A West Nile virus nonlocal model with free boundaries and seasonal succession

TL;DR

This paper develops a mathematical model for West Nile virus spread incorporating nonlocal dispersal, free boundaries, and seasonal effects, providing criteria for infection spread or vanishing and analyzing long-term dynamics.

Contribution

It introduces a novel nonlocal free boundary model with seasonal succession for West Nile virus, extending existing results and analyzing eigenvalues to determine spreading or vanishing.

Findings

Long-term behavior depends on generalized eigenvalues.

Warm season duration correlates with infection risk.

Initial infection size influences spread outcomes.

Abstract

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.