The present study introduces a new bivariate distribution based on the Sushila distribution to model bivariate lifetime data in presence of a cure fraction, right- censored data and covariates. The new bivariate probability distribution was obtained using a methodology used in the reliability theory based on fatal shocks, usually used to build new bivariate models. Additionally, the cure rate was introduced in the model based on a generalization of standard mixture models extensively used for the univariate lifetime case. The inferences of interest for the model parameters are obtained under a Bayesian approach using MCMC (Markov Chain Monte Carlo) simulation methods to generate samples of the joint posterior distribution for all parameters of the model. A simulation study was developed to study the inferential properties of the new methodology.The proposed methodology also was applied to analyze a set of real medical data obtained from a retrospective cohort study that aimed to assess specific clinical conditions that affect the lives of patients with diabetic retinopathy. For the discrimination of the proposed model with other usual models used in the analysis of bivariate survival data, some Bayesian techniques of model discrimination were used and the model validation was verified from usual Cox-Snell residuals, which allowed us to identify the adequacy of the proposed bivariate cure rate model.