273. SIR SEIR
Compartmental models of epidemic spread. The mathematical foundation of mainstream epidemiology.
273.1. SIR model
Three stocks:
- : Susceptible
- : Infected
- : Recovered (immune)
Equations:
with:
- : transmission rate (contacts × probability per contact)
- : recovery rate ( = mean infectious period)
- (total population, conserved)
273.2. Basic reproduction number
The most-cited epidemic metric:
Interpretation: average number of secondary infections from one infected person in a fully-susceptible population.
- : epidemic grows
- : epidemic dies out
- : marginal — slow spread
For COVID-19 wild-type: . For measles: . For influenza: .
273.3. Herd immunity threshold
The fraction of population that must be immune to halt spread:
For : 67% needs to be immune. For (measles): 94%.
Vaccines / prior infection drive immune fraction up; once you cross the threshold, sustained transmission is impossible (in this simple model).
273.4. SEIR: add Exposed
Adds an Exposed (infected but not yet contagious) compartment:
with = mean incubation period.
For COVID-19, SEIR is more realistic (significant incubation period).
273.5. Behavior
Epidemic curve typically S-shaped (in cumulative cases) and bell-shaped (in new cases per day):
- Exponential growth phase:
- Peak when (= herd-immunity threshold)
- Decay phase
273.6. Variants
- SIRS: lose immunity over time (return ); endemic equilibrium possible
- SIS: no immunity at all (return ); endemic equilibrium with constant
- Age-structured: multiple age compartments with different mixing rates
- Spatial: include geographic structure
- Network: contacts on a graph; heterogeneous mixing
273.7. Where used
- Public health planning: vaccination targets, lockdown timing
- Pharmaceutical R&D: market size estimation for vaccines / antivirals
- Risk modeling: insurance, business continuity
- Behavioral economics: viral information spread (memes, news)
273.8. Limitations
- Well-mixed assumption: real contacts are heterogeneous
- Constant parameters: real changes with seasonality, behavior, variants
- Deterministic: small populations need stochastic models (Gillespie, branching processes)
273.9. See also
- Logistic Growth — similar S-curve dynamics
- Stocks and Flows — modeling language
- Feedback Loops — the underlying R+B structure