Multiphasic viral decline after potent treatment.

<p>After initiation of HAART, the plasma viral load undergoes a multiphasic decay and declines to below the detection limit (e.g., 50 RNA copies/mL) of standard assays after several months. A low level of viremia below 50 copies/mL may persist in patients for many years despite apparently effective antiretroviral treatment. Intermittent viral blips with transient HIV-1 RNA above the limit of detection are usually observed in well-suppressed patients.</p

Numerical simulations of the homeostasis model (Eq. (7)) and sensitivity tests of several parameters.

<p>The system is at steady state and at drug is applied. <b>A, D, G</b> and <b>J</b>: the latent reservoir size; <b>B, E, H</b> and <b>K</b>: viral load; <b>C, F, I</b> and <b>L</b>: the ratio of to , i.e., the relative contributions to the latent reservoir persistence from ongoing viral replication and latently infected cell proliferation. <b>A, B</b> and <b>C</b>: the carrying capacity of total latently infected cells is . We use different proliferation rates: (blue solid), (green dash-dotted), and (red dashed). The black solid line represents the detection limit. <b>D, E</b> and <b>F</b>: is fixed. Different carrying capacities of the total latently infected cells are used: (green dashed), (blue solid), (red dash-dotted). <b>G, H</b> and <b>I</b>: we use different fractions of infections that result in latency: (red dashed), (blue solid), and (black dotted). <b>J, K</b> and <b>L</b>: we use different drug efficacies: (red dashed), (blue solid), (black dotted). and the carrying capacity are fixed for the last two rows. The other parameter values used are listed in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000533#pcbi-1000533-t001" target="_blank">Table 1</a>.</p

Relative contributions of ongoing viral replication and latent cell activation.

<p><b>A and B:</b> the effects of ongoing viral replication (influenced by the overall drug efficacy) on the latent reservoir and viral load in the model given by Eq. (6). Different drug efficacies are used: (red dashed line) and (blue solid line). Ongoing viral replication is only a minor contributor to the stability of the latent reservoir and low-level persistent viremia, as indicated by the minor effect of changing drug efficacy from to . <b>C and D:</b> relative contributions of ongoing viral replication ( was fixed) and latent cell activation to the latent reservoir and viral persistence. <b>C:</b> the ratio of to , and <b>D:</b> the ratio of to . We chose . The other parameter values used are listed in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000533#pcbi-1000533-t001" target="_blank">Table 1</a>.</p

Schematic representation of the model with latently infected cell activation (Eq. (4)).

<p>Following encounter with cell-specific antigens, latently infected cells are activated and undergo programmed clonal expansion and contraction. A number of activated latently infected cells transition to the productive class and produce virions, whereas another small fraction of activated cells revert back to the latent state, providing a mechanism to replenish the latent reservoir.</p

Stochastic simulations of the model with programmed expansion and contraction (Eq. (4)).

<p>The model with programmed expansion and contraction of latently infected cells can generate viral blips with reasonable amplitude and duration. , . Column <b>A: </b>. Activated latently infected cells divide about times over an interval <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000533#pcbi.1000533-Perelson1" target="_blank">[4]</a>,<a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000533#pcbi.1000533-Hammer1" target="_blank">[6]</a> days. No statistically significant decay of the latent reservoir is observed. Column <b>B: </b>. The latent reservoir decays at a very slow rate. This realization shows a half-life of months. Column <b>C: </b>. Activated cells divide about times over the same time interval. The latent reservoir decays more quickly than it does in <b>B</b>, corresponding to a half-life of roughly months. The other parameter values used are listed in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000533#pcbi-1000533-t001" target="_blank">Table 1</a>. The blue horizontal line represents the detection limit of 50 RNA copies/mL.</p

Summary of stochastic simulations of the model, Eq. (4), with programmed expansion and contraction of latently infected cells.

<p>Abbreviations: ave (average), min (minimum), max (maximum). Values above brackets are the average values over 30 simulation runs. Values in brackets are the ranges. There are 5 antigenic activations within 300 days. When or , viral blip emerges each time activation occurs. When , not every activation generates a viral blip. In some simulations with or , the latent reservoir size is predicted to increase and hence has no half-life.</p

Numerical simulations of Eq. (4) with different duration and frequency of activation.

<p>We fixed the proliferation rate of activated cells to be . Column <b>A: </b>, . No statistically significant decay of the latent reservoir is observed. Column <b>B: </b>, . The latent reservoir decays at a very slow rate. Column <b>C: </b>, . In this realization, there are 8 activations in 300 days. The latent reservoir decays more quickly than in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000533#pcbi-1000533-g003" target="_blank">Figure 3C</a>. The other parameter values used are the same as those in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000533#pcbi-1000533-g003" target="_blank">Figure 3</a>. The blue horizontal line represents the detection limit of 50 RNA copies/mL.</p

Sensitivity tests on several parameters when studying the relative contributions using model (6).

<p>The upper panels: the latent reservoir size; the middle panels: viral load; and the lower panels: the ratio of the relative contributions, i.e., the ratio of to . In column <b>A</b>, we use different activation rates: (blue solid), (red dashed), and (purple dotted). There is no change in the ratio of relative contributions. In column <b>B</b>, we use different fractions of new infections that result in latency: (blue solid), (red dashed), and (purple dotted). In column <b>C</b>, we use different reversion rates to latency: (blue solid), (red dashed), and (purple dotted). The other parameter values used are the same as those in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1000533#pcbi-1000533-g007" target="_blank">Figure 7</a>.</p

Variables, parameters and values used in models and simulations.

<p>Variables, parameters and values used in models and simulations.</p

A Hepatitis C Virus Infection Model with Time-Varying Drug Effectiveness: Solution and Analysis

<div><p>Simple models of therapy for viral diseases such as hepatitis C virus (HCV) or human immunodeficiency virus assume that, once therapy is started, the drug has a constant effectiveness. More realistic models have assumed either that the drug effectiveness depends on the drug concentration or that the effectiveness varies over time. Here a previously introduced varying-effectiveness (VE) model is studied mathematically in the context of HCV infection. We show that while the model is linear, it has no closed-form solution due to the time-varying nature of the effectiveness. We then show that the model can be transformed into a Bessel equation and derive an analytic solution in terms of modified Bessel functions, which are defined as infinite series, with time-varying arguments. Fitting the solution to data from HCV infected patients under therapy has yielded values for the parameters in the model. We show that for biologically realistic parameters, the predicted viral decay on therapy is generally biphasic and resembles that predicted by constant-effectiveness (CE) models. We introduce a general method for determining the time at which the transition between decay phases occurs based on calculating the point of maximum curvature of the viral decay curve. For the parameter regimes of interest, we also find approximate solutions for the VE model and establish the asymptotic behavior of the system. We show that the rate of second phase decay is determined by the death rate of infected cells multiplied by the maximum effectiveness of therapy, whereas the rate of first phase decline depends on multiple parameters including the rate of increase of drug effectiveness with time.</p></div