Viruses spreading over the Internet can cause significant damage and the loss of network security. On the other hand, the anti-virus process also plays an important part affecting the dynamics of the virus spreading. The spreading dynamics of most viruses depend on the underlying network topology. While much research attention has been paid in developing the antivirus software/tools, the dynamics and propagating model of the virus and anti-virus spreading in the topology-aware networks is neither well understood, nor thoroughly studied. To remedy this deficiency, we model and analyze the spreading characteristics of viruses as coexisting with the anti-virus spreading process in the two-layer small-world topology. Applying the fluid analysis, we derive the analytical solutions to the two-layer model. The simulations experiments confirm the validity of our fluid analyses in characterizing both virus and anti-virus spreading dynamics.