  In this paper, a mathematical analysis of the global dynamics of a viral
infection model in vivo is carried out. We study the dynamics of a hepatitis C
virus (HCV) model, under therapy, that considers both extracellular and
intracellular levels of infection. At present most mathematical modeling of
viral kinetics after treatment only addresses the process of infection of a
cell by the virus and the release of virions by the cell, while the processes
taking place inside the cell are not included. We prove that the solutions of
the new model with positive initial values are positive, exist globally in time
and are bounded. The model has two virus-free steady states. They are
distinguished by the fact that viral RNA is absent inside the cells in the
first state and present inside the cells in the second. There are basic
reproduction numbers associated to each of these steady states. If the basic
reproduction number of the first steady state is less than one then that state
is asymptotically stable. If the basic reproduction number of the first steady
state is greater than one and that of the second less than one then the second
steady state is asymptotically stable. If both basic reproduction numbers are
greater than one then we obtain various conclusions which depend on different
restrictions on the parameters of the model. Under increasingly strong
assumptions we prove that there is at least one positive steady state (infected
equilibrium), that there is a unique positive steady state and that the
positive steady state is stable. We also give a condition under which every
positive solution converges to a positive steady state. This is proved by
methods of Li and Muldowney. Finally, we illustrate the theoretical results by
numerical simulations.
