Dynamics of the HTLV-I stochastic model with CTL immune response and distributed delays

Abstract Human T-cell Leukemia Virus Type I (HTLV-I) is a retrovirus that primarily spreads through cell-to-cell transmission, yet its transmission mechanisms remain incompletely understood, particularly regarding the roles of immune delay and stochastic effects in the infection process. Developing a more comprehensive mathematical model is therefore essential for gaining deeper insights into the dynamics of HTLV-I transmission. This paper introduces an integrated model to study the infection dynamics of HTLV-I, aiming to address gaps in our understanding of its transmission mechanisms. Compared to previous studies, our model incorporates key advancements, including the consideration of immune delay caused by actively infected cells, the inclusion of stochastic effects among different cell types, and the comprehensive integration of cell-to-cell transmission mechanisms, making it more biologically realistic. By constructing appropriate Lyapunov functions, we establish relevant conditions and use them to prove the uniqueness, positivity, and global existence of solutions, the existence of a stationary Markov process, and the eventual extinction of latently and actively infected cells. To corroborate our theoretical results, we perform numerical simulations employing biologically relevant parameter values, which elucidate the influence of cell-to-cell transmission and critical parameters on the infection dynamics.