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Abstract Theoretical analysis of epidemic dynamics has attracted significant attention in the aftermath of the COVID–19 pandemic. In this article, we study dynamic instabilities in a spatiotemporal compartmental epidemic model represented by a stochastic system of coupled partial differential equations (SPDE). Saturation effects in infection spread–anchored in physical considerations–lead to strong nonlinearities in the SPDE. Our goal is to study the onset of dynamic, Turing–type instabilities, and the concomitant emergence of steady–state patterns under the interplay between three critical model parameters–the saturation parameter, the noise intensity, and the transmission rate. Employing a second–order perturbation analysis to investigate stability, we uncover both diffusion–driven and noise–induced instabilities and corresponding self–organized distinct patterns of infection spread in the steady state. We also analyze the effects of the saturation parameter and the transmission rate on the instabilities and the pattern formation. In summary, our results indicate that the nuanced interplay between the three parameters considered has a profound effect on the emergence of dynamical instabilities and therefore on pattern formation in the steady state. Moreover, due to the central role played by the Turing phenomenon in pattern formation in a variety of biological dynamic systems, the results are expected to have broader significance beyond epidemic dynamics.
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This content will become publicly available on April 1, 2027
Anisotropic Diffusion and the Emergence of Instabilities and Self-Organized Pattern Formation in Spatiotemporal Epidemic Dynamics
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.
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- Award ID(s):
- 2140405
- PAR ID:
- 10667567
- Publisher / Repository:
- American Society of Mechanical Engineers
- Date Published:
- Journal Name:
- ASME Letters in Dynamic Systems and Control
- Volume:
- 6
- Issue:
- 2
- ISSN:
- 2689-6117
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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