The dynamics of a delay model of hepatitis B virus infection with logistic hepatocyte growth

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.

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.

Effect of cellular regeneration and viral transmission mode on viral spread

Illness negatively affects all aspects of life and one major cause of illness is viral infections. Some viral infections can last for weeks; others, like influenza (the flu), can resolve quickly. During infections, uninfected cells can replicate in order to replenish the cells that have died due to the virus. Many viral models, especially those for short-lived infections like influenza, tend to ignore cellular regeneration since many think that uncomplicated influenza resolves much faster than cells regenerate. This research accounts for cellular regeneration, using an agent-based framework, and varies the regeneration rate in order to understand how cell regeneration affects viral infection dynamics under assumptions of different modes of transmission. We find that although the general trends in peak viral load, time of viral peak, and chronic viral load as regeneration rate changes are the same for cell-free or cell-to-cell transmission, the changes are more extreme for cell-to-cell transmission due to limited access of infected cells to newly generated cells.

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.

Acute hepatitis B virus infection model within the host incorporating immune cells and cytokine responses

We formulate and analyze a within-host hepatitis B viral mathematical model for hepatitis B in the acute phase of infection. The model incorporates hepatocytes, hepatitis B virus, immune system cells and cytokine dynamics using a system of ordinary differential equations. We use the model to demonstrate the trends of the hepatitis B infection qualitatively without the effects of immune cells and cytokines. Using these trends, we tested the effects of incorporating the immune cells only and immune cells with cytokine responses at low and high inhibitions on the hepatitis B virus infection. Our results showed that it is impossible to have the immune cells work independently from cytokines when there is an acute hepatitis B virus infection. Therefore, our results suggest that incorporating immune cells and cytokine dynamics in the acute hepatitis B virus infection stage delays infection in the hepatocytes and excluding such dynamics speeds up infection during this phase. Results from this study are useful in developing strategies for control of hepatocellular carcinoma which is caused by hepatitis B virus infection.

ANALYSIS OF A STOCHASTIC HBV INFECTION MODEL WITH NONLINEAR INCIDENCE RATE

Considering that environmental factors, diet, subconscious mind and other uncertainties play an important role in the process of delaying and treating diseases, we propose, in this paper, an amended Hepatitis B virus (HBV) model with stochastic perturbation, and investigate the longtime dynamics of this stochastic model. First, if the basic reproductive number of the corresponding deterministic model is less than 1, some sufficient conditions for almost surely exponentially stable in the sense of the infected cells and free virus are established, and the stationary probability density function of the uninfected sell is also obtained. Further, some sufficient conditions for the existence of the stationary distribution are obtained for the basic reproductive number more than 1. In addition, oscillatory behaviors of this model about the equilibrium of the corresponding deterministic model are discussed. Finally, numerical simulations demonstrate the main theoretical results and show stochastic virus model has more dynamic behaviors relative to its corresponding deterministic model. Theoretical results and numerical simulations imply that the intensity and “type (divided into positive and negative)” of white noise play very important roles in the treatment of infectious disease, which can make the disease more and more repetitive and unpredictable. Of course, comfortable environment, reasonable diet, optimistic mood and other positive uncertainty factors have active effects on the treatment and delaying of diseases, but not the converse.

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.

Dynamics of a diffusion-driven HBV infection model with capsids and time delay

In this paper, a diffusive hepatitis B virus (HBV) infection model with a discrete time delay is presented and analyzed, where the spatial mobility of both intracellular capsid covered HBV DNA and HBV and the intracellular delay in the reproduction of infected hepatocytes are taken into account. We define the basic reproduction number [Formula: see text] that determines the dynamical behavior of the model. The local and global stability of the spatially homogeneous steady states are analyzed by using the linearization technique and the direct Lyapunov method, respectively. It is shown that the susceptible uninfected steady state is globally asymptotically stable whenever [Formula: see text] and is unstable whenever [Formula: see text]. Also, the infected steady state is globally asymptotically stable when [Formula: see text]. Finally, numerical simulations are carried out to illustrate the results obtained.

Comparisons of forecasting for hepatitis in Guangxi Province, China by using three neural networks models

This study compares and evaluates the prediction of hepatitis in Guangxi Province, China by using back propagation neural networks based genetic algorithm (BPNN-GA), generalized regression neural networks (GRNN), and wavelet neural networks (WNN). In order to compare the results of forecasting, the data obtained from 2004 to 2013 and 2014 were used as modeling and forecasting samples, respectively. The results show that when the small data set of hepatitis has seasonal fluctuation, the prediction result by BPNN-GA will be better than the two other methods. The WNN method is suitable for predicting the large data set of hepatitis that has seasonal fluctuation and the same for the GRNN method when the data increases steadily.

Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions

A within-host viral infection model with both virus-to-cell and cell-to-cell transmissions and three distributed delays is investigated, in which the first distributed delay describes the intracellular latency for the virus-to-cell infection, the second delay represents the intracellular latency for the cell-to-cell infection, and the third delay describes the time period that viruses penetrated into cells and infected cells release new virions. The global stability analysis of the model is carried out in terms of the basic reproduction number R0. If R0≤1, the infection-free (semi-trivial) equilibrium is the unique equilibrium and is globally stable; if R0>1, the chronic infection (positive) equilibrium exists and is globally stable under certain assumptions. Examples and numerical simulations for several special cases are presented, including various within-host dynamics models with discrete or distributed delays that have been well-studied in the literature. It is found that the global stability of the chronic infection equilibrium might change in some special cases when the assumptions do not hold. The results show that the model can be applied to describe the within-host dynamics of HBV, HIV, or HTLV-1 infection.

Mathematical models of the interrelated dynamics of hepatitis D and B

The hepatitis delta virus (HDV) is a rarest form of viral hepatitis, but has the worst outcomes for patients.It is a subviral satellite dependent on coinfection with hepatitis B (HBV) to replicate within the host liver.To date, there has been little to no modeling effort for HDV. Deriving and analyzing such a mathematical model poses difficulty as it requires the inclusion of (HBV). Here we begin with a well-studied HBV model from the literature and expand it to incorporate HDV. We investigate two models, one with and one without infected hepatocyte replication. Additionally, we consider treatment by the drug lamivudine. Comparison of model simulations with experimental results of lamivudine treatment indicate that infected cell proliferation may play a significant role in chronic HDV infection. Our results also shed light on several questions surrounding HDV and illustrate the need for more data.

Global stability analysis of humoral immunity virus dynamics model including latently infected cells

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 R₀ and the humoral immune response activation number R₁ 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 R₀ ≤ 1, then the infection-free equilibrium is globally asymptotically stable (GAS), if R₁ ≤ 1 < R₀, then the chronic-infection equilibrium without humoral immune response is GAS, and if R₁ > 1, then the chronic-infection equilibrium with humoral immune response is globally asymptotically stable. These results are further illustrated by numerical simulations.

Modeling and simulating dynamics of complete- and poor-response chronic hepatitis B chinese patients for adefovir and traditional chinese medicine plus adefovir therapy

ChiCTR-TRC-11001263 study was the first large-scale double-blind randomized placebo-controlled traditional Chinese medicines (TCMs) and adefovir (ADV) antihepatitis B virus (HBV) infection trial in the world. A total of 560 hepatitis B e antigen- (HBeAg-) positive Chinese patients with chronical HBV were randomly classified, in 1 : 1 ratio, into two groups: experimental group (EXG) receiving TCMs + ADV and controlled group (CTG) receiving ADV + TCM-placebo treatment for 48 weeks. This paper introduces two models to model and simulate the evolutions of dynamics for the complete-response patients and the poor-response patients in EXG and CTG, respectively. The stimulated mean HBV DNA and alanine aminotransferase (ALT) levels were close to the patients' experimental data. Analysis and simulations suggest that the activated patients' immune functions by TCMs + ADV may not only clear infected hepatocytes, but also clear HBV, which made the complete-response patients' mean serum HBV DNA levels in EXG reduce rapidly 12 weeks' earlier than the ones in CTG. One can assume that both the TCMs and ADV have the function of preventing complete-response patients' infected hepatocytes from being injured by cytotoxic T lymphocytes (CTLs); the patients' activated immune cells may also block HBV replications.

Optimal Antiviral Treatment Strategies of HBV Infection Model with Logistic Hepatocyte Growth

This study considers an optimal therapy strategy for HBV infection by incorporating two controls laws into a previous hepatitis B viral infection model with logistic hepatocyte growth. Our goal is to maximize the number of healthy cells and to minimize the cost of the therapy. In this context, the existence of an optimal control is proved. The optimal control is obtained by solving the optimality system which was composed of three nonlinear ODEs with initial conditions and three nonlinear adjoint ODEs with transversality conditions. The results were analysed and interpreted numerically using MATLAB.

Global analysis of a simple parasite-host model with homoclinic orbits

In this paper, a simple parasite-host model proposed by Ebert et al.(2000) is reconsidered. The basic epidemiological reproduction number of parasite infection (R0) and the basic demographic reproduction number of infected hosts (R1) are given. The global dynamics of the model is completely investigated, and the existence of heteroclinic and homoclinic orbits is theoretically proved, which implies that the outbreak of parasite infection may happen. The thresholds determining the host extinction in the presence of parasite infection and variation in the equilibrium level of the infected hosts with R0 are found. The effects of R0 and R1 on dynamics of the model are considered and we show that the equilibrium level of the infected host may not be monotone with respect to R0. In particular, it is found that full loss of fecundity of infected hosts may lead to appearance of the singular case.

Mathematical Modeling of Viral Zoonoses in Wildlife

Zoonoses are a worldwide public health concern, accounting for approximately 75% of human infectious diseases. In addition, zoonoses adversely affect agricultural production and wildlife. We review some mathematical models developed for the study of viral zoonoses in wildlife and identify areas where further modeling efforts are needed.

MICROBIAL PEST CONTROL: A MATHEMATICAL MODEL

The traditional method for controlling pests is the application of chemical pesticides. Growing concern on the negative effects of chemicals has encouraged the development of alternatives. Inundatively and inoculatively applied microbial control agents (virus, bacteria, fungi, and entomopathogenic nematodes) have been developed as alternative control methods of a wide variety of pests. A mathematical model for microbial control of pests is formulated in this paper. The dynamical characteristics of the system are studied. The role of time-delay has been discussed. Numerical simulations are carried out to illustrate the analytical findings. Biological implications have been discussed.

Numerical diagnoses of superinfection in chronic hepatitis B viral dynamics

OBJECTIVES

Fluctuation profile has been observed in chronic hepatitis B patients who are untreated or interrupt therapy. A mathematical model and its parameters could be used to diagnose the assumption of superinfection of hepatocytes and to understand the causes for the spontaneous fluctuation pattern of HBV DNA loads in chronically infected patients.

METHODS

We propose a new conceptual model in terms of chemical kinetics, which is based on the assumption that hepatocytes can be superinfected with hepatitis B virus (HBV). Minimizing the sum of squares of the deviations, we fitted the model to the HBV DNA trajectories from clinical data and obtained the model parameters.

RESULTS

The model with the fitted parameters can capture the tendency of HBV DNA trajectories. The mean value of the fitted number of virions that enter a single hepatocyte at the beginning stage of an invasion is 2.10 ± 0.18. The dynamics patterns may correlate with the clinical phenotypes of patients and the value of clinical parameters, such as α-fetoprotein, hepatitis B e-antigen, hepatitis B e-antibody, total bilirubin and alanine transaminase.

CONCLUSIONS

The superinfection scenario is possible in HBV infection and it may induce HBV DNA fluctuation in the host.

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

The aim of this work is to investigate a new mathematical model that describes the interactions between Hepatitis B virus (HBV), liver cells (hepatocytes), and the adaptive immune response. The qualitative analysis of this as cytotoxic T lymphocytes (CTL) cells and the antibodies. These outcomes are (1) a disease free steady state, which its local stability is characterized as usual by R (0) < 1, (2) and the existence of four endemic steady states when R (0) > 1. The local stability of these steady states depends on functions of R (0). Our study shows that although we give conditions of stability of these steady states, not all conditions are feasible. This rules out the local stability of two steady states. The conditions of stability of the two other steady states (which represent the complete failure of the adaptive immunity and the persistence of the disease) are formulated based on the domination of CTL cells response or the antibody response.

Dynamical behavior of a hepatitis B virus transmission model with vaccination

Hepatitis B virus (HBV) infection is a globally health problem. In 2005, the WHO Western Pacific Regional Office set a goal of reducing chronic HBV infection rate to less than 2% among children five years of age by 2012, as an interim milestone towards the final goal of less than 1%. Many countries made some plans (such as free HBV vaccination program for all neonates in China now) to control the transmission HBV. We develop a model to explore the impact of vaccination and other controlling measures of HBV infection. The model has simple dynamical behavior which has a globally asymptotically stable disease-free equilibrium when the basic reproduction number R(0)< or =1, and a globally asymptotically stable endemic equilibrium when R(0)>1. Numerical simulation results show that the vaccination is a very effective measure to control the infection and they also give some useful comments on controlling the transmission of HBV.

Rich dynamics of a hepatitis B viral infection model with logistic hepatocyte growth

Chronic hepatitis B virus (HBV) infection is a major cause of human suffering, and a number of mathematical models have examined within-host dynamics of the disease. Most previous HBV infection models have assumed that: (a) hepatocytes regenerate at a constant rate from a source outside the liver; and/or (b) the infection takes place via a mass action process. Assumption (a) contradicts experimental data showing that healthy hepatocytes proliferate at a rate that depends on current liver size relative to some equilibrium mass, while assumption (b) produces a problematic basic reproduction number. Here we replace the constant infusion of healthy hepatocytes with a logistic growth term and the mass action infection term by a standard incidence function; these modifications enrich the dynamics of a well-studied model of HBV pathogenesis. In particular, in addition to disease free and endemic steady states, the system also allows a stable periodic orbit and a steady state at the origin. Since the system is not differentiable at the origin, we use a ratio-dependent transformation to show that there is a region in parameter space where the origin is globally stable. When the basic reproduction number, R (0), is less than 1, the disease free steady state is stable. When R (0) > 1 the system can either converge to the chronic steady state, experience sustained oscillations, or approach the origin. We characterize parameter regions for all three situations, identify a Hopf and a homoclinic bifurcation point, and show how they depend on the basic reproduction number and the intrinsic growth rate of hepatocytes.