Computation of the basic reproduction numbers for reaction-diffusion epidemic models.

Area of Science:

• Mathematical epidemiology
• Computational mathematics
• Dynamical systems

Background:

• Reaction-diffusion models are crucial for understanding spatial epidemic spread.
• Calculating basic reproduction numbers (R0) is essential for epidemic analysis.
• Connecting partial differential equation (PDE) models to ordinary differential equation (ODE) models simplifies analysis.

Purpose of the Study:

• To develop a computational approach for calculating R0 in k-dimensional reaction-diffusion epidemic models.
• To analyze the relationship between R0 in PDE epidemic models and their corresponding ODE systems.
• To identify conditions under which R0 values are equivalent between PDE and ODE epidemic models.

Main Methods:

• Formulation of k-dimensional reaction-diffusion epidemic models derived from autonomous ODE systems.
• Application of matrix theory to analyze the basic reproduction numbers.
• Development of a computational framework for R0 calculation and analysis.
• Numerical simulations to validate analytical findings.

Main Results:

• A computational method for determining R0 in reaction-diffusion epidemic models is presented.
• Matrix theory reveals a direct relationship between R0 in PDE and ODE models.
• In several key scenarios, the R0 values for the PDE models are shown to be identical to those of their associated ODE models.
• Numerical examples confirm the analytical results, demonstrating the validity of the findings.

Conclusions:

• The study provides an effective computational approach for analyzing R0 in complex reaction-diffusion epidemic models.
• The equivalence of R0 between PDE and ODE models in specific cases offers significant analytical simplification.
• These findings contribute to a better understanding of epidemic dynamics and control strategies.