COMPLEX FRACTIONAL-ORDER HIV DIFFUSION MODEL BASED ON AMPLITUDE EQUATIONS WITH TURING PATTERNS AND TURING INSTABILITY

Abstract

A weakly nonlinear analysis provides a system constituting amplitude equations and its related analysis is capable of predicting parameter regimes with different patterns expected to co-exist in dynamical circumstances that exhibit complex fractional-order system characteristics. The Turing mechanism of pattern formation as a result of diffusion-induced instability of the homogeneous steady state is concerned with unpredictable conditions. The Turing instability caused by fractional diffusion in a Human Immunodeficiency Virus model has been addressed in this study. It is important that the effect of the Human Immunodeficiency Virus to the immune system can be modeled by the interaction of uninfected cells, unhealthy cells, virus particles and antigen-specific. Initially, all potential equilibrium points are defined and the stability of the interior equilibrium point is then evaluated using the Routh–Hurwitz criteria. The conditions for Turing instability are obtained by local equilibrium points with stability analysis. In the neighborhood of the Turing bifurcation point, weakly nonlinear analysis is employed to deduce the amplitude equations. After applying amplitude equations, it is observed that this system has a very rich dynamical behavior. The constraints for the formation of the patterns like a hexagon, spot, mixed and stripe patterns are identified for the amplitude equations by dynamic analysis. Furthermore, by using the numerical simulations, the theoretical results are verified. Within this framework, this study through the dynamical behavior of the complex system perspective and bifurcation point based on the viral death rate can provide the basis for several researchers working on Human Immunodeficiency Virus model through various aspects. Accordingly, the Turing bifurcation point and weakly nonlinear analysis employed within the complex fractional-order dynamics addressed herein are highly relevant experimentally since the related effects can be studied and applied concerning different mathematical, physical, engineering and biological models.