# Evaluating the Effectiveness of COVID-19 Bluetooth-Based Smartphone Contact Tracing Applications
## 4 Epidemic Model  
In this section, we introduce our model to evaluate the effect of contact tracing on the diffusion and control of an epidemic outbreak. We use a susceptible, infected, recovered (SIR) epidemic model with quarantine considering a population of \(N\) individuals We consider a closed system with no natural births and deaths, as the time scale is small enough to allow skipping this variation on the population. We assume, as all deterministic models do, a homogeneity on the contacts. If we want to consider the heterogeneity of the contacts, we should use a stochastic one, like the one presented by the authors in [17]. As shown in this paper, when considering an average number of contacts and tracing intensity (i.e., a mean-field approximation), a stochastic model can be transformed into a deterministic model. This continuous model is useful in assessing the dynamics of the epidemic and the impact of the various quarantine methods, taking into account greater populations. The accuracy of this continuous model depends not only on the homogeneity and distribution of the contacts patterns but also on the number of individuals considered (population), becoming more accurate when there is a high number of nodes. We consider in our model a population of \(N\) individuals that can be in the following classes, representing the number of individuals on each class: \(S\), susceptible individuals (not infected); \(I\), infected individuals; \(R\), individuals recovered from the infection; \(Q_{I}\) infected individuals that have been detected (or traced) and as a result quarantined; \(Q_{S}\), susceptible individuals that are quarantined after being traced; and \(Q_{T}\), infected individuals that have been detected and are being traced. Finally, as for most of the epidemic models, the time unit is one day. For reference to the notation used in the model see Table 1.  
We can obtain the transmission rate \(\beta\) as the product of the average number of contacts with infected individuals at the time unit \(t\) (i.e., during a day), \(k\); and the transmission probability of the disease, \(b\). This value \(b\) depends on the type of disease. The infected individuals recover after \(1/\gamma\) days, being \(\gamma\) the recovery rate. All these values are related to the basic reproductive number as \(R_{0}=kb/\gamma\). Therefore, \(b\) can be obtained using some simple parameters of the disease, such as the reproductive number \(R_{0}\), the transmission rate \(\gamma\) and the contact rate \(k\). \(R_{0}\) is one of the most significant figures in epidemiology, representing the number of people that one infected person will pass the virus on to, on average. If \(R_{0}>1\), the infection will start spreading, but not if \(R_{0}<1\). In general terms, the greater the \(R_{0}\) value, the harder it is to contain the outbreak. In order to determine the transition rates between classes we need to consider the detection and quarantine measures to be taken. Firstly, we assume that when an infected individual is detected, she/he is immediately quarantined. Then, using contact tracing, her/his previous contacts are analyzed to detect individuals having a high probability of being infected, which also are quarantined. As detailed in the previous section, in our model we also consider that not all real contacts will be traced positively. This efficiency depends on several factors: the percentage of people using the mobile application and their willingness to check their risk, and the precision on detecting a physical contact (false positives and negatives). As detailed in the previous section, we consider two different fractions of traced contacts to be quarantined: \(q_{i}\) for infected individuals and \(q_{s}\) for susceptible individuals, as expressed in Equations (2) and (3). Thus, we consider the following transitions between classes:  
* \(S\to I\): Infected individuals who are not traced and consequently, not quarantined.