Asymptotic profiles of a nonlocal dispersal SIS epidemic model with Neumann boundary condition

Abstract

This article studies a nonlocal dispersal susceptible-infected-susceptible (SIS) epidemic model with Neumann boundary condition in spatially heterogeneous environment, where the spatial movement of individuals is described by dσm∫ΩJσ(x−y)[u(y)−u(x)]dy,σ is the scaling of the size of the support of dispersal kernel J, and m represents the cost parameter. We firstly derive the basic reproduction number R0 for this model, and show that 1−R0 has the same sign as the generalised principal eigenvalue of a linear nonlocal operator. We also investigate asymptotic profiles of R0 with respect to σ,m and diffusion rates. Next, the existence and uniqueness of endemic equilibrium state of this model is obtained as R0>1. Finally, we consider the impact of small dispersal spread on the persistence criteria for disease. Particularly, our results illustrate that, even though ∫Ωβ(x)dx>∫Ωγ(x)dx, there exists the possibility of disease control only when m∈[0,2)andσ→0+.