[Paper Review] Distribution of outbreak sizes for SIR disease in finite populations

Joel C. Miller|arXiv (Cornell University)|Jul 11, 2019

Plant Virus Research Studies25 references4 citations

TL;DR

This paper derives an efficient method to compute the final outbreak size distribution for SIR diseases in finite populations, using a triangular linear system based on the probability generating function (PGF) of the offspring distribution. The key contribution is a computationally tractable framework that enables inference of transmission parameters from observed outbreak sizes, despite identifiability challenges due to similar reproductive numbers across different distributions.

ABSTRACT

We consider the spread of a Susceptible-Infected-Recovered (SIR) disease through finite populations and derive an expression for the final size distribution. Our derivation allows arbitrary distributions of the number of transmissions caused by an infected individual. We show how this calculation can be used to infer parameters of the infectious disease through observations in multiple small populations. The inference suffers from some identifiability difficulties, and it requires many observations to distinguish between parameter combinations that correspond to the same reproductive number.

Motivation & Objective

• To derive an exact analytical expression for the final outbreak size distribution in finite populations under SIR dynamics with arbitrary transmission distributions.
• To enable parameter inference of the offspring distribution from observed outbreak sizes in small populations.
• To address identifiability issues in estimating transmission parameters when multiple distributions yield the same basic reproductive number $\mathcal{R}_0$.
• To extend the framework to include multiple introductions and generation-specific infection probabilities.
• To demonstrate that $\mathcal{R}_0$ can be reliably estimated even when the true offspring distribution shape is misspecified.

Proposed method

• Formulate the outbreak size distribution as a solution to a lower-triangular linear system $\mathsf{C}\vec{q} = \vec{1}$, where $\mathsf{C}$ depends on the PGF $\mu(x)$ of the offspring distribution.
• Define coefficients $c_{k,M} = \left[\mu\left(\frac{M-1}{N-1}\right)\right]^{-k} \prod_{j=1}^{k-1} \frac{M-j}{N-j}$ for efficient recursive computation.
• Use the equivalence between SIR dynamics and random directed multigraphs with Poisson-like out-degrees to frame the problem in graph-theoretic terms.
• Incorporate external transmissions via a modified PGF $\chi(x)$, adjusting the coefficient formula to include $\chi(M/N)$ in the denominator.
• Derive generation-specific infection probabilities using higher-order derivatives of $[\mu(x)]^{i_g}$, conditioned on susceptible and infected counts.
• Apply the framework to simulated epidemics to test inference accuracy and assess identifiability of parameter combinations.

Experimental results

Research questions

• RQ1 Can the final outbreak size distribution in a finite SIR population be computed exactly for arbitrary offspring distributions?
• RQ2 How accurately can transmission parameters be inferred from observed outbreak sizes in small populations?
• RQ3 To what extent do different offspring distributions with the same $\mathcal{R}_0$ produce indistinguishable outbreak size distributions?
• RQ4 How does observing multiple generations improve parameter inference compared to final size data alone?
• RQ5 What are the limitations of assuming homogeneous susceptibility when inferring transmission parameters from final size data?

Key findings

• The final outbreak size distribution is computed efficiently via a lower-triangular linear system $\mathsf{C}\vec{q} = \vec{1}$, enabling fast numerical solution.
• The coefficients $c_{k,M}$ can be computed recursively, significantly reducing computational cost compared to direct matrix inversion.
• Despite identifiability issues, $\mathcal{R}_0$ can be accurately inferred from outbreak size data, even when the true offspring distribution is misspecified.
• Overdispersion in transmission is difficult to detect in small populations due to low epidemic probability precision, requiring many observations.
• Heterogeneous susceptibility biases inference when assuming homogeneous susceptibility, as it affects final size more than epidemic probability.
• The model remains robust in predicting $\mathcal{R}_0$ in the large-population limit, where final size depends only on $\mathcal{R}_0$.