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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. 

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. 

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. 

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. 

The COVID-19 pandemic highlighted the need for improved epidemic spread forecasting, a critical precursor for developing optimal control measures for spread mitigation. Well-recognized shortcomings in computing basic and effective reproduction numbers (R0, Re)-fundamental metrics for forecasting-underscore the need for new methods for estimating them from available data. We present a novel computational framework for estimating reproduction numbers from empirical spread data. The framework is derived from a mechanistic, spatiotemporal, Partial Differential Equation (PDE) model of epidemic spread utilizing mathematical results from PDE epidemic models. Forecasts of spatiotemporal effective reproduction number Re using the framework are found to be in excellent agreement with COVID-19 spread trends for Hamilton County, Ohio, USA, for three distinct periods. Furthermore, the forecasts are shown to align with corresponding reproduction numbers computed independently using the Wallinga-Teunis and Cori retrospective methods used in epidemiology. In summary, the results establish the validity of the framework and indicate applicability to future epidemics- especially for regions such as counties and for timeframes extending in weeks -even during dynamic phases when obtainable real-time infection spread data will likely be sparse.