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Vaccination, immune memory and herd immunity
Active and passive immunity, vaccine types, primary and secondary responses, boosters, the herd immunity threshold 1 − 1/R₀ and the history of smallpox eradication. With a random-number transmission model.
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Vaccination gives active immunity: the body makes its own antibodies and memory cells. It takes longer to develop than passive immunity (antibodies received from someone else, for example across the placenta), but passive immunity fades within weeks or months.[1][4]
Vaccine antigens can be live-attenuated (weakened) or killed microbes, subunits such as surface pieces, toxoids (toxins treated to be harmless) or genetic instructions (DNA or RNA) for the body to make the antigen.[1][2]
The vaccine produces a mild primary response, which takes days or even weeks. Antigen-specific memory B and T cells remain. On real exposure the secondary response is faster, and memory B cells that become plasma cells secrete tens to hundreds of times more antibody.[4][5]
One dose often gives partial protection. Live-attenuated vaccines such as MMR usually need two doses; non-live vaccines usually need three or more, plus boosters that mimic repeat exposure.[1][5]
Vaccines are thoroughly tested and monitored. Common side effects are local soreness for 2–3 days or a high temperature for 1–2 days. Live-attenuated vaccines can be dangerous for people with a weakened immune system.[3][1]
R₀ is the number of people one case infects on average: for measles about 15. The herd immunity threshold is approximately 1 − 1/R₀: for measles this gives 1 − 1/15 ≈ 0.93, so about 93% must be immune; the vaccination target is 95%.[8][7]
No vaccine gives 100% protection: vaccinated people can still be infected but are far less likely to become seriously ill. High coverage protects those who cannot be vaccinated.[2][1][7]
In 1796 Jenner used cowpox material to protect James Phipps against smallpox, a disease that killed at least 1 in 3 of those infected. After an intensified WHO programme from 1967, smallpox was declared eradicated in 1980.[6]
Try it: Modelling herd immunity with random numbers
You need
- Two ten-sided dice (or a random number table, or a spreadsheet with =RANDBETWEEN(0,99))
- Calculator
- Results table and graph paper
Safety
No hazards: this is a paper or spreadsheet model. Treat the topic sensitively: do not ask pupils about their own or their family's vaccination status.
What to do
- Model a disease with R₀ = 4: each infected person meets 4 contacts. Calculate the predicted threshold 1 − 1/4.
- Choose a vaccine coverage, for example 0%, 50%, 75% or 90%. For each contact, roll a number 00–99: if it is below the coverage (e.g. below 50 for 50%), that contact is immune; otherwise they become infected.
- Start with 1 case (generation 0). Work out generations 1 to 5, rolling for every contact of every new case. Stop a run early if the number of new cases passes 100.
- Repeat each coverage at least 3 times (more if you use a spreadsheet). Record the total cases after 5 generations and the mean for each coverage.
- Plot mean total cases against coverage. Compare where the outbreak stops growing with your predicted threshold, and calculate the effective reproduction number R = R₀ × (1 − coverage).
What happens, and why?
Each case on average infects R₀ × (1 − coverage) people. If that number is above 1, cases grow each generation; if it is below 1, the chain dies out. With R₀ = 4 the threshold is 1 − 1/4 = 0.75, so outbreaks shrink above about 75% coverage. Random rolls give different results each time, so repeats and means are needed. The model assumes a 100%-effective vaccine and random mixing, which real populations do not have.[8][7]
Play: Primary or secondary response?
Sort each statement: does it describe the primary response, the secondary response, or both?
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Quiz
Other ages: Ages 6–7 · Ages 10–11