Mathematical modeling and real-time dynamics identification of the mean arterial blood pressure (MAP) response of a patient to vasoactive drug infusion can provide a reliable tool for automated drug administration and therefore, reduce the emergency costs and significantly benefit the patient's MAP regulation in an intensive care unit. To this end, a dynamic first-order linear parameter-varying (LPV) model with varying parameters and varying input delay is considered to capture the MAP response dynamics. Such a model effectively addresses the complexity and the intra- and inter-patient variability of the physiological response. We discretize the model and augment the state vector with model parameters as unknown states of the system and a Bayesian-based multiple-model square root cubature Kalman filtering (MMSRCKF) approach is utilized to estimate the model time-varying parameters. Since, unlike the other model parameters, the input delay cannot be captured by a random-walk process, a multiple-model module with a posterior probability estimation is implemented to provide the delay identification. Validation results confirm the effectiveness of the proposed identification algorithm both in simulation scenarios and also using animal experiment data.