Stability and Dynamical Analysis of Generalized Tumor-Immune Interaction Model with Conformable Fractional Derivative

2021 ◽  
Author(s):  
Rimpi Pal ◽  
Afroz ◽  
Ayub Khan ◽  
MOHMAD AUSIF PADDER
Keyword(s):  
Stability Analysis ◽  
Fixed Points ◽  
Fractional Order ◽  
Interaction Model ◽  
Tumor Model ◽  
Graphical Analysis ◽  
Dynamical Analysis ◽  
Complex Behaviour ◽  
The Stability ◽  
Two Parameters

Abstract Fractional order tumor-immune interaction models are being frequently used for understanding the complex behaviour of immune system and tumor growth. In this paper, a generalized fractional order tumor-immune interaction model has been developed by introducing immunotherapy (IL2) as third variable in the model. The study of generalized model is done by using conformable fractional order derivative. The stability analysis is done for both fractional order tumor model and its conformable fractional order version. By considering some biological fixed points for both versions of the model, the stability analysis around these fixed points shows that both the systems are stable at some fixed point under some stability conditions, which are defined in the model analysis. The numerical and graphical analysis is also done for both the systems by varying two parameters and keeping other parameters fixed for better understanding the dynamics of proposed model.

Numerical analysis of fractional-order tumor model

2015 ◽  
Vol 08 (05) ◽  
pp. 1550069 ◽  
Author(s):  
Ayesha Sohail ◽  
Sadia Arshad ◽  
Sana Javed ◽  
Khadija Maqbool
Keyword(s):  
Mathematical Model ◽  
Immune Response ◽  
Fractional Order ◽  
Interleukin 2 ◽  
Tumor Model ◽  
Graphical Analysis ◽  
System A ◽  
The Stability ◽  
The Mathematical Model ◽  
Limiting Values

In this paper, the tumor-immune dynamics are simulated by solving a nonlinear system of differential equations. The fractional-order mathematical model incorporated with three Michaelis–Menten terms to indicate the saturated effect of immune response, the limited immune response to the tumor and to account the self-limiting production of cytokine interleukin-2. Two types of treatments were considered in the mathematical model to demonstrate the importance of immunotherapy. The limiting values of these treatments were considered, satisfying the stability criteria for fractional differential system. A graphical analysis is made to highlight the effects of antigenicity of the tumor and the fractional-order derivative on the tumor mass.

A Survey of Fractional-Order Neural Networks

2017 ◽  
Author(s):  
Shuo Zhang ◽  
YangQuan Chen ◽  
Yongguang Yu
Keyword(s):  
Neural Networks ◽  
Stability Analysis ◽  
Fractional Order ◽  
Computational Science ◽  
General Introduction ◽  
Analysis Methods ◽  
Recent Trends ◽  
The Stability

In this paper, the literature of fractional-order neural networks is categorized and discussed, which includes a general introduction and overview of fractional-order neural networks. Various application areas of fractional-order neural networks have been found or used, and will be surveyed and summarized such as neuroscience, computational science, control and optimization. Recent trends in dynamics of fractional-order neural networks are presented and discussed. The results, especially the stability analysis of fractional-order neural networks, are reviewed and different analysis methods are compared. Furthermore, the challenges and conclusions of fractional-order neural networks are given.

scholarly journals Complex Canard Explosion in a Fractional-Order FitzHugh–Nagumo Model

2019 ◽  
Vol 29 (08) ◽  
pp. 1950111 ◽  
Author(s):  
Mohammed-Salah Abdelouahab ◽  
René Lozi ◽  
Guanrong Chen
Keyword(s):  
Dynamical System ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Search Algorithm ◽  
Two Dimensional ◽  
Mixed Mode Oscillations ◽  
The Stability ◽  
Two Parameters ◽  
Complex Phenomena ◽  
Autonomous Dynamical System

This article investigates the complex phenomena of canard explosion with mixed-mode oscillations, observed from a fractional-order FitzHugh–Nagumo (FFHN) model. To rigorously analyze the dynamics of the FFHN model, a new mathematical notion, referred to as Hopf-like bifurcation (HLB), is introduced. HLB provides a precise definition for the change between a fixed point and an [Formula: see text]-asymptotically [Formula: see text]-periodic solution of the fractional-order dynamical system, as well as the stability of the FFHN model and the appearance of the HLB. The existence of canard oscillations in the neighborhoods of such HLB points are numerically investigated. Using a new algorithm, referred to as the global-local canard explosion search algorithm, the appearance of various patterns of solutions is revealed, with an increasing number of small-amplitude oscillations when two key parameters of the FFHN model are varied. The numbers of such oscillations versus the two parameters, respectively, are perfectly fitted using exponential functions. Finally, it is conjectured that chaos could occur in a two-dimensional fractional-order autonomous dynamical system, with the fractional order close to one. After all, the article demonstrates that the FFHN model is a very simple two-dimensional model with an incredible ability to present the complex dynamics of neurons.

Dynamical analysis of fractional-order differentiated Cournot triopoly game

2020 ◽  
Vol 26 (15-16) ◽  
pp. 1367-1380
Author(s):  
Abdulrahman Al-khedhairi
Keyword(s):  
Stability Analysis ◽  
Fractional Order ◽  
Complex Dynamics ◽  
Equilibrium Points ◽  
Sufficient Conditions ◽  
Dynamical Analysis ◽  
Periodic Forcing ◽  
Nash Equilibrium Point ◽  
Dynamical Behaviors ◽  
Theoretical Results

The objective of the article is to study the dynamics of the proposed fractional-order Cournot triopoly game. Sufficient conditions for the existence and uniqueness of the triopoly game solution are obtained. Stability analysis of equilibrium points of the fractional-order game is also discussed. The conditions for the presence of Nash equilibrium point along with its global stability analysis are studied. The interesting dynamical behaviors of the arbitrary-order Cournot triopoly game are discussed. Moreover, the effects of seasonal periodic forcing on the game’s behaviors are examined. The 0–1 test is used to distinguish between regular and irregular dynamics of system behaviors. Numerical analysis is used to verify the theoretical results that are obtained, and revealed that the nonautonomous fractional-order model induces more complicated dynamics in the Cournot triopoly game behavior and the seasonally forced game exhibits more complex dynamics than the unforced one.

scholarly journals Dynamical analysis of autonomous system in entropic cosmology

2016 ◽  
Vol 94 (5) ◽  
pp. 458-465
Author(s):  
Yu Li
Keyword(s):  
Fixed Points ◽  
Autonomous System ◽  
Constant Term ◽  
Accelerated Expansion ◽  
Dynamical Analysis ◽  
Stability Properties ◽  
Expansion Of The Universe ◽  
Dynamical Properties ◽  
The Stability ◽  
The Universe

The entropic cosmology model is an alternative method to explain the accelerated expansion of the universe. In this paper, we discuss the dynamical system in two types of entropic cosmology model: Λ(t) type and bulk viscous type. We found that the stability properties of fixed points are affected by the H2 term, while the H term and constant term have no influence on stability properties of fixed points. We also found that the dynamical properties of the C-version model are the same as the H-version model.

scholarly journals Stability Analysis for a Fractional HIV Infection Model with Nonlinear Incidence

2015 ◽  
Vol 2015 ◽  
pp. 1-11 ◽  
Author(s):  
Linli Zhang ◽  
Gang Huang ◽  
Anping Liu ◽  
Ruili Fan
Keyword(s):  
Population Dynamics ◽  
Stability Analysis ◽  
Hiv Infection ◽  
Numerical Simulations ◽  
Fractional Order ◽  
Nonnegative Solution ◽  
Infection Model ◽  
Nonlinear Incidence ◽  
Hiv Infection Model ◽  
The Stability

We introduce the fractional-order derivatives into an HIV infection model with nonlinear incidence and show that the established model in this paper possesses nonnegative solution, as desired in any population dynamics. We also deal with the stability of the infection-free equilibrium, the immune-absence equilibrium, and the immune-presence equilibrium. Numerical simulations are carried out to illustrate the results.

Stability analysis of conformable fractional-order nonlinear systems depending on a parameter

2020 ◽  
Vol 26 (2) ◽  
pp. 287-296
Author(s):  
O. Naifar ◽  
G. Rebiai ◽  
A. Ben Makhlouf ◽  
M. A. Hammami ◽  
A. Guezane-Lakoud
Keyword(s):  
Stability Analysis ◽  
Nonlinear Systems ◽  
Fractional Order ◽  
Feedback Controller ◽  
Fractional Order Systems ◽  
The Stability

AbstractIn this paper, the stability of conformable fractional-order nonlinear systems depending on a parameter is presented and described. Furthermore, The design of a feedback controller for the same class of conformable fractional-order systems is introduced. Illustrative examples are given at the end of the paper to show the effectiveness of the proposed results.

scholarly journals Numerical Analysis of a Fractional Order Discrete Prey – Predator System with Functional Response

2018 ◽  
Vol 7 (4.10) ◽  
pp. 681 ◽  
Author(s):  
A. George Maria Selvam ◽  
R. Janagaraj
Keyword(s):  
Numerical Analysis ◽  
Fixed Points ◽  
Functional Response ◽  
Fractional Order ◽  
Periodic Oscillations ◽  
Numerical Examples ◽  
Discrete Equations ◽  
Predator Interactions ◽  
The Stability ◽  
Predator System

This study presents numerical examples of Discrete Fractional Order Prey Predator interactions with Functional Response. The process of discretization is applied and the version of discrete equations is obtained. Fixed points are determined and the stability around the fixed points is analyzed. Also the theoretical analysis has been verified from the numerical simulations, which help better understanding of the proposed system. Rich dynamics of system is exhibited by Bifurcation diagram and Periodic Oscillations for suitable parameters values.  

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