ORIGINAL ARTICLE On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes Manh Tuan Hoang 1 • Oluwaseun Francis Egbelowo 2 Received: 29 September 2019 / Accepted: 7 January 2020 / Published online: 23 January 2020 Ó Sociedad Matemática Mexicana 2020 Abstract Very recently, a hepatitis B epidemic model with saturated incidence rate has been proposed and analyzed in Khan et al. (Chaos Solitons Fractals 124:1–9, 2019). The local asymptotic stability of the disease endemic equilibrium (DEE) point of model has been established theoretically but its global asymptotic stability has not been studied. In this paper, we present a mathematically rigorous analysis for the global asymptotic stability of the DEE point of the model. More precisely, we prove that if the DEE point exists, then it is not only locally asymptotically stable but also globally asymptotically stable. Furthermore, we present an alternative proof for the global stability of the disease-free equilibrium point. The main result is that we obtain the complete global stability of the hepatitis B virus model. Besides, we construct nonstandard finite difference (NSFD) schemes preserving the essential qualitative properties of the continuous model. These properties include the posi- tivity of solutions and the stability of the model. Dynamical properties of the pro- posed schemes are rigorously investigated by mathematical analyses and numerical simulations. Finally, numerical simulations are performed to confirm the validity of the theoretical results as well as the advantages of the NSFD schemes. Numerical simulations also indicate that the NSFD schemes are appropriate and effective to solve the continuous model. Meanwhile, the standard finite difference schemes such as the Euler scheme, the classical fourth-order Runge–Kutta scheme cannot preserve the essential properties of the continuous model. Consequently, they can generate numerical approximations which are completely different from the solutions of the model. Keywords Hepatitis B epidemic model  Global asymptotic stability  Lyapunov function  Nonstandard finite difference schemes  Numerical simulations Mathematics Subject Classification 37N25  37B25  37M05  65L12 Extended author information available on the last page of the article 123 Boletín de la Sociedad Matemática Mexicana (2020) 26:1113–1134 https://doi.org/10.1007/s40590-020-00275-2