# How Long Will the COVID-19 Epidemic Last? Napkin Calculation

Reference:  (DIVA, pdf) The reproductive number of COVID-19 is higher compared to SARS coronavirus. Contained within the article is a permalink:  http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-168415 . Quoting,

“Teaser:Our review found the average Ro for 2019-nCoV to be 3.28…”

We’re going to do a napkin calculation, closely related to what scientists call a power-of-10 job. There is no claim of accuracy.  Even if we wanted to be sophisticated, there are things in the way of accuracy. The basic reproduction number, R_o, will not be known with any accuracy until this is all history. It’s one of the weaknesses of epidemiology, better at analyzing the past than predicting the future. Nor is there any reason to assume that R_o will be constant with the seasons, and subtle seasonal changes in social behavior.

For this napkin calculation, assume that

•  The number of people in the U.S. who are infected is actually 10 times the reported figure of 700 as of 6 a.m on 3/10/2020. The number is 7000.
• Having COVID-19 confers long lasting immunity.
• R_o, which applies to a population that has never seen COVID-19, is constant.
• The epidemic is not stopped by other means, such as quarantine, drugs, or vaccination.
• R_o = 3.78, a little higher than the reference. It is, after all, an estimate.

We want to know R, which unlike R_o takes into account herd immunity. When a  certain percentage, a majority of the population, has had COVID, with or without symptoms, R drops below 1.  Here we go:

• U.S. population: 327 million
• Percentage of the population to have experienced COVID, for R to drop below 1:  73.5% = 240 million.
• The generation time is the average time it takes an infected individual to infect others. Example: If a person is infected on March  1, and infects 4 more people before recovery, at March 7, 14, 21, and 28, the generation time is 17.5 days.
• If we assume that COVID keeps infecting with R=3.78,  that implies, starting with 7000 infected, 8 generations to reach 240 million.
• The above is not correct; R decreases as time goes on, so more than 8 generations are required. But for simplicity, to keep it a napkin calculation, let’s go with it.
• Assume the generations time is three weeks.
• 8 generations = 24 weeks.
• Now apply a correction for the gradual decrease of R as the number of people who haven’t had it declines: Multiply the above by the magic number e= 2.71828.
• This gives 65 weeks till the reproduction number drops below 1.

In  the world of this very rough estimate, the epidemic takes a downturn 65 weeks from present. It assumes  no modifications by medicine and public health, and a lot of mostly avoidable human suffering.

But there you have it. If this were the pre-scientific world of past plagues, and the pathogen had characteristics similar to COVID-19, it would be close to the truth.

Now you can panic. But do it in style.