Effective Reproduction Number
Actual average infections per case when part of the population is already immune.
Formula
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Example
R₀ = 3, 40 % immune (S = 0.6) → Rₑ = 1.8
How it works
The effective reproduction number tells you how fast an infection is spreading in a population where some people are no longer available to catch it, usually because they are immune or already infected. It comes from taking the disease’s basic spread potential, R₀, and scaling it by the fraction still susceptible, S, because only those people can be newly infected. If S drops, each case has fewer possible targets, so the average number of secondary cases falls in direct proportion.
When to use it
Use this formula in outbreak monitoring, vaccination planning, and public-health modelling when a disease is circulating in a population that is not fully susceptible. It is especially useful for estimating whether transmission is likely to grow, hold steady, or decline.
Worked example, step by step
Given: R₀ = 4.2 S = 0.35 Step 1: Substitute into the formula Rₑ = R₀ × S Rₑ = 4.2 × 0.35 Step 2: Multiply Rₑ = 1.47 Final answer: The effective reproduction number is 1.47, so each infected person is expected to infect about 1.47 others on average in the current population.
Common mistakes to avoid
- Using the percentage immune instead of the susceptible fraction S
- Confusing Rₑ with R₀ and ignoring immunity in the population
- Assuming Rₑ is fixed even though it changes as susceptibility changes over time
Did You Know?
When Rₑ drops below 1, the outbreak shrinks — this is the goal of every vaccination and lockdown strategy.
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More in Epidemiology & Public Health
View allIncidence Rate
BasicRate at which new cases of a disease appear in a population over a given period.
Prevalence
BasicProportion of a population that has a particular condition at a point in time.
Basic Reproduction Number
IntermediateAverage number of secondary infections produced by one infected individual in a fully susceptible population.
Herd-Immunity Threshold
IntermediateMinimum fraction of the population that must be immune to prevent sustained spread.
SIR Model — Susceptible
AdvancedRate of change of the susceptible population in the SIR compartmental model.
SIR Model — Infected
AdvancedRate of change of the infected population — new infections minus recoveries.