^{*}

Conceived and designed the experiments: LR ASP. Performed the experiments: LR. Analyzed the data: LR. Contributed reagents/materials/analysis tools: LR. Wrote the paper: LR ASP.

The authors have declared that no competing interests exist.

Although potent combination therapy is usually able to suppress plasma viral loads in HIV-1 patients to below the detection limit of conventional clinical assays, a low level of viremia frequently can be detected in plasma by more sensitive assays. Additionally, many patients experience transient episodes of viremia above the detection limit, termed viral blips, even after being on highly suppressive therapy for many years. An obstacle to viral eradication is the persistence of a latent reservoir for HIV-1 in resting memory ^{+} T cells. The mechanisms underlying low viral load persistence, slow decay of the latent reservoir, and intermittent viral blips are not fully characterized. The quantitative contributions of residual viral replication to viral and the latent reservoir persistence remain unclear. In this paper, we probe these issues by developing a mathematical model that considers latently infected cell activation in response to stochastic antigenic stimulation. We demonstrate that programmed expansion and contraction of latently infected cells upon immune activation can generate both low-level persistent viremia and intermittent viral blips. Also, a small fraction of activated T cells revert to latency, providing a potential to replenish the latent reservoir. By this means, occasional activation of latently infected cells can explain the variable decay characteristics of the latent reservoir observed in different clinical studies. Finally, we propose a phenomenological model that includes a logistic term representing homeostatic proliferation of latently infected cells. The model is simple but can robustly generate the multiphasic viral decline seen after initiation of therapy, as well as low-level persistent viremia and intermittent HIV-1 blips. Using these models, we provide a quantitative and integrated prospective into the long-term dynamics of HIV-1 and the latent reservoir in the setting of potent antiretroviral therapy.

Current combination therapy can suppress viral loads in HIV-1-infected individuals to below the detection limit of standard commercial assays. However, it cannot eradicate the virus from patients. HIV-1 can generally be identified in resting memory ^{+} T cells and persists in patients on potent treatment for a long time. These latently infected cells decay slowly, but can produce new virions when activated by relevant antigens. Many patients experience transient episodes of viremia, or blips, even though they have “undetectable” plasma viral loads for many years. Here, we develop a new mathematical model describing latently infected cell activation upon random antigenic stimulation. Using the model, we show that programmed expansion and contraction of latently infected cells upon activation can generate both low viral load persistence and viral blips. Occasional replenishment of the latent reservoir may explain the different decay kinetics of the reservoir observed in clinical practice. We also show that a model with homeostatic proliferation of latently infected cells can explain persistence of low-level virus, stability of the latent reservoir, and emergence of viral blips. These results provide novel insights into the long-term virus dynamics and could have implications for the treatment of HIV-1 infection.

Following initiation of highly active antiretroviral therapy (HAART) the plasma viral load declines with a rapid first phase, followed by a slower second phase (

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.

Another line of evidence for HIV-1 persistence is the observation of transient episodes of viremia (“blips”) above the detection limit in patients on HAART (

The management of HIV-1 infection requires a further understanding of the mechanisms underlying low viral load persistence, stability of the latent reservoir, and occurrence of intermittent viral blips, as well as the relationships between them. We approach this through mathematical modeling. Many models, as surveyed in

In this paper, we further study latently infected cell activation in response to antigenic stimulation by extending the models in

A basic model of latent cell activation was initially developed to examine the cell populations contributing to the second-phase viral decline after administration of both reverse transcriptase (RT) and protease inhibitors

There is only one positive steady state viral load of Eq. (1):

Both

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.

Let

As suggested by

We employ a basic “on-off” model, which has previously been used to describe the

Although

We choose the overall drug efficacy

Variable/Parameter | Value | Description | Reference |

- | Target T cells | - | |

- | Latently infected cells | - | |

- | Resting latently infected cells | - | |

- | Activated latently infected cells | - | |

- | Productively infected cells | - | |

- | Viral load | - | |

Recruitment rate of susceptible cells | |||

Death rate of susceptible cells | |||

Infection rate | |||

0.85 | Overall drug efficacy | see text | |

Fraction resulting in latency | |||

Death rate of latently infected cells | |||

Rate of transition from latently to | see text | ||

productively infected cells | |||

Death rate of productively infected cells | |||

Burst size | |||

Clearance rate of free virus | |||

Density-dependent mortality | |||

Power in density-dependent mortality function | |||

Viral production rate | |||

varied | Proliferation rate of activated cells | see text | |

Activation rate of latent cells | see text | ||

Death rate of activated cells | see text | ||

Reversion rate to latency | |||

Base death rate of activated cells | |||

varied | Maximum proliferation rate of latent cells | see text | |

varied | Carrying capacity density of latent cells | see text | |

see text | Expansion function | - | |

see text | Rapid contraction function | - |

Since we are interested in the dynamics of the third-phase viral decline during treatment, we choose the initial viral load to be

Homann et al.

A recent experimental study by Chomont et al.

We choose a small base value for the transition rate,

In order to maintain the latent cell pool during potent drug therapy, we choose the proliferation rate

The carrying capacity (i.e., the maximum sustainable population) of latently infected cells during therapy is unknown. Assuming

The simulation with an initial T cell count

Numerical simulations of model (4) show that programmed expansion and contraction of latently infected cells upon occasional antigenic stimulation can robustly generate intermittent viral blips with reasonable amplitude and duration, without seriously depleting the latent reservoir (

The model with programmed expansion and contraction of latently infected cells can generate viral blips with reasonable amplitude and duration.

An interesting result is that the amplitude of viral blips is inversely correlated with the decay of the latent reservoir. Based on model (4), viral blips originate from activation of latently infected cells into the productive class. It was initially thought that this activation would deplete the latent reservoir quickly in HAART-treated patients because

We ran stochastic simulations of the model 30 times, recorded the number and amplitudes of viral blips, and calculated the half-life of the decay of the latent reservoir based on the change in the latent reservoir size in 300 days. The summary statistics on our simulations is given in

Parameter value | Ave number of blips over [0, 300] days | Min blip amplitude (copies/mL) | Max blip amplitude (copies/mL) | Ave blip amplitude (copies/mL) | Change in the latent reservoir size over 300 days | Half-life of the latent reservoir decay (months) |

5 | 186 [140, 362] | 693 [541, 877] | 394 [298, 522] | −0.5% [−19%, +26%] | ||

5 | 168 [113, 308] | 524 [346, 680] | 334 [263, 446] | −14% [−32%, +5.5%] | 46 [18, -] | |

3.7 [2, 5] | 61 [50, 94] | 93 [71, 111] | 74 [63, 98] | −65% [−67%, −62%] | 6.6 [6.3, 7.3] |

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

The fraction of resting latently infected cells that are activated by antigenic stimulation remains largely unknown. Due to the heterogeneity of latently infected cells with respect to the antigens they respond to, it is likely that a very small fraction of latently infected cells are activated by a particular antigen. We tested model predictions (Eq. (4)) with different activation rates

The proliferation rate of activated cells,

In addition to changing the proliferation rate

We fixed the proliferation rate of activated cells to be

In summary, occasional activation of latently infected cells upon stochastic antigen encounter is able to produce a large quantity of activated T cells temporarily, and thereby generate intermittent viral blips. The blip amplitude/frequency is inversely correlated with the decay of the latent reservoir. Using different potentials of activated T cells to divide during the initial clonal expansion phase or different duration or frequency of antigenic stimulation enables us to generate the different decay characteristics of the latent reservoir observed in different clinical studies

We have assumed a density-dependent mortality rate for productively infected cells in the model given by Eq. (4) in order to maintain a low steady state viral load when antigen is absent. The reason that viral loads decrease very quickly in the absence of activation in this model is that activated cells decline quickly to an extremely low level during the contraction phase, with not enough cells entering the productive stage. Even when we choose a smaller death rate of activated cells, for example,

With

The model is able to generate viral blips as well as low-level persistent viremia. The low-level viral load is maintained by a low level of activated latently infected cells during the second slower contraction phase in the latent cell response. In the first row,

The decay of the latent reservoir, the amplitude of viral blips, and the viral load below the limit of detection are not largely influenced by the effectiveness of the treatment as long as the overall drug efficacy is beyond a threshold value,

We further compare the relative contributions of ongoing viral replication and latent cell activation to the latent reservoir and viral persistence. In

We have also performed sensitivity tests on several parameters when studying the relative contributions. The ratio of

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

As shown in previous sections, occasional activation of latently infected cells upon antigen encounter can transiently produce a large number of activated cells, a small part of which can revert to the latent state and hence replenish the latent reservoir. In fact, several other sources might also reseed the latent cell pool and contribute to a stable latent reservoir: (1) homeostatic proliferation of

The homeostasis model can robustly describe the multiphasic viral decline following initiation of combination antiretroviral treatment, and maintain both low-level persistent viremia and the latent reservoir during therapy.

The system is at steady state and at

We also plot the level of latently infected cells and viral load with different

We further examine the relative contributions to the latent reservoir persistence from ongoing viral replication and latently infected cell proliferation (i.e., the ratio of

The model with homeostatic proliferation of latently infected cells can also generate viral blips given intermittent bursts of activation of latently infected cells (i.e., increasing

A Poisson process with an average waiting time of 2 months is used to model the random encounter between latently infected cells and antigens. We assume the total body carrying capacity of latently infected cells is

The frequency of viral blips is also affected by the renewal potential since viral blips come directly from activation of latently infected cells. In the case of a weak renewal potential (

Although several months of HAART is usually able to reduce the viral load in HIV-infected patients successfully to below the detection limit of standard assays, 50 RNA copies/mL, a low level of virus can still be detected in plasma by more sensitive assays

However, even if HAART is potent enough to block all new infections of susceptible T cells, virus may still be released from a stable reservoir composed of latently infected

The mechanism for the stability of the latent reservoir in the setting of HAART remains controversial. The observation that intensification of antiretroviral therapy can accelerate the decay of the latent reservoir in some patients

Most well-suppressed patients with plasma HIV-1 RNA below the detection limit of 50 copies/mL demonstrate transient episodes of viremia above the limit (viral blips)

In many mathematical models, the steady state viral load is very sensitive to small changes of drug efficacy and thus they cannot robustly describe the low viral load persistence during HAART

Our model can robustly maintain the stability of the latent reservoir and meanwhile generate viral blips with reasonable duration and amplitude in infected individuals in the setting of HAART. We hypothesize that latently infected cells act similar to other memory cells and experience programmed proliferation and contraction upon antigenic stimulation by their recall antigens. During the response, a portion of activated latently infected cells transition into the productive class and generate viral blips. In the meanwhile, a small fraction of activated cells revert back to the resting state, providing a potential to replenish the latent reservoir. An interesting result is that this model can reconcile the divergent estimates of the decay rate of the latent reservoir in the literature. The half-life of the reservoir decay is largely determined by the frequency and duration of antigenic stimulation and by how many times the resultant activated latently infected cells proliferate during the latent cell response. In addition, we observe that assuming activated T cells remain at a low level after the rapid contraction phase (the biphasic decay model, i.e., Eq. (6)) can maintain a low level of viremia. This suggests that latently infected cell activation solely can maintain low-level viremic persistence and produce intermittent viral blips simultaneously, with the latent reservoir occasionally replenished. Model simulations show that the levels of persistent viremia and latently infected cells are not correlated with HAART potency, which suggests that low viral load persistence and the stability of the latent reservoir need not arise from ongoing active replication during HAART.

The conclusion that ongoing viral replication is a minor factor contributing to viral and the latent reservoir persistence is consistent with the results of recent studies

Motivated by the observation that latently infected cells have the potential to renew themselves when stimulated by their previously encountered antigens, a much simpler phenomenological model with homeostatic proliferation of latently infected cells was proposed to study viral persistence and HIV-1 blips. The idea that the stability of the latent reservoir can be maintained by homeostatic proliferation of latently infected cells is also supported by a very recent experimental study

Considering that the latent reservoir consists of heterogeneous mixture of latently infected T-cell clones that respond differently to different antigens, our models can be generalized to account for the heterogeneity of latently infected cells. For example, we can extend the homeostasis model by including multiple subpopulations of latently infected cells, with each subpopulation having a different transition rate

Given that the latent reservoir has been identified as a major barrier to virus eradication with current combination therapy

Unfortunately, as viral levels are driven down, say below 1 RNA copy/mL, and latently infected cells become rare, it becomes impossible to follow the dynamics of these populations. Therefore, we must rely on mathematical models to make inferences about the end game in viral eradication. Here we have presented a set of models that agree with much of our knowledge about low-level viral persistence, the latent reservoir, and viral blips. Direct experimental tests of these models would involve the examination of the latently infected cell response when cells are activated by specific antigens. In addition, all of our models suggest that there is an inverse relationship between the decay of the latent reservoir and the frequency (or amplitude) of viral blips. Thus, more accurate and frequent data on the latent reservoir size, the number of viral blips and their amplitudes also can be used to test our models.

We thank Rob de Boer and Ruy Ribeiro for useful discussions on T cell proliferation, and three referees for their suggestions that improved this manuscript.

^{+}T cells.