We investigate a stochastic eco-epidemiological framework where disease transmission occurs within the prey population and interacts with the predators under environmental noise. By employing stochastic Lyapunov methods and Khasminskii’s theory, we establish persistence-extinction conditions and confirm the existence of an ergodic stationary distribution. Furthermore, we determine the threshold parameters that govern the long-term dynamics of the system. Theoretical results are supported through numerical simulations which illustrate how variations in key biological and noise parameters affect the coexistence and extinction of species. The study highlights the significant influence of stochastic perturbations on disease dynamics and provides useful insights into the stability of eco-epidemiological systems under random environmental effects.

This work investigates the exponential synchronization of Clifford-valued coupled neural networks (NNs) subject to time-varying delays within the framework of time scales, using a unified matrix measure method. To reduce the control burden and enhance practical applicability, an aperiodically intermittent control strategy is designed, which activates over irregular time intervals rather than continuously. The time-scale setting offers a unified treatment of both continuous time and discrete time models. Without decomposing the Clifford-valued terms, synchronization criteria are directly established in the Clifford algebra setting. Through the matrix measure approach combined with Lyapunov analysis, a collection of sufficient criteria is obtained to guarantee global exponential synchronization. Furthermore, a numerical illustration is presented to validate the accuracy and effectiveness of the proposed theoretical findings across continuous, discrete, and hybrid domains.

We investigate a variable-order discrete fractional mathematical model to assess the impact of diabetes and its associated complications. The main findings demonstrate the solvability and stability of the proposed system. Additionally, we present various numerical results to evaluate the proposed model effectively. Finally, comparison results emphasize the advantages of using fractional derivatives.

In this paper, we investigate general solutions and periodic solutions of quaternion difference equations (QDCEs) with variable coefficients. To begin with, we provide general solutions for linear homogeneous QDCEs (LHQDCEs), an algorithm for computing the fundamental matrix and its properties, and derive general solutions for linear nonhomogeneous QDCEs (LNHQDCEs) using the variation of constants formula as well as for semilinear QDCEs (SLQDCEs) using the fixed-point theorem. Secondly, the conditions that ensure the existence of periodic solutions for LHQDCEs are presented, thereafter, periodic solutions of LNHQDCEs under different conditions are derived using the Green function and adjoint system, respectively. Moreover, we establish the existence and uniqueness of periodic solutions of SLQDCEs. Finally, several examples are presented to demonstrate the correctness of the theoretical results.

This paper addresses an attraction–repulsion chemotaxis system governed by Neumann boundary conditions within a bounded domain Ω⊂R3 that has a smooth boundary. The primary focus of the study is the chemotactic response of a species (cell population) to two competing signals. We establish the existence and uniqueness of a weak solution to the system by analyzing the solvability of an approximate problem and utilizing the Leray–Schauder fixed-point theorem. By deriving appropriate a priori estimates, we demonstrate that the solution of the approximate problem converges to a weak solution of the original system. Additionally, we conduct computational studies of the model using the finite element method. The accuracy of our numerical implementation is evaluated through error analysis and numerical convergence, followed by various numerical simulations in a two-dimensional domain to illustrate the dynamics of the system and validate the theoretical findings.

We investigate a class of piecewise variable-order fractional differential equations with impulsive and nonlo cal conditions in Banach space. The nonhomogeneous term in the proposed system is given in terms of variable kernel which has flexibility property. We formulate appropriate equivalent integral equations to the considered evolution problem, then we show the solvability results by using mainly fractional calculus and fixed point techniques. Further, we study Hyers-Ulam stability analysis by adapting suitable conditions. The concerned area has numerous applications in those evolution processes and phenomenon, where abrupt changes occur. At the end, we support our obtained theory by illustrative and computational example.

In this paper, we investigate the existence and nonexistence of weak solutions to an ultra-parabolic differential inequality posed on the interval (0, 1), under an inhomogeneous Dirichlet boundary condition. The considered problem involves the singular Bessel operator d2 dx2 + alpha x d dx and a nonlinearity of the form x-6|u|p. Our approach is based on nonlinear capacity estimates and an appropriate choice of test functions.

We formulate an epidemic mathematical model with multiple delays to explore the dynamics of the disease. First, we introduce the SIR model that incorporates vaccination and quarantine components in conjunction with consideration of multiple time delays. Additionally, Hopf bifurcation appears when the delay reaches a critical value, as demonstrated in our stability analysis, especially when two-time delays are simultaneously considered bifurcation parameters, producing oscillatory dynamics. The local stability of the endemic and disease-free equilibrium points (DFE) is studied in detail for all possible delay configurations. We also perform a sensitivity analysis of the basic reproduction number (BRN) concerning various model parameters. As part of the control measures, the effect of public awareness through educational campaigns is taken into account. An optimal control framework is then developed to minimize disease spread and control expenses, and the corresponding control approach is elaborated. The resolution of this optimization problem is presented, accompanied by numerical simulations to illustrate the analytical findings.

We investigate the dynamics of the hepatitis B virus by integrating variable-order calculus and discrete analysis. Specifically, we utilize the Caputo variable-order difference operator in this study. To establish the existence and uniqueness results of the model, we employ a fixed-point technique. Furthermore, we prove that the model exhibits bounded and positive solutions. Additionally, we explore the local stability of the proposed model by determining the basic reproduction number. Finally, we present several numerical simulations to illustrate the richness of our results.

The mathematical modeling of infectious diseases plays a vital role in understanding and predicting disease transmission, as underscored by recent global outbreaks; to delve deep into the dynamic of infectious disease considering latent period presciently is inevitable as it bridges the gap between realistic nature and mathematical modeling. This study extended the classical Susceptible–Infected–Recovered (SIR) model by incorporating vaccination strategies during incubation. We introduced multiple time delays to an account incubation period to capture realistic disease dynamics better. The model is formulated as a system of delay differential equations that describe the transmission dynamics of diseases such as polio or COVID-19, or diseases for which vaccination exists. Critical aspects of the study include proving the positivity of the model’s solutions, calculating the basic reproduction number (R0) using next-generation matrix theory, and identifying disease-free and endemic equilibrium points. The local stability of these equilibria is then analyzed using the Routh–Hurwitz criterion. Due to the complexity introduced by the delay components, we examine the stability by studying the roots of a fourth-degree exponential polynomial. The effects of educational campaigns and vaccination efficacy are also investigated as control measures. Furthermore, an optimization problem is formulated, based on Pontryagin’s maximum principle, to minimize the number of infections and associated intervention costs. Numerical simulations of the delay differential equations are conducted, and a modified Runge–Kutta method with delays is used to solve the optimal control problem. Finally, we present a few simulation results to illustrate the analytical findings.

This article introduces a discrete-time fractional variable order over a SEIQR model, incorporated for COVID-19. Initially, we establish the well-possedness of solution. Further, the disease-free and the endemic equilibrium points are determined. Moreover, the local asymptotic stability of the model is analyzed. We develop a novel discrete fractional optimal control problem tailored for COVID-19, utilizing a discrete mathematical model featuring a variable order fractional derivative. Finally, we validate the reliability of these analytical findings through numerical simulations and offer insights from a biological perspective.

We examine a nonlinear dynamical model that depicts the interaction between cancerous cells and an oncolytic virus. For best modelling the disease, we use the Caputo fractional derivative in piecewise approaches. By employing piecemeal techniques, we treat a compartment in the body that contains infectious and non-infectious cells. More precisely, the solvability and Ulam-Hyers (U-H) stability results are considered using standard concepts. Further, to support our investigation with numerical results, we apply the Euler method to develop an approximation solution. It connected with numerous graphical representations of the system using various arbitrary ordering and varying values of the isolation parameters. Here we remark that the multi-step behavior that certain problems exhibit, is one of important issues naturally. This paper introduces the idea of piecewise derivative with the goal of modeling real-world issues that follow multiples processes. With the help of the used approach, we investigate the cancer disease model and its transmission dynamical behavior with crossover effect.

Humans have been affected by various epidemic diseases, mostly are airborne and exhibit high transmission rates. Given these nature properties, quarantine measures are essential to control the spread of the diseases effectively. Motivated by this fact, and due to the successful use of mathematical modeling, we investigate a SIR model with quarantine and vaccination compartments. This model uses a system of fractional differential equations (SIQVR-based) with specific parameters to track the dynamics of model variables. We examine the well-posedness and boundedness results via standard tools. An effective threshold parameter ℛ_0 is determined using a generation matrix and equilibrium points of the model are obtained. To effectively manage the transmission of infection within the outlined model, we employ the strategy of optimal control. This approach involves implementing control measures and interventions guided by mathematical optimization techniques to minimize the spread of disease. These control strategies may encompass vaccination campaigns, quarantine protocols, social distancing measures and other preventive actions. Further, to evaluate the effectiveness of proposed model and the applied optimal control strategy, we conduct a series of numerical simulations. Computational results involve running the model under different scenarios, considering a range of parameters and meticulously analyzing the resulting outcomes.

In light of the pressing issue of climate change and escalating global carbon emissions, this study investigates the efficacy of blue carbon ecosystems - namely, mangroves, sea-grasses, and tidal marshes - in carbon sequestration and climate mitigation. Focused on formulating actionable strategies, we examine these ecosystems' capacity to serve as natural carbon sinks under nutrient stress conditions. Utilizing a sophisticated mathematical model that incorporates both deterministic and stochastic elements, we simulate the carbon and nitrogen cycling within these ecosystems to capture the complexity and variability of natural processes. Our equilibrium analysis identifies critical thresholds at which these ecosystems optimally function as carbon sinks. Furthermore, our sensitivity analysis highlights key parameters such as nutrient availability and hydrodynamic conditions that significantly influence these thresholds. Results indicate that management practices focusing on nutrient regulation could enhance the carbon sequestration potential of these ecosystems. The insights derived from this research not only deepen our understanding of the role of blue carbon in climate mitigation but also offer concrete recommendations for policymakers and conservationists to optimize these natural resources in combating global warming.

We introduce an epidemic disease reaction–diffusion model to study the transmission of the varicella-zoster virus in both space and time. More precisely, we present a system of partial differential equations with the Neumann boundary conditions (NBC) concerned to model the evolution of the virus. Firstly, the wellposedness results of the model are studied using the semigroup theory. Then, the boundedness of the solutions is also derived. Further, the basic reproduction number (BRN) for the proposed model is determined using the eigenvalue problem. Moreover, asymptotic profiles of the equilibrium points of the susceptible and infected compartments of the model are investigated. Finally, the advantage of the spatiotemporal model and the above theoretical results are validated with numerical experiments.

Abstract This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

We herein report a new class of impulsive fractional stochastic differential systems driven by mixed fractional Brownian motions with infinite delay and Hurst parameter H^∈(1/2,1). Using fixed point techniques, a q-resolvent family, and fractional calculus, we discuss the existence of a piecewise continuous mild solution for the proposed system. Moreover, under appropriate conditions, we investigate the approximate controllability of the considered system. Finally, the main results are demonstrated with an illustrative example.

We establish a class of nonlinear fractional differential systems with distributed time delays in the controls and impulse effects. We discuss the controllability criteria for both linear and nonlinear systems. The main results required a suitable Gramian matrix defined by the Mittag–Leffler function, using the standard Laplace transform and Schauder fixed-point techniques. Further, we provide an illustrative example supported by graphical representations to show the validity of the obtained abstract results.