Partial differential equation about the prevalence of a chronic disease in the presence of duration dependency

TL;DR

This paper derives a PDE to model the prevalence of chronic diseases considering duration dependency, enabling estimation of incidence rates from cross-sectional data, which extends previous models that ignored duration effects.

Contribution

It introduces a novel PDE framework that incorporates duration dependency into the illness-death model for chronic diseases, allowing more accurate prevalence and incidence estimation.

Findings

Derived a PDE accounting for duration dependency in disease prevalence

Enabled calculation of age-specific incidence from cross-sectional surveys

Extended previous models that neglected duration effects

Abstract

The illness-death model of a chronic disease consists of the states 'Normal', 'Disease' and 'Death'. In general, the transition rates between the states depend on three time scales: calendar time, age and duration of the chronic disease. Previous works have shown that the age-specific prevalence of the chronic disease can be described by differential equations if the duration is negligible. This article derives a partial differential equation (PDE) in the presence of duration dependency. As an important application, the PDE allows the calculation of the age-specific incidence from cross-sectional surveys.