Yibeltal Adane Terefe

Analysis and Dynamically Consistent Numerical Schemes for the SIS Model and Related Reaction Diffusion Equation

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 th...

A MATHEMATICAL MODEL FOR EBOLA EPIDEMIC WITH SELF-PROTECTION MEASURES

The South African Research Chairs Initiative of the Department of Science and Technology and National Research Foundation: SARChI Chair in Mathematical Models and Methods in Bioengineering and Biosciences. TB and YAT acknowledge the support, in part, of DST-NRF Centre of Excellence in Mathematical and Statistical Sciences (CoE-MaSS).

Global stability for the continuous and discrete SIS-diffusion epidemiological models

Abstract A susceptible-infectious-susceptible reaction-diffusion equation is used to model the spatial spread of a disease. It is shown that the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when the basic reproduction number is less than or equal to 1 and unstable when it is greater than 1. In the latter case, there exists an endemic equilibrium (EE) which is GAS. We construct nonstandard finite difference (NSFD) schemes which theoretically and computationally replicate the stability properties of the equilibria.