Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Integrating Household Dynamics in Stochastic Epidemic Modeling: An SDE Approach to the SIR Framework

Houda Yaqine, Christiane Fuchs

TL;DR

The paper addresses how household structure influences epidemic dynamics by embedding household-level interactions into a stochastic SIR framework via dual SDEs for individuals and households. It derives a fundamental threshold through a multi-type branching process near the disease-free equilibrium, yielding a numerically computable basic reproduction number $\mathcal{R}_0 = \rho(\mathbf{M})$ that depends primarily on public contacts near DFE. Through extensive simulations, the authors show that incorporating households alters epidemic timing and peak intensity, with transmission decomposed into public and household pathways and a Sobol-based sensitivity analysis identifying a key, highly active subpopulation as a critical intervention target. The study also demonstrates substantial stochastic variability in outbreak trajectories, especially during early and peak phases, underscoring the importance of uncertainty quantification for public health planning. Overall, the framework provides a rigorous, data-informed tool for evaluating interventions that target household transmission and for guiding policy decisions in structured populations.

Abstract

Understanding infectious disease spread remains a critical public health challenge, particularly given the interplay between household dynamics and community transmission patterns. Traditional epidemiological models often oversimplify these dynamics by treating populations as homogeneous, failing to capture crucial household-level interactions that can significantly impact disease spread. This paper introduces a new stochastic differential equation model extending the SIR framework by capturing the randomness in disease spread and incorporating household structure and heterogeneous mixing patterns. The model divides the population into groups based on age and household size, includes subpopulation-targeted lockdown parameters and constructs detailed contact matrices accounting for both public and within-household interactions. Through the approximation of Markov jump processes by branching processes near the disease free equilibrium, we derive the basic reproduction number of our model and conduct global sensitivity analysis using Sobol indices to identify influential factors. Our simulations reveal that incorporating household structure leads to substantially different predictions compared to traditional models, particularly in epidemic timing and peak intensity. The stochastic framework captures important variations in outbreak trajectories overlooked by deterministic approaches, especially during early and peak phases. This work contributes to both mathematical epidemiology and practical public health planning by providing a sophisticated mathematical understanding of how population structure and randomness influence disease dynamics, offering insights for intervention strategies where household transmission plays a significant role.

Integrating Household Dynamics in Stochastic Epidemic Modeling: An SDE Approach to the SIR Framework

TL;DR

The paper addresses how household structure influences epidemic dynamics by embedding household-level interactions into a stochastic SIR framework via dual SDEs for individuals and households. It derives a fundamental threshold through a multi-type branching process near the disease-free equilibrium, yielding a numerically computable basic reproduction number that depends primarily on public contacts near DFE. Through extensive simulations, the authors show that incorporating households alters epidemic timing and peak intensity, with transmission decomposed into public and household pathways and a Sobol-based sensitivity analysis identifying a key, highly active subpopulation as a critical intervention target. The study also demonstrates substantial stochastic variability in outbreak trajectories, especially during early and peak phases, underscoring the importance of uncertainty quantification for public health planning. Overall, the framework provides a rigorous, data-informed tool for evaluating interventions that target household transmission and for guiding policy decisions in structured populations.

Abstract

Understanding infectious disease spread remains a critical public health challenge, particularly given the interplay between household dynamics and community transmission patterns. Traditional epidemiological models often oversimplify these dynamics by treating populations as homogeneous, failing to capture crucial household-level interactions that can significantly impact disease spread. This paper introduces a new stochastic differential equation model extending the SIR framework by capturing the randomness in disease spread and incorporating household structure and heterogeneous mixing patterns. The model divides the population into groups based on age and household size, includes subpopulation-targeted lockdown parameters and constructs detailed contact matrices accounting for both public and within-household interactions. Through the approximation of Markov jump processes by branching processes near the disease free equilibrium, we derive the basic reproduction number of our model and conduct global sensitivity analysis using Sobol indices to identify influential factors. Our simulations reveal that incorporating household structure leads to substantially different predictions compared to traditional models, particularly in epidemic timing and peak intensity. The stochastic framework captures important variations in outbreak trajectories overlooked by deterministic approaches, especially during early and peak phases. This work contributes to both mathematical epidemiology and practical public health planning by providing a sophisticated mathematical understanding of how population structure and randomness influence disease dynamics, offering insights for intervention strategies where household transmission plays a significant role.
Paper Structure (37 sections, 3 theorems, 82 equations, 12 figures, 3 tables)

This paper contains 37 sections, 3 theorems, 82 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Model Formulation
  3. Assumptions and Notation
  4. Contact Matrices
  5. Contacts in the public
  6. Contacts within households
  7. SDEs for health states of individuals
  8. SDEs for health states of households
  9. Multi-Type Branching Process Approximation
  10. Linearization Near the Disease-Free Equilibrium
  11. Linearization of Contact Rates
  12. Linearization of Transition Rates
  13. Construction of the Multi-Type Branching Process
  14. Mean Offspring Matrix and Basic Reproduction Number
  15. Simulation Studies
  16. ...and 22 more sections

Key Result

Theorem 1

Let $\kappa: \mathbb{R}_+ \to \mathbb{R}_+$ be a continuous, strictly increasing, concave function satisfying $\kappa(0)=0$ and Suppose the drift coefficient $\bm{\mu}$ and diffusion coefficient $\bm{\sigma}$ satisfy for all $\bm{y}, \bm{y'} \in \mathbb{R}^d$ and all $t \in [t_0, T]$. Then the SDE admits pathwise unique solutions.

Figures (12)

  • Figure 1: Model structure with two types of connections. Solid arrows represent the physical state transitions (infection and recovery). Dashed lines depict the interaction between scales: individual prevalence drives household risk, which in turn reshapes individual transmission dynamics.
  • Figure 2: Examples of social contact patterns with four age categories ($0-15$ years, $16-29$ years, $30-55$ years, $56+$ years) and for four different household sizes ($1$ person, $2$ persons, $3$ persons, $4$ persons). (a) A matrix $\bm{C^p}$ for contacts in the public, illustrating interactions across various subpopulations in public environments. (b) A matrix $\bm{C^h}$ for contacts within households, highlighting the interaction patterns among household members. The matrices appear similar across household size categories due to the exemplary choice of uniform household size distributions and within-household contact proportions for all household types.
  • Figure 3: Markov jump process for subpopulation $k$ showing transitions through susceptible ($S_k$), infected ($I_k$), and recovered ($R_k$) states. Infection occurs via public or household transmission, with the household pathway incorporating a contact intensity multiplier (Equation \ref{['contact_intensity_multiplier']}).
  • Figure 4: Comparison of time-to-peak distributions across four epidemic models (U: Unstructured, A: Age only, H: Household only, and AH: Age and household). Each violin displays the distribution of peak times from $500$ unique parameter combinations $(\alpha,\beta)$, all of which were constrained to produce an $80\%$ final epidemic size. The distributions are shown on a logarithmic scale (base 10). Black diamonds represent the mean value for each model, while individual data points are overlaid in gray to show the raw distribution. The household-based models (H, mean 5.4 days; AH, mean 5.9 days) peak earlier than the non-household models (U, mean 8.5 days; A, mean 9.2 days).
  • Figure 5: Decomposition of public ($F^p$, solid) and household ($F^h$, dashed) transmission forces over time (final size 80%). Left: total transmission across all subpopulations. Right: transmission dynamics by household size. Vertical dashed lines mark epidemic peaks. Percentages show relative contributions at peak, demonstrating that household transmission dominates in larger households while public transmission prevails in smaller households.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Theorem 1: Yamada-Watanabe Condition Mao2008
  • Remark 1
  • Theorem 2: Maximal Local Solution Mao2008
  • Remark 2
  • Lemma 1: Global Existence and Positive Invariance of $\mathcal{D}$
  • proof