How this instrument works
The SIR model splits a population into three compartments — Susceptible, Infected, Recovered — and describes how people move between them during an epidemic. Its central input is R0, the basic reproduction number: how many new cases one infectious person produces, on average, in a population with no immunity at all. From R0 alone, two closed-form results fall out of the model's equations: the herd immunity threshold (the immune share needed to stop sustained spread) and the peak infected fraction (the highest simultaneous share of the population infected at once, assuming everyone starts susceptible).
This is general epidemiological mathematics, not built around any single disease or era. Kermack and McKendrick derived the underlying equations in 1927, decades before any of the diseases people commonly plug an R0 into today existed as a public concern. The two formulas here work for measles (R0 commonly cited around 12–18), seasonal influenza (R0 near 1.3), or any other pathogen with a published reproduction-number estimate — the model doesn't know or care which disease supplied the number.
Both outputs are the model's known analytic solution points, not a day-by-day simulation. The full SIR system is normally solved as a time-stepped curve showing infections rising and falling over weeks or months; herd immunity threshold and peak infected fraction are values read directly off that same system's equations without needing to run the simulation. The peak-infected formula assumes an entirely susceptible starting population — no prior immunity, no vaccination already underway — which is the standard simplifying assumption behind this closed form, so treat it as an upper-bound scenario rather than a forecast for a population with any existing protection.
- Find or estimate the disease's basic reproduction number, R0, from a published source.
- Enter Basic reproduction number (R0) — the model requires a value greater than 1.
- Read Herd immunity threshold (%) — the population share that must be immune to stop sustained spread.
- Read Peak infected fraction (%) — the model's estimated highest simultaneous share infected, assuming a fully susceptible start.
Worked example — R0 = 3, a moderately transmissible pathogen
Enter an R0 of 3, a figure in the range cited for moderately transmissible respiratory pathogens. Herd immunity threshold = (1 − 1/3) × 100 = 66.67% — two-thirds of the population needs immunity before sustained spread stops. Peak infected fraction = (1 − 1/3 − ln(3)/3) × 100 = 30.05%, the model's estimated highest simultaneous share infected if nobody starts out immune.
A second case shows the formula scaling correctly at the other end of the range: measles, with R0 commonly cited around 15, gives a herd immunity threshold of (1 − 1/15) × 100 = 93.33%. That independently lines up with the roughly 95% measles vaccination coverage target cited in public-health elimination guidance — a strong sanity check that this closed-form threshold matches real-world policy built on the same math, even though the two numbers come from entirely separate sources.
Questions
Is this tool specific to COVID-19 or any one pandemic?
No. The SIR model and both formulas here are general epidemiology mathematics from 1927, decades before any currently circulating pathogen was identified. Enter any disease's R0 — measles, seasonal flu, a hypothetical pathogen for a class assignment — and the same two equations apply. Nothing in the formula references COVID-19 or any specific outbreak.
What is R0 and where do I find a value for one?
R0, the basic reproduction number, is the average number of new infections one infectious person causes in a population where nobody has immunity yet. Published estimates exist for most well-studied diseases in the epidemiology literature — this page cites a few (measles around 12–18, seasonal influenza around 1.3) as starting references, but exact figures vary by study, population density, and contact patterns, so treat any single number as an estimate.
Why does herd immunity threshold use R0 alone and not vaccine effectiveness?
This formula gives the naive threshold assuming perfect, permanent immunity in whoever is immune — it doesn't adjust for a vaccine that's only partially effective or immunity that wanes over time. To account for an imperfect vaccine, public health analyses typically divide this threshold by the vaccine's effectiveness, which pushes the required coverage higher; that adjustment isn't built into this instrument.
Does peak infected fraction mean the total share who will eventually get sick?
No — it's the highest share infected at any single moment, not the cumulative total over the whole epidemic. The total ever infected (epidemic final size) is a separate quantity in SIR theory, calculated from a different transcendental equation that this instrument doesn't solve. Peak infected fraction only tells you how bad the worst moment gets, not how many people are affected in total.
Why does the calculator require R0 greater than 1?
At R0 of exactly 1, each case produces on average one more case, and below 1, cases decline on their own — neither scenario sustains an epidemic, so the herd-immunity and peak-infection formulas (which describe growth and containment of sustained spread) don't have a meaningful answer. The calculator blocks R0 values of 1 or less rather than returning a number that wouldn't mean anything.
Is this the same as the day-by-day epidemic curve I've seen in other SIR tools?
No. Many SIR tools run a step-by-step simulation showing infections rising and falling across days or weeks. This instrument instead evaluates the same underlying equations at their known closed-form solution points — herd immunity threshold and peak infected fraction — which requires no simulation loop and gives the two headline numbers a full time-stepped run would eventually converge on for a fully susceptible starting population.
References
- Kermack & McKendrick 1927 (1991 reprint), Bull Math Biol — original SIR model (PubMed)
- Fine, Eames & Heymann 2011, Clin Infect Dis — Herd Immunity: A Rough Guide (PubMed)
- Guerra et al. 2017, Lancet Infect Dis — basic reproduction number of measles (PubMed)
- Biggerstaff et al. 2014, BMC Infect Dis — reproduction number of influenza (PubMed)
Read this first: This instrument computes a screening figure from population formulas — it is not a diagnosis, and it cannot see the whole picture a clinician can. Use it to inform a conversation, not to replace one.