The SIR Model: Using Math to Predict an Outbreak
In the fall of 1918, a new strain of influenza swept across the globe, eventually infecting roughly 500 million people - about a third of the world's population at the time - and killing an estimated 50 million. Doctors had no way to predict how far or how fast it would spread. Today, epidemiologists use a set of equations called the SIR model to do exactly that: turn the messy chaos of a disease outbreak into numbers they can graph, test, and use to plan a response.
What You'll Learn
- Define the three compartments of the SIR model: Susceptible, Infected, and Recovered - Calculate and interpret the basic reproduction number, R0 - Explain how the herd immunity threshold is calculated from R0 - Describe at least one real-world limitation of the SIR model
Three Groups: S, I, and R
The SIR model sorts an entire population into three 'compartments.' S stands for Susceptible - people who can still catch the disease. I stands for Infected - people currently sick and able to spread it. R stands for Recovered (or Removed) - people who are now immune or have died. The total population, N = S + I + R, stays constant throughout the model. People move from S to I based on how often susceptible and infected people contact each other, multiplied by the transmission rate (called beta). People move from I to R based on the recovery rate (called gamma) - roughly how long a person stays infectious before recovering.
The Reproduction Number, R0
R0 (pronounced 'R-naught') equals beta divided by gamma. It tells you the average number of new infections one sick person causes in a population where everyone is still susceptible. R0 varies enormously by disease: measles has an R0 between 12 and 18, making it one of the most contagious diseases known. Seasonal flu sits around 1.2 to 1.4. The original strain of SARS-CoV-2 in 2020 had an estimated R0 of about 2 to 3, while the later Delta variant reached roughly 5 to 8. If R0 is greater than 1, each sick person infects more than one other person on average, so the outbreak grows. If R0 is below 1, the outbreak shrinks and eventually dies out.
Herd Immunity: When the Curve Bends Down
Epidemiologists calculate the herd immunity threshold - the fraction of a population that needs to be immune to stop an outbreak from growing - using the formula 1 minus 1 divided by R0. For measles, with R0 around 15, that works out to about 93 percent of people needing immunity, which is why public health targets for the MMR vaccine are set so high. For a disease with R0 of 3, only about 67 percent of the population needs immunity to bring the outbreak under control.
The basic SIR model assumes everyone mixes with everyone else equally, immunity lasts forever, and no one is born or dies of other causes during the outbreak. Real populations violate all three assumptions, which is why researchers build more detailed versions, like SEIR (which adds an Exposed stage) or agent-based models that simulate individual people.
A disease has R0 = 4. What fraction of the population must be immune to reach the herd immunity threshold?
In the SIR model, a person moves from the Infected compartment to the Recovered compartment based on which rate?
Build a Mini Outbreak Simulation
Using 20 dice (or 20 slips of paper) to represent a population of 20 people, start with 1 person 'infected.' Each round, roll to decide which susceptible people get infected (for example, any die showing a 6 next to an infected person's die) and which infected people recover (any infected die showing a 1 or 2). Track and graph how many people are infected each round for 10 rounds. Then rerun the simulation cutting the infection chance in half, to represent social distancing, and compare the two curves' peak height and timing.
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