Reed–Frost Model

Reed–Frost model is a discrete-time, stochastic chain-binomial epidemic model formulated in 1928 by biostatistician Lowell J. Reed and epidemiologist Wade Hampton Frost at the Johns Hopkins University School of Hygiene and Public Health. Designed originally as a classroom teaching exercise using colored wooden beads rolling through a wooden trough to simulate chance contacts, and later published mathematically by Helen Abbey (1952) and Norman T. J. Bailey (1957), the Reed–Frost model provides the fundamental probabilistic framework in epidemiology and biostatistics for analyzing transmission in small, finite populations such as households, classrooms, hospital wards, and shipboard cohorts. Whereas deterministic compartmental models in epidemiology treat population counts as continuous variables, the Reed–Frost model captures integer case counts, multiple exposure avoidance, and stochastic epidemic extinction.

Generation-Time Discretization

The Reed–Frost model divides calendar time into discrete steps t = 0, 1, 2, …, G, where the length of one time step equals the fixed serial interval or generation time of the pathogen (combining a latent period during which a newly infected host is non-infectious followed by a brief, uniform period of infectiousness). At generation t, a closed population of size N is partitioned into integer counts of Susceptible (S_t), Infectious (I_t), and Recovered/Immune (R_t) individuals:

1. Independent Pairwise Escape Probability (q = 1 − p): Let p (0 < p < 1) denote the probability that a given susceptible individual makes an effective infectious contact with a specific infectious individual during a single generation, and let q = 1 − p denote the complementary probability of escaping infection from that single infectious individual.
2. Compound Escape from Multiple Infectious Hosts (q^I_t): Assuming contacts between different pairs of individuals occur independently during generation t, a susceptible person exposed simultaneously to I_t infectious individuals escapes infection from all of them only if they escape the first (q), the second (q), and so on across all I_t infectious cases. By the multiplication rule for independent events, the total probability of escaping infection during generation t is q^I_t = (1 − p)^I_t.
3. Single-Generation Infection Probability: Consequently, the probability π_t that a given susceptible individual acquires infection during generation t is the complement of escaping all I_t cases:

The Chain-Binomial Transition Distribution

Because each of the S_t susceptible individuals independently faces infection probability π_t = 1 − q^I_t, the number of newly infected cases in generation t + 1 follows a Binomial probability distribution conditioned on (S_t, I_t):

Writing out the exact probability mass function for observing i_t+1 new cases gives:

After generation t concludes, all I_t cases recover with permanent immunity, updating the state variables deterministically via:

A complete epidemic trajectory (i₀, i₁, i₂, …, i_k = 0) is a finite Markov chain of conditional binomial draws—hence the name chain-binomial model—whose joint likelihood is the product of the single-step binomial probabilities across all generations.

Enumerating Exact Household Transmission Chains

In household cohort studies of measles, influenza, and SARS-CoV-2, the parameter p represents the household secondary attack rate (SAR)—the probability that a single index case (i₀ = 1) infects a specific susceptible household contact during the index case's infectious period. Consider a household with s₀ = 2 susceptible contacts and i₀ = 1 index case:

• Chain (1 → 0): Zero secondary cases occur (S_∞ = 2, total cases = 1) with probability q².
• Chain (1 → 1 → 0): Exactly 1 contact is infected in generation 1 (2*pq*), and the remaining 1 susceptible escapes in generation 2 (q), yielding probability 2p q².
• Chain (1 → 1 → 1 → 0): 1 contact is infected in generation 1 (2*pq*), and that secondary case infects the remaining contact in generation 2 (p), yielding probability 2p² q.
• Chain (1 → 2 → 0): Both contacts are infected directly by the index case in generation 1, yielding probability p².

Summing the two distinct chains that result in all 3 household members becoming infected (1 → 1 → 1 → 0 and 1 → 2 → 0) gives Pr(Final Size = 3) = 2p² q + p². Notice that without observing generation-by-generation symptom onset dates, observing 2 secondary cases in a household mixes direct index-case transmission (p²) with tertiary household transmission (2p²q)—enabling maximum-likelihood estimation of p even from final-size cross-sectional serosurveys (Longini & Koopman, 1982).

Connection to the Basic Reproduction Number (R₀)

In a population of size N = S₀ + 1 where p is small, the expected number of secondary infections generated by a single index case is basic reproduction number R₀ = S₀ p. Using the Taylor approximation 1 − p ≈ exp(−p) = exp(−R₀ / S₀), the single-generation infection probability becomes:

When R₀ > 1, the stochastic Reed–Frost model yields a bimodal final-size distribution: either the outbreak stutters to extinction in the first few generations (with branching-process extinction probability z_ext satisfying z = exp[R₀(z − 1)]), or it takes off to produce a major epidemic wave centered around the deterministic Kermack–McKendrick final size.

Why did Lowell Reed and Wade Hampton Frost never publish their own model in a journal?

Reed and Frost developed the chain-binomial equations in 1928 strictly as an pedagogical lecture tool for their Epidemiology I course at Johns Hopkins, using colored wooden beads (green for susceptibles, red for infectious, purple for immune) and a mechanical chute with blocking shields to demonstrate that realistic bell-shaped epidemic curves arise purely from chance contact and susceptible depletion. Its mathematical properties were published by their student Helen Abbey in 1952 after Frost's death.

How does the Reed–Frost model prevent the probability of infection from exceeding 100%?

In naive discrete-time linear models that set new infections proportional to p S_t I_t, the per-susceptible infection probability p I_t exceeds 1.0 whenever I_t > 1 / p, which is mathematically impossible because a person cannot be infected more than once in the same generation. By computing the complement of the compound escape probability, 1 − (1 − p)^I_t, the Reed–Frost formula naturally accounts for overlapping exposures and remains bounded between 0 and 1 for any number of infectious individuals.

1. Greenwood, M. (1931). "On the statistical measure of infectiousness." Journal of Hygiene, 31(3), 336–351. https://doi.org/10.1017/S002217240001086X
2. Abbey, H. (1952). "An examination of the Reed-Frost theory of epidemics." Human Biology, 24(3), 201–233. https://pubmed.ncbi.nlm.nih.gov/12990130/
3. Frost, W. H. (1976). "Some conceptions of epidemics in general (posthumous lecture notes from 1928)." American Journal of Epidemiology, 103(2), 141–151. https://doi.org/10.1093/oxfordjournals.aje.a112212
4. Fine, P. E. M. (1977). "A commentary on the mechanical analogue to the Reed-Frost epidemic model." American Journal of Epidemiology, 106(2), 87–100. https://doi.org/10.1093/oxfordjournals.aje.a112449
5. Longini, I. M., & Koopman, J. S. (1982). "Household and community transmission parameters from final distributions of infections in households." Biometrics, 38(1), 115–126. https://doi.org/10.2307/2530294