# Dynamic Infection Spread Model Based Group Testing
## 1 Introduction  
The group testing idea, introduced by [PERSON] in their seminal work [1], is an efficient approach to the detection of the prevalence of a certain infection in the test samples of a group of individuals. The group testing approach is based on the idea of dividing the individuals into groups, mixing the collected test samples within each group and testing those mixed samples. This way, a negative test result implies that every test sample included in that mixed sample is negative, while a positive test result implies that there is at least one positive sample in the mixed sample: each test acts like a logical OR operation of the test samples included in the mixed sample. [PERSON]'s original algorithm divides the population into disjoint groups and performs group tests by mixing the samples within each group. Subsequently, depending on the test results, positive groups are further tested individually to identify the status of every individual in the population. After [PERSON]'s work in [1], various adaptive (tests performed in multiple steps) and non-adaptive (tests performed in a single step) grouping testing algorithms have been proposed, capacity of the group testing problem has been studied for a variety of system models and family of algorithms, and extended analyses have been conducted for different regimes for the total number of infections in the population [2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12; 13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23]. Most of the classical group testing problems have considered either combinatorial or i.i.d. probabilistic settings. Out of \(n\) individuals, in the combinatorial setting, some fixed number of individuals, \(k\), are assumed to be infected and the problem is to identify the infected set that is uniformly randomly selected from the set of all size \(k\) subsets of \(n\) individuals with the minimum number of tests, while in the i.i.d. probabilistic group testing models, each individual is assumed to be randomly infected, with some fixed probability, independently. In the classical system models, under various assumptions, it is proven that the advantage of group testing over individual testing is considerable mostly in the scenarios where the infection prevalence rate in the population is not high [5; 15; 16] and the advantage gets diminished as the infection prevalence rate gets higher within the population. More recently, there has been an increasing focus on modified system models, where practical considerations on the system models have significantly improved the performance of the group testing systems. Ref. [24] considers a probabilistic model where each individual is randomly infected with an unequal probability, independently. Ref. [25] studies non-identical infection probability distributions and proposes a novel adaptive group testing algorithm. Refs. [26; 27] consider correlation between infection status of individuals, and Refs. [28; 29; 30; 31] further model the correlations by considering community structures among the individuals. Although these extended system models are practical and resemble real-life scenarios, in reality, the testing and infection identification processes are dynamic in nature, especially for contagious diseases. Instead of a static, single-shot identification procedure as in the classical group testing setup, testing and identification need to be done over a long time period while the infection keeps spreading and the infection status of the individuals are dynamically changing. Furthermore, rather than minimizing the required number of tests for perfect (or near perfect with vanishing error) identification of the infection status of a population as in the classical group testing setups, in most practical scenarios, the limit on the testing capacity might be fixed and the objective might be to identify as many infections as possible with the limited testing capacity.