the SEIR model
/ ess-ee-eye-ar /
Many real diseases have a hidden waiting room: after you catch the virus you are not yet contagious, because it needs days to build up inside you. The SEIR model adds exactly this delay. It inserts a fourth bin, E for Exposed — people who have been infected but are not yet able to infect others. So the flow becomes S to E to I to R: susceptible, then incubating, then infectious, then removed. That one extra compartment makes the model honest about the latency we feel in almost every infection.
The equations are S' = -beta S I, E' = beta S I - sigma E, I' = sigma E - gamma I, and R' = gamma I. Compared with SIR, the new infections beta S I now pour into E rather than straight into I. The exposed then 'mature' into infectious at rate sigma, so 1/sigma is the average latent (incubation) period. The infectious recover at rate gamma as before, with 1/gamma the infectious period. The disease must now pass through E before it can spread, which slows the takeoff and reshapes the whole curve.
Why bother with E? Because the latent period changes the timing dramatically. The epidemic grows more slowly at first and peaks later, which matters enormously when you are racing to prepare hospitals or roll out a vaccine. Importantly, the latent stage does NOT change the basic reproduction number itself — R0 still comes out as beta times the susceptible pool times the infectious duration 1/gamma — but it changes the SPEED of growth and the doubling time. SEIR was the workhorse for diseases like measles, SARS, and COVID-19, precisely because their incubation periods are too long to ignore.
It is still a smooth, well-mixed, deterministic picture. Real incubation times vary from person to person (the model uses one average rate sigma), people change behaviour as they learn of the outbreak, and early on, when only a handful are infected, chance dominates and a continuous model can mislead. SEIR adds realism in one specific dimension — latency — without pretending to be the whole truth.
Suppose the incubation period averages 5 days, so sigma = 1/5 = 0.2, and the infectious period averages 10 days, so gamma = 0.1. A newly infected person sits in E for about 5 days, silently incubating, then moves to I and spreads the disease for about 10 days before recovering — exactly the staggered timeline SIR cannot represent.
The E compartment delays a person's infectiousness, slowing the early rise of the epidemic.
Adding the E compartment slows the epidemic's growth but does not by itself lower R0; an outbreak with a long incubation period can still infect just as many people in the end — it simply takes longer to get there.