Global dynamics of cell mediated immunity in viral infection models with distributed delays

In this paper, we investigate global dynamics for a system of delay differential equations which describes a virus-immune interaction in \textit{vivo}. The model has two distributed time delays describing time needed for infection of cell and virus replication. Our model admits three possible equilibria, an uninfected equilibrium and infected equilibrium with or without immune response depending on the basic reproduction number for viral infection $R_{0}$ and for CTL response $R_{1}$ such that $R_{1}<R_{0}$. It is shown that there always exists one equilibrium which is globally asymptotically stable by employing the method of Lyapunov functional. More specifically, the uninfected equilibrium is globally asymptotically stable if $R_{0}\leq1$, an infected equilibrium without immune response is globally asymptotically stable if $R_{1}\leq1<R_{0}$ and an infected equilibrium with immune response is globally asymptotically stable if $R_{1}>1$. The immune activation has a positive role in the reduction of the infection cells and the increasing of the uninfected cells if $R_{1}>1$.Comment: 16 pages, accepted by Journal of Mathematical Analysis and Application

Impaired immune evasion in HIV through intracellular delays and multiple infection of cells

With its high mutation rate, HIV is capable of escape from recognition, suppression and/or killing by CD8(+) cytotoxic T lymphocytes (CTLs). The rate at which escape variants replace each other can give insights into the selective pressure imposed by single CTL clones. We investigate the effects of specific characteristics of the HIV life cycle on the dynamics of immune escape. First, it has been found that cells in HIV-infected patients can carry multiple copies of proviruses. To investigate how this process affects the emergence of immune escape, we develop a mathematical model of HIV dynamics with multiple infections of cells. Increasing the frequency of multiple-infected cells delays the appearance of immune escape variants, slows down the rate at which they replace the wild-type variant and can even prevent escape variants from taking over the quasi-species. Second, we study the effect of the intracellular eclipse phase on the rate of escape and show that escape rates are expected to be slower than previously anticipated. In summary, slow escape rates do not necessarily imply inefficient CTL-mediated killing of HIV-infected cells, but are at least partly a result of the specific characteristics of the viral life cycle

Mathematical model for HIV and CD4+ cells dynamics in vivo

Published by International Electronic Journal of Pure and Applied Mathematics Volume 6 No. 2 2013, 83-103Mathematical models are used to provide insights into the mechanisms and dynamics of the progression of viral infection in vivo. Untangling the dynamics between HIV and CD4+ cellular populations and molecular interactions can be used to investigate the effective points of interventions in the HIV life cycle. With that in mind, we develop and analyze a stochastic model for In-Host HIV dynamics that includes combined therapeutic treatment and intracellular delay between the infection of a cell and the emission of viral particles. The unique feature is that both therapy and the intracellular delay are incorporated into the model. We show the usefulness of our stochastic approach towards modeling combined HIV treatment by obtaining probability generating function, the moment structures of the healthy CD4+ cell, and the virus particles at any time t and the probability of virus clearance. Our analysis show that, when it is assumed that the drug is not completely effective, as is the case of HIV in vivo, the predicted rate of decline in plasma HIV virus concentration depends on three factors: the initial viral load before therapeutic intervention, the efficacy of therapy and the length of the intracellular delay.Mathematical models are used to provide insights into the mechanisms and dynamics of the progression of viral infection in vivo. Untangling the dynamics between HIV and CD4+ cellular populations and molecular interactions can be used to investigate the effective points of interventions in the HIV life cycle. With that in mind, we develop and analyze a stochastic model for In-Host HIV dynamics that includes combined therapeutic treatment and intracellular delay between the infection of a cell and the emission of viral particles. The unique feature is that both therapy and the intracellular delay are incorporated into the model. We show the usefulness of our stochastic approach towards modeling combined HIV treatment by obtaining probability generating function, the moment structures of the healthy CD4+ cell, and the virus particles at any time t and the probability of virus clearance. Our analysis show that, when it is assumed that the drug is not completely effective, as is the case of HIV in vivo, the predicted rate of decline in plasma HIV virus concentration depends on three factors: the initial viral load before therapeutic intervention, the efficacy of therapy and the length of the intracellular delay

Multiscale Modeling of Influenza A Virus Infection Supports the Development of Direct-Acting Antivirals

Influenza A viruses are respiratory pathogens that cause seasonal epidemics with up to 500,000 deaths each year. Yet there are currently only two classes of antivirals licensed for treatment and drug-resistant strains are on the rise. A major challenge for the discovery of new anti-influenza agents is the identification of drug targets that efficiently interfere with viral replication. To support this step, we developed a multiscale model of influenza A virus infection which comprises both the intracellular level where the virus synthesizes its proteins, replicates its genome, and assembles new virions and the extracellular level where it spreads to new host cells. This integrated modeling approach recapitulates a wide range of experimental data across both scales including the time course of all three viral RNA species inside an infected cell and the infection dynamics in a cell population. It also allowed us to systematically study how interfering with specific steps of the viral life cycle affects virus production. We find that inhibitors of viral transcription, replication, protein synthesis, nuclear export, and assembly/release are most effective in decreasing virus titers whereas targeting virus entry primarily delays infection. In addition, our results suggest that for some antivirals therapy success strongly depends on the lifespan of infected cells and, thus, on the dynamics of virus-induced apoptosis or the host's immune response. Hence, the proposed model provides a systems-level understanding of influenza A virus infection and therapy as well as an ideal platform to include further levels of complexity toward a comprehensive description of infectious diseases

Mathematical modelling of internal HIV dynamics

We study a mathematical model for the viral dynamics of HIV in an infected individual in the presence of HAART. The paper starts with a literature review and then formulates the basic mathematical model. An expression for R0, the basic reproduction number of the virus under steady state application of HAART, is derived followed by an equilibrium and stability analysis. There is always a disease-free equilibrium (DFE) which is globally asymptotically stable for R0 1 then some simulations will die out whereas others will not. Stochastic simulations suggest that if R0 > 1 those which do not die out approach a stochastic quasi-equilibrium consisting of random uctuations about the non-trivial deterministic equilibrium levels, but the amplitude of these uctuations is so small that practically the system is at the non-trivial equilibrium. A brief discussion concludes the paper

Role of delay differential equations in modelling low level HIV viral load

Over the past 30 years, HIV has infected over 60 million people, with almost half succumbing to AIDS-related illnesses.While antiretroviral therapy, used to significantly reduce within-host HIV replication, was available within 10 years of the discovery of HIV/AIDS, it is only within the last 10 years that it has become truly effective and universally accessible. However, there are problems with this therapy, not least that it must be administered indefinitely , but is expensive and highly toxic. Furthermore, as therapy reaches more resource-limited regions, continual access can not be guaranteed, resulting in therapy interruptions. This, coupled with a significant cost reduction by systematically interrupting therapy, means a set of models which can account for both treatment events need to be developed, as numerous models exist for therapy introduction, but those for therapy removal are limited. Thus a set of delay differential models are designed, which account for previously overlooked important features of intracellular delay and HIV latency. Incorporation of these features requires additional model components, leading to a rapid increase in complexity. To combat this complexity issue, dimensional analysis is introduced, as a novel method of identifying key components to model function, thus allowing significant reduction in parameter space. Based on these developed models, a number of existing and potential treatment interruption regimes are investigated, with a best practice regime suggested

GLOBAL ANALYSIS OF CELL INFECTION AND VIRUS PRODUCTION ON HIV-1 DYNAMICS

More than one sub-type of HIVs have been identified. This raises an issue of co-infections by multiple strains of HIVs. In this thesis, we propose two mathematical models, one ignoring intracellular delay and the other incorporating the delay, to describe the interactions of the populations of CD4+cells and two HIV stains. By nature, the two strains compete for CD4+ cells to invade for their own replications. By analyzing the two models, we find that the models demonstrate threshold dynamics: if the overall basic reproduction number Ro \u3c 1, then the infection free equilibrium is globally asymptotically stable; when R0 \u3e 1, then the competition exclusion principle generically holds in the sense that, except for the critical case R\ = R2 \u3e 1 where /?, is the individual basic reproduction number for strain i, all biologically meaningful solutions will converge to the single infection equilibrium representing the winning of the strain that has greater individual basic reproduction number. Numerical simulations are also performed to illustrate the theoretical results. The results on the model with delay also show that the basic reproduction number will be over calculated if the cellular delay is ignored