Dynamics and asymptotic properties of a partially degenerate reaction-diffusion model for West Nile virus

A partially degenerate reaction-diffusion model was proposed to describe the transmission of the West Nile virus between mosquitoes and birds, in which a general incidence rate and spatial heterogeneity were introduced to capture the complexity of disease spread. First, the well-posedness of the model was rigorously verified, and the existence of a global attractor was also proved. Furthermore, using the spectral theory of the next generation operator, the basic reproduction number Formula: see text was defined and its variational formulation was derived. Analytical results showed that when Formula: see text, the disease-free steady state was globally asymptotically stable. Conversely, when Formula: see text, the disease was uniformly persistent and the model admits at least one endemic steady state. In the spatially homogeneous case, the global asymptotic stability of the endemic steady state was proven by employing the Lyapunov function techniques. Moreover, for Formula: see text = 1 and constant diffusion coefficients, the disease-free steady state remained globally asymptotically stable. We also examined the asymptotic behavior of Formula: see text as the diffusion coefficients of the exposed and infected birds tend to zero or infinity, and established the monotonicity of Formula: see text with respect to these diffusion coefficients. Numerical simulations were conducted to validate the theoretical results and thoroughly analyze the impact of key parameters, particularly the diffusion coefficients, on the spatiotemporal distribution of the disease.