that infects about 3% of the US population and thus death totals, while horrible, are lower than if the majority were infected.

From their FAQ:
3. On the other side of the spectrum we've got models like the Imperial College 3/16 model, that explores a scenario where we fail to suppress the pandemic and the first wave sweeps through a majority of the population. These have enormous death tolls.
4. Imperial College's model predicts 2.2 million US deaths without mitigation, and 1.1-1.2 million deaths even with mitigation efforts. This is a completely different scenario than that projected in the IHME model, and I view it sort of as a model of a worst-case timeline.
5. So we've got best-case models that predict on the order of 100,000 US deaths and worst-case models that predict on the order of 1,000,000 US deaths.

Where are the middle case models? Why aren't we seeing those?
6. I was talking with @RS_McGee and @evokerr about this, and we see the answer as residing not in investigators' choices to model the extremes as in the nature of epidemic trajectories.

You can think of best-case and worst-case models as possible timelines that can unfold.
7. Imagine rolling a ball along a ridgeline. It *might* track right down the center of the ridge the whole way, but far more likely it will end up on one side of the ridge or the other.

Pretty subtle differences in starting position and angle make a big difference in outcome.
8. Epidemics work a bit like that. Either you manage to suppress the epidemic and end up on one side of the ridge, or you lose control of it and end up on the other. There is some middle ground, but not much.
9. @RS_McGee (follow him, if you're not already!) put together this figure based on a simple deterministic SEIR model with parameters.

This shows the fraction of the population that is infected at the end of the first wave, as a function of R0 (after interventions, if any).
10. (Note that this is *not* a time course. The x axis is R0 and the y axis shows the fraction infected at the end of the first wave).

In this diagram, 1% of the population is infected at time zero, and other parameters are in the right ballpark for #COVID19.
11. What we see from this diagram is that for the majority of R0 values, we end up a best case scenario with below 5% of the population infected, or in a worst case scenario with over 50% infected.

To get in between that, you need to roll the ball all the way down the ridgeline.
12. I think this is why we tend to see models at two extremes and not a lot of intermediate middle-case results.

It's also why I am comfortable in talking, loosely, about a best-case and a worst-case timeline. Epidemics tend to pull into one of those two attractors.
13. This can be valuable to keep in mind when looking at uncertainty ranges for models.

Do the uncertainty ranges include both sides of the ridgeline? If not, why not? What assumptions are being made?
14. It's perfectly reasonable to model just one timeline, just one side of the ridge, and try to understand what the world would look like if we end up there.

But when making policy recommendations, one has to take the scope and intent of the model into account.

/fin
i. Addendum: there may be one factor that tends to keep the ball on the ridge, so to speak, and move us toward middle-ground outcomes: human behavior.

When the epidemic starts to get out of control, public health authorities and individuals alike take more extreme measures.
ii. That will drop R0 back toward 1.

When the epidemic starts to wane because we've successfully reduced R0 below 1, people will be more inclined to loosen up on social distancing, raising R0 again toward 1.
iii. So we could jitter along the ridgeline, about to topple in one direction, then the other, before behavioral shifts correct course to keep us on the ridge. It's not clear to me whether the divergent dynamics of epidemics will outweigh the homeostatic effect of human behavior.

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