Abstract:
We design and investigate the reliability of various nonstandard nite di erence (NSFD)
schemes for the SIS epidemiological model in three di erent settings.
For the classical SIS model, we construct two new NSFD schemes which faithfully
replicate the property of the continuous model of having the parameter R0, the basic
reproduction number, as a threshold to determine the stability properties of equilibrium
points: the disease-free equilibrium (DFE) is globally asymptotically stable (GAS) when
R0 1; it is unstable when R0 > 1 and there appears a unique GAS endemic equilibrium
(EE) in this case. These schemes also preserve the positivity and boundedness properties
of solutions of the classical SIS model. The schemes are further used to derive NSFD
schemes for the SIS-di usion model which constitutes the second setting of the study. The
designed NSFD schemes are dynamically consistent with the global asymptotic stability of
the disease-free equilibrium for R0 1 and the instability of the disease-free equilibrium
for R0 > 1. In the latter case, the schemes replicate the global asymptotic stability of the
endemic equilibrium. Positivity and boundedness properties of solutions of the SIS-di usion
model are also preserved by the NSFD schemes.
In a third step, the classical SIS model is extended into a SIS-Volterra integral equation
model in which the contact rate is a function of fraction of infective individuals and allows
a distributed period of infectivity. The qualitative analysis is now based on two threshold
parameters Rc
0 1 Rm0
. The system can undergo the backward bifurcation phenomenon
as follows. The DFE is the only equilibrium and it is GAS when R0 < Rc
0; there exists
only one EE, which is GAS when R0 > Rm0
with the DFE being unstable when R0 > 1;
for Rc
0 < R0 < 1, the DFE is locally asymptotically stable (LAS) and coexists with at
least one LAS endemic equilibrium. We design a NSFD scheme and prove theoretically and
computationally that it preserves the above-stated stability properties of equilibria as well
as positivity and boundedness of the solutions of the continuous model.
