Thursday, April 02, 2020

When was the Lombardy Lockdown?

Two posts in one day! 

Wuhan has a nice long time series which gives a good test of the model and the fitting. Which it seemed to pass pretty well. Next on the hit-list was Lombardy, which has had the second-longest lockdown. Jules and I thought (yes this was a prior decision) that it was more logical to consider Lombardy as a region rather than Italy as a whole, as this one area was so far ahead of the rest of the country and also had substantially different policies in place with restrictions coming in at an earlier date, Actually, defining the date of lockdown wasn't that easy in this case. According to the Wikipedia page, the restrictions started on 21 Feb, were extended around the end of Feb and early March, and then went national around 8 March. While it may seem inappropriate to consider highly localised restrictions, if they covered the majority of the outbreak, then this will dominate the behaviour of the total exponential growth until the rest catches up. So it really isn't that obvious (without doing a lot more detailed investigation) what date to use. I'll try a range.

8 March is when the lockdown was imposed firmly across Lombardy (there's a later date of 11 March that applies nationally) and perhaps this is the biggest effect on R. Though some things took place earlier in March, like some schools and universities being shut down. Here are results when I assume a date of 8 March for the change in R: sort of looks like there's a bend, and the model makes some effort to fit it, but tends to overshoot a bit. Marginally narrowed compared to the prior which was N(1.5,1) remember. And nudged down a bit to lower values but still likely greater than 1. Though some people will eye up those data points and the median of the result and say the model is being unreasonably pessimistic. Time will tell. I'll come back to this in a minute.

But first, back on 22 Feb there was a local shutdown around the area of the initial outbreak. If I try this date, on the assumption that even though it's small, control in this area might dominate the overall growth, then I get the following: 

Amusing but nonsense. Rt is actually pushed up compared to my prior guess, and while that is not impossible, what really appears to be happening is that the model is trying to fit the initial uncontrolled fast growth phase with this post-intervention value. It really seems to be saying that the 22 Feb interventions did not have a strong effect.

So, how about some time in between? Choosing the mid-point of 1 March gives the following:

 It's a decent fit, Rt is pushed down and constrained a bit. Of course post-hoc playing around with the intervention date isn't really in the spirit of objective scientific testing and research, but I never claimed to be doing objective scientific research so that's ok. There is obviously a risk that I've overfitted the intervention date but it looks to me like this simulation provides a pretty good explanation of what we can see in the data.

As with Wuhan, 3-4 weeks of data after the intervention date is showing some evidence of a reduction in R0, probably below even the prior assumed mean of 1.5, though maybe not below 1. This seems reasonably encouraging news at least in terms of avoiding the worst extreme of a completely uncontrolled explosive epidemic, though it may mean the controls have to be in place for a while.


William M. Connolley said...

I must have misunderstood something... how are your R>1 estimates post-intervention compatible with the graphs trending down? I thought R>1 implied growth.

James Annan said...

At R=1.2 (say) you hit the herd immunity threshold around 20% total infected/recovered.

(precisely, 1-1/R is the herd immunity threshold as you can see from the basic logistic equation dx/dt = Rx(1-x))

Everett F Sargent said...

Same question as WMC's, wrt Rt>1. What else is changing longer term in your model. Thanks.

James Annan said...

um...and the point is they are already close to that 20% level, or at least reach it quickly.

Of course this has implications for removal of controls....cos then the effective R value increases again and herd immunity no longer applies.

James Annan said...

The "± 0.9" is doing a bit of work there...there is quite a large chunk with R<1 too. Note also that the log scale squashes the peak in perhaps a slightly misleading manner (but does a much better job of showing the exponential parts)

Everett F Sargent said...


Sorry about that one, WMC's was the only post when I started my post, I am rather slow when it comes to posting stuff (grammar, spelling and keyboard miscues - it is part of my early schooling or lack thereof). We crossed in midstream as it were.

Ben McMillan said...

I think Lombardy is tricky because both herd immunity and the effects of lockdown are playing a strong role: I imagine they will be difficult to disentangle.

In other words, I think you could fiddle with the (not very well-known) mortality rate and move these curves around quite a bit, without adjusting Rt.

James Annan said...

Possible but it seems a bit coincidental that herd immunity is starting to play a big role just at the point that a lockdown would take effect. There is a lot of uncertainty but I don't think the really low mortality rates are that credible - wishful thinking I suspect.