Taming nonlocal delay in a spatiotemporal brucellosis model: A Lyapunov approach to global attractivity

This paper investigates a spatiotemporal brucellosis model with two-stage susceptible structure, infection-age structure, spatial diffusion, environmental transmission, and nonlocal delay. The two-stage susceptible structure distinguishes susceptible lambs and susceptible adult sheep, while the infection-age structure is used to derive a Green-function-weighted nonlocal delay describing the movement of infected sheep during the latent period. The basic reproduction number R0 is derived by using the next-generation infection operator. When R0>1, we prove the existence and uniqueness of a positive spatially homogeneous endemic equilibrium and establish the uniform persistence of the infected components. Furthermore, by constructing a novel Lyapunov functional adapted to the Green-function-weighted delay, we prove the global attractivity of the endemic equilibrium for all initial histories with nontrivial infected components. Numerical simulations illustrate the threshold role of R0, the convergence toward the endemic equilibrium, and the different effects of direct and environmental transmission.