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 2.5^{6} = 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.