We consider a West Nile virus model with nonlocal diffusion and free boundaries, in the form of a cooperative evolution system that can be viewed as a nonlocal version of the free boundary model of Lin and Zhu (2017 J. Math. Biol. 75 1381-1409. The model is a representative of a class of 'vector-host' models. We prove that this nonlocal model is well-posed, and its long-time dynamical behaviour is characterised by a spreading-vanishing dichotomy. We also find the criteria that completely determine when spreading and vanishing can happen, revealing some significant differences from the model in Lin and Zhu (2017 J. Math. Biol. 75 1381-1409. It is expected that the nonlocal model here may exhibit accelerated spreading (see remark 1.4 part (c)), a feature contrasting sharply to the corresponding local diffusion model, which has been shown by Wang et al (2019 J. Math. Biol. 79 433-466) to have finite spreading speed whenever spreading happens. Many techniques developed here are applicable to more general cooperative systems with nonlocal diffusion.

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This paper deals with the spreading speed of the disease described by a partially degenerate and cooperative epidemic model with free boundaries. We show that the spreading speed is determined by a semi-wave problem. To find such a semi-wave solution, we prove the existence of a monotone solution to a reduced ODE by an upper and lower solution approach. And then we establish the uniqueness of the semi-wave solution via the sliding method. It is demonstrated that the precise asymptotic spreading speed is less than the minimal speed of traveling waves.