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Research

Real-time Estimates in Early Detection of SARS

Simon Cauchemez,*† Pierre-Yves Boëlle,*†‡ Christl A. Donnelly,§ Neil M Ferguson,§ Guy Thomas,*†‡ Gabriel M Leung,¶ Anthony J Hedley,¶ Roy M Anderson,§ and Alain-Jacques Valleron*†‡
*Institut National de la Santé et de la Recherche Médicale, Paris, France; †Université Pierre et Marie Curie, Paris, France; ‡Assistance Publique–Hôpitaux de Paris, Paris, France; §Imperial College, London, United Kingdom; and ¶University of Hong Kong, Hong Kong Special Administrative Region, People's Republic of China

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Appendix

Statistical Framework

Denoting n_t the number of cases with onset at day t and X_t the number of cases they infected, the reproduction number R_t is simply the ratio X_t /n_t defined for n_t>0. Here, we define a method to obtain the predictive distribution of R_t given the available data at day T, where data I(T) = {n_t}_{0< t <T} are the daily counts of incident onsets, assuming that the density w(.) of the generation interval is known. We will make use of the decomposition X_t = X_t^-(T) + X_t⁺(T), where the number of secondary cases X_t from cases with onset at day t has been split in those with onset before T (X_t^-(T)) (early secondary cases), and those with onset after T (X_t⁺(T)) (late secondary cases).

The construction of a global estimator is carried out in 3 stages. First, we consider the problem of right censoring, under the assumption that the exact chain of transmission has been observed until day T. In this situation, X_t^-(T) is observed while X_t⁺(T) is censored and must be predicted, conditional on X_t^-(T) and n_t, to allow computation of the predictive distribution of R_t. Second, when the exact chain of transmission has not been observed, the number of early secondary cases, X_t^-(T), is not available. Following the recommendations of Wallinga and Teunis (1), we show that it is possible to compute the distribution of X_t^-(T) given I(T). Finally, the conditional distributions of X_t⁺(T) given X_t^-(T), n_t and X_t^-(T) given I(T) are combined to derive the distribution of the reproduction number R_t conditional on I(T). All distributions presented are conditional to the number n_t of symptom onsets at day t, but notation is omitted for the sake of clarity.

Distribution of X_t⁺(T) | X_t^-(T)

We assume that X_t is Poisson distributed with mean n_t l_t and choose a vague gamma prior distribution for l_t with shape parameter a = 10^-5 and rate b = 10^-5.

Conditional on X_t, the number X_t^-(T) of early secondary cases is binomial with parameters X_t, W_tT, where W_tT is the probability that the generation interval is <T – t. It follows that X_t^-(T) | l_t is Poisson distributed with mean n_t l_t W_tT. The same argument would show that X_t⁺(T) | l_t is Poisson with mean n_t l_t (1 – W_tT). Given l_t, X_t⁺(T) and X_t^-(T) are independent so that

With Bayes' theorem,

where

Eventually, we obtain that the distribution X_t⁺(T) | X_t^-(T) = y is negative binomial with parameters p = (n_tW_tT + b)/(n_t + b), m = y + a and probability

Distribution of X_t^-(T) | I(T)

In practice, the exact realization of X_t^-(T) is unknown, and inference must be based on I(T) alone. Wallinga and Teunis (1) have shown that the probability that a case detected at day k<T has been infected by a case detected at day t<T is

where 1{.} is the indicator function. The distribution of X_t^-(T) given I(T) is a sum of independent binomial distributions

X_t^-(T) | I(T)    ~    ∑_{k <T} Bin(n_k,p_tk)

This probability distribution may be determined numerically.

Distribution of R_t | I(T)

Using the decomposition in early and late secondary cases, we obtain

After calculation, we find that the expectation and variance of X_t | I(T) are functions of the expectation and variance of X_t^-(T) | I(T) alone, derived with the method of Wallinga and Teunis:

 For the vague prior we specified for l_t, we obtain

As expected, the average proportion of secondary cases detected before T is W_tT. The first term of the variance is related to our imperfect knowledge of the realization of X_t^-(T) while the second term is related to the natural randomness of X_t⁺(T). We stress that when the lag between day t and day T is large (i.e., W_tT »1), our estimates are similar to those of Wallinga and Teunis for complete epidemics.

Given X_t | I(T), the derivation of the predictive distribution of the reproduction number R_t is straightforward considering the deterministic relation R_t = X_t/n_t.

Appendix Reference

1. Wallinga J, Teunis P. Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures. Am J Epidemiol. 2004;160:509–16.