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Home/Biophysics, Fluids & Geoscience/SIR Vaccination Threshold

SIR Vaccination Threshold

Herd-immunity threshold simulator: R_eff = R0(1-v), SIR outbreak curves, peak infected, and final size vs vaccine coverage.

SIR and vaccination

2.7
0.18 1/d
0.42
0.004

Measured values

R_eff1.566
Herd threshold63.0%
Peak infected4.63%
Outbreak size35.17%

Vaccinated individuals are placed in R at t=0. The threshold is a population-level idealization and does not include clustering, waning immunity, or heterogeneity.

Live graphs

About this model

This SIR simulator places vaccinated individuals in the removed/immune class at t=0 and integrates the remaining susceptible and infected groups. The effective reproduction number is R_eff=R0(1-v), and the ideal herd-immunity threshold is v_c=1-1/R0. The side plot shows how final outbreak size changes with vaccine coverage.

Who it's for: Epidemiology, public health, mathematical biology, and differential equations courses.

Key terms

  • SIR model
  • Vaccination
  • Herd immunity
  • Effective reproduction number
  • Final size
  • Outbreak threshold

How it works

SIR herd-immunity threshold simulator with R_eff = R0(1-v), outbreak curves, and final size vs vaccine coverage.

Key equations

Ṡ = −βSI, İ = βSI − γI, Ṙ = γI
R_eff = R0(1−v), herd threshold v_c = 1 − 1/R0

Frequently asked questions

Why does vaccination reduce R_eff?
In the homogeneous SIR model, vaccination removes a fraction v of potential transmission targets, leaving only 1-v susceptible.
Does the threshold guarantee no cases?
No. It is a population-average threshold for sustained spread, not a guarantee against introductions or clustered outbreaks.