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.

We use the pointwise Lipschitz constant to define an upper Lyapunov exponent for maps on metric spaces different to that given by Kifer [‘Characteristic exponents of dynamical systems in metric spaces’, Ergodic Theory Dynam. Systems3(1) (1983), 119–127]. We prove that this exponent reduces to that of Bessa and Silva on Riemannian manifolds and is not larger than that of Kifer at stable points. We also prove that it is invariant along orbits in the case of (topological) diffeomorphisms and under topological conjugacy. Moreover, the periodic orbits where this exponent is negative are asymptotically stable. Finally, we estimate this exponent for certain hyperbolic homeomorphisms.