Lecture 7 - Vaccination and Herd Immunity

What is herd immunity?

A reduced probability of infection for susceptible individuals because immune individuals are present in the population.

How does vaccination create herd immunity?

It decreases the number of susceptible hosts and therefore decreases opportunities for transmission.

Does vaccination only benefit vaccinated individuals?

No. If vaccination reduces transmission, unvaccinated susceptible individuals can receive indirect protection.

What is the critical vaccination proportion pc?

The proportion of the population that must be immunized so that the effective reproductive ratio is reduced to 1.

What is the critical vaccination coverage equation?

pc = 1 − 1/R₀.

What happens to pc as R₀ increases?

pc increases.

Why does a pathogen with a high R₀ require greater vaccination coverage?

Each infected host generates more secondary infections, so a larger fraction of the population must be immune to interrupt transmission.

What happens if vaccination coverage is greater than pc?

The effective reproductive ratio becomes less than 1 and sustained transmission should fail under the model assumptions.

What happens if vaccination coverage is below pc?

Transmission is reduced, but the pathogen may still be able to persist.

What is the equation for the reproductive ratio after vaccination?

R′ = R₀(1 − p).

What does p represent?

The proportion of the population that is immunized.

What does 1 − p represent?

The fraction remaining susceptible, assuming perfect vaccination.

If R₀ = 5 and p = 0.6, what is R′?

R′ = 5(0.4) = 2, so transmission can still increase under the simple model.

If R₀ = 5, what is pc?

pc = 1 − 1/5 = 0.8, or 80%.

If R₀ = 2, what is pc?

pc = 1 − 1/2 = 0.5, or 50%.

Why does a higher R₀ produce a higher vaccination threshold?

Because a pathogen with greater transmission ability requires more immune hosts to bring its effective reproduction below 1.

What happens to unvaccinated people when vaccination coverage increases?

They may receive indirect protection because there are fewer susceptible hosts available for transmission.

Why doesn't vaccination necessarily need to reach 100%?

If enough people are immune to reduce R′ below 1, transmission can no longer sustain itself under the model.

What assumption is built into the basic pc equation?

The population is assumed to mix homogeneously and the vaccine is effectively immunizing the vaccinated individuals.

Why might real vaccination coverage need to exceed the theoretical pc?

Vaccines are not 100% effective and populations do not necessarily mix homogeneously.

What is non-homogeneous mixing?

A situation in which hosts do not interact randomly and equally with all other hosts.

Why did non-homogeneous mixing matter in the rinderpest eradication case?

Transmission was concentrated among particular herds and areas, so targeting vaccination toward important transmission zones could be more effective than vaccinating the same fraction randomly.

How does vaccination affect average age at infection?

By reducing transmission, vaccination can increase the average age at which susceptible individuals become infected.

Why can increasing age at infection sometimes have unintended consequences?

For diseases that are more severe at older ages, delaying infection can potentially shift infections toward ages with greater disease consequences.

What is the relationship between R₀ and average age at infection in the lecture?

For the specified type-I survivorship assumption, R₀ = L/A, where L is lifespan and A is average age at infection.

If average age at infection increases, what happens to R₀ under R₀ = L/A?

R₀ decreases if L stays constant.

What happens to average age at infection as vaccination coverage approaches pc?

It approaches the expected lifespan L under the model.