Spatiotemporal patterns and control of a fractional-order SIRS reaction–diffusion process over complex networks

Population distribution frequently exhibits a network structure during disease transmission, with its dynamics influenced by past transmission data. Consequently, investigating fractional-order reaction–diffusion epidemic models within the framework of complex networks holds paramount importance. This study introduces a fractional-order susceptible–infected–recovered–susceptible (SIRS) reaction–diffusion system embedded in a network, emphasizing the analysis of pattern formation. Theoretically, we scrutinize how fractional-order and Laplacian eigenvalues affect Hopf bifurcation and spatiotemporal patterns. Based on the theoretical evidence, extensive numerical simulations are employed to elucidate the spatiotemporal dynamics of epidemics, focusing on the topological implications of complex networks. Notably, our findings reveal that fractional order tends to suppress the emergence of spatiotemporal patterns within networks. Remarkable differences in wave propagation patterns are observed between deterministic and random network configurations. Furthermore, the conditions conducive to formation of spatiotemporal patterns are contingent upon the type of network. Specifically, it is more feasible to generate such patterns within Watts–Strogatz (WS) networks compared to Erdös–Rényi (ER) random networks or scale-free (SF) networks. The fractional-order proportional derivative (PD) control strategy can effectively control the generation of spatiotemporal patterns. These findings may offer novel insights into comprehending the spatiotemporal dynamics of epidemics within complex networks.