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    • Mathematical models play an important role in understanding

    2022-05-09

    Mathematical models play an important role in understanding and predicting the dynamic process of HBV infection. In the 1990s, Nowak et al. proposed a basic model to analyze the effect of antiviral treatment on reducing viral loads (Nowak et al., 1996). Since then, many different mathematical models have been developed according to different biological mechanisms. Some models studied the interaction between antiviral treatment and HBV by considering drug efficacy or pharmacokinetics (Nowak et al., 1996; Murase and Sasaki, 2005; Eikenberry et al., 2009; Sypsa et al., 2005). Besides, some models investigated the effect of time delay or spatial diffusion, which took into account the time from infection to the release of free virus particles and free movement of virus particles in the liver respectively (Gourley et al., 2008; Huang et al., 2010; Guo and Cai, 2011; Wang et al., 2008). In particular, some models focused on understanding the effect of humoral immunity or CTL-mediated cellular immunity based on the body’s defense or immune system (Yousfi et al., 2011; Fisicaro et al., 2009; Bertoletti and Ferrari, 2003; Perelson, 2002). Furthermore, some studies described the interaction of multi-types of infected hepatocytes with different copies of cccDNA (Ciupe et al., 2007; Li et al., 2014). Stochastic models which took into account the random factors were also investigated (Moneim et al., 2009; Xie et al., 2017; Luzyanina and Bocharov, 2014; Liu et al., 2017). These models have greatly enriched our understanding of HBV infection dynamics. This paper aims to provide a quantitative reference for clinical treatment of Viomycin synthesis virus. The structure of this paper is as follows. In Section 2, we review the viral dynamics of HBV infection by using differential equation models. In Section 3, we introduce some commonly used parameter estimation methods and the related theory of model selection. The application of these methods are also illustrated through several specific examples. In Section 4, we make a brief summary and propose three future research programs for HBV infection.
    Viral dynamic models of HBV infection In order to better understand the dynamic process of HBV infection, in recent years, a variety of dynamic models including deterministic and stochastic models for HBV infection have been established (Murase and Sasaki, 2005; Sypsa et al., 2005; Guo and Cai, 2011; Perelson, 2002; Ciupe et al., 2007; Bellecave et al., 2009). Specifically, deterministic models explore the effect of various biological mechanisms, including antiviral therapy, CTL-mediated immune response, multi-types of infected hepatocytes, time delay and spatial diffusion. Stochastic models describe the random fluctuation of infection rate and reveal the impact of noise on stochastic HBV kinetics. Next, we discuss the differential equation models of HBV infection by considering each of the above biological factors and illustrate the new research results in recent years.
    Parameter estimation for HBV viral dynamic models
    Summary and discussion Although various models have been developed to characterize the dynamic process of HBV infection, there are still some remaining problems that cannot be explained. As a result of long-term antiviral therapy, the mechanism about the rapid emergence of drug resistance has not yet been fully revealed. The corresponding study of drug resistance with multiple strains in HBV is rarely reflected. Here, we propose a HBV model with multiple strains based on the model in Rong et al. (2010). By dividing HBV particles into drug-sensitive strain and drug-resistant strain , the model is given byHere, and represent the infected hepatocytes which are infected by and , respectively. The remaining parameters have the same meanings as above. Through exploring the coexistence condition for and , this model may provide a theoretical reference for understanding the mechanism of drug resistance. In addition, whether the within-host dynamics of HBV infection will affect the macroscopic propagation of between-host has not yet been studied in detail, which is also a future research direction. Here, we propose a coupled HBV infection model based on the model in Feng et al. (2012) which is given bywhere the transmission coefficient is a monotonically increasing saturation function of viral load . Here, , and represent the number of susceptible individuals, recovered individuals and chronic HBV carriers at time respectively, is the total population, is the birth rate, and are respectively the natural mortality and disease-related mortality, is the neonatal vaccine coverage rate, is the natural recovery rate, is the vertical transmission rate from mother to infant, is the immune protective rate of adults, is the conversion rate from acute to chronic infected individuals. This coupled model may generate new dynamic properties or threshold conditions, which may reveal more complex biological phenomena of hepatitis B.