SurvMODE: Survival Modelling via Ordinary Differential Equations

Overview

This repository contains the R and Julia code used to reproduce the two case studies in:

Christen, J.A. and Rubio, F.J. (2026). Hazard-based distributional regression via ordinary differential equations. Statistical Methods in Medical Research, in press. DOI: 10.1177/09622802251412840 | Preprint (arXiv)

Traditional hazard-based survival models specify the baseline hazard through a parametric distribution. This paper proposes an alternative approach in which the hazard function is defined implicitly as the solution to an ordinary differential equation (ODE), enabling flexible distributional regression for survival data. The framework is illustrated through two real-data case studies and supports both maximum likelihood (R) and Bayesian posterior sampling (Julia) inference.

Repository structure

SurvMODE/
├── R code/
│   ├── Example1_2A.html     # Rendered tutorial with code and outputs (Case Study I)
│   └── ...                  # R scripts for Case Study I
└── Julia code/
    ├── rotterdamJulia.html  # Rendered tutorial with code and outputs (Case Study II)
    └── ...                  # Julia scripts for Case Study II

Case studies

	Case Study I	Case Study II
Language	R	Julia
Inference	MLE and Bayesian	MLE and Bayesian
Dataset	Ipilimumab data	Rotterdam breast cancer data
Tutorial	Example1_2A.html	rotterdamJulia.html

Requirements

R (Case Study I). Install the required packages by inspecting the library() calls at the top of the R scripts. Key dependencies are likely to include:

install.packages(c("survival", "deSolve", "numDeriv"))

Julia (Case Study II). The Julia scripts require Julia 1.x. Install the required packages by running the following in the Julia REPL:

using Pkg
Pkg.add(["DifferentialEquations", "Turing", "Distributions", "DataFrames"])

Note: verify exact dependencies against the using statements at the top of each script, as the lists above may be incomplete.

Related resources

• HazReg — R package for parametric hazard-based regression models (overall and relative survival)
• HazReg.jl — Julia implementation of the same
• ODESurv — further R code for survival modelling via ODEs

Citation

If you use this code, please cite:

@article{christen:2026,
  author  = {Christen, J.A. and Rubio, F.J.},
  title   = {Hazard-based distributional regression via ordinary differential equations},
  journal = {Statistical Methods in Medical Research},
  year    = {2026},
  note    = {in press},
  doi     = {10.1177/09622802251412840}
}