SI model
The SI model divides a closed population into susceptible individuals \(S(t)\) and infectious individuals \(I(t)\). Infection is permanent: there is no recovery compartment.
\[S+I=N.\]Model equations
\[\frac{dS}{dt}=-\beta\frac{SI}{N},\qquad \frac{dI}{dt}=\beta\frac{SI}{N}.\]The transmission term \(\beta SI/N\) represents the rate at which susceptible individuals become infectious under homogeneous mixing.
Interpretation
Because infected individuals never leave the infectious class, infection generally spreads through the susceptible population when both \(S\) and \(I\) are positive.
Key idea. The SI model is the simplest compartmental transmission model and is appropriate only when removal or recovery can reasonably be ignored.