Asymptotic profiles of a spatial vector-borne disease model with Fokker–Planck-type diffusion

Abstract

This paper is concerned with a spatially heterogeneous vector-borne disease model that follows the Fokker–Planck-type diffusion law. One of the significant features in our model is that Fokker–Planck-type diffusion is used to characterize individual movement, which poses new challenges to theoretical analysis. We derive for the first time the variational characterization of basic reproduction ratio $R_0$ for the model under certain conditions and investigate its asymptotic profiles with respect to the diffusion rates. Furthermore, via overcoming the difficulty of the associated elliptic eigenvalue problem, the asymptotic behaviors of endemic equilibrium for the model are discussed. Our results imply that whether rapid or slow movement of susceptible and infected individuals are conducive to disease control depends on the degree of disease risk in the habitat. Numerically, we verify the theoretical results and detect that Fokker–Planck-type diffusion may amplify the scale of disease infection, which in turn increases the complexity of disease transmission by comparing the impacts of distinct dispersal types on disease dynamics.