This paper introduces a general framework for survival analysis based on ordinary differential equations (ODE). Specifically, this framework unifies many existing survival models, including proportional hazards models, linear transformation models, accelerated failure time models, and time-varying coefficient models as special cases. Such a unified framework provides a novel perspective on modeling censored data and offers opportunities for designing new and more flexible survival model structures. Further, the aforementioned existing survival models are traditionally estimated by procedures that suffer from lack of scalability, statistical inefficiency, or implementation difficulty. Based on well-established numerical solvers and sensitivity analysis tools for ODEs, we propose a novel, scalable, and easy-to-implement general estimation procedure that is applicable to a wide range of models. In particular, we develop a sieve maximum likelihood estimator for a general semi-parametric class of ODE models as an illustrative example. We also establish a general sieve M-theorem for bundled parameters and show that the proposed sieve estimator is consistent and asymptotically normal, and achieves the semi-parametric efficiency bound. The finite sample performance of the proposed estimator is examined in simulation studies and a real-world data example.
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