Anisotropic Diffusion and the Emergence of Instabilities and Self-Organized Pattern Formation in Spatiotemporal Epidemic Dynamics

Abstract

Abstract. Diffusive epidemic dynamics is typically studied assuming uniform disease spread, which is represented by isotropic diffusion operators in partial differential equation (PDE) models. However, anisotropic diffusion more accurately represents random epidemic spread biased by preferred directions, such as scenarios involving street grids, transport routes, and natural barriers. Investigating the effects of anisotropic diffusion using bifurcation analysis and numerical simulations, we find that directional mobility can lead to the emergence of new classes of self-organized epidemic spread patterns. Specifically, the results illustrate the effects of anisotropic diffusion on the onset of Hopf and Turing instabilities, as well as the consequent steady-state pattern formation. We present bifurcation diagrams that (1) establish the presence of Hopf and Turing bifurcations and (2) provide the contrast in the onset of Turing bifurcations between the isotropic and anisotropic diffusion cases. Additionally, studying the formation of steady-state Turing patterns, we find spatially symmetric and homogeneous equilibrium patterns in the isotropic diffusion case. However, the symmetry in the patterns is broken in the anisotropic cases, which yield striped patterns stretched in specific spatial directions dependent on the magnitudes of the anisotropic diffusion coefficients. In summary, the results uncover the role of anisotropic diffusion in triggering Hopf and Turing-type instabilities and self-organized pattern formation in a spatiotemporal, reaction–diffusion PDE epidemic model. The results are expected to be broadly significant beyond epidemic dynamics, since anisotropic diffusion generically represents the diffusive dynamics of collectives whose mobility is characterized by directional bias.