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Linear parameter-varying (LPV) systems with uncertainty in time-varying delays are subject to performance degradation and instability. In this line, we investigate the stability of such systems invoking an input-output stability approach. By considering explicit bounds on the delay rate and time-varying delay uncertainty, the scaled small-gain theorem is adopted to form an interconnected time-delay LPV system with input and output vectors of the auxiliary system introduced for the uncertain dynamics. For such an interconnected time-delay LPV system subject to external disturbances, a Lyapunov-Krasovskii functional (LKF) is constructed whose derivative is augmented with the terms resulted from the descriptor method. Then, stability conditions and a prescribed induced L2-norm in terms of the disturbance rejection performance are derived in a convex linear matrix inequalities (LMIs) setting. Subsequently, a congruent transformation enables us to compute a gain-scheduled state-feedback controller for a class of LPV systems with an uncertain time-varying delay. As a benchmark, we examine the automated mean arterial blood pressure (MAP) control in an individual with hypotension where the MAP response dynamics to drug infusion is characterized in a time-delay LPV representation. Finally, the closed-loop simulation results are provided to demonstrate the provided methodology's performance.