Silent events such as the first detectable HIV infection, the onset of Type 2 diabetes and prostate cancer progression are often ascertained by diagnostic tests and/or self-reports that are scheduled periodically. In such applications, we only observe the time to the event of interest to lie between the times of last negative and the first positive tests, resulting in interval-censored observations. In addition, in some medical studies, a substantial proportion of participants may experience the events before the study, so-called prevalent cases, or participants may never experience the event, that is regarded as non-susceptible cases (or indolent cancer or long-term survivor). In this dissertation, I develop mixture models for the analysis of heterogeneous survival data subject to interval-censoring. The first chapter of this dissertation is motivated by a study of the effects of maternal and infant antiretroviral therapy on the sensitivity of DNA PCR diagnostic tests in detecting HIV infection in infants born to HIV-positive mothers. We apply a mixture model to evaluate the association of a set of predictors with an interval-censored time to first detectable DNA PCR test, while accounting for the subset of infants who test positive at birth. The mixture model is applied to data from the Pediatric AIDS Collaborative Transmission Study and the Women and Infants Transmission Study to evaluate the effects of maternal/infant antiretroviral therapy in HIV subtype B infected mother-infant pairs. In Chapter 2, we propose a parametric mixture model for interval censored time to event outcomes, while relaxing the commonly used proportional hazards assumption. The proposed model is applied to data collected in the National Health and Nutrition Examination Survey to evaluate risk factors of Type 2 diabetes. Chapter 3 is motivated by a Canary Prostate Active Surveillance Study (PASS) where the time to cancer progression (i.e., biopsy upgrade) is of primary interest. We propose a semiparametric mixture model to handle misclassification of progressed cancer at baseline and non-susceptible cases (or, indolent cancer). In addition, we account for imperfect diagnostic tests at each visit and risk factors that change over time in the proposed model. Extensive simulation studies are conducted to assess the performance of the proposed approaches with/without mixture components. The proposed approach is applied to the Canary Prostate Active Surveillance Study to evaluate the effects of factors on the risk of cancer progression and estimate the indolent fraction under a range of sensitivity rates of biopsy.