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The main target of both human immunodeficiency virus type 1 (HIV-1) and human T-lymphotropic virus type I (HTLV-I) is the CD4+ T cell which is considered the key player in the immune system. Moreover, HIV-1 has another target that is the macrophages. The present paper aims to formulate and develop a mathematical model to analyze the interaction of two viruses, HIV-1 and HTLV-I with the immune system. We determine a bounded domain for the concentrations of the model's compartments. We discuss the dynamical behavior of the model and analyze the existence and stability of the system's steady states. The global asymptotic stability of all steady states is proven by utilizing the Lyapunov method. We also demonstrate the dynamical behavior of the system numerically. The significant impact of macrophages on the HTLV-I/HIV-1 co-infection dynamics is discussed. Our developed model will contribute to the understanding of HTLV-I/HIV-1 co-infection dynamics and help to choose different treatment strategies against HIV-1 and HTLV-I. [ABSTRACT FROM AUTHOR]

This paper investigates the stochastic HTLV-I infection model with CTL immune response, and the corresponding deterministic model has two basic reproduction numbers. We consider the nonlinear CTL immune response for the interaction between the virus and the CTL immune cells. Firstly, for the theoretical needs of system dynamical behavior, we prove that the stochastic model solution is positive and global. In addition, we obtain the existence of ergodic stationary distribution by stochastic Lyapunov functions. Meanwhile, sufficient condition for the extinction of the stochastic system is acquired. Reasonably, the dynamical behavior of deterministic model is included in our result of stochastic model when the white noise disappears. [ABSTRACT FROM AUTHOR]