Structural identifiability of linear-in-parameter parabolic PDEs through auxiliary elliptic operators

Area of Science:

• Mathematical Biology
• Computational Biology
• Systems Biology

Background:

• Parameter identifiability is crucial for interpreting biological data using mathematical models.
• Existing theory for structural identifiability of partial differential equations is limited.

Purpose of the Study:

• To present a new approach for structural identifiability analysis of parabolic partial differential equations.
• To address limitations in current theoretical frameworks for parameter identifiability.

Main Methods:

• Framing identifiability as an existence and uniqueness problem for related elliptic equations.
• Utilizing the Fredholm alternative for homogeneous equations.
• Analyzing cases with linear and nonlinear reaction terms, including boundary and initial conditions.

Main Results:

• Established unconditional identifiability for homogeneous equations using the Fredholm alternative.
• Identified conditions leading to non-identifiability due to specific initial and boundary conditions.
• Demonstrated that uniqueness in auxiliary elliptic equations corresponds to identifiability, especially for nonlinear reaction terms.

Conclusions:

• The novel approach provides a framework for advancing the theory of structural identifiability for partial differential equations.
• The findings have implications for practical identifiability, particularly in spatial biological problems.
• This perspective enables the application of well-developed analysis tools to complex biological models.