Collocation of Next-Generation Operators for Computing the Basic Reproduction Number of Structured Populations

Area of Science:

• Mathematical Biology
• Numerical Analysis
• Operator Theory

Background:

• The basic reproduction number is crucial for understanding disease spread in structured populations.
• Computing this number often involves complex, infinite-dimensional operators.
• A novel collocation approach has been proposed for this computation.

Purpose of the Study:

• To provide a comprehensive theoretical and numerical analysis of the collocation approach for computing the basic reproduction number.
• To establish the mathematical foundations for discretizing the problem.
• To rigorously assess the accuracy and convergence properties of the method.

Main Methods:

• Analysis of theoretical properties of infinite-dimensional operators associated with population dynamics models.
• Proof of operator compactness under mild regularity conditions.
• Detailed error and convergence analyses of the collocation method.
• Numerical validation through diverse test cases.

Main Results:

• The relevant operators are proven to be compact, enabling reformulation as an eigenvalue problem.
• The collocation method achieves the expected spectral accuracy.
• Numerical tests confirm the analytical findings and reveal method-specific characteristics.

Conclusions:

• The collocation approach is theoretically sound and numerically accurate for computing the basic reproduction number in structured populations.
• The method's spectral accuracy is rigorously demonstrated.
• The findings provide a validated computational tool for epidemiological and ecological modeling.