The population is split into four compartments — Susceptible, Exposed (infected but not yet infectious), Infectious, Recovered/immune — and moves between them by numerically integrating the real coupled SEIR differential equations every step (4th-order Runge–Kutta, dt = 0.25 day):
dS/dt = −βSI/N
dE/dt = βSI/N − σE
dI/dt = σE − γI
dR/dt = γI
Vaccination coverage moves that fraction of the population directly into R before the first case, so it can never be infected and can never transmit — it lowers the effective susceptible pool the outbreak has to burn through.
R₀ = β/γ is the basic reproduction number: how many people one case infects in a fully susceptible population. The critical vaccination fraction that stops sustained transmission follows directly from it:
p_crit = 1 − 1/R₀
Nothing about the collapse of the curve above p_crit is scripted — it is what the same ODEs produce once so much of S(0) has been pre-emptied into R that βS(0)I/N can no longer sustain σE − γI > 0. Push coverage past p_crit and the infectious curve you see is the genuine numerical output of that integration, not a canned animation.
- β (transmission rate) — contacts per day × probability of transmission per contact; higher β raises R₀ and the height/speed of the epidemic curve.
- 1/σ (incubation period) — average days a case spends exposed but not yet infectious, before it starts contributing to β·S·I/N.
- 1/γ (infectious period) — average days a case remains infectious before recovering; shorter periods raise γ and lower R₀ for the same β.
- Vaccination coverage — fraction of the population immunized before the outbreak starts (an idealized 100%-effective vaccine, for clarity of the mechanic).