# Dynamic SAFFRON: Disease Control Over Time via Group Testing
## 2 System Model  
There are \(n\) individuals in the system and their infection status, which we denote by \(U_{i}(t)\) for individual \(i\) at time \(t\), changes over discrete time instances. At any given time, there are three disjoint groups in the population: susceptible individuals who can become infected (S), non-isolated infections who have not been detected (I) and isolated infections who have been detected by performed tests and isolated indefinitely (R). \(U_{i}(t)\) can take values 0 (susceptible), 1 (non-isolated infection) and 2 (isolated infection), representing the infection state of the individual \(i\) at time \(t\). At \(t=0\), we introduce the initial i.i.d. infections to the system: each individual is infected with probability \(p\), independently. We introduce three random processes: \(\alpha(t)\) denotes the number of susceptible individuals, \(\lambda(t)\) denotes the number of non-isolated infections and \(\gamma(t)\) denotes the isolated infections, at time \(t\). Each discrete time instance \(t\geq 1\), consists of two consecutive phases: infection spread phase and testing phase. During the infection spread phase, each non-isolated and infected individual can infect each susceptible individual with probability \(q\), independently. Here, since each susceptible individual can be infected by each non-isolated and infected individual, this can be modeled as the realization of \(\alpha(t)\lambda(t)\) independent Bernoulli random variables with parameter \(q\), for each time \(t\geq 1\). Following the infection spread phase, the testing phase starts in which the testing capacity is limited to a given, fixed number, \(T\). Thus, the aim of designing the group testing algorithm is to construct \(T\times n\) binary test matrices that specify the testing pools. Let \(X(t)\) denote the binary test matrix for the tests at time \(t\), and \(X_{ij}(t)\) denote the \(i\)th row and \(j\)th column of \(X(t)\), where \(X_{ij}(t)\) is 1 if the test sample of \(j\)th individual is mixed in the \(i\)th mixed test sample, and 0 otherwise. Let \(y(t)\) denote the test result vector of the tests performed at time \(t\) and \(y_{i}(t)\) denote the \(i\)th test result at time \(t\). Then, we have  
\[y_{i}(t)=\bigvee_{j\in[n]}X_{ij}(t)\mathds{1}_{\{U_{j}(t)=1\}},\quad i\in[T] \tag{1}\]  
We assume that results of the tests that are performed at time \(t\) are available before the infection spread phase at time \(t+1\), i.e., \(y(t)\) vector is known while designing the binary test matrix \(X(t+1)\). Moreover, when infected individuals are detected during the testing phase at time \(t\), they are immediately isolated and thus, prevented from spreading the infection for the rest of the testing process, i.e., \(U_{i}(t)=2\) for all times \(t\geq t^{\prime}\), if the individual \(i\) is found to be infected by the tests performed at time \(t^{\prime}\). Note that the only possible infection state changes in our system are either from susceptible individuals to non-isolated infections, which happen via infection spread from non-isolated and infected individuals to susceptible individuals during the infection spread phase at each time, or from non-isolated infections to isolated infections, which happen via infection detection during the testing phase at each time instance. A group testing policy \(\pi\) is a scheme that specifies the algorithm to construct binary test matrices \(X(t)\) for each time instance \(t\geq 1\), until the infection is under control. Since the source of the infection state evolution in our dynamic model is the non-isolated infections, we define the disease control time \(\bar{t}\) to be the first time when there are no non-isolated infections remaining in the population, i.e., \(\bar{t}=\min\{t\mid\lambda(t)=0\}\). After time \(\bar{t}\), every individual in the population is either susceptible or isolated, i.e., \(U_{i}(t)=0\) or \(U_{i}(t)=2\) for \(i\in[n]\) and for all \(t\geq\bar{t}\).