Group testing is a technique that can reduce the number of tests needed to identify infected members in a population, by pooling together multiple diagnostic samples. Despite the variety and importance of prior results, traditional work on group testing has typically assumed independent infections. However, contagious diseases among humans, like SARS-CoV-2, have an important characteristic: infections are governed by community spread, and are therefore correlated. In this paper, we explore this observation and we argue that taking into account the community structure when testing can lead to significant savings in terms of the number of tests required to guarantee a given identification accuracy. To show that, we start with a simplistic (yet practical) infection model, where the entire population is organized in (possibly overlapping) communities and the infection probability of an individual depends on the communities (s)he participates in. Given this model, we compute new lower bounds on the number of tests for zero-error identification and design community-aware group testing algorithms that can be optimal under assumptions. Finally, we demonstrate significant benefits over traditional, community-agnostic group testing via simulations using both noiseless and noisy tests. Shorter versions of this article, which contained a subset of the material, were presented in the work by Nikolopoulos et al. (2021, 2021).