Flu at Ysgol Bryn Celyn: the model pack

1. The situation

Ysgol Bryn Celyn is a secondary school with 1,000 pupils. In November, 10 pupils come back from a residential trip with flu. The head teacher wants to know how far it could spread, and whether vaccination or better hygiene would make a difference. The school’s science department has built a simple model to find out.

2. Facts about flu

• Flu (influenza) is a communicable disease. It is caused by a virus, which is a type of pathogen.
• It spreads by aerosol: tiny droplets from coughs, sneezes and talking. It also spreads by contact, from hands and surfaces.
• After someone has had flu, or has been vaccinated, memory cells help the immune system respond quickly if the same virus gets in again. In this model, those pupils cannot catch it again this term.
• Public Health Wales says school-aged children have the highest rate of flu infection. Most pupils in Wales are offered a nasal spray flu vaccine at school.
• The flu vaccine does not protect every person who has it, and how well it works changes from year to year.

3. Key words

A note on R: R = 3 is a round number chosen to keep this model simple. Estimates for seasonal flu in the whole population are usually lower, often between 1 and 2, but flu can spread faster in crowded places such as schools.

4. Model 1: a pupil’s first try

Model 1 rule: new cases in the next generation = new cases in this generation × 3. Start with 10. Keep going.

5. Model 2: the science department’s model

Inputs: school size = 1,000 · R = 3 · starting cases = 10 · vaccinated pupils = you are told in each scenario.

1. Rule 1 (set up): generation 0 has 10 new cases and a total of 10. Susceptible = 1,000 − vaccinated pupils − 10.
2. Rule 2: new cases in the next generation = last generation’s new cases × R × susceptible ÷ 1,000. Use the susceptible number from the row above.
3. Rule 3: round down to a whole number. You cannot have part of a pupil ill.
4. Rule 4: if the answer is bigger than the number of susceptible pupils left, use the number left.
5. Rule 5: new susceptible = old susceptible − new cases. New total = old total + new cases.
6. Rule 6: repeat Rules 2 to 5 for the next generation until the new cases reach 0.

Worked row, so you can check you have the method. Scenario with 500 pupils vaccinated. Generation 0: 10 new cases, susceptible = 1,000 − 500 − 10 = 490, total = 10. Generation 1: 10 × 3 × 490 ÷ 1,000 = 14.7, round down to 14. Susceptible = 490 − 14 = 476. Total = 10 + 14 = 24.

6. The model in a spreadsheet

A pupil typed Model 2 into a spreadsheet. The inputs are in cells G1 (R) and G2 (school size). They typed the formulas into row 3 and then filled them down. It does not work.

What the pupil typed in row 3 (formula view):

What the spreadsheet shows (value view):

Clue: when a formula is filled down one row, every relative cell reference in it moves down one row too. A $ sign, as in $G$1, stops a reference moving.

7. Model 2 results the department has already run

Scenario B: 500 pupils vaccinated (50%), R = 3.

Generation 15: 0 new cases. The outbreak is over. 309 pupils infected in total.

Scenario D: no vaccination, but a hand-washing and “catch it, bin it, kill it” campaign lowers R to 2.

The outbreak is over after generation 11. 869 pupils infected in total.

Scenario E: 500 pupils vaccinated and the hygiene campaign, R = 2.

The outbreak is over after generation 9. 55 pupils infected in total.

8. A real-style outbreak to test the model against

Last winter, flu spread through Ysgol Glan Morfa, another school with 1,000 pupils. Very few of its pupils had been vaccinated. The school nurse counted new cases in blocks of 3 days, which matches one generation of the model. Pupils who were ill were sent home.

The results in Section 7 and the Glan Morfa figures are also in flu-model-scenarios.csv, for use in a spreadsheet.