Search Results (51)

According to the World Health Organization (WHO), globally, cervical cancer ranks as the fourth most common cancer in women, with around 660,000 new cases in 2022. In the same year, about 94 percent of the 350,000 deaths caused by cervical cancer occurred in low- and middle-income countries. This paper focuses on the dynamics of HPV by modeling the interactions between four compartments, as follows: S(t), the number of susceptible females; I(t), females infected with HPV; X(t), females infected with HPV but not yet affected by cervical cancer (CCE); and V(t), females infected with HPV and affected by CCE. A compartmental model is formulated to analyze the progression of HPV, ensuring all key mathematical properties, such as existence, uniqueness, positivity, and boundedness of the solution. The equilibria of the model, such as the HPV-free equilibrium and HPV-present equilibrium, are analyzed, and the basic reproduction number, $R_0$, is computed using the next-generation matrix method. Local and global stability of these equilibria are rigorously established to understand the conditions for disease eradication or persistence. Sensitivity analysis around the reproduction number is carried out using partial derivatives to identify critical parameters influencing $R_0$, which gives insights into effective intervention strategies. With appropriate positivity, boundedness, and numerical stability, a new stochastic non-standard finite difference (NSFD) scheme is developed for the proposed model. A comparison analysis of solutions shows that the NSFD scheme is the most consistent and reliable method for a stochastic fractional delay model. Graphical simulations are presented to provide visual insights into the development of the disease and lend the results to a more mature discourse. This research is crucial in highlighting the mathematical rigor and practical applicability of the proposed model, contributing to the understanding and control of HPV progression. Full article

This paper introduces a stable non-standard finite difference (NSFD) method to solve time-fractional singularly perturbed convection–diffusion problems. The fractional derivative in time is defined in the Caputo sense. The proposed method shows high efficiency when applied using a uniform mesh and can be easily extended to a Shishkin mesh in the spatial domain. We discuss error estimates to demonstrate the convergence of the numerical scheme. Additionally, various numerical examples are presented to illustrate the behavior of the solution for different values of the perturbation parameter $ϵ$ and the order of the fractional derivative. Full article

This paper presents the results of experimental and numerical analysis of fracture mechanics testing of ductile metallic materials using a non-standard procedure with PRNT (pipe ring notched tensile) ring-shaped specimens, introduced in previous publications through analysis of 3D-printed polymer rings. The main focus of this research is the determination of the values of the plastic geometry factor η_pl since the specimen is not a standard one. Toward this aim, the finite element software package Simulia Abaqus was applied to evaluate the J-integral (by using the domain integral method) and the F-CMOD curve so that the plastic geometry factor η_pl can be evaluated for different values of the ratio of crack length to specimen width (a₀/W = 0.45 ÷ 0.55). In this way, a procedure and the possibility of practical implementation on the thin-walled pipelines are established. Full article

Cassava is the sixth most important food crop worldwide and the third most important source of calories in the tropics. More than 800 million people depend on this plant’s tubers and sometimes leaves. To protect cassava crops and the livelihoods depending on them, we developed a stochastic fractional delayed model based on stochastic fractional delay differential equations (SFDDEs) to analyze the dynamics of cassava mosaic disease, focusing on two equilibrium states, the state of being absent from cassava mosaic disease and the state of being present with cassava mosaic disease. The basic reproduction number and sensitivity of parameters were estimated to characterize the level beyond which cassava mosaic disease prevails or declines in the plants. We analyzed the stability locally and globally to determine the environment that would ensure extinction and its persistence. To support the theoretical analysis, as well as the reliable results of the model, the present study used a nonstandard finite difference (NSFD) method. This numerical method not only improves the model’s accuracy but also guarantees that cassava mosaic probabilities are positive and bounded, which is essential for the accurate modeling of the cassava mosaic processes. The NSFD method was applied in all the scenarios, and it was determined that it yields adequate performance in modeling cassava mosaic disease. The ideas of the model are crucial for exploring key variables, which affect the scale of cassava mosaic and the moments of intervention. The present work is useful for discerning the mechanism of cassava mosaic disease as it presents a solid mathematical model capable of determining the stage of cassava mosaic disease. Full article

Nonstandard finite-difference (NSFD) methods, pioneered by R. E. Mickens, offer accurate and efficient solutions to various differential equation models in science and engineering. NSFD methods avoid numerical instabilities for large time steps, while numerically preserving important properties of exact solutions. However, most NSFD methods are only first-order accurate. This paper introduces two new classes of explicit second-order modified NSFD methods for solving n-dimensional autonomous dynamical systems. These explicit methods extend previous work by incorporating novel denominator functions to ensure both elementary stability and second-order accuracy. This paper also provides a detailed mathematical analysis and validates the methods through numerical simulations on various biological systems. Full article

Two novel crossover models for breast cancer that incorporate $\Psi$-Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion and the crossover model for breast cancer that incorporates Atangana–Baleanu Caputo fractal variable-order fractional derivatives, fractal fractional-order derivatives, and variable-order fractional stochastic derivatives driven by variable-order fractional Brownian motion are presented here, where we used a simple nonstandard kernel function $\Psi \left(t\right)$ in the first model and a non-singular kernel in the second model. Moreover, we evaluated our models using actual statistics from Saudi Arabia. To ensure consistency with the physical model problem, the symmetry parameter $\zeta$ is introduced. We can obtain the fractal variable-order fractional Caputo and Caputo–Katugampola derivatives as special cases from the proposed $\Psi$-Caputo derivative. The crossover dynamics models define three alternative models: fractal variable-order fractional model, fractal fractional-order model, and variable-order fractional stochastic model over three-time intervals. The stability of the proposed model is analyzed. The $\Psi$-nonstandard finite-difference method is designed to solve fractal variable-order fractional and fractal fractional models, and the Toufik–Atangana method is used to solve the second crossover model with the non-singular kernel. Also, the nonstandard modified Euler–Maruyama method is used to study the variable-order fractional stochastic model. Numerous numerical tests and comparisons with real data were conducted to validate the methods’ efficacy and support the theoretical conclusions. Full article

This work deals with a semilinear system of parabolic partial differential equations (PDEs) on an unbounded domain, related to environmental pollution modeling. Although we study a one-dimensional sub-model of a vertical advection–diffusion, the results can be extended in each direction for any number of spatial dimensions and different boundary conditions. The transformation of the independent variable is applied to convert the nonlinear problem into a finite interval, which can be selected in advance. We investigate the positivity of the solution of the new, degenerated parabolic system with a non-standard nonlinear right-hand side. Then, we design a fitted finite volume difference discretization in space and prove the non-negativity of the solution. The full discretization is obtained by implicit–explicit time stepping, taking into account the sign of the coefficients in the nonlinear term so as to preserve the non-negativity of the numerical solution and to avoid the iteration process. The method is realized on adaptive graded spatial meshes to attain second-order of accuracy in space. Some results from computations are presented. Full article

In this article, we study the semi-linear two-dimensional reaction–diffusion equation with Dirichlet boundaries. A reliable numerical scheme is designed, coupling the nonstandard finite difference method in the time together with the Galerkin in combination with the compactness method in the space variables. The aforementioned equation is analyzed to show that the weak or variational solution exists uniquely in specified space. The a priori estimate obtained from the existence of the weak or variational solution is used to show that the designed scheme is stable and converges optimally in specified norms. Furthermore, we show that the scheme preserves the qualitative properties of the exact solution. Numerical experiments are presented with a carefully chosen example to validate our proposed theory. Full article

The therapeutic potential of delta9-tetrahydrocannabinol (THC), a primary cannabinoid in the cannabis plant, has led to its development into oral medical products for treating various conditions. However, THC, being a psychoactive substance, can lead to addiction if taken in inappropriate amounts. Thus, studying the pharmacokinetics of THC is crucial for understanding how the drug behaves in the body after administration. This study aims to develop a multi-compartmental model to investigate the pharmacokinetics of medical THC and its metabolites after oral administration. Using the law of mass action, the model was converted into ordinary differential equations (ODEs) to describe the rate of concentration changes of THC and its metabolites in each compartment. The nonstandard finite difference (NSFD) method was then applied to construct numerical solution schemes, which were implemented in MATLAB along with estimated pharmacokinetic rate constants. The results demonstrate that the simulation curves depicting the plasma concentration–time profiles of THC and 11-hydroxy-THC (THC-OH) closely resemble actual data samples, indicating the model’s accuracy. Moreover, the model predicts the pharmacokinetics of THC and its metabolites in various tissues. Consequently, this model serves as a valuable tool for enhancing our understanding of the pharmacokinetics of THC and its metabolites, guiding dosage adjustments, and determining administration durations for oral medical THC. Full article

We develop a mathematical model for the SARAS-CoV-2 double variant transmission characteristics with variant 1 vaccination to address this novel aspect of the disease. The model is theoretically examined, and adequate requirements are derived for the stability of its equilibrium points. The model includes the single variant 1 and variant 2 endemic equilibria in addition to the endemic and disease-free equilibria. Various approaches are used for the global and local stability of the model. For both strains, we determine the basic reproductive numbers ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$. To investigate the occurrence of the layers (waves), we expand the model to include some analysis based on the second-order derivative. The model is then expanded to its stochastic form, and numerical outcomes are computed. For numerical purposes, we use the nonstandard finite difference method. Some error analysis is also recorded. Full article

In order to reduce the failure accidents caused by the insufficient strength of fracturing string joints, theoretical calculation and string design methods were adopted to conduct finite element calculations on commonly used long circular threads. The distribution laws of stress and contact pressure of long round threads were obtained, a non-standard special thread was designed, and a finite element model of the joint of the casing was established. Considering different make-up torques, tensile loads, and tensile torque loads within a certain range, the stress variation law of the special casing threaded joint under this design size was analyzed. Finally, the stress and contact pressure variation law on the threaded tooth was analyzed under different structures, working conditions, and wall thickness parameters. The thread strength and sealing function were compared under various parameters. The results showed that the smaller the wall thickness of the joints, the greater the contact pressure at the threaded tooth. Among them, the contact pressure of the external threaded tooth is too high, and is prone to the sticking phenomenon. The distribution of contact pressure in the middle section is relatively reasonable. Compared with the original structure, the new structure significantly reduces the contact pressure at the head and tail ends of the threaded connection, reducing the risk of sticking. Full article

The Huxley equation, which is a nonlinear partial differential equation, is used to describe the ionic mechanisms underlying the initiation and propagation of action potentials in the squid giant axon. This equation, just like many other nonlinear equations, is often very difficult to analyze because of the presence of the nonlinearity term, which is always very difficult to approximate. This paper aims to design a reliable scheme that consists of a combination of the nonstandard finite difference in time method, the Galerkin method and the compactness methods in space variables. This method is used to show that the solution of the problem exists uniquely. The a priori estimate from the existence process is applied to the scheme to show that the numerical solution from the scheme converges optimally in the $L^2$ as well as the $H^1$ norms. We proceed to show that the scheme preserves the decaying properties of the exact solution. Numerical experiments are introduced with a chosen example to validate the proposed theory. Full article

This paper focuses on examining numerical solutions for fractional-order models within the context of the coupled multi-space Korteweg-de Vries problem (CMSKDV). Different types of kernels, including Liouville-Caputo fractional derivative, as well as Caputo-Fabrizio and Atangana-Baleanu fractional derivatives, are utilized in the examination. For this purpose, the nonstandard finite difference method and spectral collocation method with the properties of the Shifted Vieta-Lucas orthogonal polynomials are employed for converting these models into a system of algebraic equations. The Newton-Raphson technique is then applied to solve these algebraic equations. Since there is no exact solution for non-integer order, we use the absolute two-step error to verify the accuracy of the proposed numerical results. Full article

Suspended waterproof curtains combined with pumping wells are the primary method for controlling groundwater levels in foundation pits within soft soil areas. However, there is still a lack of a systematic approach to predict the groundwater drawdown within the foundation pit caused by the influence of these suspended curtains. In order to investigate the variation of groundwater level within the excavation during dewatering processes, the finite difference method is employed to analyze the seepage characteristics of foundation pits with suspended waterproof curtains. Basing on the concept of equivalent well, this study examines the coupled effects of aquifer anisotropy (k_i), aquifer thickness (M_i), well screen length (l_i), and the depth of waterproof curtain embedment on the seepage field distortion. A characteristic curve is established for standard conditions, which exposes the blocking effect of the curtain on the amount of groundwater drawdown in the pit. Additionally, correction coefficients are proposed for non-standard conditions, which, in turn, results in a prediction formula with a wider range of applicability. Comparative analysis between the calculated predictions and the field observation data from an actual foundation pit project in Zhuhai City validates the feasibility of the quantitative prediction method proposed in this research, which also provides a 21% safety margin. Full article

For decades, understanding the dynamics of infectious diseases and halting their spread has been a major focus of mathematical modelling and epidemiology. The stochastic SIRS (susceptible–infectious–recovered–susceptible) reaction–diffusion model is a complicated but crucial computational scheme due to the combination of partial immunity and an incidence rate. Considering the randomness of individual interactions and the spread of illnesses via space, this model is a powerful instrument for studying the spread and evolution of infectious diseases in populations with different immunity levels. A stochastic explicit finite difference scheme is proposed for solving stochastic partial differential equations. The scheme is comprised of predictor–corrector stages. The stability and consistency in the mean square sense are also provided. The scheme is applied to diffusive epidemic models with incidence rates and partial immunity. The proposed scheme with space’s second-order central difference formula solves deterministic and stochastic models. The effect of transmission rate and coefficient of partial immunity on susceptible, infected, and recovered people are also deliberated. The deterministic model is also solved by the existing Euler and non-standard finite difference methods, and it is found that the proposed scheme forms better than the existing non-standard finite difference method. Providing insights into disease dynamics, control tactics, and the influence of immunity, the computational framework for the stochastic SIRS reaction–diffusion model with partial immunity and an incidence rate has broad applications in epidemiology. Public health and disease control ultimately benefit from its application to the study and management of infectious illnesses in various settings. Full article