Cover lesson · Maths · Year 11 · Advanced

Mathematical modelling of epidemics

DCF: Data and computational thinking → Problem-solving and modelling

Name: Class: Date:

Cover teacher — you need no subject knowledge

Print: this sheet, one per pupil. Pupils build the model in Excel or Google Sheets. If there are no computers, pupils complete the paper table on page 2 with a calculator — the recurrence relations are written out for them and each row only needs four multiplications. Either route meets the criteria.

Run: 5 min read the brief · 10 min set up the parameter block and row for Day 0 · 15 min write the recurrences into row 3 and fill down to Day 30 (or complete 14 rows on paper) · 10 min plot or sketch the three curves and find the peak · 10 min change β and write the two-paragraph evaluation.

Collect: this sheet. Spreadsheets are saved as sir-model-INITIALS in the class folder. Nothing goes outside school.

What you are doing

A closed community of 1000 people — say a boarding school or a small Welsh island community — has five infectious people in it on Day 0. You are going to build the standard SIR model of that outbreak, run it for 30 days, find the day the outbreak peaks, and then test what happens if people cut their contacts. You will finish by judging how much the model can honestly be trusted.

Step by step

5 min — Read the model definition below until you can say in your own words what S, I and R each count.
10 min — Set up the parameter block and the Day 0 row exactly as printed.
15 min — Type the five recurrence formulae into row 3 and fill down to Day 30. On paper, complete 14 rows instead.
10 min — Plot S, I and R against day as a line graph (or sketch it in the frame). Ring the peak of I and record the day and the value.
10 min — Change β from 0.40 to 0.20, record what happens to the peak, then write your evaluation.

The model

Everybody in the community is in exactly one of three compartments, and the three always add up to N:

S — susceptible: not yet infected, could catch it.
I — infectious: has it now and can pass it on.
R — removed: recovered and immune, so out of the chain.

Parameters

Cell	Name	Value	Meaning
H1	N	1000	Total population, which never changes.
H2	β (beta)	0.40	Transmission rate: the average number of people one infectious person would infect per day if everyone else were susceptible.
H3	γ (gamma)	0.10	Recovery rate: the fraction of infectious people who recover each day. 0.10 means a 10-day average illness.

The basic reproduction number is R₀ = β ÷ γ. Work it out for these values before you start — it tells you whether the outbreak grows at all.

The recurrence relations

Write n for the day number. New cases on day n depend on how many infectious people there are and what fraction of the population is still susceptible:

new infections on day n = β × S(n) × I(n) / N new recoveries on day n = γ × I(n) S(n+1) = S(n) − new infections I(n+1) = I(n) + new infections − new recoveries R(n+1) = R(n) + new recoveries

Round every value to 1 decimal place as you go so that your table can be checked row by row.

The spreadsheet formulae

Cell	What goes in it
A2 : D2	Day 0 starting values, typed in: 0, 995, 5, 0
E2
F2
A3
B3
C3
D3

Column E is new infections, column F is new recoveries. Use $H$1, $H$2 and $H$3 so the parameters stay locked when you fill down, and wrap each result in ROUND(…,1).

The paper table

Day 0 is done for you. Each row needs: E = 0.4 × S × I ÷ 1000, F = 0.1 × I, then carry the three totals down. Check every row: S + I + R must still be 1000.0.

Day	S	I	R
0	995.0	5.0	0.0
1
2
3
4
5
6
7
8
9
10
11
12
13
14

Your graph

Day on the horizontal axis, number of people on the vertical axis, three labelled curves. Ring the peak of I.

Findings

Question	Your answer
R₀ = β ÷ γ =
On which day does I reach its highest value?
How many people are infectious on that day?
How many are still susceptible at the peak? What do you notice about that figure compared with N ÷ R₀?
With β = 0.20 instead: new R₀, new peak day, new peak size

Evaluation

Two paragraphs. First: what the model predicts and what halving β does to the peak. Second: name two assumptions in this model that are not true of a real community, and say which direction each one biases the prediction.

Success criteria

I turned a written description of an outbreak into a set of recurrence relations with defined variables and parameters.
My model runs itself: each day is calculated from the day before by formula, not typed in by hand.
I tested the model by checking that S + I + R stays equal to N on every single row.
I used the model to answer a question it was built for — when the peak falls and how big it is.
I changed one parameter, recorded the effect, and judged the limits of the model instead of treating its output as fact.

If you finish early

Add a vaccination parameter v that moves a fixed number of people straight from S to R on Day 0. What is the smallest starting vaccinated fraction that stops I ever rising above its Day 0 value? Compare your answer with 1 − 1/R₀.
Split I into "mildly ill" and "in hospital" with 4% of new infections going to hospital. Add a hospital-bed capacity line at 15 beds and find the first day the model breaches it.
The real epidemic curve of a disease is never this smooth. List three reasons, and say which of them a larger population would fix and which it would not.