In my previous article I presented empirical evidence supporting the claim that “lockdowns” as of Thursday, May 7th are no longer a rational response, and suggested that they are likely politically motivated. I’d like to continue that critique with some additional data as the talking points begin to shift. I will also clarify a few points that were either unclear from my first essay or which I had intentionally left out to maintain the focus on the question of “lockdowns.”
We have moved on from the time in March and early April when it was reasonable to be concerned about PPE for nurses and the health care system’s ability to handle new cases. Since that initial panic has subsided, the public is starting to question if extreme measures to force distancing, such as school closures, are worth the damage to society. To keep the public afraid, it has become politically necessary to create a new panic, this time in the form of “spikes” of Covid cases.
The concern about a resurgence of new cases is not irrational in the slightest, indeed it is a completely valid worry. However, as I will argue, imprecise terminology – e.g. “spike” or “surge” – is confusing the public about the situation.
Given the prevalence of the virus and asymptomatic spread, cases will inevitably rise, no matter what we do. The correct question is not whether to wait until there are zero new cases, but whether the rate of new cases is sustainable for the healthcare system.
The language being used to describe the inevitable increase in cases is imprecise and allows news anchors and politicians to spin the facts to suit their own purposes. Therefore, it is prudent to look at exactly what “spikes” or a “surge” might mean objectively, and then assess whether this is actually happening.
Let’s Talk About The Exponential Function
One recurring issue, easily exploited by talking heads, is the public’s complete and utter lack of understanding of the exponential function, and poor mathematics education in general. A brief primer: a value is exponentially increasing if it is is doubling at some constant interval. Exponential growth means not only is something increasing, it’s increasing at an increasing rate. When graphed, the exponential function looks like an “upward curve”:
Note that different growth rates yield different looking graphs. However, these are effectively the same function with a different scaling factor applied.
In the beginning, we saw cases double every 4.2 – 4.4 days, depending on which model you used. That’s exponential growth and it’s very, very bad. Officials were correctly terrified at this stage. Why? Because if cases double every 4.2 days, 1 case turns into 1 million in only 20 doubling periods, or roughly twelve weeks.
2^20 = 1,048,576
And before we knew anything about Covid, we could have reasonably thought that 1 million cases would mean 1 million men, women, and children requiring special ventilators and a team of doctors to keep them alive. That would very quickly overwhelm our hospitals, so we constructed emergency field hospitals in New York to handle the overflow. Thankfully, CDC’s guidelines and the public’s effort through distancing did not allow this situation to come to pass. Many emergency field hospitals were shut down and most of them never used.
In other words, we “flattened the curve”, although it’s not quite “flat” just yet. In fact, there are three phases of the “curve” that we should expect. One interesting way to model diseases is with the Gompertz function. The first phase is exponential growth (1), the second is a linear phase (2), and the final is the logistic phase (3).
In the beginning, we see the exponential term dominating. In the middle, we see a linear growth, which is the case in the U.S. as a whole now. In the tail end, we see values approaching an asymptote. I don’t expect that many readers will want to dig into the math, so I’ll just emphasize a few key points relevant to our discussion:
- The exponential growth phase (1) is when we see the big “surge” or “spike” in number of new cases, and the disease is out of control
- The linear phase (2) sees many new cases every day, but the rate of growth is not increasing
- The logistic phase (3) sees few new cases every day, and the rate of growth is decreasing
- In every phase, there are new cases every day
That last point is really important in regard to our discussion. Even in the long-tail logistic phase at the end, there are still new cases every day. However, that number will be getting lower every day. When it’s low enough that individual cases can be traced effectively, the crisis will be over.
The U.S. is clearly now in the latter part of the linear phase, although this should really be determined by county, since there are some areas like New York that are clearly ahead of other areas, like Los Angeles. This is clear evidence that some combination of distancing, “lockdown”, and following CDC sanitation guidelines has worked. How do we know that it worked? Must we trust politicians and news anchors to tell us that it worked? No, in fact, we can use our knowledge of the exponential function to verify that fact against the publicly available data from Johns Hopkins University. Here are the numbers in the U.S. graphed by date. This graph ends at May 10th.
If you compare that to the Gompertz curve above, you can visually verify that indeed the U.S. is toward the end of the linear phase. As we can see plainly, the current situation does not show signs of exponential growth.
We can see this even more clearly by looking at the numbers in logarithmic scale. If there is exponential growth happening, we would see it show up as linear in the log scale. If you have been following the press conferences this is exactly what Dr. Birx has been pleading for people to do for months.
So what exactly does this mean? Think of it this way. The more “flat” this graph looks, the less exponential the growth is. You can see how in March it was linear, meaning the the exponential term was dominating. That’s why public officials were raising the alarm in March. We had cases doubling every 4.2-4.4 days. That translates into about an 18% increase per day, because:
1.18 ^ 4.2 ≈ 2
We had to do something to get out of exponential growth and into the linear phase. We have been in the linear phase since mid April. The daily growth of new cases was down from around 18% in March to 1.469% as of May 10. You can calculate this yourself by just taking the total reported cases for May 10th and dividing by the total reported cases for May 9th:
1366794 / 1347003
That number has been dropping since I stopped manually calculating it back around April 25th. Remember, this value is supposed to be 18% (i.e. 1.18) if we’re in exponential growth mode.
Let’s Define “Spike”
It has become a common talking point for the media to throw out, with no evidence, statements like: “COVID-19 cases are spiking all over the country”. What does this statement mean? The author is not looking at infection rates or anything, but rather commenting on one specific incident in a remote area, then generalizing it to the entire country. The statement that “cases are spiking all over the country” is not an accurate assessment of what the data indicates about what is happening at a national level.
I am not interested in politically motivated reasoning. I want to see reality with clarity and then reach conclusions based on those facts. We need more precision of language so when we say there’s a “spike” we know what we are talking about in terms of the growth rate.
So let’s try to define what a “spike”means in the context of our discussion above. What we’re really trying to get at with a word like “surge” or “spike” is – what if exponential growth returns? That would mean that the rate of growth begins to increase.
So what would a “spike” mean given this more precise definition? Once again, it helps to look at the data in logarithmic scale. Here are the U.S. new cases viewed in log scale. As before, if this rate is increasing, we expect that this will be a linear function. If it isn’t increasing, then we’re not experiencing a “surge” or “spike” of new cases. A “flat” graph here is good because that means that the rate of growth is the same.
The above graphs show the new case in the U.S. per day through May 10th. Newer data might be available depending on when you’re reading this. You can view the data yourself in both modes – standard and logarithmic – and compare countries with this nice tool.
As of May 10th, which is the newest data I had available when I wrote this, we have irrefutable evidence that exponential growth is not happening anymore – at least, when we look at a national level. The U.S. has definitely been in the linear phase for some time. It’s not a matter of opinion, this is an empirical fact that we can take as a given. The doctors know this. The CDC knows this. Anybody with access to the internet should know this. There is no new surge, and cases are not “spiking across the country.”
And I’m going to repeat myself because I suspect people don’t get this: this doesn’t mean there are no new cases – it just means that the rate of new cases is stable.
But Could There Be A Surge?
The idea of a new exponential growth outbreak happening is a reasonable concern, as counties begin to lift restrictions and businesses start to open. After all, we are changing the rules, and any change creates the possibility of circumstances changing. However, it is not inevitable, as news media is reporting – and it is certainly not already happening at a national level, as we just verified. Whether it will in the future is purely speculative.
But let’s speculate for a moment, shall we? Surely we’ve earned it after all that math.
The return to exponential growth only seems likely to occur in small pockets – e.g., a county that never really experienced the original “surge” in the first place. Every city in the country already has distancing measures in place and the public knows the drill. “Lockdown” has not been shown to be directly responsible for controlling infection rates, so I don’t expect that removing shelter orders will make a difference alone. Widespread exponential growth on a national scale seems like it could only happen if all restrictions and sanitization efforts were immediately lifted and everyone stopped following those protocols overnight.
It seems more likely that the linear phase would be extended as restrictions are lifted. New cases would remain at a known, sustainable pace, and people would be able to get on with their lives and start earning a paycheck again. People in vulnerable groups, like nursing homes, would be “locked down” to avoid impacting those most likely to have a severe case. Basically, we could balance out the economic damage by extending the time it would take before the rate of new infections starts to drop.
We know from our discussion above that the really dangerous part of this was the exponential growth phase – which ended sometime in mid April. During this phase, we could expect to see our 1 case tur to 1 million in just twelve weeks. That’s a lot of new sick people in a short span of time. But let’s also not forget that there will always be new cases no matter how we proceed. This is a different statement from saying that everyone will get eventually infected. The important thing is not if there are new cases, but whether they are occurring at an increasing (i.e. exponential) rate – and, of course, how well we can treat them. If we get into a situation where the number of new cases is actually declining, that means we’re out of the linear phase and into the long-tail logistic phase – but there will still be new cases, even in the logistic phase. Because that’s how asymptotes work.
The reason that’s important is that if we know we’re in a linear phase, then we know with some certainty we will have at most X new cases this week, which means we need Y masks and gloves, etc. It yields a situation where we can make accurate predictions about load on hospitals which means that hospitals can be open for business and we don’t need to take extreme measures to reduce incoming patient rates, like an indefinite nation-wide “lockdown.”
Another issue with saying “cases are spiking all over the country” is that the implication is that we should shut down at a national level when certain areas have an outbreak. The U.S. is not one single dense urban area like Hong Kong, nor does it consist mostly of dense urban centers in close proximity, like South Korea. Quite the opposite, in fact – at least half of the counties in the U.S. are rural and semi-rural regions, and except for parts of the east coast, U.S. cities aren’t clustered together. Should we shut down NYC again because some town in Iowa had an outbreak?
The public is being confused by these messages about “spikes” and national implications. I was on a video chat with a friend in another state the other day and mentioned that I had gone hiking with a friend – totally okay, in my state. “It doesn’t sound like you’re socially distancing,” he chided, and implied that I was putting his family in danger. I live in Texas and he’s in the Midwest, literally 1,000 miles away. The fear-mongering about threats of “spikes” leads people to believe in magical thinking, that what happens 1,000 miles away will directly endanger their families.
Regional planning makes much more sense now, given the geography of the U.S. and the fact that we have very limited movement between urban areas now. If counties do enter into exponential growth mode, those counties would need to be controlled with a heavier hand.
In other words, “all over the country” should be replaced with, “in specific, known parts of the country.”
A Note About Political Motivations
The U.S. has become a deeply divided partisan political environment over the last five to ten years. Conspiratorial thinking and motivated reasoning becomes normalized under such circumstances, as each partisan bloc tries to smear the other to steal swing voters in the next election. Partisan rhetoric replaces principles, and independent thinking becomes misconstrued as just more political jockeying. The problem is exacerbated by media firms turning to engagement marketing models as traditional sources of revenue, such as subscriptions, continue to decline. The sociological effects of these trends are increasingly irrational public debate which inhibits civic decision-making.
I do not personally have any such political motivations. I have been inspired to write these articles by precisely the opposite of a political motive: I want to bring discourse back to objective reality. Making decisions about complex issues requires some consensus about what is real and what is rhetorical, i.e. “spin.” Our consensus reality has become bogged down by sophistry and engagement-driven sound-byte journalism, so our only choice is to reconstruct consensus reality from the source.
A good starting point for objective reality is data, because it is boring and pleasantly resistant to sensationalization efforts.