# Dynamic Infection Spread Model Based Group Testing
## 6 Conclusions  
In this paper, we considered a dynamic infection spread model over discrete time, inspired by the SIR model, widely used in the modelling of contagious infections in populations. Instead of recovered individuals in the system, we considered isolated infections, where infected individuals can be identified and isolated via testing. In our system model, infection status of the individuals are random processes, rather than random variables such as the infection status of the individuals in the classical group testing problems. In parallel with the dynamic configuration of our system, we considered dynamic group testing algorithms: At each time instance, after the infection is spread by infected individuals to the susceptible individuals randomly, a given limited number of (possibly group) tests are performed to identify and isolate infected individuals. This dynamic infection spread and identification system is more challenging than the classical group testing problem setup, since negative identifications are not finalized and can change over time while only the positive identifications are isolated for the rest of the process. We analyzed the performance of dynamic testing algorithms by providing approximation results for the expected number of susceptible individuals (that have never gotten infected) when the infection is brought under control where all infections are identified and isolated, for symmetric and converging algorithms. Then, we proposed two dynamic algorithms: dynamic individual testing algorithm and dynamic Dorfman type group testing algorithm. We considered the weak versions of these algorithms and used our general result to provide lower bounds on the expected number of susceptible individuals for these two algorithms. We compared the average identification performance of these two algorithms by deriving conditions when one algorithm outperforms the other. In our simulations, we implemented both the original and weak versions of the proposed algorithms and also simulated and compared the theoretical approximation results that we derived, for three different sets of parameters and we demonstrated various possible scenarios. Our work is unique in that the disease spread in our dynamic system is due to limited testing capacity as opposed to delay in obtaining (unlimited) test results in the existing literature.  
Our numerical results suggest that the performance of the proposed algorithms strongly depends on the initial state of the system as well as the other system parameters. Each algorithm can perform better than the other one depending on the system parameters. In this sense, system model assumptions and system parameters play critical roles in the algorithm performances, as expected. This observation leads to the following future directions:  
* Further dynamic group testing algorithms can be proposed and their performance under varying system parameters can be analyzed.
* Variations of the proposed dynamic system model can be studied.
* Varying performance metrics can be proposed and studied for varying objectives, such as faster disease spread control time, rather than the number of susceptible individuals at the steady state.  
**Author Contributions:** Conceptualization, B.A. and S.U.; methodology, B.A. and S.U.; software, B.A. and S.U.; validation, B.A. and S.U.; formal analysis, B.A. and S.U.; investigation, B.A. and S.U.; resources, B.A. and S.U.; data curation, B.A. and S.U.; writing--original draft preparation, B.A. and S.U; writing--review and editing, B.A. and S.U.; visualization, B.A. and S.U.; supervision, S.U.; project administration, S.U.; funding acquisition, S.U. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
Data Availability Statement: Not applicable.  
**Conflicts of Interest:** The authors declare no conflict of interest.