Fractional diffusion induced pattern formation in an epidemic model with nonlinear incidence rate

The study of pattern formation in spatial fractional reaction-diffusion systems has garnered widespread attention in recent years. Particularly, it has been observed that during epidemic outbreaks, population movement aligns more with Lévy flights rather than short-distance random walks. Building upon this observation, this paper delves into the pattern dynamics of a spatial fractional susceptible-infected (SI) model with nonlinear infection rates. Specifically, we emphasize the necessity of nonlinear incidence for generating Turing patterns by comparing it with bilinear incidence. Critical conditions for Hopf bifurcation and Turing instability are determined using stability theory and bifurcation theory. Additionally, by employing central manifold reduction theory, we develop amplitude equations for two-dimensional (2D) and three-dimensional (3D) Turing patterns near the Turing bifurcation point, facilitating a comprehensive exploration of their stability and associated patterns. Numerical simulations reveal six distinct types of 2D patterns, all highlighting the impact of spatial fractional exponents on pattern formation. For 3D patterns, various configurations are observed, including tubular, planar sheet, and spherical droplet patterns. Our findings clarify how fractional diffusion drives self-organization in disease spread, suggesting that the resulting patterns may represent infection hotspots. Consequently, identifying and quantifying these patterns provides robust theoretical insight into high-risk region evolution and supports spatially targeted intervention strategies.