compartmental-models

Overview

compartmental-models covers the workhorse models of infectious-disease dynamics: SIR, SEIR, and their extensions. It focuses on building models that inform decisions — estimating R0, projecting under interventions, and communicating uncertainty — rather than on mathematical novelty. A model is a decision tool; its value is measured in better decisions, not in fit statistics.

When to use

• Estimating R0/Rt from early outbreak data.
• Projecting epidemic trajectories under different intervention scenarios.
• Evaluating interventions: vaccination coverage, NPIs, test-and-trace.
• Understanding herd-immunity thresholds and final-size relations.
• Fitting models to case/hospitalization data with proper uncertainty.
• Communicating model results and limits to decision-makers.

Core concepts

• SIR structure. Susceptible → Infectious → Recovered, with transmission rate β and recovery rate γ. dS/dt = −βSI/N, dI/dt = βSI/N − γI, dR/dt = γI. Simple, interpretable, and the foundation everything else extends.
• SEIR and extensions. Add Exposed (latent period) for SEIR; add compartments for asymptomatic (A), hospitalized (H), quarantined (Q), vaccinated (V), or waning immunity (SIRS) as the question demands. Every compartment needs a justification — complexity must earn its keep in better decisions.
• R0, Re, Rt. R0 = β/γ: expected secondary cases in a fully susceptible population. Re/Rt: effective reproduction number given immunity and interventions. Rt > 1 means growing. Estimate from the early exponential growth rate (r) via the generation-interval distribution — R0 ≈ 1 + r×Tg for simple models, but use the proper estimator (EpiEstim/cori for Rt).
• Herd immunity threshold. 1 − 1/R0 — the immune fraction at which Re < 1 without interventions. It's a threshold for decline, not eradication, and it assumes homogeneous mixing — heterogeneity (superspreading, structured contacts) changes it substantially.
• Homogeneous mixing is the big lie. Real populations have age structure, households, networks, and superspreading. Age-structured contact matrices (POLYMOD-style) are the minimum realism for policy models; overdispersion (k parameter) matters for outbreak probability and control strategy.
• Fitting. Fit to the most reliable data stream (hospitalizations/deaths over cases, which depend on testing). Use Bayesian inference (MCMC) or likelihood-based methods with proper observation models (negative binomial for overdispersed counts, reporting delays modeled explicitly). Fit to multiple streams jointly when possible.
• Forecasting discipline. Short-term forecasts (1-4 weeks) with quantified uncertainty (prediction intervals, ensemble approaches); scenario projections (not predictions) for policy — "if X, then Y" with X explicit. Evaluate past forecasts honestly (coverage of prediction intervals).
• Intervention modeling. Vaccination (reduce susceptibility/infectiousness by coverage × efficacy), NPIs (reduce β or contacts), test-trace-isolate (reduce infectious period or remove infecteds). Model the intervention as actually implementable — 100% compliance scenarios are fantasies.

Practical workflow

1. Question. What decision does this model inform? (Not "model the epidemic" — a specific choice with options.)
2. Structure. Simplest model that captures the decision-relevant mechanism; justify each compartment.
3. Parameters. Literature values with sources and ranges; estimate the rest from data; document every number.
4. Fit. Bayesian/MCMC with observation model; check convergence, posterior predictive fits, and sensitivity to priors.
5. Scenarios. Baseline + intervention options with explicit assumptions; present as ranges across parameter uncertainty, not point trajectories.
6. Validate. Backtest on held-out periods; compare against simple baselines (a naive model that beats your SIR is telling you something).
7. Communicate. Assumptions up front, uncertainty as intervals, scenarios labeled as scenarios. Never present a single trajectory as "the prediction."

Example (Python sketch):

from scipy.integrate import odeint
def seir(y, t, N, beta, sigma, gamma):
    S, E, I, R = y
    return [-beta*S*I/N, beta*S*I/N - sigma*E, sigma*E - gamma*I, gamma*I]

Common pitfalls

• Fitting to case counts without modeling testing/reporting changes.
• Ignoring reporting delays (recent data are always incomplete — nowcasting needed).
• Homogeneous-mixing models for policy without age/behavior structure.
• Presenting scenarios as predictions.
• Overconfident narrow intervals (models are usually overconfident — widen honestly).
• Parameters from a different pathogen/population without adjustment.
• No validation against held-out data or simple baselines.