Each dot is a person wandering inside the arena. When an infectious person (red) comes within contact range of a susceptible one (blue) there is a per-contact chance of transmission set by the transmission risk slider — lower values model better hand-washing, mask-wearing and surface hygiene. Vaccinated people (cyan) and those who already recovered (green) cannot catch the disease again, so as vaccination coverage and natural recovery raise the immune share of the population, chains of transmission run out of susceptible people to reach — herd immunity. Social distancing shrinks the contact radius directly, cutting the number of close encounters per day regardless of how contagious the pathogen is.
dI/dt = β·S·I/N − γ·I
R(t) = β/γ · (S/N)
herd immunity threshold: S/N < γ/β
- Vaccination coverage — fraction of the population immune from day one; raising it shrinks the pool the infection can reach before it even starts spreading.
- Transmission risk — probability an infectious contact actually passes the disease on; hygiene and distancing both act on this same number in real life.
- Social distancing — reduces the contact radius, so fewer susceptible people fall inside an infectious person's reach each frame.
- Effective R(t) — the average number of new infections each currently-infectious person is producing right now; above 1 the outbreak is still growing, below 1 it is dying out.
Real-world relevance: this is the same logic behind public-health guidance — vaccines and hygiene both lower how many susceptible people an infection can reach, and once enough of the population is immune the chain of transmission collapses on its own before everyone has been exposed.