SIR epidemic model simulations

Project description

This project simulates the spread of an epidemic using the SIR (Susceptible-Infected-Recovered) model. The system of ordinary differential equations (ODEs) was solved numerically using a fourth-order Runge-Kutta integrator (RK4). A sensitivity analysis was performed varying the infection rate ( $\beta$ ) and the recovery rate ( $\gamma$ ).

Mathematical Model

The SIR model describes the time evolution of three compartments:

• $S(t)$: susceptible individuals
• $I(t)$: infected individuals
• $R(t)$: recovered individuals

The dynamics are governed by:

$$ \frac{dS}{dt} = -\beta \frac{SI}{N} $$

$$ \frac{dI}{dt} = \beta \frac{SI}{N} - \gamma I $$

$$ \frac{dR}{dt} = \gamma I $$

where:

• $\beta$ = infection rate
• $\gamma$ = recovery rate
• $N = S + I + R$ = total population

Sensitivity is computed as:

$$ S_p = \left| \frac{Y(p + \Delta p) - Y(p)}{Y(p)} \cdot \frac{p}{\Delta p} \right| $$

where $Y$ is a model output and $p$ is a parameter.

Repository files

• SIR.ipynb – Jupyter notebook.