Modeling the adaptive immune response in HBV infection. J Math Biol. 2011;63:933-957. [PMID: 21234570 DOI: 10.1007/s00285-010-0397-x]

Dynamical analysis of a general delayed HBV infection model with capsids and adaptive immune response in presence of exposed infected hepatocytes

The aim of this paper is to develop and investigate a novel mathematical model of the dynamical behaviors of chronic hepatitis B virus infection. The model includes exposed infected hepatocytes, intracellular HBV DNA-containing capsids, uses a general incidence function for viral infection covering a variety of special cases available in the literature, and describes the interaction of cytotoxic T lymphocytes that kill the infected hepatocytes and the magnitude of B-cells that send antibody immune defense to neutralize free virions. Further, one time delay is incorporated to account for actual capsids production. The other time delays are used to account for maturation of capsids and free viruses. We start with the analysis of the proposed model by establishing the local and global existence, uniqueness, non-negativity and boundedness of solutions. After defined the threshold parameters, we discuss the stability properties of all possible steady state constants by using the crafty Lyapunov functionals, the LaSalle's invariance principle and linearization methods. The impacts of the three time delays on the HBV infection transmission are discussed through local and global sensitivity analysis of the basic reproduction number and of the classes of infected states. Finally, an application is provided and numerical simulations are performed to illustrate and interpret the theoretical results obtained. It is suggested that, a good strategy to eradicate or to control HBV infection within a host should concentrate on any drugs that may prolong the values of the three delays.

Effect of color cross-correlated noise on the growth characteristics of tumor cells under immune surveillance

Based on the Michaelis-Menten reaction model with catalytic effects, a more comprehensive one-dimensional stochastic Langevin equation with immune surveillance for a tumor cell growth system is obtained by considering the fluctuations in growth rate and mortality rate. To explore the impact of environmental fluctuations on the growth of tumor cells, the analytical solution of the steady-state probability distribution function of the system is derived using the Liouville equation and Novikov theory, and the influence of noise intensity and correlation intensity on the steady-state probability distributional function are discussed. The results show that the three extreme values of the steady-state probability distribution function exhibit a structure of two peaks and one valley. Variations of the noise intensity, cross-correlation intensity and correlation time can modulate the probability distribution of the number of tumor cells, which provides theoretical guidance for determining treatment plans in clinical treatment. Furthermore, the increase of noise intensity will inhibit the growth of tumor cells when the number of tumor cells is relatively small, while the increase in noise intensity will further promote the growth of tumor cells when the number of tumor cells is relatively large. The color cross-correlated strength and cross-correlated time between noise also have a certain impact on tumor cell proliferation. The results help people understand the growth kinetics of tumor cells, which can a provide theoretical basis for clinical research on tumor cell growth.

Global dynamics of multiple delays within-host model for a hepatitis B virus infection of hepatocytes with immune response and drug therapy

In this paper, a mathematical model describing the hepatitis B virus (HBV) infection of hepatocytes with the intracellular HBV-DNA containing capsids, cytotoxic T-lymphocyte (CTL), antibodies including drug therapy (blocking new infection and inhibiting viral production) with two-time delays is studied. It incorporates the delay in the productively infected hepatocytes and the delay in an antigenic stimulation generating CTL. We verify the positivity and boundedness of solutions and determine the basic reproduction number. The local and global stability of three equilibrium points (infection-free, immune-free, and immune-activated) are investigated. Finally, the numerical simulations are established to show the role of these therapies in reducing viral replication and HBV infection. Our results show that the treatment by blocking new infection gives more significant results than the treatment by inhibiting viral production for infected hepatocytes. Further, both delays affect the number of infections and duration i.e. the longer the delay, the more severe the HBV infection.

Innate Immunity, Inflammation, and Intervention in HBV Infection

Hepatitis B virus (HBV) infection is still one of the most dangerous viral illnesses. HBV infects around 257 million individuals worldwide. Hepatitis B in many individuals ultimately develops hepatocellular carcinoma (HCC), which is the sixth most common cancer and the third leading cause of cancer-related deaths worldwide. The innate immunity acts as the first line of defense against HBV infection through activating antiviral genes. Along with the immune responses, pro-inflammatory cytokines are triggered to enhance the antiviral responses, but this may result in acute or chronic liver inflammation, especially when the clearance of virus is unsuccessful. To a degree, the host innate immune and inflammatory responses dominate the HBV infection and liver pathogenesis. Thus, it is crucial to figure out the signaling pathways involved in the activation of antiviral factors and inflammatory cytokines. Here, we review the interplay between HBV and the signal pathways that mediates innate immune responses and inflammation. In addition, we summarize current therapeutic strategies for HBV infection via modulating innate immunity or inflammation. Characterizing the mechanisms that underlie these HBV-host interplays might provide new approaches for the cure of chronic HBV infection.

Novel stochastic dynamics of a fractal-fractional immune effector response to viral infection via latently infectious tissues

In this paper, the global complexities of a stochastic virus transmission framework featuring adaptive response and Holling type II estimation are examined via the non-local fractal-fractional derivative operator in the Atangana-Baleanu perspective. Furthermore, we determine the existence-uniqueness of positivity of the appropriate solutions. Ergodicity and stationary distribution of non-negative solutions are carried out. Besides that, the infection progresses in the sense of randomization as a consequence of the response fluctuating within the predictive case's equilibria. Additionally, the extinction criteria have been established. To understand the reliability of the findings, simulation studies utilizing the fractal-fractional dynamics of the synthesized trajectory under the Atangana-Baleanu-Caputo derivative incorporating fractional-order α and fractal-dimension ℘ have also been addressed. The strength of white noise is significant in the treatment of viral pathogens. The persistence of a stationary distribution can be maintained by white noise of sufficient concentration, whereas the eradication of the infection is aided by white noise of high concentration.

Global Dynamics of a Stochastic Viral Infection Model with Latently Infected Cells

In this paper, we study the global dynamics of a stochastic viral infection model with humoral immunity and Holling type II response functions. The existence and uniqueness of non-negative global solutions are derived. Stationary ergodic distribution of positive solutions is investigated. The solution fluctuates around the equilibrium of the deterministic case, resulting in the disease persisting stochastically. The extinction conditions are also determined. To verify the accuracy of the results, numerical simulations were carried out using the Euler–Maruyama scheme. White noise’s intensity plays a key role in treating viral infectious diseases. The small intensity of white noises can maintain the existence of a stationary distribution, while the large intensity of white noises is beneficial to the extinction of the virus.

A quantitative systems pharmacology model for acute viral hepatitis B

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Mechanistic model characterizing acute immune response and HBV system interactions.

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Key role of the cellular and regulatory response triggering hepatitis B chronicity.

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Modelling framework to easily incorporate and explore additional biological mechanisms.

Hepatitis B liver infection is caused by hepatitis B virus (HBV) and represents a major global disease problem when it becomes chronic, as is the case for 80–90% of vertical or early life infections. However, in the vast majority (>95%) of adult exposures, the infected individuals are capable of mounting an effective immune response leading to infection resolution. A good understanding of HBV dynamics and the interaction between the virus and immune system during acute infection represents an essential step to characterize and understand the key biological processes involved in disease resolution, which may help to identify potential interventions to prevent chronic hepatitis B.

In this work, a quantitative systems pharmacology model for acute hepatitis B characterizing viral dynamics and the main components of the innate, adaptive, and tolerant immune response has been successfully developed. To do so, information from multiple sources and across different organization levels has been integrated in a common mechanistic framework. The final model adequately describes the chronology and plausibility of an HBV-triggered immune response, as well as clinical data from acute patients reported in the literature. Given the holistic nature of the framework, the model can be used to illustrate the relevance of the different immune pathways and biological processes to ultimate response, observing the negligible contribution of the innate response and the key contribution of the cellular response on viral clearance. More specifically, moderate reductions of the proliferation of activated cytotoxic CD8+ lymphocytes or increased immunoregulatory effects can drive the system towards chronicity.

Understanding the antiviral effects of RNAi-based therapy in HBeAg-positive chronic hepatitis B infection

The RNA interference (RNAi) drug ARC-520 was shown to be effective in reducing serum hepatitis B virus (HBV) DNA, hepatitis B e antigen (HBeAg) and hepatitis B surface antigen (HBsAg) in HBeAg-positive patients treated with a single dose of ARC-520 and daily nucleosidic analogue (entecavir). To provide insights into HBV dynamics under ARC-520 treatment and its efficacy in blocking HBV DNA, HBsAg, and HBeAg production we developed a multi-compartmental pharmacokinetic-pharamacodynamic model and calibrated it with frequent measured HBV kinetic data. We showed that the time-dependent single dose ARC-520 efficacies in blocking HBsAg and HBeAg are more than 96% effective around day 1, and slowly wane to 50% in 1-4 months. The combined single dose ARC-520 and entecavir effect on HBV DNA was constant over time, with efficacy of more than 99.8%. The observed continuous HBV DNA decline is entecavir mediated, the strong but transient HBsAg and HBeAg decays are ARC-520 mediated. The modeling framework may help assess ongoing RNAi drug development for hepatitis B virus infection.

Approximate solutions for HBV infection with stability analysis using LHAM during antiviral therapy

Hepatitis B virus (HBV) is a life-threatening virus that causes very serious liver-related diseases from the family of Hepadnaviridae having very rare qualities resembling retroviruses. In this paper, we analyze the effect of antiviral therapy through mathematical modeling by using Liao's homotopy analysis method (LHAM) that defines the connection between the target liver cells and the HBV. We also examine the basic nonlinear differential equation by LHAM to get a semi-analytical solution. This can be a very straight and direct method which provides the appropriate solution. Moreover, the local and global stability analysis of disease-free and endemic equilibrium is done using Lyapunov function. Mathematica 12 software is used to find out the solutions and graphical representations. We also discuss the numerical simulations up to sixth-order approximation and error analysis using the same software.

BATF Interference Blocks Th17 Cell Differentiation and Inflammatory Response in Hepatitis B Virus Transgenic Mice

BACKGROUND

B cell-activating transcription factor (BATF) contributes to Th17 cell differentiation and pathological inflammatory responses.

AIMS

This study explored BATF as a regulator of Th17 differentiation in normal and hepatitis B virus (HBV) transgenic mice.

METHODS

Normal mice were divided into control, short hairpin RNA (shRNA) scramble, and shRNA BATF groups. HBV transgenic mice were divided into control, entecavir, shRNA scramble, entecavir + vector control, entecavir + shRNA scramble, shRNA BATF, and entecavir + shRNA BATF groups. Serum concentrations of AST, ALT, HBV-DNA, BATF, IL-17, and IL-22 and Th17 cell frequencies in the liver were compared among the groups. Correlations of serum HBV surface antigen (HBsAg), e-antigen (HBeAg), and core antigen (HBcAg) concentrations with BATF mRNA expression and the proportion of Th17 cells in the livers of HBV transgenic mice were also analyzed.

RESULTS

Serum AST, ALT, BATF, IL-17, and IL-22 concentrations and Th17 cell proportions were higher in HBV transgenic mice relative to normal controls. Positive correlations of the HBcAg concentration with BATF mRNA and the proportion of Th17 cells were observed in HBV transgenic mice. BATF interference reduced the proportion of Th17 cells and serum IL-17 and IL-22 concentrations and led to obvious downregulation of AST, ALT, BATF, IL-17, and IL-22 expression and a reduced proportion of Th17 cells when combined with entecavir.

CONCLUSION

HBV markedly upregulated BATF expression and promoted Th17 cell activation. By contrast, BATF interference significantly impeded the proliferation of Th17 cells and secretion of IL-17 and IL-22 while alleviating hepatic lesions.

The review of differential equation models of HBV infection dynamics

Understanding the infection and pathogenesis mechanism of hepatitis B virus (HBV) is very important for the prevention and treatment of hepatitis B. Mathematical models contribute to illuminate the dynamic process of HBV replication in vivo. Therefore, in this paper we review the viral dynamics in HBV infection, which may help us further understand the dynamic mechanism of HBV infection and efficacy of antiviral treatment. Firstly, we introduce a family of deterministic models by considering different biological mechanisms, such as, antiviral therapy, CTL immune response, multi-types of infected hepatocytes, time delay and spatial diffusion. Particularly, we briefly describe the stochastic models of HBV infection. Secondly, we introduce the commonly used parameter estimation methods for HBV viral dynamic models and briefly discuss how to use these methods to estimate unknown parameters (such as drug efficacy) through two specific examples. We also discuss the idea and method of model identification and use a specific example to illustrate its application. Finally, we propose three new research programs, namely, considering HBV drug-resistant strain, coupling within-host and between-host dynamics in HBV infection and linking population dynamics with evolutionary dynamics of HBV diversity.

Mathematical Analysis and Treatment for a Delayed Hepatitis B Viral Infection Model with the Adaptive Immune Response and DNA-Containing Capsids

We model the transmission of the hepatitis B virus (HBV) by six differential equations that represent the reactions between HBV with DNA-containing capsids, the hepatocytes, the antibodies and the cytotoxic T-lymphocyte (CTL) cells. The intracellular delay and treatment are integrated into the model. The existence of the optimal control pair is supported and the characterization of this pair is given by the Pontryagin's minimum principle. Note that one of them describes the effectiveness of medical treatment in restraining viral production, while the second stands for the success of drug treatment in blocking new infections. Using the finite difference approximation, the optimality system is derived and solved numerically. Finally, the numerical simulations are illustrated in order to determine the role of optimal treatment in preventing viral replication.

Stability of latent pathogen infection model with adaptive immunity and delays

In this paper we propose and analyze a pathogen dynamics model with antibody and Cytotoxic T Lymphocyte (CTL) immune responses. We incorporate latently infected cells and three distributed time delays into the model. We show that the solutions of the proposed model are nonnegative and ultimately bounded. We derive four threshold parameters which fully determine the existence and stability of the five steady states of the model. Using Lyapunov functionals, we established the global stability of the steady states of the model. The theoretical results are confirmed by numerical simulations.

Mathematical model of immune response to hepatitis B

A new detailed mathematical model for dynamics of immune response to hepatitis B is proposed, which takes into account contributions from innate and adaptive immune responses, as well as cytokines. Stability analysis of different steady states is performed to identify parameter regions where the model exhibits clearance of infection, maintenance of a chronic infection, or periodic oscillations. Effects of nucleoside analogues and interferon treatments are analysed, and the critical drug efficiency is determined.

A discrete-time analog for coupled within-host and between-host dynamics in environmentally driven infectious disease

In this paper, we establish a discrete-time analog for coupled within-host and between-host systems for an environmentally driven infectious disease with fast and slow two time scales by using the non-standard finite difference scheme. The system is divided into a fast time system and a slow time system by using the idea of limit equations. For the fast system, the positivity and boundedness of the solutions, the basic reproduction number and the existence for infection-free and unique virus infectious equilibria are obtained, and the threshold conditions on the local stability of equilibria are established. In the slow system, except for the positivity and boundedness of the solutions, the existence for disease-free, unique endemic and two endemic equilibria are obtained, and the sufficient conditions on the local stability for disease-free and unique endemic equilibria are established. To return to the coupling system, the local stability for the virus- and disease-free equilibrium, and virus infectious but disease-free equilibrium is established. The numerical examples show that an endemic equilibrium is locally asymptotically stable and the other one is unstable when there are two endemic equilibria.

Modelling the Impact of Cell-To-Cell Transmission in Hepatitis B Virus

Cell-free virus is a well-recognized and efficient mechanism for the spread of hepatitis B virus (HBV) infection in the liver. Cell-to-cell transmission (CCT) can be a more efficient means of virus propagation. Despite experimental evidence implying CCT occurs in HBV, its relative impact is uncertain. We develop a 3-D agent-based model where each hepatocyte changes its viral state according to a dynamical process driven by cell-free virus infection, CCT and intracellular replication. We determine the relative importance of CCT in the development and resolution of acute HBV infection in the presence of cytolytic (CTL) and non-CTL mechanisms. T cell clearance number is defined as the minimum number of infected cells needed to be killed by each T cell at peak infection that results in infection clearance within 12 weeks with hepatocyte turnover (HT, number of equivalent livers) ≤3. We find that CCT has very little impact on the establishment of infection as the mean cccDNA copies/cell remains between 15 to 20 at the peak of the infection regardless of CCT strength. In contrast, CCT inhibit immune-mediated clearance of acute HBV infection as higher CCT strength requires higher T cell clearance number and increases the probability of T cell exhaustion. An effective non-CTL inhibition can counter these negative effects of higher strengths of CCT by supporting rapid, efficient viral clearance and with little liver destruction. This is evident as the T cell clearance number drops by approximately 50% when non-CTL inhibition is increased from 10% to 80%. Higher CCT strength also increases the probability of the incidence of fulminant hepatitis with this phenomenon being unlikely to arise for no CCT. In conclusion, we report the possibility of CCT impacting HBV clearance and its contribution to fulminant hepatitis.

Global stability of a delayed virus dynamics model with multi-staged infected progression and humoral immunity

In this paper, we propose a nonlinear virus dynamics model that describes the interactions of the virus, uninfected target cells, multiple stages of infected cells and B cells and includes multiple discrete delays. We assume that the incidence rate of infection and removal rate of infected cells are given by general nonlinear functions. The model can be seen as a generalization of several humoral immunity viral infection model presented in the literature. We derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to establish the existence and global stability of the three equilibria of the model. We study the global asymptotic stability of the equilibria by using Lyapunov method. We perform some numerical simulations for the model with specific forms of the general functions and show that the numerical results are consistent with the theoretical results.

Global dynamics for discrete-time analog of viral infection model with nonlinear incidence and CTL immune response

In this paper, a discrete-time analog of a viral infection model with nonlinear incidence and CTL immune response is established by using the Micken non-standard finite difference scheme. The two basic reproduction numbers R 0 and R 1 are defined. The basic properties on the positivity and boundedness of solutions and the existence of the virus-free, the no-immune, and the infected equilibria are established. By using the Lyapunov functions and linearization methods, the global stability of the equilibria for the model is established. That is, when R 0 ≤ 1 then the virus-free equilibrium is globally asymptotically stable, and under the additional assumption ( A 4 ) when R 0 > 1 and R 1 ≤ 1 then the no-immune equilibrium is globally asymptotically stable and when R 0 > 1 and R 1 > 1 then the infected equilibrium is globally asymptotically stable. Furthermore, the numerical simulations show that even if assumption ( A 4 ) does not hold, the no-immune equilibrium and the infected equilibrium also may be globally asymptotically stable.

Global properties of nonlinear humoral immunity viral infection models

In this paper, we consider two nonlinear models for viral infection with humoral immunity. The first model contains four compartments; uninfected target cells, actively infected cells, free virus particles and B cells. The second model is a modification of the first one by including the latently infected cells. The incidence rate, removal rate of infected cells, production rate of viruses and the latent-to-active conversion rate are given by more general nonlinear functions. We have established a set of conditions on these general functions and determined two threshold parameters for each model which are sufficient to determine the global dynamics of the models. The global asymptotic stability of all equilibria of the models has been proven by using Lyapunov theory and applying LaSalle's invariance principle. We have performed some numerical simulations for the models with specific forms of the general functions. We have shown that, the numerical results are consistent with the theoretical results.

In silico single cell dynamics of hepatitis B virus infection and clearance

The progression of acute hepatitis B virus (HBV) to chronic infection or clearance is highly dependent on the host immune response composed of cytolytic (CTL) and non-cytolytic (non-CTL) effects. Cytolytic processes induce hepatocyte killing while non-CTL processes inhibit intracellular replication. Both effects are widely recognized and accepted. However, there are uncertainties about the assistance provided by either the loss of covalently circular closed DNA (cccDNA) during cell proliferation or the emergence of refractory cells to immune mediated clearance. We developed an agent-based mathematical model and tested the relative roles of different mechanisms of the immune system in the clearance of acute HBV infection. HBV viremia clearance time and hepatocyte turnover (HT) were used as the two major criteria in determining reasonable outcomes. Modelling results in 90% of cells containing between 1 and 17 cccDNA copies and normally distributed at the peak of infection. Variations in p36 levels, responsible for determining export of virions or recirculation to amplify cccDNA numbers, have a much greater impact on mean cccDNA level/cell at peak viremia than virus infectivity and cccDNA half-life. A strong CTL effect alone failed to clear infection with HT ≈ 10. Acute infection clearance was possible with combined CTL and non-CTL effects along with complete loss of intracellular viral components during cell proliferation resulting in the desired range of HT (0.7-1). The emergence of cells refractory to infection can reduce HT by up to 90%. However their impact was less effective than complete loss of intracellular viral components during cell proliferation.

CONCLUSION

the existence of refractory cells is not necessary when there is complete loss of intracellular quantities during cell proliferation but is essential with only partial clearance.

Antibody responses during hepatitis B viral infection

Hepatitis B is a DNA virus that infects liver cells and can cause both acute and chronic disease. It is believed that both viral and host factors are responsible for determining whether the infection is cleared or becomes chronic. Here we investigate the mechanism of protection by developing a mathematical model of the antibody response following hepatitis B virus (HBV) infection. We fitted the model to data from seven infected adults identified during acute infection and determined the ability of the virus to escape neutralization through overproduction of non-infectious subviral particles, which have HBs proteins on their surface, but do not contain nucleocapsid protein and viral nucleic acids. We showed that viral clearance can be achieved for high anti-HBV antibody levels, as in vaccinated individuals, when: (1) the rate of synthesis of hepatitis B subviral particles is slow; (2) the rate of synthesis of hepatitis B subviral particles is high but either anti-HBV antibody production is fast, the antibody affinity is high, or the levels of pre-existent HBV-specific antibody at the time of infection are high, as could be attained by vaccination. We further showed that viral clearance can be achieved for low equilibrium anti-HBV antibody levels, as in unvaccinated individuals, when a strong cellular immune response controls early infection.

CD4+CD25+ regulatory T cells and different states of HBV infection

CD4⁺CD25⁺ regulatory T cells are a recently discovered subset of CD4⁺ T cell population that mediates immune suppression. Recent studies suggested that regulatory T cells are closely associated with the prognosis of different states of HBV infection. Here we review the types, mechanisms of action and immunophenotypes of CD4⁺CD25⁺ regulatory T cells, as well as their relationship with different states of HBV infection.

Stability Analysis of an Improved HBV Model with CTL Immune Response

To better understand the dynamics of the hepatitis B virus (HBV) infection, we introduce an improved HBV model with standard incidence function, cytotoxic T lymphocytes (CTL) immune response, and take into account the effect of the export of precursor CTL cells from the thymus and the role of cytolytic and noncytolytic mechanisms. The local stability of the disease-free equilibrium and the chronic infection equilibrium is obtained via characteristic equations. Furthermore, the global stability of both equilibria is established by using two techniques, the direct Lyapunov method for the disease-free equilibrium and the geometrical approach for the chronic infection equilibrium.