# Dynamic Infection Spread Model Based Group Testing
## 3 System Model  
We consider a population of \(n\) individuals whose infection statuses change over time. The time dimension \(t\) is discrete in our system, i.e., \(t\in\{0,1,2,\ldots\}\). Similar to the classical discrete SIR model, the population consists of three distinct subgroups: susceptible individuals who are not infected but can get infected by infected individuals (S), infected individuals who can infect the susceptible individuals (I), and isolated individuals who were infected, have been detected via performed tests and isolated indefinitely (R) (These are called recovered (R) individuals in the SIR model; we call them isolated individuals. As they are isolated indefinitely, they are recovered eventually). Let \(U_{i}(t)\) denote the infection status of individual \(i\) at time \(t\), where 1 represents being infected, 0 represents not being infected and 2 represents being isolated. At the beginning (\(t=0\)), we introduce the initial infections in the system, independently with probability \(p\), where \(U_{i}(0)\) is a Bernoulli random variable with parameter \(p\). Random variables \(U_{i}(0)\) are mutually independent for \(i\in[n]\). Let \(\alpha(t)\) denote the number of susceptible individuals at time \(t\), \(\lambda(t)\) denote the number of non-isolated infected individuals at time \(t\) and \(\gamma(t)\) denote the number of isolated individuals at time \(t\). Starting from \(t=1\), each time instance consists of two phases: infection spread phase and testing phase, in the respective order. Infection Spread Phase: Infected individuals spread the infection to the susceptible members of the population. At each time instance, starting from \(t=1\), the infection spreads independently across the individuals: Each infected individual can infect each susceptible individual with probability \(q\), independent across both infected individuals and susceptible individuals. Isolated individuals cannot infect others and their infection status cannot change after they are isolated. Thus, probability of the event that individual \(i\) gets infected by another individual \(j\) at time \(t\geq 1\) is equal to  
\[qP\big{(}U_{j}(t-1)=1,U_{i}(t-1)=0\big{)}\quad for\quad i,j\in[n]. \tag{1}\]  
Testing Phase: At each time instance starting from \(t=1\), \(T\) tests can be performed to the individuals. Note that the testing capacity \(T\) is a given parameter and thus, in contrast to the classical group testing systems, we do not seek to minimize the number of performed tests for full identification of the infection status of the population but aim to efficiently perform \(T\) tests at each time instance to identify and isolate as many infections as possible to control the infection spread. Here, performed tests can be group tests, and we define the \(T\times n\) binary test matrices, \(\mathbf{X}(t)\), which specify the pooling scheme for the tests at each time \(t\). For each time instance \(t\geq 1\), we have the test result vectors \(y(t)\), which are equal to  
\[y_{i}(t)=\bigvee_{j\in[n]}\mathbf{X}_{ij}(t)\mathds{1}_{\{U_{j}(t)=1\}},\quad i \in[T] \tag{2}\]  
where \(y_{i}(t)\) denotes the \(i\)th test result at time \(t\), \(\mathbf{X}_{ij}(t)\) denotes the \(i\)th row, \(j\)th column of the test matrix \(\mathbf{X}(t)\). Note that, since the previous test matrices and test results are available while designing these test matrices, \(\mathbf{X}(t)\) can depend on the previous test results \(y(t^{\prime})\) for \(t^{\prime}<t\). We assume that when tests are performed at some time instance \(t^{\prime}\), the test results \(y(t^{\prime})\) will be available before the infection spread phase at time \(t^{\prime}+1\). Thus, after the test results are available, detected infections are isolated immediately, i.e., if the \(i\)th individual is detected to be infected during the testing phase at time \(t^{\prime}\), then \(U_{i}(t^{\prime})=2\).