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2013 | 23 | 2 | 357-372
Tytuł artykułu

An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We formulate and analyze an unconditionally stable nonstandard finite difference method for a mathematical model of HIV transmission dynamics. The dynamics of this model are studied using the qualitative theory of dynamical systems. These qualitative features of the continuous model are preserved by the numerical method that we propose in this paper. This method also preserves the positivity of the solution, which is one of the essential requirements when modeling epidemic diseases. Robust numerical results confirming theoretical investigations are provided. Comparisons are also made with the other conventional approaches that are routinely used for such problems.
Rocznik
Tom
23
Numer
2
Strony
357-372
Opis fizyczny
Daty
wydano
2013
otrzymano
2012-03-21
poprawiono
2012-12-17
Twórcy
  • Department of Mathematics and Applied Mathematics, University of the Western Cape, Bellville 7535, South Africa
  • DST/NRF Centre of Excellence in Epidemiological Modelling and Analysis, Stellenbosch University, Stellenbosch 7600, South Africa
  • Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, South Africa
Bibliografia
  • Allen, L. (2007). An Introduction to Mathematical Biology, Prentice-Hall, Englewood Cliffs, NJ.
  • Anguelov, R., Lubuma, J.-S. and Shillor, M. (2009). Dynamically consistent nonstandard finite difference schemes for continuous dynamical systems, Discrete and Continuous Dynamical Systems: Supplement 61: 34-43.
  • Arenas, A., No, J.M. and Cortés, J. (2008). Nonstandard numerical method for a mathematical model of RSV epidemiological transmission, Computers and Mathematics with Applications 56(3): 670-678.
  • Bacaer, N., Ouifki, R., Pretorius, C., Wood, R. and Williams, B. (2008). Modeling the joint epidemics of TB and HIV in a South African township, Journal of Mathematical Biology 57(4): 557-593.
  • Brauer, F. and Castillo-Chavez, C. (2001). Mathematical Models in Population Biology and Epidemiology, Springer-Verlag, New York, NY.
  • Dimitrov, D. and Kojouharov, H. (2005). Nonstandard finite-difference schemes for general two-dimensional autonomous dynamical systems, Applied Mathematics Letters 18(7): 769-774.
  • Dimitrov, D. and Kojouharov, H. (2006). Positive and elementary stable nonstandard numerical methods with applications to predator-prey models, Journal of Computational and Applied Mathematics 189(1): 98-108.
  • Dimitrov, D. and Kojouharov, H. (2007). Stability-preserving finite-difference methods for general multi-dimensional autonomous dynamical systems, International Journal of Numerical Analysis Modeling 4(2): 280-290.
  • Karcz-Dulęba, I. (2004). Asymptotic behaviour of a discrete dynamical system generated by a simple evolutionary process, International Journal of Applied Mathematics and Computer Science 14(1): 79-90.
  • Gumel, A., McCluskey, C. and van den Driessche, P. (2006). Mathematical study of a staged-progression HIV model with imperfect vaccine, Bulletin of Mathematical Biology 68(8): 2105-2128.
  • Gumel, A., Patidar, K. and Spiteri, R. (2005). Asymptotically consistent nonstandard finite difference methods for solving mathematical models arising in population biology, in R. Mickens (Ed.), Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore, pp. 385-421.
  • Ibijola, E., Ogunrinde, R. and Ade-Ibijola, O. (2008). On the theory and applications of new nonstandard finite difference methods for the solution of initial value problems in ordinary differential equations, Advances in Natural and Applied Sciences 2(3): 214-224.
  • Jódar, L., Villanueva, R., Arenas, A. and González, G. (2008). Nonstandard numerical methods for a mathematical model for influenza disease, Mathematics and Computers in Simulation 79(3): 622-633.
  • Kadalbajoo, M., Patidar, K. and Sharma, K. (2006). ε-uniformly convergent fitted methods for the numerical solution of the problems arising from singularly perturbed general DDEs, Applied Mathematics and Computation 182(1): 119-139.
  • Kouche, M. and Ainseba, B. (2010). A mathematical model of HIV-1 infection including the saturation effect of healthy cell proliferation, International Journal of Applied Mathematics and Computer Science 20(3): 601-612, DOI: 10.2478/v10006-010-0045-z.
  • Lubuma, J.-S. and Patidar, K. (2006). Uniformly convergent non-standard finite difference methods for self-adjoint singular perturbation problems, Journal of Computational and Applied Mathematics 191(2): 229-238.
  • Lubuma, J.-S. and Patidar, K. (2007a). Non-standard methods for singularly perturbed problems possessing oscillatory/layer solutions, Applied Mathematics and Computation 187(2): 1147-1160.
  • Lubuma, J.-S. and Patidar, K. (2007b). Solving singularly perturbed advection reaction equation via non-standard finite difference methods, Mathematical Methods in the Applied Sciences 30(14): 1627-1637.
  • Lubuma, J.-S. and Patidar, K. (2007c). ε-uniform non-standard finite difference methods for singularly perturbed nonlinear boundary value problems, Advances in Mathematical Sciences and Applications 17(2): 651-665.
  • Mickens, R. (2007). Calculation of denominator functions for nonstandard finite difference schemes for differential equations satisfying a positivity condition, Numerical Methods for Partial Differential Equations 23(3): 672-691.
  • Mickens, R. and Ramadhani, I. (1994). Finite-difference schemes having the correct linear stability properties for all finite step-sizes III, Computer in Mathematics with Applications 27(4): 77-84.
  • Mickens, R. and Smith, A. (1990). Finite difference models of ordinary differential equations: Influence of denominator functions, Journal of the Franklin Institute 327(1): 143-145.
  • Munyakazi, J. and Patidar, K. (2010). Higher order numerical methods for singularly perturbed elliptic problems, Neural, Parallel & Scientific Computations 18(1): 75-88.
  • Patidar, K. (2005). On the use of nonstandard finite difference methods, Journal of Difference Equations and Applications 11(8): 735-758.
  • Patidar, K. (2008). A robust fitted operator finite difference method for a two-parameter singular perturbation problem, Journal of Difference Equations and Applications 14(12): 1197-1214.
  • Patidar, K. and Sharma, K. (2006a). Uniformly convergent nonstandard finite difference methods for singularly perturbed differential difference equations with delay and advance, International Journal for Numerical Methods in Engineering 66(2): 272-296.
  • Patidar, K. and Sharma, K. (2006b). ε-uniformly convergent non-standard finite difference methods for singularly perturbed differential difference equations with small delay, Applied Mathematics and Computation 175(1): 864-890.
  • Rauh, A., Brill, M. and Günther, C. (2009). A novel interval arithmetic approach for solving differential-algebraic equations with ValEncIA-IVP, International Journal of Applied Mathematics and Computer Science 19(3): 381-397, DOI: 10.2478/v10006-009-0032-4.
  • Rauh, A., Minisini, J. and Hofer, E.P. (2009). Verification techniques for sensitivity analysis and design of controllers for nonlinear dynamic systems with uncertainties, International Journal of Applied Mathematics and Computer Science 19(3): 425-439, DOI: 10.2478/v10006-009-0035-1.
  • Villanueva, R., Arenas, A. and Gonzalez-Parra, G. (2008). A nonstandard dynamically consistent numerical scheme applied to obesity dynamics, Journal of Applied Mathematics, Article ID 640154, DOI: 10.1155/2008/640154.
  • Xu, C., Liao, M. and He, X. (2011). Stability and Hopf bifurcation analysis for a Lotka-Volterra predator-prey model with two delays, International Journal of Applied Mathematics and Computer Science 21(1): 97-107, DOI: 10.2478/v10006-011-0007-0.
  • Zhai, G. and Michel, A.M. (2004). Generalized practical stability analysis of discontinuous dynamical systems, International Journal of Applied Mathematics and Computer Science 14(1): 5-12.
Typ dokumentu
Bibliografia
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