Between catching a disease and spreading it there is a wait. SEIR names that wait: an Exposed compartment — infected but not yet infectious — inserted between S and I. The same total attack; the same R₀; a later, lower peak. Down the center the parameters go in, four coupled ODEs integrate, and the delayed curve comes out. The blue team builds and defends it; the red team tries to break it. A model with named assumptions — not medical advice.
source R. M. Anderson & R. M. May, Infectious Diseases of Humans: Dynamics and Control (Oxford Univ. Press, 1991) — the standard reference for the compartmental SEIR form. No stable free full text; see the book record. Rendered, not quoted.
SEIR splits the population N = S+E+I+R and moves it by mass-action:
dS/dt = −β·S·I/N
dE/dt = +β·S·I/N − σ·E
dI/dt = +σ·E − γ·I
dR/dt = +γ·I
β transmission, 1/σ the latent period (E→I), 1/γ the infectious period (I→R). Only I transmits; E is infected-but-quiet. The four flows are the only way mass moves — every term that leaves one box enters the next, so N is conserved exactly (window 7 re-checks it).
Live conservation for the current run: —
Take the-sir-model and insert one box. In SIR a susceptible becomes infectious the instant they are infected. Real diseases have an incubation — measles ~10 days, COVID ~5 — during which the host carries the pathogen but sheds none.
Add that latent stage and SIR becomes SEIR. The exposed all eventually become infectious, so R₀ = β/γ is unchanged — but the peak is pushed back and flattened. Each sphere is the next one's premise: SIR is the frame SEIR delays.
The blue team's live check: re-integrate SEIR and SIR at the reference parameters and confirm the invariants — conservation, the delayed peak, and equal final size (R₀ intact). If red makes E infectious, the final sizes split and this badge turns red.
Feed the model three rates and a starting seed. Everything else is derived:
| symbol | meaning | drives |
|---|---|---|
| β | transmission rate | speed of spread |
| 1/σ | latent period (days) | the delay E→I |
| 1/γ | infectious period (days) | recovery I→R |
| N, I₀ | population, seed | scale |
R₀ = β/γ. Assumptions declared: one well-mixed population, constant rates, no births/deaths, no age or spatial structure. A model — not a forecast, not advice.
RK4, fixed step Δt=0.1 d over 200 d. The SIR curve (same β, γ) is drawn dashed for comparison — watch the peak slide right as you lengthen the latent period.
What the machine proves, live: mass is conserved (S+E+I+R=N to 1e-9); the latent stage delays the peak versus SIR and flattens the early exponential growth (rSEIR < β−γ); and the final attack rate is identical to SIR — the same z solving z = 1 − e−R₀z — because R₀ = β/γ is untouched by the wait.
The blue witness (left) confirms these live; the red team (right) tries to make them false.
SEIR is a lens, not the territory. It tells you the shape of a delay, not a date. Fitting it to real data recovers effective β, σ, γ that fold in behaviour and reporting — quantities that drift week to week. This panel is a teaching model; it is not medical advice.
"The E box makes the outbreak bigger." Cut. It does not. Final size depends only on R₀=β/γ; the latent stage leaves total infections unchanged — it moves the peak in time, lower and later, never larger.
"Exposed people spread the disease." Cut. By definition E is infected-but-not-infectious. Presymptomatic shedding is real, but that is a modelling choice you'd add explicitly — do it silently and R₀ is wrong (see window 6).
"1/σ is the incubation period you observe." Kept, corrected. The latent (infectiousness) period and the clinical incubation (symptom) period differ; SEIR's σ governs the former.
The red team's move: let the Exposed transmit too — sneak a +β·S·E/N term into the flow. Now every case infects during both E and I, so R₀ inflates past β/γ and the final size blows up. The blue witness (window 7) is integrating both models and watching.
Add exposed transmission and the SEIR final size overshoots SIR's — the witness recomputes, the attack rates split, and the badge turns red. Nothing is faked; the attack is real and it is caught.