# Evaluating the Effectiveness of COVID-19 Bluetooth-Based Smartphone Contact Tracing Applications
## 3 Smartphone-Based Contact Tracing
### Characterizing Contact Tracing Effectiveness  
There are three main aspects of current contact tracing application that can impact their effectiveness: precision, utilization and speed. In this section, we parametrize the impact of these aspects on the fraction of possible traced contacts and the tracing time, which will be used in the epidemic model. Precision is fundamental in detecting real contacts. We are not talking about detecting the contacts that have effectively produced an infection, which would depend on many other factors. Instead, precision is measured as the capacity of detecting the physical contacts that the health authorities determine that supposes an exposure to a significant risk of infection. For example, if health authorities determine that contacts that last at least 15 min in a range of 2 m are an exposure to risk (as detailed in [40]), the mobile application should detect only these contacts. Nevertheless, Bluetooth technology is far from being perfect, and it has been shown that many factors can affect the correct computation of distances [32]. The goal of contact tracing is to precisely detect the defined risky physical contacts, that is, a positive contact. As was previously said, Bluetooth technology for detecting exposure contacts is not precise enough (for example, when the distance is wrongly measured), and there might be some contacts missing (false negatives), and some other contacts added that should not have been (false positives). We measure this precision using the following ratios: the false negative ratio, \(FN\), which corresponds to the probability of failing to detect a positive contact; and the false positive ratio \(FP\), as the ratio over the number of positives. Note that the false positive ratio is usually defined based on the proportion of all negatives that yields a (false) positive. Nevertheless, in this case, there is no clear way to define all real negative contacts. Considering that, in a day, the number of true positive contacts are \(TP\), we can obtain the whole number of contacts generated by the tracing application as \((FP+(1-FN))TP\). That is, we add the false positives and the true positives \((1-FN)\). We define the relative error as:  
\[\epsilon=\frac{(FP+FN)}{(1+FP)} \tag{1}\]  
Figure 1: Centralized vs. decentralized contact tracing approaches: (**a**) when two users’ smartphones are in range, they exchange their anonymous key codes; (**b**) when an individual is detected as positive, he/she notifies the application of his/her this new status; in the centralized approach (**c**), the smartphone uploads its key and also the other keys gathered from its previous contacts; (**d**) the possible risk contacts are detected using the centralized servers; in the decentralized approach (**e**), the smartphone only uploads its key; and (**f**) all keys of detected positives are downloaded by the application and the matching is performed locally. which ranges from 0 (no error) to 1 (no real contact is detected). The second aspect that impacts on the effectiveness is the utilization (adoption) of the application. In other words, how many people would use a contact tracing application? This question has not a simple answer and depends on many factors. As an example, Singapore's official contact tracing app, TraceTogether, was only used by 20% of the population [6]. Based on the ratio of utilization \(U\) of the contact tracing application we can estimate the number of contacts traced. Assume that, throughout a day, the average number of physical contacts of an individual is \(k\). Detecting a contact requires that two individuals use the application. Thus, the probability of detecting a contact will be simply \(U\times U\) (both individuals used the application), and the average number of contacts traced by individuals will be \(U^{2}k\). For example, considering the contact tracing ratio of utilization of 0.2, we have that only 4% of the contacts are detected. Therefore, in order to be effective, the utilization of the applications must be increased.