Updated 12 October 2026 · By Pete Bromfield, former Head of Maths with IB examining experience

IA idea · Differential equations & dynamics

How fast would flu spread through my school? An SIR model

AI HLAA HL Ambitious Also in: Health & biology, Statistics

Research question

How do the infection and recovery rates in an SIR model affect the peak number of students ill at once in a school of my school's size, and what vaccination level would keep the peak below 5%?

Make it yours, then agree it with your teacher.

Next step: plan this IA free — the planner opens with this question filled in. For feedback on your own draft, the IA package (€39 once) gives you 5 AI feedback credits and every annotated exemplar in full.

Why it makes a good exploration

The SIR model became famous during COVID-19. Building one for your own school, solving it with Euler's method, and asking what vaccination level prevents a big outbreak is serious applied mathematics with a clear decision at the end.

The mathematics you'll need

Coupled differential equations dS/dt, dI/dt, dR/dt
Euler's method in a spreadsheet; effect of step size
The basic reproduction number R₀ = β/γ
Sensitivity analysis on parameters
Herd-immunity threshold 1 − 1/R₀

Where the data comes from

Use published values of R₀ and infectious period for influenza from public-health sources; optionally compare with anonymised absence counts from your school (with permission).

WHO Global Health Observatory — 1,000+ health indicators by country (immunisation, mortality, risk factors).

Real data for this idea

Primary source: WHO Global Health Observatory: FluNet (global influenza surveillance). Weekly positive flu tests by country: use one season to judge whether your parameters give a sensible peak. Licence: CC BY-NC-SA 3.0 IGO: non-commercial use only, credit WHO, share alike.
Readable chart: Our World in Data: Influenza Data Explorer. Use it to see the pattern; download your numbers from the primary source or the chart's own download. Cite as: Our World in Data, ‘Influenza Data Explorer’, CC BY 4.0. Original source: World Health Organization, FluNet.

WHO data is licensed for non-commercial use only (CC BY-NC-SA 3.0 IGO): fine for a school project, and credit WHO. Links checked October 2026; we link to the data and never copy it. More free data sets.

A possible outline

Explain the compartments and the assumptions (well-mixed population).
Choose and justify parameters.
Solve numerically and find the peak.
Vary vaccination levels to find the threshold.
Reflect: schools are not well mixed; weekends and classes matter.

Pitfalls that cost marks

Parameters with no source.
Euler's method with a step too large, giving negative numbers.
Presenting the model as a prediction for your real school.

Showing personal engagement

Model your own school's size and structure.
Split the school into year groups with different mixing.
Compare the model with a real outbreak in the news.

See Criterion C: personal engagement for what examiners look for.

Which course is it for?

Course	Fit	Maths to lean on
AA SL	Not a natural fit	The core technique sits in the AI course or at HL; an AA SL student could use it only as clearly explained new mathematics.
AA HL	Good fit	Coupled differential equations dS/dt, dI/dt, dR/dt; Euler's method in a spreadsheet; effect of step size
AI SL	Not a natural fit	The mathematics is mainly AA or HL (calculus or proof beyond AI SL); an AI SL version would need a data-driven, technology-based approach.
AI HL	Good fit	Coupled differential equations dS/dt, dI/dt, dR/dt; Euler's method in a spreadsheet; effect of step size

Level: Ambitious. Suits confident students; expect to learn some mathematics on your own. See how the IA differs between AA and AI, SL and HL.

How this idea reaches the top bands

Reflection (D)

For this idea, start with the weak points: parameters with no source; and euler's method with a step too large, giving negative numbers. Say how each one affects your answer.

Use of mathematics (E)

HL: coupled differential equations dS/dt, dI/dt, dR/dt; euler's method in a spreadsheet; effect of step size; the basic reproduction number R₀ = β/γ; sensitivity analysis on parameters; herd-immunity threshold 1 − 1/R₀, each explained in your own words and carried out correctly.

How differential equations explorations reach the top bands in general: Differential equations & dynamics IA ideas. Guides: A · B · C · D · E · self-grade sheet.

Taking it further

Add a latent (Exposed) compartment for an SEIR model and compare peak timing.

Extending it for HL

Add an HL technique the question genuinely needs: what an HL differential equations exploration adds.

See a complete IA, marked

Is it cheaper to leave the heating on low overnight? A two-temperature model of my bedroom (AI HL) is a complete differential equations exploration, marked on every criterion.

Frequently asked questions

Which IB Maths course is this IA idea for?

It works for AI HL and AA HL. We rate it Ambitious: it suits confident students, who should expect to learn some mathematics on their own.

What mathematics does this IA use?

Coupled differential equations dS/dt, dI/dt, dR/dt, euler's method in a spreadsheet; effect of step size, the basic reproduction number R₀ = β/γ, sensitivity analysis on parameters and herd-immunity threshold 1 − 1/R₀.

Where can I get data for this IA?

Use published values of R₀ and infectious period for influenza from public-health sources; optionally compare with anonymised absence counts from your school (with permission). Checked sources and tools: WHO Global Health Observatory and Our World in Data.

Turn this idea into your IA

The planner opens with “How fast would flu spread through my school? An SIR model” filled in: the question, the maths and the data sources.

Plan this IA free

Your plan is kept in this browser. A free account keeps it with your drafts on any device: Join free · Sign in

Also: free IA self-check · A–E checklist (PDF) · have a draft? AI feedback or a review with Pete (€100)

Writing it up: how to write a Maths IA, step by step · structure and format · mistakes that cost marks · IA modelling, step by step