# Evaluating the Effectiveness of COVID-19 Bluetooth-Based Smartphone Contact Tracing Applications
## 4 Epidemic Model
In general, the transmission rate, that is the rate in which the infection is transmitted from one infected individual to one susceptible individual, is formed from the product of the number of contacts per time unit \(k\), and the probability to transmit the disease \(b\); hence, newly infected individuals are generated with a whole rate \(kbI\frac{S}{N}\). Note that this rate is for all infected individuals. Since in this transition we are only considering the infected ones that are not traced, the previous rate is multiplied by \((1-q_{i})\), that is the fraction of non-traced contacts, so that the transition rate is \((1-q_{i})kbI\frac{S}{N}\). The other fraction of infected individuals is considered in transition \(S\to Q_{T}\). * \(S\to Q_{T}\): Infected individuals who are being traced and go directly to the tracing state. When infected individuals are detected (that is, when they are in class \(Q_{T}\)) their previous \begin{table}
\begin{tabular}{c c} \hline
**Symbol** & **Definition** \\ \hline \(N\) & Population \\ \(S\), \(I\), \(R\) & Susceptible, Infected and Recovered classes \\ \(Q_{S}\) & Class of susceptible individuals in quarantine by contact tracing \\ \(Q_{I}\) & Class of infected individuals detected and quarantined. \\ \(Q_{T}\) & Class of infected individuals detected, quarantined and being traced. \\ \(Q_{a}\) & People quarantined by contact tracing. \\ \(R_{0}\) & Basic/effective reproductive number (\(R_{0}=kb/\gamma=\beta/\gamma\)) \\ \(k\) & Average contacts per individual and time unit (days). \\ \(b\) & Probability to transmit the disease. \\ \(\delta\) & Detection rate of infected individuals \\ \(\beta\) & Transmission rate \\ \(\gamma\) & Recovery rate (that is, \(1/\gamma\) = days to recover) \\ \(1/\tau_{Q}\) & Average duration of the quarantine \\ \(1/\tau_{Q}^{\prime}\) & Average duration of the remaining quarantine (that is, \(1/\tau_{Q}-1/\tau_{T}\)) \\ \(1/\tau_{T}\) & Average tracing time (also \(TT\)) \\ \(FN\) & False negative ratio \\ \(FP\) & False positive ratio \\ \(U\) & Ratio of user using the application \\ \(RC\) & Ratio of users checking for exposure \\ \(q_{i}\),\(q_{s}\) & Estimated fraction of traced contacts (normalized) \\ \hline \end{tabular}
\end{table} 
Table 1: Notation table. contacts that are in the susceptible class are traced, and some of them are quarantined. Therefore, this transition considers only the traced contacts, that is the fraction \(q_{s}\), so the rate is \(q_{i}kbI\frac{S}{N}\). It refers to the individuals that may have been infected during the day. * \(S\to Q_{S}\): Non-infected individuals traced and quarantined. The transition rate is \(q_{s}k(1-b)I\frac{S}{N}\), considering that \((1-b)\) is the probability of not transmitting the disease. In this transition rate, the fraction \(q_{s}\) considers the effect of false positives, which can increase the number of people unnecessarily quarantined. * \(I\to Q_{T}\): Detection of infected individuals, triggering the tracing of their previous contacts using the application while being quarantined. In our model, infected individuals are detected, traced and quarantined with \(\delta\) rate, staying in quarantine for an average time of \(1/\tau_{Q}\) days. Prior to going to the final quarantine class \(Q_{I}\), these individuals stay during a short time \(1/\tau_{T}\) in class \(Q_{T}\), for tracing their previous contacts. This time \(1/\tau_{T}\) is the tracing time (\(TT\)) that models the necessary time to trace the contacts, allowing to compare the fast-tracing centralized approach versus the slower-tracing decentralized approach. * \(Q_{T}\to Q_{I}\): This transition supposes the end of the tracing. After the tracing time \(1/\tau_{T}\) they move to the \(Q_{I}\) class, to finish their quarantine. * \(Q_{I}\to R\): End of quarantine and infection. In this class, individuals continue their quarantine for the remaining quarantine time, which is obtained as \(1/\tau_{T}^{\tau}=1/\tau_{Q}-1/\tau_{T}\), and, finally, they are considered recovered and move to the \(R\) class afterward.