A statistical method can be used for early monitoring of the effect of disease control measures.

We propose a Bayesian statistical framework for estimating the reproduction number

The reproduction number

If all incident cases could be traced to their index cases, estimating the reproduction number would simply be a matter of counting secondary cases. However, if tracing information is incomplete or ambiguous, modeling or statistical approaches are required. For example, a mathematical model for disease transmission fitted to available data can provide estimates of

In this report, we show how to estimate the reproduction number in an ongoing epidemic, which will account for yet unobserved secondary cases. The method is applied to data from the 2003 SARS outbreak in Hong Kong (

We propose a Bayesian statistical framework for real-time inference on the temporal pattern of the reproduction number of an epidemic. Here, the reproduction number _{t}_{t}_{t}_{t}_{t}_{t}_{t}

Assume that we would like to compute the daily values _{t}_{t}_{t}

A 3-step construct is necessary. We first predict the eventual number of late secondary cases (as yet unobserved), for cases reported at day _{t}

The method was retrospectively used to analyze the SARS outbreak in Hong Kong. The data consisted of the dates of symptom onset of the 1,755 case-patients who were detected in Hong Kong in 2003 (

Using simulations, we explored the ability of the method to quickly detect the effect of control measures. Five hundred epidemics were simulated with the following characteristics. During the first 20 days of the epidemics, the theoretical reproduction number was 3. Control measures were implemented at day 20. In a first scenario, control measures were completely effective (no transmission occurred after day 20). In a second scenario, the theoretical reproduction number after control measures were implemented was 0.7. Details on the simulations are available from the corresponding author.

In a simulation study, the bias and precision of the real-time estimator were investigated in situations in which the theoretical reproduction number remained constant with time. We also evaluated the effect of the length of the generation interval on the results. Detailed information can be obtained from the corresponding author.

_{t}

Application of real-time estimation to the severe acute respiratory syndrome outbreak in Hong Kong. A) Data. B–F) Expectation (solid lines) and 95% credible intervals (dashed lines) of the real-time estimator of _{t}

After a lag of 2 days, the 95% credible intervals were wide and displayed an undesirable feature: they sharply decreased to 0 as soon as no cases had been observed for 2 consecutive days (

With lags >5 days, the trends of expected values were relatively similar, with a peak around day 20, a decreasing trend after this date, and the expectation of _{t}_{t}_{t}

In _{t}_{20}_{t}_{t}

Average expectation of the temporal pattern of _{t}

Our statistical framework provided real-time estimates of the reproduction number of an epidemic, and thus quickly showed the impact of control measures. In simulations of SARS-like diseases, the derived estimator detected the decrease of _{t}

In theory, the method could be applied to communicable diseases with the following characteristics: 1) no asymptomatic cases; 2) no underreporting; 3) knowledge of the generation interval. The list of communicable diseases that could be monitored is therefore relatively large, although it does not include diseases such as influenza, for which the proportion of asymptomatic or unreported cases may be large. In practice, the delay until estimates of the reproduction number become reliable will depend critically on the generation interval distribution. For SARS, when the reproduction number was constant over time, our real-time estimates were almost unbiased after only 1 day. With the original estimator of Wallinga and Teunis (

We assumed that the distribution of the generation interval was known and remained unchanged during the course of the outbreak. In practice, however, this distribution is derived from a subset of traced cases. If the subset is small, e.g., the case at the beginning of an emerging disease outbreak, uncertainty will be large. Furthermore, the generation interval may decrease during the course of the outbreak because of quicker interventions, leading to possible bias in the estimates of

The approach smoothed the temporal pattern of the reproduction number, leading to overestimation of

The method has a natural real-time implementation in which 1) a first estimate of the reproduction number is available after a lag that depends on the generation interval, and 2) while the epidemic goes on, the estimate is consolidated, and its credible interval narrows. Incorporation of such a statistical estimation framework into real-time surveillance of future infectious disease outbreaks would enhance the ability of epidemiologists to provide timely advice to public health policymakers.

Denoting _{t}_{t}_{t}_{t} /n_{t}_{t}_{t}_{t}}_{0< t <T}_{t}_{t}^{-}(T)_{t}^{+}(T)_{t}_{t}^{-}(T)_{t}^{+}(T)

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}^{-}(T)_{t}^{+}(T)_{t}^{-}(T)_{t}_{t}_{t}^{-}(T)_{t}^{-}(T)_{t}^{+}(T)_{t}^{-}(T)_{t}_{t}^{-}(T)_{t}_{t}

We assume that _{t}_{t} λ_{t}_{t}^{-5} and rate β = 10^{-5}.

Conditional on _{t}_{t}^{-}(T)_{t}_{tT}_{tT}_{t}^{-}(T)_{t}_{t} λ_{t} W_{tT}_{t}^{+}(T)_{t}_{t} λ_{t} (1 – W_{tT})_{t}_{t}^{+}(T)_{t}^{-}(T)_{t}^{+}(T)_{t}^{-}(T)_{t}W_{tT} + β)/(n_{t} + β

In practice, the exact realization of _{t}^{-}(T)_{t}^{-}(T)_{t}^{-}(T)_{k <T}_{k}_{tk}

Using the decomposition in early and late secondary cases, we obtain_{t}_{t}^{-}(T)_{t}

As expected, the average proportion of secondary cases detected before _{tT}_{t}^{-}(T)_{t}^{+}(T)_{tT}

Given _{t}_{t}_{t}_{t}_{t}

The work in Hong Kong was supported in part by a commissioned grant from the Research Fund for the Control of Infectious Diseases of the Health, Welfare and Food Bureau of the Hong Kong SAR Government.

Mr Cauchemez is a doctoral student at University Pierre et Marie Curie, Paris, France. He develops statistical methods to analyze transmission of infectious diseases using incomplete information.