SIS model
The SIS model describes an infection for which an individual can recover but does not acquire lasting immunity. After recovery, the individual becomes susceptible again and may be reinfected.
\[S(t)=\text{number of susceptible individuals},\qquad I(t)=\text{number of infectious individuals}.\]For a closed population,
\[S(t)+I(t)=N.\]Why does this model make sense?
There are two competing biological processes. Infection moves individuals from \(S\) to \(I\), while recovery moves individuals from \(I\) back to \(S\).
Under homogeneous mixing, the fraction of the population that is infectious is \(I/N\). A susceptible individual therefore experiences infection at rate \(\beta I/N\). With \(S\) susceptible individuals, the total infection rate is
\[\beta\frac{SI}{N}.\]If each infectious individual recovers at rate \(\gamma\), then with \(I\) infectious individuals the total recovery rate is
\[\gamma I.\]The susceptible population loses people through infection and gains them through recovery:
\[\boxed{\frac{dS}{dt}=-\beta\frac{SI}{N}+\gamma I}.\]The infectious population gains exactly those infections and loses exactly those recoveries:
\[\boxed{\frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I}.\]When is an SIS model useful?
The SIS structure is useful for infections where recovery does not give substantial lasting immunity and reinfection is possible. A commonly used example is gonorrhoea: after treatment or clearance, a person can become susceptible and infected again. SIS-type models are also used as simplified descriptions of other infections when immunity is negligible on the time scale being studied.
When is it not appropriate?
SIS is not appropriate when recovery produces important immunity that should be represented explicitly. For an infection where recovered individuals remain immune for the period of interest, an SIR model is more suitable. If immunity is temporary and is lost only after some time, an SIRS model is more realistic because it includes a separate recovered state before individuals return to susceptibility.
SIS also does not include a latent or exposed stage. If there is an important delay between becoming infected and becoming infectious, an exposed compartment may be needed.
The epidemic threshold
At the beginning of an outbreak, almost everyone is susceptible, so \(S/N\approx1\). The infectious equation is then approximately
\[\frac{dI}{dt}\approx(\beta-\gamma)I.\]Therefore infection initially grows when \(\beta>\gamma\) and declines when \(\beta<\gamma\). This leads to the basic reproduction number
\[\boxed{R_0=\frac{\beta}{\gamma}}.\]If \(R_0>1\), infection can initially grow and, in this simple deterministic model, a positive endemic equilibrium exists. If \(R_0<1\), infection declines toward extinction.