Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model

Martín-Vaquero, J.; Dios, Araceli Queiruga; Rey, Ángel Martín del; Encinas, Ascensión Hernández; Guillén, Jose Diamantino Hernández; Sánchez, Gerardo Rodríguez

doi:10.1016/j.cam.2017.03.031

Cited by 24 publications

(12 citation statements)

References 21 publications

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“…Several authors have used discrete Lyapunov functions to study the global behavior of numerical solutions generated by non-standard finite difference schemes (NSFD), see [19,[106][107][108][109][110]. For the study of the global stability of the critical points of the numerical scheme (34) it is necessary to use the Lyapunov functions and Equation (28).…”

Section: Global Stability Of the Nsfd Numerical Schemementioning

confidence: 99%

“…Numerous mathematical models represented by means of a system of differential equations, with or without delay, have been discretized by means of the non-standard finite difference method proposed by Ronald Mickens, see [24][25][26][27][28][29][30][31][32][33][34]. Their use is mainly because they are very effective in preserving certain qualitative properties of the original differential equations and the convergence, consistency and stability of their solutions have been demonstrated, see [35][36][37][38][39][40][41][42][43][44][45][46].…”

Section: Introductionmentioning

confidence: 99%

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Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay

et al. 2021

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We propose a mathematical model based on a set of delay differential equations that describe intracellular HIV infection. The model includes three different subpopulations of cells and the HIV virus. The mathematical model is formulated in such a way that takes into account the time between viral entry into a target cell and the production of new virions. We study the local stability of the infection-free and endemic equilibrium states. Moreover, by using a suitable Lyapunov functional and the LaSalle invariant principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable. In addition, we designed a non-standard difference scheme that preserves some relevant properties of the continuous mathematical model.

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Section: Global Stability Of the Nsfd Numerical Schemementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay

et al. 2021

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“…The scheme was the predictor-corrector type, but it did not satisfy some features of the model. Some more work on the non-standard finite difference method and compact numerical schemes can be seen in Yanga (2016); Raza et al 2021;Martin-Vaquero et al 2018;Arenas and Gilberto Gonz'alez-Parra, Benito M. Chen-Charpentier, 2017 ;Muhammad Rafiq et al 2021;Arif et al 2019;Nawaz andArif 2021 Nawaz et al 2021a). A Caputo fractional derivative-based COVID-19 pandemic model has been formulated in Mohammad et al (2021).…”

Section: Introductionmentioning

confidence: 99%

Development of Explicit Schemes for Diffusive SEAIR COVID-19 Epidemic Spreading Model: An Application to Computational Biology

Nawaz

Arif

Ashraf

2021

Iran J Sci Technol Trans Sci

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In this contribution, a first-order time scheme is proposed for finding solutions to partial differential equations (PDEs). A mathematical model of the COVID-19 epidemic is modified where the recovery rate of exposed individuals is also considered. The linear stability of the equilibrium states for the modified COVID-19 model is given by finding its Jacobian and applying Routh–Hurwitz criteria on characteristic polynomial. The proposed scheme provides the first-order accuracy in time and second-order accuracy in space. The stability of the proposed scheme is given using the von Neumann stability criterion for standard parabolic PDEs. The consistency for the proposed scheme is also given by expanding the involved terms in it using the Taylor series. The scheme can be used to obtain the condition of getting a positive solution. The stability region of the scheme can be enlarged by choosing suitable values of the contained parameter. Finally, a comparison of the proposed scheme is made with the existing non-standard finite difference method. The results indicate that the non-standard classical technique is incapable of preserving the unique characteristics of the model’s epidemiologically significant solutions, whereas the proposed approaches are capable of doing so. A computational code for the proposed discrete model scheme may be made available to readers upon request for convenience.

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“…This motivates the problem of increasing the order of accuracy of NSFD schemes. Recently, some higher order NSFD schemes were constructed [14,23,24]. They are based on the combination of nonstandard schemes and Richardson's extrapolation.…”

Section: Introductionmentioning

confidence: 99%

Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

Dang¹,

Hoang²

2019

International Journal of Computer Mathematics

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In this paper, we construct explicit nonstandard Runge-Kutta (ENRK) methods which have higher accuracy order and preserve two important properties of autonomous dynamical systems, namely, the positivity and linear stability. These methods are based on the classical explicit Runge-Kutta methods, where instead of the usual h in the formulas there stands a function ϕ(h). It is proved that the constructed methods preserve the accuracy order of the original Runge-Kutta methods. The numerical simulations confirm the validity of the obtained theoretical results. difference schemes; Positive nonstandard finite difference methods; Elementary stable. Our first objective is to construct difference schemes preserving the linear stability of the equilibrium points of System (1) for all finite step-size h > 0. These schemes are called also elementary stable [3,7,8]. It should be emphasized that standard finite difference schemes cannot preserve properties of the differential equations for any step-sizes h > 0, including the linear stability. Mickens called this phenomenon numerical instability [25].The construction of elementary stable difference schemes play especially important role in numerical solution of differential equations and numerical simulation of nonlinear dynamical systems. In 2005, Dimitrov and Kojouharov [7] proposed a method for constructing elementary stable NSFD methods for general two-dimensional autonomous dynamical systems. These NSFD methods are based on the explicit and implicit Euler and the second-order Runge-Kutta methods. Later, in 2007 these results are extented for the general n-dimensional dynamical systems, namely, NSFD schemes preserving elementary stability are constructed based on the θ-methods and the second-order Runge-Kutta methods [8]. One important action in the construction of the elementary stable NSFD schemes is the replacement of the standard denominator function ϕ(h) = h by the nonstandard denominator function, which is Email addresses: dangquanga@cic.vast.vn (Quang A Dang),

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Variable step length algorithms with high-order extrapolated non-standard finite difference schemes for a SEIR model

Cited by 24 publications

References 21 publications

Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay

Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay

Development of Explicit Schemes for Diffusive SEAIR COVID-19 Epidemic Spreading Model: An Application to Computational Biology

Positive and elementary stable explicit nonstandard Runge-Kutta methods for a class of autonomous dynamical systems

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