A Note on Quantifying the Contributions of Incidence Functions in Spatio-Temporal Epidemic Models

Area of Science:

• Epidemiology
• Mathematical Biology
• Computational Science

Background:

• Reaction-diffusion models are crucial for understanding infectious disease spread in space and time.
• The choice of incidence function significantly impacts epidemic model accuracy.
• Systematic methods for selecting incidence functions are currently lacking.

Purpose of the Study:

• To develop a theoretical framework for selecting optimal incidence functions in epidemic models.
• To interpret incidence function selection as a data-driven optimization problem.
• To enhance the accuracy of disease transmission modeling.

Main Methods:

• Formulating incidence function selection as a PDE-constrained optimization problem.
• Determining optimal weights for a convex combination of incidence functions.
• Establishing Fréchet differentiability of the parameter-to-state operator.
• Deriving first-order optimality conditions using an adjoint system.
• Applying the Landweber iteration method for numerical illustration.

Main Results:

• A novel framework for selecting incidence functions based on observational data.
• Mathematical proof of Fréchet differentiability for the parameter-to-state operator.
• Derivation of optimality conditions for incidence function selection.
• Numerical validation demonstrating the framework's effectiveness.

Conclusions:

• The proposed framework provides a systematic approach to incidence function selection in epidemic modeling.
• This method enhances modeling accuracy and supports disease prevention strategies.
• The framework offers a valuable mathematical tool for public health analysis.