On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes
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ORIGINAL ARTICLE
On the global asymptotic stability of a hepatitis B epidemic model and its solutions by nonstandard numerical schemes Manh Tuan Hoang1
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Oluwaseun Francis Egbelowo2
Received: 29 September 2019 / Accepted: 7 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 positivity of solutions and the stability of the model. Dynamical properties of the proposed 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
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M. T. Hoang, O. F. Egbelowo
1 Introduction It is well known that the hepatitis B is one of the most devastating diseases in the world. It is a viral infection that attacks the liver and can cause both acute and chronic diseases. The hepatitis B is a major health problem, and the most serious type of viral hepatitis. This disease leads to cirrhosis, liver cancer, or liver failure if not detected and managed. It is a big global health issue. There are over 350 million chronic HBV carriers and 0.6 million fatalities annually from HBV-related liver disease or hepatocellular carcinoma [29, 30, 32]). Therefore, an efficient technique to predict and eradicate the hepatitis B virus is very needed. Several mathematical models for HBV have been devel
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