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Communicated by M. EfendievIn this paper, a mathematical model for HIV-1 infection with antibody, cytotoxic T-lymphocyte immune responses and Beddington-DeAngelis functional response is investigated. The stability of the infection-free and infected steady states is investigated. The basic reproduction number R 0 is identified for the proposed system. If R 0 < 1, then there is an infectionfree steady state, which is locally asymptotically stable. Further, the infected steady state is locally asymptotically stable for R 0 > 1 in the absence of immune response delay. We use Nyquist criterion to estimate the length of the delay for which stability continues to hold. Also the existence of the Hopf bifurcation is investigated by using immune response delay as a bifurcation parameter. Numerical simulations are presented to justify the analytical results. Stability analysis of the infection-free steady state (E 0 )If R 0 < 1, then an infected cell produces less than one new infected cell over the course of its infectious period. Hence, the infection cannot persist. Now, we state and prove the following theorem related to R 0 < 1 for system (3).Theorem 2.1 If R 0 < 1, then the infection-free steady state E 0 is locally asymptotically stable. ProofThe infection-free steady state E 0 .x 0 , y 0 , v 0 , w 0 , z 0 / of the system (3) is given by E 0 D d , 0, 0, 0, 0 .

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