Maths IA · Modelling · topic 39 of 40

Rumour spread: how does the contact rate matter? · IB Maths IA idea: guide, sample graph and step-by-step plan

stretch topic: keep the model small

A rumour crosses a school in days; a disease takes weeks. Same maths, different numbers. You're going to build the spread yourself and see what really matters.

Difficulty●●● Stretch
KitA spreadsheet or Python, plus your GDC
TimeAbout 14 to 18 hours
FitsAA and AI, HL only
  • ✓ you like building a model from rules
  • ✓ you can code or use a spreadsheet with random numbers
  • ✓ you want differential equations to explain something real

The big idea

Model a closed group where each person has not heard it, is spreading it, or has stopped. Each day a spreader meets some people and passes it on with some chance. A random simulation shows what happens in practice; a differential equation such as dI/dt = βI(1 − I/N) shows what happens on average. Connect the two, and test how the speed changes with the contact rate. State and test every assumption: well mixed, fixed contacts, nobody forgets. This is an HL topic: solving or numerically approximating the equations is the maths criterion E rewards.

Make it yours: pick one twist

Copying a classic won't show your choices. Pick one of these (or invent your own) and your IA becomes yours.

Swipe for more twists →

Add stopping

People lose interest (or recover) at rate γ, which gives an SIR model.

→ Two or three coupled equations; the spread peaks and falls; you can find the peak and the final size.
Vary the contact rate

Run the simulation at several contact rates.

→ A graph of spread speed against contact rate, and a test of whether doubling contacts doubles speed.
Solve the simple model exactly

Separate variables and use partial fractions for the no-stopping case.

→ An exact answer to check the simulation against. At AA HL this is the core of the maths.
Use a numerical method

Euler's method with several step sizes, or an improved method.

→ Error and convergence become part of the analysis.
Add structure

Two groups (for example two year groups) who mix differently.

→ A system of equations to set up, and a comparison with the one-group model.
Bring in real data

Absence records, or published flu data.

→ You see how far the model bends towards reality and where it fails.

Your Maths IA research question: run the check

define 'speed' in the RQ

Write your own question first. Then make sure it ticks every box:

  • States which process you model (a rumour, a disease) and the group it spreads in, with the assumptions listed.
  • Names the one quantity you vary (such as the contact rate) and the one you measure (such as the time to reach half the group).
  • Says how the simulation and the equations will be compared.
  • Defines what 'speed' means: time to half the group, the largest daily rise, or something else.
  • Keeps the model small enough that you can explain every assumption.

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The model

A simple flow: people move from 'not heard' to 'spreading' to 'stopped'. The dots show what one simulated day does: spreaders meet random people.

SIRβγwho has heard it, who is passing it on?β = contacts per day x chance it passes onγ: how fast interest fadesclosed class: N stays fixed

How your Maths IA could be laid out

The usual order for a topic like this, what each part needs to show, and roughly how much of your 12 to 20 pages it takes. The words are all yours.

Introduction and aim~1.5 pages
Your question; why this rumour or disease; what you expect contact rate to do and why.
Assumptions and the model~2.5 pages
The rules in words, then as equations; every symbol defined with units; assumptions listed and justified.
The simulation~2 pages
How it works (a short flow chart or list of rules); how many runs; how you handle randomness. Code in an appendix, not the body.
The equations~3 pages
HL: solve by separation of variables and partial fractions, or approximate with a stated method and step size, and show the error.
Comparing simulation and equations~2 pages
Same parameters, same graph; explain the differences (early randomness, small numbers).
Varying the contact rate~2 pages
Your investigation: speed against contact rate, with a clear definition of speed.
Conclusion~1 page
Answer the question with numbers and say how far the model can be trusted.
Reflectionbuilt in, plus ~0.5 page
Not one paragraph at the end: a comment after each result or choice (a poor fit, a surprise, a decision you changed), then a specific evaluation of the model's limits.
Bibliographynot counted
Every source for data, theory and any code or software used.

What your Maths IA graph could look like

bars = spread between runs ✓

Made-up values to show the form of a strong graph, not the answer: a rumour in a class of 200. Yours will look different. example only

02468101214020406080100120140160180200Time t / daysPeople who know it, I (out of N = 200)equation: dI/dt = βI(1 − I/N), β = 0.8contact rate halved: β = 0.4simulation: mean of 20 runs, spread shownwhere is the spread fastest?what is dI/dt there?ILLUSTRATIVE · MADE-UP DATA
  • Time in days and a count out of N; say N on the axis.
  • Simulation points are the mean of several runs. The bars show the spread between runs, not a measurement error.
  • The equation and a changed contact rate sit on one set of axes, so the effect is visible.
  • The steepest part of the S-curve is where the spread is fastest: compare it with what the equation predicts.
  • Another graph to consider: the daily rise against I for the simulated data. For this equation it should be a parabola peaking at I = N/2.

Does a smaller step size help?

how small a step do you need?

If you use Euler's method, show the error shrinking as the step shrinks, and say which step you chose and why. Here the value at day 8 is checked against the exact solution. The step needed is smaller than you might guess.

Step 2 days72.1%Step 1 day44.8%Step 0.5 days21.8%Step 0.25 days10.1%Step 0.1 days3.8%step size matters: say how you chose it% error in I at day 8 compared with the exact solutionILLUSTRATIVE · MADE-UP NUMBERS

Your game plan

do the 20-person trial by hand!

Tick steps off as you go. We'll remember on this device.

0 of 10 done

  1. Choose the storyRumour or disease, and in what group. Write down who mixes with whom and what counts as 'spreading'.
  2. Write the rulesList every assumption: group size, contacts per day, chance of passing it on, when someone stops. Say which you will test.
  3. Trial run (1 lesson)Simulate 20 people by hand with dice or a spreadsheet. Do the rules make sense? Does the spread always start the same way?
  4. Build the simulationFix the rules, run many repeats and record the mean and the spread. Keep the code for an appendix.
  5. Set up the equationsLink β to your simulation rules (contacts per day and chance of passing it on). Define every symbol and unit.
  6. Solve or approximateSolve exactly if you can. If you use a numerical method, state the step size and show the error shrinking.
  7. ComparePlot simulation and equation together. Explain the differences instead of tuning parameters to hide them.
  8. Vary the contact rateRun your investigation. Define 'speed' before you measure it.
  9. Write, then cutWrite in layout order. Explain the rules in words and mathematics, not screenshots of code.
  10. Check before your teacher's draftCheck every assumption is stated, every symbol defined, graphs titled with units, and reflection runs through the piece.

Maths you'll need

Traps to avoid in this Maths IA

  • A simulation with no stated rules: if the reader cannot see the assumptions, they cannot judge the results.
  • Running the simulation once. Random spread needs many runs, averaged, with the spread shown.
  • Tuning β until the curve matches, then saying the model 'agrees'.
  • Euler's method with one step size and no error check.
  • Predicting the real world from a model that assumes everyone mixes equally.
  • Parameters (β, γ) with no units or meaning.
  • Code pasted as screenshots instead of explained in words and mathematics.

More Maths IA ideas like this

Make sure it's yours

This guide shows how a topic like this is usually approached. It is not an IA. Your question, choices, data, analysis and words must be your own: change something, tell your teacher where the idea came from, and follow your school's rules on outside help.

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