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The paper deals with a West Nile virus (WNv) model, where the nonlocal diffusion is introduced to characterize a long-range dispersal, the free boundary is used to describe the spreading front, and seasonal succession accounts for the effect of the warm and cold seasons. The well-posedness of the model is firstly given, its long-term dynamical behaviours are investigated and depend on the generalized eigenvalues of the corresponding linear operator. For the spatial-independent WNv model with seasonal succession, the generalized eigenvalues are calculated and new properties are found. For the WNv nonlocal model with seasonal succession, the generalized eigenvalues are discussed. We then develop the indexes to the case with the free boundary and further use these indexes to judge whether spreading or vanishing happens. The criteria extends known results for the case with the nonlocal diffusion and the case with the free boundary. Moreover, the generalized eigenvalues reveal that there exists positive correlation between the duration of the warm season and the risk of infection. The index of the nonlocal free boundary problem, which depends on the time t, determines the spreading or vanishing of WNv. Moreover, the initial infection length, the initial infection scale and the spreading ability to the new area play an important role for the long time behavior of the solution.

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This paper is concerned with a West Nile virus (WNv) model on a growing domain, which accounts for habitat expansion of mosquitoes because of climate warming. We aim to understand the relationship of the growing rate and the transmission risk of WNv. The basic reproduction number, which is related to the growing rate and diffusion rate, is introduced through spectral theory. The conditions to determine whether the virus vanishes or spreads are deduced.The obtained results reveal that domain growth leads to increased risk of infection, and is detrimental to the control and prevention of WNv. To verify the feasibility of our analytical results on the long time behavior of WNv, some numerical simulations are given.