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SIR Epidemic Model Calculator

Health

Quick answer

The SIR epidemic calculator simulates how an infection spreads through a population with the classic susceptible–infected–recovered model, from population size, initial cases, R₀ and infectious period, and gives the peak, its timing and the final share infected.

days
%
Peak number infected at once
23,405
Day of the peak
45
Total infected by the end
89,311
Share of the population infected
89.3 %
Herd immunity threshold
60 %
Effective R at the start
2.5
Still infected at the end
0

The SIR model, introduced by Kermack and McKendrick in 1927, captures the basic shape of an epidemic: exponential growth, a peak when the susceptible share falls below 1/R₀, and decline. Real epidemics are shaped by behaviour, interventions and population structure. Everything is calculated in your browser, so nothing you type is sent to our server.

How to use the SIR Epidemic Model Calculator

  1. Population
  2. Initially infected
  3. Basic reproduction number (R₀)
  4. Infectious period
  5. Days to simulate
  6. Share immune at the start

Everything runs in your browser, so results appear instantly and nothing you type is sent to a server.

How it works

Enter the population, starting infections, R₀, infectious period, simulated days and any share vaccinated at the start.

Tips

  • Use counts and person-time from the same population and period.
  • Report confidence intervals with rates and ratios.
  • Check whether figures are crude or age-adjusted before comparing.
  • Larger samples give more stable estimates.

Common mistakes to avoid

  • Confusing incidence (new cases) with prevalence (all cases).
  • Comparing rates per 1,000 with rates per 100,000.
  • Reading a relative risk without the baseline risk.

Health note: this calculator is for education and screening only and is not a diagnosis or treatment advice. Discuss your results with a qualified clinician.

Formula

dS/dt = −β S I ÷ N · dI/dt = β S I ÷ N − γ I · dR/dt = γ I β = R₀ × γ, γ = 1 ÷ infectious period Herd immunity threshold = 1 − 1 ÷ R₀

Worked example

Inputs — Population: 100,000; Initially infected: 10; Basic reproduction number (R₀): 2.5; Infectious period: 7 days; Days to simulate: 365; Share immune at the start: 0 %.
Results — Peak number infected at once = 23,405; Day of the peak = 45; Total infected by the end = 89,311; Share of the population infected = 89.3 %; Herd immunity threshold = 60 %; Effective R at the start = 2.5.

Key terms

SIR
Susceptible–Infected–Recovered.

Frequently asked questions

What does the SIR model assume?

A well-mixed population, lifelong immunity after recovery and constant transmission.

Why does the epidemic peak?

When the susceptible share falls below 1/R₀, each case infects fewer than one person.

Why do more people get infected than the herd immunity threshold?

Momentum: cases already infected keep spreading after the threshold is passed.

Sources

  1. Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proc R Soc Lond A, 1927

Cite this calculator

Speedy Calculator. (2026). SIR Epidemic Model Calculator. Speedy Calculator. https://speedycalculator.com/calculators/health/viral-infection-sir/