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In this paper, we formulate a within‐host dynamics model for HIV/HTLV‐I co‐infection under the influence of cytotoxic T lymphocytes (CTLs). The model incorporates silent HIV‐infected CD4+T cells and silent HTLV‐infected CD4+T cells. The model includes two routes of HIV transmission, virus to cell (VTC) and cell to cell (CTC). It also incorporates two modes of HTLV‐I transmission, horizontal transmission via direct CTC contact and vertical transmission through mitotic division of Tax‐expressing HTLV‐infected cells. The model takes into account five types of distributed‐time delays. We analyze the model by proving the nonnegativity and boundedness of the solutions, calculating all possible equilibria, deriving a set of key threshold parameters, and proving the global stability of all equilibria. The global asymptotic stability of all equilibria is established by utilizing Lyapunov function and LaSalle's invariance principle. We present numerical simulations to justify the applicability and effectiveness of the theoretical results. In addition, we discuss the effect of HTLV‐I infection on the HIV dynamics and vice versa. [ABSTRACT FROM AUTHOR]