scholarly journals A new mathematical model for Zika virus transmission

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shahram Rezapour ◽  
Hakimeh Mohammadi ◽  
Amin Jajarmi
Keyword(s):  
Mathematical Model ◽  
Fractional Order ◽  
Zika Virus ◽  
Fixed Point Theory ◽  
Equilibrium Points ◽  
Virus Transmission ◽  
Point Theory ◽  
Euler Method ◽  
Order Derivative ◽  
The Stability

Abstract We present a new mathematical model for the transmission of Zika virus between humans as well as between humans and mosquitoes. In this way, we use the fractional-order Caputo derivative. The region of the feasibility of system and equilibrium points are calculated, and the stability of equilibrium point is investigated. We prove the existence of a unique solution for the model by using the fixed point theory. By using the fractional Euler method, we get an approximate solution to the model. Numerical results are presented to investigate the effect of fractional derivative on the behavior of functions and also to compare the integer-order derivative and fractional-order derivative results.

scholarly journals A study on the AH1N1/09 influenza transmission model with the fractional Caputo–Fabrizio derivative

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shahram Rezapour ◽  
Hakimeh Mohammadi
Keyword(s):  
Fractional Order ◽  
Fixed Point Theory ◽  
Equilibrium Points ◽  
Point Theory ◽  
Euler Method ◽  
Transmission Model ◽  
Order Model ◽  
Influenza Transmission ◽  
The Stability ◽  
Disease Free

Abstract We study the SEIR epidemic model for the spread of AH1N1 influenza using the Caputo–Fabrizio fractional-order derivative. The reproduction number of system and equilibrium points are calculated, and the stability of the disease-free equilibrium point is investigated. We prove the existence of solution for the model by using fixed point theory. Using the fractional Euler method, we get an approximate solution to the model. In the numerical section, we present a simulation to examine the system, in which we calculate equilibrium points of the system and examine the behavior of the resulting functions at the equilibrium points. By calculating the results of the model for different fractional order, we examine the effect of the derivative order on the behavior of the resulting functions and obtained numerical values. We also calculate the results of the integer-order model and examine their differences with the results of the fractional-order model.

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 SEIR epidemic model for COVID-19 transmission by Caputo derivative of fractional order

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shahram Rezapour ◽  
Hakimeh Mohammadi ◽  
Mohammad Esmael Samei
Keyword(s):  
Epidemic Model ◽  
Fixed Point Theory ◽  
Equilibrium Points ◽  
Real Data ◽  
Point Theory ◽  
Euler Method ◽  
Simulation Based ◽  
The World ◽  
Feasibility Region ◽  
The Stability

Abstract We provide a SEIR epidemic model for the spread of COVID-19 using the Caputo fractional derivative. The feasibility region of the system and equilibrium points are calculated and the stability of the equilibrium points is investigated. We prove the existence of a unique solution for the model by using fixed point theory. Using the fractional Euler method, we get an approximate solution to the model. To predict the transmission of COVID-19 in Iran and in the world, we provide a numerical simulation based on real data.

scholarly journals Modeling and analysis of fractional order Zika model

2021 ◽  
Vol 7 (3) ◽  
pp. 3912-3938
Author(s):  
Muhammad Farman ◽  
◽  
Ali Akgül ◽  
Sameh Askar ◽  
Thongchai Botmart ◽  
...  
Keyword(s):  
Mathematical Model ◽  
Fixed Point ◽  
Control Strategy ◽  
Zika Virus ◽  
Fixed Point Theory ◽  
Point Theory ◽  
Fractional Operator ◽  
Modeling And Analysis ◽  
Virus Model ◽  
Positivity And Boundedness

<abstract> <p>We propose mathematical model for the transmission of the Zika virus for humans spread by mosquitoes. We construct a scheme for the Zika virus model with Atangna-Baleanue Caputo sense and fractal fractional operator by using generalized Mittag-Leffler kernel. The positivity and boundedness of the model are also calculated. The existence of uniquene solution is derived and stability analysis has been made for the model by using the fixed point theory. Numerical simulations are made by using the Atangana-Toufik scheme and fractal fractional operator with a different dimension of fractional values which support the theoretical outcome of the proposed system. Developed scheme including simulation will provide better understanding in future analysis and for control strategy regarding Zika virus.</p> </abstract>

scholarly journals Modeling, analysis and numerical solution to malaria fractional model with temporary immunity and relapse

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Attiq ul Rehman ◽  
Ram Singh ◽  
Thabet Abdeljawad ◽  
Eric Okyere ◽  
Liliana Guran
Keyword(s):  
Fractional Order ◽  
Equilibrium Points ◽  
Order Derivative ◽  
Fractional Order Derivative ◽  
Modeling Analysis ◽  
Proposed Model ◽  
Adomian Decomposition ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Temporary Immunity

AbstractThe present paper deals with a fractional-order mathematical epidemic model of malaria transmission accompanied by temporary immunity and relapse. The model is revised by using Caputo fractional operator for the index of memory. We also recommend the utilization of temporary immunity and the possibility of relapse. The theory of locally bounded and Lipschitz is employed to inspect the existence and uniqueness of the solution of the malaria model. It is shown that temporary immunity has a great effect on the dynamical transmission of host and vector populations. The stability analysis of these equilibrium points for fractional-order derivative α and basic reproduction number $\mathcal{R}_{0}$ R 0 is discussed. The model will exhibit a Hopf-type bifurcation. The two control variables are introduced in this model to decrease the number of populations. Mandatory conditions for the control problem are produced. Two types of numerical method via Laplace Adomian decomposition and Runge–Kutta of fourth order for simulating the proposed model with fractional-order derivative are presented. To validate the mathematical results, numerical simulations, sensitivity analysis, convergence analysis, and other important studies are given. The paper is finished with some conclusions and discussion.

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 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 Class of Fractional-Order Chaotic or Hyperchaotic Systems in the Context of the Caputo Fractional-Order Derivative

2020 ◽  
Vol 2020 ◽  
pp. 1-15
Author(s):  
Ndolane Sene ◽  
Ameth Ndiaye
Keyword(s):  
Lyapunov Exponents ◽  
Fractional Order ◽  
Initial Conditions ◽  
Equilibrium Points ◽  
Phase Portraits ◽  
Order Derivative ◽  
Order System ◽  
Fractional Order Systems ◽  
Fractional Order Derivative ◽  
The Stability

In this paper, we consider a class of fractional-order systems described by the Caputo derivative. The behaviors of the dynamics of this particular class of fractional-order systems will be proposed and experienced by a numerical scheme to obtain the phase portraits. Before that, we will provide the conditions under which the considered fractional-order system’s solution exists and is unique. The fractional-order impact will be analyzed, and the advantages of the fractional-order derivatives in modeling chaotic systems will be discussed. How the parameters of the model influence the considered fractional-order system will be studied using the Lyapunov exponents. The topological changes of the systems and the detection of the chaotic and hyperchaotic behaviors at the assumed initial conditions and the considered fractional-order systems will also be investigated using the Lyapunov exponents. The investigations related to the Lyapunov exponents in the context of the fractional-order derivative will be the main novelty of this paper. The stability analysis of the model’s equilibrium points has been focused in terms of the Matignon criterion.

A malaria model with Caputo–Fabrizio and Atangana–Baleanu derivatives

2021 ◽  
pp. 2150013
Author(s):  
Hamadjam Abboubakar ◽  
Pushpendra Kumar ◽  
Norodin A. Rangaig ◽  
Sachin Kumar
Keyword(s):  
Fixed Point Theory ◽  
Point Theory ◽  
Euler Method ◽  
Lyapunov Theory ◽  
Stability And Convergence ◽  
Uniqueness Of Solutions ◽  
Fractional Models ◽  
Number Formula ◽  
The Stability ◽  
Malaria Model

In this paper, we study two fractional models in the Caputo–Fabrizio sense and Atangana–Baleanu sense, in which the effects of malaria infection on mosquito biting behavior and attractiveness of humans are considered. Using Lyapunov theory, we prove the global asymptotic stability of the unique endemic equilibrium of the integer-order model, and the fractional models, whenever the basic reproduction number [Formula: see text] is greater than one. By using fixed point theory, we prove existence, and conditions of the uniqueness of solutions, as well as the stability and convergence of numerical schemes. Numerical simulations for both models, using fractional Euler method and Adams–Bashforth method, respectively, are provided to confirm the effectiveness of used approximation methods for different values of the fractional-order [Formula: see text].

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