An HBV model with diffusion and time delay

In this paper, a hepatitis B virus (HBV) model with spatial diffusion and saturation response of the infection rate is investigated, in which the intracellular incubation period is modelled by a discrete time delay. By analyzing the corresponding characteristic equations, the local stability of an infected steady state and an uninfected steady state is discussed. By comparison arguments, it is proved that if the basic reproductive number is less than unity, the uninfected steady state is globally asymptotically stable. If the basic reproductive number is greater than unity, by successively modifying the coupled lower-upper solution pairs, sufficient conditions are obtained for the global stability of the infected steady state. Numerical simulations are carried out to illustrate the main results.     

具免疫应答和细胞内部时滞的HIV-1感染模型的稳定性分析

考虑了CTLs免疫应答和细胞内部时滞建立HIV-1感染的数学模型.对模型的无感染平衡点全局稳定性进行了分析,对CTLs未激活和CTLs已激活的感染平衡点给出了局部稳定的充分条件.数值模拟支持了得到的理论结果.     

Dynamics of a delay differential equation model of hepatitis B virus infection

We formulate and systematically study the global dynamics of a simple model of hepatitis B virus in terms of delay differential equations. This model has two important and novel features compared to the well-known basic virus model in the literature. Specifically, it makes use of the more realistic standard incidence function and explicitly incorporates a time delay in virus production. As a result, the infection reproduction number is no longer dependent on the patient liver size (number of initial healthy liver cells). For this model, the existence and the component values of the endemic steady state are explicitly dependent on the time delay. In certain biologically interesting limiting scenarios, a globally attractive endemic equilibrium can exist regardless of the time delay length.     

A mathematical model of cell-to-cell spread of HIV-1 that includes a time delay

 We consider a two-dimensional model of cell-to-cell spread of HIV-1 in tissue cultures, assuming that infection is spread directly from infected cells to healthy cells and neglecting the effects of free virus. The intracellular incubation period is modeled by a gamma distribution and the model is a system of two differential equations with distributed delay, which includes the differential equations model with a discrete delay and the ordinary differential equations model as special cases. We study the stability in all three types of models. It is shown that the ODE model is globally stable while both delay models exhibit Hopf bifurcations by using the (average) delay as a bifurcation parameter. The results indicate that, differing from the cell-to-free virus spread models, the cell-to-cell spread models can produce infective oscillations in typical tissue culture parameter regimes and the latently infected cells are instrumental in sustaining the infection. Our delayed cell-to-cell models may be applicable to study other types of viral infections such as human T-cell leukaemia virus type 1 (HTLV-1). Received: 18 November 2000 / Published online: 28 February 2003 RID="*" ID="*" Research was partially supported by the NSERC and MITACS of Canada and a start-up fund from the College of Arts and Sciences at the University of Miami. On leave from Dalhousie University, Halifax, Nova Scotia, Canada. Current address: Department of Mathematics, Clarke College, Dubuque, Iowa 52001, USA Key words or phrases: HIV-1 – Cell-to-cell spread – Time delay – Stability – Hopf bifurcation – Periodicity     

The impact of maturation delay of mosquitoes on the transmission of West Nile virus

We formulate and analyze a delay differential equation model for the transmission of West Nile virus between vector mosquitoes and avian hosts that incorporates maturation delay for mosquitoes. The maturation time from eggs to adult mosquitoes is sensitive to weather conditions, in particular the temperature, and the model allows us to investigate the impact of this maturation time on transmission dynamics of the virus among mosquitoes and birds. Numerical results of the model show that a combination of the maturation time and the vertical transmission of the virus in mosquitoes has substantial influence on the abundance and number of infection peaks of the infectious mosquitoes.     

Apoptosis in virus infection dynamics models

In this paper, on the basis of the simplified two-dimensional virus infection dynamics model, we propose two extended models that aim at incorporating the influence of activation-induced apoptosis which directly affects the population of uninfected cells. The theoretical analysis shows that increasing apoptosis plays a positive role in control of virus infection. However, after being included the third population of cytotoxic T lymphocytes immune response in HIV-infected patients, it shows that depending on intensity of the apoptosis of healthy cells, the apoptosis can either promote or comfort the long-term evolution of HIV infection. Further, the discrete-time delay of apoptosis is incorporated into the pervious model. Stability switching occurs as the time delay in apoptosis increases. Numerical simulations are performed to illustrate the theoretical results and display the different impacts of a delay in apoptosis.     

一类含分布时滞革新传播模型的稳定性

建立了一类含分布时滞的革新传播模型dU（t）/dt=-（α＋βA（t））U（t）-pU（t）＋p,dA（t）/dt=∫＋∞ 0 αE（τ）U（t-τ）dτ＋βU（t）A（t）-（p＋k）A（t）。研究了分布时滞对传播过程的影响,讨论了正平衡点的存在性和唯一性及其局部与全局的渐近稳定性,当分布时滞的核函数取δe^-δτ时,证明了正平衡点是绝对渐近稳定的。     

Success of prophylactic antiviral therapy for SARS-CoV-2: Predicted critical efficacies and impact of different drug-specific mechanisms of action

Repurposed drugs that are safe and immediately available constitute a first line of defense against new viral infections. Despite limited antiviral activity against SARS-CoV-2, several drugs are being tested as medication or as prophylaxis to prevent infection. Using a stochastic model of early phase infection, we evaluate the success of prophylactic treatment with different drug types to prevent viral infection. We find that there exists a critical efficacy that a treatment must reach in order to block viral establishment. Treatment by a combination of drugs reduces the critical efficacy, most effectively by the combination of a drug blocking viral entry into cells and a drug increasing viral clearance. Below the critical efficacy, the risk of infection can nonetheless be reduced. Drugs blocking viral entry into cells or enhancing viral clearance reduce the risk of infection more than drugs that reduce viral production in infected cells. The larger the initial inoculum of infectious virus, the less likely is prevention of an infection. In our model, we find that as long as the viral inoculum is smaller than 10 infectious virus particles, viral infection can be prevented almost certainly with drugs of 90% efficacy (or more). Even when a viral infection cannot be prevented, antivirals delay the time to detectable viral loads. The largest delay of viral infection is achieved by drugs reducing viral production in infected cells. A delay of virus infection flattens the within-host viral dynamic curve, possibly reducing transmission and symptom severity. Thus, antiviral prophylaxis, even with reduced efficacy, could be efficiently used to prevent or alleviate infection in people at high risk.     

A model of HIV-1 pathogenesis that includes an intracellular delay

Mathematical modeling combined with experimental measurements have yielded important insights into HIV-1 pathogenesis. For example, data from experiments in which HIV-infected patients are given potent antiretroviral drugs that perturb the infection process have been used to estimate kinetic parameters underlying HIV infection. Many of the models used to analyze data have assumed drug treatments to be completely efficacious and that upon infection a cell instantly begins producing virus. We consider a model that allows for less then perfect drug effects and which includes a delay in the initiation of virus production. We present detailed analysis of this delay differential equation model and compare the results to a model without delay. Our analysis shows that when drug efficacy is less than 100%, as may be the case in vivo, the predicted rate of decline in plasma virus concentration depends on three factors: the death rate of virus producing cells, the efficacy of therapy, and the length of the delay. Thus, previous estimates of infected cell loss rates can be improved upon by considering more realistic models of viral infection.     

Transformation by Murine Sarcoma Virus: Fixation (Deoxyribonucleic Acid Synthesis) and Development

Transformation of rat embryo cells by murine sarcoma virus (MSV) was contingent upon synthesis of deoxyribonucleic acid (DNA) during the first 12 hr of infection. Inhibition of DNA synthesis by thymidine (20 mm) or cytosine arabinoside (0.1 mm) resulted in the protection of cells from transformation by MSV. Transient suppression of DNA synthesis prior to infection or after a 12-hr delay had little effect on subsequent transformation, emphasizing the critical time period in in which DNA synthesis was necessary for intracellular fixation of the viral genome. These results are similar to those previously described for Rous sarcoma virus. Development of transformed cells after viral fixation was shown to be influenced by cellular density. Under conditions which allowed fixation of virus in confluent cellular monolayers, less than 20% of these cells developed into transformed foci.     

A simple macroscopic model for the diffusion and adsorption kinetics of r-adenovirus

The diffusion of viruses toward cells is a limiting step of the infection process. To be modeled correctly, this step must be evaluated in combination with the adsorption of the virus to the cell surface, which is a rapid but reversible step. In this paper, the recombinant adenovirus (rAd) diffusion and its adsorption to 293S cells in suspension were both measured and modeled. First, equilibrium experiments permitted to determine the number of receptors on the surface of 293S (R(T) = 3,500 cell(-1)) and the association constant (K(A) = 1.9 x 10(11) M(-1)) for rAd on these cells based on a simple monovalent adsorption model. Non-specific binding of the virus to the cell surface was not found to be significant. Second, total virus particle degradation rates between 5.2 x 10(-3) and 4.0 x 10(-2) min(-1) were measured at 37 degrees C in culture medium, but no significant virus degradation was observed at 4 degrees C. Third, free viral particle disappearance rates from a mixed suspension of virus and cells were measured at different virus concentrations. Experimental data were compared to a phenomenological dynamic model comprising both the diffusion and the adsorption steps. The diffusion to adsorption ratio, a fitted parameter, confirmed that the contact process of a virus with a cell is indeed diffusion controlled. However, the characteristic diffusion time constants obtained, based on a reversible adsorption model, were eightfolds smaller than those reported in the literature, based on diffusion models that assume irreversible adsorption.     

Diffusion barriers for ADP in the cardiac cell

The regulation of mitochondrial respiration in the intact heart may differ from that of isolated mitochondria if intracellular diffusion is restricted. Here we consider which factors may hinder diffusion in vivo and, based on computational analysis, design a reverse engineering approach to estimate the role of diffusional resistance in mitochondrial regulation from an experiment on the intact heart. Computational analysis of respiration measurements on skinned heart fibers shows that the outer mitochondrial membrane does not hinder diffusion enough to cause ADP gradients of tens of micromolars. A diffusion model further shows that the mesoscale structure of the myofibrillar space also does not hinder diffusion appreciably. However, ADP gradients are suggested by the measured activation time of oxidative phosphorylation and may be caused by diffusion restriction of other intracellular structures or the in vivo microstructure of networks of physically interacting proteins. Based on computational modeling we propose an experiment on the intact heart that allows to estimate the effective diffusion restriction between ATP producing and consuming sites in the cardiac cell.     

Mathematical model of influenza A virus production in large-scale microcarrier culture

A mathematical model that describes the replication of influenza A virus in animal cells in large-scale microcarrier culture is presented. The virus is produced in a two-step process, which begins with the growth of adherent Madin-Darby canine kidney (MDCK) cells. After several washing steps serum-free virus maintenance medium is added, and the cells are infected with equine influenza virus (A/Equi 2 (H3N8), Newmarket 1/93). A time-delayed model is considered that has three state variables: the number of uninfected cells, infected cells, and free virus particles. It is assumed that uninfected cells adsorb the virus added at the time of infection. The infection rate is proportional to the number of uninfected cells and free virions. Depending on multiplicity of infection (MOI), not necessarily all cells are infected by this first step leading to the production of free virions. Newly produced viruses can infect the remaining uninfected cells in a chain reaction. To follow the time course of virus replication, infected cells were stained with fluorescent antibodies. Quantitation of influenza viruses by a hemagglutination assay (HA) enabled the estimation of the total number of new virions produced, which is relevant for the production of inactivated influenza vaccines. It takes about 4-6 h before visibly infected cells can be identified on the microcarriers followed by a strong increase in HA titers after 15-16 h in the medium. Maximum virus yield Vmax was about 1x10(10) virions/mL (2.4 log HA units/100 microL), which corresponds to a burst size ratio of about 18,755 virus particles produced per cell. The model tracks the time course of uninfected and infected cells as well as virus production. It suggests that small variations (<10%) in initial values and specific rates do not have a significant influence on Vmax. The main parameters relevant for the optimization of virus antigen yields are specific virus replication rate and specific cell death rate due to infection. Simulation studies indicate that a mathematical model that neglects the delay between virus infection and the release of new virions gives similar results with respect to overall virus dynamics compared with a time delayed model.     

HIV treatment models with time delay

The present paper shows possible effects of antiretroviral treatment on the dynamics of the spread of the disease of human immunodeficiency virus infection in a population of varying size. By introducing time delays, we model the latency period and the delayed onset of positive treatment effects in the patients. The Hopf bifurcation and stability behaviour of the delay differential-equation model are analysed and simulations for different scenarios depending on the size of the treatment-induced delay are presented, and the results are discussed in detail.     

Dynamics of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells

Stilianakis and Seydel (Bull. Math. Biol., 1999) proposed an ODE model that describes the T-cell dynamics of human T-cell lymphotropic virus I (HTLV-I) infection and the development of adult T-cell leukemia (ATL). Their model consists of four components: uninfected healthy CD4+ T-cells, latently infected CD4+ T-cells, actively infected CD4+ T-cells, and ATL cells. Mathematical analysis that completely determines the global dynamics of this model has been done by Wang et al. (Math. Biosci., 2002). In this note, we first modify the parameters of the model to distinguish between contact and infectivity rates. Then we introduce a discrete time delay to the model to describe the time between emission of contagious particles by active CD4+ T-cells and infection of pure cells. Using the results in Culshaw and Ruan (Math. Biosci., 2000) in the analysis of time delay with respect to cell-free viral spread of HIV, we study the effect of time delay on the stability of the endemically infected equilibrium. Numerical simulations are presented to illustrate the results.     

In vitro experimental infection of primary duck hepatocyte cultures with duck hepatitis B virus.

Duck hepatitis B virus (DHBV) obtained from the serum of congenitally infected ducks was used to infect primary duck hepatocyte cultures 1 to 4 days after plating. Virus replication was demonstrated by the appearance, beginning at 2 days after infection, of intracellular covalently closed-circular and single-stranded DHBV DNA replicative intermediates which were not present in the inoculating virus preparation. With increasing time after infection there was further amplification of intracellular relaxed circular, covalently closed-circular, and single-stranded DHBV DNA. Cultures of primary duck hepatocytes are competent for infection with DHBV only during the first 4 days of culture. Synthesis of DHBV core antigen and DHBV surface antigen was detected by immunofluorescence in 10% of the hepatocytes in culture. De novo synthesis and release of infectious virus was also demonstrated. Therefore, all stages of viral replication were carried out by these experimentally infected primary hepatocyte cultures. This system makes it possible to study DHBV replication in vitro.     

Understanding hepatitis C viral dynamics with direct-acting antiviral agents due to the interplay between intracellular replication and cellular infection dynamics

The current paradigm for modeling viral kinetics and resistance evolution after treatment initiation considers only the level of circulating virus and cellular infection (CI model), while the intra-cellular level is disregarded. This model was successfully used to explain HIV dynamics and Hepatitis C virus (HCV) dynamics during interferon-based therapy. However, in the new era of direct-acting antiviral agents (DAAs) against HCV, viral kinetics is characterized by a more rapid decline of the wild-type virus as well as an early emergence of resistant strains that jeopardize the treatment outcome. Although the CI model can be extended to describe these new kinetic patterns, this approach has qualitative and quantitative limitations. Instead, we suggest that a more appropriate approach would consider viral dynamics at the cell infection level, as done currently, as well as at the intracellular level. Indeed, whereas in HIV integrated DNA serves as a static replication unit and mutations occur only once per infected cell, HCV replication is deeply affected by DAAs and furthermore processes of resistance evolution can occur at the intra-cellular level with a faster time-scale.We propose a comprehensive model of HCV dynamics that considers both extracellular and intracellular levels of infection (ICCI model). Intracellular viral genomic units are used to form replication units, which in turn synthesize genomic units that are packaged and secreted as virions infecting more target cells. Resistance evolution is modeled intra-cellularly, by different genomic- and replication-unit strains with particular relative-fitness and drug sensitivity properties, allowing for a rapid resistance takeover.Using the ICCI model, we show that the rapid decline of wild-type virus results from the ability of DAAs to destabilize the intracellular replication. On the other hand, this ability also favors the rapid emergence, intracellularly, of resistant virus. By considering the interaction between intracellular and extracellular infection we show that resistant virus, able to maintain a high level of intracellular replication, may nevertheless be unable to maintain rapid enough de novo infection rate at the extracellular level. Hence this model predicts that in HCV, and contrary to our experience with HIV, the emergence of productively resistant virus may not systematically prevent from a viral decline in the long-term. Thus, the ICCI model can explain the transient viral rebounds observed with DAA treatment as well as the viral resistance found in most patients with viral relapse at the end of DAA combination therapy.     

一类具有CTL免疫和时滞的HTLV-I传染病模型的动力学性质(英文)

主要研究了一类具有CTL免疫和时滞的HTLV-I传染的数学模型.通过构造Lyapunov泛函,分别证明了当R₀≤1,R₁≤10,R1>1时,系统（1.1）的无病平衡点E₀,无免疫平衡点E₁及地方病平衡点E₂是全局吸引的.