In this paper, we introduce a continuous-time competing virus model with a shared resource. We say that the system is in the healthy state if all the agents are healthy, and the shared resource is not contaminated. If the epidemic remains persistent, and the shared resource is contaminated, we say that the system is in the endemic state. First of all we show, under appropriate assumptions, that the model we introduce is well-posed. Secondly, we establish sufficient conditions for exponential (resp. asymptotic) eradication of a virus. Thirdly, for the single-virus case with a shared resource, we establish conditions that lead to existence of an endemic equilibrium. Finally, we provide a necessary and sufficient condition for uniqueness of the healthy state.