In a dumb model where people aren’t infectious on day 1-5, and then on day 6-8 infect one person each day under normal conditions and a third of a person each day under lockdown conditions, you get this graph (with the lockdown starting on 4-1):
When I change Rt to be 0.85 (so on each infectious day it’s 0.28 people), you get this graph:
One thing you’ll notice about the graph is the waviness of the total cases line. Even though the new infections plummet on 4-1 because the rate dropped, there’s still 8 days of ‘infectious case’ growth until it starts to drop, because those people were infected before the lockdown (but didn’t become infectious until later), and the same effect happens for each bulge. In a more realistic setting, you’d expect Rt to drop as smoothly as people gradually raise their defenses, which is probably less sudden than a step change.
In case you’re wondering, I think the magic here is mostly being done by the serial interval. This is what it looks like if people are infectious for three days, but those are instead days 1, 2, and 3:
There seems to be no reason for the effect of the interventions that have been made to hit one this closely, other than shear coincidence.
Sure; proximity to 1 is surprising, but I didn’t predict what Rt would be, I observed it, and so unless you have a really strong prior here it makes sense that my posterior is mostly peaked on the observation.
I agree it is moderately surprising that the measures people have employed so far have only gotten the Rt down to 1, instead of lower, but perhaps this is because they aren’t using masks or confining cats or whatever turns out to have been important.
Rt=0.85 with serial interval of 6-8 does look almost like a straight line for the relevant time period. Given that the actual data is noisy (probably beyond simple Poisson variation, with various reporting effects), it may be compatible with that explanation (without, for instance, needing to hypothesize stranger reporting artifacts that would systematically keep the reported deaths nearly constant). Though the linear plots of world case and death counts at https://www.worldometers.info/coronavirus/ do still look very straight to me.
As a more general point, it’s not entirely satisfactory to say that you made an observation and got Rt approximately one, so that’s just what it is. The simple model would be that initially R0 was something greater than one (otherwise we’d never have heard of this virus) as a result of viral characteristics, human behaviour, weather, etc. - it could be 1.3, could be 4.7, etc. - and then we changed our behaviour, and so Rt became something smaller than R0 - maybe a lot smaller, maybe a little smaller, hard to tell. There’s no reason in this model that it should end up really close to one, except by chance. If it seems to be really close to one, then alternative models become more plausible—such as a model in which testing or hospital limits somehow lead to reported cases or deaths saturating at some upper limit (regardless of the real numbers), or in which the transmission mechanism is something completely different from what we think—since in these models there may be a good reason why the apparent Rt should be close to one.
As a more general point, it’s not entirely satisfactory to say that you made an observation and got Rt approximately one, so that’s just what it is.
I suspect we agree. That is, there’s both a general obligation to consider other causal models that would generate your observations (“do we only observe this because of a selection effect?”), and a specific obligation that R0=1 in particular has a compelling alternate generator (“fixed testing capacity would also look like this”).
Where I think we disagree is that in this case, it looks to me like we can retire those alternative models by looking at other data (like deaths), and be mildly confident that the current R0 is approximately 1, and then there’s not a ‘puzzle’ left. It’s still surprising that it’s 0.85 (or whatever) in particular, but in the boring way that any specific number would be shocking in its specificity; to the extent that many countries have a R0 of approximately 1, it’s because they’re behaving in sufficiently similar ways that they get sufficiently similar results.
In a dumb model where people aren’t infectious on day 1-5, and then on day 6-8 infect one person each day under normal conditions and a third of a person each day under lockdown conditions, you get this graph (with the lockdown starting on 4-1):
When I change Rt to be 0.85 (so on each infectious day it’s 0.28 people), you get this graph:
One thing you’ll notice about the graph is the waviness of the total cases line. Even though the new infections plummet on 4-1 because the rate dropped, there’s still 8 days of ‘infectious case’ growth until it starts to drop, because those people were infected before the lockdown (but didn’t become infectious until later), and the same effect happens for each bulge. In a more realistic setting, you’d expect Rt to drop as smoothly as people gradually raise their defenses, which is probably less sudden than a step change.
In case you’re wondering, I think the magic here is mostly being done by the serial interval. This is what it looks like if people are infectious for three days, but those are instead days 1, 2, and 3:
Sure; proximity to 1 is surprising, but I didn’t predict what Rt would be, I observed it, and so unless you have a really strong prior here it makes sense that my posterior is mostly peaked on the observation.
I agree it is moderately surprising that the measures people have employed so far have only gotten the Rt down to 1, instead of lower, but perhaps this is because they aren’t using masks or confining cats or whatever turns out to have been important.
Thanks for the interesting graphs!
Rt=0.85 with serial interval of 6-8 does look almost like a straight line for the relevant time period. Given that the actual data is noisy (probably beyond simple Poisson variation, with various reporting effects), it may be compatible with that explanation (without, for instance, needing to hypothesize stranger reporting artifacts that would systematically keep the reported deaths nearly constant). Though the linear plots of world case and death counts at https://www.worldometers.info/coronavirus/ do still look very straight to me.
As a more general point, it’s not entirely satisfactory to say that you made an observation and got Rt approximately one, so that’s just what it is. The simple model would be that initially R0 was something greater than one (otherwise we’d never have heard of this virus) as a result of viral characteristics, human behaviour, weather, etc. - it could be 1.3, could be 4.7, etc. - and then we changed our behaviour, and so Rt became something smaller than R0 - maybe a lot smaller, maybe a little smaller, hard to tell. There’s no reason in this model that it should end up really close to one, except by chance. If it seems to be really close to one, then alternative models become more plausible—such as a model in which testing or hospital limits somehow lead to reported cases or deaths saturating at some upper limit (regardless of the real numbers), or in which the transmission mechanism is something completely different from what we think—since in these models there may be a good reason why the apparent Rt should be close to one.
I suspect we agree. That is, there’s both a general obligation to consider other causal models that would generate your observations (“do we only observe this because of a selection effect?”), and a specific obligation that R0=1 in particular has a compelling alternate generator (“fixed testing capacity would also look like this”).
Where I think we disagree is that in this case, it looks to me like we can retire those alternative models by looking at other data (like deaths), and be mildly confident that the current R0 is approximately 1, and then there’s not a ‘puzzle’ left. It’s still surprising that it’s 0.85 (or whatever) in particular, but in the boring way that any specific number would be shocking in its specificity; to the extent that many countries have a R0 of approximately 1, it’s because they’re behaving in sufficiently similar ways that they get sufficiently similar results.