  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.
