Dynamics of the nonlocal diffusive vector-disease model with delay and spatial heterogeneity

In this paper, we study the dynamics of a general nonlocal delayed reaction–diffusion–advection vector-disease model with spatial heterogeneity. The existence of the nonconstant positive steady state is identified by means of the Rabinowitz’s bifurcation theory. Further, the stability of the positive steady state and the Hopf bifurcation are analyzed in terms of the eigenvalue distribution associated with the steady state. Moreover, the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution are obtained by the center manifold reduction and normal form theory with the weighted inner product. It is found that in spatial heterogeneity, the large diffusion can change the stability of the infected host population due to the combined effects of the nonlocal dispersal and time delay.