SIRS and SEIRS models
SIRS and SEIRS models are used when recovery provides immunity, but that immunity is temporary rather than permanent. After some time in the recovered class, individuals can become susceptible again and may be reinfected.
SIRS: adding loss of immunity to SIR
The SIR model has the flow
\[S\longrightarrow I\longrightarrow R.\]If immunity can later be lost, we add one more transition:
\[\boxed{S\longrightarrow I\longrightarrow R\longrightarrow S}.\]Where does the new term come from?
Let \(\omega\) be the per-capita rate at which immunity is lost. If there are \(R\) recovered individuals, the total rate at which they return to susceptibility is
\[\omega R.\]This is an outflow from \(R\) and an equal inflow to \(S\). Therefore the SIRS equations are
\[\boxed{\frac{dS}{dt}=-\beta\frac{SI}{N}+\omega R},\]\[\boxed{\frac{dI}{dt}=\beta\frac{SI}{N}-\gamma I},\]\[\boxed{\frac{dR}{dt}=\gamma I-\omega R}.\]Under the usual constant-rate assumption, the mean duration of immunity is
\[\boxed{\frac{1}{\omega}}.\]Why does waning immunity change the dynamics?
In the basic SIR model, recovered individuals remain in \(R\), so infection permanently removes people from the susceptible pool. In SIRS, the flow \(R\rightarrow S\) continually replenishes susceptibility.
This means that even after an epidemic has reduced the susceptible population, new susceptible individuals can gradually become available. Depending on the parameters and other model assumptions, infection may persist at an endemic level or recurrent epidemic behaviour may occur rather than the population experiencing only a single outbreak.
SEIRS: also including an exposed stage
If there is an important delay between infection and infectiousness, we add the exposed compartment \(E\). The cycle becomes
\[\boxed{S\longrightarrow E\longrightarrow I\longrightarrow R\longrightarrow S}.\]The SEIRS equations follow directly from the four flows:
\[\boxed{\frac{dS}{dt}=-\beta\frac{SI}{N}+\omega R},\]\[\boxed{\frac{dE}{dt}=\beta\frac{SI}{N}-\sigma E},\]\[\boxed{\frac{dI}{dt}=\sigma E-\gamma I},\]\[\boxed{\frac{dR}{dt}=\gamma I-\omega R}.\]SIS, SIR, SIRS, SEIR or SEIRS?
The choice depends on the biology that must be represented. SIS returns individuals directly from infectious to susceptible when lasting immunity is absent. SIR keeps recovered individuals protected over the modelled period. SIRS allows that protection to wear off. SEIR adds a delay from infection to infectiousness, while SEIRS includes both that delay and waning immunity.
When are SIRS and SEIRS useful?
These structures are useful when reinfection occurs because protection after infection declines over time. They can be used as simplified frameworks for infections where immunity is not permanent. For diseases such as COVID-19, models may include waning immunity and reinfection, although realistic models often require additional features such as vaccination, variants, age structure or multiple immunity levels.
The choice between SIRS and SEIRS then depends mainly on whether the delay between infection and infectiousness needs to be represented explicitly.