Early estimation of the reproduction number in the presence of imported cases: pandemic influenza H1N1-2009 in New Zealand

Abstract

We analyse data from the early epidemic of H1N1-2009 in New Zealand, and estimate the reproduction number R. We employ a renewal process which accounts for imported cases, illustrate some technical pitfalls, and propose a novel estimation method to address these pitfalls. Explicitly accounting for the infection-age distribution of imported cases and for the delay in transmission dynamics due to international travel, R was estimated to be (95% confidence interval: 107,1.47). Hence we show that a previous study, which did not account for these factors, overestimated R. Our approach also permitted us to examine the infection-age at which secondary transmission occurs as a function of calendar time, demonstrating the downward bias during the beginning of the epidemic. These technical issues may compromise the usefulness of a well-known estimator of R--the inverse of the moment-generating function of the generation time given the intrinsic growth rate. Explicit modelling of the infection-age distribution among imported cases and the examination of the time dependency of the generation time play key roles in avoiding a biased estimate of R, especially when one only has data covering a short time interval during the early growth phase of the epidemic.

Figure 1. The daily incidence of H1N1-2009 in New Zealand from April to September 2009.

Only confirmed cases are shown. White bars represent local cases (i.e. locally transmitted cases without overseas travel), black bars represent imported cases. Vertical solid lines indicate the last calendar date of each month. The vertical dashed line is at June 22, the date on which the control policy switched from a containment to a management phase.

Figure 2. Observed (black) and predicted (grey) cumulative numbers of confirmed locally transmitted cases.

Predicted values represent conditional expectations given by where is the cumulative number of cases at day , and is the maximum likelihood estimate of the growth rate.

Figure 3. Transmission dynamics of H1N1-2009 in New Zealand.

A: Observed daily incidence of imported (black) and local (grey) cases from 28 May to 22 June 2009. We examine only confirmed cases during the containment phase. B: Discretised distribution of the generation time. Mean and variance are assumed to be 2.70 days and 1.21 days, respectively. C: Observed (black) and predicted (grey) numbers of local confirmed cases. Predicted values represent conditional expectations derived from our proposed model, which includes adopting a negative binomial offspring distribution. D: Sensitivity of the estimated reproduction number to the mean generation time, over the range days. Whiskers extend to the upper and lower 95% confidence intervals based on the profile likelihood.

Figure 4. Assessment of the distribution of generation time.

A: The mean generation time as a function of calendar time. B: The variance of the generation time as a function of time. C: Sensitivity of population entropy to mean generation time. D: Sensitivity of population entropy to the variance of the generation time. The horizontal dashed line in A, and the vertical dashed line in C, represents the assumed mean generation time, 2.70 days; which is fixed in B and D. The horizontal dashed line in B, and the vertical dashed line in D, represents the assumed variance, 1.21 days; which is fixed in A and C.