Basic reproduction number (R0) and herd immunity

What it is

The basic reproduction number (R0) is the average number of secondary cases that one case produces in a fully susceptible population. The effective reproduction number (Rt) is the same average at a given moment, with part of the population immune or control measures in place. If R is above 1 the epidemic grows; below 1, it dies out. The herd immunity threshold, 1 − 1/R0, is the share of immune people needed for each case to cause fewer than one new case.

Formula and reading

• Herd immunity threshold: H = 1 − 1/R0.
• Critical vaccine coverage: Vc = (1 − 1/R0) / E, where E is vaccine effectiveness (0 to 1).
• Rt ≈ R0 × s, where s is the share of the population still susceptible.
• Reading: R above 1, growth; equal to 1, stable endemic level; below 1, the outbreak dies out.

Worked example

1. Data (made up): a respiratory infection has R0 = 5; the available vaccine has 90 % effectiveness.
2. Herd immunity threshold: 1 − 1/5 = 1 − 0.2 = 0.8, that is, 80 % of the population must be immune.
3. Coverage needed: 0.80 / 0.90 = 0.889; rounding up, 89 % of the population vaccinated.
4. Current situation: coverage is 70 %; immune = 0.70 × 0.90 = 0.63 (63 %).
5. Susceptible: 1 − 0.63 = 0.37 (37 %).
6. Effective R: 5 × 0.37 = 1.85; since it is above 1, one imported case can start an outbreak.
7. To bring R below 1, fewer than 20 % must be susceptible: 5 × 0.20 = 1.
8. Reading: at 70 % coverage each case infects almost 2 people on average; raising coverage to 89 % breaks the chain.

Markers

• R0
• Rt
• 1 − 1/R0
• Herd immunity

How to read it

• Every unvaccinated child adds susceptibles: coverage below the threshold in one municipality allows outbreaks even if the national average is high.
• Herd immunity protects those who cannot be vaccinated (very young infants, immunosuppressed people) only if the rest of the population is vaccinated.

Common pitfalls

• Using the threshold as the coverage target without correcting for vaccine effectiveness: it underestimates the coverage needed.
• Treating R0 as a biological constant: it changes with contact density and control measures.
• Mixing up R0 (fully susceptible population) and Rt (the current situation).

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Elsewhere in DurgiPedia

• Herd immunity and outbreak control
• Colombia's PAI schedule (which vaccines, at what age)
• Vaccine types: live, inactivated, subunit, toxoid, mRNA, vector

Connections

← All of epidemiology

Educational content on epidemiological and statistical methods; the examples use made-up numbers to teach the calculation. Not a clinical or legal guide: current Colombian rules are in the official sources.

Sources: Coronavirus disease (COVID-19): Herd immunity, lockdowns and COVID-19 (World Health Organization, 2020); Principles of Epidemiology | Lesson 1 - Section 11 (Centers for Disease Control and Prevention, 2012).

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Representative model, for educational purposes only.