R0basic Reproduction Number Calculator
Compute r0basic reproduction number using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
R0 = beta x c x D
R0 (basic reproduction number) is the product of beta (transmission probability per contact), c (average number of contacts per unit time), and D (duration of infectiousness). When R0 > 1, an epidemic occurs. The herd immunity threshold is 1 - 1/R0. The SIR model uses differential equations: dS/dt = -beta*c*S*I/N, dI/dt = beta*c*S*I/N - gamma*I, dR/dt = gamma*I.
Worked Examples
Example 1: Seasonal Influenza Outbreak
Problem:A flu strain has transmission probability 0.02 per contact, average 15 contacts/day, and 5-day infectious period. Calculate R0 and herd immunity threshold.
Solution:R0 = beta x c x D = 0.02 x 15 x 5 = 1.50\nHerd immunity threshold = 1 - 1/R0 = 1 - 1/1.50 = 0.333 = 33.3%\nDoubling time = D x ln(2)/ln(R0) = 5 x 0.693/0.405 = 8.6 days\nFinal epidemic size (solving numerically): ~58.3% of population
Result:R0: 1.50 | Herd immunity: 33.3% | Doubling time: 8.6 days | ~58% ultimately infected
Example 2: Highly Contagious Disease
Problem:A disease has transmission probability 0.05, 20 contacts/day, 10-day infectious period in a population of 50,000.
Solution:R0 = 0.05 x 20 x 10 = 10.0\nHerd immunity threshold = 1 - 1/10 = 90%\nHerd immunity count = 0.90 x 50,000 = 45,000 people\nDoubling time = 10 x ln(2)/ln(10) = 10 x 0.301 = 3.0 days\nFinal epidemic size: ~99.99% (virtually entire population)
Result:R0: 10.0 | 90% herd immunity needed (45,000 people) | Doubles every 3 days
Frequently Asked Questions
What is the basic reproduction number (R0)?
The basic reproduction number, denoted R0 (pronounced 'R-naught'), is the average number of secondary infections caused by a single infected individual in a completely susceptible population. It is one of the most important metrics in epidemiology for understanding the transmission potential of an infectious disease. If R0 > 1, each infected person infects more than one other person on average, leading to epidemic growth. If R0 < 1, the infection will die out over time. R0 is calculated as the product of transmission probability per contact, contact rate, and duration of infectiousness.
What is the difference between R0 and effective reproduction number (Re)?
R0 represents the transmission potential in a fully susceptible population with no interventions, while the effective reproduction number (Re or Rt) accounts for existing immunity and control measures in the population. Re = R0 x S/N, where S is the number of susceptible individuals and N is the total population. As people become immune through infection or vaccination, S decreases and Re drops below R0. Public health interventions like social distancing, masking, and quarantine also reduce Re by lowering the contact rate. An epidemic is growing when Re > 1 and declining when Re < 1.
References
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy