Sensitivity of contact-tracing for COVID-19 in Thailand: a capture-recapture application

Our data stems from Thailand’s regular COVID-19 CT operations. Figure 1 presents a flowchart of the contact tracing process undertaken by the local communicable disease control units (CDCU) and joint investigation teams (JIT) from DDC. Once the patient is diagnosed as being infected with SARS-CoV-2, so called the confirmed case, contact tracing will be conducted to obtain the list of contacts. The identified contacts are classified as either high-risk contacts or low-risk contact following investigation guidelines [13]. High-risk contact is defined as a contact who is more likely to contract the virus through exposure to respiratory secretions of the confirmed case while not wearing PPE according to standard precautions. Low-risk contact is defined as a contact who is less likely to contract the virus from the confirmed case. This includes contacts who have not met the definition for high-risk contact. Only high-risk contacts were quarantined in the designated places and basic demographic information such as age, sex, and nationality were collected and recorded in the contact form. Our data set comprises the period 11 January 2020 to 30 June 2020. A total of 352 cases were identified through contact tracing system leading to 6359 high risk contacts and 4299 low risk contacts.

We are interested in deriving an estimate of the unknown true number of COVID-19 cases with contacts that entered the CT mechanism. This would address question 1 (Q1) as above. The tracing of contacts is likely to lead to the identification of secondary infections for a subset of index cases. This data informs question 2 (Q2). For both questions, the data can be binned into the number of index cases with one listed (Q1) or infected (Q2) contact (\(f_1\)), two listed or infected contacts (\(f_2\)), and so on up to the number of index cases with the maximum number of listed or infected contacts (\(f_m\)). Here, \(f_0\), the frequency of index cases with unobserved contacts (for Q1) or unobserved infected contacts (for Q2) is unknown and the target of the inference. Statistically, the identification process leads to a zero-truncated count distribution of cases with at least one listed or infected contact, i.e. with positive integers (ones, twos, threes, etc.), but no zeros. By applying CRC approaches, we can infer \(f_0\), the number of unobserved cases with at least one listed or infected contact.

For both questions, we fit parametric models (Poisson, Negative Binomial, and Geometric) to the observed counts using the maximum likelihood method. Then, the smallest Akaike and Bayesian Information Criterion (AIC and BIC, respectively) are used for model selection. After estimating model parameters, we can estimate \(f_0\) aswhere n is the observed sample size and \(p_0\) is the estimated probability of missing an index case with non-zero contacts as computed from the models. The population size estimator \({\hat{N}}=n+{\hat{f}}_0\) follows.

In addition to the model-based estimators we consider two further alternatives for comparison purposes: the Turing’s estimator [14] and Chao’s lower bound estimator [15]. Turing’s estimator is formulated under a homogeneous Poisson distribution with parameter \(\lambda\). Let \(p_0\) be the probability of zero count or missing an observation. We haveThe estimate of \(p_0\) can be calculated from observed frequencies as followswhere \(S = \sum _{i=0}^m if_i\). Thus, Turing’s estimator for estimating the population size is given byChao (1987) suggested a mixed Poisson model with \(p_i = \int _0^{\inf } \frac{e^{-\lambda }\lambda ^i}{i!}g(\lambda )d\lambda\) for i = 0, 1, 2, ... and arbitrary density \(g(\lambda )\) [15]. Chao’s estimator incorporates not only the unobserved heterogeneity in the Poisson parameter but also leads to a very simple nonparametric estimator by applying the Cauchy–Schwarz inequality to the lower bound for the probability of a not observed eventReplacing these probabilities by observed frequencies, the lower bound for the estimate of zero counts is computed as \({\hat{f}}_0 \ge f_1^2/(2f_2)\). As a result, Chao’s lower bound estimator for the population size isClearly, (6) uses only part of the available information, \(f_1\) and \(f_2\), as opposed to Turing estimator that uses all the information in the sample by means of S. In addition, a mixing distribution \(g(\lambda )\) is not required to be specified and estimated showing the non-parametric nature of this estimator.

To estimate 95% confidence intervals (95% CIs), we use resampling techniques as described in the CRC literature [16, 17]. Suppose that \({\hat{N}}\) is the estimated size under a fitted model. Then, we generate B samples of size \({\hat{N}}\) using the fitted model and its estimated parameter(s). For each sample, all zeros are truncated and the size estimate \({\hat{N}}_b\) computed, for each of the samples b = 1, 2,..., B leading to a sample of estimates \({\hat{N}}_1\), \({\hat{N}}_2\),..., \({\hat{N}}_B\). We choose B = 10,000 to minimize bootstrap simulation random error, and then use two methods towards CI construction:

In the period 4 Jan 2020 to 30 June 2020, 3171 cases were confirmed (Fig. 2). Of those, 352 (11.1%) index cases were followed through CT leading to the identification of subsequent contacts for 341 of them. Among these 341 index cases with non-zero contacts from which 6,359 high risk contacts were listed, there were 44 index cases with one contact (\(f_1\) = 44), 22 with two contacts (\(f_2\) = 22), 24 with three contacts (\(f_3\) = 24), and so on. Table 1 shows the complete distribution of index cases with traced contacts for the first 50 index cases. For infected contacts, the complete distribution is as follows: index cases with one infected contact (\(f_1\) = 16), two infected contacts (\(f_2\) = 9), three infected contacts (\(f_3\) = 4), and four infected contacts (\(f_4\) = 1).

Of the 341 index cases with at least one contact, 196 (57.48%) were males and 145 (42.52%) were females. At a 5% level of significance, there was sufficient evidence to conclude that there was a difference between the proportions of these contacts from male and female (goodness-of-fit Chi-square test with P-value = 0.007). See more details of the goodness-of-fit test in [18]. The median age was 37 years (mean = 39.62, interquartile range (IQR) = 28–50, min = 0.3, max = 83). The statistics of age by gender are given in the following:

These showed median age for males was significantly greater than that of females (Wilcoxon signed-rank test with P-value = 0.004) [18]. The vast majority of cases (290, 85.04%) cases were Thai. Meanwhile, 51 (14.96%) were foreign nationals: 26 cases (7.62%) from China, 5 cases (1.46%) from Japan, 4 cases (1.17%) from Denmark, and 61 cases from other locations.

From 341 index cases with non-zero contacts noted before, 30 (8.8%) index cases had at least one infected contact. The median age of this set of index cases was 44 years (mean \(= 42.87\), IQR \(= 29.25\)–56, min \(= 6\) and max \(= 80\)). Summary statistics of age by gender are concluded as follows:

Furthermore, almost all index cases with infected contacts were Thai (28 cases, 93.33%). We also show summary statistics for the set of 30 index cases with infected contacts in Table 2.

A zero-truncated Poisson regression (best fitting model for this reduced dataset (30 observations)) showed no significant covariate effects on the number of contacts per index case with at least one secondary case. For the larger dataset of the number of contacts per index case, the best fitting model (a zero-truncated negative binomial regression) also showed no significant effects of the covariates. These results support that no consideration of observed heterogeneity in our capture-recapture models was required. The logistic regression showed no significant covariate effects either.

The application of a zero-inflated negative binomial model to the full distribution of all index cases (n = 352 that includes all cases detected through the CT mechanism and their contacts; we note that for 311 of such cases there were zero infected contacts, hence the use of a zero-inflated model) allows the estimation of the average number of secondary infections over the course of the outbreak that equates to an average effective reproductive number (RE = 0.14; 95% CI: 0.09–0.22), and dispersion parameter (k = 0.1). We note that our estimate of the reproduction number applies to the entirety of the period under study and is sensitive to the implementation of several public and health social measures in country at different times.

As seen in Fig. 3, the distribution of index cases with contacts presents a long tail. Clearly, this long-tailed distribution is fitted a lot better by the negative binomial than by the Poisson distribution and the geometric distribution (see Fig. 4). The best fit (in effect addressing Q1) is given by a zero-truncated negative binomial model (Table 3) leading to an estimate of unobserved index cases with contacts of \({\hat{f}}_0\) = 98.18, and an estimated size of the overall count of index cases with contacts of \({\hat{N}}\) = 439.18. In the appendix we derive population size estimators for Turing and Chao approaches for the chosen negative binomial distribution, and in Table 4 we present the results of the three estimators including 95% confidence intervals. Note that Chao’s estimator is slightly higher than the other two, indicating potential residual heterogeneity. However, this might be also still within random error variation as the confidence intervals in Table 4 express. Using the estimated \(f_0\) from the zero-truncated negative binomial model, we estimate the CT sensitivity to detect index cases with contacts as 341/(341 + 98) = 0.776 or 77.6%.

Next, we address Q2. As can be seen in Fig. 5, the best fitting model is given by the zero-truncated Poisson model (Table 5) with \({\hat{\lambda }}\) = 1.126. Table 6 provides the estimated frequency of index cases with infected but unobserved contacts for the zero truncated Poisson \({\hat{f}}_0 = \frac{n e^{-{\hat{\lambda }}}}{1-e^{-{\hat{\lambda }}}}\) = 14.37, and those from the Turing and Chao approaches for reference. Using the estimated \(f_0\) from the zero truncated Poisson model we estimated the sensitivity of contact tracing to detect index cases with infected contacts as 30/(30 + 14) = 0.676 or 67.6%.

For each index case, the number of observed contacts allowed to derive a count distribution which has then been modelled parametrically. Using the best fitting model, the number of index cases with unobserved contacts could be determined and, thus, the completeness of CT. Clearly, the estimate of the frequency of index cases with undetected contacts depends on the model of choice. Hence, we also considered alternative estimators including those of Chao and Turing which weaken the assumption of the chosen model. Chao’s estimator allows for heterogeneity in the parameter of the probability model whereas Turing’s estimator avoids maximum likelihood estimation. If these alternative approaches lead to substantially different estimates of the size, the choice of the model might be questionable. In all our analyses, the approaches led to similar size estimates. We have also considered whether the distribution was affected by the observed heterogeneity as captured by the available covariates gender, age, or nationality. A generalized linear model analysis (using Poisson and logistic regression) showed no significant association to any of these covariates. Hence, we did not consider a stratified capture-recapture modeling. This is not to say that these variables have no effect on the sensitivity of CT, just that for our dataset such predictors did not show any significance in the unobserved number of index cases. We note that a recent study on EVD showed different patterns in the number of contacts and the probability of zero contacts between two well-defined waves in DRC, and suggested possible improvements in CT as teams become more accustomed over time [11]. In our case, there was no clear break in the time series of cases to support such analysis. However, comparing first and subsequent waves of cases in Thailand would be feasible.

We assumed a closed population which is a reasonable assumption under lock-down conditions, and typically met in these kinds of applications by steering the observational window to be small enough. We also assumed independence in the observation (sampling) of index cases. This would be typically violated if these would occur in clusters. Heterogeneity and clustering work in the same way so that Chao’s lower bound estimator would still be a conservative approach to the estimation of completeness. In all cases, the parametric modelling and Chao’ estimator have returned similar findings which supports our assumption of independence.

Several countries have used different CT mechanisms, e.g., traditional CT, use of CCTV systems, mobile applications, for the purpose of identifying contacts. In such situations, multi-list CRC models might merit study to assess multiple identification of contacts by more than one data stream.

Hook and Regal (1995) stated that the application of CRC methods had very little impact in the public health arena. In other words, their policy value might be small [22]. Providing more informative outputs with indication of where under-reporting is occurring, and what population groups might be more affected would increase the policy value [23]. However, our limited dataset did not present significant heterogeneity to inform such questions. Richer datasets would be required to that effect. A related challenge is the timing of these types of evaluations, with their retrospective nature also limiting their policy value. More real-time applications of CRC across the operational units engaged in the deployment of CT would merit study. These studies might support the identification and quantification of the impact of operational constraints (e.g., size of contact tracing teams, experienced processes and teams) in the sensitivity of CT. Such efforts to extract more value from CT data might provide additional stimulus to strengthen this critical and neglected public health capacity.