The latest Gallup Poll reports that 35% of Americans would refuse to receive an effective FDA-approved vaccine, if it were available today. That’s a 65% rate of persons who would in theory receive it. But some of those persons might not end up getting the vaccine, because of various circumstances in their lives.

So the question is raised: What percentage of the population must be vaccinated to reach the herd immunity threshold (HIT)?

Herd immunity is a protective effect whereby an unvaccinated individual, who has no natural immunity from antibodies (i.e. from a prior infection of that disease), receives a type of protection because the persons around the individual are immune. They could be immune from having had the disease before, or from a vaccine. Then the Herd Immunity Threshold (HIT) is the level of immunity from vaccinations AND prior infection resulting in antibodies needed to reach herd immunity.

Here’s the HIT formula: V=(1-(1/R0))/E

The effectiveness of the vaccine: E=(1-(1/R0))/V

Determination of R0: R0=-1/(E*V-1)

So simple. V is the percentage of persons who must be either Vaccinated OR have natural immunity from recovering from the disease.

E is the effectiveness of the vaccine, as a percent.

R0 is the “basic reproduction number” also called R-nought or R-zero (or R-maru in Japan), which is the number of persons, on average, who catch the disease from each infected person. If you have Covid-19 and you infect 3 other persons, that’s an R0 of 3.0. And since it is an average, you can have a fraction, such as 2.2 or 4.7. Once the R0 falls below 1.0, the disease dies out because the number of persons subsequently infected decreases continually.

If 1,000 persons are infected, and the R0 is 1.0, then each infected person passes the disease on to one other person, keeping the number infected at 1,000. If the R0 is 0.9, then those 1000 persons next infect only 900, and the 900 then next infect only 810, and so on, until only one person is left with the disease. (Then you send that one person to a deserted island in the middle of the ocean, and everyone is safe.)

Now, here is the problem. Vaccines are not 100% effective. And the R-nought for a contagious disease can be so high that the vaccine fails to reach the HIT. The result is that the vaccine only slows down the spread of the virus, it doesn’t stop it. And what happens next is that more and more persons become infected until the number who develop natural immunity add to the percent protected by vaccine to raise the level of V until it is sufficiently high for the above stated formulae to reach the R0 value. This is because natural immunity plus vaccination has the effect of reducing R0.

At 65% of the population willing to be vaccinated we have a problem. You will never vaccinate every person willing to be vaccinated, simply because life is complicated and some persons will not end up getting the vaccine who are willing. So 65% willing might be 52 to 58.5% actually vaccinated, assuming only 80 to 90% of those willing actually receive the vaccine. And those values require a highly effective vaccine to reach R0.

But Covid-19 vaccines are unlikely to be highly effective. The virus, SARS-CoV-2, is particularly good at fending off the immune system. In fact, S2 actually attacks the immune system directly, as explained in this article. So it is unlikely that we will have a 90% effective vaccine.

The yearly flu vaccine is 40 to 60% effective — when they choose the right strains to put in the vaccine. When they fail, the effectiveness falls as low as 19%.

If 65% of the population are willing to be vaccinated, but only 80% of that number actually are vaccinated, you have 52% protected by vaccination. Then we add the number of persons protected by antibodies (after recovering from infection), which is about 5 million persons out of 330 million it the U.S. or 1.5%, and we have 53.5% protected (V).

Now V is reduced by E, the effectiveness of the vaccine. In effect, we are also reducing the value we added for those who have antibodies because they have recovered from Covid. But as far as we know, those naturally-induced antibodies have about the same degree of protection as the vaccine-induced antibodies. So we don’t need a separate calculation for reduction of effectiveness. (Whew!)

Another factor, which will be left out of the calculation, is that persons protected by recovering from Covid-19 are not equally dispersed in the population with those protected by vaccination. And, in fact, those more likely to get Covid-19 are also more likely to spread it. For example, the grocery store clerk exposed to hundreds of persons a day, or the healthcare worker exposed to dozens of confirmed cases a day. They get the disease more readily, and then hopefully are protected, making them less likely to spread the disease. This has an exaggerated effect, in a good way, giving society a higher degree of protection once those most readily infected have recovered.

A study by Temime et al., “Estimating R0 OF SARS-CoV-2 in Healthcare Settings,” found that the R0 in hospitals and clinics is higher than in the general population. This is because infected healthcare workers are in contact not only with many persons, but with persons vulnerable to infection.

Put quite simply, those who are vulnerable to infection by either contact with many infected persons or contact with very many persons who might be infected are also likely to spread the disease in the other direction, from them rather than to them. Once they are infected and have recovered, they also give society a higher degree of contribution to herd immunity than other persons. And this is just not easily put into a formula.

Returning to what we CAN calculate, if the R0 is a mere 2.2, the effectiveness required of the vaccine, in the scenario of only 52.5% of the population actually being vaccinated, would be over 100%, which is impossible. The meaning of the result, then, is that the vaccine will not produce herd immunity. And that means the disease will continue to spread, but at a slower rate. This occurs because there are enough non-vaccinated persons such that even if everyone vaccinated is protected (100% effective vaccine), you still don’t reduce the R0 to 1 or less.

The solution to that problem is to require persons to be vaccinated. A mandatory vaccine will raise the number of persons vaccinated so that the HIT is reached, thereby protecting persons who refuse vaccination as well as those who cannot be vaccinated because of immune system problems.

Before you panic about a mandatory vaccine, the suggestion is this: Anyone who objects to the vaccine on religious or conscience reasons may fill out a form, one side of one page, and be exempt. The reason for the form is so that there is a record that you object, thereby preventing you from being repeatedly asked to get vaccinated, and preventing your employer or anyone else from discriminating against you. This would require a law that not only makes the vaccine mandatory but permits exemptions and prevents discrimination. Those who don’t want to fill out the paperwork can pay a fine of $100, which is to be used only for a narrow subset of Covid-19 purposes, not including vaccines or any type of research (as there are moral or religious objections to those).

If the semi-mandatory plan stated above does not work, then the fine needs to be raised, and the requirements for exemption need to be narrowed. But we should never have an absolutely mandatory vaccination program, as people have freedom of conscience and religion.

What happens if HIT is never reached?

The rate at which the disease now spreads, after a vaccination program fails to reach HIT, is determined by this formula:

R0′ =-1/(E*V-1)

Yes, that is R-nought-prime.

When the formula gives us a value for R0′ which is less than R0, the actual R-nought in the population, we don’t have herd immunity, but we slow the progression of the disease. How much is it slowed? The new R0 is R0” (double prime), which is arrived at by simple division: R0/R0′ =R0”

Example: 60% of the population is vaccinated, and the vaccine is 70% effective. Those are likely numbers, more or less. And let’s say we reach a point where 2% of the population has recovered and have the same protection from antibodies that vaccinated persons have from the vaccine. That gives us a value of 62% for V, and 70% for E. If the R0 in the population is 1.767 or lower, then HIT is achieved. That is because the calculated R0′ is 1.767.

If R0 in the population is 2.7, then we are only 65.4% of the way toward HIT (R0’/R0). But what is the continued spread of the disease after this vaccination program? It is R0 divided by R0′. So in the example, 2.7/1.767 gives us 1.528. And that is the new basic reproduction number, the rate at which the disease will now spread, after vaccination. So the spread of the disease is slower, but it is not stopped.

Conclusion

Currently, the R0 is estimated at between 2.2 and 2.7, with some researchers claiming an R0 as high as 4.7 to 6.6 [2]. At an R0 of 2.2, if the vaccine is 70% effective, then 78% of the population would need to be vaccinated in order to reach Herd Immunity Threshold. At an R0 of 2.7, if 70% of the population were vaccinated, a vaccine would need an unlikely 90% effectiveness to reach the HIT. Overall, the result of this analysis is that HIT is unlikely to be reached by vaccination alone. Reducing the R0 by wearing masks, social distancing, and restrictions on gatherings is required, along with a high percentage of the population accepting a vaccine, in order to reach HIT and stem the spread of the disease.

Ronald L. Conte Jr.

Covid.us.org

*Note: The author of this article is not a doctor, nurse, or healthcare provider, and this article does not offer medical advice.*

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1. Gustin, M., et al. “Estimating R0 OF SARS-CoV-2 in Healthcare Settings.” (2020). Study Link

2. Sanche, Steven, et al. “The novel coronavirus, 2019-nCoV, is highly contagious and more infectious than initially estimated.” arXiv preprint arXiv:2002.03268 (2020). Study Link.