Acquired Herd Immunity Via Vaccination

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.

Covid Seroprevalence Population Survey in Slovakia

Over the weekend of 31 Oct – 1 Nov, Slovakia conducted antigen diagnostic tests on 3.65 million people — two-thirds of the national population. Of those tested, 1.06 percent came up positive.

The methodology deployed in this study has come under criticism, but to my knowledge it’s the only such nationwide population survey that’s been conducted anywhere in the world. As such, it provides an opportunity to evaluate my quick-and dirty algorithm for converting death rates into infection rates.

Here we go:

Death is a lagging indicator of infection, each death resulting from an infection that would have been contracted 2 or 3 weeks earlier.

The fatality rate for covid, inferred from various seroprevalence surveys, is around 0.7 percent in the US. The fatality rate is age-dependent: nations with older populations will have a fatality rate higher than 0.7%. The median age of the US population is 38 years; that of Slovakia, 41 years — 3 years older than the US. So, using the data-based conversion ratio I’ve devised based on age-adjusted mortality data, the Slovakian covid fatality rate  = 0.7% x 1.13 = 0.9 percent.

In order to estimate the number of people infected over a given two-day testing interval, I’d need to work backward from the number of people who died 2 or 3 weeks later. Assume that, on average, a person who’s been infected with covid will test positive for 14 days. So, those who tested positive over the 2-day weekend would comprise a cohort of people who’d been infected up to 15 days before the tests were administered. Therefore,  in using deaths as a lagging indicator of infections, I’d need a 15-day count of covid deaths recorded two to four weeks following the antigen survey.

It’s been only three days since the antigen survey was conducted; therefore I have to project future death rates based on current rates and recent trends, then work backward from there to estimate infections for this past weekend.

In Slovakia, 43 people died of covid during the 15 days between October 18 and November 2. During that same 15-day interval, the number of covid cases increased by about 70 percent from the preceding 15-day interval. Assuming case counts are correlated strongly with infection rates, then we could project a 15-day death count starting 2 weeks from now to be around three times the most recent count: 43 x 3 = 129 deaths.

To estimate infections from deaths, divide deaths by the fatality rate. For Slovakia, it’s 129/.009 = 14,300 infections. That’s the estimated number of people in Slovakia who’d have been infected over the past weekend.

How does my algorithmic estimate compare with the population antigen survey? Poorly: my lagging-indicator estimate of around 14 thousand infections comes nowhere near the antigen survey result of 38 thousand test-positives. That’s kind of surprising, given that the algorithm has worked quite well for both the US and Europe. Is it the algo’s fault, or the antigen survey methods, or the death count reporting accuracy? As they say in the Discussion section of scientific publications, more research is needed…

ADDENDUM: I think the death count is suspect. To date Slovakia has recorded a total of nearly 63.6K cases but only 235 deaths: that’s a cumulative death rate of 0.4 percent. In contrast, the death rate in the US is = 2.4%; in France it’s 2.5%; UK = 4.4%. Applying an algo to an undercounted death rate can’t help but result in an undercounted infection rate.