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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 January 1, 2026
Instabilities and Pattern Formation in Epidemic Spread Induced by Nonlinear Saturation Effects and Ornstein–Uhlenbeck Noise
Abstract We analytically study the emergence of instabilities and the consequent steady-state pattern formation in a stochastic partial differential equation (PDE) based, compartmental model of spatiotemporal epidemic spread. The model is characterized by: (1) strongly nonlinear forces representing the infection transmission mechanism and (2) random environmental forces represented by the Ornstein–Uhlenbeck (O–U) stochastic process which better approximates real-world uncertainties. Employing second-order perturbation analysis and computing the local Lyapunov exponent, we find the emergence of diffusion-induced instabilities and analyze the effects of O–U noise on these instabilities. We obtain a range of values of the diffusion coefficient and correlation time in parameter space that support the onset of instabilities. Notably, the stability and pattern formation results depend critically on the correlation time of the O–U stochastic process; specifically, we obtain lower values of steady-state infection density for higher correlation times. Also, for lower correlation times the results approach those obtained in the white noise case. The analytical results are valid for lower-order correlation times. In summary, the results provide insights into the onset of noise-induced, and Turing-type instabilities in a stochastic PDE epidemic model in the presence of strongly nonlinear deterministic infection forces and stochastic environmental forces represented by Ornstein–Uhlenbeck noise.
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- PAR ID:
- 10574182
- Publisher / Repository:
- ASME
- Date Published:
- Journal Name:
- ASME Letters in Dynamic Systems and Control
- Volume:
- 5
- Issue:
- 1
- ISSN:
- 2689-6117
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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