Poisson Network SIR Epidemic Model

TL;DR

Traditional SIR models neglect explicit contact network structure, limiting their ability to capture transmission heterogeneity. This paper develops a Poisson network SIR using a configuration-model framework and dynamical survival analysis, deriving an exact pairwise closure for Poisson degree and performing inference via Hamiltonian Monte Carlo on Ebola data from the 2018–2020 DRC outbreak. The network model yields a near-realistic fit, reveals a highly skewed average degree (mean about $\mu \approx 40$, mode around $25$) and a network reproduction number $\tilde{\mathcal R}_0 \approx 1.07$, while illustrating that for large $\mu$ the network dynamics approximate the classical SIR curves. This approach provides a scalable, network-aware method for inferring contact patterns and outbreak size, bridging network-based dynamics with traditional mass-action models to inform targeted interventions in real-world epidemics.

Abstract

We extend the classical Susceptible-Infected-Recovered (SIR) model to a network-based framework where the degree distribution of nodes follows a Poisson distribution. This extension incorporates an additional parameter representing the mean node degree, allowing for the inclusion of heterogeneity in contact patterns. Using this enhanced model, we analyze epidemic data from the 2018-20 Ebola outbreak in the Democratic Republic of the Congo, employing a survival approach combined with the Hamiltonian Monte Carlo method. Our results suggest that network-based models can more effectively capture the heterogeneity of epidemic dynamics compared to traditional compartmental models, without introducing unduly overcomplicated compartmental framework.

Figures (4)

• Figure 1: SIR Dynamics on Network. Blue nodes represent susceptible individuals, while red and pink ones represent the initially infected and secondarily infected individuals, respectively. The black node indicates a removed individual. Dashed half-edges connect uniformly at random to form solid edges.
• Figure 2: Approximating Network SIR Model. A simple example illustrating the result of Proposition 1. For large mean degree distribution $\mu$ (here at least 40), the network curves for infected are seen to get close to the one corresponding to the classical SIR model (lowest curve). For this particular example the values of the parameters $(\tilde{\beta},\tilde{\gamma},\rho)$ are taken from the first column of Table \ref{['tab:1']} in Section \ref{['sec:3']} below.
• Figure 3: Model Fit. Histograms of onset and recovery (hospitalization) times overlaid with Poisson SIR model fit, computed at the means of the posterior parameter values (solid lines) with the corresponding 95% credible intervals (shaded regions).
• Figure 4: Posterior Parameter Densities. The posterior distributions of $\tilde{\beta}$, $\tilde{\gamma}$, $\mu$ and $\tilde{{\cal R}}_0$. The posterior distribution of average degree ($\mu$) is seen to be right skewed but with mode below 25.

Theorems & Definitions (2)

• Theorem 1
• Proposition 1