Misspecification of the generation time distribution and its impact on Rt estimates in structured populations

Abstract

Due to its ability to summarise 'real-time' epidemic behaviour, the time-dependent reproduction number, Rt, is a useful metric for tracking pathogen transmission and quantifying the effects of interventions during infectious disease outbreaks. The predominant models underlying inferred Rt trajectories are renewal equations, their success owing in part to the relatively few assumptions they require. One necessary assumption is the generation time distribution, which summarises the time periods between infections in infector-infectee transmission pairs. This distribution is typically assumed to be the same across all members of a population. In reality, however, it may vary systematically between population groups. In this study, we consider two Rt inference frameworks based on renewal equation models: one for a single, homogeneous group and another accounting for a structured population. We compare the estimates of Rt generated by the two models and investigate, both analytically and through simulations, under which conditions the conclusions drawn from these modelling paradigms differ. We also demonstrate a methodology for selecting the generation time for the one-group model that correctly encapsulates variations between different population groups; this allows us to use a renewal framework for a one-group model to infer Rt when, in fact, the population is structured. Finally, we use real epidemic data to demonstrate that practical Rt estimates can differ depending on whether the underlying model is the one-group model or the multi-group model. Our results motivate the need for rigorous collection of detailed epidemic data and consideration of differences between population groups to improve the accuracy of Rt estimates that are used to guide public health policy responses.

Figures (14)

• Figure 1: Group-specific generation time distributions used in our analyses. Generation time distributions are shown for a two-group toy population when: (left-panel) they match up to day $5$ in their unnormalised form, (right-panel) they are different throughout. These generation times follow a discretised gamma distribution with mean and standard deviation as described in Table \ref{['tab:generation-time-comparison']}.
• Figure 2: $R_t$ estimates obtained from the one- and multi-group models match in the long-term when the contact matrix does not vary temporally. The mean and 95% confidence region of the overall $R_t$ trajectories are inferred using the multi-group model (red lines) and one-group model (light blue lines) for a range of model parameters as described in Table \ref{['tab:comparison-conditions']}, when the generation time for the one-group model is calculated appropriately (as described in the text). The simulated disease incidence data (top panels) underlying the $R_t$ estimates are generated using a two-group model.
• Figure 3: $R_t$ estimates obtained from the one- and multi-group models diverge when the contact matrix varies temporally. The mean and 95% confidence region of the overall $R_t$ trajectories inferred using the multi-group model (red lines) and the one-group model (light blue lines), when the generation time for the one-group model is correctly calculated but the contact matrix changes in time. The underlying disease incidence data are generated using a two-group population and the group-specific daily disease incidence data are shown in the top panels of the figure.
• Figure 4: $R_t$ estimates obtained using the one-group and multi-group models from disease incidence data from the 2009 Japan A/H1N1 epidemic. (Top panel) Disease incidence data for the A/H1N1 Japan outbreak in $2009$. The numbers of recorded cases are split across two population groups: 0-19 (blue bars) and $20+$ years old (red bars). (Bottom panel) Inferred mean trajectories and $95\%$ confidence regions of the group-specific reproduction numbers (yellow lines) and overall $R_t$ inferred from the multi-group renewal equation model (red lines) and the one-group model (blue lines). The epidemic dataset consists of data for $20$ days, starting from $9^\text{th}$ May 2009.
• Figure S1: Disease incidence data generated using a two-group model. These data are used to infer the trajectory of the overall $R_t$ in Figure \ref{['Same serial interval']} for both the one-group and the multi-group renewal models for a range of changed model parameters as described in Table \ref{['tab:comparison-conditions']}. The same generation time distribution is used for both population groups. The group-specific mean and standard deviation for the generation time distributions as described in Table \ref{['tab:generation-time-comparison']}.