SIR model
The SIR model separates the population into susceptible \(S\), infectious \(I\), and removed or recovered \(R\) compartments.
\[S\longrightarrow I\longrightarrow R.\]Equations
\[\frac{dS}{dt}=-\beta\frac{SI}{N},\]\[\frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I,\]\[\frac{dR}{dt}=\gamma I.\]The infectious population initially increases when transmission into \(I\) exceeds recovery from \(I\).
Epidemic threshold
At the beginning of an outbreak in a nearly fully susceptible population, the basic reproduction number is \(R_0=\beta/\gamma\).
Key idea. The SIR model captures depletion of susceptible individuals and permanent removal from infectiousness.