Worked example · Maths · Year 11
DCF: Data and computational thinking → Problem-solving and modelling
A completed Year 11 SIR model with N = 1000, S(0) = 995, I(0) = 5, β = 0.40 and γ = 0.10, every value rounded to 1 decimal place. You can check any row against the row above it with a calculator — the numbers below are exactly what those recurrences produce. Show it after pupils have built their own model, not before.
The written recurrences have become working formulae with the parameters absolutely referenced, so changing H2 re-runs the whole outbreak — first and second success criteria.
R0 = β ÷ γ = 0.40 ÷ 0.10 = 4. R0 > 1, so the outbreak will grow. One infectious person passes it to four others while almost everyone is still susceptible, and the illness lasts 1 ÷ 0.10 = 10 days on average.
Interprets the parameters before trusting any output — the start of the fifth criterion.
| Day | S | I | R | New inf. | New rec. | S+I+R |
|---|---|---|---|---|---|---|
| 0 | 995.0 | 5.0 | 0.0 | 2.0 | 0.5 | 1000.0 |
| 1 | 993.0 | 6.5 | 0.5 | 2.6 | 0.7 | 1000.0 |
| 2 | 990.4 | 8.4 | 1.2 | 3.3 | 0.8 | 1000.0 |
| 3 | 987.1 | 10.9 | 2.0 | 4.3 | 1.1 | 1000.0 |
| 4 | 982.8 | 14.1 | 3.1 | 5.5 | 1.4 | 1000.0 |
| 5 | 977.3 | 18.2 | 4.5 | 7.1 | 1.8 | 1000.0 |
| 6 | 970.2 | 23.5 | 6.3 | 9.1 | 2.4 | 1000.0 |
| 7 | 961.1 | 30.2 | 8.7 | 11.6 | 3.0 | 1000.0 |
| 8 | 949.5 | 38.8 | 11.7 | 14.7 | 3.9 | 1000.0 |
| 9 | 934.8 | 49.6 | 15.6 | 18.5 | 5.0 | 1000.0 |
| 10 | 916.3 | 63.1 | 20.6 | 23.1 | 6.3 | 1000.0 |
| 11 | 893.2 | 79.9 | 26.9 | 28.5 | 8.0 | 1000.0 |
| 12 | 864.7 | 100.4 | 34.9 | 34.7 | 10.0 | 1000.0 |
| 13 | 830.0 | 125.1 | 44.9 | 41.5 | 12.5 | 1000.0 |
| 14 | 788.5 | 154.1 | 57.4 | 48.6 | 15.4 | 1000.0 |
Check Day 0 to Day 1 by hand: new infections = 0.4 × 995 × 5 ÷ 1000 = 1.99 → 2.0; new recoveries = 0.1 × 5 = 0.5. So S(1) = 995.0 − 2.0 = 993.0, I(1) = 5.0 + 2.0 − 0.5 = 6.5, R(1) = 0.0 + 0.5 = 0.5. Check Day 11 to Day 12: 0.4 × 893.2 × 79.9 ÷ 1000 = 28.546… → 28.5 and 0.1 × 79.9 = 7.99 → 8.0, giving S = 864.7, I = 79.9 + 28.5 − 8.0 = 100.4, R = 26.9 + 8.0 = 34.9.
The final column is the model's own test: S + I + R stays at 1000.0 on every row, which it must because nobody enters or leaves the community — third success criterion.
Uses the model to answer the question it was built for: the peak and its timing — fourth success criterion.
R0 = 4.
Peak day: day 24. Peak size: I = 424.7, so roughly 425 of the 1000 people are infectious at once.
Susceptible at the peak: S = 210.1 on day 24, and S = 253.0 the day before. N ÷ R0 = 1000 ÷ 4 = 250, so S crosses 250 between day 23 and day 24 — exactly when I stops rising. That is not a coincidence: I only grows while βSI/N is bigger than γI, which rearranges to S > Nγ/β = N ÷ R0.
With β = 0.20: R0 = 0.20 ÷ 0.10 = 2. The peak moves to day 53 and falls to I = 159.6, with S = 489.7 still susceptible at that point (and N ÷ R0 = 500, which S has just dropped below). Halving the transmission rate has cut the peak by about 62% and pushed it back 29 days.
One parameter changed, the effect recorded numerically, and the theory used to predict where the turning point should be — fifth success criterion.
With β = 0.40 the model says the outbreak takes off slowly — only 14 people are infectious after four days — then rises steeply to a peak of about 425 simultaneous cases on day 24 before burning out as the susceptible pool empties. Halving β to 0.20 does not prevent the outbreak, because R0 is still 2, but it flattens it: the peak drops from 424.7 to 159.6 and arrives on day 53 instead of day 24. That extra month is the whole argument for reducing contact early — the same number of beds copes with a far smaller peak.
Two assumptions make the prediction unreliable. First, the model assumes homogeneous mixing: every person is equally likely to meet every other person each day. Real communities cluster into forms, households and friendship groups, which slows transmission between clusters, so this model will tend to overstate the peak and make it earlier than reality. Second, it assumes β and γ are constant. In a real outbreak people change their behaviour once they hear about it, and public health measures arrive, so β falls over time; a constant β again overstates the total number infected. A third, smaller issue is that the model is continuous — it happily reports 424.7 infectious people, when the true count must be a whole number, and with only five initial cases random chance could plausibly have ended the outbreak on day 1.
Judges the limits of the model and states the direction of each bias, rather than reporting the output as a fact.