SIR Epidemic Model

SIR Epidemic Model

The SIR model divides a population into three compartments:

• S (Susceptible): can get infected
• I (Infected): currently infectious
• R (Recovered): immune

Equations

$\frac{dS}{dt} = -\beta \cdot \frac{S \cdot I}{N}$ $\frac{dI}{dt} = \beta \cdot \frac{S \cdot I}{N} - \gamma \cdot I$ $\frac{dR}{dt} = \gamma \cdot I$

• $\beta$: transmission rate (contacts per day $\times$ probability of transmission)
• $\gamma$: recovery rate (1/days infectious)
• $N = S + I + R$: total population (constant)

Key Metric: Basic Reproduction Number

$\mathcal{R}_0 = \frac{\beta}{\gamma}$

• $\mathcal{R}_0 > 1$: epidemic grows (each infected person infects more than one)
• $\mathcal{R}_0 < 1$: epidemic dies out

COVID-19 had $\mathcal{R}_0 \approx 2$–$3$. Measles has $\mathcal{R}_0 \approx 12$–$18$.

Herd Immunity Threshold

An epidemic can't grow if enough people are immune. The threshold:

$\text{fraction immune needed} = 1 - \frac{1}{\mathcal{R}_0}$

For $\mathcal{R}_0 = 3$: need $1 - 1/3 = 67\%$ immune.

Your Task

Implement sir(S0, I0, R0, beta, gamma, t_end, n) using Euler's method. Return (S, I, R) at time t_end.