275. SIR SEIR

Compartmental models of epidemic spread. The mathematical foundation of mainstream epidemiology.

275.1. SIR model

Three stocks:

• $𝑆$: Susceptible
• $𝐼$: Infected
• $𝑅$: Recovered (immune)

Equations:

with:

• $𝛽$: transmission rate (contacts × probability per contact)
• $𝛾$: recovery rate ($\frac{1}{𝛾}$ = mean infectious period)
• $𝑁=𝑆+𝐼+𝑅$ (total population, conserved)

275.2. Basic reproduction number ${𝑅}_0$

The most-cited epidemic metric:

Interpretation: average number of secondary infections from one infected person in a fully-susceptible population.

• ${𝑅}_0>1$: epidemic grows
• ${𝑅}_0<1$: epidemic dies out
• ${𝑅}_0=1$: marginal — slow spread

For COVID-19 wild-type: ${𝑅}_0\approx 2.5-3$. For measles: ${𝑅}_0\approx 12-18$. For influenza: ${𝑅}_0\approx 1-2$.

275.3. Herd immunity threshold

The fraction of population that must be immune to halt spread:

For ${𝑅}_0=3$: 67% needs to be immune. For ${𝑅}_0=18$ (measles): 94%.

Vaccines / prior infection drive immune fraction up; once you cross the threshold, sustained transmission is impossible (in this simple model).

275.4. SEIR: add Exposed

Adds an Exposed (infected but not yet contagious) compartment:

with $\frac{1}{𝜎}$ = mean incubation period.

For COVID-19, SEIR is more realistic (significant incubation period).

275.5. Behavior

Epidemic curve typically S-shaped (in cumulative cases) and bell-shaped (in new cases per day):

• Exponential growth phase: $𝐼\left(𝑡\right)\approx {𝐼}_0{𝑒}^{\left(𝛽-𝛾\right)𝑡}$
• Peak when $𝑆\left(𝑡\right)=\frac{𝑁}{{𝑅}_0}$ (= herd-immunity threshold)
• Decay phase

275.6. Variants

• SIRS: lose immunity over time (return $𝑅\to 𝑆$); endemic equilibrium possible
• SIS: no immunity at all (return $𝐼\to 𝑆$); endemic equilibrium with constant $𝐼$
• Age-structured: multiple age compartments with different mixing rates
• Spatial: include geographic structure
• Network: contacts on a graph; heterogeneous mixing

275.7. 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)

275.8. Limitations

• Well-mixed assumption: real contacts are heterogeneous
• Constant parameters: real ${𝑅}_0$ changes with seasonality, behavior, variants
• Deterministic: small populations need stochastic models (Gillespie, branching processes)

275.9. See also

• Logistic Growth — similar S-curve dynamics
• Stocks and Flows — modeling language
• Feedback Loops — the underlying R+B structure