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Household epidemic models revisited

Frank Ball, Tom Britton, Peter Neal

TL;DR

The paper advances a generalized stochastic household epidemic framework defined by $(X_G,X_L)$, capturing global and local infection contacts, and introduces a swapped-contact variant with probability $p$ for local-to-global transitions. It derives rigorous large-population limits, including a central limit theorem for the final size of a major epidemic and fixed-point characterizations of the limiting outbreak proportion, while giving explicit variance formulas and practical computation guidance. The authors prove monotonicity results: the probability of a major outbreak increases with both household size $h$ and swap probability $p$ for any $(X_G,X_L)$, and the limiting final size $z^{(h,p)}$ increases in $h$ and $p$ when the pgf of $X_L$ is log-convex (as in traditional mixed-Poisson household models); they also present counterexamples when this condition fails. The methodology hinges on enhanced embedding via Sellke constructions, branch- ing-process approximations, and careful variance analysis, complemented by numerical illustrations that validate asymptotic results and reveal nuanced behaviors. The work also discusses extensions to unequal households and multi-type structures, showing the broader relevance of the embedding approach to complex two-level mixing epidemics.

Abstract

We analyse a generalized stochastic household epidemic model defined by a bivariate random variable $(X_G, X_L)$, representing the number of global and local infectious contacts that an infectious individual makes during their infectious period. Each global contact is selected uniformly among all individuals and each local contact is selected uniformly among all other household members. The main focus is when all households have the same size $h \geq 2$, and the number of households is large. Large population properties of the model are derived including a central limit theorem for the final size of a major epidemic, the proof of which utilises an enhanced embedding argument. A modification of the epidemic model is considered where local contacts are replaced by global contacts independently with probability $p$. We then prove monotonicity results for the probability of the major outbreak and the limiting final fraction infected $z$ (conditioned on a major outbreak). a) The probability of a major outbreak is shown to be increasing in both $h$ and $p$ for any distribution of $X_L$. b) The final size $z$ increases monotonically with both $h$ and $p$ if the probability generating function (pgf) of $X_L$ is log-convex, which is satisfied by traditional household epidemic models where $X_L$ has a mixed-Poisson distribution. Additionally, we provide counter examples to b) when the pgf of $X_L$ is not log-convex.

Household epidemic models revisited

TL;DR

The paper advances a generalized stochastic household epidemic framework defined by , capturing global and local infection contacts, and introduces a swapped-contact variant with probability for local-to-global transitions. It derives rigorous large-population limits, including a central limit theorem for the final size of a major epidemic and fixed-point characterizations of the limiting outbreak proportion, while giving explicit variance formulas and practical computation guidance. The authors prove monotonicity results: the probability of a major outbreak increases with both household size and swap probability for any , and the limiting final size increases in and when the pgf of is log-convex (as in traditional mixed-Poisson household models); they also present counterexamples when this condition fails. The methodology hinges on enhanced embedding via Sellke constructions, branch- ing-process approximations, and careful variance analysis, complemented by numerical illustrations that validate asymptotic results and reveal nuanced behaviors. The work also discusses extensions to unequal households and multi-type structures, showing the broader relevance of the embedding approach to complex two-level mixing epidemics.

Abstract

We analyse a generalized stochastic household epidemic model defined by a bivariate random variable , representing the number of global and local infectious contacts that an infectious individual makes during their infectious period. Each global contact is selected uniformly among all individuals and each local contact is selected uniformly among all other household members. The main focus is when all households have the same size , and the number of households is large. Large population properties of the model are derived including a central limit theorem for the final size of a major epidemic, the proof of which utilises an enhanced embedding argument. A modification of the epidemic model is considered where local contacts are replaced by global contacts independently with probability . We then prove monotonicity results for the probability of the major outbreak and the limiting final fraction infected (conditioned on a major outbreak). a) The probability of a major outbreak is shown to be increasing in both and for any distribution of . b) The final size increases monotonically with both and if the probability generating function (pgf) of is log-convex, which is satisfied by traditional household epidemic models where has a mixed-Poisson distribution. Additionally, we provide counter examples to b) when the pgf of is not log-convex.

Paper Structure

This paper contains 30 sections, 14 theorems, 174 equations, 6 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model and main results
  3. The general household epidemic model
  4. Relation to traditional household epidemic models
  5. The $\mathcal{E}_{n,h} (X_G, X_L,p)$ model described as an $\mathcal{E}_{n,h} (X_G^\prime, X_L^\prime)$ model
  6. Main results for the general household epidemic model
  7. Counter examples to Theorem \ref{['thm:finalsize']} when $\sigma_L^2 < \mu_L$
  8. An example where final size decreases with household size
  9. An example where moving local to global contacts lead to smaller final size
  10. Numerical illustrations
  11. Accuracy of asymptotic approximations
  12. Exploring model behaviour
  13. Central limit theorem proof
  14. Introduction
  15. Model description
  16. ...and 15 more sections

Key Result

Theorem 2.1

Suppose that $R_\ast >1$, and that there exists $m \geq 1$ such that $m_n=m$ for all sufficiently large $n$, and $a > 0$ such that ${\rm E} [X_G^{2+a}] <\infty$. Let $z>0$ be given by eq:z and $\rho$ be the unique solution in $[0,1)$ of Then where the random variable $Z$ has probability mass function Furthermore, there exists $0 < \sigma^2 < \infty$ given by eq:sigma, such that

Figures (6)

  • Figure 1: Histograms of 100,000 simulations of the fraction of the population infected in $\mathcal{E}_{n,2} (X_G, X_L)$ when $X_G \sim {\rm Po}(1)$ and $X_L \sim {\rm Po}(1)$ independently, for population sizes $N=nh=250, 500, 1,000$ and $2,000$, with a normal approximation superimposed; see text for details.
  • Figure 2: Histograms of 100,000 simulations of the fraction of the population infected in a major outbreak in $\mathcal{E}_{n,2} (X_G, X_L)$ when $X_G \sim {\rm Po}(1)$ and $X_L \sim {\rm Po}(1)$ independently, for population sizes $N=nh=250, 500, 1,000$ and $2,000$, with a normal approximation superimposed; see text for details.
  • Figure 3: Graphs of the fraction of the population infected by a major outbreak, $z^{(h,p)}$, against $p$ for different choices of household size $h$ and distribution of $(X_G, X_L)$.
  • Figure 4: Graphs of the scaled variance, $\sigma^{(h,p)}$, of the fraction of the population infected by a major outbreak against $p$ for different choices of household size $h$ and distribution of $(X_G, X_L)$.
  • Figure 5: Graphs of $z^{(h,p)}$ (left panel) and $\sigma^{(h,p)}$ (right panel) when $X_G \sim {\rm Bin}(2, \frac{3}{4})$ and $X_L \sim {\rm Bin}(2, \frac{3}{4})$ independently.
  • ...and 1 more figures

Theorems & Definitions (26)

  • Theorem 2.1
  • Theorem 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Corollary 4.1
  • ...and 16 more