SIS model
The SIS model represents infections for which recovery does not produce lasting immunity. Individuals move from susceptible to infectious and then return to susceptible.
\[S\longrightarrow I\longrightarrow S.\]Equations
\[\frac{dS}{dt}=-\beta\frac{SI}{N}+\gamma I,\]\[\frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I.\]Here \(\beta\) controls transmission and \(\gamma\) is the recovery rate.
Threshold
For this simple model, \(R_0=\beta/\gamma\). If \(R_0>1\), a positive endemic equilibrium can exist; if \(R_0<1\), infection cannot persist deterministically.
Key idea. SIS dynamics allow repeated infection because recovered individuals immediately re-enter the susceptible class.