Browsing by Author "Lubuma, Jean M.S."

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    Analysis and Dynamically Consistent Numerical Schemes for the SIS Model and Related Reaction Diffusion Equation
    (2011-10) Lubuma, Jean M.S.; Mureithi, Eunice; Terefe, Yibeltal A.
    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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    Nonstandard Discretizations of the SIS Epidemiological Model with and without Diffusion
    (2013-12) Lubuma, Jean M.S.; Mureithi, Eunice; Terefe, Yibeltal
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    Stability Analysis and Dynamics Preserving Nonstandard Finite Difference Schemes for a Malaria Model
    (2013-03) Anguelov, Roumen; Dumont, Yves; Lubuma, Jean M.S.; Mureithi, Eunice
    When both human and mosquito populations vary, forward bifurcation occurs if the basic reproduction number R 0 is less than one in the absence of disease-induced death. When the disease-induced death rate is large enough, R 0 = 1 is a subcritical backward bifurcation point. The domain for the study of the dynamics is reduced to a compact and feasible region, where the system admits a specific algebraic decomposition into infective and non-infected humans and mosquitoes. Stability results are extended and the possibility of backward bifurcation is clarified. A dynamically consistent nonstandard finite difference scheme is designed.