Asymptotic behavior of the basic reproduction ratio for periodic reaction-diffusion systems

TL;DR

This paper analyzes how the basic reproduction ratio in periodic reaction-diffusion systems behaves asymptotically as diffusion coefficients become very small or large, providing insights into disease spread modeling.

Contribution

It establishes the continuity of the basic reproduction ratio and characterizes its asymptotic limits for large and small diffusion coefficients in periodic systems.

Findings

Reproduction ratio is continuous with respect to parameters.

Asymptotic limits of the principal eigenvalue are characterized.

Behavior of positive periodic solutions is described for large diffusion.

Abstract

This paper is devoted to the study of asymptotic behavior of the basic reproduction ratio for periodic reaction-diffusion systems in the case of small and large diffusion coefficients. We first establish the continuity of the basic reproduction ratio with respect to parameters by developing the theory of resolvent positive operators. Then we investigate the limiting profile of the principal eigenvalue of an associated periodic eigenvalue problem for large diffusion coefficients. We then obtain the asymptotic behavior of the basic reproduction ratio as the diffusion coefficients go to zero and infinity, respectively. We also investigate the limiting behavior of positive periodic solution for periodic and cooperative reaction-diffusion systems with the Neumann boundary condition when the diffusion coefficients are large enough. Finally, we apply these results to a reaction-diffusion model of Zika virus transmission.