Part 4: Stochastic Differential Equations

An SDE combines deterministic drift with continuous random diffusion. This pathway builds the idea carefully, derives epidemic models from biological events, simulates independent paths, and checks both numerical and interpretive validity.

Stage 1 · Understand stochastic change

Foundations of an SDE

Move from expected deterministic change to drift, diffusion and Brownian increments.

01From deterministic change to stochastic changeExpected change versus random realised change.02Drift, diffusion and biological meaningDirection, fluctuation scale, units and event-rate derivation.03Brownian motion and normal random incrementsContinuous Brownian paths and independent normal increments.04Simulating ΔW = √Δt ZProgram and verify square-root time scaling.

Stage 2 · Build SDE models

Numerical method and epidemic construction

Implement Euler–Maruyama, then derive SIS, SIR and SEIR diffusions from biological events.

05Euler–Maruyama method from scratchCombine drift and Brownian diffusion changes step by step.06Constructing a stochastic SIS modelDerive one-dimensional drift and diffusion from two events.07Constructing a stochastic SIR modelUse shared event noise to conserve population.08Constructing a stochastic SEIR modelAdd latent progression and a state-change matrix.

Stage 3 · Simulate and summarise

One path, ensembles and uncertainty

Separate possible histories from empirical distributions and correctly labelled uncertainty summaries.

09Programming one SDE trajectoryPackage inputs, randomness, storage and diagnostics.10Programming many independent SDE trajectoriesVectorise independent paths on a common grid.11Mean paths, variance and uncertainty bandsDistinguish path spread, percentiles and Monte Carlo error.

Stage 4 · Evaluate and communicate

Comparison, numerical validity and reporting

Assess when the approximation is useful, diagnose its numerical behaviour and communicate results.

12Comparing deterministic, CTMC and SDE resultsCompare matched models and their extinction behaviour.13Time-step choice, negative values and model conditionsCheck boundaries, refinement and coefficient conditions.14Path plots, density plots, bands, histograms and tablesMatch stochastic questions to accurate displays.

Coverage rule

Lessons 1–4 explain the stochastic ingredients. Lessons 5–8 construct numerical and biological models. Lessons 9–11 simulate and summarise paths. Lessons 12–14 compare, validate and communicate results. Later lessons reuse earlier code but focus only on their stated new purpose.