Modeling the Hazard Function with Non-linear Systems in Dynamical Survival Analysis

TL;DR

A new statistical framework for modeling and simulating hazard functions governed by higher-order ODEs is proposed, allowing risk to depend on both its current level, its rate of change, and time, and is evaluated through simulation studies and illustrated using real world survival data.

Abstract

Hazard functions play a central role in survival analysis, offering 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 new statistical framework for modeling and simulating hazard functions governed by higher-order ODEs, allowing risk to depend on both its current level, its rate of change, and time. This class of models captures complex time dependent risk behaviors relevant to survival analysis and reliability studies. We develop a simulation procedure by reformulating the higher order ODE as a system of nonlinear first order equations solved numerically, with failure times generated via cumulative hazard inversion. Likelihood based 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 world survival data, where oscillatory hazard dynamics capture temporal risk patterns beyond standard monotone models.

Figures (12)

• Figure 1: Flowchart illustrating the simulation framework for survival models driven by higher-order dynamical ODEs. The hazard function is modeled using a higher-order ODE, reformulated as a system of first-order equations, numerically solved to obtain the cumulative hazard, and inverted to generate synthetic failure times.
• Figure 2: Hazard functions $h(t)$ and survival functions $S(t)$ for the damped oscillatory model under the underdamped ($\alpha = 0.5, \ \beta = 1, \ \gamma = 0.2, \ h_0 = 0.1, \ v_0 = 0.3$; blue), critically damped ($\alpha = 2, \ \beta = 1, \ \gamma = 0.2, \ h_0 = 0.1, \ v_0 = 0.3$; red), and overdamped ($\alpha = 3, \ \beta = 1, \ \gamma = 0.2, \ h_0 = 0.1, \ v_0 = 0.3$; black) cases.
• Figure 3: Hazard functions $h(t)$ and survival functions $S(t)$ for logistic hazard dynamics between first-order (black), delayed (blue), and damped second-order logistic (red) hazard models ($r=0.8,\; K=1,\; \tau=1.2,\; \zeta=0.5,\; h_0=0.1,\; v_0=0.2$).
• Figure 4: Hazard function $h(t)$ and survival function $S(t)$ for the sinusoidal hazard model ($h_0 = 0.1,\ v_0 = 0.2, \ \omega = 0.2\pi, \ c = 0.6$).
• Figure 5: Hazard function $h(t)$ and survival function $S(t)$ for the exponential hazard model ($\alpha = 0.1,\ v_0 = -0.1$).

Theorems & Definitions (4)

• Theorem 1
• Theorem 2
• proof
• proof