Chapter 10 - Pattern formation induced by fractional-order diffusive model of COVID-19

Abstract

Differential equations with complex order fractional derivatives enable the regulation of complicated fractional systems. Within this scale, fractional calculus unfolds the fundamental mechanisms and multi-scale dynamic phenomena in biological tissues. It is viable that weakly nonlinear analysis represents a system that includes amplitude equations and the analysis that corresponds with it, which allows the prediction of the various patterns of parameter regimes likely to coexist in complicated dynamical and transient circumstances. A weakly nonlinear analysis creates a system comprising amplitude equations in dynamical contexts with fractional-order system characteristics, and its associated analyses are useful for predicting parameter regimes of several patterns that are expected to coexist. The development of patterns as a result of Turing instability in the homogeneous steady state is known to be a Turing process addressing unpredictability in numerous contexts. In a COVID-19 model, we investigate the Turing instability produced by fractional diffusion. To that purpose, positive equilibrium points are first specified, and then the Routh-Hurwitz criteria are used to assess the positive equilibrium point's stability. Local equilibrium points and stability analysis are employed to find the conditions for Turing instability. The amplitude equations near the Turing bifurcation point are deduced using weakly nonlinear analysis. After the application of amplitude equations, this structure has manifested highly rich dynamical properties. Regarding the amplitude equations, the dynamic analysis determines the conditions for the development of patterns such as spot, hexagon, stripe, and mixed patterns. Moreover, the theoretical effects are confirmed using numerical simulations. Within this context, this analysis, which looks into the system's dynamical behavior and the bifurcation point centered on the death rate, will serve as a leverage for further studies in different disciplines concerning COVID-19 model through the lenses of distinct viewpoints. Based on modeling as regards complex and heterogeneous materials, fractional system ensures the formation of patterns by identifying the specific and required significant attributes of complexity that convey information in terms of dynamical behavior. As a result of the analyses which reveal the highly complex connection between COVID-19 and fractional-order diffusion, the Turing bifurcation point and weakly nonlinear analysis used in the fractional-order dynamics discussed in this study are critically important on a quantitative basis owing to the fact that the results can be applied and extended to a variety of statistical, physical, engineering, biological and other related further models.