SIR Model Calculator (Infectious Disease)
HealthEstimate herd immunity threshold, peak infection fraction, and final attack rate from the basic reproduction number (R0) using the classic SIR epidemic model.
Herd Immunity Threshold
66.67%
Estimated Peak Infected Fraction
30.05%
Estimated Final Attack Rate
94.05%
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.
How to use this SIR Model calculator
- Enter the basic reproduction number (R0) for the scenario you're exploring.
- Read the Herd Immunity Threshold, Peak Infected Fraction, and Final Attack Rate instantly.
- Try different R0 values to see how sensitively these outputs respond to changes in transmissibility.
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
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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