Narrow Results

This paper introduces a novel nonlinear fractal fractional model to comprehensively analyze influenza epidemics. By integrating fractional calculus and considering nonlinear interactions among individuals, the model utilizes the Fractal-Fractional (FF) operator. This operator, combining fractal and fractional calculus, establishes a unique framework for investigating influenza virus propagation dynamics and potential vaccination strategies. Amid growing concerns over influenza outbreaks, fractional derivatives are employed to address intricate challenges. The proposed model sheds light on virus spread dynamics and countermeasures. The integration of the FF operator enriches analysis, while the application of the fixed point theory of Schauder and Banach demonstrates solution existence and uniqueness. Model validation employs numerical simulations with MATLAB12̂ and the Adams–Bashforth method, confirming its ability to capture influenza propagation dynamics accurately. Ulam–Hyers stability techniques ensure the model’s reliability. Beyond its scientific contributions, the model underscores the significance of studying influenza epidemics via mathematical modeling in understanding disease dynamics and guiding effective intervention strategies. Through a synergy of mathematical innovation and epidemiological insights, this study establishes a robust foundation for more effective strategies against influenza epidemics.

This paper investigates the use of weak convergence in Stratonovich stochastic differential equations (SDEs), shifting the focus from the robust convergence techniques previously employed. We introduce a novel application of the trivial coupling method within the weak convergence framework, specifically addressing non-invertible equations. Our approach simplifies the handling of random scenarios and computational tasks, with potential applications spanning physics, biology, and engineering. We provide a detailed account of the method, including its theoretical background and practical implementation using MATLAB. Our results confirm the validity of our approach, demonstrating its effectiveness even with degenerate diffusion coefficients. This advancement in weak convergence strategies offers new insights and practical solutions for complex systems and opens avenues for further research.

Wireless sensor networks have been extensively studied for their potential applications in both civil and military domains. However, due to their inherent resource constraints, sensor nodes are highly susceptible to cyber threats, particularly worm attacks, which pose significant security challenges. This study explores the dynamics of potential worm attacks within wireless sensor networks using a compartmental epidemic model. The model analyzes the temporal evolution of worm propagation while effectively capturing both spatial and temporal aspects. We determine the reproduction number and equilibrium of the system, with the local stability assessed using the Jacobian matrix. The linear growth and Lipschitz conditions are used to establish the existence and uniqueness of the solution. Furthermore, the Hyers–Ulam stability of the proposed model is also evaluated within the context of the fractal-fractional operator. Finally, a numerical method is developed to investigate the dynamic behavior of the wireless sensor network model under fractal-fractional orders, providing valuable insights into its robustness and security against potential worm attacks.

In this work, we formulate a two-strain model of dengue infection, incorporating the possibility of co-infection through integer and Caputo-type fractional derivatives. We present the foundational theory and results related to fractional operators for the analysis of the proposed dengue model. We establish the positivity and boundedness of the solution to validate the model. It is demonstrated that the solution to the proposed model exists and is unique. The basic reproduction number, denoted by ${{ℛ}}_c$, is determined, and the local asymptotic stability of the dengue-free equilibrium point is established when ${{ℛ}}_c<1$. We also prove the existence of an endemic equilibrium and establish the global stability of solutions to the fractional model in the Ulam–Hyers sense. Additionally, we construct a numerical scheme and prove its stability under certain conditions. Several numerical simulations are performed to validate our analytical results, and we examine the impact of various input factors on the solution pathways of the system through these simulations.

This paper focuses on flow structures and thermal fields of the Carreau–Yasuda (CY) nanofluid model through a two-dimensional, wavy, complicated vertical asymmetrical conduit filled with porous elements. Formulations of the viscous dissipation in the case of CY nanofluids are derived and nonlinear radiation flux as well as joule heating are examined. Buongiorno’s nanofluid approach, which involves Brownian motion and thermophoresis aspects is considered. The electrical conductivity of the suspension is considered as a variable where it depends upon the ambient temperature and concentration distributions and the Joule heating impacts are not neglected. The approach of solving the problem is contingent upon converting the system to dimensionless form then the lubrication approach with low magnetic Reynold numbers is applied. Numerical solutions are found for the resultant system, and wide ranges are considered for Weissenberg number We, non-Newtonian parameter n and Darcy number $D_a$, namely, $0\le \mathrm{\text{We}}\le 2$, $-0.5\le n\le 1.5$ and $0\le D_a\le 1.6$, respectively. The major results indicated that gradients of the pressure are higher in case of shear thickening $(n>1)$ comparing to in the instance of shear thinning $(n<1)$. Also, the velocity is enhanced, close to the channel’s lowest portion, as the Weissenberg number is growing. The variable electrical conductivity gives a higher mass transfer rate compared to the constant property.

In this paper, we present a compartmental mathematical model of hepatitis B virus with optimal control strategies. First, we formulate the model applying the optimal control techniques which use control variables in the form of isolation, educational campaign and vaccination. We derive the conditions under which it is optimal to eradicate the disease and examine the impact of possible vaccination treatment strategies on disease transmission. When such an elimination is impossible, we use the techniques of Pontryagin’s Maximum Principle to derive the necessary conditions for the optimal control problem. The numerical results show that some effective vaccination and control can reduce the disease spread in the community.

In this paper, a mathematical model for malaria-dysentery co-infection was formulated in order to study and examine its dynamic relationship in the presence of malaria and dysentery preventive and treatment measures. First, analysis of the single infection steady states was done and then the basic reproduction number was obtained. Furthermore, investigation into the existence and stability of equilibria carried out. The single infection models were found to exhibit the possibility of backward bifurcation. Thereafter, the impact of malaria on the dynamics of dysentery is further investigated. Second, incorporating time-dependent controls, using Pontryagin’s Maximum Principle, the necessary conditions for the optimal control of the disease was derived. It is found that malaria infection may be associated with an increased risk of dysentery. Also, that dysentery infection may be associated with an increased risk for malaria. Therefore, to effectively control malaria, the malaria intervention strategies by policy makers must at the same time it also includes effective prevention and control measures for dysentery. Policy makers should take efforts on preventive strategies in combating dysentery and malaria.

We present a dynamical model for the particle-cluster aggregation process of marine snow. The model consists of only two parts, i.e. single cells and one type of aggregates. The parameters are settling and decomposition of aggregates; growth and aggregation of single cells. There are developed numerical results of the system.

In this paper, new model on novel coronavirus disease (COVID-19) with four compartments including susceptible, exposed, infected, and recovered class with fractal-fractional derivative is proposed. Here, Banach and Leray–Schauder alternative type theorems are used to establish some appropriate conditions for the existence and uniqueness of the solution. Also, stability is needed in respect of the numerical solution. Therefore, Ulam–Hyers stability using nonlinear functional analysis is used for the proposed model. Moreover, the numerical simulation using the technique of fundamental theorem of fractional calculus and the two-step Lagrange polynomial known as fractional Adams-Bashforth (AB) method is proposed. The obtained results are tested on real data of COVID-19 outbreak in Malaysia from 25 January till 10 May 2020. The numerical simulation of the proposed model has performed in terms of graphs for different fractional-order $q$ and fractal dimensions $p$ via number of considered days of disease spread in Malaysia. Since COVID-19 transmits rapidly, perhaps, the clear understanding of transmission dynamics of COVID-19 is important for countries to implement suitable strategies and restrictions such as Movement Control Order (MCO) by the Malaysian government, against the disease spread. The simulated results of the presented model demonstrate that movement control order has a great impact on the transmission dynamics of disease outbreak in Malaysia. It can be concluded that by adopting precautionary measures as restrictions on individual movement the transmission of the disease in society is reduced. In addition, for such type of dynamical study, fractal-fractional calculus tools may be used as powerful tools to understand and predict the global dynamics of the mentioned disease in other countries as well.

Today, most of the real physical world problems can be modeled with variable-coefficient KdV–modified KdV (vcKdV–mKdV) equation. Besides, the solution methods and their reliabilities are the most important. Therefore, a high precision numerical method is always needed. In this paper, Fourier spectral method is applied to solve the space fractional generalized vcKdV–mKdV equation and the influence of fractional orders on numerical solution of the space fractional generalized vcKdV–mKdV equation is investigated. Numerical simulations in different situations of equation are conducted, including the propagation and interaction of the generalized ball-type, kink-type and periodic-depression solitons. From the numerical experiments pondered here and compared with the other methods, it is found that the numerical solutions match well with the exact solutions, which demonstrate that the Fourier spectral method is a satisfactory and efficient algorithm.

This study is conducted to solve a nonlinear biological prey–predator system (NBPPS) using a novel design of the Levenberg–Marquardt backpropagation approach (LMBA). The LMBA-based supervised neural networks (SNNs) deal with three kinds of sample data, training, validation, and testing. The percentages for these data to solve three different cases of the NBPPS are selected: for training 75%, validation 10%, and testing 15%, respectively. The numerical performances of the Adams method are used for the reference dataset to solve the NBPPS. The obtained form of the numerical solutions of the NBPPS based on the SNNs along with LMBA is used to reduce the functions of mean square error (MSE). For the correctness, competence, and effectiveness of the proposed SNNs along with LMBA, the numerical procedures are proficient based on the proportional schemes and analyses in terms of MSE results, correlation, error histograms, and regression.

The human immunodeficiency virus (HIV) is a major global public health issue and causes millions of deaths around the globe. The most severe phase of HIV infection is known as AIDS. In recent years, a number of mathematical models based on classical integer-order derivative have been developed to analyze the insight dynamics of HIV/AIDS. This paper presents the transmission dynamics of HIV/AIDS using fractional order (FO) and a fractal-fractional order compartmental model with the power-law kernel. In the first phase, the proposed model is formulated using the Caputo-type fractional derivative. The basic properties such as the solution positivity and existence as well as uniqueness of the fractional model are presented. The equilibria and the basic reproductive number ${{ℛ}}_0$ are evaluated. Further, using fractional stability concepts the stability of the model (both local and global) around the equilibrium is presented in the disease-free case. In addition, the fractional model is solved numerically, and the graphical results with many values of $q_1$ are shown. In the second phase, the concept of a fractal-fractional (FF) operator is applied to obtain a more generalized model that addresses the dynamics of HIV/AIDS. The uniqueness and existence of the solutions of the FF-based model are shown via the Picard–Lindelof approach while the modified Adams–Bashforth method is utilized to present the numerical solution. Detailed numerical simulations are presented for various values fractional as well as the fractal orders, $q_1$ and $q_2,$ respectively. The graphical results reveal that the FF-based model provides biologically more feasible results than the models in fractional and classical integer-order cases.

This paper is related to some qualitative results about the existence and uniqueness of a solution to a third-order problem by using a fixed point approach. Haar technique is applied for numerical solution of a third-order linear integro-differential equation (IDE) with initial conditions. In IDE, the third-order derivative is computed by Haar functions, and the integration is used to get the expression of second- and first-order derivatives, as well as an approximate solution. Some examples from the literature are used to verify the validity of the proposed method. Error analysis is performed. Also, comparison between the exact and numerical solutions at different collocation points (CPs) is derived. The convergence rate is recorded taking different numbers of CPs, which is approximately equal to 2.

The purpose of this work is to investigate the dynamical behavior of a measles disease model adopting fractional and fractal–fractional operators with Mittag-Leffler kernel using two distinct numerical algorithms. First, we discuss the measles model in a fractional framework with Atangana–Baleanu–Caputo derivative and examine some fundamental mathematical assumptions of the considered model. We implement fixed-point theory to explore the existence and uniqueness of model solutions. Next, we apply the novel fractal–fractional concept with Atangana–Baleanu derivative to the measles model and reveal that the model has unique solution. We present the approximate results for the proposed models with graphical illustrations. The results are presented with various choices of fractal and fractional orders. The system behavior to various biological parameters is also investigated. In addition, we compare the considered operators using novel numerical schemes that take into account different $\varrho$ values.

A mathematical model of progressive disease of the nervous system also called multiple sclerosis (MS) is studied in this paper. The proposed model is investigated under the concept of the fractal-fractional order derivative (FFOD) in the Caputo sense. In addition, the tools of nonlinear functional analysis are applied to prove some qualitative results including the existence theory, stability, and numerical analysis. For the recommended results of the existence theory, Banach and Krassnoselski’s fixed point theorems are used. Additionally, Hyers–Ulam (HU) concept is used to derive some results for stability analysis. Additionally, for numerical illustration of approximate solutions of various compartments of the considered model, the modified Euler method is utilized. The aforementioned results are displayed graphically for various values of fractal-fractional orders.

Geometrically nonlinear analysis of functionally graded circular plates subjected to mechanical and thermal loads is carried out in this paper. The Green–Lagrange strain tensor in its entirety is used in the analysis. The locally effective material properties are evaluated using homogenization method which is based on the Mori–Tanaka scheme. In the case of thermally loaded plates, the temperature variation through the thickness is determined by solving a steady-state heat transfer (i.e. energy) equation. As an example, a functionally gradient material circular plate composed of zirconium and aluminum is used and results are presented in graphical form.

We outline an approximate solution for the numerical simulation of two-phase fluid flows with a relative velocity between the two phases. A unified two-phase flow model is proposed for the description of the gas–liquid processes which leads to a system of hyperbolic differential equations in a conservative form. A numerical algorithm based on a splitting approach for the numerical solution of the model is proposed. The associated Riemann problem is solved numerically using Godunov methods of centered-type. Results show the importance of the Riemann problem and of centered schemes in the solution of the two-phase flow problems. In particular, it is demonstrated that the Slope Limiter Centered (SLIC) scheme gives a low numerical dissipation at the contact discontinuities, which makes it suitable for simulations of practical two-phase flow processes.

In the present study, a finite element formulation is presented to investigate vibration response of elastic plates in contact with a fluid medium. The fluid is assumed to be incompressible and inviscid, and the impermeability condition of the plate is taken into account. The classical plate theory (CPT), first-order shear deformation plate theory (FSDT), and Reddy third-order shear deformation plate theory (RSDT) are considered for the kinematic description of the solid medium and the simplified Navier–Stokes equations are used as the governing equations for the fluid medium. For each plate theory, a coupled set of finite element equations is derived. The effect of the fluid pressure is considered as an added mass and its effect on natural frequencies and mode shapes is investigated through several numerical simulations by varying the boundary conditions.

In this paper, a fractional-order model which describes the human immunodeficiency type-1 virus (HIV-1) infection is presented. Numerical solutions are obtained using a generalized Euler method (GEM) to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. We show that the model established in this paper possesses non-negative solutions. Comparisons between the results of the fractional-order model, the results of the integer model and the measured real data obtained from 10 patients during primary HIV-1 infection are presented. These comparisons show that the results of the fractional-order model give predictions to the plasma virus load of the patients better than those of the integer model.

In this paper, an SEIVR epidemic model with generalized incidence and preventive vaccination is considered. First, we formulate the model and obtain its basic properties. Then, we find the equilibrium points of the model, the disease-free and the endemic equilibrium. The stability of disease-free and endemic equilibrium is associated with the basic reproduction number . If the basic reproduction number , the disease-free equilibrium is locally as well as globally asymptotically stable. Moreover, if the basic reproduction number , the disease is uniformly persistent and the unique endemic equilibrium of the system is locally as well as globally asymptotically stable under certain conditions. Finally, the numerical results justify the analytical results.