Non-standard discretizations of SIS epidemiological model with or without diffusion
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Date
2014
Journal Title
Journal ISSN
Volume Title
Publisher
Contemporary Mathematics, Mathematics of Continuous and Discrete Dynamical System
Abstract
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.