theory.toys – SIR Epidemic Dynamics

Initial Conditions

Population (N): 1000 Initially infected (I₀): 10 Initially recovered (R₀): 0

Disease Parameters

Transmission rate (β): 0.50 Recovery rate (γ): 0.10

R₀ (basic reproduction number): 5.00
R₀ = β/γ. If R₀ > 1, epidemic spreads.

Simulation

Duration: 200 days

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The SIR Model

The SIR model divides a population into three compartments:

S (Susceptible): Individuals who can contract the disease
I (Infected): Individuals who are currently infected and can transmit
R (Recovered): Individuals who have recovered and are immune

Differential Equations

dS/dt = -βSI/N — Susceptible individuals become infected
dI/dt = βSI/N - γI — New infections minus recoveries
dR/dt = γI — Infected individuals recover

Key Parameters

β (beta): Transmission rate. Higher values mean more contagious disease.
γ (gamma): Recovery rate. The average infectious period is 1/γ days.
R₀ = β/γ: Basic reproduction number. The average number of secondary infections from one infected individual in a fully susceptible population.

Epidemic Threshold

R₀ > 1: Epidemic occurs. Infection spreads through the population.
R₀ < 1: Disease dies out. Each infected person infects less than one other.
Herd immunity threshold: 1 - 1/R₀. The fraction of the population that must be immune to prevent epidemic spread.

Model Dynamics

The infection curve rises, peaks when dI/dt = 0 (at S = γ/β = N/R₀), then declines as susceptibles are depleted. The final epidemic size depends on R₀ and initial conditions. Not everyone gets infected—the epidemic ends when there are insufficient susceptibles, not when everyone is infected.