The <SEIR model> inserts a latent exposed state:
$$
S\xrightarrow{\lambda(t)}E\xrightarrow{\sigma}I\xrightarrow{\gamma}R.
$$
The expected time from infection through the end of infectiousness is $d=1/\sigma+1/\gamma$; the duration during which transmission occurs is $1/\gamma$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 207 / 2 / d / ii / Solution

The linear <SEIR model> with a constant susceptible-to-exposed intensity is
$$
\dot S=-q_{SE}S,
\qquad
\dot E=q_{SE}S-q_{EI}E,
\qquad
\dot I=q_{EI}E-q_{IR}I,
\qquad
\dot R=q_{IR}I.
$$
Adding the four equations confirms <conservation of population>: $S+E+I+R=N$.

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 207 / 2 / d / i / Solution

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{wiki=Compartmental_models_in_epidemiology#The_SEIR_model}

The SEIR model divides a population into susceptible, exposed, infectious, and recovered compartments.


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