We propose a pathogen dynamics model with CTL immune response and both pathogenic and cellular infections. Both actively infected cells and latently infected cells are incorporated into the model. The infected-susceptible and pathogen-susceptible infection rates are given by saturated incidence. Three distributed time delays are considered. The existence and global stability of the equilibria are determined by two threshold parameters, the basic reproduction number and the CTL response activation number. The global stability of the three equilibria are proven using Lyapunov method. We solve the system of delay differential equations numerically to support the theoretical results.