Multi-state models describe a process where individuals move among a series of states over time. They are increasingly popular in a wide range of applications in biostatistics. For instance, breast cancer, HIV and ageing problems. There are two types of effects when describing the hazards for change of status: fixed effects and random effects. For the fixed-effects multi-state model, the characteristics of individuals are usually considered as covariates, such as age and gender. However, there is still some unobserved heterogeneity, which can be taken into account as random effects. Models with both fixed effects and random effects in survival analysis are called frailty models. A large number of papers discusses parametric univariate frailties in multi-state models. This study presents both parametric and non-parametric frailty models. For the parametric frailty model, both univariate and bivariate frailties in multi-state models are discussed, in which frailties follow several common distributions. In particular, the contribution of this study is to apply a bivariate gamma-distributed frailty in the multi-state model for the interval-censored data, in order to describe the unobserved heterogeneity and investigate the correlation between two transition hazards. Model validation and prediction are discussed as well. In the application, we illustrate both fixed-effect models and frailty models for a cardiac allograft vasculopathy study and a cognitive impairment process
