# Evaluating the Effectiveness of COVID-19 Bluetooth-Based Smartphone Contact Tracing Applications
## 4 Epidemic Model
* \(I\to R\): Non detected infected individuals recovered from the disease. This transition considers those infected individuals not detected, that is, asymptomatic ones which have not been tested. The rate is simply the recovery rate \(\gamma\). * \(Q_{S}\to S\): End of quarantine for susceptible individuals. After being quarantined, individuals in the \(Q_{S}\) class return to the susceptible class, after staying in quarantine for \(1/\tau_{Q}\) days. Based on these transitions, the model is defined by the following equations:  
\[ \begin{split} S^{\prime}&=-(1-q_{i})\frac{kbIS}{N} -q_{i}\frac{kbIS}{N}-q_{s}\frac{k(1-b)IS}{N}+\tau_{Q}Q_{S}\\ I^{\prime}&=(1-q_{i})\frac{kbIS}{N}-\delta I- \gamma I\\ R^{\prime}&=\gamma I+\tau_{Q}Q_{I}\\ Q_{S}^{\prime}&=q_{s}\frac{k(1-b)IS}{N}-\tau_{Q}Q _{S}\\ Q_{I}^{\prime}&=\tau_{T}Q_{T}-\tau_{Q}^{\prime}Q _{I}\\ Q_{T}^{\prime}&=\delta I+q_{i}\frac{kbIS}{N}- \tau_{T}Q_{T}\end{split}  \tag{4}\]  
The time has been omitted in all the classes. For example, for class \(I\), \(I^{\prime}=dI(t)/dt\) and \(I=I(t)\)). Note also that we have chosen not to simplify some mathematical expressions to clearly differentiate the transitions between classes. Note that, in the case that the tracing time would be greater than one, \(q_{i}\) and \(q_{s}\) values must be normalised considering the whole average tracing time. The idea is to distribute these fractions of traced contacts among the tracing time, using the following expressions: \(q_{i}^{n}=q/(1/\tau_{T})=q_{i}\tau_{T}\) and \(q_{s}^{n}=q_{s}\tau_{T}\). That is, if the tracing time is longer (as in the decentralized model), it is because it will take more time to trace back the prior contacts. In epidemic management and control, it is crucial to reduce the number of undetected infected individuals who may contribute to the rapid spread of the infection (this is known as one of the leading causes of COVID-19's rapid spread). This goal can be obtained by increasing the detection ratio (for example, by incrementing the number of tests even for asymptomatic individuals), and also by increasing the number of traced individuals. It is easy to see from the equations of the previous model,that the condition to control an outbreak is when the number of infected individuals \(I\) decreases, that is, when \(I^{\prime}\) is negative. Therefore, the condition for \(I^{\prime}\) to be negative in expression (4) is:  
\[(1-q_{i})\frac{S}{N}R_{0}\gamma<\delta+\gamma \tag{5}\]  
considering that \(kb=R_{0}\gamma\). Using this expression the threshold for an epidemic outbreak can be easily obtained. This threshold depends on \(R_{0}\) (the basic reproductive number) and the ratio of susceptible individuals (\(S/N\)). Note that this threshold depends on \(q_{i}\) and not on \(q_{d}\). The reason is that the spread of the infection can only be stopped by the detection and quarantine of infected individuals. This also means that less precise contact tracing will increase the number of susceptible quarantined individuals. That is, considering the same number of traced quarantined people, a less precise tracing will be less effective. If we also want to evaluate the impact of the application utilization (\(U\) and \(RC\)), combining this Equation (5) with the value of \(q_{i}\) defined in Equation (2) we can obtain the threshold values of \(RC\) depending on \(U\) as:  
\[RC=\frac{1-\frac{N}{S}\frac{\delta+\gamma}{R_{0}\gamma}}{(1-FN)\cdot U^{2}} \tag{6}\]  
Finally, as Equation (4) do not have an algebraic solution, a numerical solution is needed (for example, using the Euler's method or more efficient and precise numerical methods such as Matlab's function ode45, which is the one used in this paper). For solving this model, we consider an initial value of \(R(0)\) and \(I(0)\) (that is, the number of recovered and infected individuals at the beginning of an outbreak). Note, that in most of our experiments these values \(R(0)\), \(I(0)\) are 0, as we consider an initial outbreak with no people infected or immunized.