Teaching objectives
How does a virus spread from one person to thousands? This lab uses COVID-19 as its through-line to help secondary-school students build, step by step, the real mathematical models used by epidemiologists. It starts with a grid of people infecting each other at random and ends with three crisis-management missions — flu A, Ebola, and COVID — where students decide when to lock down, how many vaccines to distribute, and who should get them.
What you'll learn
Students develop the ability to move from a qualitative observation ("the curve rises fast") to a quantitative model fitted to real data.
- Students discover that contact density determines whether an epidemic occurs: below ~50 % density the disease dies out on its own (percolation threshold).
- Comparing the cellular automaton with real COVID data shows that without movement the curve is linear, and that adding a displacement radius turns it exponential.
- The SAIR model (asymptomatic individuals who spread the disease without knowing it) is introduced, along with its effect on propagation speed.
- Students calculate by hand the discrete differential equations ΔS = −β·S·I/N, ΔR = γ·I, ΔI = −ΔS − ΔR and then fit β and γ to real COVID data.
- The basic reproductive number R₀ is explored, and students experiment with vaccination, lockdown, and mask-wearing to keep the peak number of infections below specific thresholds.
- In the social-networks room, topologies are compared (regular, random, scale-free) and students decide who to vaccinate in a real network with only 3 doses.
Key mathematical ideas
- A cellular automaton on a grid models local spreading; the 4-cell neighborhood defines who can be infected at each step.
- Percolation: there is a critical density threshold above which the infection reaches the entire system.
- The SIR equations are a finite-difference system that discretizes the rates of transmission (β) and recovery (γ).
- R₀ = β / γ: if R₀ > 1 there is an epidemic; herd immunity requires vaccinating at least 1 − 1/R₀ of the population.
- In a scale-free network, a few highly connected nodes (hubs) dramatically accelerate spreading; vaccinating them first is more efficient than random vaccination.
Room-by-room contents
Room 1 · Welcome to the model
Introduction to the SIR cellular automaton: a grid of people with three states (healthy, infected, immune) and a contagion probability β. The student launches the simulation and observes how the first outbreak spreads.
Student tasks
- Press Start and observe how the initial red cell infects its neighbours each day.
- Identify the three colours on the grid: green (S), red (I) and grey (R).
Room 2 · Density matters
The concept of population density is introduced: with a grid that is not completely full, some cells are empty. The student discovers the percolation threshold (~50 %) below which the disease does not reach everyone.
Student tasks
- Move the density slider and observe when the epidemic dies out before spreading.
- Find the approximate density value at which the disease can no longer reach the whole population.
Room 3 · Which curve fits reality?
The first 25 days of COVID-19 in Spain are shown alongside three candidate curves: linear, polynomial and exponential. The student identifies which one fits best and reasons why.
Student tasks
- Visually compare the three curves with the real data.
- Select the best-fitting curve and explain why contagion dynamics are exponential.
Room 4 · Our automaton vs. real data
The static automaton (no movement) is overlaid in red on the real COVID curve in yellow. The discrepancy is striking: the automaton peaks much later and at a lower scale, which motivates introducing movement.
Student tasks
- Observe the difference between the automaton curve and the real COVID curve.
- Reflect and write at least one reason that explains the discrepancy.
Room 5 · Adding movement
Infected individuals can move within a radius before spreading the disease. There are two sliders: contagion probability β and movement radius. Only one of them changes the shape of the curve from linear to exponential.
Student tasks
- Move the β slider and observe the effect on the curve.
- Move the radius slider and check which of the two transforms the shape of the curve.
Room 6 · Asymptomatics and isolation
The SAIR model is introduced: infected individuals first go through an asymptomatic phase (they spread the disease and move without knowing it) and then isolate when symptoms appear. The longer the asymptomatic phase, the higher the peak.
Student tasks
- Adjust the slider for the duration of the asymptomatic phase.
- Observe how a longer asymptomatic period drives up the peak of infections.
Room 7 · The SIR model, day by day
The three discrete differential equations of the SIR model are presented: ΔS = −β·S·I/N, ΔR = γ·I, ΔI = −ΔS − ΔR. The student manually calculates the first two days and then the simulator completes the curve.
Student tasks
- Calculate the values of S, I and R for day 1 using the given formulae.
- Calculate day 2 and compare with the simulator's result.
Room 8 · Fitting the model to COVID data
With the real COVID-19 data from Spain as a reference, the student adjusts parameters β and γ of the yellow SIR curve until the mean error drops below 550 cases/day.
Student tasks
- Move the β slider and observe how the peak rises or falls.
- Adjust γ to align the width of the curve with the data.
- Achieve a mean error below 550 cases/day.
Room 9 · R₀ and the epidemic threshold
R₀ is introduced as the average number of transmissions per infected individual. The student manipulates β and the duration of the contagious period until R₀ reaches ≈1, the critical threshold between extinction and epidemic.
Student tasks
- Move the β and infectious-days sliders to explore how R₀ changes.
- Bring R₀ as close as possible to 1 to unlock Continue.
Room 10 · Vaccination: flatten the curve
Initial immunity (vaccinated individuals) is added and a yellow dotted line appears at 15 % infected. The student raises the vaccination percentage until the peak stays below that line.
Student tasks
- Gradually raise the Vaccinated slider and observe how the peak falls.
- Find the minimum vaccination percentage that keeps the peak below 15 %.
Room 11 · How much vaccination is needed?
Three diseases with very different R₀ values: flu (R₀ ≈ 1.5), COVID (R₀ ≈ 3) and measles (R₀ ≈ 15). For each one, the student finds the minimum vaccination coverage that extinguishes the epidemic.
Student tasks
- For each disease, adjust vaccine coverage until the outbreak is extinguished.
- Compare the three thresholds and relate them to the formula 1 − 1/R₀.
Room 12 · Lockdown
A fraction of the population stays home (no movement, but they can still be infected by neighbours). The challenge is to keep the peak below 30 % using the lightest possible lockdown.
Student tasks
- Raise the lockdown level and observe the fall in the peak.
- Find the minimum percentage of the population under lockdown that keeps the peak ≤ 30 %.
Room 13 · Masks
Masks reduce β: β_effective = β · (1 − usage). The student must keep the peak ≤ 25 % with the minimum mask usage, seeking the balance between effectiveness and social cost.
Student tasks
- Adjust the mask-usage slider and see how β_effective and the peak change.
- Find the minimum usage that keeps the peak below 25 %.
Room 14 · Social networks: the power of the hub
The grid is abandoned in favour of contact graphs. The student compares three topologies (regular, random, scale-free) and discovers why scale-free networks are the most vulnerable; then decides whom to vaccinate with only 3 doses.
Student tasks
- Launch the outbreak across all three topologies and compare the speed of spread.
- Choose the 3 nodes to vaccinate in the real network to minimise damage.
Room 15 · Mission 1 — flu A outbreak
First open-ended mission: an unknown virus appears in a corner of the map. The student has a budget and several tools (vaccination, lockdown, masks) and must keep deaths below the set limit.
Student tasks
- Analyse the initial outbreak and design an intervention strategy with the available budget.
- Keep deaths below the threshold by the end of the simulation.
Room 16 · Mission 2 — Ebola outbreak
Second mission with a pathogen that has different parameters (higher mortality, lower airborne transmissibility). The same tools as in the previous mission are used, but the optimal strategy is different.
Student tasks
- Adapt the strategy learned in Room 15 to the new Ebola profile.
- Complete the mission while keeping deaths within the limit.
Room 17 · Mission 3 — COVID-19 (early days)
Final mission: reproduce the scenario of the early days of COVID-19 with real parameters and the lab's tools. Closes the journey by connecting mathematical modelling with a concrete historical event.
Student tasks
- Apply everything learned (β, γ, R₀, vaccination, lockdown, masks) to contain the outbreak.
- Complete the mission as efficiently as possible.
Rooms to project
The most striking ones to show and discuss in class.