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Combinatorial group testing (CGT) is used to identify defective items from a set of items by grouping them together and performing a small number of tests on the groups. Recently, group testing has been used to design efficient COVID-19 testing, so that resources are saved while still identifying all infected individuals. Due to test waiting times, a focus is given to non-adaptive CGT, where groups are designed a priori and all tests can be done in parallel. The design of the groups can be done using Cover-Free Families (CFFs). The main assumption behind CFFs is that a small number $d$ of positives are randomly spread across a population of $n$ individuals. However, for infectious diseases, it is reasonable to assume that infections show up in clusters of individuals with high contact (children in the same classroom within a school, households within a neighbourhood, students taking the same courses within a university, people seating close to each other in a stadium). The general structure of these communities can be modeled using hypergraphs, where vertices are items to be tested and edges represent clusters containing high contacts. We consider hypergraphs with non-overlapping edges and overlapping edges (first two examples and last two examples, respectively). We give constructions of what we call structure-aware CFF, which uses the structure of the underlying hypergraph. We revisit old CFF constructions, boosting the number of defectives they can identify by taking the hypergraph structure into account. We also provide new constructions based on hypergraph parameters.