Epidemic Models Sir: Concepts, Formulas and Examples

The SIR model is the simplest mathematical description of an epidemic. It divides a population into Susceptible, Infectious and Recovered groups, with equations for how people move between them as the disease spreads and burns out.

It explains public health's most famous number: R₀, the basic reproduction number. The SIR equations show why R₀ > 1 means an outbreak grows, why epidemics end before everyone is infected, and how vaccination stops spread.

Key Concepts

• S(t), I(t), R(t) — numbers of susceptible, infectious and recovered people at time t. The total N = S + I + R stays constant.
• Transmission rate β — how fast susceptibles become infected per infectious contact (units: 1/time).
• Recovery rate γ — how fast infectious people recover (units: 1/time); 1/γ is the average infectious period.
• R₀ (basic reproduction number) — average number of new infections caused by one infectious person in a fully susceptible population.
• Herd immunity threshold — the fraction of the population that must be immune so an outbreak cannot grow.

Formulas

• dS/dt = −βSI/N, dI/dt = βSI/N − γI, dR/dt = γI — the SIR equations. Infection needs contact between S and I; recovery removes people from I at rate γ. Assumes a well-mixed population with no births, deaths or migration.
• R₀ = β/γ — derived from the equations: each infectious person infects βS/N ≈ β people per day for 1/γ days. If R₀ > 1, infections initially grow exponentially.
• Herd immunity threshold = 1 − 1/R₀ — the immune fraction at which the effective reproduction number drops to 1. For R₀ = 3, this is 1 − 1/3 ≈ 67%.
• Final size relation — the epidemic ends with some susceptibles never infected; the total infected fraction solves a transcendental equation, always less than 100%.

Worked Example

A flu-like illness: β = 0.3 per day, γ = 0.1 per day (average infectious period 10 days), population N = 10,000, starting with I₀ = 10 infectious.

1. Step 1: R₀ = β/γ = 0.3 / 0.1 = 3. Each case causes 3 new ones — the outbreak will grow.
2. Step 2: Initial growth rate ≈ (β − γ) = 0.2 per day, so cases multiply by e^0.2 ≈ 1.22 daily at first.
3. Step 3: Herd immunity threshold = 1 − 1/3 = 0.667, so about 6,667 people must be immune (by infection or vaccine) to stop growth.
4. Step 4: The epidemic peaks when S/N = 1/R₀ = 1/3 — that is, when susceptibles fall to about 3,333 — and then declines even though 3,333 people never got sick.

Common Misconceptions

• "The epidemic stops when everyone has been infected." No. It stops when enough people are immune that each case causes fewer than one new case — typically well before 100% infection, as the final-size relation shows.
• "R₀ is fixed for a virus." It depends on behaviour too: β includes contact rates, so masks, distancing and ventilation lower R₀ without changing the virus.
• "Recovered means immune forever." The basic SIR model assumes lifelong immunity. For diseases with waning immunity, the SIRS variant lets recovered people become susceptible again.

Real-World Applications

• Vaccination targets — herd immunity thresholds from R₀ estimates set national vaccination goals.
• Outbreak forecasting — public-health teams fit SIR-type models to early case data to predict peak hospital demand.
• Policy evaluation — comparing fitted β before and after interventions measures whether restrictions actually worked.

The SIR model distills an epidemic into three boxes and two rates, yet yields R₀, the growth threshold, herd immunity and the outbreak's final size. It is the foundation of all fancier epidemic models — simple mathematics guiding life-and-death decisions.