**Pfizer vaccine and implication for herd immunity**

*The encouraging results of Pfizer’s vaccine against Covid-19 has renewed the idea of herd immunity. Central to the idea is the question “how many people in the general community need to be vaccinated to create herd immunity?” This note will answer that question.*

By now, most people probably know what is *herd immunity* (also referred to as *herd protection*). Briefly, the idea of herd immunity is based on the conjecture that when a large proportion of people is immune to an infectious disease, these people will provide indirect protection to those not immune to the disease (see illustration). So, if 70% of a population is immune to, say, SARS-Cov-2, then 7 out of every 10 people who contact someone with the disease will not be infected. Consequently, the disease will not spread further and the epidemic is controlled. Nice idea.

There are basically two ways to create a herd immunity in the general population. One way is to let the epidemic take its course, which literally means the virus is allowed to spread as much as possible. This approach was once considered by the UK government, and needless to say, it is *very* controversial.

Another way to create herd immunity is by vaccination. The beauty is that vaccines can create immunity without causing illness or adverse effects. Moreover, vaccines can protect the whole population from an infectious disease, including those who cannot be vaccinated (e.g., newborns). This vaccination is an ideal approach to achieve herd protection.

The critical question is: *how many people in the community needed to be vaccinated*? The answer to this question is dependent on two factors: vaccine efficacy (*VE*) and the **reproduction number **(*R0*). I have already explained the meaning of VE [1]. Now, *R0* is a very important parameter in any epidemic because it reflects the degree of transmission of a virus in the community. Briefly, *R0* is the average number of secondary infections caused by a primary case. For instance, *R0* = 2 means that each new case will infect 2 more cases (assuming, of course, that everyone around the new case was susceptible).

OK, then the proportion of people need to be vaccinated (T) to create a herd immunity is formally defined as:

T = (1–1 / *R0*) / *VE*

Now, we tentatively know that VE (vaccine efficacy) is 90%. We also know from multiple studies that R0 = 3.17 [2]. Then, substituting the two numbers into the equation, we get:

T = (1–1 / 3.17) / 0.90 = 0.76.

In other words, 76% of a population need to be vaccinated to achieve herd immunity. Now, R0 can be different from one population to another, and the following table shows the required proportion of vaccination for various R0 and VE:

As can be seen from the above table, for a reproduction number, the higher the vaccine efficacy the lower the proportion of population need to be vaccinated. On the other hand, for a vaccine efficacy, the required percent of vaccination proportionally increases with the reproduction number.

The next question is how long will the protection last? The duration of protection does influence the calculation [3] that I presented earlier. However, at present, I think it is fair to say that for Pfizer vaccine, we don’t know yet.

In summary, the vaccine efficacy of 90% and the reproduction number of 3.17 imply that approximately 75% of people in the community need to be vaccinated to achieve herd immunity.

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[1] https://tuanvnguyen.medium.com/the-real-meaning-of-90-efficacy-for-pfizer-vaccine-77499dec6e1c

[2] Dong et al. A Meta Analysis for the Basic Reproduction Number of COVID-19 with Application in Evaluating the Effectiveness of Isolation Measures in Different Countries. *Journal of Data Science* 18 (3), 496–510.

[3] Heffernan JM and Keeling MJ. Implications of vaccination and waning immunity. *Proc R Soc Ser B.* 2009; 276: 2071–2080