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2014 | 24 | 3 | 635-646

Tytuł artykułu

An unconditionally positive and global stability preserving NSFD scheme for an epidemic model with vaccination

Treść / Zawartość

Warianty tytułu

Języki publikacji

EN

Abstrakty

EN
In this paper, a NonStandard Finite Difference (NSFD) scheme is constructed, which can be used to determine numerical solutions for an epidemic model with vaccination. Here the NSFD method is employed to derive a set of difference equations for the epidemic model with vaccination. We show that difference equations have the same dynamics as the original differential system, such as the positivity of the solutions and the stability of the equilibria, without being restricted by the time step. Our proof of global stability utilizes the method of Lyapunov functions. Numerical simulation illustrates the effectiveness of our results.

Rocznik

Tom

24

Numer

3

Strony

635-646

Opis fizyczny

Daty

wydano
2014
otrzymano
2013-05-31
poprawiono
2013-11-15
poprawiono
2014-03-08

Twórcy

autor
  • Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai Shangdong 264209, PR China
autor
  • Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai Shangdong 264209, PR China
autor
  • Department of Mathematics, Harbin Institute of Technology (Weihai), Weihai Shangdong 264209, PR China

Bibliografia

  • Alexander, M.E., Summers, A.R. and Moghadas, S.M. (2006). Neimark-Sacker bifurcations in a non-standard numerical scheme for a class of positivity-preserving ODEs, Proceedings of the Royal Society, A: Mathematical, Physical and Engineering Sciences 462(2074): 3167-3184.
  • Anderson, R. and May, R. (1991). Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford/New York, NY.
  • Anguelo, V.R. and Lubuma, J. (2003). Nonstandard finite difference method by nonlocal approximation, Mathematics and Computers in Simulation 61(3): 465-475.
  • Arenas, A., Moraño, J. and Cortés, J. (2008). Non-standard numerical method for a mathematical model of RSV epidemiological transmission, Computers and Mathematics with Applications 56(3): 670-678.
  • Bruggeman, J., Burchard, H., Kooi, B.W. and Sommeijer, B. (2007). A second-order, unconditionally positive, mass-conserving integration scheme for biochemical systems, Applied Numerical Mathematics 57(1): 36-58.
  • Chen, M. and Clemence, D. (2006). Stability properties of a nonstandard finite difference scheme for a hantavirus epidemic model, Journal of Difference Equations and Applications 12(12): 1243-1256.
  • Chinviriyasit, S. and Chinviriyasit, W. (2010). Numerical modelling of an SIR epidemic model with diffusion, Applied Mathematics and Computation 216(2): 395-409.
  • Dimitrov, D.T. 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. (2007). Stability-preserving finite-difference methods for general multi-dimensional autonomous dynamical systems, International Journal of Numerical Analysis and Modeling 4(2): 280-290.
  • Dimitrov, D. and Kojouharov, H. (2008). Nonstandard finite difference methods for predator-prey models with general functional response, Mathematics and Computers in Simulation 78(1): 1-11.
  • Ding, D., Ma, Q. and Ding, X. (2013). A non-standard finite difference scheme for an epidemic model with vaccination, Journal of Difference Equations and Applications 19(2): 179-190.
  • Dumont, Y. and Lubuma, J.M.-S. (2005). Non-standard finite-difference methods for vibro-impact problems, Proceedings of the Royal Society, A: Mathematical, Physical and Engineering Sciences 461(2058): 1927-1950.
  • Enatsu, Y., Nakata, Y. and Muroya, Y. (2010). Global stability for a class of discrete SIR epidemic models, Mathematical Biosciences and Engineering 7: 347-361.
  • Enszer, J.A. and Stadtherr, M.A. (2009). Verified solution method for population epidemiology models with uncertainty, International Journal of Applied Mathematics and Computer Science 19(3): 501-512, DOI: 10.2478/v10006-009-0040-4.
  • Gumel, A. (2002). A competitive numerical method for a chemotherapy model of two HIV subtypes, Applied Mathematics and Computation 131(2): 329-337.
  • Hildebrand, F. (1968). Finite Difference Equations and Simulations, Prentice-Hall, Englewood Cliffs, NJ.
  • Jódar, L., Villanueva, R., Arenas, A. and Gonz´alez, G. (2008). Nonstandard numerical methods for a mathematical model for influenza disease, Mathematics and Computers in Simulation 79(3): 622-633.
  • Jang, S. (2007). On a discrete west Nile epidemic model, Computational and Applied Mathematics 26(3): 397-414.
  • Jang, S. and Elaydi, S. (2003). Difference equations from discretization of a continuous epidemic model with immigration of infectives, Canadian Applied Mathematics Quarterly 11(1): 93-105.
  • Jansen, H. and Twizell, E. (2002). An unconditionally convergent discretization of the SEIR model, Mathematics and Computers in Simulation 58(2): 147-158.
  • 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.
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  • Mickens, R. (2000). Advances in the Applications of Nonstandard Finite Difference Schemes, World Scientific, Singapore.
  • Mickens, R. (2002). Nonstandard finite difference schemes for differential equations, Journal of Difference Equations and Applications 8(9): 823-847.
  • Mickens, R. (2005). Dynamic consistency: A fundamental principle for constructing nonstandard finite difference schemes for differential equations, Journal of Difference Equations and Applications 11(7): 645-653.
  • Moghadas, S., Alexander, M. and Corbett, B.D.and Gumel, A. (2003). A positivity preserving Mickens-type discretization of an epidemic model, Journal of Difference Equations and Applications 9(11): 1037-1051.
  • Moghadas, S. and Gumel, A. (2003). A mathematical study of a model for childhood diseases with non-permanent immunity, Journal of Computational and Applied Mathematics 157(2): 347-363.
  • Muroya, Y., Nakata, Y., Izzo, G. and Vecchio, A. (2011). Permanence and global stability of a class of discrete epidemic models, Nonlinear Analysis: Real World Applications 12(4): 2105-2117.
  • Obaid, H., Ouifki, R. and Patidar, K.C. (2013). An unconditionally stable nonstandard finite difference method applied to a mathematical model of HIV infection, International Journal of Applied Mathematics and Computer Science 23(2): 357-372, DOI: 10.2478/amcs-2013-0027.
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  • Sundarapandian, V. (2003). An invariance principle for discrete-time nonlinear systems, Applied Mathematics Letters 16(1): 85-91.

Typ dokumentu

Bibliografia

Identyfikatory

Identyfikator YADDA

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