Spectral Theory of Fractional Cooperative Systems and Threshold Dynamics in Epidemic Models

Spectral analysis has long been recognized as a fundamental tool for studying the existence, uniqueness, and qualitative behavior of solutions to semilinear elliptic and parabolic equations, as well as their long-time dynamics. In modern mathematics, fractional Laplacians are widely used to model nonlocal or long-range diffusion processes arising in biology, including anomalous movement, long-distance dispersal, and Levy-flight migration of organisms, cells, and epidemics. In this paper, we employ the spectral fractional Laplacian introduced by Caffarelli and Stinga (2016) to develop the eigentheory for a cooperative system describing an infectious epidemic process and to analyze its long-term behavior. Using Fredholm theory and related analytical techniques, building in part on ideas of Lam and Lou (2016), we establish a sharp criterion ensuring the existence and simplicity of the principal eigenvalue, together with variational characterizations and consequences for the validity of maximum principles. We further derive the asymptotic behavior of the principal eigenvalue with respect to diffusion coefficients, fractional orders, and domain scaling, complementing recent developments by Zhao and Ruan (2023) and Feng, Li, Ruan, and Xin (2024). As an application of this spectral framework, we prove the existence, uniqueness, and threshold-type long-time dynamics of solutions to an endemic reaction-diffusion system with fractional diffusion, providing a perspective that differs from earlier approaches such as Hsu and Yang (2013). Our results contribute to the growing interaction between spectral theory and nonlocal analysis, in line with recent advances in the area.