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SIR Model Calculator (Infectious Disease)

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Estimate herd immunity threshold, peak infection fraction, and final attack rate from the basic reproduction number (R0) using the classic SIR epidemic model.

Reviewed by the thecalcu.com team · Last updated July 6, 2026

1.0120

Herd Immunity Threshold

66.67%
Estimated Peak Infected Fraction
30.05%
Estimated Final Attack Rate
94.05%

This calculator computes your Herd Immunity Threshold, Estimated Peak Infected Fraction, Estimated Final Attack Rate from the values you enter.

Inputs
Basic Reproduction Number (R0)
Outputs
Herd Immunity ThresholdEstimated Peak Infected FractionEstimated Final Attack Rate

What is a SIR Model?

The SIR Model Calculator applies the classic Susceptible-Infected-Recovered epidemiological model to estimate three key epidemic characteristics, herd immunity threshold, peak infected fraction, and final attack rate, from a single input: the basic reproduction number, R0.

For a related, gathering-specific estimate, see the Event Risk Calculator.

Why Use a SIR Model Calculator?

The SIR model is foundational to epidemiology, and its key results can be derived directly from R0 using closed-form and numerically solvable equations. This calculator makes those classic results accessible without needing to run a full differential equation simulation.

Who Should Use This Calculator?

  • Students of epidemiology or applied mathematics learning the classic SIR model's key results.
  • Public health communicators illustrating how R0 drives outbreak dynamics.
  • Researchers and writers wanting a quick reference calculation for a given R0 value.
  • Anyone curious about the mathematical relationship between R0 and epidemic scale.

What Insights Does the SIR Model Calculator Give You?

The Herd Immunity Threshold shows what fraction of the population would need immunity for transmission to stop growing. The Estimated Peak Infected Fraction shows the highest proportion of the population infected at any single point in an unmitigated epidemic. The Estimated Final Attack Rate shows the total proportion of the population eventually infected over the full course of an unmitigated epidemic.

How to use this SIR Model calculator

  1. Enter the basic reproduction number (R0) for the scenario you're exploring.
  2. Read the Herd Immunity Threshold, Peak Infected Fraction, and Final Attack Rate instantly.
  3. Try different R0 values to see how sensitively these outputs respond to changes in transmissibility.
Show formula & methodology ↓Show less ↑

Formula & Methodology

Herd Immunity Threshold = 1 − 1/R0

Peak Infected Fraction = 1 − 1/R0 − ln(R0)/R0

Final Attack Rate (z) is solved from the final-size relation z = 1 − e^(−R0·z) using iterative approximation.

Worked example, an R0 of 3:

Herd Immunity Threshold = 1 − 1/3 = 66.7%

Peak Infected Fraction = 1 − 1/3 − ln(3)/3 ≈ 30.6%

Final Attack Rate solved from z = 1 − e^(−3z) ≈ 94%

Frequently Asked Questions

What is the SIR model?
The SIR model is a classic mathematical framework in epidemiology that divides a population into three compartments, Susceptible, Infected, and Recovered, and describes how people move between them over the course of an epidemic based on the basic reproduction number, R0.
What is R0 (the basic reproduction number)?
R0 represents the average number of new infections caused by one infected person in a fully susceptible population with no immunity or interventions in place. An R0 above 1 means an outbreak tends to grow, while below 1 it tends to shrink.
What is herd immunity threshold?
Herd immunity threshold is the proportion of a population that needs to be immune (through prior infection or vaccination) for sustained transmission to no longer be self-sustaining, calculated as 1 minus 1 divided by R0 in the classic SIR model.
What is peak infected fraction?
Peak infected fraction is the highest proportion of the population infected at any single point during an unmitigated epidemic, calculated from a closed-form solution of the SIR differential equations based on R0 alone.
What is final attack rate?
Final attack rate is the total proportion of the population that becomes infected over the entire course of an unmitigated epidemic, derived from the SIR model's final-size relation, which is solved numerically for a given R0.
Does this calculator model interventions like vaccination or distancing?
No, this calculator models a basic, unmitigated SIR epidemic based purely on R0, without vaccination, behavior change, or seasonal effects, which would all reduce the effective reproduction number and lower these estimates in reality.
Why do these outputs only depend on R0?
The classic SIR model's key structural results, herd immunity threshold, peak infected fraction, and final attack rate, can each be expressed as closed-form or numerically solvable functions of R0 alone, assuming the population starts almost entirely susceptible, which is what this calculator assumes.
Is this calculator specific to any particular disease?
No, this is a general mathematical model that applies the same equations regardless of the specific illness, as long as an appropriate R0 estimate is used as the input.
How accurate is the SIR model for real-world outbreaks?
The basic SIR model is a simplified starting point in epidemiology; real-world outbreaks are also shaped by population structure, behavior changes, interventions, seasonality, and other factors not captured in this basic version, so results should be treated as illustrative rather than predictive.
How does this relate to the Event Risk Calculator?
The [Event Risk Calculator](/event-risk-calculator/) estimates the chance of an infectious person at a single gathering from a current case rate, while this calculator models the broader trajectory of an entire epidemic from its basic reproduction number, both are epidemiology tools addressing different scales of the same topic.
Also known as
SIR epidemic model calculatorherd immunity threshold calculatorR0 calculatorbasic reproduction number calculator