Comparative estimation of the reproduction number for pandemic influenza from daily case notification data

Abstract

The reproduction number, R, defined as the average number of secondary cases generated by a primary case, is a crucial quantity for identifying the intensity of interventions required to control an epidemic. Current estimates of the reproduction number for seasonal influenza show wide variation and, in particular, uncertainty bounds for R for the pandemic strain from 1918 to 1919 have been obtained only in a few recent studies and are yet to be fully clarified. Here, we estimate R using daily case notifications during the autumn wave of the influenza pandemic (Spanish flu) in the city of San Francisco, California, from 1918 to 1919. In order to elucidate the effects from adopting different estimation approaches, four different methods are used: estimation of R using the early exponential-growth rate (Method 1), a simple susceptible-exposed-infectious-recovered (SEIR) model (Method 2), a more complex SEIR-type model that accounts for asymptomatic and hospitalized cases (Method 3), and a stochastic susceptible-infectious-removed (SIR) with Bayesian estimation (Method 4) that determines the effective reproduction number Rt at a given time t. The first three methods fit the initial exponential-growth phase of the epidemic, which was explicitly determined by the goodness-of-fit test. Moreover, Method 3 was also fitted to the whole epidemic curve. Whereas the values of R obtained using the first three methods based on the initial growth phase were estimated to be 2.98 (95% confidence interval (CI): 2.73, 3.25), 2.38 (2.16, 2.60) and 2.20 (1.55, 2.84), the third method with the entire epidemic curve yielded a value of 3.53 (3.45, 3.62). This larger value could be an overestimate since the goodness-of-fit to the initial exponential phase worsened when we fitted the model to the entire epidemic curve, and because the model is established as an autonomous system without time-varying assumptions. These estimates were shown to be robust to parameter uncertainties, but the theoretical exponential-growth approximation (Method 1) shows wide uncertainty. Method 4 provided a maximum-likelihood effective reproduction number 2.10 (1.21, 2.95) using the first 17 epidemic days, which is consistent with estimates obtained from the other methods and an estimate of 2.36 (2.07, 2.65) for the entire autumn wave. We conclude that the reproduction number for pandemic influenza (Spanish flu) at the city level can be robustly assessed to lie in the range of 2.0-3.0, in broad agreement with previous estimates using distinct data.

Protecting residential care facilities from pandemic influenza

“…We simulated a baseline scenario and three plans that included NPIs for a range of pandemic severities (4,6,15) represented by the basic reproduction number, R 0 (Fig. 1, all values displayed in the figure are listed together with 95% confidence intervals in supporting information (SI) Tables S1 and S2).…”

Estimating functions and the generalized method of moments

“…This is the case, for example, with many deterministic models of system dynamics in application areas such as epidemic modelling [4]. Such models are, however, routinely calibrated by minimizing some measure of discrepancy between model outputs and observations and, unless they provide a perfect fit to the observations, one is often interested in making some statements about model parameter uncertainty.…”

Transmissibility of the 1918 pandemic influenza in Montreal and Winnipeg of Canada

“…It is seen that the variances of the estimates of the growth rate and reproduction number for Montreal are bigger than ones for Winnipeg. There are several different methods for estimating the reproduction number based on number of reported cases 8 . Also, it is obvious that the assumption about the latent and infectious periods of the 1918 pandemic influenza affect the estimation of the reproduction number 34 .…”

“…A measure of transmissibility and of the stringency of control measures required to stop an epidemic is determined by the magnitude of the reproductive number R. 1 Estimates of R for 1918 pandemic influenza vary widely from city to city, 8 , 22 because R depends on not only the duration of the latent and infectious periods but also the infectious agent and the host population; for example, estimates for measles vary between rural and urban populations 1 . The median estimated R for 1918 Spanish flu in 45 US cities was 2·7 with interquartile range (2·3, 3·4), 22 based on the serial interval of 6 days 19 .…”