The Dynamics of a Delay Model of Hepatitis B Virus Infection with Logistic Hepatocyte Growth

Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, United States.
Mathematical biosciences and engineering: MBE (Impact Factor: 0.84). 05/2009; 6(2):283-99. DOI: 10.3934/mbe.2009.6.283
Source: PubMed


Chronic HBV affects 350 million people and can lead to death through cirrhosis-induced liver failure or hepatocellular carcinoma. We analyze the dynamics of a model considering logistic hepatocyte growth and a standard incidence function governing viral infection. This model also considers an explicit time delay in virus production. With this model formulation all model parameters can be estimated from biological data; we also simulate a course of lamivudine therapy and find that the model gives good agreement with clinical data. Previous models considering constant hepatocyte growth have permitted only two dynamical possibilities: convergence to a virus free or a chronic steady state. Our model admits a third possibility of sustained oscillations. We show that when the basic reproductive number is greater than 1 there exists a biologically meaningful chronic steady state, and the stability of this steady state is dependent upon both the rate of hepatocyte regeneration and the virulence of the disease. When the chronic steady state is unstable, simulations show the existence of an attracting periodic orbit. Minimum hepatocyte populations are very small in the periodic orbit, and such a state likely represents acute liver failure. Therefore, the often sudden onset of liver failure in chronic HBV patients can be explained as a switch in stability caused by the gradual evolution of parameters representing the disease state.

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    • "These prompt many researchers to study mathematical modelling and model analysis of the interaction between the host cells and viruses such as human immunodeficiency virus (HIV) (see e.g. [3–7,9,15,16,18,22,24]), hepatitis B virus (HBV) [2] [12], hepatitis C virus (HCV) [14] [20] [21], human T cell leukemia [11] and dengue virus [23], etc. There are many benefits from mathematical models of viral infection including: (i) they provide important quantitative insights into viral dynamics in vivo, (ii) they can improve diagnosis and treatment strategies which raise hopes of patients infected with viruses, (iii) they can be used to estimate key parameter values that control the infection process. "
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    ABSTRACT: In this paper, we propose and analyse a virus dynamics model with humoral immune response including latently infected cells. The incidence rate is given by Beddington-DeAngelis functional response. We have derived two threshold parameters, the basic infection reproduction number [Formula: see text] and the humoral immune response activation number [Formula: see text] which completely determined the basic and global properties of the virus dynamics model. By constructing suitable Lyapunov functions and applying LaSalle's invariance principle we have proven that if [Formula: see text], then the infection-free equilibrium is globally asymptotically stable (GAS), if [Formula: see text], then the chronic-infection equilibrium without humoral immune response is GAS, and if [Formula: see text], then the chronic-infection equilibrium with humoral immune response is globally asymptotically stable. These results are further illustrated by numerical simulations.
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    • "it can be detected in non hepatic tissues (kidney, pancreas or skin). Hepatitis B is endemic in Africa [8] [32] [1] [23] [49] (see also the references cited therein) and some models have been constructed in order to understand its dynamic with Ordinary Differential Equations (with or without time delay) or based on stochastic processes [5] [10] [20] [32] [16] [45] [30] [56] or using partial differential equations models [55] [53]. "

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    • "Hepatitis B is endemic in Africa [4] [14] [18] (see also the references cited therein) and some models have been constructed in order to understand its dynamic with ODE deterministic (with delay or not) or stochastic processes [1] [2] [5] [6] [7] [14] [21] or partial differential equations [19] [20]. Studies like [11] [12] [16] [19] [20] recognized the importance of the age factor in the dynamics of infectious diseases like hepatitis B (see [20] for a good description). "
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