In biomedical studies, paired survival data arise naturally when two event times are observed within the same subject. Existing statistical models seldom accommodate both cure fractions and complex dependence structures. In this paper, we propose a novel bivariate cure frailty-copula model for paired survival data with a cure fraction. By incorporating a zero-inflated gamma frailty, the proposed framework simultaneously accommodates a cure fraction and continuous unobserved heterogeneity among uncured subjects. Dependence between cure statuses is modeled naturally via an odds-ratio parameter, while dependence between survival times conditional on frailty is captured through a copula. To provide interpretable measures of overall dependence in the presence of cure, we derive population-level rank correlation coefficients, namely tie-adjusted versions of Kendall's tau and Spearman's rho. For suitable choices of baseline marginal survival functions and copulas, the joint survival function admits a closed-form expression, enabling maximum likelihood estimation and likelihood ratio testing. Simulation studies demonstrate satisfactory finite-sample performance and favorable comparisons with existing bivariate cure models. In a real-data analysis, the proposed model is favored by information criteria. An R package, curecopula, implementing the proposed methods is publicly available on GitHub.

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