Mathematical modelling, analysis and numerical simulation of social media addiction and depression.

We formulate a mathematical model of social media addiction and depression (SMAD) in this study. Key aspects, such as social media addiction and depression disease-free equilibrium point (SMADDFEP), social media addiction and depression endemic equilibrium point (SMADEEP), and basic reproduction number (R0), have been analyzed qualitatively. The results indicate that if R0 < 1, the SMADDFEP is locally asymptotically stable. The global asymptotic stability of the SMADDFEP has been established using the Castillo-Chavez theorem. On the other hand, if R0 > 1, the unique endemic equilibrium point (SMADEEP) is locally asymptotically stable by Lyapunov theorem, and the model exhibits a forward bifurcation at R0 = 1 according to the Center Manifold theorem. To examine the model's sensitivity, we calculated the normalized forward sensitivity index and conducted a Partial Rank Correlation Coefficient (PRCC) analysis to describe the influence of parameters on the SMAD. The numerical results obtained using the Fourth-order Runge-Kutta (RK-4) scheme show that increasing the number of addicted individuals leads to an increase in the number of depressed individuals.

A new investigation on fractionalized modeling of human liver.

This study focuses on improving the accuracy of assessing liver damage and early detection for improved treatment strategies. In this study, we examine the human liver using a modified Atangana-Baleanu fractional derivative based on the mathematical model to understand and predict the behavior of the human liver. The iteration method and fixed-point theory are used to investigate the presence of a unique solution in the new model. Furthermore, the homotopy analysis transform method, whose convergence is also examined, implements the mathematical model. Finally, numerical testing is performed to demonstrate the findings better. According to real clinical data comparison, the new fractional model outperforms the classical integer-order model with coherent temporal derivatives.

A scale conjugate neural network approach for the fractional schistosomiasis disease system.

This study presents the numerical solutions of the fractional schistosomiasis disease model (SDM) using the supervised neural networks (SNNs) and the computational scaled conjugate gradient (SCG), i.e. SNNs-SCG. The fractional derivatives are used for the precise outcomes of the fractional SDM. The preliminary fractional SDM is categorized as: uninfected, infected with schistosomiasis, recovered through infection, expose and susceptible to this virus. The accurateness of the SNNs-SCG is performed to solve three different scenarios based on the fractional SDM with synthetic data obtained with fractional Adams scheme (FAS). The generated data of FAS is used to execute SNNs-SCG scheme with 81% for training samples, 12% for testing and 7% for validation or authorization. The correctness of SNNs-SCG approach is perceived by the comparison with reference FAS results. The performances based on the error histograms (EHs), absolute error, MSE, regression, state transitions (STs) and correlation accomplish the accuracy, competence, and finesse of the SNNs-SCG scheme.

Heuristic computing with active set method for the nonlinear Rabinovich-Fabrikant model.

The current study shows a reliable stochastic computing heuristic approach for solving the nonlinear Rabinovich-Fabrikant model. This nonlinear model contains three ordinary differential equations. The process of stochastic computing artificial neural networks (ANNs) has been applied along with the competences of global heuristic genetic algorithm (GA) and local search active set (AS) methodologies, i.e., ANNs-GAAS. The construction of merit function is performed through the differential Rabinovich-Fabrikant model. The results obtained through this scheme are simple, reliable, and accurate, which have been calculated to optimize the merit function by using the GAAS method. The comparison of the obtained results through this scheme and the conventional reference solutions strengthens the correctness of the proposed method. Ten numbers of neurons along with the log-sigmoid transfer function in the neural network structure have been used to solve the model. The values of the absolute error are performed around 10-07 and 10-08 for each class of the Rabinovich-Fabrikant model. Moreover, the reliability of the ANNs-GAAS approach is observed by using different statistical approaches for solving the Rabinovich-Fabrikant model.

Optical soliton solutions of the coupled Radhakrishnan-Kundu-Lakshmanan equation by using the extended direct algebraic approach.

The analytical soliton solutions place a lot of value on birefringent fibres. The major goal of this study is to generate novel forms of soliton solutions for the Radhakrishnan-Kundu-Lakshmanan equation, which depicts unstable optical solitons that arise from optical propagations using birefringent fibres. The (presumably new) extended direct algebraic (EDA) technique is used here to extract a large number of solutions for RKLE. It gives soliton solutions up to thirty-seven, which essentially correspond to all soliton families. This method's ability to determine many sorts of solutions through a single process is one of its key advantages. Additionally, it is simple to infer that the technique employed in this study is really straightforward yet one of the quite effective approaches to solving nonlinear partial differential equations so, this novel extended direct algebraic (EDA) technique may be regarded as a comprehensive procedure. The resulting solutions are found to be hyperbolic, periodic, trigonometric, bright and dark, combined bright-dark, and W-shaped soliton, and these solutions are visually represented by means of 2D, 3D, and density plots. The present study can be extended to investigate several other nonlinear systems to understand the physical insights of the optical propagations through birefringent fibre.

Fractional Modeling of Cancer with Mixed Therapies.

BACKGROUND: Cancer is the biggest cause of mortality globally, with approximately 10 million fatalities expected by 2020, or about one in every six deaths. Breast, lung, colon, rectum, and prostate cancers are the most prevalent types of cancer. METHODS: In this work, fractional modeling is presented which describes the dynamics of cancer treatment with mixed therapies (immunotherapy and chemotherapy). Mathematical models of cancer treatment are important to understand the dynamical behavior of the disease. Fractional models are studied considering immunotherapy and chemotherapy to control cancer growth at the level of cell populations. The models consist of the system of fractional differential equations (FDEs). Fractional term is defined by Caputo fractional derivative. The models are solved numerically by using Adams-Bashforth-Moulton method. RESULTS: For all fractional models the reasonable range of fractional order is between ß = 0.6 and ß = 0.9. The equilibrium points and stability analysis are presented. Moreover, positivity and boundedness of the solution are proved. Furthermore, a graphical representation of cancerous cells, immunotherapy and chemotherapy is presented to understand the behaviour of cancer treatment. CONCLUSIONS: At the end, a curve fitting procedure is presented which may help medical practitioners to treat cancer patients.

Analysis of Dengue Transmission Dynamic Model by Stability and Hopf Bifurcation with Two-Time Delays.

BACKGROUND: Mathematical models reflecting the epidemiological dynamics of dengue infection have been discovered dating back to 1970. The four serotypes (DENV-1 to DENV-4) that cause dengue fever are antigenically related but different viruses that are transmitted by mosquitoes. It is a significant global public health issue since 2.5 billion individuals are at risk of contracting the virus. METHODS: The purpose of this study is to carefully examine the transmission of dengue with a time delay. A dengue transmission dynamic model with two delays, the standard incidence, loss of immunity, recovery from infectiousness, and partial protection of the human population was developed. RESULTS: Both endemic equilibrium and illness-free equilibrium were examined in terms of the stability theory of delay differential equations. As long as the basic reproduction number (R0) is less than unity, the illness-free equilibrium is locally asymptotically stable; however, when R0 exceeds unity, the equilibrium becomes unstable. The existence of Hopf bifurcation with delay as a bifurcation parameter and the conditions for endemic equilibrium stability were examined. To validate the theoretical results, numerical simulations were done. CONCLUSIONS: The length of the time delay in the dengue transmission epidemic model has no effect on the stability of the illness-free equilibrium. Regardless, Hopf bifurcation may occur depending on how much the delay impacts the stability of the underlying equilibrium. This mathematical modelling is effective for providing qualitative evaluations for the recovery of a huge population of afflicted community members with a time delay.

Predictive dynamical modeling and stability of the equilibria in a discrete fractional difference COVID-19 epidemic model.

The SARSCoV-2 virus, also known as the coronavirus-2, is the consequence of COVID-19, a severe acute respiratory syndrome. Droplets from an infectious individual are how the pathogen is transmitted from one individual to another and occasionally, these particles can contain toxic textures that could also serve as an entry point for the pathogen. We formed a discrete fractional-order COVID-19 framework for this investigation using information and inferences from Thailand. To combat the illnesses, the region has implemented mandatory vaccination, interpersonal stratification and mask distribution programs. As a result, we divided the vulnerable people into two groups: those who support the initiatives and those who do not take the influence regulations seriously. We analyze endemic problems and common data while demonstrating the threshold evolution defined by the fundamental reproductive quantity R 0 . Employing the mean general interval, we have evaluated the configuration value systems in our framework. Such a framework has been shown to be adaptable to changing pathogen populations over time. The Picard Lindelöf technique is applied to determine the existence-uniqueness of the solution for the proposed scheme. In light of the relationship between the R 0 and the consistency of the fixed points in this framework, several theoretical conclusions are made. Numerous numerical simulations are conducted to validate the outcome.

Design of a fractional-order atmospheric model via a class of ACT-like chaotic system and its sliding mode chaos control.

Investigation of the dynamical behavior related to environmental phenomena has received much attention across a variety of scientific domains. One such phenomenon is global warming. The main causes of global warming, which has detrimental effects on our ecosystem, are mainly excess greenhouse gases and temperature. Looking at the significance of this climatic event, in this study, we have connected the ACT-like model to three climatic components, namely, permafrost thaw, temperature, and greenhouse gases in the form of a Caputo fractional differential equation, and analyzed their dynamics. The theoretical aspects, such as the existence and uniqueness of the obtained solution, are examined. We have derived two different sliding mode controllers to control chaos in this fractional-order system. The influences of these controllers are analyzed in the presence of uncertainties and external disturbances. In this process, we have obtained a new controlled system of equations without and with uncertainties and external disturbances. Global stability of these new systems is also established. All the aspects are examined for commensurate and non-commensurate fractional-order derivatives. To establish that the system is chaotic, we have taken the assistance of the Lyapunov exponent and the bifurcation diagram with respect to the fractional derivative. To perform numerical simulation, we have identified certain values of the parameters where the system exhibits chaotic behavior. Then, the theoretical claims about the influence of the controller on the system are established with the help of numerical simulations.

Orthonormal piecewise Vieta-Lucas functions for the numerical solution of the one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations.

INTRODUCTION: Recently, a new family of fractional derivatives called the piecewise fractional derivatives has been introduced, arguing that for some problems, each of the classical fractional derivatives may not be able to provide an accurate statement of the consideration problem alone. In defining this kind of derivatives, several types of fractional derivatives can be used simultaneously. OBJECTIVES: This study introduces a new kind of piecewise fractional derivative by employing the Caputo type distributed-order fractional derivative and ABC fractional derivative. The one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations are defined using this piecewise fractional derivative. METHODS: A new class of basis functions called the orthonormal piecewise Vieta-Lucas (VL) functions are defined. Fractional derivatives of these functions in the Caputo and ABC senses are computed. These functions are utilized to construct two numerical methods for solving the introduced problems under non-local boundary conditions. The proposed methods convert solving the original problems into solving systems of algebraic equations. RESULTS: The accuracy and convergence order of the proposed methods are examined by solving several examples. The obtained results are investigated, numerically. CONCLUSION: This study introduces a kind of piecewise fractional derivative. This derivative is employed to define the one- and two-dimensional piecewise fractional Galilei invariant advection-diffusion equations. Two numerical methods based on the orthonormal VL polynomials and orthonormal piecewise VL functions are established for these problems. The numerical results obtained from solving several examples confirm the high accuracy of the proposed methods.

A novel fractional model for the projection of households using wealth index quintiles.

Forecasting household assets provides a better opportunity to plan their socioeconomic activities for the future. Fractional mathematical models offer to model the asset-holding data into a piece of scientific evidence in addition to forecasting their future value. This research focuses on the development of a new fractional mathematical model based on the wealth index quintile (WIQ) data. To accomplish the objective, we used the system of coupled fractional differential equations by defining the fractional term with the Caputo derivative and verified it with the stability tests considering the steady-state solution. A numerical solution of the model was obtained using the Adams-Bashforth-Moulton method. To validate the model, we used real-time data obtained from the household series of surveys in Punjab, Pakistan. Different case studies that elucidate the effect of quintiles on the population are also presented. The accuracy of results between real-world and simulated data was compared using absolute and relative errors. The synchronization between the simulated results and real-time data verifies the formulation of the fractional WIQ model. This fractional model can be utilized to predict the approximation of the asset-holding of the households. Due to its relative nature, the model also provides the opportunity for the researchers to use the WIQs of their respective regions to forecast the households' socioeconomic conditions.

Numerical treatments for the optimal control of two types variable-order COVID-19 model.

In this paper, a novel variable-order COVID-19 model with modified parameters is presented. The variable-order fractional derivatives are defined in the Caputo sense. Two types of variable order Caputo definitions are presented here. The basic reproduction number of the model is derived. Properties of the proposed model are studied analytically and numerically. The suggested optimal control model is studied using two numerical methods. These methods are non-standard generalized fourth-order Runge-Kutta method and the non-standard generalized fifth-order Runge-Kutta technique. Furthermore, the stability of the proposed methods are studied. To demonstrate the methodologies' simplicity and effectiveness, numerical test examples and comparisons with real data for Egypt and Italy are shown.

Analytical results for positivity of discrete fractional operators with approximation of the domain of solutions.

We study the monotonicity method to analyse nabla positivity for discrete fractional operators of Riemann-Liouville type based on exponential kernels, where $ \left({}_{{c_0}}^{C{F_R}}\nabla^{\theta} \mathtt{F}\right)(t) > -\epsilon\, \Lambda(\theta-1)\, \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1) $ such that $ \bigl(\nabla \mathtt{F}\bigr)(c_{0}+1)\geq 0 $ and $ \epsilon > 0 $. Next, the positivity of the fully discrete fractional operator is analyzed, and the region of the solution is presented. Further, we consider numerical simulations to validate our theory. Finally, the region of the solution and the cardinality of the region are discussed via standard plots and heat map plots. The figures confirm the region of solutions for specific values of $ \epsilon $ and $ \theta $.

The development of sustainable IoT E-waste management guideline for households.

The introduction of new technology, such as the Internet of Things (IoT), entails a growth in digital devices, which could ultimately result in a high amount of electronic trash (e-waste) production if they are not appropriately managed. Apart from that, the regulation on possible "IoT E-waste" generation is yet to be regulated, probably due to the new development and implementation of IoT globally. Hence, this paper proposed a Sustainable IoT E-waste Management guideline for households. This guideline could assist government agencies and policymakers in their strategies, planning, development, and implementation of a sustainable household IoT e-waste management initiatives in Malaysia. This study is an exploratory study that adopts a qualitative case study research method. The guideline was developed based on the Integrated Sustainable Waste Management (ISWM) Model. This guideline contributes to Malaysia's sustainability agenda in reducing carbon emissions intensity towards 2030 by 45 percent.

A delayed plant disease model with Caputo fractional derivatives.

We analyze a time-delay Caputo-type fractional mathematical model containing the infection rate of Beddington-DeAngelis functional response to study the structure of a vector-borne plant epidemic. We prove the unique global solution existence for the given delay mathematical model by using fixed point results. We use the Adams-Bashforth-Moulton P-C algorithm for solving the given dynamical model. We give a number of graphical interpretations of the proposed solution. A number of novel results are demonstrated from the given practical and theoretical observations. By using 3-D plots we observe the variations in the flatness of our plots when the fractional order varies. The role of time delay on the proposed plant disease dynamics and the effects of infection rate in the population of susceptible and infectious classes are investigated. The main motivation of this research study is examining the dynamics of the vector-borne epidemic in the sense of fractional derivatives under memory effects. This study is an example of how the fractional derivatives are useful in plant epidemiology. The application of Caputo derivative with equal dimensionality includes the memory in the model, which is the main novelty of this study.

Oscillation result for half-linear delay difference equations of second-order.

In this paper, we obtain the new single-condition criteria for the oscillation of second-order half-linear delay difference equation. Even in the linear case, the sharp result is new and, to our knowledge, improves all previous results. Furthermore, our method has the advantage of being simple to prove, as it relies just on sequentially improved monotonicities of a positive solution. Examples are provided to illustrate our results.

New classifications of monotonicity investigation for discrete operators with Mittag-Leffler kernel.

This paper deals with studying monotonicity analysis for discrete fractional operators with Mittag-Leffler in kernel. The $ \nu- $monotonicity definitions, namely $ \nu- $(strictly) increasing and $ \nu- $(strictly) decreasing, are presented as well. By examining the basic properties of the proposed discrete fractional operators together with $ \nu- $monotonicity definitions, we find that the investigated discrete fractional operators will be $ \nu^2- $(strictly) increasing or $ \nu^2- $(strictly) decreasing in certain domains of the time scale $ \mathbb{N}_a: = \{a, a+1, \dots\} $. Finally, the correctness of developed theories is verified by deriving mean value theorem in discrete fractional calculus.

Boger nanofluid: significance of Coriolis and Lorentz forces on dynamics of rotating fluid subject to suction/injection via finite element simulation.

This study briefings the roles of Coriolis, and Lorentz forces on the dynamics of rotating nanofluids flow toward a continuously stretching sheet. The nanoparticles are incorporated because of their unusual qualities like upgrade the thermal transportation, which are very important in heat exchangers, modern nanotechnology, electronics, and material sciences. The primary goal of this study is to improve heat transportation. Appropriate similarity transformations are applied for the principal PDEs to transform into nonlinear dimensionless PDEs. A widely recognized Numerical scheme known as the Finite Element Method is employed to solve the resultant convective boundary layer balances. Higher input in the solvent fraction parameter has a rising effect on the primary velocity and secondary velocity magnitude, and decreasing impact on the distributions of temperature. It is seen that growing contributions of the Coriolis, and Lorentz forces cause to moderate the primary and secondary velocities, but the temperature and concentration functions show opposite trend. The concentration, temperature, and velocities distributions for suction case is prominently than that of injection case, but inverse trend is observed for local Nusselt and Sherwood numbers. These examinations are relevant to the field of plastic films, crystal growing, paper production, heat exchanger, and bio-medicine.

Design of neuro-swarming computational solver for the fractional Bagley-Torvik mathematical model.

This study is to introduce a novel design and implementation of a neuro-swarming computational numerical procedure for numerical treatment of the fractional Bagley-Torvik mathematical model (FBTMM). The optimization procedures based on the global search with particle swarm optimization (PSO) and local search via active-set approach (ASA), while Mayer wavelet kernel-based activation function used in neural network (MWNNs) modeling, i.e., MWNN-PSOASA, to solve the FBTMM. The efficiency of the proposed stochastic solver MWNN-GAASA is utilized to solve three different variants based on the fractional order of the FBTMM. For the meticulousness of the stochastic solver MWNN-PSOASA, the obtained and exact solutions are compared for each variant of the FBTMM with reasonable accuracy. For the reliability of the stochastic solver MWNN-PSOASA, the statistical investigations are provided based on the stability, robustness, accuracy and convergence metrics.