In this paper, we develop an evolutionary variational inequality model of the Internet with multiple classes of traffic and demonstrate its utility through the formulation and solution of a time-dependent Braess paradox. The model can handle time-dependent changes in demand as a consequence of developing news stories, following, for example, natural disasters or catastrophes or major media events. The model can also capture the time-varying demand for Internet resources during a regular weekday with its more regular rhythm of work and breaks. In addition, the model includes time-varying capacities on the route flows due to, for example, government interventions or network-type failures.