Mr. Brightside Peeks Over the Plateau

Something different seems to be happening.

First, some background. Corona’s R0 — the virus’s basic reproduction rate in the wild when unrestrained by mitigation — is around 2.5: each infected person spreads the disease to an average of 2.5 other people. Assume that an infected person remains contagious for around 10 days. Divide the contagion interval by the R0: 10/2.5 = 4 days: that’s the basic doubling rate — the number of days it takes for the total number of people in the population who have been infected to double. Doubling the total number of corona diagnoses every 4 days: that’s what the US upswing looked like in early April, before sheltering in place and social distancing were widely enacted.

The goal of mitigation is to reduce the effective reproduction rate, or Rt. When it’s at 1, then each infected person infects one other person before recovering or dying. When Rt=1, the doubling rate is 10/1 or 10 days — just equal to the 10-day average interval of contagion. So as newly infected people are added to the total, the same number of previously infected people stop being contagious. Viral spread through the population becomes a zero-sum game: that’s the plateau. US mortality hit the 10-day doubling plateau around the middle of April. And there it stayed as Trump and the Republican governors agitated for liberation and the citizenry started getting antsy. This lapse of discipline seemed to portend disaster, the epidemic jumping from steady state to accelerated contagion, illness, hospitalization, death.

Here’s the paradox: even as the country has begun opening up again, the numbers have begun looking better.

Rates of confirmed diagnoses can be misleading, because they’re contingent on how many tests are being administered. Deaths are harder to manipulate. While there is a lot of variation, the average daily death toll appears to have decreased, from around 2,000 through most of April to around 1,500 during the first half of May. Over the past week deaths have increased less than 2% per day. That rate of increase, compounded daily, projects to a doubling rate of 35 days (102% x 1035 days = 200%) and an Rt of 0.3 (R0/35 days). This is the way it’s supposed to work once the Rt drops below 1: infections and deaths come down the other side of the plateau until eventually the epidemic extinguishes itself.

At this rate, how long would it take for the wave to wash itself out, with new infections and deaths approaching zero? Here’s a quick-and-dirty Simulation A:

  • Assumptions at baseline: contagion interval = 10 days, Rt = 0.3, daily new infection rate = 160,000.
  • In 10 days, those 160,000 cases will themselves have infected 160K x 0.3 = 48K people. But by then those original 160K cases will no longer be contagious, resulting in 160K – 48K = 112K fewer active cases in the population.
  • 20 days from now, those 48K newly infected people will in turn have passed the virus on to 48 x 0.3 = 14.4K people.
  • By 30 days new cases will have dropped to 14.4K x 0.3 = 4.3K daily.
  • 40 days from now — around the beginning of July — daily new cases will be running at 4.3K x 0.3 = 1.3K. Given a mortality rate of 1%, the death toll would have dropped to 13 per day nationwide.

Under Scenario A, maybe 20,000 people would die of corona in the US over the next six weeks, but that’s a lot better than the 80,000 who have died over the past six weeks. And that would be the end of it — the first wave would have washed itself out. These are the sorts of projections that the IHME has been generating, though they repeatedly have had to adjust their numbers upward as the plateau persisted. Stagewise reopening guidelines are predicated on new infections and deaths decreasing, making it possible to relax incrementally and safely the rigorous lockdown levels of mitigation.

So what happens now, with Trump and company undermining their own epidemiologists’ recommendations and with so many parts of the country jumping the gun? It depends on the extent to which mitigation efforts are relaxed. Relatively strict enforcement has dropped the virus’s effective reproduction rate from 2.5 to around 0.3. If Wild West anything-goes laxity prevails and the reproduction rate jumps above 1, then infection rates and deaths will climb again and we’ll be back where we started. Suppose we split the difference, and the Rt settles in at around 0.8. Simulation B:

  • Over the next 10 days, today’s 160,000 newly infected cases will have infected 160K x 0.8 = 128K people.
  • By the beginning of July (40 days from now): 160K x 0 .84 = 65.5K new cases daily.
  • By around the end of the year (210 days from now) there would be 160K x 0.821 = 1.4K new cases and 14 new deaths daily. The first wave would be extinguished.

In Scenario B, relaxing remediation from strict to moderate compliance would subject our population to an additional  8 months of illness, but at steadily decreasing rates. Over those 8 months maybe another 7 million will be infected and another 70 thousand people will die before the wave is extinguished. Scenario B’s toll in human suffering and death would be considerably worse than Scenario A, but it’s still a lot better than the unbridled contagion that preceded lockdown.

By the time the wave of contagion dissipates in Scenario B, around 5 percent of the national population will have been infected and 160,000 will have died — that turns out to be pretty darned close to the current IHME projection of around 150,000 covid deaths. It’s tragic to be sure, but nothing like the casualties that would mount up if contagion were to spread unchecked until herd immunity is reached.  The results modeled in Scenario B could conceivably be achieved even without deploying the more targeted armamentarium of extensive diagnostic testing and case isolation and social contact tracing. Any success in implementing those methods of abatement could further reduce the effective reproduction rate and accelerate the wave’s dissipation.

Is this moderate-case scenario realistic? Looking at the images on the Internet you’d think that pent-up demand for shopping and haircuts and barhopping and beachgoing will overshoot pre-pandemic baseline levels of in-your-face social proximity. But that’s just clickbait and agitprop. Most people are wary; most expect to maintain social distancing as best they can even as they go back to school and work; most of the old and the health-compromised expect to keep their distance until a vaccine becomes available.

There are regional variations. It’s widely reported that deaths have been going down in the hardest-hit states of NY, NJ, and MA, while for the remaining states the upward trend persists. But the upticks are relatively small, and as a percentage of total deaths they’re declining. In my state of North Carolina daily deaths have been increasing by 3.2%, which projects to a 22-day doubling rate and an Rt of 0.45. That means the wave will recede more slowly, allowing for a smaller margin for error in easing the social distancing protocols. On the other hand, the wave isn’t as tall here — current per-capita rates of new diagnoses and deaths in NC are lower than the national average — so all else equal it wouldn’t take as long to get the NC levels back down to sea level.

The next 3 weeks should tell the tale, as the effects of increased mobility and social proximity and resumed business as usual propagate through the population and the viral contagion cycle runs its course.


One thought on “Mr. Brightside Peeks Over the Plateau

  1. News from 21 May:

    Sweden has revealed that despite adopting more relaxed measures to control coronavirus, only 7.3% of people in Stockholm had developed the antibodies needed to fight the disease by late April. The figure, which Sweden’s Public Health Authority confirmed to CNN, is roughly similar to other countries that have data and well below the 70-90% needed to create “herd immunity” in a population.


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