In this study, a system of ordinary differential equations used for modeling pathogen-specific CD8 cell immune response is studied. The positivity and the limitation of the model solutions are proven. The equilibrium points in the cases with the presence and the absence of the pathogen are determined and their stabilities are investigated. Furthermore, the essential reproduction number is obtained for the system. The conditions of the reproduction number for the stability of the equilibrium are shown and numerical examples are given for supporting the analytical results.