There are a lot of models of disease transmission out there, I hope to do multiple blog posts on them! But for now I want to point out a simple “back of envelope” calculation may give as good an estimate on the (scary) number of deaths as far more complicated models. All you have to do is do some simple arithmetic combined with the notion of herd immunity . Herd immunity is a fundamental concept in epidemiology, it means that enough people are immune so that the disease will no longer grow exponentially. I promise to do a post on herd immunity and how you can calculate at what percentage of the population herd immunity happens. (Turns out, though it uses only simple arithmetic, you can prove a theorem that lets you calculate at what percentage of the population herd immunity occurs. Because it only uses simple arithmetic plus some insightful thinking, I think that the proof of the “threshold theorem” as it is called, is a triumph of the mathematical way of thinking.)
Hint of things to come-because we use the threshold theorem number here: First off, recall that the fundamental number r0 is how many people an infected person will infect on average. The higher r0, the faster exponential growth (see my Lilly Pond blog entry) occurs. The threshold theorem says herd immunity occurs when the percentage of immune people is > 1-(1/r0). This means that if you assume Covid-19 has an r0 of 3, herd immunity occurs when more than 2/3rds of the population is immune.
As I write this on May 3, 2020, there are 65,735 confirmed deaths. Absent a proven therapy that reduces infection rates or a vaccine, the likeliest scenario is that there will continue to be waves upon waves of infection – well until enough people have had the disease so that we reach herd immunity. I believe with the current level of testing, no truly effective drugs and the lack of effective contact tracing, there is simply no escaping this. And, unfortunately, to be blunt, it’s not something one hears often enough on the news. There are about 330 million people in the United States. Herd immunity, using an r0 of 3 occurs (roughly) when 220 million people are immune.
Now we get to the back of envelope calculation. The CDC reports as of May 3, 2020 there are about 1.12 million cases. Given how bad testing is, this is horribly off. When one does sampling on, for example, pregnant women entering the hospital in a city like NYC or a random sample of a hard hit population, one gets estimates of between 10% and 20% of the population has been infected. But since that percentage is likely true only in hard hit areas, it seems reasonable to me to assume that between 5 and 10% of the population has been infected. Assuming as I mentioned that:
- No drug is discovered that lowers the risk of death significantly
- That everyone who gets infected is now immune (which hasn’t yet been proven – just to be even scarier)
- Herd immunity occurs at 67%
we will have between 6.7x and 13.4x (67/10, 67/5) more deaths. That gives us a back of envelope calculation of between 440,000 and 880,00 deaths. I’m not a religious person, but sometimes the phrase “God help us” slips out.
Added: someone asked me after reading a first draft whether there is anything that could change my analysis. The answer is yes, as I mentioned above in passing, three things come to mind. The most likely, is that we find a drug that significantly lowers the risk of death. The next most likely is we start doing both quick testing and effective (and quick) contact tracing. That could by itself bring the r0 in the modified population (which is usually called rt) to be < 1! Finally, because each successive wave takes time, if we lengthen the time between wave by social distancing, using masks etc., we gain more time to develop a vaccine. My scenario makes the assumption that no vaccine exists before herd immunity develops and r0 stays at 3, good testing and contact tracing lowers that significantly! But to reiterate, if we end up relying only on the development of herd immunity, absent a drug to lower the risk of death significantly, that means way way too many deaths.)
Further Added: clarified that testing and contact tracing changes the original notion of r0 in the modified population to what is usually called rt.