Fractional-order Dynamics Research Articles

Dynamic analysis of mean-field and fractional-order epidemic vaccination strategies by evolutionary game approach

Individual's perception of the participant in a vaccine program reflects their intrinsic appreciation of the trade-off between vaccine behavior, risk of infection, and memory effect. Here we present a mean-field approximation and fractional-order model embedded with the evolutionary game theory (EGT) process for susceptible-vaccinated-infected-recovered (SVIR) epidemic dynamics, where the two types of immunity, artificial and natural immunity, are considered. Besides the well-known vaccination game models, we successfully establish a theoretical approach of fractional-order dynamics for vaccination games in which epidemic spread and individual decisions are supposed to evaluate social behavior. The EGT governs the strategy adoption, while the fractional-order process directs the memory effect state. Our analytical forecasts are validated by numerical simulation of the finite difference method and Adams-Bashforth-Moulton algorithm for the mean-field and Caputo fractional-order derivative that assesses various graphs at varying parameters. Our results show that an effective vaccine may meaningfully minimize the risk of infection. However, despite vaccines' obvious impacts and advantages on a community, many individuals choose not to vaccinate for various reasons, leading to anti-vaccination groups and vaccine apprehension. In this aspect, people are interested in gaining natural immunity. Notably, the individual's risk perception is fundamental for controlling the infection, while the fractional-order dynamics mainly dictate the degree of freedom with memory effect. Higher fractional order with EGT leads to an improved vaccination intake, while a cheaper and reliable vaccine ensures to lessening of contagious disease. The proposed model captures relevant disease behavior and the memory effect of a synchronized pandemic event, emphasizing the underlying role of social schemes.

A Semi-Analytical Method to Investigate Fractional-Order Gas Dynamics Equations by Shehu Transform

This work aims at a new semi-analytical method called the variational iteration transformation method for solving nonlinear homogeneous and nonhomogeneous fractional-order gas dynamics equations. The Shehu transformation and the iterative technique are applied to solve the suggested problems. The proposed method has an advantage over existing approaches because it does not require additional materials or computations. Four problems are used to test the authenticity of the proposed method. Using the suggested method, the solution proves to be more accurate. The proposed method can be implemented to solve many nonlinear fractional order problems because it has a straightforward implementation.

A Reliable Way to Deal with Fractional-Order Equations That Describe the Unsteady Flow of a Polytropic Gas

In this paper, fractional-order system gas dynamics equations are solved analytically using an appealing novel method known as the Laplace residual power series technique, which is based on the coupling of the residual power series approach with the Laplace transform operator to develop analytical and approximate solutions in quick convergent series types by utilizing the idea of the limit with less effort and time than the residual power series method. The given model is tested and simulated to confirm the proposed technique’s simplicity, performance, and viability. The results show that the above-mentioned technique is simple, reliable, and appropriate for investigating nonlinear engineering and physical problems.

New LADRC scheme for fractional-order systems

Active Disturbance Rejection Control (ADRC) is a very promising recent concept for the design of controllers to control systems with uncertainties. Based on an extended state observer, this control structure allows both external disturbances and modeling uncertainties to be actively rejected. The use of the active rejection control of disturbances is gaining more and more interest among the system control community. This is mainly due to its easy design, undemanding implementation and robustness. In this paper, three solutions using the ARDC structure to control fractional-order system are discussed. Two methods taken from the literature are briefly introduced, namely, the fractional-order dynamics rejection scheme (FODRS) and the fractional linear active disturbance rejection control (FLADRC). The basic ideas and technical formulations of these two are presented. A third method, based on the robustness performance introduced by fractional-order controllers, is also proposed in this paper. The proposed design method of the set-point tracking controller is based on the Bode’s ideal transfer function (Bode’s ITF). This will improve the robustness of the control scheme.

Study of fractional order dynamics of nonlinear mathematical model

This manuscript is devoted to establishing some theoretical and numerical results for a nonlinear dynamical system under Caputo fractional order derivative. Further, the said system addresses an infectious disease like COVID-19. The proposed system involves natural death rates of susceptible, infected and recovered classes respectively. By using nonlinear analysis feasible region and boundedness have been established first in this study. Global and Local stability analysis along with basic reproduction number have also addressed by using the next generation matrix method. Upon using the fixed point approach, existence and uniqueness of the approximate solution for the mentioned problem has also investigated. Some stability results of Hyers-Ulam (H-U) type have also discussed. Further for numerical treatment, we have exercised two numerical schemes including modified Euler method (MEM) and nonstandard finite difference (NSFD) method. Further the two numerical schemes have also compared with respect to CPU time. Graphical presentations have been displayed corresponding to different fractional order by using some real data.

Preliminary Observations on the Novel Approaches in Fuzzy Conformable Differential Calculus

This paper presents a novel extension of fuzzy calculus by integrating it with conformable calculus to introduce the fuzzy conformable derivative, a mathematical tool designed to handle the complexities of systems characterized by both uncertainty and fractional-order dynamics. The study begins by defining the fuzzy conformable derivative of order Ψ, which combines the ability of fuzzy calculus to model vagueness with the flexibility of conformable derivatives to capture non-integer order behaviors. This concept is further extended to higher orders, specifically the second order (2Ψ) and arbitrary order pΨ, enabling the modeling of more complex, multi-order dynamics in fuzzy-valued functions. Key contributions include the formal definitions of these derivatives, the establishment of their properties, and the development of operational rules that extend classical calculus operations to fuzzy systems. Additionally, the paper demonstrates how these derivatives can be applied to solve fuzzy differential equations and approximate functions using Taylor series expansions. While the fuzzy conformable derivative offers a powerful framework for modeling complex systems, the study also identifies certain limitations, such as potential unboundedness of solutions and the non-adherence to classical mathematical laws in higher-order cases. Overall, this work provides a comprehensive approach to differentiating fuzzy-valued functions in systems where uncertainty and fractional-order dynamics play a critical role. The proposed methods open up new avenues for research and application in fields such as control systems, economics, and engineering, where traditional calculus methods may not suffice. Future research directions include refining these methods, exploring computational techniques, and applying the framework to a broader range of real-world problems.

Coexisting multiple firing behaviors of fractional-order memristor-coupled HR neuron considering synaptic crosstalk and its ARM-based implementation

The crosstalk between synapses has a far-reaching impact on the specificity of synaptic communication in human brain. Therefore, the influence of synaptic crosstalk on the dynamic behaviors of neural network cannot be ignored. In this paper, we investigate the dynamics of fractional-order (FO) memristor-coupled Hindmarsh-Rose (HR) neuron model considering synaptic crosstalk. Firstly, the equilibrium points and stability are investigated via qualitative analysis, from which it can be obtained that the number and stability of equilibrium points have diversity for different crosstalk strength parameters. Secondly, the global coexistence of multiple firing patterns is reflected by phase diagrams, Lyapunov exponent spectrums and bifurcation diagrams. The local attraction basins reveal the multi-stability phenomenon related to the initial value. The parameter related firing behaviors are described by the two-parameter bifurcation diagram. In particular, the model shows the offset boosting control method for a single variable. Then, spectral entropy (SE) and C0 complexity chaos diagram are used to observe the change of system complexity when two parameters change simultaneously. Finally, the digital implementations based on ARM are given to verify the consistency with the numerical simulation results.

Analysis of fractional-order dynamics of dengue infection with non-linear incidence functions

The infection of dengue is a devastating mosquito-borne infection around the globe that affects human health, social and economic sectors in low-income areas. Therefore, policymakers and health experts are trying to point out better policies to reduce these losses and provide better information for the development of vaccination and medication. Here, we formulated a compartmental model for the transmission phenomena of dengue fever with nonlinear forces of infection through fractional derivative. We established several results related to the solution of our dengue model by using the basic properties of fractional calculus. We determined the basic reproduction number of our fractional-order system, symbolized by [Formula: see text]. We established the local asymptotic stability of the infection-free equilibrium of our dengue system for [Formula: see text], and proved that the infection-free equilibrium is globally asymptotically stable without vaccination. The threshold dynamics [Formula: see text] is tested through partial rank correlation coefficient method to notice the importance of parameters in the transmission of dengue infection. In addition, we have shown the impact of memory on the basic reproduction number numerically with the variation of different parameters. We conclude that the biting rate, recruitment rate of mosquitoes and index of memory are the most sensitive factors, which can effectively lower the level of dengue fever. The dynamical behavior of the proposed fractional system is presented through a numerical scheme to explore the overall transmission process. We predict that the fractional-order model can explore more accurately and preciously the intricate dengue disease transmission model rather than the integer-order derivative.

On Solutions of Fractional-Order Gas Dynamics Equation by Effective Techniques

In this work, the novel iterative transformation technique and homotopy perturbation transformation technique are used to calculate the fractional-order gas dynamics equation. In this technique, the novel iteration method and homotopy perturbation method are combined with the Elzaki transformation. The current methods are implemented with four examples to show the efficacy and validation of the techniques. The approximate solutions obtained by the given techniques show that the methods are accurate and easy to apply to other linear and nonlinear problems.

Design of an Analog Time-Varying Audio Cryptography System Based on Sliding Mode Synchronization of Non-identical Chaotic Systems Described with Time-Delayed Fractional-Order Dynamics

Design of an Analog Time-Varying Audio Cryptography System Based on Sliding Mode Synchronization of Non-identical Chaotic Systems Described with Time-Delayed Fractional-Order Dynamics

Fractional-order dynamics of Chagas-HIV epidemic model with different fractional operators

<abstract><p>In this research, we reformulate and analyze a co-infection model consisting of Chagas and HIV epidemics. The basic reproduction number $ R_0 $ of the proposed model is established along with the feasible region and disease-free equilibrium point $ E^0 $. We prove that $ E^0 $ is locally asymptotically stable when $ R_0 $ is less than one. Then, the model is fractionalized by using some important fractional derivatives in the Caputo sense. The analysis of the existence and uniqueness of the solution along with Ulam-Hyers stability is established. Finally, we solve the proposed epidemic model by using a novel numerical scheme, which is generated by Newton polynomials. The given model is numerically solved by considering some other fractional derivatives like Caputo, Caputo-Fabrizio and fractal-fractional with power law, exponential decay and Mittag-Leffler kernels.</p></abstract>

Modelling fractional-order dynamics of COVID-19 with environmental transmission and vaccination: A case study of Indonesia

<abstract><p>SARS-CoV-2 is the newly emerged infectious disease that started in Wuhan, China, in early December 2019 and has spread the world over very quickly, causing severe infections and death. Recently, vaccines have been used to curtail the severity of the disease without a permanent cure. The fractional-order models are beneficial for understanding disease epidemics as they tend to capture the memory and non-locality effects for mathematical models. In the present study, we analyze a deterministic and fractional epidemic model of COVID-19 for Indonesia, incorporating vaccination and environmental transmission of the pathogen. Further, the model is fitted to Indonesia's active cases data from 1 June 2021 to 20 July 2021, which helped determine the model parameters' value for our numerical simulation. Mathematical analyses such as boundedness, existence and uniqueness, reproduction number, and bifurcation were presented. Numerical simulations of the integer and fractional-order model were also carried out. The results obtained from the numerical simulations show that an increase in the contact rate of the virus transmission from the environment leads to an increase in the spread of SARS-CoV-2. In contrast, an increase in the vaccination rate negatively impacts on our model basic reproduction number. These results envisage here are essential for the control and possibly eradicate COVID-19 in Indonesia.</p></abstract>

Controllability of Flow-Conservation Transportation Networks with Fractional-Order Dynamics

In this paper, we adapt the fractional derivative approach to formulate the flow-conservation transportation networks, which consider the propagation dynamics and the users’ behaviors in terms of route choices. We then investigate the controllability of the fractional-order transportation networks by employing the Popov-Belevitch-Hautus rank condition and the QR decomposition algorithm. Furthermore, we provide the exact solutions for the full controllability pricing controller location problem, which includes where to locate the controllers and how many controllers are required at the location positions. Finally, we illustrate two numerical examples to validate the theoretical analysis.

On the analysis of Caputo fractional order dynamics of Middle East Lungs Coronavirus (MERS-CoV) model

The current paper deals with the transmission of MERS-CoV model between the humans populace and the camels, which are suspected to be the primary source for the infection. The effect of time MERS-CoV disease transmission is explored using a non-linear fractional order in the sense of Caputo operator in this paper. The considered model is analyzed for the qualitative theory, uniqueness of the solution are discussed by using the Banach contraction principle. Stability analysis is investigated by the aid of Ulam-Hyres (UH) and its generalized version. Finally, we show the numerical results with the help of generalized Adams-Bashforth-Moulton Method (GABMM) are used for the proposed model, for supporting our analytical work.

Observer-based robust controller design for fractional-order systems with stochastic perturbations: A diffusive representation approach

We investigate the fractional-order systems that are perturbed by stochastic input to achieve stabilization via sliding mode control (SMC) approach. It is assumed that the system states are unknown and there is uncertainty and time-delay in the system. We utilize the diffusive representation of the stochastic fractional-order dynamics to transform the system into an integer-order system perturbed by Brownian motion. Provided that some linear matrix inequalities (LMIs) are feasible, it is proven that the estimation error system is stochastically stabilized and the overall closed-loop system is stable in probability. A numerical simulation shows the effectiveness of the results.

Distributed Policy Evaluation with Fractional Order Dynamics in Multiagent Reinforcement Learning

The main objective of multiagent reinforcement learning is to achieve a global optimal policy. It is difficult to evaluate the value function with high-dimensional state space. Therefore, we transfer the problem of multiagent reinforcement learning into a distributed optimization problem with constraint terms. In this problem, all agents share the space of states and actions, but each agent only obtains its own local reward. Then, we propose a distributed optimization with fractional order dynamics to solve this problem. Moreover, we prove the convergence of the proposed algorithm and illustrate its effectiveness with a numerical example.

A computationally efficient adaptive robust control scheme for a quad-rotor transporting cable-suspended payloads

This article proposes a computationally efficient adaptive robust control scheme for a quad-rotor with cable-suspended payloads. Motion of payload introduces unknown disturbances that affect the performance of the quad-rotor controlled with conventional schemes, thus novel adaptive robust controllers with both integer- and fractional-order dynamics are proposed for the trajectory tracking of quad-rotor with cable-suspended payload. The disturbances acting on quad-rotor due to the payload motion are estimated by utilizing adaptive laws derived from integer- and fractional-order Lyapunov functions. The stability of the proposed control systems is guaranteed using integer- and fractional-order Lyapunov theorems. Overall, three variants of the control schemes, namely adaptive fractional-order sliding mode (AFSMC), adaptive sliding mode (ASMC), and classical Sliding mode controllers (SMC)s) are tested using processor in the loop experiments, and based on the two performance indicators, namely robustness and computational resource utilization, the best control scheme is evaluated. From the results presented, it is verified that ASMC scheme exhibits comparable robustness as of SMC and AFSMC, while it utilizes less sources as compared to AFSMC.

Distributed solving Sylvester equations with fractional order dynamics

Distributed solving Sylvester equations with fractional order dynamics

COMPLEX FRACTIONAL-ORDER HIV DIFFUSION MODEL BASED ON AMPLITUDE EQUATIONS WITH TURING PATTERNS AND TURING INSTABILITY

A weakly nonlinear analysis provides a system constituting amplitude equations and its related analysis is capable of predicting parameter regimes with different patterns expected to co-exist in dynamical circumstances that exhibit complex fractional-order system characteristics. The Turing mechanism of pattern formation as a result of diffusion-induced instability of the homogeneous steady state is concerned with unpredictable conditions. The Turing instability caused by fractional diffusion in a Human Immunodeficiency Virus model has been addressed in this study. It is important that the effect of the Human Immunodeficiency Virus to the immune system can be modeled by the interaction of uninfected cells, unhealthy cells, virus particles and antigen-specific. Initially, all potential equilibrium points are defined and the stability of the interior equilibrium point is then evaluated using the Routh–Hurwitz criteria. The conditions for Turing instability are obtained by local equilibrium points with stability analysis. In the neighborhood of the Turing bifurcation point, weakly nonlinear analysis is employed to deduce the amplitude equations. After applying amplitude equations, it is observed that this system has a very rich dynamical behavior. The constraints for the formation of the patterns like a hexagon, spot, mixed and stripe patterns are identified for the amplitude equations by dynamic analysis. Furthermore, by using the numerical simulations, the theoretical results are verified. Within this framework, this study through the dynamical behavior of the complex system perspective and bifurcation point based on the viral death rate can provide the basis for several researchers working on Human Immunodeficiency Virus model through various aspects. Accordingly, the Turing bifurcation point and weakly nonlinear analysis employed within the complex fractional-order dynamics addressed herein are highly relevant experimentally since the related effects can be studied and applied concerning different mathematical, physical, engineering and biological models.

Stability analysis and optimal control of Covid-19 pandemic SEIQR fractional mathematical model with harmonic mean type incidence rate and treatment

In the present work, we investigated the transmission dynamics of fractional order SARS-CoV-2 mathematical model with the help of Susceptible S(t), Exposed E(t), Infected I(t), Quarantine Q(t), and Recovered R(t). The aims of this work is to investigate the stability and optimal control of the concerned mathematical model for both local and global stability by third additive compound matrix approach and we also obtained threshold value by the next generation approach. The author’s visualized the desired results graphically. We also control each of the population of underlying model with control variables by optimal control strategies with Pontryagin’s maximum Principle and obtained the desired numerical results by using the homotopy perturbation method. The proposed model is locally asymptotically unstable, while stable globally asymptotically on endemic equilibrium. We also explored the results graphically in numerical section for better understanding of transmission dynamics.