SuppressionStrategy: Difference between revisions
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The blue line is the ratio of smoothed case counts (from the [https://coronavirus.data.gov.uk/ UK Covid-19 dashboard]) at day $$d$$ and $$d+14$$, where $$d$$ is the x-coordinate on this graph. This effectively brings it forward by 7 days which should roughly account for incubation+testing+reporting delays. This is naturally in units of growth ($$\lambda$$) rather than $$R$$, but I'm making the y-axis $$R$$ rather than growth rate because I think people are more used to $$R$$. The conversion formula used here is $$R=(1+\lambda/\beta)^\alpha$$, where $$\alpha=1.99$$ and $$\beta=0.336$$. This is what $$r(\lambda)$$ (see above) works out as if the mean generation time, $$X$$, is distributed as $$\Gamma(\alpha,\beta)$$, using the (shape,rate) notation. These values of $$\alpha$$, $$\beta$$ were fitted to the SAGE estimates (green line), which is possible because they publish both $$R$$ and growth rate. So even if this conversion formula isn't quite right, it should therefore make the blue and green lines comparable. In any case when $$R$$ is close to $$1$$, $$R-1$$ is going to be close to proportional to the growth rate. | The blue line is the ratio of smoothed case counts (from the [https://coronavirus.data.gov.uk/ UK Covid-19 dashboard]) at day $$d$$ and $$d+14$$, where $$d$$ is the x-coordinate on this graph. This effectively brings it forward by 7 days which should roughly account for incubation+testing+reporting delays. This is naturally in units of growth ($$\lambda$$) rather than $$R$$, but I'm making the y-axis $$R$$ rather than growth rate because I think people are more used to $$R$$. The conversion formula used here is $$R=(1+\lambda/\beta)^\alpha$$, where $$\alpha=1.99$$ and $$\beta=0.336$$. This is what $$r(\lambda)$$ (see above) works out as if the mean generation time, $$X$$, is distributed as $$\Gamma(\alpha,\beta)$$, using the (shape,rate) notation. These values of $$\alpha$$, $$\beta$$ were fitted to the SAGE estimates (green line), which is possible because they publish both $$R$$ and growth rate. So even if this conversion formula isn't quite right, it should therefore make the blue and green lines comparable. In any case when $$R$$ is close to $$1$$, $$R-1$$ is going to be close to proportional to the growth rate. | ||
The blue line probably overestimates $$R$$ because testing has been steadily increasing, so we're likely to be finding a greater proportion of cases as time goes on. In particular | The blue line probably overestimates $$R$$ because testing has been steadily increasing, so we're likely to be finding a greater proportion of cases as time goes on. In particular in April there was a sudden explosion in available tests, which may account for the blue line being at $$R~=1$$ then when the true value was probably more like $$0.8$$. At a guess, later blue line entries are closer to their true values. | ||
[https://www.imperial.ac.uk/medicine/research-and-impact/groups/react-study/real-time-assessment-of-community-transmission-findings/ The REACT 1 survey] provides another estimate of $$R$$, but is not shown here. This tends to give slightly more moderate values of $$R$$, meaning closer to $$1$$. As far as I can see they are using a dodgy estimate of the mean generation time distribution. On [https://www.medrxiv.org/content/10.1101/2020.09.11.20192492v2.full.pdf page 4 of this paper] they state they use $$X\sim\Gamma(2.29,2.29/6.29)$$ derived from [https://www.medrxiv.org/content/10.1101/2020.03.03.20028423v3 Bi et al], but the latter is an estimate of the serial interval distribution not the mean generation time, so will have the right mean but a larger (possibly considerably larger) variance. (I also think it may be slightly dubious to use an estimate from Jan-Feb 2020 in Shenzhen and apply it to Western countries with different cultures, age demographics and suppression measures. It may be OK as a starting point estimate, but I'm not sure that they revisit it. One might expect that suppression measures will tend to reduce the mean and variance of $$X$$ as more symptomatic than asymptomatic people isolate themselves.) | [https://www.imperial.ac.uk/medicine/research-and-impact/groups/react-study/real-time-assessment-of-community-transmission-findings/ The REACT 1 survey] provides another estimate of $$R$$, but is not shown here. This tends to give slightly more moderate values of $$R$$, meaning closer to $$1$$. As far as I can see they are using a dodgy estimate of the mean generation time distribution. On [https://www.medrxiv.org/content/10.1101/2020.09.11.20192492v2.full.pdf page 4 of this paper] they state they use $$X\sim\Gamma(2.29,2.29/6.29)$$ derived from [https://www.medrxiv.org/content/10.1101/2020.03.03.20028423v3 Bi et al], but the latter is an estimate of the serial interval distribution not the mean generation time, so will have the right mean but a larger (possibly considerably larger) variance. (I also think it may be slightly dubious to use an estimate from Jan-Feb 2020 in Shenzhen and apply it to Western countries with different cultures, age demographics and suppression measures. It may be OK as a starting point estimate, but I'm not sure that they revisit it. One might expect that suppression measures will tend to reduce the mean and variance of $$X$$ as more symptomatic than asymptomatic people isolate themselves.) | ||
Revision as of 17:09, 8 December 2020
About
Description
The purpose of this is to take a broad look from a certain zoomed-out point of view at what kind of suppression strategies might be best. Here "suppression" is meant to mean that you intend to maintain infections in a low-level holding pattern until there is some favourable change in circumstances such as widespread immunity from a vaccine. I'm not trying to argue here against either infection-induced herd immunity or against zero-Covid/elimination strategies, just looking at what kinds of steady-state suppression strategies might work better than others. As of November 2020, most countries in the world are following a suppression strategy of some sort.
The reason I'm writing this is that when Covid-19 came on the scene in early 2020 I had a slight hunch that the best strategy of this form would be to alternate between strong and weak suppression. The discussion here is meant to explore the plausibility of this from a simplified point of view - it does not claim to prove it.
The abstraction here is that you just look at what values of the basic reproduction number, $$R$$, are best to aim at. This is an idealisation of the process of controlling a disease and ignores all sorts of things like how $$R$$ differs between regions and subcommunities, how to actually achieve a particular value of $$R$$, what the generation time is, how efficient it is to swap between different suppression states, what the effect of suppression on the distribution of infectivity time, to what extent the concept of $$R$$ actually makes sense, and so on, yet I would like to pursue it to see if there is any useful intuition to be obtained at this level. Is it better to keep $$R$$ hovering around 1, or is it better to mix strong suppression phases with relaxed phases? Should there be more than two sorts of such phases? Recently (as of November 2020) this has become topical again in the UK as we are trading off a relaxation over the Christmas period with some extra suppression before and after.
A suppression strategy means keeping the average growth rate of the disease equal to zero (or less). This is similar to keeping the multiplicative average $$R$$ at 1, though not quite the same (see below). For example if you alternate $$R=4$$ for 1 week and $$R=\frac12$$ for 2 weeks then the net effect over three weeks will be similar to an $$R$$ of $$4\times\frac12\times\frac12 = 1$$. This is because, to first approximation, $$R$$ is related to the growth rate $$\lambda$$ and the average time for an infected person to infect another (the mean generation time), $$T$$ by
\[R=e^{\lambda T}.\]
But because the time taken to infect someone is not a constant, a better formula is
\[R=r(\lambda)=\mathbb E[e^{-\lambda X}]^{-1},\]
where $$X$$ is the time to infection for each infectee, considered as a random variable. (Note, the above expectation uses a different probability space from that considered next.)
Consider the reproduction rate $$R$$ to be a random variable satisfying $$0<R\le R_0$$. Our goal (imagining ourselves to be the government) is to choose the best distribution for $$R$$. Here $$R_0$$ is the usual reproduction number in the absence of any distancing measures, which serves as an upper bound for $$R$$ under distancing. For example, $$\mathbb P(1<R<2)=1/4$$ would mean that we want be spending $$1/4$$ of the time with $$R$$ between $$1$$ and $$2$$. Let us suppose there is some utility function $$U(R)$$, increasing in $$R$$, that represents the benefit to society of maintaining a constant reproduction of $$R$$. Additive constants don't matter to $$U()$$, so it may be helpful to think of $$U(R)$$ negative and increasing up to $$U(R_0)=0$$. $$R=R_0$$ represents normal life in the absence of a pandemic, while negative $$U(R)$$ for $$R<R_0$$ represents the level of pain at implementing various levels of distancing/lockdown. Note that $$U(R)$$ is the utility when you fix the reproduction number at $$R$$; the possibility of varying this is what we're going to look at here and is represented by a mixture of different values of $$U()$$. For example, the strategy described in the preceding paragraph would have value $$\frac13(U(4)+2U(\frac12))$$. It seems reasonable that there is some version of the function $$U()$$ that represents reality to some extent, though of course it will be hard to agree upon any particular version of $$U()$$. We shall therefore try to see what we can deduce that doesn't depend on a precise $$U()$$, starting with observations that are more-or-less completely $$U()$$-independent, and ending with speculation that depends on a broad form of $$U()$$.
For these purposes it's more convenient to work with the growth rate $$\lambda$$, which we can consider as a random variable $$\Lambda$$ related to $$R$$ by $$R=r(\Lambda)$$. However, we shall also keep $$R$$ in the picture because it's probably more intuitive to people now as we've had experience of reproduction rates in the news. Using growth rate as the random variable, we can define $$V(\lambda)=U(r(\lambda))$$ and formulate the problem like this:
We wish to maximise
\[\mathbb E[V(\Lambda)]\]
subject to
\[\mathbb E[\Lambda]=0.\]
$$V()$$ is some increasing function and $$\Lambda\le\Lambda_0$$, where $$\Lambda_0$$ is defined by $$r(\Lambda_0)=R_0$$. This is meant to capture the time-independent aspect of the problem of medium/long term control of the virus: decide how long you want to spend at different lockdown levels, and factor out "implementation details" of how/when to switch from one level to another. Obviously in reality such "details" are very important and may in practice be a sticking point but still I think it's interesting to consider what kinds of distributions are likely to make sense.
Note that we don't consider the possibility of $$\mathbb E[\Lambda]<0$$ even though this would also result in disease suppression, because $$V()$$ is increasing so if $$-\infty<E[\Lambda]<0$$ we may as well increase $$\Lambda$$ a bit to improve $$\E[V(\Lambda)]$$ and maintain suppression. And $$\mathbb E[\Lambda]=-\infty$$ is a zero-Covid strategy (leading to extinction of the disease) which we are not considering here, so in particular we must have $$\mathbb P(\Lambda=-\infty)=\mathbb P(R=0)=0$$.
Only two values of $$R$$ are needed
Equivalently, only two values of $$\Lambda$$ are needed. Suppose that $$\Lambda$$ only takes finitely many values: $$\mathbb P(\Lambda=\lambda_i)=p_i$$ for $$i=1,\ldots, n$$, where $$p_i>0$$ and $$0