The SIR Model
How does a disease spread through a population? The SIR model shows that it all comes down to one number: R₀.
💡 The Key Insight
If each infected person infects more than one other person on average (R₀ > 1), the disease spreads exponentially. If R₀ < 1, it dies out. That's it. Everything else—lockdowns, vaccines, masks—is about pushing R₀ below 1.
Understanding the Model
The model divides a population into three groups that people move through over time:
Susceptible (S)
People who can catch the disease. At the start, almost everyone is susceptible.
Infected (I)
People who have the disease and can spread it. This is what we want to minimize.
Recovered (R)
People who had the disease and are now immune. They can't get it or spread it again.
The Two Key Parameters
β Infection Rate (beta)
How many people does one infected person successfully infect per day? Higher β = more contagious disease. Social distancing and masks reduce β.
γ Recovery Rate (gamma)
What fraction of infected people recover each day? If γ = 0.1, the average person is sick for 10 days (1/γ). Better treatment increases γ.
The Magic Number: R₀
R₀ = β / γ
Average number of people one infected person will infect
- ▸ R₀ > 1: Each person infects more than one other → epidemic grows exponentially
- ▸ R₀ < 1: Each person infects less than one other → epidemic dies out
- ▸ R₀ = 1: Epidemic stays constant (endemic)
Try It Yourself
Adjust the parameters and watch how the epidemic unfolds. Try to find the threshold where R₀ = 1.
Parameters
Contacts per day that transmit infection
1/γ = 10.0 days to recover
Epidemic spreads. Each person infects 4.0 others.
Click "Run Simulation" to start
Show the math
dS/dt = -βSI (susceptible becoming infected)
dI/dt = βSI - γI (new infections minus recoveries)
dR/dt = γI (infected recovering)
R₀ = β/γ (average secondary infections per case)
Real-World R₀ Values
| Disease | R₀ | Notes |
|---|---|---|
| Measles | 12-18 | Extremely contagious, airborne |
| COVID-19 (Original) | 2-3 | Lockdowns reduced effective contact rate below threshold |
| COVID-19 (Omicron) | 8-12 | High R₀ + immune evasion; vaccines still reduced severity |
| Seasonal Flu | 1.2-1.4 | Barely above threshold |
| Ebola | 1.5-2.5 | Deadly but not very contagious |
🎮 Sound Familiar?
If you've played Plague Inc., you've been manipulating this model. Evolving "transmission" symptoms increases β. The "cure research" is humanity trying to increase γ and develop immunity. Your goal as the plague? Keep R₀ above 1 long enough to infect everyone before they can push it below 1.
What This Model Doesn't Capture
The SIR model is elegantly simple, but reality is messier:
- • No incubation period — Real diseases have a delay before you're infectious (SEIR model adds this)
- • Homogeneous mixing — Assumes everyone can meet everyone (network models fix this)
- • No age structure — Different age groups have different contact rates and outcomes
- • Constant parameters — β and γ don't change, but behavior does
- • Permanent immunity — Assumes recovered = immune forever (SEIRS allows reinfection)