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Recurrent event and terminal event data commonly arise in clinical and observational studies. To evaluate the efficacy of a treatment effect for both types of events, a composite endpoint has been used as a possible assessment, particularly when faced with high costs and a longer follow-up study. To model recurrent event processes complicated by the existence of a terminal event, joint frailty modeling has been typically employed. In this study, the objective was to develop some target-driven response adaptive randomization strategies using a composite endpoint based on joint frailty modeling. We first implemented a balanced randomized design and then investigated the response adaptive randomization. The former is intuitively first adopted while the latter is expected to be desirable and ethical in terms of allocating more subjects to the more effective treatment. The results show that the proposed procedures using a composite endpoint are capable of reducing the number of trial participants who receive inferior treatment while simultaneously reaching a desired optimal target as compared to a balanced randomized design. The R shiny application for calculating the sample size and allocation probabilities is also available. Finally, two clinical trials were used as pilot datasets to introduce the proposed procedures.