Stochastic epidemic models: a survey

This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic model properties (relying on a large community) are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.

💡 Research Summary

The paper provides a comprehensive survey of stochastic epidemic models, focusing primarily on the classic stochastic SIR framework and its extensions. It begins with a brief historical overview, tracing the evolution from early deterministic models (e.g., Kermack‑McKendrick) to modern stochastic formulations. The deterministic SIR model is revisited through its differential equations, highlighting the central role of the basic reproduction number (R_0 = \lambda/\gamma). While the deterministic approach predicts a major outbreak when (R_0>1) and a minor one when (R_0<1), it fails to capture random fluctuations that are crucial in small populations or when the epidemic is seeded by only a few initial infectives.

To address these limitations, the authors define the “standard stochastic SIR epidemic model.” In a closed, homogeneously mixing population of size (n), each infectious individual makes contacts at a Poisson rate (\lambda); each contact with a susceptible results in immediate infection. Infectious periods are i.i.d. with mean (1/\gamma). Two common choices for the infectious period are considered: exponential (yielding a continuous‑time Markov chain) and deterministic (the Reed‑Frost model), the latter being analytically tractable via an Erdős‑Rényi random graph representation.

Exact results are limited to the final epidemic size. By exploiting Wald’s identity for total infection pressure and the interchangeability of individuals, the paper derives a recursive formula for the distribution of the total number of infections beyond the initial (m) cases. Closed‑form expressions for the time evolution are unavailable, but asymptotic approximations become accurate when (n) is large. In this regime, the early phase of the epidemic behaves like a branching process with mean offspring number (R_0). If the branching process is super‑critical ((R_0>1)), there is a positive probability of a large outbreak; conditional on a large outbreak, the final size satisfies the deterministic balance equation (1-z = (1-\varepsilon)e^{-R_0 z}) with Gaussian fluctuations of smaller order.

Temporal dynamics are decomposed into three phases: (i) an initial “seed” phase lasting (O(\log n)) where stochastic effects dominate, (ii) a main phase of order (O(1)) where the deterministic trajectory is followed, and (iii) a terminal “fade‑out” phase of order (O(\log n)). Consequently, the total duration of a major outbreak scales as (O(\log n)).

The paper then shows how stochastic models facilitate inference. By observing the final size distribution or the early exponential growth rate, one can estimate (R_0) and (\gamma) and attach confidence intervals via likelihood, bootstrap, or Bayesian methods. The stochastic framework also naturally yields the probability of extinction, which is essential for evaluating control measures.

Several extensions are surveyed. Multi‑type models allow heterogeneous contact rates (\lambda_{ij}) and recovery rates (\gamma_i); the overall (R_0) becomes the dominant eigenvalue of the next‑generation matrix. Household models distinguish within‑household and between‑household transmission, often modeled as a Reed‑Frost process inside cliques and a Poisson process across cliques, capturing the amplification effect of close contacts. Endemic (steady‑state) models incorporate demographic turnover, leading to a non‑zero equilibrium prevalence that can be analyzed via stochastic differential equations. Finally, models with births, deaths, and migration (dynamic populations) are briefly mentioned.

In the concluding discussion, the authors argue that stochastic epidemic models provide richer insight than deterministic counterparts, especially for (a) quantifying uncertainty in small or early outbreaks, (b) estimating key epidemiological parameters with associated variability, and (c) assessing intervention strategies such as vaccination. They highlight open research directions, including unified frameworks that combine multi‑type, network, and spatial heterogeneities, and real‑time Bayesian updating using modern surveillance data. Overall, the survey underscores the centrality of stochastic modeling in contemporary infectious‑disease epidemiology.