Heterogeneity in the onwards transmission risk between local and imported cases affects practical estimates of the time-dependent reproduction number

Abstract

During infectious disease outbreaks, inference of summary statistics characterizing transmission is essential for planning interventions. An important metric is the time-dependent reproduction number (R_t), which represents the expected number of secondary cases generated by each infected individual over the course of their infectious period. The value of R_t varies during an outbreak due to factors such as varying population immunity and changes to interventions, including those that affect individuals' contact networks. While it is possible to estimate a single population-wide R_t, this may belie differences in transmission between subgroups within the population. Here, we explore the effects of this heterogeneity on R_t estimates. Specifically, we consider two groups of infected hosts: those infected outside the local population (imported cases), and those infected locally (local cases). We use a Bayesian approach to estimate R_t, made available for others to use via an online tool, that accounts for differences in the onwards transmission risk from individuals in these groups. Using COVID-19 data from different regions worldwide, we show that different assumptions about the relative transmission risk between imported and local cases affect R_t estimates significantly, with implications for interventions. This highlights the need to collect data during outbreaks describing heterogeneities in transmission between different infected hosts, and to account for these heterogeneities in methods used to estimate R_t. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

Keywords: COVID-19; SARS-CoV-2; branching processes; imported cases; mathematical modelling; reproduction number.

Figure 1.

A disease incidence time-series dataset can be generated by different combinations of transmission risks from imported and local cases. In the first scenario (bottom left), observed cases are mostly due to infections by imported cases, whereas in the second scenario (bottom right), observed cases are mostly due to infections by local cases. In the bottom panels, red arrows represent infections generated by imported cases and black arrows represent infections generated by local cases. An individual who is infected by an imported case is classified as a local case, since they have themselves been infected locally. Despite the same overall incidence, the two scenarios shown correspond to different risks of sustained local transmission (the risk of sustained local transmission is higher in the second scenario—bottom right), with implications for public health measures. (Online version in colour.)

Figure 2.

Inference of the local reproduction number (R_t) under different assumptions about the relative transmission risk from imported and local cases. (a) The COVID-19 incidence time-series datasets used in our main analyses, for Ontario (i), New South Wales (ii) and Victoria (iii). Black bars represent the daily numbers of local cases, and pink bars represent the daily numbers of imported cases. (b) Inferred R_t values for different assumed values of the relative transmission risk from an imported case compared with a local case (ε). The grey horizontal line represents the threshold R_t = 1, and shaded regions represent the 95% central credible interval of the R_t estimates. (Online version in colour.)

Figure 3.

Implications of differences in the assumed relative transmission risk from imported and local cases on policymaking. (a) Inferred mean R_t values for different values of the relative transmissibility of imported cases compared with local cases (ε). (b) Dates on which the estimated values of R_t cross policy-relevant thresholds (in scenarios where the thresholds are crossed at some stage in the outbreak; otherwise dates are not plotted). For Ontario (i), the date shown represents the first date when the estimated R_t value is above one and remains above one for the remainder of the time period considered (until 20 April 2020). This represents the first date when the outbreak is not inferred to be under control for the remainder of the time period. For New South Wales (ii) and Victoria (iii), the date shown represents the first date on which the estimated R_t value is below one and remains so for the remainder of the time period considered (until 13 April 2020). This represents the first date on which the outbreak could be concluded as being under control for the remainder of the time period. (c) The proportion of the time periods considered for which the inferred R_t values are above one (so the outbreak is not inferred to be under control). In (b,c), results are shown for the mean values of the posterior for R_t (grey), and well as for the 2.5th (yellow dotted) and 97.5th (green dotted) percentile values of the posterior for R_t (which span the 95% central credible interval). (Online version in colour.)

Figure 4.

Inference of the local reproduction number (R_t) for estimated values of the relative transmission risk from imported and local cases. (a) The COVID-19 incidence time-series datasets used in our main analyses, for Hong Kong (i) and Hainan Province, China (ii). Black bars represent the daily numbers of local cases, and pink bars represent the daily numbers of imported cases. (b) Inferred R_t values for different assumed values of the relative transmission risk from an imported case compared with a local case (ε), for Hong Kong (i) and Hainan Province (ii). The grey horizontal line represents the threshold R_t = 1, and shaded regions represent the 95% central credible interval of the R_t estimates. The values ε = 0.2 for Hong Kong and ε = 0.785 for Hainan were estimated from alternative data sources, as described in the text. (Online version in colour.)