Epidemic Spread

How fast does an infection spread, how many people are sick at the peak, and how many catch it in the end? This is the classic SIR model: everyone is Susceptible, Infected or Recovered. Add vaccination to see how much it takes to stop the outbreak, and how many cases it prevents.

EpidemiologySIR modelR₀Vaccination

Inputs

Input examples

R₀ is the number of people one sick person infects in a population where nobody is immune. The infectious period is how long someone stays contagious. Vaccinated people are protected with the vaccine’s efficacy: 90% efficacy means 9 out of 10 vaccinated people are immune. Leave the vaccinated share at 0 for an outbreak without vaccines.

The disease

Basic reproduction number R₀ — people infected by one case

Infectious period D (days)

The population

Population size N

Infected at the start

Days to show

Vaccination — optional

Vaccinated share of the population (%)

Vaccine efficacy (%)

How an Epidemic Grows, and How Vaccines Stop It

At the start almost everyone is susceptible, so each case infects about R₀ others and the number of sick people grows exponentially. As more people recover and become immune, each case finds fewer people to infect. The epidemic peaks once a share of 1 − 1/R₀ is immune; the momentum carries it on, so in the end more people are infected than this herd-immunity threshold.

Vaccination makes people immune before the outbreak. Every immune person blocks a chain of infection, so once enough people are immune the effective reproduction number R_e drops below 1 and the disease cannot take off — protecting even those who could not be vaccinated. The more contagious the disease, the higher the bar: about 67% for R₀ = 3, around 95% for measles, and more with a vaccine that is not fully effective.

R₀ > 1the outbreak grows

1 − 1/R₀herd immunity threshold

R_e < 1outbreak cannot grow

below the barstill fewer cases

Epidemic Curve

SIR model

susceptible

infected (sick now)

recovered

sick without vaccination

Result

Enter the values above

Cases vs. Vaccination Coverage

final size

share infected in the end

coverage that stops it

Key Relations

Basic and Effective Reproduction Number

β is the number of people one sick person infects per day when everybody is susceptible; times the days they are contagious gives R₀. Only people who are still susceptible can pass the infection on.

Example:

Herd Immunity and Critical Coverage

The number of sick people stops growing when S/N falls to 1/R₀. Vaccinating more than v_c keeps R_e below 1 from the start; below it every extra vaccination still prevents cases.

Example:

Peak and Final Size

The peak height follows from the SIR equations without solving them step by step; the share s∞ that never gets infected solves the final-size equation, found numerically.

Example:

Sources & Verification

References

W. O. Kermack & A. G. McKendrick, “A contribution to the mathematical theory of epidemics”, Proceedings of the Royal Society of London A 115(772), 700–721 (1927).
M. J. Keeling & P. Rohani, Modeling Infectious Diseases in Humans and Animals, Princeton University Press, 2008 — Ch. 2 (Introduction to Simple Epidemic Models) and the chapter “Controlling Infectious Diseases”.