This paper explores the impact of prey group cooperation on predator-prey dynamics through a novel mathematical model incorporating a Caputo fractional derivative and gestation delay. Solutions' existence, uniqueness, and boundedness of solutions are verified within the framework. The stability analysis indicates that the coexistence equilibrium point is globally stable and that periodic oscillations are caused by the Hopf bifurcation. Our results reveal a critical link between model order, prey refuge rate, and cooperation level. As the model order decreases or the prey refuge rate and cooperation level diminish, the system transitions from unstable to stable behavior. These findings suggest that while strong memory (represented by a higher model order) hinders stable coexistence, weaker memory (lower order) can promote it. This study highlights the significance of incorporating memory effects and prey behavior into predator-prey models for a more comprehensive understanding of population dynamics.

This study offers a comprehensive analysis of infectious disease transmission, focusing specifically on HIV transmission among populations classified by biological sex. The primary objective of the project is to develop a sex-specific compartmental model that incorporates memory-dependent dynamics and time lags to more accurately depict the temporal evolution of infections. The work utilised bifurcation analysis to systematically assess the stability properties of the disease-free and endemic equilibrium points. A sensitivity study clarifies the epidemiological factors that significantly impact transmission intensity in the community. The analytical conclusions are validated through computational experiments, with outcomes illustrated in clear graphical representations. The results emphasise that timely identification of infection, along with tailored antiretroviral medication, significantly improves the likelihood of epidemic control. The authors advocate for further research that integrates more intricate transmission channels and ground model outputs with real epidemiology data crucial measures for enhancing intervention frameworks in light of evolving disease transmission dynamics.

INTRODUCTION AND OBJECTIVE:Diabetes is a chronic metabolic disorder characterized by higher blood glucose levels due to insufficient insulin production, weakened insulin action, or both. This study aims to construct and analyze a fractional-order mathematical model using the Anatanga-Baleanu derivative to investigate how glucose levels in the human body are controlled through interactions with insulin and glucagon. The objective is to thoroughly understand these interactions, which could improve current treatments and medications. METHODS:A fractional-order diabetes model has been developed. Fixed points and their Ulam-Hyers stability are analyzed to ensure the mathematical reliability of the model. The qualitative dynamics are examined to capture glucose-insulin-glucagon interactions in detail. The explicit fractional Euler method is utilized to demonstrate the efficiency of the model for numerical results. Also, the existence and uniqueness of the solutions are obtained. Finally, numerical simulations validate the significance of the results, underscoring their relevance to diabetes research and treatment strategies. RESULTS:The findings reveal the dynamic interactions among glucose, insulin, and glucagon in diabetes, showing that higher parameter values increase peak amplitudes and prolong transients. The fractional-order glucose-insulin model, with greater flexibility than classical derivatives, captures memory effects that influence diabetes progression and highlights potential therapeutic targets for improved disease management. CONCLUSION:This study provides a deeper understanding of diabetes through a detailed analysis of the proposed fractional-order glucose-insulin interactive model by addressing its well-posedness, steady-state behavior, and Ulam-Hyers stability. It bridges theoretical models and experimental observations, clarifying insulin-glucagon interactions. Further, it provides a robust framework for exploring glucose regulation mechanisms and may guide researchers and clinicians in developing more effective therapeutic strategies for diabetes and related metabolic disorders.

The fractional‐order Atangana–Baleanu in Caputo sense fractional derivative (ABC) approach to the smoking illness model, which covers the impacts of externally and internally affected smokers on smoking, is introduced in this work. The model's authenticity has been investigated further in terms of its benefits and drawbacks. Model solution systems constructed using fixed point theory and repeated techniques are present and distinctive. The suggested approach was used to do numerical simulations to show the consequences of partial layout modifications and to back up theoretical conclusions.

This paper investigates predator-prey interactions to explore the complex dynamics arising from the Allee effect on prey populations, incorporating an Ivlev functional response. The analysis focuses on the occurrence and stability of equilibrium points. The study demonstrated that the system undergoes period-doubling (PD) and Neimark-Sacker (NS) bifurcations at the positive fixed point, utilizing center-manifold analysis and bifurcation theory. The chaotic behaviors of the system were identified using bifurcation diagrams and maximum Lyapunov exponent graphs. The system's bifurcating and fluctuating behavior can be regulated using OGY control methods. Bifurcations in a discrete predator-prey model within a coupled network were examined. Numerical simulations demonstrated that chaotic behavior emerges in complex dynamical networks once the coupling strength parameter reaches a critical value. Furthermore, the Euler-Maruyama method was utilized for stochastic simulations to explore the system under environmental uncertainty, taking into account diverse environmental scenarios. All theoretical findings related to stability, bifurcations, and chaotic transitions in the coupled network were validated through numerical simulations. This study highlights the value of discrete models in capturing the full range of potential dynamics in ecological systems, as they can uncover complex phenomena - such as bifurcations and chaos - that may remain hidden in continuous-time frameworks. Such insights are crucial for advancing our understanding of ecological processes and for accurately modeling real-world species interactions.

Compartmental models, in particular, are essential to epidemiology because they provide a basic mathematical framework for comprehending such systems and can be used to anticipate the evolution of diseases, illustrate epidemic consequences and facilitate public health interventions. For low-immune people, a mathematical model is created utilizing IL2 and anti-PD-L-1 inhibitors, suggesting that boosting the immune system with antibody cells can enhance it. Using a hybrid fractional-order derivative, the model is further transformed into a fractional-order system. The purpose of this work is to analyze fractional models to have a better understanding of how the order of the fractional derivative influences the spread of lung cancer. A new system, TCDIL(2)Z, is examined for stability, boundedness, positivity and uniqueness. The system's global stability is also investigated using Lyapunov's first derivative functions. Fractional-order differential equations are solved using the Laplace Adomian Decomposition method, and tables and graphs are included to improve the precision of the numerical findings. Simulations are conducted to identify control situations after detection and treatment. According to the study, the fractional-order lung cancer model can better visualize the dynamics of the disease since it shows a memory effect, in contrast to the classical model. Additionally, it shows that fractional-order derivations have greater reliability than classical order in the explanation of bodily approaches. This research will aid in understanding the spread of the disease and developing control strategies based on justified outcomes.

In this work, the dynamical behavior of the cholera epidemic model is examined by using a fractal fractional technique with an exponential decay kernel with treatment in society. System analysis with results from the theory of fixed points is carried out both qualitatively and quantitatively. The existence and biological viability of the system are verified by analyzing the global derivative, linear growth, and Lipschitz criteria, ensuring that the model remains biologically plausible and consistent with real-world ecological dynamics. Additionally, the unique solution’s positivity and boundedness are discussed. Using wave analysis to construct the first and second derivatives for the Lyapunov function based on the equilibrium point and reproductive number, the proposed model’s global stability is built. A sensitivity analysis is carried out to determine how various parameters affect the fractional order model. Numerical simulations are derived using a two-step Lagrange polynomial technique with an insight into the exponential decay kernel with a fractional operator. Results validate theoretical and experimental findings by employing local as well as non-singular kernels at various fractional order values and fractal dimensions to show the strong memory effect. The model’s analytical properties are determined, and simulations illustrate potential methods for lowering the disease’s endemic levels in the community, which would improve human health.

This study investigates the application of the Caputo derivative with a variable fractional order to time-dependent models of Ordinary Differential Equations (ODEs), aiming to enhance the simulation accuracy of dynamic systems characterized by complex, nonlinear temporal behaviors. The proposed approach provides a more refined understanding and predictive capability for non-constant real-world phenomena, contributing to the development of advanced scientific and engineering solutions. The research centers on a variable-order Lotka-Volterra predator-prey model, employing the Arzela`-Ascoli and Schaefer fixed point theorems to establish the existence of solutions, and the Banach fixed point theorem to demonstrate their uniqueness. Numerical analyses are conducted to compare the proposed model with its integer-order, fractional-order, and variable-order counterparts, utilizing various time-varying and constant delay functions. The findings validate the efficacy of the proposed method in accurately modeling dynamic systems.

This study gives us an expression of non-integer order in mathematics via fractional Caputo operator just for the broadcast development of different emotions under emergencies due to any situation or some disease. In this work, we fear the effects of COVID-19 in panic situations, considering incidence data by using power law kernels under a fractal fractional operator. The effects of the emotion that causes COVID-19 are also evaluated locally and globally using stability. Based on the fractional order model of COVID-19 viral infection, equilibrium points devoid of illness, well-posedness, uniqueness, and biological viability of solutions are all demonstrated. The effects of the COVID-19 model’s sensitivity analysis with treatment were also investigated. Unique solution and picards stability of iterative scheme verified by using the fixed point theory concept. To discover the solution of the fractional order system and evaluate the effect of fractional parameters, an advanced numerical approach is applied. In the simulation, all classes are shown to have convergent properties and to hold their positions over time, which accurately depicts how COVID-19 infection behaves in practice. We find a more comparable outcome when comparing non-integer orders to integer orders, which supports the non-integer order’s position. This model’s tools seem to be reasonably strong and capable of creating the predicted theoretical conditions for the problem.

The Human Immunodeficiency Virus (HIV) attacks particular immune system cells such as Tcells (primarily CD+4T cells) and triggers lifetime severe sickness with a prolonged incubation period. This study develops and analyzes a novel mathematical model to understand the spread of the virus, using real-world data reported cases in Taiwan from 2000 to 2023. The mathematical properties of the model, such as existence, uniqueness, positivity, and boundedness, are rigorously examined to ensure reliability. Equilibrium points are determined, and their stability is analyzed to understand the long-term behavior of the disease. The fundamental reproduction number is obtained using the next generation approach. Sensitivity analysis is performed using different variables as response functions each time, employing Latin Hypercube Sampling and Partial Rank Correlation Coefficient with 200 iterations. Theoretical results are validated using numerical simulations and graphically display the impacts of different model parameters. Results indicate that reducing contact with infected individuals and accelerating disease management interventions can significantly lower the burden of infection.

Schistosomiasis is assumed to be one of the deadliest and overlooked endemic diseases found in tropical and sub-tropical regions, which is increasing at a concerning rate annually. The World Health Organization (WHO) has introduced a new roadmap for neglected tropical diseases from 2021 to 2030, with the elimination of schistosomiasis from all endemic countries as a global objective. In this paper, we develop a compartment model to represent the disease transmission and to understand the impact of parameters including public health awareness, number of parasite eggs, miracidia emergence rate, snail control parameter, and efficacy of pesticides on the dynamics of disease transmission. The basic reproduction number, R0 is calculated using the next-generation matrix method, which serves as a crucial indicator of infection risks. The local stability of disease-free equilibrium and endemic equilibrium is established analytically using R0. Through sensitivity analysis, the key parameters are identified that significantly affect R0. Based on our model analysis, we hypothesize that massive public health awareness along with moderate snail control with efficacious molluscicides (pesticides) is the most sustainable intervention strategy to eradicate the disease from the tropical and sub-tropical regions.

This study examines the discrete-time dynamics of a predator-prey model incorporating an Ivlev functional response with an Allee effect. Through rigorous algebraic analysis, we establish the occurrence of period-doubling (PD) and Neimark–Sacker (NS) bifurcations in the positive phase space. Using the center-manifold theorem and bifurcation theory, we provide a theoretical framework to understand these bifurcations. Numerical simulations confirm our findings, illustrating chaotic behavior through phase portraits, period-12 orbits, invariant closed curves, and chaotic attractors. Lyapunov exponents further validate the chaotic nature of the system, emphasizing the influence of parameter variations on dynamic transitions. To mitigate chaos, we implement an OGY control strategy, effectively stabilizing trajectories around an unstable equilibrium. Furthermore, we extend the analysis to a coupled predator-prey network, revealing that chaotic behavior emerges when the coupling strength exceeds a critical threshold. Stochastic simulations, employing the Euler-Maruyama method, account for environmental uncertainties and explore system behavior under varying ecological conditions. This research advances the understanding of non-linear predator-prey interactions, bifurcations, and chaotic transitions in isolated and networked systems while demonstrating effective strategies for controlling instability in ecological dynamics.

The numerical simulation of biological processes with non-integer ordering is attracting an increasing amount of interest from scientists and academics. Traditional biological systems can be presented in a fixed order, but fractional-order derivative systems are not considered stable orders. When the fractional derivative has a non-fixed order, it becomes more useful for simulating real-world problems. In this paper, we aim to study the dynamics of a novel technique that we propose, implement, and use in a radiation model for the treatment of cancer. We present some intriguing results for the cancer treatment fractal fractional model in the context of this innovative operator. Research has been done on the cancer model in both qualitative and quantitative manners. The first and second derivatives of the Lyapunov function are used to analyze the stability of the cancer fractal fractional model. Using the linear growth theory, the existence of a unique solution has been derived under the FFM. Lagrangian-piece-wise interpolation has been used to obtain numerical results for various fractal-fractional operators. The fractal fractional model was used to simulate the treatment process of three patients. Different values of fractional order [Formula: see text], fractal dimension [Formula: see text], and other parameter values have been used to show the graphs. Additionally, we looked at how radiation changed both healthy cells and malignant cells over time. The study confirmed the effectiveness of radiation medicine against populations as well as the occurrence of the memory effect during [Formula: see text] and [Formula: see text] transitions from 1. A biological process requires fractal-fractional processes which provide superior modeling capabilities compared to traditional fractional operators as well as classical operators. This research brings novel significance through its implementation of fractal-fractional operators as they provide a superior approach to model cancer treatment processes by better representing biological system complexities. Standard modeling systems cannot reproduce both important memory dynamics together with non-local communication patterns which play essential roles in cancer development and treatment analysis. The implementation of fractal-fractional derivatives enables our model to produce a more realistic representation of cancer cell and healthy cell radiotherapy responses throughout time. Our study has upgraded theoretical cancer dynamic analysis and developed optimized treatment methods for customization purposes. Wider understanding of cancer cell reactions to treatments enables healthcare providers to adopt personalized strategies that produce superior recovery outcomes for their patients. The model acquires stability strength through Lyapunov functions analysis to create a solid scientific foundation in oncology research.

This paper presents a novel mathematical model for diabetes transmission and progression, called SDC, which captures the dynamics of susceptible, uncomplicated, and complicated diabetes cases. Further, the study proposes a fractional-order technique for the diabetes epidemic model to observe the dynamics of disease in society with complications and non-complications. We treated positiveness, boundedness of solutions, and positively invariant regions for this. Through the fixed-point theory, the effect of global derivatives is explored. Also, the uniqueness and existence of solutions are verified. The stability analysis of the model employing the Lyapunov method approach applicable for first and second-order derivative tests is carried out, revealing how the collapse of equilibrium points is influenced by the propagation rate in the disease dynamics. Sensitivity analysis is used to identify the key parameters that drive the propagation rate. For the proposed study using a two-step Lagrange polynomial, we construct the comparative results with a generalized version of the Mittag-Leffler kernel through simulation of various orders of fractional derivative for α . Through numerical simulations, the theoretical findings are illustrated, highlighting the potential of the SDC model for informing effective diabetes management strategies. Overall, this study contributes to the development of the field of controlling diabetes transmission and providing control measures for its progression.

This work investigates the dynamical properties of a discrete-time predator–prey system obtained from a continuous model using the piecewise constant argument method. We study the existence and stability of fixed points. It is shown that the system undergoes period-doubling and Neimark-Sacker bifurcations at the positive fixed point. Further numerical examples confirm these analytical findings and elucidate how variations in parameter values may result in transitions from stability to chaos and even species extinction. The findings emphasize the increased intricacy of discrete-time models in comparison with continuous-time models. Specifically, the discrete-time system not only undergoes bifurcations but also includes chaotic dynamics, underlining the subtle and multifaceted nature of ecological interactions in discrete time. This work emphasizes the need to use discrete models to accurately represent a wider variety of dynamic behaviors in ecological systems, such as bifurcations and chaos, which are often ignored in continuous-time models. For better knowledge of ecological dynamics and the correct modeling of actual ecological interactions, these insights are essential.

The multiple sequence alignment (MSA) problem is a central issue in bioinformatics, with significant applications in the evolutionary analysis of biological sequences, identification of conserved regions, protein structure prediction, gene annotation, and function prediction. MAFFT is currently one of the most widely used tools for solving the MSA problem. This article aims to improve the limitations of MAFFT in solving the MSA problem by using a multi-objective evolutionary algorithm (MOEA) to enhance the quality of the iterative search. To this end, a new three-objective model is defined, and a novel MOEA is proposed to optimize this model. Four mutation and two crossover operators are designed to generate promising offspring individuals during the evolutionary process. To evaluate the effectiveness of the proposed method, computational experiments are conducted on 60 instances randomly selected from the BAliBASE 3.0 database. The results show that the proposed method can improve the quality of the iterative search and output alignment of MAFFT in terms of the Q and TC metric scores. This study provides new insights into how to better utilize existing tools to solve the MSA problem and offers a promising direction for improving the quality of their output alignments in future research.

Lumpy skin disease (LSD) is a potentially deadly viral disease of cattle that causes nodules on the skin, fever, and swollen lymph nodes. If left untreated, LSD can result in severe economic losses owing to reduced milk production, weight loss, and even death. To monitor the increasing prevalence of LSD and develop cost-effective prevention strategies, disease modeling is a crucial tool. We propose a fractional-order mathematical model for LSD that captures the interval between an individual contracting the infection and the onset of symptoms using the Mittag-Leffler kernel. The proposed system undergoes a qualitative study, including a quantitative investigation and a well-posedness assessment. We create endemic and disease-free equilibrium points and analyze the proposed system’s stability. The major reproductive number condition is derived for a newly developed system to verify the rate of spread of the lumpy virus. The first derivative tests are utilized using the Lyapunov function for the analysis of global stability. In addition, fixed-point theory and the Lipschitz condition are employed to verify that the precise solution is bounded and unique, which are the key properties to verify the newly developed model. To analyze the impact of the fractional operator using numerical simulations and highlight the influence of the disease through various parameters, a two-step Lagrange polynomial is utilized based on the generalized Mittag-Leffler kernel for approximate solutions. The model’s robustness in predicting an infectious disease like LSD, which can assist managers in taking preventive measures, such as gathering adequate human, pharmacological, and logistical resources.

Biological studies have shown that macrophages play a dual role in cancer dynamics. Through the secretion of different factors, they can modulate the tumor microenvironment and directly interact with tumor cells. Active macrophages, more precisely tumor‐associated macrophages (TAM), can secrete substances that increase the proliferation and survival of tumor cells, thus promoting tumor growth. This study explores the dynamic interactions between tumor carcinogenesis and macrophage activation through a newly developed mathematical model using the Caputo sense fractional operator. The main focus is to analyze the influence of macrophage polarization on tumor progression and cancer development. To precisely capture the dynamic behavior of macrophage‐tumor interactions, the model is verified for qualitative analysis. The fixed points of the model are obtained and their stability analysis is carried out. It is started by explaining the essential components of the model and then assuring its mathematical reliability. The Adams–Bashforth–Moulton method (ABM) is also discussed to show its applicability and efficiency for the numerical results of the developed model. The findings shed light on cancer progress, potential analyses, and the influence of public information. The outcome of the study offers a more detailed understanding of possible treatment targets by highlighting the delicate balance between macrophage pro‐ and anti‐tumor effects. The outcomes might improve public health policy, healthcare budget apportionment, and contest tumors and related infections. Further, it improves the understanding of tumor‐immune system interactions and lays the solid basis for cancer treatment research. Finally, numerical simulations make our outcomes more significant.

This paper explores the dynamic properties of a discrete-time commensalism system using the forward Euler method based on a continuous model. It investigates the existence and stability of all possible trivial, boundary, and positive fixed points of the system. Additionally, it demonstrates that the system undergoes period-doubling bifurcation at the positive fixed point. Numerical simulation results are provided to support the theoretical analysis. The study suggests that harvesting plays a crucial role in stabilizing the population sizes of both species. The system can maintain stability when a significant portion of the stock is available for harvesting. However, as the accessible stock decreases, the system becomes more susceptible to changes, ultimately leading to destabilization and the potential collapse of the ecological community.

The current best-known performance guarantees for the extensively studied Traveling Salesman Problem (TSP) of determinate approximation algorithms is 32, achieved by Christofides' algorithm 47 years ago. This paper investigates a new generalization problem of the TSP, termed the Minimum-Cost Bounded Degree Connected Subgraph (MBDCS) problem. In the MBDCS problem, the goal is to identify a minimum-cost connected subgraph containing n=|V| edges from an input graph G=(V,E) with degree upper bounds for particular vertices. We show that for certain special cases of MBDCS, the aim is equivalent to finding a minimum-cost Hamiltonian cycle for the input graph, same as the TSP. To appropriately solve MBDCS, we initially present an integer programming formulation for the problem. Subsequently, we propose an algorithm to approximate the optimal solution by applying the iterative rounding technique to solution of the integer programming relaxation. We demonstrate that the returned subgraph of our proposed algorithm is one of the best guarantees for the MBDCS problem in polynomial time, assuming P≠NP. This study views the optimization of TSP as finding a minimum-cost connected subgraph containing n edges with degree upper bounds for certain vertices, and it may provide new insights into optimizing the TSP in future research.