Part 5: Moment Equations

Moment equations describe the evolving centre, spread and dependence of stochastic epidemic variables. This pathway begins with simulated samples, derives exact identities, introduces closure only when necessary, and ends with validated probability applications.

Stage 1 · Measure stochastic outcomes

Random variables and empirical moments

Define epidemic variables and estimate their centre, spread and dependence from independent simulations.

01Random variables and simulated epidemic samplesDistinguish variables, realisations, samples and empirical distributions.02Expectation and the sample meanEstimate a model expectation using independent sample means.03Second moments, variance and covarianceMeasure spread and joint variation between epidemic quantities.04Estimating moments from Monte Carlo simulationsSeparate intrinsic variance from finite-simulation error.

Stage 2 · Derive and close moment equations

From transition rates to a finite system

Derive exact low-order identities, identify nonlinear hierarchy and state closure assumptions.

05Deriving first-moment equationsDerive expected change from event sizes and rates.06Deriving second-moment equationsUse squared jump changes to derive variance dynamics.07Why nonlinear models produce unclosed momentsSee how SIS nonlinearity raises moment order.08Moment-closure approximations and assumptionsCompare normal and lognormal closures with simulations.

Stage 3 · Program, solve and validate

Closed moment ODEs

Translate retained moments into code, solve them and assess closure accuracy.

09Programming moment equations as an ODE systemBuild a solver-compatible retained-moment function.10Solving moment equations numericallyIntegrate closed equations and validate reconstructed variance.11Expected infectious population and its varianceInterpret burden, spread and relative variability.12Comparing moment solutions with simulationsValidate mean and variance against matched CTMC samples.

Stage 4 · Convert moments into risk information

Probabilities, capacity and communication

Add explicit distributional assumptions, validate probability estimates and communicate results.

13Approximate extinction and outbreak probabilitiesUse continuity-corrected normal approximations cautiously.14Hospital-capacity exceedance probabilitiesTransform prevalence moments into critical-care demand risk.15Mean, variance, covariance and probability visualisationsMatch each quantity to an accurate display and table.

Coverage rule

Lessons 1–4 establish empirical moments. Lessons 5–8 derive and close equations. Lessons 9–12 program, solve and validate them. Lessons 13–15 add explicitly assumption-dependent probability and capacity applications. Each later lesson reuses earlier mathematics only to teach its stated new purpose.