Pros and cons of estimating the reproduction number from early epidemic growth rate of influenza A (H1N1) 2009

Abstract

Background: In many parts of the world, the exponential growth rate of infections during the initial epidemic phase has been used to make statistical inferences on the reproduction number, R, a summary measure of the transmission potential for the novel influenza A (H1N1) 2009. The growth rate at the initial stage of the epidemic in Japan led to estimates for R in the range 2.0 to 2.6, capturing the intensity of the initial outbreak among school-age children in May 2009.

Methods: An updated estimate of R that takes into account the epidemic data from 29 May to 14 July is provided. An age-structured renewal process is employed to capture the age-dependent transmission dynamics, jointly estimating the reproduction number, the age-dependent susceptibility and the relative contribution of imported cases to secondary transmission. Pitfalls in estimating epidemic growth rates are identified and used for scrutinizing and re-assessing the results of our earlier estimate of R.

Results: Maximum likelihood estimates of R using the data from 29 May to 14 July ranged from 1.21 to 1.35. The next-generation matrix, based on our age-structured model, predicts that only 17.5% of the population will experience infection by the end of the first pandemic wave. Our earlier estimate of R did not fully capture the population-wide epidemic in quantifying the next-generation matrix from the estimated growth rate during the initial stage of the pandemic in Japan.

Conclusions: In order to quantify R from the growth rate of cases, it is essential that the selected model captures the underlying transmission dynamics embedded in the data. Exploring additional epidemiological information will be useful for assessing the temporal dynamics. Although the simple concept of R is more easily grasped by the general public than that of the next-generation matrix, the matrix incorporating detailed information (e.g., age-specificity) is essential for reducing the levels of uncertainty in predictions and for assisting public health policymaking. Model-based prediction and policymaking are best described by sharing fundamental notions of heterogeneous risks of infection and death with non-experts to avoid potential confusion and/or possible misuse of modelling results.

Figure 1

Temporal distribution of confirmed cases of influenza A (H1N1) 2009 virus infection in Japan from May to July 2009 (n = 3,480). All the confirmed cases were diagnosed by RT-PCR. The horizontal axis represents the date of onset. Cases are stratified by (A) age and (B) travel history. Here "cases with travel history" are associated with overseas travel within 10 days preceding onset of illness and those with such a history are referred to as imported cases in our analysis.

Figure 2

Simple extrapolation of the exponential growth of cases. Two exponential fits are compared with the observed number of confirmed cases. Exponential fit 1 employs the data set from 5 May to 17 May during which clusters of cases in a few high schools fuelled the epidemic. Exponential fit 2 draws the best fit to the data from 29 May to 14 July representing the spread of influenza into the wider population. The growth rates for fits 1 and 2 are estimated at 0.37 and 0.08 per day, respectively.

Figure 3

Model prediction. Observed (dots) and predicted (lines) age-specific numbers of confirmed cases as a function of onset time are compared. The prediction on day t was conditioned on observations from days 0 to (t-1).

Figure 4

Parameter estimates and sensitivity analysis. Panel A examines the sensitivity of the reproduction number to different mean lengths of the generation time ranging from 2.1 to 3.3 days. Panel B shows the estimate of the age-specific relative susceptibility. The expected value of susceptibility for those aged 20-39 years was taken as the reference. In both panels, the whiskers extend to the upper and lower 95% confidence intervals based on the profile likelihood.