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Published: 30 December 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 244-251; https://doi.org/10.53391/mmnsa.2022.020
In this paper, we focus on modelling the glucose-insulin system for the purpose of estimating glucagon, insulin, and glucose in the liver in the internal organs of the human body. A three-compartmental mathematical model is proposed. The model parameters are estimated using a nonlinear inverse optimization problem and data collected in Chad. In order to identify insulin and glucose in the liver for type 2 diabetic patients, the Sampling Importance Resampling (SIR) particle filtering algorithm is used and implemented through discretization of the developed mathematical model. The proposed mathematical model allows further investigation of the dynamic behavior of hepatic glucose, insulin, and glucagon in internal organs for type 2 diabetic patients. During periods of hyperglycemia (i.e., after meal ingestion), whereas insulin secretion is increased, glucagon secretion is reduced. The results are in agreement with empirical and clinical data and they are clinically consistent with physiological responses.
Published: 29 December 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 228-243; https://doi.org/10.53391/mmnsa.2022.019
For the sake of human health, it is crucial to investigate infectious diseases including HIV/AIDS, hepatitis, and others. Worldwide, the recently discovered new coronavirus (COVID-19) poses a serious threat. The experimental vaccination and different COVID-19 strains found around the world make the virus' spread unavoidable. In the current research, fractional order is used to study the dynamics of a nonlinear modified COVID-19 SEIR model in the framework of the Caputo-Fabrizio fractional operator with order b. Fixed point theory has been used to investigate the qualitative analysis of the solution respectively. The well-known Laplace transform method is used to determine the approximate solution of the proposed model. Using the COVID-19 data that is currently available, numerical simulations are run to validate the necessary scheme and examine the dynamic behavior of the various compartments of the model. In order to stop the pandemic from spreading, our findings highlight the significance of taking preventative steps and changing one's lifestyle.
Published: 28 December 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 221-227; https://doi.org/10.53391/mmnsa.2022.018
In this article, we consider a class of systems of differential equations with multiple delays. We define a transform that reformulates the system with delays into a singular linear system of differential equations whose coefficients are non-square constant matrices, and the number of their columns is greater than the number of their rows. By studying only the singular system, we provide a form of solutions for both systems.
Published: 27 December 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 211-220; https://doi.org/10.53391/mmnsa.2022.017
In this paper, we use an effective method which is the rational sine-Gordon expansion method to present new wave simulations of a governing model. We consider the (1+1)-dimensional conformable Fisher equation which is used to describe the interactive relation between diffusion and reaction. Various types of solutions such as multi-soliton, kink, and anti-kink wave soliton solutions are obtained. Finally, the physical behaviours of the obtained solutions are shown by 3D, 2D, and contour surfaces.
Published: 1 December 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 197-210; https://doi.org/10.53391/mmnsa.2022.016
We propose a new epidemic model to study the coinfection dynamics of COVID-19 and bacterial pneumonia, which is the first model in the literature used to describe mathematically the interaction of these two diseases while considering two infection ways for pneumonia: community-acquired and hospital-acquired transmission. We show that the existence and local stability of equilibria depend on three different parameters, which are interpreted as the basic reproduction numbers of COVID-19, bacterial pneumonia, and bacterial population in the hospital. Numerical simulations are performed to complement our theoretical analysis, and we show that both diseases can persist if the basic reproduction number of COVID-19 is greater than one.
Published: 26 June 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 108-116; https://doi.org/10.53391/mmnsa.2022.009
In this article, a mathematical model of the COVID-19 pandemic with control parameters is introduced. The main objective of this study is to determine the most effective model for predicting the transmission dynamic of COVID-19 using a deterministic model with control variables. For this purpose, we introduce three control variables to reduce the number of infected and asymptomatic or undiagnosed populations in the considered model. Existence and necessary optimal conditions are also established. The Grünwald-Letnikov non-standard weighted average finite difference method (GL-NWAFDM) is developed for solving the proposed optimal control system. Further, we prove the stability of the considered numerical method. Graphical representations and analysis are presented to verify the theoretical results.
Published: 29 September 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 187-196; https://doi.org/10.53391/mmnsa.2022.015
The aim of the paper is to study a cancer model based on anti-angiogenic therapy and radiotherapy. A set-valued analysis is carried out to control the tumor and carrying capacity of the vasculature, so in order to reverse tumor growth and augment tumor repair. The viability technique is used on an augmented model to solve the control problem. Obtained control is a selection of set-valued map of regulation and reduces tumor volume to around zero. A numerical simulation scheme with graphical representations and biological interpretations are given.
Published: 28 September 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 177-186; https://doi.org/10.53391/mmnsa.2022.014
The purpose of this paper is to find approximate solutions to the fractional telegraph differential equation (FTDE) using Laplace transform collocation method (LTCM). The equation is defined by Caputo fractional derivative. A new form of the trial function from the original equation is presented and unknown coefficients in the trial function are computed by using LTCM. Two different initial-boundary value problems are considered as the test problems and approximate solutions are compared with analytical solutions. Numerical results are presented by graphs and tables. From the obtained results, we observe that the method is accurate, effective, and useful.
Published: 25 September 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 164-176; https://doi.org/10.53391/mmnsa.2022.013
This article deals with a Caputo fractional-order viral model that incorporates the non-cytolytic immune hypothesis and the mechanism of viral replication inhibition. Firstly, we establish the existence, uniqueness, non-negativity, and boundedness of the solutions of the proposed viral model. Then, we point out that our model has the following three equilibrium points: equilibrium point without virus, equilibrium state without immune system, and equilibrium point activated by immunity with humoral feedback. By presenting two critical quantities, the asymptotic stability of all said steady points is examined. Finally, we examine the finesse of our results by highlighting the impact of fractional derivatives on the stability of the corresponding steady points.
Published: 16 September 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 147-163; https://doi.org/10.53391/mmnsa.2022.012
This work investigates the complex Ginzburg-Landau equation (CGLE) with Kerr law in nonlinear optics, which represents soliton propagation in the presence of a detuning factor. The φ^6-model expansion approach is used to find optical solitons such as dark, bright, singular, and periodic as well as the combined soliton solutions to the model. The results presented in this study are intended to improve the CGLE's nonlinear dynamical characteristics, it might also assist in comprehending some of the physical implications of various nonlinear physics models. The hyperbolic sine, for example, appears in the calculation of the Roche limit and gravitational potential of a cylinder, while the hyperbolic cotangent appears in the Langevin function for magnetic polarization. The current research is frequently used to report a variety of fascinating physical phenomena, such as the Kerr law of non-linearity, which results from the fact that an external electric field causes non-harmonic motion of electrons bound in molecules, which causes nonlinear responses in a light wave in an optical fiber. The obtained solutions' 2-dimensional, 3-dimensional, and contour plots are shown.
Published: 4 September 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 1-8; https://doi.org/10.53391/mmnsa.2022.021
The basic principle of this study is to obtain various solutions to the (1+1) dimensional Mikhailov-Novikov-Wang integrable equation (MNWIE). For this purpose, the generalized exponential rational function method (GERFM) is applied to this equation. Thus, several trigonometric functions, hyperbolic functions, and dark soliton solutions to the studied equation are acquired. In this way, some new solutions to the equation that have not been presented before have been obtained. In addition, 2D and 3D graphics of the acquired solutions are drawn for specific values. The obtained results and the graphic drawings of the results have been provided by using Wolfram Mathematica 12.
Published: 10 August 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 127-146; https://doi.org/10.53391/mmnsa.2022.011
In the presence of one auxiliary variable and two auxiliary variables, we analyze various exponential estimators. The ranks of the auxiliary variables are also connected with the study variables, and there is a linkage between the study variables and the auxiliary variables. These ranks can be used to improve an estimator's accuracy. The Optional Randomized Response Technique (ORRT) and the Quantitative Randomized Response Technique are two techniques we utilize to estimate the sensitive variables from the population mean (QRRT). We used the scrambled response technique and checked the proposed estimators up to the first-order of approximation. The mean square error (MSE) equations are obtained for all the proposed ratio exponential estimators and show that our proposed exponential type estimator is more efficient than ratio estimators. The expression of mean square error is obtained up to the first degree of approximation. The empirical and theoretical comparison of the proposed estimators with existing estimators is also be carried out. We have shown that the proposed optional randomized response technique and quantitative randomized response model are always better than existing estimators. The simulation study is also carried out to determine the performance of the estimators. Few real-life data sets are also be applied in support of proposed estimators. It is observed that our suggested estimator is more efficient as compared to an existing estimator.
Published: 30 June 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 117-126; https://doi.org/10.53391/mmnsa.2022.010
The main objective of this work is to introduce and define the concept of s-type m-preinvex function and derive the new sort of Hermite-Hadamard inequality via the newly discussed idea. Furthermore, to enhance the quality of paper, we prove two new lemmas and in order to these lemmas, we attain some extensions of Hermite-Hadamard-type inequality in the manner of newly explored definition. The concepts and tools of this paper may invigorate and revitalize for additional research in this mesmerizing and absorbing field of mathematics.
Published: 24 June 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 88-107; https://doi.org/10.53391/mmnsa.2022.008
This study proposes a novel mathematical model of COVID-19 and its qualitative properties. Asymptotic behavior of the proposed model with local and global stability analysis is investigated by considering the Lyapunov function. The mentioned model is globally stable around the disease-endemic equilibrium point conditionally. For a better understanding of the disease propagation with vaccination in the population, we split the population into five compartments: susceptible, exposed, infected, vaccinated, and recovered based on the fundamental Kermack-McKendrick model. He's homotopy perturbation technique is used for the semi-analytical solution of the suggested model. For the sake of justification, we present the numerical simulation with graphical results.
Published: 19 June 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 73-87; https://doi.org/10.53391/mmnsa.2022.007
A new concept in the transmutation of distribution applying variable transmuting function has been conceived. Test examples with power function by quadratic and cubic transmutations have been demonstrated by the applications of the error-function and standard logistic function variable transmuting functions. The efficiency and properties of the new approach by numerical examples addressing the rate constants of the transmuting functions and the shape parameter of the test power function have been demonstrated. An additional example with a quadratic transmutation of the exponential distribution through the error function as a variable transmuting parameter has been developed.
Published: 29 May 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 59-72; https://doi.org/10.53391/mmnsa.2022.006
The current paper investigates a newly developed model for Hepatitis-B infection in sense of the Atangana-Baleanu Caputo (ABC) fractional-order derivative. The proposed technique classifies the population into five distinct categories, such as susceptible, acute infections, chronic infections, vaccinated, and immunized. We obtain the Ulam-Hyers type stability and a qualitative study of the corresponding solution by applying a well-known principle of fixed point theory. Furthermore, we establish the deterministic stability of the proposed model. For the approximation of the ABC fractional derivative, we use a newly proposed numerical method. The obtained results are numerically verified by MATLAB 2020a.
Published: 31 March 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 48-58; https://doi.org/10.53391/mmnsa.2022.01.005
This paper addresses the mathematical modelling of aircraft landing gear based on the shock absorber system’s dynamics and examination of results depending on different touchdown scenarios and design parameters. The proposed methodology relies on determining an analytical formulation of the shock absorber system’s equation of motion, modelling this formulation on the model-based environment (Matlab/Simulink), and integrating with an accurate aircraft nonlinear dynamic model to observe the performance of landing gear in different touchdown or impact velocities. A suitable landing performance depends on different parameters which are related to the shock absorber system’s working principle. There are three subsystems of the main system which are hydraulic, pneumatic, and tire systems. Subsystems create a different sort of forces and behaviors. The air in the pneumatic system is compressed by the impact effect so it behaves like a spring and creates pneumatic or air spring force so the most effective parameter in this structure is determined as initial air volume. Hydraulic oil in the receptacle of the hydraulic system flow in an orifice hole when impact occurs so it behaves as a damper and creates damping or hydraulic force. The same working principle is acceptable for the air in the tire. The relationship between tire and ground creates a friction force based on dynamic friction coefficient depending on aircraft dynamics. As a result of this study effect of the impact velocity and initial air volume parameters on the system are examined and determined by optimization according to maximum initial load limits of aircraft and displacement of strut and tire surface.
Published: 16 February 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 41-47; https://doi.org/10.53391/mmnsa.2022.01.004
A three-dimensional system is introduced in this paper and its local stability is analyzed. Our study establishes the validity and uniqueness of the linear feedback control for the proposed system and proves its existence and uniqueness. The numerical simulation algorithm described by Atanackovic and Stankovic is finally applied. The analytical results are analyzed and the dynamics of the system are explored in more detail.
Published: 27 January 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 26-40; https://doi.org/10.53391/mmnsa.2022.01.003
Several models based on discrete and continuous fields have been proposed to comprehend residential criminal dynamics. This study introduces a two-dimensional model to describe residential burglaries diffusion, employing Lèvy flights dynamics. A continuous model is presented, introducing bidimensional fractional operator diffusion and its differences with the 1-dimensional case. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even fractional derivative order is modified. We also provide qualitative evidence that steady-state approximation in one dimension by series expansion is insufficient to capture similar original system behavior. At least for the case where series coefficients have a linear relationship with derivative order. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even if fractional derivative order is modified. Two dynamic regimes emerge in maximum and total attractiveness magnitude as a result of fractional derivative changes, these regimes can be understood as considerations about different urban environments. Finally, we add a Law enforcement component, embodying the "Cops on dots" strategy; in the Laplacian diffusion dynamic, global attractiveness levels are significantly reduced by Cops on dots policy but lose efficacy in Lèvy flight-based diffusion regimen. The four-step Preditor-Corrector method is used for numerical integration, and the fractional operator is approximated, getting the advantage of the spectral methods to approximate spatial derivatives in two dimensions.
Published: 26 January 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 13-25; https://doi.org/10.53391/mmnsa.2022.01.002
In this paper, we consider the constructive equations of the fractional second-grade fluid. The considered fluid model is described by the Caputo derivative. The problem consists to determine the exact analytical solution using the Laplace transform method. The influence of the order of the used fractional operator has been presented in this paper. We also analyze the influence of the Prandtl number in the dynamics of the temperature distribution according to the variation of the order of the Caputo derivative. The impact of the second-grade parameter and the Grashof number in the dynamics of the velocity has been presented and discussed. The influences of the parameters used in the modeling have been interpreted in terms of a fractional context. In general, it is shown that the order of the fractional operator influences the diffusivity of the considered fluid. This influence can cause an increase or decrease in the temperature and velocity distributions. The main findings of the paper have been illustrated using the graphical representations of the considered distributions according to the order of the fractional operator.
Published: 22 January 2022
Mathematical Modelling and Numerical Simulation With Applications, Volume 2, pp 1-12; https://doi.org/10.53391/mmnsa.2022.01.001
Published: 21 December 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 102-111; https://doi.org/10.53391/mmnsa.2021.01.010
In this paper, we simulate an epidemic model of cholera disease in the sense of generalized Liouville-Caputo fractional derivative. We provide the results related to the existence of a unique solution by using some well-known theorems. Numerical solutions of the given model are derived by using two different numerical methods along with their importance. A number of graphs are plotted to understand the given cholera disease dynamics. The main motivation to do this research is to understand the given disease dynamics as well as the efficiency of both methods which are very recent to the literature.
Published: 18 December 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 95-101; https://doi.org/10.53391/mmnsa.2021.01.009
This paper focuses on introducing a two-dimensional discrete-time chemical model and the existence of its fixed points. Also, the one and two-parameter bifurcations of the model are investigated. Bifurcation analysis is based on numerical normal forms. The flip (period-doubling) and generalized flip bifurcations are detected for this model. The critical coefficients identify the scenario associated with each bifurcation. To confirm the analytical results, we use the MATLAB package MatContM, which performs based on the numerical continuation method. Finally, bifurcation diagrams are presented to confirm the existence of flip (period-doubling) and generalized flip bifurcations for the glycolytic oscillator model that gives a better representation of the study.
Published: 16 December 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 84-94; https://doi.org/10.53391/mmnsa.2021.01.008
Published: 13 December 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 67-83; https://doi.org/10.53391/mmnsa.2021.01.007
Published: 26 November 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 56-66; https://doi.org/10.53391/mmnsa.2021.01.006
Published: 4 September 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 32-43; https://doi.org/10.53391/mmnsa.2021.01.004
Published: 15 August 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 1-10; https://doi.org/10.53391/mmnsa.2021.01.001
Published: 21 August 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 11-23; https://doi.org/10.53391/mmnsa.2021.01.002
Published: 30 September 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 44-55; https://doi.org/10.53391/mmnsa.2021.01.005
Published: 24 August 2021
Mathematical Modelling and Numerical Simulation With Applications, Volume 1, pp 24-31; https://doi.org/10.53391/mmnsa.2021.01.003