SIR Epidemic Model

The SIR (Susceptible–Infected–Recovered) model is a classic compartmental framework for infectious disease dynamics. This simulator uses normalized fractions with S + I + R = 1, representing the entire population divided into three groups: susceptible individuals who can still catch the disease, infected individuals who can transmit it, and recovered (or removed) individuals who are no longer infectious and are assumed immune. The ordinary differential equations are S′ = −β S I, I′ = β S I − γ I, and R′ = γ I, where β is the transmission rate combining contact frequency and probability of transmission per contact, and γ is the recovery rate (inverse of the typical infectious period). The basic reproduction number ℛ₀ = β/γ describes how many secondary infections one typical infective would cause in a fully susceptible population; when ℛ₀ > 1, an outbreak can grow initially. A standard threshold argument shows that growth slows once the susceptible fraction drops near S_crit ≈ 1/ℛ₀, and in a homogeneous well-mixed model with no demography, a herd-immunity reference level is often quoted as an immune fraction of about 1 − 1/ℛ₀, above which the effective reproduction number would fall below one if susceptibles were the only constraint. The simulator integrates the ODEs with a fixed-step fourth-order Runge–Kutta method, renormalizes S + I + R after each step to correct drift, plots S, I, and R versus time, marks the peak of I(t), and draws a dashed horizontal line at y = 1 − 1/ℛ₀ when ℛ₀ > 1. Sliders allow β, γ, initial infected fraction I(0), and initial immune fraction R(0). The model omits births, deaths, seasonality, spatial structure, age structure, stochasticity, and waning immunity—standard extensions appear in SEIR models and network epidemiology.