SEIR model
The SEIR model extends the SIR model by separating infection from becoming infectious. After a susceptible individual is infected, that individual first enters an exposed stage before becoming infectious.
\[S(t)=\text{susceptible},\quad E(t)=\text{exposed},\quad I(t)=\text{infectious},\quad R(t)=\text{recovered or removed}.\]Here, exposed means infected but not yet infectious in this basic SEIR model. For a closed population,
\[S(t)+E(t)+I(t)+R(t)=N.\]Why is the exposed compartment needed?
In the SIR model, a person effectively moves from susceptible to infectious immediately after infection. For many infections this is too simple. There can be a period after infection during which the pathogen is developing inside the host but the person is not yet infectious.
The SEIR model represents this delay by
\[S\longrightarrow E\longrightarrow I.\]Building the equations from the flows
New infections move susceptible individuals into the exposed class at rate
\[\beta\frac{SI}{N}.\]Therefore,
\[\boxed{\frac{dS}{dt}=-\beta\frac{SI}{N}}.\]The exposed class gains new infections and loses individuals as they become infectious:
\[\boxed{\frac{dE}{dt}=\beta\frac{SI}{N}-\sigma E}.\]The infectious class gains individuals from \(E\) and loses them through recovery or removal:
\[\boxed{\frac{dI}{dt}=\sigma E-\gamma I}.\]The recovered/removed class receives those leaving the infectious class:
\[\boxed{\frac{dR}{dt}=\gamma I}.\]Again, each equation can be read as rate in − rate out.
What do \(\sigma\) and \(\gamma\) mean?
The parameter \(\sigma\) is the per-capita rate of progression from exposed to infectious. Under the usual constant-rate assumption, the mean time spent in \(E\) is
\[\boxed{\frac{1}{\sigma}}.\]Similarly, \(\gamma\) is the recovery/removal rate and the corresponding mean infectious period under the same assumption is
\[\boxed{\frac{1}{\gamma}}.\]What difference does the exposed stage make?
The exposed compartment introduces a delay between new infections and the appearance of new infectious individuals. As a result, changes in transmission are not reflected immediately in \(I(t)\). This can change the timing of epidemic growth and the epidemic peak compared with a model that assumes immediate infectiousness.
Notice that exposed individuals are already infected even though they are not yet contributing to transmission in this basic formulation. Therefore, observing only the infectious population can give an incomplete picture of infections already present in the system.
When is an SEIR model useful?
SEIR is useful when the delay from infection to infectiousness is important for the question being studied. It has been widely used for infections with a meaningful latent stage, including models of COVID-19 and measles.
Whether SEIR is appropriate depends on the biological detail required. For some diseases, individuals may become infectious before symptoms appear, or infectiousness may vary through several stages. Those situations may require additional compartments rather than a single exposed class.
When is a simpler or more detailed model better?
If the delay between infection and infectiousness is negligible for the purpose of the study, an SIR model may be sufficient. If immunity is temporary, an SEIRS-type model can allow recovered individuals eventually to become susceptible again. Age structure, vaccination, hospitalisation, disease-induced death or multiple infectious stages can also be added when they are important to the modelling question.