# COVID Thoughts: Is 50 cases/100,000 over 28 days reasonable?

~675 words, ~4 min reading time

Disclaimer: I’m not an epidemiologist. However, economists are pretty good with mathematical models, and I’m sticking to some of the simplest epidemiology models here.

As I write this, cases in Ohio are tanking. Our 7 day average for cases is literally 80% below its high. Hospitalizations are also tumbling. Things are getting better.

But, the CDC says we should continue to be cautious? Why? Because we’ve not gotten below 50 cases/100,000 population over a 28 day period. This got me thinking: where did this standard come from? Is it actually reasonable?

The reality is this: since COVID is going to be endemic, it’s not going away. Immunity isn’t perfect (whether from previous infection or from vaccination), and it doesn’t last forever (again, whether from previous infection or from vaccination). So, there is always going to be some underlying number of people that will be infected with COVID. So, is that number less than 50 per 100,000?

We can try to answer this question with a simple SIR model with a few tweaks. I set the model up with a population of 100,000. I started with 1 infected person, and R0 of 8 – which is on the low end of what I’ve heard estimated for the Omicron variant. I set a 14 day average recovery time (so, 1/14 infected people get over their infection each day), and a 180 day average immunity time (so, 1/180 previously immune people become susceptible each day). To keep things simple, I assumed the virus didn’t kill anyone. (Since I was looking for a “per 100K” number, having people die would make the math too hard.) Then, I had the thing simulate 2 years.

At the end of 2 years, the virus had basically stabilized. We’ll call this the “steady-state”.

And 6,314 people were infected. So, each day, 1/14 of those recovered and the same number of people took their place – that’s 451 new infections each day, which adds up to that 6,314/100,000 in a 2 week period – or 12,628/100,000 in a 28 day period.

So, how are we going to hit 50 cases/100K population?

Let’s think about this a bit more. Naturally, we could just stop testing. Since cases are based on positive tests, not actual infections, if we just test < 0.5% of cases, we can hit 50/100K cases over a 28 day period.

What about vaccination? Here, I assumed natural infection and natural immunity. Can vaccination get us to 50 infections/100k as a stable endemic equilibrium?

Handling this was a bit tricky. My first inclination was to assume some kind of vaccination rate and level of vaccine efficacy and booster schedule. Then, I realized that was adding a lot of parameters when there was a simpler path forward.

Let’s just look at the ideal case. Suppose that there’s a segment of the population that keeps up on their boosters and that the vaccine is 100% perfect. (Obviously not realistic! But, I’m just setting a benchmark here!) In that case, the R0 of 8 applies if the population is all unvaccinated. However, the presence of people with immunity drives down the actual ability of the virus to spread.

Fairly simple math shows that an immunity rate of 87.5%+ results in the virus being slowly eradicated. So, if we had 87.5% of the population vaccinated with a perfect vaccine, then the virus would go away completely. Anything less, and it will linger, but be rare. So, let’s try some vaccination rates.

In the US, our vaccination rate is about 65% for “fully vaccinated” – that would result in a stable number of cases around 1,623/100k – with a two week recovery period, that means 3,246/100k new cases every 28 days. About 60 x the CDC target. What vaccination rate would be need to stabilize with peak infections around 50/100k over a 28 day period? A vaccination rate of 87.16% – if the vaccine is perfect. So, LESS THAN 0.5% away from the rate needed for eradication.

Of course, if the vaccine isn’t totally effective then the proportion of the population vaccinated would need to be higher to allow for the reality that the vaccine won’t be effective for everyone – and if the vaccine isn’t at least 87.16% effective, then 50 infections/100k over a 28 day period is literally impossible. (A quick Google search found an NEJM article claiming that one vaccine they tested was ~70% effective against omicron.)

All to say: I’d really like to see how the CDC got this 50/100K case number as a target, as it doesn’t even seem possible to me, unless we just give up on testing a large proportion of symptomatic cases.