A reaction–diffusion epidemic model with incubation period in almost periodic environments

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In this paper, we propose and study an almost periodic reaction–diffusion epidemic model in which disease latency, spatial heterogeneity and general seasonal fluctuations are incorporated. The model is given by a spatially nonlocal reaction–diffusion system with a fixed time delay. We first characterise the upper Lyapunov exponent λ* for a class of almost periodic reaction–diffusion equations with a fixed time delay and provide a numerical method to compute it. On this basis, the global threshold dynamics of this model is established in terms of λ* It is shown that the disease-free almost periodic solution is globally attractive if λ* < 0, while the disease is persistent if λ* > 0. By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.