Effects of diffusion and advection on endemic steady states in a partially degenerate West Nile virus model

Jie Xing; Lin Wang; Jie Xing; Lin Wang

doi:10.3934/math.2026950

AIMS Mathematics

2026, Volume 11, Issue 8: 23588-23605. doi: 10.3934/math.2026950

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Effects of diffusion and advection on endemic steady states in a partially degenerate West Nile virus model

Jie Xing ^1,2,
Lin Wang ^{3
,
,}

1.
School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541002, Guangxi, China
2.
Center for Applied Mathematics of Guangxi (GUET), Guilin 541002, Guangxi, China
3.
School of Science, Jimei University, Xiamen 361021, Fujian, China

Received: 15 June 2026 Revised: 16 July 2026 Accepted: 27 July 2026 Published: 03 August 2026
MSC : 35K57, 35B40, 92D25

This paper studies endemic steady states of a partially degenerate West Nile virus model in a bounded one-dimensional heterogeneous habitat. The infected bird population is subject to diffusion and directional advection, whereas the infected mosquito density is determined at steady state by a local host-vector transmission balance. This leads to an elliptic-algebraic steady-state problem rather than a fully diffusive system. We first reduce the system to a scalar elliptic boundary value problem and prove the existence and uniqueness of a positive endemic steady state by the method of upper and lower solutions. We then examine how the diffusion rate and the advection rate affect the endemic spatial profile. In the advection-dominated regime, including large advection and vanishing diffusion with fixed positive advection, the infected bird density forms a downstream boundary layer, and the limiting boundary value is determined by a scalar equation. In the large-diffusion regime, the infected bird density converges to a positive spatially homogeneous state and the infected mosquito density is obtained from the local transmission relation. Numerical simulations are presented to illustrate these diffusion-advection effects on the spatial distribution of infection.
- West Nile virus model,
- partially degenerate system,
- endemic steady states,
- advection effect,
- boundary layer
Citation: Jie Xing, Lin Wang. Effects of diffusion and advection on endemic steady states in a partially degenerate West Nile virus model[J]. AIMS Mathematics, 2026, 11(8): 23588-23605. doi: 10.3934/math.2026950

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Abstract

This paper studies endemic steady states of a partially degenerate West Nile virus model in a bounded one-dimensional heterogeneous habitat. The infected bird population is subject to diffusion and directional advection, whereas the infected mosquito density is determined at steady state by a local host-vector transmission balance. This leads to an elliptic-algebraic steady-state problem rather than a fully diffusive system. We first reduce the system to a scalar elliptic boundary value problem and prove the existence and uniqueness of a positive endemic steady state by the method of upper and lower solutions. We then examine how the diffusion rate and the advection rate affect the endemic spatial profile. In the advection-dominated regime, including large advection and vanishing diffusion with fixed positive advection, the infected bird density forms a downstream boundary layer, and the limiting boundary value is determined by a scalar equation. In the large-diffusion regime, the infected bird density converges to a positive spatially homogeneous state and the infected mosquito density is obtained from the local transmission relation. Numerical simulations are presented to illustrate these diffusion-advection effects on the spatial distribution of infection.

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