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.

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.

scholarly journals Control and Synchronization of the Fractional-Order Lorenz Chaotic System via Fractional-Order Derivative

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ping Zhou ◽  
Rui Ding
Keyword(s):  
Fractional Order ◽  
Chaotic System ◽  
Stability Theory ◽  
Chaos Synchronization ◽  
Equilibrium Points ◽  
Order Derivative ◽  
Fractional Order Systems ◽  
Fractional Order Derivative ◽  
Lorenz Chaotic System ◽  
Control And Synchronization

The unstable equilibrium points of the fractional-order Lorenz chaotic system can be controlled via fractional-order derivative, and chaos synchronization for the fractional-order Lorenz chaotic system can be achieved via fractional-order derivative. The control and synchronization technique, based on stability theory of fractional-order systems, is simple and theoretically rigorous. The numerical simulations demonstrate the validity and feasibility of the proposed method.

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 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.

Stability analysis of a fractional-order cancer model with chaotic dynamics

2021 ◽  
pp. 2150046
Author(s):  
Parvaiz Ahmad Naik ◽  
Jian Zu ◽  
Mehraj-ud-din Naik
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Chaotic Behavior ◽  
Equilibrium Points ◽  
Three Dimensional ◽  
Cancer Model ◽  
Effector Cells ◽  
Proposed Model ◽  
Uniqueness Of The Solution ◽  
Theoretical Results

In this paper, we develop a three-dimensional fractional-order cancer model. The proposed model involves the interaction among tumor cells, healthy tissue cells and activated effector cells. The detailed analysis of the equilibrium points is studied. Also, the existence and uniqueness of the solution are investigated. The fractional derivative is considered in the Caputo sense. Numerical simulations are performed to illustrate the effectiveness of the obtained theoretical results. The outcome of the study reveals that the order of the fractional derivative has a significant effect on the dynamic process. Further, the calculated Lyapunov exponents give the existence of chaotic behavior of the proposed model. Also, it is observed from the obtained results that decrease in fractional-order [Formula: see text] increases the chaotic behavior of the model.

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.

scholarly journals A Non-Integer Variable Order Mathematical Model of Human Immunodeficiency Virus and Malaria Coinfection with Time Delay

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-13 ◽  
Author(s):  
A. A. M. Arafa ◽  
M. Khalil ◽  
A. Sayed
Keyword(s):  
Time Delay ◽  
Fractional Order ◽  
Order Derivative ◽  
Variable Order ◽  
Fractional Order Derivative ◽  
Proposed Model ◽  
Immunodeficiency Virus ◽  
Predictor Corrector ◽  
Disease Free ◽  
Hiv Aids

The purpose of this paper is to propose a variable fractional-order model with a constant time delay of the coinfection of HIV/AIDS and malaria. The proposed model describes the interaction between HIV/AIDS and malaria. This model is presented by using variable fractional-order derivative which is an extension of the constant fractional-order derivative to explain a certain pattern in the development of infection of several patients. The presented model has been solved numerically via the predictor-corrector scheme. The local and global stability conditions of the disease-free equilibrium are investigated. Also, numerical simulations are presented for different variable fractional-order derivatives in Caputo sense.

scholarly journals A Complete Model of Crimean-Congo Hemorrhagic Fever (CCHF) Transmission Cycle with Nonlocal Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Hakimeh Mohammadi ◽  
Mohammed K. A. Kaabar ◽  
Jehad Alzabut ◽  
A. George Maria Selvam ◽  
Shahram Rezapour
Keyword(s):  
Fractional Derivative ◽  
Fractional Order ◽  
Fixed Point Theory ◽  
Equilibrium Points ◽  
Hemorrhagic Fever ◽  
Order Derivative ◽  
Common Disease ◽  
Fractional Order Derivative ◽  
Crimean Congo Hemorrhagic Fever ◽  
Caputo Fractional Order Derivative

Crimean-Congo hemorrhagic fever is a common disease between humans and animals that is transmitted to humans through infected ticks, contact with infected animals, and infected humans. In this paper, we present a boxed model for the transmission of Crimean-Congo fever virus. With the help of the fixed-point theory, our proposed system model is investigated in detail to prove its unique solution. Given that the Caputo fractional-order derivative preserves the system’s historical memory, we use this fractional derivative in our modeling. The equilibrium points of the proposed system and their stability conditions are determined. Using the Euler method for the Caputo fractional-order derivative, we calculate the approximate solutions of the fractional system, and then, we present a numerical simulation for the transmission of Crimean-Congo hemorrhagic fever.

scholarly journals Fractional-Order Derivative Model of Rift Valley Fever in Urban Peridomestic Cycle

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Saul C. Mpeshe
Keyword(s):  
Fractional Order ◽  
Disease Transmission ◽  
Rift Valley ◽  
Rift Valley Fever ◽  
Reproduction Number ◽  
Order Derivative ◽  
Valley Fever ◽  
Fractional Order Derivative ◽  
Exposed Population ◽  
The Stability

Rift Valley fever is a zoonotic disease which is mainly transmitted by mosquitoes and has potential to affect humans and animals. To gain some understanding on its dynamics in an urban peridomestic cycle, a fractional-order derivative model is formulated and analysed. The basic reproduction number ℛ 0 is computed and used in analysing the stability of disease when an outbreak occurs. Numerical simulations are performed in order to the variation of each population at order α = 1,0.75 , 0.5 , and 0.25. Results from simulations show that there is an increase in susceptible and exposed population in both human and mosquitoes as the value of α decreases. The infected population decreases with a decrease in the value of α . However, a rapid increase in susceptible mosquitoes is observed just after the first 30 days and a rapid decrease in infected human and mosquitoes after the first 30 days for α = 1 . Hence, fractional-order derivative also plays a significant role in providing insight on disease transmission and dynamics.

GENERALIZED THEORY OF MAGNETO-THERMO-VISCOELASTIC SPHERICAL CAVITY PROBLEM UNDER FRACTIONAL ORDER DERIVATIVE: STATE SPACE APPROACH

2020 ◽  
Vol 9 (11) ◽  
pp. 9769-9780
Author(s):  
S.G. Khavale ◽  
K.R. Gaikwad
Keyword(s):  
State Space ◽  
Fractional Order ◽  
Order Derivative ◽  
Laplace Inversion ◽  
State Space Approach ◽  
Spherical Region ◽  
Fractional Order Derivative ◽  
Numerical Laplace Inversion ◽  
Mathcad Software ◽  
Space Approach

This paper is dealing the modified Ohm's law with the temperature gradient of generalized theory of magneto-thermo-viscoelastic for a thermally, isotropic and electrically infinite material with a spherical region using fractional order derivative. The general solution obtained from Laplace transform, numerical Laplace inversion and state space approach. The temperature, displacement and stresses are obtained and represented graphically with the help of Mathcad software.

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