Dynamic group testing to control and monitor disease progression in a population

In the context of a pandemic like COVID-19, and until most people are vaccinated, proactive testing and interventions have been proved to be the only means to contain the disease spread. Recent academic work has offered significant evidence in this regard, but a critical question is still open: Can we accurately identify all new infections that happen every day, without this being forbiddingly expensive, i.e., using only a fraction of the tests needed to test everyone everyday (complete testing)? Group testing offers a powerful toolset for minimizing the number of tests, but it does not account for the time dynamics behind the infections. Moreover, it typically assumes that people are infected independently, while infections are governed by community spread. Epidemiology, on the other hand, does explore time dynamics and community correlations through the well-established continuous-time SIR stochastic network model, but the standard model does not incorporate discrete-time testing and interventions. In this paper, we introduce a "discrete-time SIR stochastic block model" that also allows for group testing and interventions on a daily basis. Our model can be regarded as a discrete version of the continuous-time SIR stochastic network model over a specific type of weighted graph that captures the underlying community structure. We analyze that model w.r.t. the minimum number of group tests needed everyday to identify all infections with vanishing error probability. We find that one can leverage the knowledge of the community and the model to inform nonadaptive group testing algorithms that are order-optimal, and therefore achieve the same performance as complete testing using a much smaller number of tests.