Skip to 0 minutes and 13 seconds JUDITH GLYNN: There’s considerable work going on to find a vaccine for Ebola, but how would a vaccine help not just the individuals who are vaccinated, but the wider population? If it is a good vaccine, it would reduce the risk of disease in the individuals who are vaccinated, but it could also reduce transmission in the community, and it is this that’s the focus of this lecture. The basic case reproduction number, R0, is the average number of secondary cases per case in a totally susceptible population. We’ve said that if R0 is greater than one, the number of cases increases. If it’s one, it’s stable. If it’s less than one, the cases decrease, and this is what we’re aiming for.

Skip to 0 minutes and 53 seconds We looked at the different ways of doing this through treatment, through barriers and distance, through isolation and quarantine. But the other thing to think about is the proportion of cases who are susceptible. R0, the reproduction number at time zero, when all are susceptible. The net case reproduction number is the reproduction number at time t and it varies with the proportion who are still susceptible. So imagine an infection moving through a population. In this case, each case is infecting two more. The R0 would be two. Everybody is susceptible, acquiring the disease, and passing it on. But supposing some people were immune, either naturally or because of a vaccine. The number of new cases is greatly reduced.

Skip to 1 minute and 43 seconds Some of the people who would otherwise have acquired the infection are immune. They don’t get ill. They don’t pass it on to other people. So R, the net reproduction number, is R0 multiplied by the proportion susceptible. Another way of reducing the reproduction number, the average number of new cases arising per case, is to decrease the proportion susceptible. This will happen naturally to some extent, immunity after infection, survivors who have a degree of immunity, but you can also reduce it through vaccination. So we can work out how many people you’d need to make immune to get the R down to one, and that would depend on the R0 that you start with.

Skip to 2 minutes and 28 seconds So for example, if the R0 was 3, to get an R of 1, you’d need to make 3 minus 1 people immune, so the proportion would be 3 minus 1 divided by 3, or 67%.

Skip to 2 minutes and 45 seconds So in general, to get the R to 1, you’d need to make R0 minus 1 people divided by R0 immune, and that number is the herd immunity threshold. Formally, the herd immunity threshold is the proportion of the population that needs to be immune for a disease to become stable, to get R to 1, so that every case, on average, only gives rise to one new case. If the proportion immune is greater than the herd immunity threshold, R becomes less than 1 and the disease decreases. If we’re trying to do this through vaccination, we have to remember that the propulsion vaccinated is not the same as the proportion immune.

Skip to 3 minutes and 25 seconds The proportion vaccinated would only be the same as the proportion immune if vaccine efficacy were 100%, but unfortunately, that’s never the case. It’s always less than 100%. That’s to say not everyone who is vaccinated becomes fully immune. The vaccine efficacy is the measure of how good a vaccine is at preventing disease. It can be calculated by looking at the rate of disease in the unvaccinated and the rate of disease in the vaccinated. So the vaccine efficacy is the rate of the disease in the unvaccinated minus the rate in the vaccinated divided by the rate in the unvaccinated. It’s the proportion of disease in the vaccinated that can be prevented by the vaccine.

Skip to 4 minutes and 6 seconds So we can put this together with the R0 to work out how many people you’d need to vaccinate to get an R of 1. So going back to our example, if the R0 is 3, the herd immunity threshold is 3 minus 1 divided by 3, or 67%. Now, suppose you had a vaccine efficacy of 90%. The proportion to be vaccinated to make 67% immune to reach the herd immunity threshold would be 0.67 divided by the vaccine efficacy, 0.9, 74%. So to get R less than 1, you’d need to vaccinate more than 74% of people. Of course, these sorts of calculations rely on several assumptions. An important one is that there’s random mixing in the population. That’s never true.

Skip to 4 minutes and 52 seconds It’s important to note also that the vaccine efficacy that’s measured in clinical trials is always better than the vaccine effectiveness in practise. In practise, there may be problems with the way it’s delivered, or the schedules not being adhered to completely, or problems with the cold chain. Nevertheless, these sorts of calculations are useful for working out the proportion of the population that would need to be vaccinated under different circumstances. In this graph, we look at the proportion of the population you’d need to vaccinate depending on the vaccine efficacy, along the bottom, and on what the R0 is.

Skip to 5 minutes and 29 seconds If we start with an R0 of 2, we look at the proportion of the population we’d need to vaccinate depending on the vaccine efficacy. You can see if you had a vaccine efficacy of 50%, you’d need to vaccinate all of the population in order to get the R down to 1. As the vaccine efficacy improves, this proportion comes down, but it’s still high. However, if you could reduce R0 by other means, through the barriers and the isolation and so on, then the proportion of the population you’d have to vaccinate decreases, and you’d get away with a vaccine efficacy that was lower and a lower proportion of the population vaccinated to get your R down to 1.

Skip to 6 minutes and 8 seconds For Ebola, there are other practical issues to consider as well as the vaccine efficacy. For there to be trials of a vaccine, there are issues of who these should be carried out on and how they would be done. When a vaccine is available, who should be vaccinated first? Should it be the whole population? How acceptable would a vaccine be? Are there likely to be issues with vaccine supply? If we’re talking about a whole population vaccine, you’d need very large numbers.

# How would vaccines help?

This lecture examines how vaccines could potentially help at the community level as well as helping the individuals who are vaccinated.

In the video and below this is explained in terms of R_{0} and R. But let’s start by thinking about the issues intuitively.

If everyone in the population is susceptible it makes sense that an infection will spread faster than if some are already immune.

The more people that are immune the harder it is for an infection to spread.

Vaccines will increase the proportion of the population that is immune, so make it harder for an infection to spread.

If a vaccine only protects a small proportion of those who are vaccinated, then more people would have to be vaccinated to get the same number protected than if the vaccine works well.

R_{0} is the average number of secondary cases per case in a totally susceptible population. As an infection spreads in a population those who have been infected and survived may be immune. The **net reproduction number, R,** is the average number of secondary cases per case when some of the population are immune.

R = R_{0} x proportion susceptible.

A vaccine will reduce the proportion susceptible and therefore reduce R. For an infection to decrease in the population, R must be less than 1, meaning that each case will give rise to less than one new case, on average. The proportion of the population that need to be made immune to bring R down to 1 is known as the **herd immunity threshold**. It depends on R_{0}:

Herd immunity threshold = (R_{0} – 1) / R_{0}

If the proportion of the population that is immune is greater than the herd immunity threshold then R < 1 and the disease will decrease.

The proportion of the population that needs to be vaccinated to reach the herd immunity threshold depends on how well the vaccine works. The **vaccine efficacy** is a measure of how well a vaccine works.

The vaccine efficacy (VE) is defined as:

VE = (rate of disease in unvaccinated – rate of disease in vaccinated) / rate of disease in unvaccinated

The proportion of the population that would need to be vaccinated to reach the herd immunity threshold is given by:

herd immunity threshold / vaccine efficacy

The less efficacious the vaccine, and the higher R_{0}, the higher the proportion of the population that would need to be vaccinated to pass the herd immunity threshold and get R < 1.

Let’s take an example.

Suppose for a new disease R_{0}=4, so that on average every case gives rise to 4 new cases in totally susceptible population.

The herd immunity threshold is reached when each case on average gives rise to one new case, so R=1. What proportion of the population needs to be made immune to get R=1?

The formula for the herd immunity threshold is (R_{0}-1)/R_{0}

So to reach the herd immunity threshold you would need to make (4–1)/4 = 75% of the population immune.

You can see that this makes sense: if 75% of the population are immune, then 25% are susceptible. If only 25% are susceptible, then each case, instead of giving rise to 4 new cases will only give rise to 25% of 4 new cases, i.e. to 1 new case, on average.

If you had a vaccine with 100% efficacy so that everyone who was vaccinated became immune then in this example you would need to vaccinate 75% of the population to reach the herd immunity threshold.

If the vaccine efficacy is less than 100% you would need to vaccinate more people to make 75% immune. If the vaccine efficacy was 80% then the proportion of the population you would need to vaccinate to make 75% immune would be 0.75/0.80, or 93%

Note that vaccine effectiveness in practice may not be as good as vaccine efficacy measured in clinical trials. Vaccines usually need to be given following particular schedules, and need to be stored in certain conditions. Many need to be kept cold, and maintaining the ‘cold chain’ from supply to administration in the population is challenging.

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