The hazard function represents one of the main quantities of interest in the analysis of survival data. We propose a general approach for modelling the dynamics of the hazard function using systems of autonomous ordinary differential equations (ODEs). This modelling approach can be used to provide qualitative and quantitative analyses of the evolution of the hazard function over time. Our proposal capitalises on the extensive literature of ODEs which, in particular, allow for establishing basic rules or laws on the dynamics of the hazard function via the use of autonomous ODEs. We show how to implement the proposed modelling framework in cases where there is an analytic solution to the system of ODEs or where an ODE solver is required to obtain a numerical solution. We focus on the use of a Bayesian modelling approach, but the proposed methodology can also be coupled with maximum likelihood estimation. A simulation study is presented to illustrate the performance of these models and the interplay of sample size and censoring. Two case studies using real data are presented to illustrate the use of the proposed approach and to highlight the interpretability of the corresponding models. We conclude with a discussion on potential extensions of our work and strategies to include covariates into our framework.

翻译：风险函数是生存数据分析中的核心关注量之一。我们提出了一种通用方法，利用自治常微分方程组对风险函数的动态变化进行建模。该建模方法可用于对风险函数随时间演化的定性及定量分析。本方案充分利用了常微分方程的丰富文献，特别地，通过使用自治常微分方程，能够建立关于风险函数动态变化的基本规则或定律。我们展示了在两种情况下如何实现所提出的建模框架：当常微分方程组存在解析解时，以及当需使用常微分方程求解器获取数值解时。本研究以贝叶斯建模方法为重点，但所提出的方法论亦可与极大似然估计相结合。通过模拟研究展示了这些模型的性能以及样本量与删失的相互作用。基于真实数据的两项案例研究阐释了所提方法的应用，并凸显了相应模型的可解释性。最后，我们讨论了本工作的潜在扩展方向，以及将协变量纳入框架的策略。