MS K710, Theoretical Biology and Biophysics, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Abstract

The new field of viral dynamics, based on within-host modeling of viral infections, began with models of human immunodeficiency virus (HIV), but now includes many viral infections. Here we review developments in HIV modeling, emphasizing quantitative findings about HIV biology uncovered by studying acute infection, the response to drug therapy and the rate of generation of HIV variants that escape immune responses. We show how modeling has revealed many dynamical features of HIV infection and how it may provide insight into the ultimate cure for this infection.

Since the discovery of HIV as the etiological agent of AIDS, numerous advances have been made in our understanding of the molecular biology, pathogenesis, and epidemiology of the virus, and the host immune response to it. Not least among these has been the knowledge obtained by mathematical analysis and within-host modeling of changes in viral load and T-cell counts after initiation of potent antiretroviral therapy in individual subjects. Indeed, modeling of the kinetics of HIV RNA under drug therapy has led to substantial insights into the dynamics and pathogenesis of HIV-1

Below we briefly review how quantitative data and modeling have contributed to the understanding of HIV biology.

A model of HIV infection

In the simplest and earliest models of viral infection, only the key players in HIV infection were taken into account _{
T
} per cell, and to be infected by free virus, according to a simple mass action infection term, that is, _{
T
}, to reflect viral effects in shortening the infected cell lifespan. Finally, free viruses are produced by infected cells at constant rate

Diagram of viral dynamics model

**Diagram of viral dynamics model.** Uninfected cells (

This simple model was shown to be able to describe the kinetics of acute HIV infection

Only a small fraction of CD4+ T cells in the periphery become infected with HIV

Modeling antiretroviral therapy

Early decay

The effects of antiretroviral therapy can easily be included in equation (1), so as to analyze the dynamics of viral decline under different therapeutic regimes. For example, reverse transcriptase inhibitors can prevent the establishment of productive infection of a cell. To model this the infection term _{RTI})_{RTI} is a number between 0 and 1 called the effectiveness of the reverse transcriptase inhibitor. Here ϵ_{RT} = 1 implies a 100% effective inhibitor. Protease inhibitors (PIs) prevent the maturation of HIV virions into infectious particles. To model PIs, the viral population is split into two populations, _{
I
} and _{
NI
}, where _{
NI
} represents immature non-infectious particles created by the action of the PI

where ϵ_{PI} is the effectiveness of the PI. This simple model has been used successfully to fit viral load data taken from individuals on antiretroviral therapy

Phases of viral decay under treatment

**Phases of viral decay under treatment.** When treatment is initiated, plasma viral load undergoes a multiphasic decay, with slower rates of viral loss as treatment progresses. One possible explanation is that there are various classes of infected cells (Figure

Second phase decay

If one follows the response to combination antiretroviral therapy (cART) for more than a week or two, one sees that the rapid first phase of decay of plasma viremia is followed by a slower second phase of decay (Figure

Irrespective of the mechanism generating the second phase, with continued cART viral levels decline below the detection limit of clinical assays (50 HIV RNA copies/ml), and with these assays one cannot determine how long the second phase lasts.

Third and fourth phase decays

Based on the rate of second phase decay, modeling suggested that 3 to 4 years of fully suppressive therapy could eliminate the cells responsible for second phase virus

Latency

HIV incorporates into the host cell genome and can establish a latent form of HIV infection, involving a small fraction of resting memory CD4+ T cells that carry integrated viral genomes

Modeling has also examined the role of early therapy in preventing the establishment of latency

So far, we have described the ‘ideal’ scenario for antiretroviral therapy, where treatment is successful in driving the viral load below detection levels. However, viral load sometimes rebounds and new viral strains with mutations that confer resistance to the drugs used in the treatment protocol are observed.

Drug resistance

When HIV infects a cell, the viral RNA genome is reverse transcribed into DNA. The reverse transcription process is error prone and results in mutation at an estimated rate of 3 × 10^{-5} per base per generation, with about two-thirds of these mutations being nucleotide substitutions ^{8} cells are infected each day, thus allowing all possible single and many double mutations to be explored each day

More recently, studies have shown that some patients derive clinical benefit from continued therapy even when the virus is resistant to the drug protocol, as the resistant virus can be less fit than wild type

Even though the introduction of antiretroviral treatment was a great success, and improved the quality of life of countless people, it has not become the panacea that was expected. For several reasons, including those alluded to above, treatment has not yet cured HIV. Thus, it has become clear that the development of a vaccine that prevents infection in the first place or ameliorates the course of disease is very important. To this end a better understanding of the clinical progression of HIV and the immune response against it is crucial. Here too, models have been playing a critical role as we discuss below.

Viral dynamics in primary infection

HIV is enormously diverse

A number of other modeling questions are of interest in the setting of acute HIV infection. Very early on, it was realized that the simple viral dynamics model in equation (1) could explain the behavior of the virus during early infection _{0}, for HIV _{0} for HIV is about 8, suggesting that each infected cell on average infects 8 others

Data fits in primary infection

**Data fits in primary infection.** The basic model of HIV dynamics (equation (1)) provides a good description for the initial period of viral infection, from the time of infection through the initial peak in viral load and subsequent control and achievement of a quasi steady-state (data from

Not every encounter with HIV/SIV results in infection; the transmission probability for HIV has been estimated as 10^{-3} to 10^{-2} per coital act

Although the modeling studies mentioned so far, dealing with a variety of different processes in HIV infection, have been quite successful in advancing our understanding of HIV biology, a puzzling aspect remains: these models do not necessarily include an explicit immune response.

Immune responses during HIV infection

The basic model given by equation (1), lacks an explicit representation of the immune response and hence has been called a target-cell limited model

While many models have included CD8 responses

Modeling antibody responses to HIV is still an undeveloped area, although some work involving data interpretation has been done

One method of trying to estimate the strength of both humoral and cell-mediated immune responses is to determine how fast the virus can escape from these responses

Immune escape at cytotoxic T lymphocyte epitopes

**Immune escape at cytotoxic T lymphocyte epitopes.** Schematic representation of the takeover of the viral population by virus mutated at a particular CTL epitope escaping the host immune response. Two different epitopes (diamond and circle symbols) are represented with different rates of population turnover. The equations shown represent the dynamics of wild-type (

While models with only two viral species are easy to analyze, simulation models have looked at much more complex situations in which there are multiple escapes

Some studies quantified the efficiency of cytotoxic T lymphocytes in killing infected cells based on the rate of escape of viral variants at specific epitopes. These results were used to compare the effect of immune pressure in macaques infected with SIV with that of humans

Balamurali

As AIDS is characterized by a loss of CD4+ T cells, much modeling work has focused on quantifying T-cell turnover during HIV infection using direct labeling as well as cell markers such as T-cell receptor excision circles. Much of this modeling literature has recently been reviewed

Future perspective

Modeling HIV dynamics has been a rich area of study that, we believe, has spearheaded the wider field of modeling in viral and immune system dynamics. Indeed, there is much more research in HIV modeling than we could possibly cover here, but we hope to have given a representative flavor of the most innovative studies.

Looking into the future, there are several areas where we believe modeling can still make important contributions. Clearly, as more quantitative data on the immune response, both cellular and humoral, become available, it will be important to include these in the mechanistic models of HIV infection. This will be especially important to explain the events during primary infection, and in the analysis of vaccine trial data in humans. These models should be able to describe not just the time evolution of the viral load, as do current models, but also the kinetics of the immune response and its effect on virus and infected cells. Another area of interest to modelers is the potential effect of cell-to-cell transmission of HIV-1

Overall, we believe that two of the most pressing questions in the HIV field are why primary infection invariably leads to chronicity, and why chronic infection leads to persistence of the virus, even in the face of very potent and durable therapy. It is unquestionable that future modeling studies of novel datasets will help us to understand both of these questions, proposing different mechanisms and hypotheses for these observations. In so doing, we might uncover new intervention strategies to help prevent or eradicate infection. The idea of curing HIV infection either, by viral eradication or functionally curing the infection by controlling it, is gaining traction

Conclusion

Mathematical analysis of HIV-1 viral dynamics and immune responses has led to a number of important insights about the dynamics and pathogenesis of HIV infection. Modeling plasma virus decay under therapy demonstrated the fast turnover of virus, explaining the potential for generation of mutants and the development of drug resistance. This early work paved the way for many collaborations between clinicians and modelers to understand the nature of the multiphasic viral decay observed in treated patients, the initial expansion of virus upon infection, the turnover of CD4+ and CD8+ T cells, the probability of single transmitted/founder viruses and many others. It is fair to say that most, if not all, of these insights would not have been possible without close interdisciplinary collaborations allowing quantitative modeling.

Acknowledgments

Portions of this work were performed under the auspices of the US Department of Energy under contract DE-AC52-06NA25396 and supported by NIH grants P20-GM103452, OD011095, AI028433, and the Center for HIV/AIDS Vaccine Immunology and Immunogen Discovery grant number UM1-AI100645-01. RMR received partial funding from the EU 7th Framework Program under grant no. PCOFUND-GA-2009-246542 and from FCT Portugal and part of his work was done while visiting Instituto de Medicina Molecular, Faculdade de Medicina da Universidade de Lisboa, Lisbon, Portugal.