Dynamical Survival Analysis for Modeling Hazard Functions with Nonlinear Systems

TL;DR

This paper introduces a flexible framework for modeling hazard functions using higher-order nonlinear ODEs, enabling the capture of complex oscillatory and nonlinear risk behaviors in survival analysis.

Contribution

It develops a novel class of nonlinear, oscillatory hazard models with interpretable dynamics, along with simulation and Bayesian inference methods for these complex systems.

Findings

The framework can model non-monotonic, oscillatory hazard functions.

Simulation studies validate the approach's ability to reproduce complex risk patterns.

Application to real data demonstrates improved fit over standard models.

Abstract

Hazard functions play a central role in survival analysis, providing insight into the underlying risk dynamics of time-to-event data, with broad applications in medicine, epidemiology, and related fields. First-order ordinary differential equation (ODE) formulations of the hazard function have been explored as extensions beyond classical parametric models. However, such approaches typically produce monotonic hazard patterns, limiting their ability to represent oscillatory behavior, nonlinear damping, or coupled growth-decay dynamics. We propose a general statistical framework for modeling and simulating hazard functions governed by higher-order ODEs, allowing the hazard to depend on both its current level, its rate of change, and time. This formulation accommodates complex temporal risk behaviors arising in a range of applications. Building on this framework, we develop a class of nonlinear and oscillatory hazard models, each associated with an interpretable dynamical mechanism and an induced survival distribution. We also present a simulation procedure for solving a system of non-linear higher-order ODEs, with failure times generated via cumulative hazard inversion. Likelihood-based Bayesian inference under right censoring is also developed, and moment generating function analysis is used to characterize tail behavior. The proposed framework is evaluated through simulation studies and illustrated using real data, demonstrating its ability to capture temporal risk patterns not well represented by standard monotone models. In contrast to existing linear ODE-based hazard models, the proposed approach accommodates nonlinear and non-equilibrium dynamics, enabling the representation of temporal risk patterns that are not well captured by first-order or linear oscillator-based formulations.