R. Akbari - Department of Mathematical Sciences‎, ‎Payame Noor University‎ ,‎P.O.Box ۱۹۳۹۵-۳۶۹۷‎ , ‎Tehran‎ ,‎Iran.
A. Vahidian Kamyad - Department of Mathematics Sciences‎ , ‎University of Ferdowsi‎, ‎Mashhad‎, ‎Iran‎.
A. A. Heydari - Research Center for Infection Control and Hand ‎Hygiene, Mashhad University Of Medical Sciences, Mashhad, Iran.
A. Heydari - Department of Mathematical Sciences‎, ‎Payame Noor University‎,‎ P. O. Box ۱۹۳۹۵-۳۶۹۷‎, ‎Tehran‎,‎ Iran‎.

‎Hepatitis B virus (HBV) infection is a major public health problem in the world today‎. ‎A mathematical model is formulated to describe the spread of hepatitis B‎, ‎which can be controlled by vaccination as well as treatment‎. ‎We study the dynamical behavior of the system with fixed control for both vaccination and treatment‎. ‎The results shows that the dynamics of the model is completely determined by the basic reproductive number R_۰. ‎if R_۰<۱, ‎the disease-free equilibrium is globally asymptotically stable by using approach that given by Kamgang and Sallet‎. ‎Then the authors prove that if R_۰>۱, ‎the disease-free equilibrium is unstable and the disease is uniformly persistent‎. ‎Furthermore‎, ‎If R_۰>۱, ‎the unique endemic equilibrium is globally asymptotically stable by using a generalization of the Poincar e-Bendixson ‎criterion.‎

Hepatitis B virus (HBV), Basic reproduction number (R_۰), ‎G‎ompound matrices, Global ‎stability.‎