scholarly journals Study of HIV Disease and Its Association with Immune Cells under Nonsingular and Nonlocal Fractal-Fractional Operator

Complexity ◽  
2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Shabir Ahmad ◽  
Aman Ullah ◽  
Ali Akgül ◽  
Manuel De la Sen
Keyword(s):  
Fractional Order ◽  
Primary Infection ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Hiv Model ◽  
Proposed Model ◽  
Lethal Infection ◽  
Uniqueness Of The Solution ◽  
Fractional Orders

HIV, like many other infections, is a severe and lethal infection. Fractal-fractional operators are frequently used in modeling numerous physical processes in the current decade. These operators provide better dynamics of a mathematical model because these are the generalization of integer and fractional-order operators. This paper aims to study the dynamics of the HIV model during primary infection by fractal-fractional Atangana–Baleanu (AB) operators. The sufficient conditions for the existence and uniqueness of the solution of the proposed model under the AB operator are derived via fixed point theory. The numerical scheme is presented by using the Adams–Bashforth method. Numerical results are demonstrated for different fractal and fractional orders to see the effect of fractional order and fractal dimension on the dynamics of HIV and CD4+ T-cells during primary infection.

scholarly journals Study of a Fractional-Order Epidemic Model of Childhood Diseases

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
Aman Ullah ◽  
Thabet Abdeljawad ◽  
Shabir Ahmad ◽  
Kamal Shah
Keyword(s):  
Fixed Point ◽  
Fractional Order ◽  
Epidemic Model ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Childhood Diseases ◽  
Proposed Model ◽  
Adomian Decomposition ◽  
Uniqueness Of The Solution

In this article, we discuss the existence and uniqueness of the solution of the fractional-order epidemic model of childhood diseases by using fixed point theory. The technique of natural transform coupled with the Adomian decomposition is used to find the solution of the proposed model. At the end of the article, the model is demonstrated with appropriate numerical and graphical description.

FRACTIONAL ORDER ANALYSIS OF HBV AND HCV CO-INFECTION UNDER ABC DERIVATIVE

Fractals ◽  
2021 ◽  
Author(s):  
HUSSAM ALRABAIAH ◽  
MATI UR RAHMAN ◽  
IBRAHIM MAHARIQ ◽  
SAMIA BUSHNAQ ◽  
MUHAMMAD ARFAN
Keyword(s):  
Mathematical Model ◽  
Fixed Point ◽  
Approximate Solution ◽  
Fractional Order ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Comparative Result ◽  
Order Analysis ◽  
The Stability ◽  
Fractional Orders

In this paper, we consider a fractional mathematical model describing the co-infection of HBV and HCV under the non-singular Mittag-Leffler derivative. We also investigate the qualitative analysis for at least one solution and a unique solution by applying the approach fixed point theory. For an approximate solution, the technique of the iterative fractional order Adams–Bashforth scheme has been implemented. The simulation for the proposed scheme has been drawn at various fractional order values lying between (0,1) and integer-order of 1 via using Matlab. All the compartments have shown convergence and stability with time. A detailed comparative result has been given by the different fractional orders, which showed that the stability was achieved more rapidly at low orders.

scholarly journals Existence Theory and Novel Iterative Method for Dynamical System of Infectious Diseases

2020 ◽  
Vol 2020 ◽  
pp. 1-11
Author(s):  
Gauhar Ali ◽  
Ghazala Nazir ◽  
Kamal Shah ◽  
Yongjin Li
Keyword(s):  
Dynamical System ◽  
Existence And Uniqueness ◽  
Integral Transform ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Point Theory ◽  
Qualitative Theory ◽  
Existence Theory ◽  
Uniqueness Of The Solution ◽  
Respective Theory

This manuscript is devoted to investigate qualitative theory of existence and uniqueness of the solution to a dynamical system of an infectious disease known as measles. For the respective theory, we utilize fixed point theory to construct sufficient conditions for existence and uniqueness of the solution. Some results corresponding to Hyers–Ulam stability are also investigated. Furthermore, some semianalytical results are computed for the considered system by using integral transform due to the Laplace and decomposition technique of Adomian. The obtained results are presented by graphs also.

scholarly journals Analysis of logistic equation pertaining to a new fractional derivative with non-singular kernel

2017 ◽  
Vol 9 (2) ◽  
pp. 168781401769006 ◽  
Author(s):  
Devendra Kumar ◽  
Jagdev Singh ◽  
Maysaa Al Qurashi ◽  
Dumitru Baleanu
Keyword(s):  
Numerical Simulation ◽  
Fixed Point ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Existence And Uniqueness ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Logistic Equation ◽  
Iterative Technique ◽  
Uniqueness Of The Solution

In this work, we aim to analyze the logistic equation with a new derivative of fractional order termed in Caputo–Fabrizio sense. The logistic equation describes the population growth of species. The existence of the solution is shown with the help of the fixed-point theory. A deep analysis of the existence and uniqueness of the solution is discussed. The numerical simulation is conducted with the help of the iterative technique. Some numerical simulations are also given graphically to observe the effects of the fractional order derivative on the growth of population.

scholarly journals Fractal–fractional and stochastic analysis of norovirus transmission epidemic model with vaccination effects

2021 ◽  
Vol 11 (1) ◽  
Author(s):  
Ting Cui ◽  
Peijiang Liu ◽  
Anwarud Din
Keyword(s):  
Epidemic Model ◽  
Fixed Point Theory ◽  
Sufficient Conditions ◽  
Stochastic Perturbation ◽  
Point Theory ◽  
Fractional Dynamics ◽  
Contagious Diseases ◽  
Proposed Model ◽  
Global Positive Solution ◽  
Definition Of

AbstractIn this paper, we investigate an norovirus (NoV) epidemic model with stochastic perturbation and the new definition of a nonlocal fractal–fractional derivative in the Atangana–Baleanu–Caputo (ABC) sense. First we present some basic properties including equilibria and the basic reproduction number of the model. Further, we analyze that the proposed stochastic system has a unique global positive solution. Next, the sufficient conditions of the extinction and the existence of a stationary probability measure for the disease are established. Furthermore, the fractal–fractional dynamics of the proposed model under Atangana–Baleanu–Caputo (ABC) derivative of fractional order “$${p}$$ p ” and fractal dimension “$${q}$$ q ” have also been addressed. Besides, coupling the non-linear functional analysis with fixed point theory, the qualitative analysis of the proposed model has been performed. The numerical simulations are carried out to demonstrate the analytical results. It is believed that this study will comprehensively strengthen the theoretical basis for comprehending the dynamics of the worldwide contagious diseases.

A conformable mathematical model for alcohol consumption in Spain

2019 ◽  
Vol 12 (05) ◽  
pp. 1950057 ◽  
Author(s):  
Aqsa Nazir ◽  
Naveed Ahmed ◽  
Umar Khan ◽  
Syed Tauseef Mohyud-Din
Keyword(s):  
Alcohol Consumption ◽  
Fractional Order ◽  
Fixed Point Theory ◽  
Sharp Decrease ◽  
Variational Iteration Method ◽  
Point Theory ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Analytical Technique ◽  
Uniqueness Of The Solution

A study on the conformable model of alcohol consumption in Spain has been presented. For the proposed model, the existence as well as the uniqueness of the solution has been discussed with the help of fixed-point theory. An analytical technique, Variational Iteration Method (VIM), has been used to obtain the solution to the governing system of differential equations. With the help of suitable plots, the role of fractional order derivative has been highlighted. For decreasing values of fractional order derivative, decrease in the number of non-consumers and non-risk consumers has been observed. By increasing the value of fractional order derivative, a sharp decrease can be seen in the compartment of risk-consumers. The agreement between the current study and the already existing studies, with ordinary derivatives, has also been pointed out.

scholarly journals On analysis of a nonlinear fractional system for social media addiction involving Atangana–Baleanu–Caputo derivative

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Jutarat Kongson ◽  
Weerawat Sudsutad ◽  
Chatthai Thaiprayoon ◽  
Jehad Alzabut ◽  
Chutarat Tearnbucha
Keyword(s):  
Social Media ◽  
Fixed Point Theory ◽  
Equilibrium Points ◽  
Point Theory ◽  
Social Media Addiction ◽  
Proposed Model ◽  
Reproduction Numbers ◽  
The Stability ◽  
Predictor Corrector ◽  
Fractional Orders

AbstractA mathematical model for the dynamic systems of $\mathbb{SMA}$ SMA involving the $\mathbb{ABC}$ ABC -fractional derivative is considered in this manuscript. We examine the basic reproduction number and analyze the stability of the equilibrium points. We prove the theoretical results of the existence and Ulam’s stability of the solutions for the proposed model using fixed point theory and nonlinear analytic techniques. Using the Adams type predictor–corrector rule for the $\mathbb{ABC}$ ABC -fractional integral operator, a numerical scheme is devised for obtaining the approximate solution of the proposed model. Different numerical plots corresponding to various fractional orders are presented. In addition, we demonstrate a numerical simulation for the transmission of social media addiction in two cases with the basic reproduction numbers greater than and less than one.

scholarly journals Mathematical Modeling and Forecasting of COVID-19 in Saudi Arabia under Fractal-Fractional Derivative in Caputo Sense with Power-Law

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 228
Author(s):  
Mdi Begum Jeelani ◽  
Abeer S. Alnahdi ◽  
Mohammed S. Abdo ◽  
Mansour A. Abdulwasaa ◽  
Kamal Shah ◽  
...  
Keyword(s):  
Saudi Arabia ◽  
Fractional Order ◽  
Power Law ◽  
Reproduction Number ◽  
Fixed Point Theory ◽  
Fractal Dimensions ◽  
Real Data ◽  
Point Theory ◽  
Modeling And Forecasting ◽  
Uniqueness Of The Solution

This manuscript is devoted to investigating a fractional-order mathematical model of COVID-19. The corresponding derivative is taken in Caputo sense with power-law of fractional order μ and fractal dimension χ. We give some detailed analysis on the existence and uniqueness of the solution to the proposed problem. Furthermore, some results regarding basic reproduction number and stability are given. For the proposed theoretical analysis, we use fixed point theory while for numerical analysis fractional Adams–Bashforth iterative techniques are utilized. Using our numerical scheme is verified by using some real values of the parameters to plot the approximate solution to the considered model. Graphical presentations corresponding to different values of fractional order and fractal dimensions are given. Moreover, we provide some information regarding the real data of Saudi Arabia from 1 March 2020 till 22 April 2021, then calculated the fatality rates by utilizing the SPSS, Eviews and Expert Modeler procedure. We also built forecasts of infection for the period 23 April 2021 to 30 May 2021, with 95% confidence.

scholarly journals Study of a Fractal-Fractional Smoking Models with Relapse and Harmonic Mean Type Incidence Rate

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Zareen A. Khan ◽  
Mati ur Rahman ◽  
Kamal Shah
Keyword(s):  
Fixed Point ◽  
Fractal Dimension ◽  
Incidence Rate ◽  
Fractional Order ◽  
Caputo Derivative ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Harmonic Mean ◽  
The Stability ◽  
Fractional Orders

This manuscript investigates fractal-fractional order smoking models with relapse and harmonic mean type incidence rate under the Caputo derivative. We derive the existence and unique results about the solution for the considered model via fixed point theory. For the stability of the considered system, Ulam-Hyers (UH) approach is used. We compute the numerical solution by using fractional Adams-Bashforth method. For the simulation of the model, we consider different values of fractional order δ and fractal dimension θ by using some real values of the parameters. The proposed scheme is used to simulate the available data for some smoking community including potential, light, and quit smokers. Various graphical presentations are given to understand the dynamics of the model at various fractional orders.

scholarly journals ON ANALYSIS OF FRACTIONAL ORDER MATHEMATICAL MODEL OF HEPATITIS B USING ATANGANA–BALEANU CAPUTO (ABC) DERIVATIVE

Fractals ◽  
2021 ◽  
pp. 2240017 ◽  
Author(s):  
ANWARUD DIN ◽  
YONGJIN LI ◽  
FAIZ MUHAMMAD KHAN ◽  
ZIA ULLAH KHAN ◽  
PEIJIANG LIU
Keyword(s):  
Hepatitis B ◽  
Fractional Order ◽  
Fixed Point Theory ◽  
Nonlinear Problems ◽  
Point Theory ◽  
Scaling Exponent ◽  
Candidate Solution ◽  
Acute Infections ◽  
Proposed Model ◽  
Adomian Polynomial

The scaling exponent of a hierarchy of cities used to be regarded as a fractional. This paper investigates a newly constructed system of equation for Hepatitis B disease in sense of Atanganaa–Baleanu Caputo (ABC) fractional order derivative. The proposed approach has five distinctive quantities, namely, susceptible, acute infections, chronic infection, immunized and vaccinated populace. By applying some well-known results of fixed point theory, we find the Ulam–Hyers type stability and qualitative analysis of the candidate solution. The deterministic stability for the proposed system is also computed. We apply well-known transform due to Laplace and decomposition techniques (LADM) and Adomian polynomial for nonlinear terms for computing the series solution for the proposed model. Graphical results show that LADM is an efficient and robust method for solving nonlinear problems.

Sign in / Sign up

Export Citation Format

Share Document