Read

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.

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.

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.

Experiment: Modelling herd immunity with random numbers

1. Model a disease with R₀ = 4: each infected person meets 4 contacts. Calculate the predicted threshold 1 − 1/4.
2. 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.
3. 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.
4. 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.
5. 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).

Coverage	R = R₀ × (1 − coverage)	Total cases, run 1	Run 2	Run 3	Mean
0%	Predict the coverage at which outbreaks stop growing.
50%	Predict the coverage at which outbreaks stop growing.
75%	Predict the coverage at which outbreaks stop growing.
90%	Predict the coverage at which outbreaks stop growing.

Questions

1. Sketch a labelled graph of antibody concentration against time after a first vaccine dose and after a later exposure to the same pathogen.

   🍎

2. Calculate the herd immunity threshold for R₀ = 2, 6 and 15. What does the pattern tell you?

3. Use your model results to describe how total cases change with coverage. Where did outbreaks stop growing, and how does that compare with 1 − 1/R₀?

4. Give two reasons why real-world herd immunity thresholds differ from the simple model.

Sources for every fact on this sheet: echoesof.science/school/vaccines/sources/

🖨 Print this sheet ← back to the lesson