Spatiotemporal dynamics in epidemic models with Lévy flights: A fractional diffusion approach

Recent field and experimental studies show that mobility patterns for humans exhibit scale-free nonlocal dynamics with heavy-tailed distributions characterized by Lévy flights. To study the long-range geographical spread of infectious diseases, in this paper we propose a susceptible-infectious-susceptible epidemic model with Lévy flights in which the dispersal of susceptible and infectious individuals follows a heavy-tailed jump distribution. Owing to the fractional diffusion described by a spectral fractional Neumann Laplacian, the nonlocal diffusion model can be used to address the spatiotemporal dynamics driven by the nonlocal dispersal. The primary focuses are on the existence and stability of disease-free and endemic equilibria and the impact of dispersal rates and fractional powers on the spatial profiles of these equilibria. A variational characterization of the basic reproduction number R₀ is obtained and its dependence on dispersal rates and fractional powers is also examined. Then R₀ is utilized to investigate the effects of spatial heterogeneity on the transmission dynamics. It is shown that R₀ serves as a threshold for determining the existence and nonexistence of an epidemic equilibrium as well as the stability of the disease-free and endemic equilibria. In particular, in low-risk regions both dispersal rates and fractional powers play a critical role and are capable of altering the threshold value. Numerical simulations were performed to illustrate the theoretical results.