Numerical Simulations of Fractional Order Nonlinear Dynamical Systems

Abstract

Mathematical models play a role in analyzing and control infectious diseases in a population. These models construction clarifies assumptions, variables and parameters, and provide conceptual insights such as thresholds and basic reproduction numbers for various infectious diseases. Some very important theories are built and tested, some quantitative speculations are made and some specific questions are answered with the help of mathematical models. This leads to a better strategy for overcoming the transmission of diseases. For the last twenty years, chaos theory has brought about a valuable association between mathematicians and researchers in bio-medical sciences. Such association has described a biomedical system with ordinary and fractional order mathematical model usually consists of a nonlinear ordinary or fractional order differential equation or system of non-linear ordinary or fractional order differential equations. The fractional order mathematical model is used to predict the behavior of corresponding bio-medical system. The model must be investigated to guarantee that it does not foresee chaos in the bio-medical system under examination, when chaos is not actually present in the system. The mathematician must further confirm that any method used to solve the fractional order mathematical model does not envisage chaos when chaos is not a feature of the bio-medical system. The contrived chaos can be avoided and stability can be retained using implicit methods instead of using explicit numerical methods. In recent years, fractional differential equations have become one of the most important topics in mathematics and have received much consideration and growing curiosity due to the options of unfolding nonlinear systems and due to their prospective applications in physics, control theory, and engineering. The generalization is obtained by changing the ordinary derivative with the fractional order derivative. The benefit of fractional differential equation systems is that they allow greater degrees of freedom and incorporate the memory effect in the model. Due to this fact, they were introduced in epidemiological modeling systems. The main reason for using integer order models was the absence of solution methods for fractional differential equations. Various applications, like in the reaction kinetics of proteins, the anomalous electron transport in amorphous materials, the dielectrical or mechanical relation of polymers, the modeling of glass forming liquids and others, are successfully performed in numerous research works. The physical and geometrical meaning of the non-integer integral containing the real and complex conjugate power-law exponent has been proposed. Since integer order differential equations cannot precisely describe the experimental and field measurement data, as an alternative approach, non-integer order differential equation models are now being widely applied. The advantage of fractional-order differential equation systems over ordinary differential equation systems is that they allow greater degrees of freedom and incorporate memory effect in the model. In other words, they provide an excellent tool for the description of memory and hereditary properties which were not taken into account in the classical integer order model. In the present research work, we developed and investigated fractional order numerical techniques for the solution of fractional order models for infectious diseases, whose fixed points will be seen to be the same as the critical points of model equations and to have the same stability properties. These techniques will numerically analyze the behavior of solution of the fractional order models, stability analysis of the steady states and threshold criteria for the epidemics. The proposed techniques may be used with arbitrarily fractional order, thus making them more economical to use when integrating for arbitrary fractional order and may preserve all the essential properties like dynamical consistency, positivity and boundedness, of the corresponding fractional order dynamical systems.