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Nonstandard numerical methods for a mathematical model for influenza disease

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  • Jódar, Lucas
  • Villanueva, Rafael J.
  • Arenas, Abraham J.
  • González, Gilberto C.

Abstract

In this paper we construct and develop two competitive implicit finite difference scheme for a deterministic mathematical model associated with the evolution of influenza A disease in human population. Qualitative dynamics of the model is determined by the basic reproduction number, R0. Numerical schemes developed here are based on nonstandard finite difference methods. Our aim is to transfer essential properties of the continuous model to the discrete schemes and to obtain unconditional stable schemes. The proposed numerical schemes have two fixed points which are identical to the critical points of the continuous model and it is shown that they have the same stability properties. Numerical simulations with different initial conditions, parameters values and time step sizes are developed for different values of parameter R0, convergence to the disease free equilibrium point when R0<1 and to the endemic equilibrium point when R0>1 are obtained independent of the time step size. These numerical integration schemes are useful since can reproduce the dynamics of original differential equations.

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  • Jódar, Lucas & Villanueva, Rafael J. & Arenas, Abraham J. & González, Gilberto C., 2008. "Nonstandard numerical methods for a mathematical model for influenza disease," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(3), pages 622-633.
    • Handle: RePEc:eee:matcom:v:79:y:2008:i:3:p:622-633
    DOI: 10.1016/j.matcom.2008.04.008
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    References listed on IDEAS

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    1. Dimitrov, Dobromir T. & Kojouharov, Hristo V., 2008. "Nonstandard finite-difference methods for predator–prey models with general functional response," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 78(1), pages 1-11.
    2. Oded Galor, 2007. "Discrete Dynamical Systems," Springer Books, Springer, edition 1, number 978-3-540-36776-5, November.
    3. Jansen, H. & Twizell, E.H., 2002. "An unconditionally convergent discretization of the SEIR model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 58(2), pages 147-158.
    4. Anguelov, Roumen & Lubuma, Jean M.-S., 2003. "Nonstandard finite difference method by nonlocal approximation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 61(3), pages 465-475.
    5. Chen-Charpentier, Benito M. & Dimitrov, Dobromir T. & Kojouharov, Hristo V., 2006. "Combined nonstandard numerical methods for ODEs with polynomial right-hand sides," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 73(1), pages 105-113.
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    Cited by:

    1. Abraham J. Arenas & Gilberto González-Parra & Jhon J. Naranjo & Myladis Cogollo & Nicolás De La Espriella, 2021. "Mathematical Analysis and Numerical Solution of a Model of HIV with a Discrete Time Delay," Mathematics, MDPI, vol. 9(3), pages 1-21, January.
    2. Xin Jiang, 2021. "Global Dynamics for an Age-Structured Cholera Infection Model with General Infection Rates," Mathematics, MDPI, vol. 9(23), pages 1-20, November.
    3. Adamu, Elias M. & Patidar, Kailash C. & Ramanantoanina, Andriamihaja, 2021. "An unconditionally stable nonstandard finite difference method to solve a mathematical model describing Visceral Leishmaniasis," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 187(C), pages 171-190.
    4. Acedo, L. & González-Parra, Gilberto & Arenas, Abraham J., 2010. "Modal series solution for an epidemic model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 389(5), pages 1151-1157.
    5. María Ángeles Castro & Miguel Antonio García & José Antonio Martín & Francisco Rodríguez, 2019. "Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems," Mathematics, MDPI, vol. 7(11), pages 1-14, November.
    6. Khalsaraei, Mohammad Mehdizadeh & Shokri, Ali & Ramos, Higinio & Heydari, Shahin, 2021. "A positive and elementary stable nonstandard explicit scheme for a mathematical model of the influenza disease," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 182(C), pages 397-410.

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