A reaction–diffusion vector-borne disease model with incubation period in almost periodic environments

Abstract

To study the multiple effects of seasonal fluctuations, spatial heterogeneity and extrinsic incubation period on vector-borne disease transmission, a nonlocal almost periodic reaction–diffusion model of vector-borne diseases is proposed and studied. We first give a characterization of the upper Lyapunov exponent λ∗ for a class of linear almost periodic reaction–diffusion systems with time delay, and provide a numerical scheme to compute it. Then we show that λ∗ is a threshold value determining the uniform persistence and extinction of our model. Specifically, the disease will die out when λ∗<0, while the disease is uniformly persistent when λ∗>0. Some numerical simulations finally are presented to verify our theoretical results, and to investigate the effects of diffusion rates, extrinsic incubation period and spatial heterogeneity on disease transmission.