Classifying the level set of principal eigenvalue for time-periodic parabolic operators and applications

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We study the effect of diffusion and frequency on the principal eigenvalue of linear time-periodic parabolic operators with zero Neumann boundary conditions. Monotonocity of the principal eigenvalue and its asymptotic behavior, as diffusion rate and frequency approach zero or infinity, are established. This leads to a classification of the topological structures of level sets for the principal eigenvalue, as a function of diffusion rate and frequency. As applications, we investigate a susceptible-infected-susceptible reaction-diffusion model in spatially heterogeneous and time-periodic environment. We characterize the parameter regions for the persistence and extinction of infectious disease by the basic reproduction number. The asymptotic profiles of endemic periodic solutions are determined when the diffusion rate of susceptible population is small. Our results suggest that fast movement of infected populations and high frequency of oscillation tend to eliminate the disease. Even if the disease persists, it can be controlled by limiting the movement of susceptible populations.