The paper studies the qualitative behavior of a set of Ordinary Differential Equations that models the dynamics of bi-virus epidemics over bilayer networks. Each layer is a weighted digraph associated with a strain of virus; the weights γz represent the rates of infection from node i ij to node j of strain z. We establish a sufficient condition on the γ’s that guarantees survival of the fittest–only one strain survives. We propose an ordering of the weighted digraphs, the ⋆-order, and show that if the weighted digraph of strain y is ⋆-dominated by the weighted digraph of strain x, then y dies out in the long run. We prove that the orbits of the ODE accumulate to an attractor. Due to the coupled nonlinear high-dimension nature of the ODEs, there is no natural Lyapunov function to study their global qualitative behavior. We prove our results by combining two important properties of these ODEs: (i) monotonicity under a partial ordering on the set of graphs; and (ii) dimension-reduction under symmetry of the graphs. Property (ii) allows us to fully address the survival of the fittest for regular graphs. Then, by bounding the epidemics dynamics for generic networks by the dynamics on regular networks, we prove the result for general networks.