Nonstandard Finite Difference Scheme Research Articles

Dynamical behavior of a hepatitis B epidemic model and its NSFD scheme

AbstractHepatitis is inflammation of the liver, and one of its types, hepatitis B, is a contagious infection. Using mathematical models, the nature of the spread of the Hepatitis B virus can be predicted. In the present paper, a hepatitis B epidemic model with a Beddington–DeAngelis type incidence rate and a constant vaccination rate is considered. Some dynamical properties of this model, such as non-negativity, boundedness character, the basic reproduction number $$\mathcal {R}_0$$ R 0 , stability nature, and the bifurcation phenomenon, are investigated. By the Bendixson theorem, it is demonstrated that the disease-free equilibrium is globally asymptotically stable. It is shown that a transcritical bifurcation phenomenon occurs when $$\mathcal {R}_0=1$$ R 0 = 1 . It is concluded that the endemic equilibrium is globally asymptotically stable when $$\mathcal {R}_0>1$$ R 0 > 1 , by utilizing Dulac’s criteria. Also, a discrete system of difference equations is obtained by constructing a non-standard finite difference (NSFD) scheme for the continuous model. It is shown that the solutions of this discrete system are dynamically consistent for all finite step sizes. The theoretical results obtained are also supported and visualized by numerical simulations. These simulations also demonstrate that the NSFD scheme produces much more efficient results than the Euler or RK4 schemes, as shown in the theoretical results obtained.

Comparative analysis of a fractional co-infection model using nonstandard finite difference and two-step Lagrange polynomial methods

In this work, Caputo fractional model for co-dynamics of two-strains (Alpha and Delta variants) of SARS-CoV-2 and tuberculosis (TB) is formulated. We investigates the basic mathematical analysis, existence of unique solution, stability analysis, sensitivity analysis and simulations of the proposed model. In the basic mathematical analysis, it is proved that the model’s solutions are non-negative, bounded and locally asymptotically stable at disease free equilibrium (DFE) point. The existence of unique solution is proved by using Banach fixed point theorem (BFPT). Stability analysis is carried out by employing Hyers–Ulam stability criteria. Sensitivity analysis is performed to assess the impact of the model parameters on the general dynamics of the model by implementing partial rank correlation coefficient (PRCC) technique. Simulations of the proposed model obtained from non-standard finite difference scheme (NSFDs) are compared with the simulations obtained from two-step Lagrange polynomial method (TLPM). It is concluded that NSFDs gives more accurate and realistic results than TLPM. To assess some control measures necessary for reducing the co-spread of both diseases, comparative simulations are carried out for the infected classes. Also, numerical assessments showed that preventive efforts against SARS-CoV-2 variants could results in the reduction of TB prevalence and the co-infection of both diseases.

Analyzing the dynamic patterns of COVID-19 through nonstandard finite difference scheme

This paper presents a novel approach to analyzing the dynamics of COVID-19 using nonstandard finite difference (NSFD) schemes. Our model incorporates both asymptomatic and symptomatic infected individuals, allowing for a more comprehensive understanding of the epidemic's spread. We introduce an unconditionally stable NSFD system that eliminates the need for traditional Runge–Kutta methods, ensuring dynamical consistency and numerical accuracy. Through rigorous numerical analysis, we evaluate the performance of different NSFD strategies and validate our analytical findings. Our work demonstrates the benefits of using NSFD schemes for modeling infectious diseases, offering advantages in terms of stability and efficiency. We further illustrate the dynamic behavior of COVID-19 under various conditions using numerical simulations. The results from these simulations demonstrate the effectiveness of the proposed approach in capturing the epidemic's complex dynamics.

Ebola virus disease model with a nonlinear incidence rate and density-dependent treatment

This paper studies an Ebola epidemic model with an exponential nonlinear incidence function that considers the efficacy and the behaviour change. The current model also incorporates a new density-dependent treatment that catches the impact of the disease transmission on the treatment. Firstly, we provide a theoretical study of the nonlinear differential equations model obtained. More precisely, we derive the effective reproduction number and, under suitable conditions, prove the stability of equilibria. Afterwards, we show that the model exhibits the phenomenon of backward-bifurcation whenever the bifurcation parameter and the reproduction number are less than one. We find that the bi-stability and backward-bifurcation are not automatically connected in epidemic models. In fact, when a backward-bifurcation occurs, the disease-free equilibrium may be globally stable. Numerically, we use well-known standard tools to fit the model to the data reported for the 2018–2020 Kivu Ebola outbreak, and perform the sensitivity analysis. To control Ebola epidemics, our findings recommend a combination of a rapid behaviour change and the implementation of a proper treatment strategy with a high level of efficacy. Secondly, we propose and analyze a fractional-order Ebola epidemic model, which is an extension of the first model studied. We use the Caputo operator and construct the Grünwald-Letnikov nonstandard finite difference scheme, and show its advantages.

Fractional epidemic model of coronavirus disease with vaccination and crowding effects

Most of the countries in the world are affected by the coronavirus epidemic that put people in danger, with many infected cases and deaths. The crowding factor plays a significant role in the transmission of coronavirus disease. On the other hand, the vaccines of the covid-19 played a decisive role in the control of coronavirus infection. In this paper, a fractional order epidemic model (SIVR) of coronavirus disease is proposed by considering the effects of crowding and vaccination because the transmission of this infection is highly influenced by these two factors. The nonlinear incidence rate with the inclusion of these effects is a better approach to understand and analyse the dynamics of the model. The positivity and boundedness of the fractional order model is ensured by applying some standard results of Mittag Leffler function and Laplace transformation. The equilibrium points are described analytically. The existence and uniqueness of the non-integer order model is also confirmed by using results of the fixed-point theory. Stability analysis is carried out for the system at both the steady states by using Jacobian matrix theory, Routh–Hurwitz criterion and Volterra-type Lyapunov functions. Basic reproductive number is calculated by using next generation matrix. It is verified that disease-free equilibrium is locally asymptotically stable if R0<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}<1$$\\end{document} and endemic equilibrium is locally asymptotically stable if R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}>1$$\\end{document}. Moreover, the disease-free equilibrium is globally asymptotically stable if R0<1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}<1$$\\end{document} and endemic equilibrium is globally asymptotically stable if R0>1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${R}_{0}>1$$\\end{document}. The non-standard finite difference (NSFD) scheme is developed to approximate the solutions of the system. The simulated graphs are presented to show the key features of the NSFD approach. It is proved that non-standard finite difference approach preserves the positivity and boundedness properties of model. The simulated graphs show that the implementation of control strategies reduced the infected population and increase the recovered population. The impact of fractional order parameter α\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha$$\\end{document} is described by the graphical templates. The future trends of the virus transmission are predicted under some control measures. The current work will be a value addition in the literature. The article is closed by some useful concluding remarks.

Exploring local and global stability of COVID-19 through numerical schemes

Respiratory sensitivity and pneumonia are possible outcomes of the coronavirus (COVID-19). Surface characteristics like temperature and sunshine affect how long the virus survives. This research article analyzes COVID-19 mathematical model behavior based on symptomatic and non-symptomatic individuals. In the reproductive model, the best result indicates the intensity of the epidemic. Our model remained stable at a certain point under controlled conditions after we evaluated a specific element. This approach is in place of traditional approaches such as Euler’s and Runge–Kutta’s. An unusual numerical approach known as the non-standard finite difference (NSFD) scheme is used in this article. This numerical approach gives us positivity. A dependable numerical analysis allowed us to evaluate different approaches and verify our theoretical results. Unlike the widely used Euler and RK4 approaches, we investigated the benefits of implementing NSFD schemes. By numerically simulating COVID-19 in a variety of scenarios, we demonstrated how our theoretical concepts work. The simulation findings support the usefulness of both approaches.

Nonstandard finite difference schemes for linear and non-linear Fokker–Planck equations

Nonstandard finite difference schemes for linear and non-linear Fokker–Planck equations

Stability analysis and numerical results for some schemes discretising 2D nonconstant coefficient advection–diffusion equations

Abstract We solve two numerical experiments described by 2D nonconstant coefficient advection–diffusion equations with specified initial and boundary conditions. Three finite difference methods, namely Lax–Wendroff, Du-Fort–Frankel and a nonstandard finite difference scheme, are derived and used to solve the two problems, whereby only the first problem has an exact solution. Stability analysis is performed to obtain a range of values of the time step size at a fixed spatial step size. We obtain the rate of convergence in space when the three methods are used to solve Problem 1. Computational times of the three algorithms are computed for Problem 1. Results are displayed for the two problems using the three methods at times T = 1.0 T=1.0 and T = 5.0 T=5.0 . The main novelty is the stability analysis, which is not straightforward as we are working with numerical methods discretising 2D nonconstant coefficient advection–diffusion equation where many parameters are involved. The second highlight is to determine the most efficient scheme from the three methods. Third, there are very few published studies on analysis and use of numerical methods to solve nonconstant coefficient advection–diffusion equations, and this is one of the very few rare articles treating such topics.

Nonstandard Finite Difference Scheme for the Epidemic Model with Vaccination

Nonstandard Finite Difference Scheme for the Epidemic Model with Vaccination

Numerical bifurcation of a delayed diffusive hematopoiesis model with Dirichlet boundary condition

UDC 517.9 Numerical bifurcation of a delayed diffusive hematopoiesis model with Dirichlet boundary condition is studied by using a nonstandard finite-difference scheme. We prove that a series of numerical Neimark–Sacker bifurcations appear at the positive equilibrium as the time delay increases. At the same time, the parameter conditions for the existence of numerical Neimark–Sacker bifurcations at positive equilibrium point are presented. Finally, we use several examples to verify the accuracy of the results.

Numerical analysis of singularly perturbed parabolic reaction diffusion differential difference equations

The authors present numerical analysis of singularly perturbed parabolic problems. The previous papers in this direction focussed on the convection-diffusion problems with regular type of boundary layers. It is very difficult and almost impossible as stated in the literature, e.g, in Chapter 14 of ‘Fitted numerical methods for singular perturbation problems: error estimates in the maximum norm for linear problems in one and two dimensions’ (1996) by Miller, John JH and O'Riordan, Eugene and Shishkin, Grigorii I, to capture singularities due to parabolic layers and develop a robust scheme for reaction-diffusion problems with general values of space shifts. The work is in progress now and this is the first paper in this direction. In this work, we are investigating such problems and develop a robust numerical scheme using Mickens's non-standard finite difference scheme and special mesh. Furthermore, numerical examples have been presented to verify the theoretical findings.

A second-order nonstandard finite difference method for a general Rosenzweig–MacArthur predator–prey model

In this paper, we consider a general Rosenzweig–MacArthur predator–prey model with logistic intrinsic growth of the prey population. We develop the Mickens’ method to construct a dynamically consistent second-order nonstandard finite difference (NSFD) scheme for the general Rosenzweig–MacArthur predator–prey model. The second-order NSFD method is based on a novel nonlocal approximation using right-hand side function weights and nonstandard denominator functions.Through rigorous mathematical analysis, we show that the NSFD method not only preserves two important and prominent dynamical properties of the continuous-time model, namely positivity and asymptotic stability independent of the values of the step size, but also is convergent of order 2. Therefore, it provides a solution to the contradiction between the dynamic consistency and high-order accuracy of NSFD methods.The proposed NSFD method improves positive and elementary stable nonstandard numerical schemes constructed in a previous work of Dimitrov and Kojouharov (2006). Moreover, the present approach can be extended to construct second-order NSFD methods for some classes of nonlinear dynamical systems.Finally, the theoretical insights and advantages of the constructed NSFD scheme are supported by some illustrative numerical simulations.

A generalized model for the population dynamics of a two stage species with recruitment and capture using a nonstandard finite difference scheme

The aim of this work is to formulate and analyze a new and generalized discrete-time population dynamics model for a two-stage species with recruitment and capture factors. This model is derived from a well-known continuous-time population dynamics model of a two-stage species with recruitment and capture developed by Ladino and Valverde and the nonstandard finite difference (NSFD) methodology proposed by Mickens. We establish positivity and asymptotic stability of the proposed discrete-time population dynamics model. As an important consequence, the population dynamics of the new discrete-time model is determined fully. Also, a set of numerical examples is conducted to illustrate the theoretical results and to demonstrate advantages of the new model. The theoretical results and numerical examples show that the proposed discrete-time model not only preserves correctly the population dynamics of the continuous one but is also easy to be implemented. However, some discrete-time models based on the standard Runge–Kutta methods fail to preserve the population dynamics of the continuous-time model. As a result, they generate numerical approximations which are not only non-negative but also unstable.

Nonstandard finite difference method for time-fractional singularly perturbed convection–diffusion problems with a delay in time

In this work, nonstandard finite difference method is presented for the numerical solution of time-fractional singularly perturbed convection–diffusion problems with a delay in time. The time-fractional derivative is considered in the Caputo sense and discretized using Crank–Nicholson technique. Then, a nonstandard finite difference scheme is constructed on a uniform mesh discretization along the spatial direction. The parameter-uniform convergence of the proposed method is proved rigorously and shown to be ɛ-uniform convergent with order of convergence O((Δt)2) along the temporal domain and M−1 along the spatial domain. Finally, the proposed scheme is validated using model examples and the computational results are in agreement with the theoretical expectation.

Extending nonstandard finite difference scheme rules to systems of nonlinear ODEs with constant coefficients

In this paper, we present a reformulation of Mickens' rules for nonstandard finite difference (NSFD) schemes to adapt them to systems of ODEs. We want to compare several methods within the class of NSFD schemes. These methods are expressed as exponential integrators which treat exactly the linear part of the equations. While the classical methods address each equation of a system separately, we propose here to address them as a whole, which involves a reformulation of Mickens' rules. This leads to enhance the results compared to the classical case, even if approximations of the matrix exponentials are used. Precisely, using Cayley–Hamilton theorem we obtain two correction factors, one of which is linked to the size of the system and the other to the nonlinear term, we are thus able to use test cases which show the impact of one or another corrector. We prove the consistency and the zero-stability of the method in the nonlinear case to conclude to the convergence of the method.

An Application of Nonstandard Finite Difference Method to a Model Describing Diabetes Mellitus and Its Complications

In this study, a mathematical model describing diabetes mellitus and its complications in a population is considered. Since standard numerical methods can lead to numerical instabilities, it aims to solve the problem using a nonstandard method. Among the nonstandard methods, nonstandard finite difference (NSFD) schemes that satisfy dynamical consistency are preferred to make the model discrete. Both continuous and discrete models are analyzed to show the stability of the model at the equilibrium points. The Schur-Cohn criterion is used to perform stability analysis at the equilibrium point of the discretized model. Thus, asymptotically stability of the model is presented. Moreover, the advantages of the NSFD method are emphasized by comparing the stability for different step sizes with classical methods, such as Euler and Runge-Kutta. It has been observed that the NSFD method is convergence for larger step sizes. In addition, the numerical results obtained by NSFD schemes are compared with the Runge–Kutta–Fehlberg (RKF45) method in graphical forms. The accuracy of the NSFD method is observed.

Mathematical Modelling of Tuberculosis and Diabetes Co-Infection Using the Non- Standard Finite Difference Scheme

One of the major health challenge facing Africa and in particular, Kenya is the risk of Tuberculosis and Diabetes. To understand the dynamics of this, a nine compartmental model for tuberculosis-diabetes co-infection is formulated. The Non-standard finite difference Scheme (NSFD) of the model is formulated from the first-order ordinary differential equations (ode) to avoid full implicit schemes that are computationally expensive. The overly small step sizes in NSFD give the user autonomy in controlling the accuracy of the results, making it suitable for disease control applications. Numerical simulations with different step sizes of the NSFD for the TB-Diabetes model are carried out to find the optimal step size, h. A comparison of the best resultant numerical simulation based on optimal h in NSDF indicates NSFD gives better results when compared with the corresponding first-order ode. The phase-plane analysis revealed that the NSFD formulated for tuberculosis and diabetes co-infection is generally asymptotically stable. Future studies should consider formulating the proposed model with varied control parameters such as medication to compare the results with those from first-order ode.

Numerical treatment of Burgers' equation based on weakly L-stable generalized time integration formula with the NSFD scheme

In this study, we present a weakly L-stable convergent time integration formula of order N1>3 (N1∈Z is odd) to solve Burgers' equation. The time integration formula for the initial value problem w′(t)=f(t,w),w(t0)=η0 is formulated using backward explicit Taylor series approximation of order (N1−1) and Hermite approximation polynomial of order (N1−2). We convert the Burgers' equation into the initial value problem using the nonstandard finite difference scheme for spatial derivatives and implement the derived integration formula. The nonstandard finite difference scheme makes it possible to choose several denominator functions. The method's convergence, stability and truncation error are also discussed. To show the correctness and effectiveness of the proposed technique, we present numerical solutions, ‖e‖2 and ‖e‖∞ error norms in several tables and figures. Additionally, the numerical outcomes are compared with the results of some existing techniques and exact solutions.

A dynamically consistent nonstandard finite difference scheme for a generalized SEIR epidemic model

This work is devoted to the proposal and analysis of a new mathematical study of the transmission dynamics of infectious diseases. First, a generalized SEIR epidemic model is presented that uses general nonlinear incidence rates to describe the ‘psychological’ effect. Then, a rigorous mathematical analysis is performed for the proposed SEIR model. We establish positivity and boundedness, calculate the basic reproduction number, determine possible equilibrium points (disease-free and endemic), and investigate their asymptotic stability properties of the SEIR model. The obtained results improve and extend a SEIR model constructed in a recent work; moreover, the proposed model is useful for studying the COVID-19 epidemic in particular and other infectious diseases in general. For the purpose of numerical simulation, the Mickens method is applied to construct a dynamically consistent non-standard finite difference (NSFD) model for the proposed SEIR epidemic model. The constructed NSFD scheme is able to provide reliable approximations that not only preserve the dynamic properties of the SEIR model for all values of the step size, but also are easy to implement. Finally, a series of illustrative numerical experiments are performed to support the theoretical findings and confirm the advantages of the NSFD scheme over some well-known standard methods.

Numerical investigation of a typhoid disease model in fuzzy environment

Salmonella Typhi, a bacteria, is responsible for typhoid fever, a potentially dangerous infection. Typhoid fever affects a large number of people each year, estimated to be between 11 and 20 million, resulting in a high mortality rate of 128,000 to 161,000 deaths. This research investigates a robust numerical analytic strategy for typhoid fever that takes infection protection into consideration and incorporates fuzzy parameters. The use of fuzzy parameters acknowledges the variation in parameter values among individuals in the population, which leads to uncertainties. Because of their diverse histories, different age groups within this community may exhibit distinct customs, habits, and levels of resistance. Fuzzy theory appears as the most appropriate instrument for dealing with these uncertainty. With this in mind, a model of typhoid fever featuring fuzzy parameters is thoroughly examined. Two numerical techniques are developed within a fuzzy framework to address this model. We employ the non-standard finite difference (NSFD) scheme, which ensures the preservation of essential properties like dynamic consistency and positivity. Additionally, we conduct numerical simulations to illustrate the practical applicability of the developed technique. In contrast to many classical methods commonly found in the literature, the proposed approach exhibits unconditional convergence, solidifying its status as a dependable tool for investigating the dynamics of typhoid disease.