R₀ = 3.00
Time Series: S, I, R over time
Phase Portrait: S vs I
Bifurcation Diagram: Peak Infection vs R₀
Final Size Relation
The SIR Model and Its Bifurcation
The SIR model: dS/dt = −βSI, dI/dt = βSI − γI, dR/dt = γI. The basic reproduction number R₀ = β/γ determines whether an epidemic occurs.
When R₀ < 1: each infected person infects fewer than one others → epidemic dies out immediately. Disease-free equilibrium is stable.
When R₀ > 1: epidemic occurs → infected fraction grows before eventually declining. The disease-free equilibrium loses stability — a transcritical bifurcation at R₀ = 1.
Final size r∞ satisfies: ln(1−r∞) = −R₀·r∞ (implicit equation, no closed form). The fraction ever infected goes from 0 (R₀≤1) to a positive value (R₀>1).