The coronavirus disease 2019 (COVID-19) pandemic has quickly become a global public health crisis unseen in recent years. It is known that the structure of the human contact network plays an important role in the spread of transmissible diseases. In this work, we study a structure aware model of COVID-19 CGEM. This model becomes similar to the classical compartment-based models in epidemiology if we assume the contact network is a Erdos-Renyi (ER) graph, i.e. everyone comes into contact with everyone else with the same probability. In contrast, CGEM is more expressive and allows for plugging in the actual contact networks, or more realistic proxies for it. Moreover, CGEM enables more precise modelling of enforcing and releasing different non-pharmaceutical intervention (NPI) strategies. Through a set of extensive experiments, we demonstrate significant differences between the epidemic curves when assuming different underlying structures. More specifically we demonstrate that the compartment-based models are overestimating the spread of the infection by a factor of 3, and under some realistic assumptions on the compliance factor, underestimating the effectiveness of some of NPIs, mischaracterizing others (e.g. predicting a later peak), and underestimating the scale of the second peak after reopening.