An official website of the United States government
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.
more »
« less
This content will become publicly available on June 1, 2027
The Emergence of Turing Instability and Pattern Formation in a Nonlinear Stochastic Spatiotemporal Epidemic Model with Reinfections
Instabilities and Turing patterns in stochastic spatiotemporal systems in which a fraction of an evolving population, after undergoing a series of dynamic transitions, returns to its original state, remain largely unexplored. Adopting an epidemic model incorporating reinfections as an exemplar of such a system, we present stability and pattern-formation analyses of the stochastic reaction-diffusion equations that represent the model. Saturation effects in epidemic spread lead to nonlinear considerations, while random environmental effects motivate a stochastic term. Turing bifurcation and the emergence of equilibrium patterns are analysed with respect to three fundamental parameters - reinfection, saturation, and noise intensity. Using higher-order stability analysis and stochastic averaging, we find the Turing instability and also uncover self–organized, distinct equilibrium patterns of infection spread. Additionally, results elucidating the effects of stochastic excitation and its intensity, as well as the competing influence of saturation and reinfection on stability and pattern formation, are presented. The results are also expected to be broadly significant beyond epidemic modelling, for studies of noise-induced instabilities and morphogenesis in spatiotemporal nonlinear dynamical systems.
more »
« less
- PAR ID:
- 10667568
- Publisher / Repository:
- Springer Nature
- Date Published:
- Journal Name:
- Acta Biotheoretica
- Volume:
- 74
- Issue:
- 2
- ISSN:
- 0001-5342
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
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.more » « less
-
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.more » « less
-
Rubin, J (Ed.)We study pattern formation in a class of high-dimensional neural networks posed on random graphs and subject to spatiotemporal stochastic forcing. Under generic conditions on coupling and nodal dynamics, we prove that the network admits a rigorous mean-field limit, resembling a Wilson--Cowan neural-field equation. The state variables of the limiting systems are the mean and variance of neuronal activity. We select networks whose mean-field equations are tractable and we perform a bifurcation analysis using as a control parameter the diffusivity strength of the afferent white noise on each neuron. We find conditions for Turing-like bifurcations in a system where the cortex is modelled as a ring, and we produce numerical evidence of noise-induced spiral waves in models with a two-dimensional cortex. We provide numerical evidence that solutions of the finite-size network converge weakly to solutions of the mean-field model. Finally, we prove a large deviation principle, which provides a means of assessing the likelihood of deviations from the mean-field equations induced by finite-size effects.more » « less
-
For many infectious diseases, including malaria and COVID-19, the host may experience more than one episode of infection, where reinfection occurs due to waning immunity. In this paper, we propose a new age-structured epidemic model to investigate the dynamics of such diseases with multiple infections. The model is based on a system of partial differential equations that describes the interplay between completely susceptible individuals, temporarily immune individuals, and infected individuals at different stages. The model incorporates both time and age-dependent variables and parameters. We derive the basic reproduction number and conduct rigorous analyses on the equilibrium solutions and their stability properties. Specifically, we study the global asymptotic stability of the disease-free equilibrium and obtain the explicit conditions for the occurrence of a backward bifurcation. Our findings could provide useful insights into the effects of disease prevention and intervention strategies such as vaccination campaigns.more » « less
