Spatiotemporal patterns in a delay-induced infectious disease model with superdiffusion

Typically, human mobility patterns exhibit distinct large-scale and long-distance features, rendering the standard Brownian motion-based reaction–diffusion modeling inadequate. Consequently, this paper introduces a novel reaction-superdiffusion infectious disease model to delve into the long-distance geographical dissemination of infectious diseases. Theoretically, we have determined the threshold conditions for Turing instability without delay, formulated the amplitude equations characterizing Turing patterns, and devised stability assessment methodologies. Additionally, we have delved into the mechanisms of Hopf and Turing–Hopf bifurcations triggered by delay, utilizing normal form theory and the center manifold theorem to scrutinize the stability and directionality of the resulting periodic solutions. The numerical simulations robustly corroborate the theoretical findings. Notably, while the superdiffusion exponent minimally impacts the pattern shape for Turing patterns, a decrement in its value conspicuously augments the area and isolation of the patches. Conversely, in wave patterns, both the superdiffusion exponent and delay act in concert to modulate the pattern's morphological evolution. Interestingly, when the selected parameters lie within the Turing–Hopf instability region, we observe spatio-temporal heterogeneous patterns distinct from Turing and spiral patterns. Our findings enrich the existing results.