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The seven-equation model is a compressible multiphase formulation that allows for phasic velocity and pressure disequilibrium. These equations are solved using a diffused interface method that models resolved multiphase flows. Novel extensions are proposed for including the effects of surface tension, viscosity, multi-species, and reactions. The allowed non-equilibrium of pressure in the seven-equation model provides numerical stability in strong shocks and allows for arbitrary and independent equations of states. A discrete equations method (DEM) models the fluxes. We show that even though stiff pressure- and velocity-relaxation solvers have been used, they are not needed for the DEM because the non-conservative fluxes are accurately modeled. An interface compression scheme controls the numerical diffusion of the interface, and its effects on the solution are discussed. Test cases are used to validate the computational method and demonstrate its applicability. They include multiphase shock tubes, shock propagation through a material interface, a surface-tension-driven oscillating droplet, an accelerating droplet in a viscous medium, and shock-detonation interacting with a deforming droplet. Simulation results are compared against exact solutions and experiments when possible.