[Paper Review] On characteristics of an ordinary differential equation and a related inverse problem in epidemiology

Ralph Brinks|arXiv (Cornell University)|Aug 27, 2012

advanced mathematical theories1 references3 citations

TL;DR

This paper formulates a one-dimensional ordinary differential equation (ODE) linking age-specific prevalence, incidence, and mortality rates in chronic disease epidemiology. It proves that deriving incidence from prevalence data is an ill-posed inverse problem, meaning small errors in prevalence can lead to arbitrarily large errors in estimated incidence, highlighting instability in such estimations despite the model's epidemiological plausibility.

ABSTRACT

In this work we examine the properties of a recently described ordinary differential equation that relates the age-specific prevalence of a chronic disease with the incidence and mortalities of the diseased and healthy persons. The equation has been used to estimate the incidence from prevalence data, which is an inverse problem. The ill-posedness of this problem is proven, too.

Motivation & Objective

• To analyze the mathematical properties of a recently proposed ODE that links age-specific prevalence with incidence and mortality rates in chronic disease epidemiology.
• To investigate the feasibility and stability of estimating incidence rates from observed prevalence data, a problem known as the inverse problem.
• To establish the ill-posedness of this inverse problem in the sense of Hadamard, demonstrating instability under small perturbations.
• To clarify the conditions under which analytical solutions exist and when numerical methods are required.

Proposed method

• Derives a one-dimensional ODE for prevalence dynamics from a three-state compartment model (Normal, Diseased, Dead) using the quotient rule on the system of ODEs for susceptible (S) and case (C) populations.
• Expresses the rate of change of prevalence p(a) as dp/da = (1−p)·(i − (m − m₀)), where i is incidence, m is overall mortality, and m₀ is mortality in the non-diseased group.
• Classifies the ODE type (linear, Riccati, or Abelian) based on available mortality information, determining the existence of analytical solutions.
• Applies Hadamard's criteria for well-posedness to prove the inverse problem is ill-posed by showing discontinuity in the inverse operator mapping prevalence to incidence.
• Uses a sequence of perturbed prevalence functions involving high-frequency sine waves to demonstrate unbounded sensitivity in incidence estimation.
• Considers model limitations, including time-homogeneity, absence of duration-dependent mortality, and irreversible disease progression, and notes extensions to time-dependent and duration-dependent models in future work.

Experimental results

Research questions

• RQ1 Can the age-specific incidence rate be reliably estimated from cross-sectional prevalence data using a differential equation model?
• RQ2 What mathematical properties does the derived ODE for prevalence dynamics possess under different assumptions about mortality data?
• RQ3 Under what conditions does the inverse problem of estimating incidence from prevalence become ill-posed?
• RQ4 How does the type of ODE (linear, Riccati, Abelian) affect the solvability and stability of incidence estimation?
• RQ5 What are the implications of high-frequency noise in prevalence data for the accuracy of derived incidence rates?

Key findings

• The ODE dp/da = (1−p)·(i − (m − m₀)) provides a mathematically sound and epidemiologically meaningful framework linking prevalence, incidence, and mortality.
• The inverse problem of estimating incidence from prevalence is ill-posed in the sense of Hadamard, as the inverse operator is discontinuous.
• Small, high-frequency perturbations in the prevalence function can lead to unbounded errors in the estimated incidence rate, demonstrating instability.
• Analytical solutions exist only when the ODE is linear; in most epidemiologically relevant cases, the ODE is Riccati or Abelian, requiring numerical solution.
• The model assumes time-homogeneity and no remission, but the core ODE remains valid even with remission if an additional term is included.
• Despite the ill-posedness, the model offers a practical pathway to estimate incidence from relatively inexpensive cross-sectional studies, though caution is needed due to instability.