One number decides an outbreak
Three compartments — Susceptible, Infectious, Recovered — plus two rates and a bit of arithmetic.
The whole shape of an epidemic follows from a single number: R₀.
Meet your population. Every face is one person. Everyone starts 🙂 susceptible. One brave soul is 🤢 infectious — patient zero. Once someone recovers, they become 😎 immune and can't catch it again. Watch what happens as we turn one knob.
The R₀ dial · how contagious is it?
Herd immunity, in one picture
Once enough of the population is immune (recovered or vaccinated), each infectious person meets fewer than one susceptible on average. The chain breaks. The magic threshold is 1 − 1/R₀:
- Flu (R₀ ≈ 1.3): about 23% need immunity
- COVID-19 ancestral (R₀ ≈ 2.5): about 60%
- Smallpox (R₀ ≈ 5): about 80%
- Measles (R₀ ≈ 15): about 93% — which is why WHO targets 95% coverage
Pre-vaccinate a fraction · then release patient zero
Time series & the peak
The classic textbook picture: S falls, I rises to a peak, R climbs to whatever the final attack rate is. Interventions can flatten the curve — lower the peak, buy time for hospitals — without necessarily changing the total infected.
S / I / R over time
Related models
- 🕸️ Networks — contagion doesn't happen in a vacuum; who is connected to whom reshapes R₀ and the final attack rate.
- 🚶 Random walks — the discrete, individual-level dynamics underneath the smooth S/I/R curves.
- 🌐 Markov chains — treat each person as a state machine (S → I → R) and the whole thing becomes a compartmental Markov model.
