🦠 Epidemic Model — SIR & Vaccination Thresholds

SIR stands for Susceptible, Infectious and Recovered — the three compartments a fixed population is divided into. Susceptible people can catch the disease, Infectious people are currently sick and can transmit it, and Recovered people have either recovered or died and can no longer be infected or infect others in this simple model. The model tracks how people flow from S to I to R over time using a pair of coupled differential equations.

Each simulated day, the number of new infections is proportional to β·S·I/N, where β is the transmission rate, S and I are the current susceptible and infectious counts, and N is the total population (400). This term is largest when both S and I are sizeable — early in an outbreak, when almost everyone is susceptible and the number of infectious people is growing — and shrinks automatically as either group empties out.

R₀, the basic reproduction number, is the average number of secondary infections one infectious person causes in a fully susceptible population. In this simulation it is linked to the transmission rate by β = R₀·γ, where γ is the recovery rate. Raising the R₀ slider therefore increases β directly, causing infections to grow faster and produce a higher, earlier epidemic peak.

Vaccination removes a percentage of the population from the susceptible pool before day zero, so those people can never be infected, directly shrinking the pool available to sustain an outbreak. Lockdown instead reduces the transmission rate β by half for as long as it is active, slowing — but not necessarily stopping — spread. The Intervention Day slider models a similar but permanent 30% cut in β once a chosen day is reached, representing measures like mask mandates or social distancing introduced partway through an outbreak.

The herd immunity threshold is the fraction of the population that needs to be immune (through vaccination or prior infection) for an epidemic to no longer be able to grow, calculated here as p_c = 1 − 1/R₀. A higher R₀ means each infectious person can reach more susceptible contacts, so a larger share of the population must be protected to break the chain of transmission — this is why the simulation shows roughly 67% for a COVID-19-like R₀ of 3 but around 93% for measles-like R₀ of 15.