Are the lockdowns responsible for declines in infection growth?

Recently, infection and death rates in Europe and the United States appear to be leveling off and even dropping.  Since this was the point of the massive social distancing measures the whole industrialized world has been taking, the natural interpretation of this would be: the measures we took are working and we are beginning to see the results.

However, some people have claimed that the leveling off of deaths is not due to strict isolation measures, but something the disease was going to do anyway.  What we are seeing, this theory claims, is a natural peak in the disease.  Lockdown measures may have slightly reduced the total number of deaths, but the behavior of the curve was going to follow the current path we are seeing anyway, more or less.  The clear implication of this theory, if it is correct, is that we should end the strict social distancing measures and the disease will dwindle away on its own.

Can we determine which interpretation of the facts fits best?  I think we can, fairly simply.

What the Prevailing Theory Expects

Given the theory that the disease will act in the standard way in which one expects an epidemic to act, what we should see is that the infection grows exponentially at first, but then rapidly shifts its growth rates after social distancing measures are put into place, in every place in which these measures are enacted.  We can visualize this infection curve using an online pandemic calculator, available here: http://gabgoh.github.io/COVID/index.html

To model our scenario, I have put in a disease with an R(0) of 3.  This is midway between earlier estimates of Covid-19's R(0), which was around 2.4, with later estimates which have put it as high as 3.87.  Then I set an intervention date about a month into the course of the disease which has the effect of reducing the R(0) to around 1: the threshold below which a disease will begin to die out.  Here's what that looked like:


Note that the resulting curve is composed of two curves, which I've marked in red and in blue.  On the left of the intervention, there is a standard exponential curve, concave up.  Right at the point of the intervention, it rapidly switches to concave down.  It still rises for a bit, but it has a shallow hump which then trails off gradually afterwards.

The exact shape of the right-hand side of the curve depends a lot on what you set the R(0) to be after the intervention.  Depending on how effective your intervention strategy is, the daily infections can die off either rather steeply, or rather slowly.  I've heard, for example, an estimation of current, post-lockdown R(0) being put at 0.62.  This is what the pandemic calculator looks like with that number instead:



I encourage my readers to go to this site and play around with the numbers yourself--if nothing else, this should cause you to have better sympathy for the shifting numbers coming from the IHME projections, because you will quickly see that small changes to the R(0) (which is the degree to which people are spreading the virus around) can have quite large changes to the final infected number.

What's Actually Happening

So now let's look at reality instead of this model.  Do we see this same sort of results in those countries that have had significant outbreaks, and then initiated strict societal interventions in order to flatten the curve?

In order to look at this, we'll pick some countries from the Worldometer site.  In order not to have our results confused by poor testing (which has been a problem in many countries), we are going to look at daily death rates rather than daily infection rates.  This curve should be the same shape as the daily infected curve, just smaller and with some time lag, since only a fraction of infected will die and since it takes time for people to progress from having the infection to dying.

Here is Spain's daily death chart:

You can see clearly the exponential growth on the left side and a clear, abrupt transition to a smoothly curved peak and gradual decay on the right.  The date of the transition appears to be somewhere around March 24th.

Here's Italy's chart:

Again, we can clearly see exponential growth on the left abruptly transitioning to a shallow hump and gradual decline to the right.

What about Germany?  In this case, the shape is less clear:

In this case, the exponential growth on the left is obvious, but it's not so obvious what's happening on the right.  Here we should realize that Germany's curve starts later than Spain's and Italy's.  The pandemic apparently reached Germany later than it did Italy and Spain, so we're not seeing the peak and trail-off yet in the death rates.  However, let's cheat with Germany and look at the daily infection rates chart--we should be able to see the effects of lockdown earlier with these numbers because of the time lag between infection and death.  Germany has been doing a lot better in testing than a lot of other countries, so maybe we can trust that their infection rate numbers are fairly reliable:

Nice!  It actually looks just like a continued form of the deaths chart from above.  More evidence, I think, that Germany's testing has been far more representative of actual infection rates than other countries' has been.

What about South Korea?  Here we have a problem that South Korea has been so on top of the pandemic, from the very beginning, that their daily death rate chart doesn't have enough data to form a recognizable curve: they just haven't had enough people die.  This is excellent, but it does mean we can't use their chart for this analysis.  However, since their epidemic control has been driven by extensive testing, we can probably do what we did for Germany and use their daily infection rates, again probably with a good degree of confidence:

OK, the smaller dataset does make the curve more patchy, but it still fits pretty well: concave up on the left and a trail-off on the right.  It does appear to me that the drop-off on the right for South Korea is more dramatic than the trail-offs we've been seeing in Europe.  This would fit with their lower overall death rate, though: the fact is, South Korea has simply had a better handle on the epidemic from the beginning.

So lastly, how is the United States doing?  First, it should be pointed out that, as opposed to all of the countries we've listed so far, the United States hasn't had one set of lockdown measures.  Different states implemented different lockdown measures at different time.  We should expect to see a bit of overlapping curve flattening from the time periods when different states were probably experiencing different disease growth rates.  Second, the United States is clearly behind Italy and Spain in the pandemic timeline, so we're likely to have small amounts of data for the right side of the curve.  Those caveats being given, here's what we see:

Fits pretty well, I'd say, given the caveats above.  Given the massive testing problems the U.S. had early on, I'm reluctant to use the daily infection rate curve, but given that our testing has been better recently, maybe we can get a better sense at least of what the right side of the curve looks like?

Still unclear, I'd say; we're still too early on.  However, it certainly doesn't invalidate the theory; I'd say it weakly confirms it.

Timeline of the Inflection Points

Now that we've seen that the shape of the curves we see in real life are matching quite well with the predicted curves for the standard theory, can we also ask the question, does the timing of the curve flattening correspond with the lockdowns?  Different countries imposed societal lockdowns at different times; if they are what is responsible for the curve flattening, we should expect to see some correlation in the timelines.

For the nations that we have looked at so far, here is a table showing the dates for which those countries imposed a nation-wide lockdown (or in South Korea's case, a nation-wide banning of large public gatherings), side-by-side with the dates at which I am seeing an inflection curve in their charts:

	Lockdown Imposed	Date of Inflection
South Korea	Feb. 21st	Feb. 27th (for infections)
Italy	March 9th	March 19th
Spain	March 14th	March 24th
Germany	March 22nd	April 2nd
United States	March 22nd (New York)	~April 4th

To me, this timeline is compelling; I don't see how anyone could look at this data and not conclude that we are seeing the results of societal changes in the infection and death rates at this point.

The Alternative Theory: The Disease is Peaking by Itself

However, let's suppose the above is not found to be convincing.  What about the alternative theory?  What would we expect to see if the disease is playing itself out, without social distancing being a major factor in the decline of the disease? 

You can see what a standard epidemic disease curve looks like by using the epidemic calculator (making the intervention meaningless by setting the post-intervention R(0) to the same as the pre-intervention R(0)):

Notice that the left and right hand sides of the curve are symmetrical: it declines as rapidly as it attacks, once the population has been saturated.  This shape does show up in real-life as well; we see this sort of shape in uncontrolled epidemics all the time.  Here's an example graph from some '70s measles outbreaks, for example:

This outbreak came in a rapid sequence of waves (measles is *extremely* infectious), which had a roughly symmetrical look to them.  Here's another example which is a collection of epidemic curves from the SARS outbreak:

https://www.who.int/csr/sars/epicurve/epiindex/en/

Here I notice that the right hand side of the curve in these charts is typically as steep to decline or steeper than is the left hand attack portion of the curve.

I find the lack of "spikiness" of the real data we are seeing difficult to square with how epidemics of very infectious diseases look like.  I don't know how proponents of this theory explain an exponential attack and a much less exponential decay.

Total Infection Counts

The real problem with this theory, though, is the total infection counts, as a percentage of the population.  If the real factor in limiting the continued growth of the disease were that it was reaching inherent limits of the population and herd immunity were kicking in, then each nation should see roughly the same total percentage of their population infected by the end.  Herd immunity works by a certain percentage of the population becoming immune, thus crippling the disease's ability to spread rapidly.

So what sort of total infection percentages are we looking at here?

South Korea has a population of 52 million people.  They have had 217 deaths so far, and their curve is completely flattened.  That's a total death rate of 0.0004% of the population.

Italy has a population of about 60 million people.  They're not done with their curve yet, but they've had 20,000 deaths so far . . . maybe we'll guess 25,000 deaths before the curve fully flattens.  That's a total death rate of 0.042% of the population.  That's over 100 times as many people as a percentage of their population than South Korea.

Spain has a population of about 47 million people.  They're also not done with their curve yet, but they look on track to total maybe about 20,000 deaths.  That's a total death rate of 0.043% . . . very similar to Italy's.

Germany has a population of about 83 million people.  They're even further behind in the timeline than Spain and Italy, but with only 3000 deaths so far, maybe we can project a full doubling and say 6000 total deaths by the time of curve flattening.  That's a total death rate of 0.0072% of the population.  This is less than 1/5th of the total numbers Italy and Spain are going towards, but more than 15 times the number South Korea is going to end up with.

The United States has a population of about 327 million people.  Again, we're back in the timeline a bit too far to project accurately, but applying the same logic as I did with Germany, we'd end up with a total of around 45,000 deaths  (I know 60,000 or so is the current best estimate, but I'm just trying to be consistent with what I did for Germany).  This would work out to a total death rate of 0.014%, which would put us somewhere in between Germany on the one hand and Spain and Italy on the other hand for total percentage infected.

These numbers are all impossible to explain by the theory that the disease is simply spreading naturally and hitting its natural peak due to herd immunity building in all the countries of the world in which it is spreading.  Why should South Korea hit that natural peak 100 times faster than Italy did?  Why should Germany hit that peak 15 times faster than Italy but only twice as fast as the United States?

Conclusion

The conclusion here is quite clear: Covid-19 is currently being limited by drastic social distancing measures (in the case of most of the world) or a combination of early testing and case management plus less drastic social distancing measures (in the case of South Korea and some others).  There is no way for naturally acquired herd immunity to explain the current decrease in rates of infection and death that we are seeing, but it is easy to explain this using the standard, accepted theory of the disease spread.  The highly different percentages of the total population that will die is therefore strictly due to the difference in promptness which these different nations implemented effective disease control measures.