The three-dimensional dynamic theory of elasticity is applied to investigate the mechanical properties of the virus capsid. An idealized model of viruses is based on the 3D boundary-value problem of mathematical physics formulated in a spherical coordinate system for the steady-state oscillation process. The virus is modeled by a hollow elastic sphere filled with an acoustic medium and located in a different acoustic medium. The stated boundary-value problem is solved with the help of the method of integral transforms and the method of the discontinuous solutions. As a result, the exact solution of the problem is obtained. The numerical calculations of the elastic characteristics of the virus are carried out.