Regularized Semiparametric Estimation for Ordinary Differential Equations.

Area of Science:

• Mathematical modeling
• Dynamical systems analysis
• Scientific computing

Background:

• Ordinary differential equations (ODEs) are crucial for modeling dynamic systems across various scientific disciplines.
• Accurate parameter estimation in ODEs is essential for understanding system behavior, but constant parameters can be overly restrictive.
• Real-world data often exhibits disturbances, leading to poor model-data agreement with fixed ODE parameters.

Purpose of the Study:

• To develop a novel regularized estimation procedure for time-varying parameters in ODE systems.
• To accommodate short-term interferences and improve model-data fit by allowing parameters to change dynamically.
• To provide theoretical guarantees and demonstrate practical applicability of the proposed method.

Main Methods:

• Proposed a regularized estimation procedure for time-varying parameters in ODEs.
• Developed a method where parameters adapt during transitions but remain constant in stable phases.
• Conducted simulation studies and derived finite-sample estimation error bounds.

Main Results:

• The proposed regularized method demonstrated robust performance in simulation studies.
• The method exhibited less parameter variation compared to non-regularized approaches.
• Finite-sample estimation error bounds were theoretically established for the new procedure.

Conclusions:

• The developed regularized estimation method effectively models time-varying parameters in ODEs, handling dynamic changes and disturbances.
• The approach offers improved accuracy and stability over traditional methods, supported by theoretical analysis and simulations.
• Successful applications to ecological (hare-lynx) and epidemiological (measles) models highlight its practical utility and meaningful results.