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

This survey paper provides a comprehensive overview of stochastic epidemic models, beginning with the classic stochastic SIR (Susceptible‑Infected‑Recovered) framework and extending to more realistic formulations. The authors first define the basic model in continuous time as a Markov chain: each susceptible individual becomes infected at rate β I/N upon contact with an infected person, and each infected individual recovers at rate γ. Exact results are derived by constructing the transition probability matrix, which yields the full distribution of the number of infected individuals over time and the final epidemic size.

When the population size N is large, the authors show that the early phase of the outbreak can be approximated by a Galton‑Watson branching process. This approximation leads to a simple expression for the probability of a major outbreak and for the expected final size, both of which depend on the basic reproduction number R₀ = β/γ. The paper rigorously proves the threshold property: if R₀ ≤ 1 the epidemic dies out with probability one, whereas if R₀ > 1 there is a positive probability of a large‑scale epidemic. As N → ∞, a central‑limit‑type result demonstrates that the final size converges to a normal distribution whose mean coincides with the deterministic SIR prediction, thereby linking stochastic and deterministic perspectives.

The impact of vaccination is examined by introducing a pre‑emptive immunisation proportion p. The effective reproduction number becomes Rₑ = R₀(1 − p), and the critical vaccination coverage p_c = 1 − 1/R₀ is derived as the threshold for herd immunity. The authors discuss extensions for imperfect vaccines (partial protection) and for age‑ or risk‑group‑specific vaccination strategies, showing how these modify the effective Rₑ and the required p_c.

Parameter inference is addressed in depth. Using observed epidemic curves, final size data, or contact tracing information, the paper outlines maximum‑likelihood estimation (MLE) and Bayesian approaches for β, γ, and consequently R₀. In the early exponential growth phase, the growth rate r satisfies r ≈ γ(R₀ − 1), allowing a simple estimator R̂₀ = 1 + r/γ. The authors also present methods for constructing confidence intervals via bootstrap or Markov‑chain Monte Carlo (MCMC) techniques, and they illustrate how to validate vaccination impact by comparing pre‑ and post‑intervention estimates of p_c.

To increase realism, several model extensions are surveyed. Multi‑type models allow individuals to belong to different categories (e.g., age groups), with a matrix of transmission rates β_{ij} and type‑specific recovery rates γ_i. This framework captures heterogeneity in contact patterns and susceptibility. Household models introduce a two‑level contact structure: a high intra‑household transmission rate κ_intra and a lower inter‑household rate κ_inter. The resulting double‑layer network produces rapid within‑household spread but slower between‑household propagation, and the paper provides analytical approximations for the probability of household‑level outbreaks and the overall epidemic threshold. Geographic or network‑based extensions incorporate mobility matrices and distance‑dependent transmission, yielding meta‑population models that can simulate spatial spread.

The survey also covers endemic disease modeling, where births, deaths, and loss of immunity are incorporated to sustain a non‑zero equilibrium prevalence. By solving the steady‑state equations for SIS or SIRS structures, the authors demonstrate that an endemic equilibrium exists if and only if R₀ > 1, and they discuss how vaccination shifts the equilibrium prevalence and alters the critical coverage needed to eradicate the disease.

In the concluding section, the authors argue that stochastic models are indispensable for capturing the randomness inherent in the early stages of an outbreak, for evaluating the probability of rare but catastrophic events, and for informing control policies under uncertainty. They advocate for the integration of real‑time data streams with Bayesian updating, the combination of stochastic epidemic models with machine‑learning techniques for parameter learning, and the development of multi‑objective optimisation frameworks that simultaneously consider vaccination, treatment, and social‑distancing interventions. The paper thus serves as both a methodological reference and a roadmap for future research in stochastic epidemic modelling.