A Bayesian modelling framework with model comparison for epidemics with super-spreading

Abstract

The transmission dynamics of an epidemic are rarely homogeneous. Super-spreading events and super-spreading individuals are two types of heterogeneous transmissibility. Inference of super-spreading is commonly carried out on secondary case data, the expected distribution of which is known as the offspring distribution. However, this data is seldom available. Here we introduce a multi-model framework fit to incidence time-series, data that is much more readily available. The framework consists of five discrete-time, stochastic, branching-process models of epidemics spread through a susceptible population. The framework includes a baseline model of homogeneous transmission, a unimodal and a bimodal model for super-spreading events, as well as a unimodal and a bimodal model for super-spreading individuals. Bayesian statistics is used to infer model parameters using Markov Chain Monte-Carlo methods. Model comparison is conducted by computing Bayes factors, with importance sampling used to estimate the marginal likelihood of each model. This estimator is selected for its consistency and lower variance compared to alternatives. Application to simulated data from each model identifies the correct model for the majority of simulations and accurately infers the true parameters, such as the basic reproduction number. We also apply our methods to incidence data from the 2003 SARS outbreak and the Covid-19 pandemic caused by SARS-CoV-2. Model selection consistently identifies the same model and mechanism for a given disease, even when using different time series. Our estimates are consistent with previous studies based on secondary case data. Quantifying the contribution of super-spreading to disease transmission has important implications for infectious disease management and control. Our modelling framework is disease-agnostic and implemented as an R package, with potential to be a valuable tool for public health.

Keywords: Bayesian modelling; Infectious disease epidemiology; Model comparison; Super-spreading; Transmission heterogenity.

Fig. 1

Typical simulation from the five models of epidemic transmission. The time series plots show incidence data I_[1:T] simulated from the Baseline, SSE, SSI, SSEB and SSIB models for R₀ = 2.0, duration T = 50.

Fig. 2

Results of inference of R₀ across the five models. Posterior estimates in each model fit to 5000 simulated datasets from the model itself for R₀ in the range [0.9, 4.0]. Dots indicate the mean of the estimated posteriors, bars represent the 95 % CI of the posterior, and the black diagonal line represents the true value used for simulation.

Fig. 3

Incidence data from New Zealand's Southland district during the SARS-CoV-2 outbreak in 2020, before and after the wedding-related super-spreading event on March 21st.

Fig. 4

Reported cases from the Waitemata district of Auckland, New Zealand during a wave of SARS-CoV-2 cases in August 2021 resulting in a Level 4 lockdown.

Fig. 5

Bar plots of the posterior model probabilities of the five models applied to the two waves of SARS-CoV-2, New Zealand in 2020 and 2021.

Fig. 6

Incidence data and bar plots of the posterior model probabilities of the five models applied to the two waves of SARS outbreaks in 2003.