As I mentioned in my posts this week, a vaccine protects us in two ways: (i) direct benefit from our our vaccination and (ii) by herd immunity – the vaccine protecting enough of the population to put an end to the virus spreading between people. There is a need for any CoVid-19 vaccine programme to produce herd immunity as the vaccine may not work in everyone and its effects may not last. In this post I consider, given our current knowledge about the possible vaccines and the behaviour of the virus, just how easy it will be to induce herd immunity.

*(A quick note to say that I have tried to make the answer to a complex question easy to follow, especially for people who are not experts in maths! Feel free to go straight to the conclusions at the end! You might find it easier to read this on an iPad or laptop rather than a mobile to take it all in – but do feedback whether it is too complicated).*

**A quick refresher on herd immunity!**

- During an epidemic we can divide the population into 3 groups
- those who are
*infected*(the red figures below)

- those who are
*susceptible*– ie have no immunity to the infection (the light blue figures below)

- those who are
*immune*-ie are protected against infection and therefore cannot pass it on to others (the green figures below)

- (Individuals can be immune either because of natural infection or because of vaccination)

- those who are
- In the picture below, when there has been no vaccination programme, the larger red figure can spread the infection to lots of the susceptible people

- Now, following a vaccination programme, together with people who have become immune naturally, the situation is as in the picture below
- The infected person has far fewer people that they can pass on the infection to. More importantly, the people who are still susceptible are less likely to come in contact with an infected person. In this picture the large red figure can only infect one other person, whilst the light blue figures are surrounded by people who are immune

- When transmission of infection effectively stops, we say that there is a state of herd immunity.
- As shown in the pictures, we don’t need for everyone to be immune to bring about herd immunity
- The proportion who need to be immune varies between viruses. The more infectious a virus, the higher the proportion needs to be

**What are the key factors that will determine how many people need to be vaccinated to achieve herd immunity with Covid-19**?

- A paper in the Lancet* on November 4
^{th}showed that is possible to calculate the answer to this question - The calculations need to consider the following factors:
- What is the rate of transmission?
- The short term efficacy of the vaccine
- How long the vaccine protection will last

- These are considered in turn below

*https://www.thelancet.com/journals/lancet/article/PIIS0140-6736(20)32318-7/fulltext

**What is the rate of transmission**?

- This is the ‘R’ we’ve been hearing so much about
- We all know that if ‘R’ is below 1.0, then the infection will die out (and theoretically no vaccination is needed)
- We can only achieve an ‘R’ of that level with very strict social distancing and other mitigation strategies (eg face masks)
- Without any mitigation strategies, the natural ‘R’ for Covid-19 is around 2.5-3.5 (each infected person, on average, infects between 2.5 and 3.5 other people)
- In my calculations, I have considered 3 possible scenarios with a vaccination programme
- We continue to adopt mitigation measures such as face masks and social distancing, accepting that the ‘R’ will fall to say 1.2, but won’t get below the magic 1.0

- Once the vaccination programme starts, those who have been vaccinated then go back to normal life, ie the transmission rate is 2.5

- As above, but a more pessimistic R of 3.5

- This is what the calculations show:

- To explain this graph, the blue bars show the percent of people who need to be vaccinated to achieve herd immunity for different values of R
- If R is 1, as explained above, we don’t need a vaccination programme as the infection will disappear in time
- If R remains as 1.2, then only around 16% of the population will need to be vaccinated to achieve herd immunity, but that means staying in some kind of lockdown until that has been achieved
- If we go back to normal life and R is as high as 3.5 then we would need to vaccinate around 70% of the population to achieve herd immunity (shown approximately by the white arrows)

**The short term efficacy of the vaccine**

- The calculations above assume the vaccine is 100% effective
- The Pfizer vaccine data suggested 90% efficacy – that might be optimistic and may not apply to all sub-groups, e.g. those who are elderly
- Obviously the lower the efficacy, the lower the proportion who are vaccinated who are actually immune
- We also do not know what the efficacy of other vaccines might be, so I have assumed that the range will be from 60% to 100%
- I have recalculated the figures from the graph above to allow for differences in the efficacy rate
- This is what I found:

- Let me help you to follow this graph*
- The orange bars could represent Pfizer’s vaccine, with its reported 90% efficacy
- The yellow bars could represent another company’s vaccine which may report 70% efficacy
- Thus, if we remain in some kind of lockdown, ie with a ‘R’ of around 1.2, then to achieve herd immunity we would need to vaccinate 18.5% with the Pfizer vaccine and 22.8% with the new vaccine (red arrows)
- If we resume normal activities and accept an ‘R’ of 3. 5, then we would need to vaccinate 79% with the Pfizer vaccine and 98% with the new vaccine (blue arrows)
**Comment: it is highly unlikely that we could achieve anything like a 98% coverage**

*It’s a bit confusing as there are two percentages here. The figures above the coloured squares show the percentage efficacy of the vaccine. The numbers on the vertical axis of the graph show the percentage of people that need to be vaccinated

**How long the vaccine protection will last**?

- This will also prove to be a challenge
- We don’t know how long the immunity reported in the initial findings of the Pfizer trial, or with any of the vaccines, will last for
- This is important as it will influence over how short a time the vaccination programme needs to be delivered. This will then impact on how long herd immunity will last for
- If herd immunity begins to be lost, then booster immunization programmes will be needed
- The Lancet paper did some fairly complex calculations and from their figures I have produced the following graph based on an assumption of
- A vaccine which is 80% effective

- An R value of 2.5

- What this graph shows is that if a vaccine is 90% effective then herd immunity will last for around 17 months. If it is 97.5% effective it will last for over 2 years. If it is only 75% effective it will last just 10 months
- Whenever that time point is reached then, as stated above, if the infection is still around a booster may be needed
- None of these calculations have considered the fact that over time a new vaccine may be required to cope with a changing strain of the virus
- However, the vaccine developers have shown they ae very nimble and should be able to adapt production of any new vaccine to cope

**Conclusions**

- Whether or not the vaccine is effective for any of us as individuals: for our continued protection and for society to return to normal, we need a vaccination programme to deliver herd immunity
- The chances of herd immunity are obviously increased the more effective the vaccine. Vaccines with lower than the current reported success for the Pfizer vaccine can achieve herd immunity – but these would need very high take up rates of vaccination
- Herd immunity will be easier to achieve if coupled with a continued stringency in adherence to mitigation actions such as mask wearing and social distancing – although this is something of a balancing act, as the aim of a successful vaccination programme is to return to normality

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## 8 replies on “What percent of the population needs to be vaccinated to end the pandemic?”

Excellent analysis & exposition of a complex scenario. Thank you Alan.

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Thanks!

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Seems a bit simplistic, as it ignores the effect of an immunization program on decreasing the value of R.

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I assume you are referring to the calculation on the duration that herd immunity will last. It would be possible to model for a changing value of R directly but that this to some extent is taken care of in the computer model. R is a measure of transmission and its decline, with the growing reduction in the proportion of susceptibles, will be the direct contributor to the duration of herd immunity. The mathematical model was published in the supplement to the Lancet paper which, you could check out

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Alan – thanks very much for the very helpful post as you made it easier to understand the complexities of the vaccines and their effectiveness

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To a non medical person, I found this very helpful to understand the process needed to get herd immunity to be able to protect a higher range of people and therefore reduce transmission. Many thanks.

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Thanks for this feedback. That is the aim of the blog and please pass on to any friend or family!

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