Joint estimation of the basic reproduction number and serial interval using Sequential Bayes

TL;DR

The paper tackles real-time joint estimation of $R_0$ and the serial interval from incidence data using a sequential Bayes framework under an SIR model. It introduces mildly informative log-Gamma marginals linked via a Gaussian copula to form a joint prior over $(R_0, \gamma)$, with $SI = 1/\gamma$, and updates the joint posterior as new data arrive. Simulation studies show improved precision and stability for $R_0$ compared to the White-Pagano estimator, while SI estimates are more sensitive to prior misspecification but can be reliable when priors are informative; results hold under model misspecification such as SEIR/SEAIR. A Canadian COVID-19 incidence analysis demonstrates practical applicability and highlights guidance to use multiple priors for SI.

Abstract

Early in an infectious disease outbreak, timely and accurate estimation of the basic reproduction number ($R_0$) and the serial interval (SI) is critical for understanding transmission dynamics and informing public health responses. While many methods estimate these quantities separately, and a small number jointly estimate them from incidence data, existing joint approaches are largely likelihood-based and do not fully exploit prior information. We propose a novel Bayesian framework for the joint estimation of $R_0$ and the serial interval using only case count data, implemented through a sequential Bayes approach. Our method assumes an SIR model and employs a mildly informative joint prior constructed by linking log-Gamma marginal distributions for $R_0$ and the SI via a Gaussian copula, explicitly accounting for their dependence. The prior is updated sequentially as new incidence data become available, allowing for real-time inference. We assess the performance of the proposed estimator through extensive simulation studies under correct model specification as well as under model misspecification, including when the true data come from an SEIR or SEAIR model, and under varying degrees of prior misspecification. Comparisons with the widely used White and Pagano likelihood-based joint estimator show that our approach yields substantially more precise and stable estimates of $R_0$, with comparable or improved bias, particularly in the early stages of an outbreak. Estimation of the SI is more sensitive to prior misspecification; however, when prior information is reasonably accurate, our method provides reliable SI estimates and remains more stable than the competing approach. We illustrate the practical utility of the proposed method using Canadian COVID-19 incidence data at both national and provincial levels.

Figures (10)

• Figure 1: Each panel shows the prior that is used for that specific week represented by a contour plot, which, except for week 1, is the posterior from the previous week. The density is shown such that the more vibrant colours indicate higher density regions. The red lines denote 95% credible intervals (CrI) for the prior used in each week, which were found using the highest density regions (HDR). Specifically, within each CrI is 95% of the density, i.e. $P((\gamma, R_0) \in CrI) = 0.95$.
• Figure 2: Plotting the L1 Differences of the medians of weeks 4, 5, and 6 under 121 cases.
• Figure 3: Joint prior density used in sequential Bayes for the well-specified, as well as the five misspecified, cases. The parameter $\gamma = 7/SI$, where SI is in days, is on the y-axis, and $R_0$ on the x-axis. The brighter colours indicate areas of higher density.
• Figure 4: Comparison of estimates of $R_0$ from the SIR, SEIR, and SEAIR datasets of both Influenza 1 and 2. Methods include the White and Pagano method, along with the proposed sequential Bayes method under the varying mispecifications of the prior listed in Table \ref{['tab: misspecifiedCases']}.
• Figure 5: Comparison of estimates of the serial interval (SI) from the SIR, SEIR and SEAIR datasets of both Influenza 1 and 2. Methods include the White and Pagano method, along with the proposed sequential Bayes method under the varying mispecifications of the prior listed in Table \ref{['tab: misspecifiedCases']}.