Some applications of phase-type distributions in recurrent events

TL;DR

This paper models recurrent events using phase-type distributions within a Markov process framework, enabling analysis of multiple recurrences, durations, and death times, with calibration on real and simulated data.

Contribution

It introduces a novel Markov process model for recurrent events with phase-type distributions, providing differential equations for transition probabilities and durations, calibrated with bootstrap methods.

Findings

Model accurately captures recurrence and death times.

Differential equations facilitate probability calculations.

Bootstrap methods provide confidence intervals.

Abstract

In this paper, the recurrent events that can occur more than one over the follow-up time have been modeled by phase-type distributions. We use the finite-state continuous-time Markov process with multi states for patients with recurrent events. The number of recurrences until time $t$, the time stay for every state and the time till death are of importances. The time till death is assumed to have a phase-type distribution (which is defined in a Markov chain environment) with interpretable parameters. The underlying continuous-time Markov chain has one absorbing state (death) and transient states to reflect recovery and disease stages. A system of differential equations is obtained to calculate the probability of various number of transitions, the conditional expected time to stay in a disease stage and the probability of transition from a stage to another. The model has been calibrated via a real and simulated datasets. The bootstrap techniques have been used to construct the confidence intervals for the parameters.