Browsing by Author "Terefe, Yibeltal"

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    Analysis and dynamically consistent numerical schemes for the SIS model and related reaction diffusion equation
    (Proceedings of the 3rd International Conference on Applied Mathematics in Technical and Natural Sciences,(AMiTaNS’11), (Albena, Bulgaria), Institute of Physics-AIP Conf. Proc., 2011) Lubuma, Jean; Mureithi, Eunice; Terefe, Yibeltal
    The classical SIS epidemiological model is extended in two directions: (a) The number of adequate contacts per infective in unit time is assumed to be a function of the total population in such a way that this number grows less rapidly as the total population increases; (b) A diffusion term is added to the SIS model and this leads to a reaction diffusion equation, which governs the spatial spread of the disease. With the parameter R0 representing the basic reproduction number, it is shown that R0 = 1 is a forward bifurcation for the model (a), with the disease-free equilibrium being globally asymptotic stable when R0 is less than 1. In the case when R0 is greater than 1, traveling wave solutions are found for the model (b). Nonstandard finite difference (NSFD) schemes that replicate the dynamics of the continuous models are presented. In particular, for the model (a), a nonstandard version of the Runge-Kutta method having high order of convergence is investigated. Numerical experiments that support the theory are provided.
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    Non-standard discretizations of SIS epidemiological model with or without diffusion
    (Contemporary Mathematics, Mathematics of Continuous and Discrete Dynamical System, 2014) Lubuma, Jean; Mureithi, Eunice; Terefe, Yibeltal
    We design and investigate the reliability of various nonstandard finite difference (NSFD) schemes for SIS-type epidemiological models. The success of the study is due to an innovative use of Mickens’ rules of complex denominator functions and nonlocal approximation of nonlinear terms. For the classical SIS-ODE model, we construct for the first time a nonstandard Runge- Kutta method, which is shown to be of order four. We also consider two new NSFD schemes which faithfully replicate the property of the continuous model of having the value R0 = 1 of the basic reproduction parameter as a forward bifurcation: the disease-free equilibrium is globally asymptotically stable when R0 < 1; it is unstable when R0 > 1 and there appears a unique locally asymptotically stable endemic equilibrium in this case. The latter schemes are further used to derive NSFD schemes that are dynamically consistent with the positivity and boundedness properties of the SIS-diffusion model for the spatial spread of disease. Numerical simulations that support the theory are provided.
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    Nonstandard Discretizations of the SIS Epidemiological Model with and without Diffusion
    (2013-12) Lubuma, Jean M.S.; Mureithi, Eunice; Terefe, Yibeltal