  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.

Analysis of PDEs