Change rates and prevalence of a dichotomous variable: simulations and applications

TL;DR

This paper introduces a simple PDE-based equation linking prevalence and change rates in a three-state model, enhancing epidemiological analysis of dichotomous health variables across various applications.

Contribution

The authors develop and validate a new PDE equation that simplifies the calculation of prevalence from change rates in a three-state health model, broadening its practical use.

Findings

Validated the PDE with simulated data on chronic disease and dementia.

Applied the equation to real-world data on smoking in Germany.

Demonstrated the equation's utility in predicting disease prevalence and planning interventions.

Abstract

Background: A common modelling approach in public health and epidemiology divides the population under study into compartments containing persons that share the same status. Here we consider a three-state model with the compartments: A, B and Dead. States A and B may be the states of any dichotomous variable, for example, Healthy and Ill, respectively. The transitions between the states are described by change rates (or synonymously: densities), which depend on calendar time and on age. So far, a rigorous mathematical calculation of the prevalence of property B has been difficult, which has limited the use of the model in epidemiology and public health. Methods: We develop an equation that simplifies the use of the three-state model. To demonstrate the broad applicability and the validity of the equation, it is applied to simulated data and real world data from different health-related topics. Results: The three-state model is governed by a partial differential equation (PDE) that links the prevalence with the change rates between the states. The validity of the PDE has been shown in two simulation studies, one about a hypothetical chronic disease and one about dementia. In two further applications, the equation may provide insights into smoking behaviour of males in Germany and the knowledge about the ovulatory cycle in Egyptian women. Conclusions: We have found a simple equation that links the prevalence of a dichotomous variable with the transmission rates in the three-state model. The equation has a broad applicability in epidemiology and public health. Examples are the estimation of incidence rates from cross-sectional surveys, the prediction of the future prevalence of chronic diseases, and planning of interventions against risky behaviour (e.g., smoking).