Reliable approximations for a hepatitis B virus model by nonstandard numerical schemes

Abstract

In this work, we propose and analyze nonstandard finite difference (NSFD) schemes for an improved hepatitis B virus (HBV) model. Dynamical properties of the constructed NSFD schemes are studied rigorously. It is proven that the NSFD schemes preserve positivity, boundedness and asymptotic stability of the HBV model for all finite step sizes. In addition, convergence and error bounds for the NSFD schemes are also investigated. The theoretical results and advantages of the NSFD schemes over some existing standard finite difference ones are supported and illustrated by a set of numerical examples. The numerical results show that some typical standard numerical schemes cannot preserve the dynamics of the continuous model for some given step sizes; consequently, they generate numerical solutions that are not only unstable but also negative. Conversely, the constructed NSFD schemes preserve the essential mathematical features of the continuous model for any finite step size; therefore, they are suitable to simulate the dynamics of the HBV model over long time intervals.