Global dynamics of an HIV-1 infection model with distributed intracellular delays

AbstractIn this paper, an HIV-1 infection model with distributed intracellular delays is investigated, where the intracellular delays account for the time the target cells are contacted by the virus particles and the time the contacted cells become actively infected meaning that the contacting virions enter cells and the time the virus has penetrated into a cell and the time the new virions are created within the cell and are released from the cell, respectively. By analyzing the characteristic equations, the local stability of an infection-free equilibrium and a chronic-infection equilibrium of the model is established. By using suitable Lyapunov functionals and LaSalle’s invariance principle, it is proved that if the basic reproduction ratio is less than unity, the infection-free equilibrium is globally asymptotically stable; and if the basic reproduction ratio is greater than unity, the chronic-infection equilibrium is globally asymptotically stable

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

A stochastic model for internal HIV dynamics

In this paper we analyse a stochastic model representing HIV internal virus dynamics. The stochasticity in the model is introduced by parameter perturbation which is a standard technique in stochastic population modelling. We show that the model established in this paper possesses non-negative solutions as this is essential in any population dynamics model. We also carry out analysis on the asymptotic behaviour of the model. We approximate one of the variables by a mean reverting process and nd out the mean and variance of this process. Numerical simulations conclude the paper

Viral Dynamics of Delayed CTL-inclusive HIV-1 Infection Model With Both Virus-to-cell and Cell-to-cell Transmissions

We consider a mathematical model that describes a viral infection of HIV-1 with both virus-tocell and cell-to-cell transmission, CTL response immune and four distributed delays, describing intracellular delays and immune response delay. One of the main features of the model is that it includes a constant production rate of CTLs export from thymus, and an immune response delay. We derive the basic reproduction number and show that if the basic reproduction number is less than one, then the infection free equilibrium is globally asymptotically stable; whereas, if the basic reproduction number is greater than one, then there exist a chronic infection equilibrium, which is globally asymptotically stable in absence of immune response delay. Furthermore, for the special case with only immune response delay, we determine some conditions for stability switches of the chronic infection equilibrium. Numerical simulations indicate that the intracellular delays and immune response delay can stabilize and/or destabilize the chronic infection equilibrium

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

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