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An unconditionally stable nonstandard finite difference method to solve a mathematical model describing Visceral Leishmaniasis

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  • Adamu, Elias M.
  • Patidar, Kailash C.
  • Ramanantoanina, Andriamihaja

Abstract

In this paper, a mathematical model of Visceral Leishmaniasis is considered. The model incorporates three populations, the human, the reservoir and the vector host populations. A detailed analysis of the model is presented. This analysis reveals that the model undergoes a backward bifurcation when the associated reproduction threshold is less than unity. For the case where the death rate due to VL is negligible, the disease-free equilibrium of the model is shown to be globally-asymptotically stable if the reproduction number is less than unity. Noticing that the governing model is a system of highly nonlinear differential equations, its analytical solution is hard to obtain. To this end, a special class of numerical methods, known as the nonstandard finite difference (NSFD) method is introduced. Then a rigorous theoretical analysis of the proposed numerical method is carried out. We showed that this method is unconditionally stable. The results obtained by NSFD are compared with other well-known standard numerical methods such as forward Euler method and the fourth-order Runge–Kutta method. Furthermore, the NSFD preserves the positivity of the solutions and is more efficient than the standard numerical methods.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:matcom:v:187:y:2021:i:c:p:171-190
    DOI: 10.1016/j.matcom.2021.02.007
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    References listed on IDEAS

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    1. Oded Galor, 1996. "Discrete Dynamical Systems," Working Papers 96-23, Brown University, Department of Economics.
    2. Biswas, Santanu, 2017. "Mathematical modeling of Visceral Leishmaniasis and control strategies," Chaos, Solitons & Fractals, Elsevier, vol. 104(C), pages 546-556.
    3. 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.
    4. Oded Galor, 2007. "Discrete Dynamical Systems," Springer Books, Springer, edition 1, number 978-3-540-36776-5, September.
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    Cited by:

    1. Pasha, Syed Ahmed & Nawaz, Yasir & Arif, Muhammad Shoaib, 2023. "On the nonstandard finite difference method for reaction–diffusion models," Chaos, Solitons & Fractals, Elsevier, vol. 166(C).
    2. Hoang, Manh Tuan, 2022. "Reliable approximations for a hepatitis B virus model by nonstandard numerical schemes," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 193(C), pages 32-56.
    3. Hoang, Manh Tuan, 2022. "Positivity and boundedness preserving nonstandard finite difference schemes for solving Volterra’s population growth model," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 199(C), pages 359-373.

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