Analysis and dynamically consistent numerical schemes for the SIS model and related reaction diffusion equation
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Date
2011
Journal Title
Journal ISSN
Volume Title
Publisher
Proceedings of the 3rd International Conference on Applied Mathematics in Technical and Natural Sciences,(AMiTaNS’11), (Albena, Bulgaria), Institute of Physics-AIP Conf. Proc.
Abstract
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.
Description
Keywords
Epidemiological models, local/global stability, nonstandard finite difference scheme, reaction diffusion