  This paper introduces an Ordinary Differential Equation (ODE) notion for
survival analysis. The ODE notion not only provides a unified modeling
framework, but more importantly, also enables the development of a widely
applicable, scalable, and easy-to-implement procedure for estimation and
inference. Specifically, the ODE modeling framework unifies many existing
survival models, such as the proportional hazards model, the linear
transformation model, the accelerated failure time model, and the time-varying
coefficient model as special cases. The generality of the proposed framework
serves as the foundation of a widely applicable estimation procedure. As an
illustrative example, we develop a sieve maximum likelihood estimator for a
general semi-parametric class of ODE models. In comparison to existing
estimation methods, the proposed procedure has advantages in terms of
computational scalability and numerical stability. Moreover, to address unique
theoretical challenges induced by the ODE notion, we establish a new 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.
