The question isn’t how many people have been infected since the beginning of the pandemic. The question is: how many are currently infected by the virus, currently carrying active virus inside their bodies?

The Assumptions

On average, a person infected with covid remains infected for 2 weeks. So the total number of infected people on any given day is the sum of those who have been newly infected each day for the prior 14 days.

The daily covid death count is the most accurate indicator of the spread of the virus. However, deaths lag behind infections by about three weeks; i.e., the average person who dies from covid was infected 3 weeks prior to dying.

Somewhere around 0.65 percent of those who’ve been infected with covid eventually die from the disease. So, divide the most recent 14-day death count by .0065 to estimate the number of people infected 21 days ago.

The daily case count in the US underestimates the daily infections. However, over the past month the change in daily case counts has fairly accurately mapped onto the change in estimated daily infections.

The Math

Over the past two weeks, November 6-20, 18,100 Americans died of covid. So, as of three weeks ago, there would have been 18,100/.0065 = 2.8 million Americans actively infected by the virus.

Over the past two weeks (11/6 – 11/20), the 14-day total new case count was 2,212K. Over the preceding three-week interval (10/15 – 10/29) the 14-day total new case count was 992K. So, the current case count is 2212/992 = 2.23 times what it was two weeks ago. Assuming proportionality of changes in case counts to changes in new infections, then the total number of infected people today is 2.23 times the number of people who were infected three weeks ago.

2.8 million infected 3 weeks ago x 2.23 = 6.24 million Americans are currently infected by covid.

The total US population is 328 million. 6.24/328 = 0.019. So, about two percent of the American population is currently infected by live covid virus.

Let’s suppose that further clinical trials validate the preliminary numbers, and the new covid vaccine turns out to be 90 percent effective. I.e., the virus will infect only 10 percent of those who would have been infected had they not been vaccinated.

That’s great for those who get the shot, which would almost surely include me. What would be the vaccine’s impact on the pandemic if not everyone gets vaccinated?

Right now around half a percent of the US population is currently covid-infected and contagious. How long would it take for the virus, left unchecked, to spread through the entire population? Covid’s estimated reproduction rate, or R0, is around 2.5; i.e., if the virus is left unchecked by preventive measures, then on average every 4 people who’ve been infected with the virus will infect 4 x 2.5 = 10 other people before they’re no longer contagious. Those 10 newly infected people will in turn infect 25 more people, and so on — a geometrically increasing rate of contagion. People who’ve been infected remain contagious for around 10 days, so every 10 days the percentage of the population infected would increase to 2.5 times its prior rate. How many ten-day infection cycles would it take for the virus to infect everyone in the population?

0.5% x 2.5⁶ = 122%

After 6 ten-day cycles — two months — of unabated contagion, everyone in the US would have been infected. Herd immunity would be “achieved” by the end of January 2021 via uncontrolled spread of the disease.

The virus isn’t totally out of control; various preventive measures — social distancing, masking up, etc. — are slowing the spread. Lately there’s been a spike in cases and a slower rise in deaths. Rt — the effective contagion rate — is now around 1.5 in the US. At this rate, all Americans will have been covid-infected after 10 10-week cycles, or 100 days. Herd immunity by mid-April 2021 — right around the time the vaccine would become widely available. Presumably behavioral restraints will tighten down and the rate of contagion will restabilize before that happens.

An effective vaccine can eventually lead to acquired herd immunity if the Rt drops below 1. Suppose a 90%-effective vaccination is approved and made widely available, while the Rt stays at its current level of 1.5. For those who take the vaccine, the Rt drops by 90%, to 0.15; for those who don’t take the vaccine, the Rt remains at 1.5. What percentage of the population would need to be vaccinated in order to bring the Rt below 1?

((1.5 x .1) x V) + (1.5 x (1 – V) < 1   –>   V > 0.33

I.e., at least a third of the population would need to be vaccinated in order to bring the effective reproduction rate below 1, eventually extinguishing community spread of the virus through acquired herd immunity. The higher the vaccination rate rises above 33 percent, the faster herd immunity can be achieved and the pandemic can be quashed.

What if, once the vaccine becomes available, people stop social distancing and wearing masks and so on? Then the Rt would return to the R0, or 2.5. Assuming 90% vaccine effectiveness and no behavioral constraints on contagion, how much of the population must be vaccinated in order to bring Rt below 1?

((2.5 x .1) x V) + (2.5 x (1 – V) < 1   –>   V > 0.67

I.e., at least two-thirds of the population would need to be vaccinated in order to bring the effective reproduction rate below 1, eventually extinguishing community viral spread through acquired herd immunity.

Bringing the Rt below 1 will be important not only for those who don’t get vaccinated. Antibodies degrade pretty rapidly over time, and so will the effectiveness of any individual’s acquired immunity via vaccination. If the virus continues to percolate through the population, people will need to be re-immunized periodically, perhaps as often as every 4 months, forever. If enough people get the shot, then the intervals between shots would be extended. Maybe, after a few rounds of widespread vaccination over the course of a couple of years, the virus, which seems not to mutate very rapidly, will effectively be extinguished from the population.