The hazard function represents one of the main quantities of interest in the analysis of survival data. We propose a general approach for parametrically 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. Although we focus on examples on Medical Statistics, the proposed framework is applicable in any context where the interest lies on estimating and interpreting the dynamics hazard function.

翻译：风险函数是生存数据分析中主要关注量之一。本文提出一种通用方法，利用自治常微分方程系统对风险函数的动态变化进行参数化建模。该建模方法可用于对风险函数随时间演化的定性与定量分析。我们的提案充分利用了常微分方程的丰富文献，特别是通过自治常微分方程建立风险函数动态变化的基本规则或定律。我们展示了如何在常微分方程组存在解析解或需要ODE求解器获得数值解的情况下实施所提出的建模框架。我们重点采用贝叶斯建模方法，但所提出的方法论也可与最大似然估计相结合。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。两个使用真实数据的案例研究展示了所提出方法的应用，并突出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策略以及将协变量纳入框架的方法。虽然我们主要关注医学统计领域的案例，但所提出的框架适用于任何需要估计和解释动态风险函数的研究场景。