This study investigates the efficacy of nonstandard finite difference (NSFD) schemes in enhancing stability of explicit SFD schemes for 1D and 2D Caputo-type time-fractional diffusion equations (TFDEs). The Caputo fractional derivative introduces a nonlocal temporal memory effect, allowing the system dynamics to depend on the entire history of the solution rather than only its current state. Unlike existing NSFD approaches for fractional problems which apply nonstandard discretizations only to integer-order terms, present work introduces a nonstandard L 1 approximation of the Caputo fractional derivative on a graded mesh. The local truncation error of this approximation is derived, and its performance is validated through numerical simulations on test examples for various choices of denominator functions. Its absolute stability on a uniform mesh is rigorously examined using the boundary locus method. Based on this framework, explicit NSFD schemes for 1D and 2D Caputo-type TFDEs are developed on a uniform mesh. Their stability is further assessed using the discrete energy method, with particular focus on expanding the stability region. The convergence of the proposed NSFD schemes is also established. Finally, numerical experiments are conducted to demonstrate the accuracy and stability advantages of the proposed methods. The results are presented through tabular and graphical illustrations.

The study introduces a mathematical model for the transmission dynamics of canine Chagas disease caused by Trypanosoma cruzi within a host–vector framework. The interaction between triatomine vectors and peridomestic dogs is described by a deterministic delay differential equation model incorporating biologically relevant time delays associated with vector development and infection processes. The vector population is structured into developmental stages, while the adult population is further stratified based on feeding behavior to reflect epidemiological relevance. The qualitative properties of the model are rigorously analyzed, including positivity, boundedness, and existence of solutions, ensuring biological feasibility. Equilibrium points are derived, and the basic reproduction number is computed using the next-generation matrix method to characterize the threshold dynamics of disease transmission. A sensitivity analysis of the reproduction number is also performed to identify key epidemiological parameters influencing disease spread. First-order and second-order nonstandard finite difference (NSFD) schemes are developed to preserve essential qualitative properties such as stability, positivity, and boundedness of the continuous model. Numerical simulations are conducted to validate the analytical findings and to investigate the influence of delay parameters, transmission rates, and discretization effects on the system dynamics. The results indicate that delay mechanisms play a significant role in reducing effective transmission and altering transient dynamics, while transmission parameters strongly influence disease persistence and spread. Overall, the proposed framework provides useful insights into the complex dynamics of canine Chagas disease and offers a reliable tool for exploring control strategies and long-term epidemiological behavior.

The growing number of infectious disease outbreaks, like the one caused by the SARS-CoV-2 virus, underscores the necessity of actuarial models that can adapt to epidemic-driven risks. Traditional life insurance frameworks often rely on static mortality assumptions that fail to capture the temporal and behavioral complexity of disease transmission. In this article, we propose an integrated actuarial framework based on the SEIARD epidemiological model. This framework enables the explicit modeling of incubation periods and disease-induced mortality. We derive key actuarial quantities, including the present value of annuity benefits, payment streams, and net premiums, based on SEIARD dynamics. We formulate a prospective reserve function and analyze its evolution throughout the course of an epidemic. Additionally, we examine the forces of infection, mortality, and removal to assess their impact on epidemic-adjusted survival probabilities. Numerical simulations implemented via a nonstandard finite difference (NSFD) scheme illustrate the model’s applicability under various parameter settings and insurance policy assumptions.

ABSTRACT In this work, we propose and analyze a new fractional-order two-stage species model with recruitment, which combines a well-known integer-order two-stage species model with the Caputo fractional derivative, to discover memory effects on population dynamics. Firstly, the positivity and boundedness of solutions are investigated by using some standard comparison results. Next, a simple approach is utilized to study stability properties of the fractional-order model. This approach is based on the Lyapunov stability theory in combination with some nonstandard techniques for fractional dynamical systems. More clearly, we use general quadratic Lyapunov candidate functions and combine them with characteristics of quadratic forms associated with real matrices to establish the stability properties. Consequently, global asymptotic stability, uniform and Mittag-Leffler stability and, therefore, population dynamics of the proposed fractional-order model are analyzed rigorously. In addition, we extend Mickens’ methodology to construct a dynamically consistent nonstandard finite difference (NSFD) scheme for the purpose of numerical simulation. It is proved that the NSFD scheme preserves the positivity and boundedness of the fractional-order model regardless of the values of the step size; moreover, it is also simple and efficient. Lastly, the theoretical results and advantages of the NSFD scheme are supported by illustrative numerical experiments.

The present research work introduces a novel computational approach to the study of a nonlinear stochastic epidemic model for skin sores and also a deterministic model. Theoretical analyzing leads to establishing the existence, positive nature, and boundedness of solutions along with the local stability of equilibria. The application of multiple numeric methods including classical Euler, Runge–Kutta, and Euler–Maruyama methods as well as a stochastic nonstandard finite difference (NSFD) approach together with the model’s dynamic behavior exploration is the main method of research for this project. The NSFD scheme stands out for its ability to give better precision and stability in numerical computations, besides being time-efficient and not dependent on the size of the time step. A complete comparison reveals the proposed scheme’s capability and consistency in seizing both deterministic and stochastic dynamics of the epidemic system. The results of the study lead to a connection between theoretical analysis and practical computation and still give considerable understanding of the reliable simulations of stochastic epidemic processes.

In this work, we present a rigorous proof that a designed second-order NSFD scheme is dynamically consistent with respect to the solution of a generalized eco-epidemiological predator-prey model. More specifically, positivity of populations, equilibrium points, trapping domain, and local stability, are maintained regardless of the time step size, i.e. the method is unconditionally stable. The design of the scheme relies on the usual nonlocal approximation of the right-hand side function while the nonstandard denominator functions are defined depending not only on the time step size but also on the state variables. We prove that resulting scheme is convergent with the desired order. The proposed methodology can be used to design second-order NSFD methods for other models similar to the predator-prey model presented in this paper. Finally, we present numerical examples that support the mathematical analysis and show the advantages of the constructed NSFD schemes.

Simplified semi-analytical solutions for dynamic responses of composite cylinders subjected to far-field underwater explosions

Respiratory syncytial virus (RSV) is a single-stranded RNA virus responsible for a wide range of respiratory tract infections, including those affecting the lungs, airways, and middle ear. Understanding its transmission dynamics remains essential for effective disease control. A bio-inspired stochastic delay model for RSV transmission is proposed and analyzed. The model's qualitative properties including positivity, boundedness, equilibrium states, and the basic reproduction number are rigorously established through well-posedness theorems. Parameter sensitivity is also examined. To investigate the system's stochastic behavior, numerical schemes such as Stochastic Euler, Runge-Kutta, and Euler-Maruyama methods are applied. However, these traditional approaches fail to fully preserve the dynamic characteristics of the model. To address these limitations, a stochastic nonstandard finite difference (NSFD) scheme with delay is developed. This approach ensures non-negativity, boundedness, consistency, and unconditional convergence, overcoming issues of instability and divergence often observed in standard stochastic numerical methods. Comparative simulations demonstrate that the NSFD method reliably reproduces the true dynamic states of the model. The proposed stochastic delayed modeling framework enhances our understanding of RSV dynamics and provides a stable computational tool for analyzing complex biological systems. The findings open new avenues for exploring nonlinear stochastic processes in epidemiological and neurobiological modeling.

Herpes simplex virus (HSV) is a widespread infection responsible for painful blisters and ulcers. According to the World Health Organization, approximately 519.5million people aged 15-49 years (13.3%) worldwide are infected with herpes simplex virus type II (HSV-II), the primary cause of genital herpes. In this study, we develop a nonlinear stochastic fractional delay differential equation (SFDDE) model to describe the transmission dynamics of HSV-II in a human population. The population is divided into susceptible [Formula: see text], exposed [Formula: see text], asymptomatic [Formula: see text], symptomatic [Formula: see text], HSV-infected [Formula: see text], and recovered [Formula: see text]compartments. The model's fundamental properties, including existence, uniqueness, positivity, and boundedness of solutions, are established. Local and global stability analyses are conducted around the HSV-free and HSV-present equilibrium points, and the basic reproduction number is derived using the next-generation matrix method along with sensitivity analysis. Numerical simulations based on a stochastic nonstandard finite difference (NSFD) scheme confirm the theoretical results and demonstrate the stability of the proposed model. These findings highlight the importance of nonlinear fractional stochastic modeling in understanding and controlling HSV-II transmission dynamics.

Hepatitis B virus (HBV) remains a major global public health concern, motivating the use of mathematical models to better understand its transmission dynamics and control strategies. In this study, a compartmental mathematical model based on a system of linear differential equation is formulated to describe the spread of HBV in a population. The basic reproduction number R0 is derived to characterized the transmission potential of the disease. Analytical results show that the disease-free equilibrium (DFE) is locally asymptotically stable when R0 < 1 and become unstable when R0 > 1, while the endemic equilibrium (EE) exists and is stable for R0 > 1. To investigate the dynamic behavior of the model numerically, both the standard finite difference (SFD) scheme and a non-standard finite difference (NSFD) scheme are implemented. The SFD scheme exhibits conditional convergence and may produce nonphysical results for certain step sizes, whereas the proposed NSFD scheme preserves essential qualitative properties of the continuous model, including positivity and stability of solutions. A stability analysis of the NSFD scheme is also presented. Numerical simulations and comparative analysis of both schemes validate the theoretical findings and demonstrate the superior performance of the NSFD method in accurately capturing the transmission dynamics of HBV.

Diabetes mellitus is a chronic, non-communicable disease that continues to pose a major global health burden. Effective management strategies require not only clinical interventions but also robust modeling frameworks to guide public health decisions. In this study, we develop a compartmental model to describe the dynamics of diabetes under both pharmacological and non-pharmacological interventions. The model is solved using a Nonstandard Finite Difference (NSFD) scheme, which preserves key properties such as positivity and boundedness, and demonstrates superior stability compared to classical numerical methods (RK4 and Forward Euler). The model parameters are estimated by fitting to annual diabetes prevalence data from the United States (2000–2022). A global sensitivity analysis using Partial Rank Correlation Coefficients (PRCC) identifies the most influential parameters driving disease dynamics. To evaluate intervention strategies, the model is extended with three time-dependent control measures, followed by a cost-effectiveness analysis to assess their relative efficiency. Our findings highlight the critical parameters influencing diabetes progression and suggest optimal intervention strategies that balance effectiveness with economic feasibility. These results provide valuable insights for improving diabetes management and support evidence-based public health planning.

ABSTRACT In this study, we investigate the fractional‐order formulations of the quadratic isothermal autocatalytic chemical system (FQIACS) and analyze their corresponding numerical solutions. Two non‐singular fractional operators are considered, namely the Liouville–Caputo and Caputo–Fabrizio derivatives. The governing fractional models are transformed into an algebraic system using a combination of the nonstandard finite difference scheme and the spectral collocation method based on Shifted Vieta–Lucas orthogonal polynomials. The resulting nonlinear algebraic systems are solved using the Newton–Raphson method. Since closed‐form analytical solutions are generally unavailable for noninteger orders, the accuracy of the proposed numerical approximations is evaluated through the residual error function. The results demonstrate the effectiveness and reliability of the developed numerical framework for handling fractional chemical kinetics involving different memory kernels.

Formulation and analysis of an implicit non-standard finite difference scheme for the Black-Scholes option pricing model

Chlamydia is a widespread sexually transmitted infection in Europe, often leading to complications such as rectal discomfort, throat inflammation, and reactive arthritis. This study presents a novel nonlinear delay differential equation model that enhances the classical SEIAISR framework to more accurately represent Chlamydia transmission dynamics. The model integrates biologically justified exponential time delays to reflect incubation periods and the delayed impact of interventions like condom use, routine screening, partner reduction, and microbiome health. We establish the existence and uniqueness of solutions using the Banach fixed point theorem and analyze the model’s dynamics by computing the basic reproduction number and studying equilibria and their stability via Lyapunov functions and Routh-Hurwitz criteria. A sensitivity analysis identifies key epidemiological drivers. For numerical simulation, we employ Euler’s method, the Runge-Kutta 4th order (RK4) method, and a specially developed non-standard finite difference (NSFD) scheme. The NSFD approach preserves critical properties such as positivity and stability, making it suitable for realistic long-term predictions. Results highlight the importance of timely interventions and show the superiority of structurepreserving numerical methods. The findings support the development of more targeted and effective strategies to reduce chlamydia transmission and complications among high-risk groups, reinforcing evidence-based decisionmaking within the healthcare system.

The primary goal of this research is to analyze the transmission dynamics of Maize Streak Virus (MSV) by means of a computational and stochastic modeling technique where the time delay and uncertainty factors in the epidemic process are vital considerations. A compartmental MSV deterministic model was established, which later got an extension to a stochastic delay differential system having five biological compartments consisting of susceptible, insecticide-treated, exposed, infected, and recovered plants. Analytical methods were employed to find the maize streak-free and endemic equilibriums and to derive the treatment reproduction number. The stability of the deterministic and stochastic systems was studied. The numerical methods used for comparison were Euler-Maruyama, stochastic Runge-Kutta, and the stochastic Nonstandard Finite Difference (NSFD) scheme, which were assessed for accuracy, stability, and computational efficiency. Theoretical results show that under some parameter values, both equilibrium points are stable in an asymptotic sense. The numerical experiments reveal that the stochastic NSFD scheme is more stable, preserves positivity better, and is independent of step size than the classical methods. Including the stochasticity captures the uncertainty associated with MSV transmission in the real world, thereby enhancing the predictive simulation's validity. The suggested stochastic NSFD model is indeed a strong computationally efficient and biologically realistic method to simulate MSV and other plant virus epidemics. The results boost our understanding and management of the agricultural disease control strategies.

ObjectivesThe primary goal of this research is to analyze the transmission dynamics of Maize Streak Virus (MSV) by means of a computational and stochastic modeling technique where the time delay and uncertainty factors in the epidemic process are vital considerations.MethodologyA compartmental MSV deterministic model was established, which later got an extension to a stochastic delay differential system having five biological compartments consisting of susceptible, insecticide-treated, exposed, infected, and recovered plants. Analytical methods were employed to find the maize streak–free and endemic equilibriums and to derive the treatment reproduction number. The stability of the deterministic and stochastic systems was studied. The numerical methods used for comparison were Euler-Maruyama, stochastic Runge–Kutta, and the stochastic Nonstandard Finite Difference (NSFD) scheme, which were assessed for accuracy, stability, and computational efficiency.Key ResultsTheoretical results show that under some parameter values, both equilibrium points are stable in an asymptotic sense. The numerical experiments reveal that the stochastic NSFD scheme is more stable, preserves positivity better, and is independent of step size than the classical methods. Including the stochasticity captures the uncertainty associated with MSV transmission in the real world, thereby enhancing the predictive simulation’s validity.ConclusionsThe suggested stochastic NSFD model is indeed a strong computationally efficient and biologically realistic method to simulate MSV and other plant virus epidemics. The results boost our understanding and management of the agricultural disease control strategies.

In the banking system, the repayment rate of loans, which is influenced by interest rates and nonperforming loans, plays an important role in the bank’s cash flow. In this paper, we propose a discrete model of deposit–loan volumes by considering the repayment rate. The proposed model involves the standard forward Euler discretization and the non-standard finite difference (NSFD) scheme. The numerical schemes of both models are explicitly defined. Both models have three fixed points, i.e., the transaction-free point, the loan-free point, and the active-transaction point. The transaction-free fixed point is unstable, while the other two are locally asymptotically stable under certain conditions. The stability of the Euler model’s fixed point depends on the stepsize h. This indicates that the NSFD model is dynamically more consistent since it does not depend on h. Numerical simulations also confirm that the stability property of the NSFD model’s fixed points does not depend on h. Meanwhile, the stability of the fixed points of the Euler model depends on h. The simulations also show that the Euler model undergoes period-doubling and Neimark–Sacker bifurcations. This is indicated by changes in the stepsize that cause the convergence of the solution to shift into oscillations or even chaos. The chaotic condition is an undesired or even avoided situation in the banking sector. High and irregular fluctuations lead to the failure of policy control and liquidity projection. We also performed a case study using weekly loan data from September 2022 to March 2025 via parameter estimation. We use two performance metrics, i.e., the coefficient of determination (R2) and the root mean square error (RMSE). Both models produce realistic parameter values and provide a good fit to the data trend, as observed visually and from R2. Based on RMSE, the NSFD model performs better than the Euler model. Moreover, the larger the h, the better the performance. These results suggest the use of the NSFD model, which has better relevance and accuracy than the Euler model.

Abstract With over 700 million cases and 6.8 million deaths worldwide by September 2023, the COVID-19 pandemic had significant adverse effects on both public health and the global economy. Research shows that the virus disproportionately affects people with pre-existing medical issues. Therefore, it is essential to comprehend the COVID-19 transmission kinetics. A gender-specific mathematical framework validated with COVID-19 data from India is presented in this research. Male and female basic reproduction numbers are calculated independently, demonstrating their impact on equilibrium point stability. To better depict the dynamics of the disease, numerical simulations are conducted utilizing the Non-Standard Finite Difference (NSFD) scheme. Sensitivity analysis emphasizes even more how important comorbidity-related factors are that impact the progression of the dynamics. The study also examines the ratios of death, recovery, and transmission rates for males and females, as well as the levels of antagonism and synergy in mortality, recovery rates and basic reproduction ratios across genders.In order to measure how comorbidities and disease outcomes interact, the study presents a novel gender-based synergy score. The findings highlight important gender-specific vulnerabilities by demonstrating a higher level of synergy in female recovery and transmission as well as male mortality (3 vs. 2.5). The findings support tailored interventions, particularly for high-risk populations with comorbidities, and highlight the importance of the butterfly effect in the context of control efforts.

Dynamic analysis on the model of commensalism symbiosis with the discretized Michaelis-Menten cropping by using different schemes to non-standard finite difference (NSFD) is the main focus in this article. The analysis is started by searching the equilibrium points with their existence terms and local stability with their stability terms. In this article, there are four equilibrium points. The points are the extinction point of both populations, the host extinction point, the commensal extinction point, and the point where both populations can coexist (the coexistence equilibrium point). The existence of a host extinction point and a point at which both populations can coexist depends on the conditions of existence that must be met. Among the four equilibrium points, the commensal extinction point and the coexistence equilibrium point are locally asymptotically stable provided that the specified stability conditions are met. In the final analysis, numerical simulations were performed using the 4th order Runge–Kutta scheme for the continuous model and the NSFD scheme for the discrete model. The results show that the NSFD scheme offers greater flexibility in choosing the integration time step to ensure convergence to a feasible solution, outperforming the 4th order Runge–Kutta scheme in this respect.

Singularly perturbed reaction-diffusion differential equations with large negative and advanced shifts are significant challenges in mathematical modeling across various scientific disciplines, particularly in control theory and neuroscience. This study addresses the numerical treatment of these complex equations where the presence of a small perturbation parameter combined with large shift terms creates boundary and interior layers that standard numerical approaches fail to capture accurately. We develop a robust nonstandard finite difference scheme (NSFDS) specifically designed to handle the difficulties arising from the singular perturbation parameter and mixed large shift arguments. Our approach incorporates a carefully constructed denominator function that maintains the essential qualitative properties of a continuous model. Through rigorous convergence analysis, we established that the proposed scheme is uniformly convergent with respect to the perturbation parameter, achieving second-order accuracy. Extensive numerical experiments are performed to validate the theoretical predictions, demonstrating the scheme’s superior convergence rate, efficiency, and ability to resolve layer phenomena without spurious oscillations, even for extremely small perturbation parameters. The results demonstrate that our method effectively resolves layer phenomena and maintains stability and accuracy even for extremely small values of the perturbation parameter, outperforming existing standard approaches as well as other existing previous results. This study provides a reliable computational framework for this challenging class of equations, significantly advancing state-of-the-art numerical methods for singularly perturbed problems with shift terms.