Timing is nothing

Recently mathematical epidemiologist Thomas House published a nice blogpost (see link) in which he argued that it was important to time population control measures in epidemics such as the current one. He presented a simple example in which acting too early in the epidemic would give worse results, just like acting too late. He helpfully presented his model code which I've translated into R and will be using here. (I haven't included the code as I can't work out how to do it neatly like he did on this blogging platform.)

My aim here is to critically examine his example. I believe that while it is technically correct, it has the potential to be catastrophically misleading.

His model is basically a SEIR model with the acronym representing susceptible, exposed, infectious, resistant/recovered (sources seem to differ on what the R really represents). Exposed means those who are infected but not yet infectious due to a latent period which is 5 days in this case. Thomas seems to have implemented this model with two consecutive stages for each of the exposed and infectious phases, giving 6 boxes in all. I'm guessing that this is in order to model the delay between the stages better than a simple 4 box scheme would give, but I could well be wrong about that. A detail that most readers probably don't care about anyway. The figures below replicate the ones Thomas plotted, so I've clearly replicated the model adequately.

Thomas' scenario which I have replicated here is where we start with a baseline of an uncontrolled epidemic, and have the option to implement a 3-week lock-down at the time of our choosing, that temporarily reduces the reproductive rate R₀ from 2.5 to 0.75. The top row of plots is a zoom into the early stages of the epidemic, the second row is the full thing. Thomas argued that timing the intervention was critical and in particular that it should not be implemented too early, because getting it right is crucial to minimising the peak. See the different lines in the left hand plots and especially that the maximum of the red curve in the lower left hand plot is lower than the others. His claim is true but I would argue that in our current situation it is also potentially misleading to a catastrophic degree.

The next graph demonstrates my point. It shows an estimate of the critical cases arising from the epidemic, which I take to be 2% of infectious cases at any moment in time. This is a very rough estimate, but the value must surely be significantly greater than the overall mortality which is generally expected to be about 1% (when properly treated). The maximum capacity of intensive care beds in the UK is about 5000 (from the Ferguson et al Imperial College report) and I've plotted this as the cyan line.

Oops.

I hope you all see the problem here. The red peaks may be lower than the others, but none of the model results are in the same postcode let alone ballpark as the capacity. If capacity was some 20 times higher (dashed line) then Thomas' analysis would be entirely correct and important. But actually, "let's wait a month or two before we do anything" is in this example a catastrophic policy and although it's still technically true that a greater proportion of cases are properly treated in his scenario (ie the area under the cyan line and also under the red curve), the vast majority are not. In the base case, 7% are treated and this rises to just over 10% in the best case. Well, while those additional 3% of victims would surely appreciate it, it's a bit like taking a band-aid to a trauma ward and fussing over where to optimally apply it.

I believe a more appropriate conclusion to draw from this analysis is that the short sharp shock of a temporary lock-down is basically useless as a strategy, and we need to develop a better solution. My intuition suggests to me that such a better strategy should be implemented as soon as possible rather than waiting for the epidemic to grow first, as I'll investigate in future posts.

This model appears to give somewhat worse results than the Ferguson et al report, which uses a vastly more detailed model with lots of different sub-populations rather than just one homogeneous bunch of people. Their results suggest only exceeding capacity by a factor of about 20 in the case of no action, versus the 40 here (ie the dashed cyan line would have to be twice as high again to cover the blue peak). Maybe they expect a higher proportion of isolated older people to escape the disease. Anyway I do not expect this model to provide precise and accurate answers, but it does look like a broadly reasonable tool.