# Dynamic SAFFRON: Disease Control Over Time via Group Testing
## 1 Introduction  
The group testing approach is proposed by [PERSON] in [1]. Its main objective is to identify the infection status of a group of individuals by performing fewer tests than the number of individuals. The main idea behind group testing is to pool the obtained test samples and to test these pooled samples, instead of testing each sample individually. By this way, when a mixed sample results as negative, it implies that every test sample included in that mixed sample is negative. On the other hand, if a mixed sample results as positive, it implies that there is at least one positive test sample among the test samples that are mixed in that mixed sample. The main objective of the group testing problem is to design the testing scheme by specifying the test pools such that the infected set is identified with a minimum number of tests. Since the group testing scheme that was proposed by [PERSON] in [1], various group testing algorithms have been studied. Mainly, two standard infection models have been considered in the majority of works: a combinatorial model where the number of infections is fixed and the infected set is uniformly randomly selected from all fixed sized subsets of all individuals, and a probabilistic model where each individual is infected with a fixed probability, independently. The group testing algorithms can be divided into several categories: adaptive group testing algorithms where the tests are performed in multiple steps and test results from previous steps can be used while designing the future test pools, and non-adaptive group testing algorithms where all of the tests are designed to be performed in a single step. The group testing algorithms can be categorized depending on their decoding error performance where zero-error group testing algorithms identify the infected set perfectly and small (vanishing) error algorithms identify the infected set with a small error that vanishes with a large population. As a variation of the test result model, noisy test results setting has been proposed and studied. For these standardsettings, various algorithms have been proposed, analyzed and the capacities have been characterized for different parameter regimes [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21]. Group testing works that are based on the standard settings have a limitation in terms of the overall performance: the advantage of group testing diminishes as the prevalence rate of the infection increases, due to the nature of the problem [5; 15]. In more recent works, the standard settings have been challenged: non-identical infection probabilities, correlations between infection status of the individuals and community structure-based available side information scenarios have been proposed and studied [22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. These practical settings prove that by challenging the standard settings, further improvements can be obtained: the group testing approach can be used efficiently even when the prevalence rate is high among the population if further available side information is used in test designs. However, the overall static nature of the group testing problem is not challenged in these works. For a given group of individuals, the aim is to identify the infected set, as in the standard settings. In a recent line of works, the static nature of the majority of group testing works has been challenged, and dynamic models are proposed [34; 35; 36; 37; 38]. In these works, SIR [39] based dynamic infection spread models are studied. Testing and identification are done over time while the infection status of the individuals also evolve. In this paper, we aim to challenge the standard, static definition of the group testing problem and approach it over a time dimension over a similar setting with our prior work [38]. By this way, we consider a setting where a group testing approach can be applied in systems that evolve over time, such as contagious diseases that spread within a population.