Dynamically consistent discrete-time SI and SIS epidemic models

Abstract

Discrete-time $SI$ and $SIS$ epidemic models are constructed by applying the nonstandard finite difference (NSFD) schemes to the differential equation models. The difference equation systems are dynamically consistent with their analog continuous-time models. The basic standard incidence $SI$ and $SIS$ models without births and deaths, with births and deaths, and with immigrations, are considered. The continuous models possess either the conservation law that the total population is a constant or the total population $N$ satisfies $N'(t)=\lambda-\mu N$ and so that $N$ approaches a constant $\lambda/\mu$ as $t$ approaches infinity. The difference equation systems via NSFD schemes preserve all properties including the positivity of solutions, the conservation law, and the local and some of the global stability of the equilibria. They are said to be dynamically consistent with the continuous models with respect to these properties. We show that a simple criterion for choosing a certain NSFD scheme such that the positivity solutions are preserved is usually an indication of an appropriate NSFD scheme.