Reed–Frost Model

A finite-population, discrete-generation epidemic model in which independent pairwise exposure risks combine into a binomial count of new cases while each infectious generation then exits.

Version

v1 · 2026-10-31 · History

Domain-specific #

14565

Domain group

Applied Sciences & Engineering

Origin domain

Medicine & Healthcare

Subdomain

Infectious Disease Modeling → Medicine & Healthcare

Aliases

Reed Frost Chain Binomial Model

Core Idea

The Reed–Frost model describes epidemic spread in a finite population in discrete infectious generations. If \(S_t\) people are susceptible, \(I_t\) are infectious, and each infectious–susceptible pair transmits independently with common probability \(p\), a susceptible escapes with probability \((1-p)^{I_t}\). The next infectious count is conditionally binomial: [I_{t+1}\mid(S_t,I_t)\sim\mathrm{Binomial}(S_t,1-(1-p)^{I_t}).] The current infectious generation then leaves that class; susceptible counts decrease by new infections. These are model assumptions, not a claim that every real contact is homogeneous and independent.^[1]

How would you explain it like I'm…

Coin-Flip Germ Tag

Imagine a game of tag where each round, every kid who is 'it' gets a chance to tag each other kid, like flipping a coin. Kids who get tagged become 'it' next round, and the old 'it' kids sit down for good. Sometimes the game spreads to almost everyone, and sometimes it stops early, even when you start the same way.

Sickness Spreading by Rounds

The Reed–Frost Model is a way to imagine how a sickness spreads through a group, one 'generation' at a time. Each sick person has the same small chance of passing it to each healthy person who can still catch it, like separate coin flips. A healthy person stays well only if they dodge every sick person that round. Those who catch it become the sick ones next round, while the earlier sick ones are done. Because it's based on chance, the same starting group can end with a big outbreak or a tiny one, and it stops when no one is sick anymore.

Chain-Binomial Epidemic Model

The Reed–Frost model is a chance-based (stochastic) model of an epidemic in a fixed-size population, moving one generation of infection at a time. Each infectious person has the same probability p of infecting each susceptible person in a generation, and these pairings are treated as independent. A susceptible person escapes one infectious person with probability 1−p, so they escape all I of them with probability (1−p)^I, and get infected with probability 1−(1−p)^I. The number of new infections is then like flipping that weighted coin for every susceptible person, and the current infectious group is removed. If no one is infectious, the process stops, even if susceptible people remain. These are simplifying assumptions of the model, not claims that real contacts are uniform.

 

The Reed–Frost Model is a discrete-generation stochastic epidemic model for a finite population. With S_t susceptible, I_t infectious, and p the per-pair transmission probability per generation, pairwise transmission events are assumed independent. A susceptible therefore escapes infection with probability q^{I_t}, where q = 1 − p, and is infected with probability 1 − q^{I_t}. Conditional on current counts, the next infectious count is binomial: I_{t+1} ~ Binomial(S_t, 1 − q^{I_t}). Susceptibles are then updated by S_{t+1} = S_t − I_{t+1}, and the current infectious generation is removed, R_{t+1} = R_t + I_t. If I_t = 0, the escape probability is 1 and the chain stops even with susceptibles remaining. The model thus generates a distribution over epidemic chains, including early extinction and variable final sizes. Its defining feature is the escape exponent equal to the current infectious count coupled to a binomial next generation; discrete time or randomness alone does not make a model Reed–Frost, and the uniform independent mixing is a modeling assumption.

Structural Signature

• Finite state: susceptible and infectious counts determine the trials and exposure sources.
• One-generation turnover: current infectious people leave the infectious class after their generation.
• Independent common pair chance: the same transmission probability applies to each modeled infectious–susceptible opportunity.
• Compounded escape and binomial update: individual escape is \((1-p)^{I_t}\); the new infectious count is drawn conditionally over susceptibles.
• Stopping condition: with no infectious people, the internal epidemic chain stops.^[1]

What It Is Not

Greenwood's chain-binomial model uses a positive infection chance that does not increase with the number infectious; the Reed–Frost escape exponent distinguishes the classic rule. Replacing the random next count with its mean gives a different deterministic summary. An extension can add external introductions or heterogeneous infectiousness without making those features part of the classic rule.^[1][2]

Examples

Household common cold. Heasman and Reid compared observed family epidemic chains with a model; their generation classifications and contact assumptions remain choices for that fit. The cited counts are episodes, not independent families.^[3]

Ship measles. Paterson and colleagues used historical ship data and a Reed–Frost simulation to examine outbreak scenarios. Incomplete records and estimated immunity limit the historical interpretation; simulated generations are not all observed events.^[4]

Reasoning and Boundary

Specify the population, initial state, generation interval and pair chance before applying the conditional transition. The model defines a discrete-time Markov chain in the finite-state setting, but a fit or simulated trajectory does not prove its independence assumptions held in the observed outbreak. Report which counts were observed, classified or simulated.^[1][3][4]

Neighborhood in Abstraction Space

Reed–Frost Model sits in a sparse region of the domain-specific corpus (95^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Unclustered & Miscellaneous (2835 abstractions)

Nearest neighbors

• Particle Filter — 0.78
• Chebyshev's Inequality — 0.77
• Branching process — 0.77
• Muller's ratchet — 0.77
• Moran Process — 0.77

Computed from structural-signature embeddings · 2026-10-31

References

[1] reedfrost project, “Model descriptions”, CDCgov repository owner, sections “General assumptions and corollaries,” “Reed-Frost,” and “Greenwood.” Formal model definition and comparison; not empirical evidence that homogeneous independent contacts occur in nature. registry ↩a ↩b ↩c ↩d

[2] Christophe Fraser et al., “Influenza Transmission in Households During the 1918 Pandemic,” American Journal of Epidemiology 174, no. 5 (2011): 505–514, https://doi.org/10.1093/aje/kwr122. See “Basic Reed-Frost model” and “An extended model.” Used here as a bounded internal-component and extension comparison, not a second whole classic example. registry ↩

[3] M. A. Heasman and D. D. Reid, “Theory and Observation in Family Epidemics of the Common Cold,” British Journal of Preventive & Social Medicine 15, no. 1 (1961): 12–16, https://doi.org/10.1136/jech.15.1.12. See printed pp. 12–15, especially first-stage Table I, generation classification, and complete-chain Table V. Scanned original available at https://pmc.ncbi.nlm.nih.gov/articles/PMC1058809/. registry ↩a ↩b

[4] Beverley J. Paterson, Martyn D. Kirk, A. Scott Cameron, Catherine D’Este, and David N. Durrheim, “Historical Data and Modern Methods Reveal Insights in Measles Epidemiology: A Retrospective Closed Cohort Study,” BMJ Open 3, no. 1 (2013): e002033, https://doi.org/10.1136/bmjopen-2012-002033. See “Reed-Frost stochastic model,” Methods, and “Stochastic measles outbreak modelling,” Results. The article contains a later corrected numerical \(R_0\) value; no numeric \(R_0\) finding is used here. registry ↩a ↩b