We consider joint analysis of event times to a recurrent and a non-recurrent event, with the event time data subject to type I interval censoring. The motivation arises from a survey study, which collected current count data for time to occurrences of fracture (recurrent event), and current status (i.e., type I interval censored) data for time to osteoporosis (non-recurrent event). The aim of the study is to examine risk factors for, and levels of association between, the recurrent and non-recurrent events. We propose a joint analysis of current count and current status data based on a joint modeling for recurrent and non-recurrent events. In the proposed framework, a non-homogeneous Poisson process is assumed for the recurrent event, a proportional hazards model is assumed for failure time of the non-recurrent event, and the two event time processes share a common gamma frailty. A semiparametric maximum likelihood estimator, together with a stable computation algorithm, is developed for the joint model. The parametric (covariate effects and frailty) and nonparametric (baseline mean and cumulative hazard functions) components of the estimator are consistent at rates of square root and cubic root of the sample size, respectively. The asymptotic normality for the parametric component of the estimator is established. The application to the survey data mentioned above shows that, female is a common strong risk factor for both fracture and osteoporosis, and times to fracture and osteoporosis are highly associated.