(R1504) Second-order Modified Nonstandard Runge-Kutta and Theta Methods for One-dimensional Autonomous Differential Equations

Nonstandard finite difference methods (NSFD) are used in physical sciences to approximate solutions of ordinary differential equations whose analytical solution cannot be computed. Traditional NSFD methods are elementary stable but usually only have first order accuracy. In this paper, we introduce two new classes of numerical methods that are of second order accuracy and elementary stable. The methods are modified versions of the nonstandard two-stage explicit Runge-Kutta methods and the nonstandard one-stage theta methods with a specific form of the nonstandard denominator function. Theoretical analysis of the stability and accuracy of both modified NSFD methods is presented. Numerical simulations that concur with the theoretical findings are also presented, which demonstrate the computational advantages of the proposed new modified nonstandard finite difference methods

Numerical treatment for a novel crossover mathematical model of the COVID-19 epidemic

This paper extends a novel piecewise mathematical model of the COVID-19 epidemic using fractional and variable-order differential equations and fractional stochastic derivatives in three intervals of time. The deterministic models are augmented with hybrid fractional order and variable order operators, while the stochastic differential equations incorporate fractional Brownian motion. To probe the behavior of the proposed models, we introduce two numerical techniques: the nonstandard modified Euler Maruyama method for the fractional stochastic model, and the Caputo proportional constant-Grünwald-Letnikov nonstandard finite difference method for the fractional and variable-order deterministic models. Several numerical experiments corroborate the theoretical assertions and demonstrate the efficacy of the proposed approaches

Investigating a Fractal–Fractional Mathematical Model of the Third Wave of COVID-19 with Vaccination in Saudi Arabia

The severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) is responsible for coronavirus disease-19 (COVID-19). This virus has caused a global pandemic, marked by several mutations leading to multiple waves of infection. This paper proposes a comprehensive and integrative mathematical approach to the third wave of COVID-19 (Omicron) in the Kingdom of Saudi Arabia (KSA) for the period between 16 December 2022 and 8 February 2023. It may help to implement a better response in the next waves. For this purpose, in this article, we generate a new mathematical transmission model for coronavirus, particularly during the third wave in the KSA caused by the Omicron variant, factoring in the impact of vaccination. We developed this model using a fractal-fractional derivative approach. It categorizes the total population into six segments: susceptible, vaccinated, exposed, asymptomatic infected, symptomatic infected, and recovered individuals. The conventional least-squares method is used for estimating the model parameters. The Perov fixed point theorem is utilized to demonstrate the solution’s uniqueness and existence. Moreover, we investigate the Ulam–Hyers stability of this fractal–fractional model. Our numerical approach involves a two-step Newton polynomial approximation. We present simulation results that vary according to the fractional orders (γ) and fractal dimensions (θ), providing detailed analysis and discussion. Our graphical analysis shows that the fractal-fractional derivative model offers more biologically realistic results than traditional integer-order and other fractional models

Analysis of Leptospirosis transmission dynamics with environmental effects and bifurcation using fractional-order derivative

Mathematical formulations are essential tool to show the dynamics that how various diseases spread in the community. Differential equations with fractional or integer order can be utilized to see the effect of the dynamics direct or indirect Leptospirosis transmission, which are analyzed with different aspects. A mathematical description and dynamical sketch of Leptospirosis with environmental effects have been studied as a result of the successful efforts of various writers. In this study, we analyzed the Leptospirosis model described using a nonlinear fractional-order differential equation that takes the environmental effects into consideration. The proposed fractional order system is investigated qualitatively as well as quantitatively to identify its stable position. Local stability of the Leptospirosis system is verified and test the system with flip bifurcation. Also system is investigated for global stability using Lyapunov first and second derivative functions. The existence, boundedness and positivity of the Leptospirosis is checked, which are the key properties for such of type of epidemic problem to identify reliable findings. Effect of global derivative is demonstrated to verify its rate of effects according to their sub-compartments. Solutions for fractional order system are derived with the help of advanced tool fractal fractional operator for different fractional values. Simulation are carried out to see symptomatic as well as a asymptomatic effects of Leptospirosis in the world wide, also show the actual behavior of Leptospirosis which will be helpful to understand the outbreak of Leptospirosis with environmental effects as well as for future prediction and control strategies