Modeling plasma virus concentration during primary HIV infection

Dynamics of a Class of HIV Infection Models with Cure of Infected Cells in Eclipse Stage

In this paper, we propose two HIV infection models with specific nonlinear incidence rate by including a class of infected cells in the eclipse phase. The first model is described by ordinary differential equations (ODEs) and generalizes a set of previously existing models and their results. The second model extends our ODE model by taking into account the diffusion of virus. Furthermore, the global stability of both models is investigated by constructing suitable Lyapunov functionals. Finally, we check our theoretical results with numerical simulations.

Order reduction for an RNA virus evolution model

A mathematical or computational model in evolutionary biology should necessary combine several comparatively fast processes, which actually drive natural selection and evolution, with a very slow process of evolution. As a result, several very different time scales are simultaneously present in the model; this makes its analytical study an extremely difficult task. However, the significant difference of the time scales implies the existence of a possibility of the model order reduction through a process of time separation. In this paper we conduct the procedure of model order reduction for a reasonably simple model of RNA virus evolution reducing the original system of three integro-partial derivative equations to a single equation. Computations confirm that there is a good fit between the results for the original and reduced models.

HIV Replication Is Not Controlled by CD8+ T Cells during the Acute Phase of the Infection in Humanized Mice

HIV replication follows a well-defined pattern during the acute phase of the infection in humans. After reaching a peak during the first few weeks after infection, viral replication resolves to a set-point thereafter. There are still uncertainties regarding the contribution of CD8⁺ T cells in establishing this set-point. An alternative explanation, supported by in silico modeling, would imply that viral replication is limited by the number of available targets for infection, i.e. CD4⁺CCR5⁺ T cells. Here, we used NOD.SCID.gc^-/- mice bearing human CD4⁺CCR5⁺ and CD8⁺ T cells derived from CD34⁺ progenitors to investigate the relative contribution of both in viral control after the peak. Using low dose of a CCR5-tropic HIV virus, we observed an increase in viral replication followed by “spontaneous” resolution of the peak, similar to humans. To rule out any possible role for CD8⁺ T cells in viral control, we infected mice in which CD8⁺ T cells had been removed by a depleting antibody. Globally, viral replication was not affected by the absence of CD8⁺ T cells. Strikingly, resolution of the viral peak was equally observed in mice with or without CD8⁺ T cells, showing that CD8⁺ T cells were not involved in viral control in the early phase of the infection. In contrast, a marked and specific loss of CCR5-expressing CD4⁺ T cells was observed in the spleen and in the bone marrow, but not in the blood, of infected animals. Our results strongly suggest that viral replication during the acute phase of the infection in humanized mice is mainly constrained by the number of available targets in lymphoid tissues rather than by CD8⁺ T cells.

A multi-scale mathematical modeling framework to investigate anti-viral therapeutic opportunities in targeting HIV-1 accessory proteins

Human immunodeficiency virus-1 (HIV-1) employs accessory proteins to evade innate immune responses by neutralizing the anti-viral activity of host restriction factors. Apolipoprotein B mRNA-editing enzyme 3G (APOBEC3G, A3G) and bone marrow stromal cell antigen 2 (BST2) are host resistance factors that potentially inhibit HIV-1 infection. BST2 reduces viral production by tethering budding HIV-1 particles to virus producing cells, while A3G inhibits the reverse transcription (RT) process and induces viral genome hypermutation through cytidine deamination, generating fewer replication competent progeny virus. Two HIV-1 proteins counter these cellular restriction factors: Vpu, which reduces surface BST2, and Vif, which degrades cellular A3G. The contest between these host and viral proteins influences whether HIV-1 infection is established and progresses towards AIDS. In this work, we present an age-structured multi-scale viral dynamics model of in vivo HIV-1 infection. We integrated the intracellular dynamics of anti-viral activity of the host factors and their neutralization by HIV-1 accessory proteins into the virus/cell population dynamics model. We calculate the basic reproductive ratio (Ro) as a function of host-viral protein interaction coefficients, and numerically simulated the multi-scale model to understand HIV-1 dynamics following host factor-induced perturbations. We found that reducing the influence of Vpu triggers a drop in Ro, revealing the impact of BST2 on viral infection control. Reducing Vif׳s effect reveals the restrictive efficacy of A3G in blocking RT and in inducing lethal hypermutations, however, neither of these factors alone is sufficient to fully restrict HIV-1 infection. Interestingly, our model further predicts that BST2 and A3G function synergistically, and delineates their relative contribution in limiting HIV-1 infection and disease progression. We provide a robust modeling framework for devising novel combination therapies that target HIV-1 accessory proteins and boost antiviral activity of host factors.

Ebola virus dynamics in mice treated with favipiravir

The polymerase inhibitor favipiravir is a candidate for the treatment of Ebola virus disease. Here, we designed a mathematical model to characterize the viral dynamics in 20 mice experimentally infected with Ebola virus, which were either left untreated or treated with favipiravir at 6 or 8days post infection. This approach provided estimates of kinetic parameters of Ebola virus reproduction, such as the half-life of productively infected cells, of about 6h, and the basic reproductive number which indicates that virus produced by a single infected cell productively infects about 9 new cells. Furthermore, the model predicted that favipiravir efficiently blocks viral production, reaching an antiviral effectiveness of 95% and 99.6% at 2 and 6days after initiation of treatment, respectively. The model could be particularly helpful to guide future studies evaluating favipiravir in larger animals.

Reactivation Kinetics of HIV-1 and Susceptibility of Reactivated Latently Infected CD4+ T Cells to HIV-1-Specific CD8+ T Cells

The "shock and kill" model of human immunodeficiency virus type 1 (HIV-1) eradication involves the induction of transcription of HIV-1 genes in latently infected CD4(+) T cells, followed by the elimination of these infected CD4(+) T cells by CD8(+) T cells or other effector cells. CD8(+) T cells may also be needed to control the spread of new infection if residual infected cells are present at the time combination antiretroviral therapy (cART) is discontinued. In order to determine the time frame needed for CD8(+) T cells to effectively prevent the spread of HIV-1 infection, we examined the kinetics of HIV transcription and virus release in latently infected cells reactivated ex vivo. Isolated resting, primary CD4(+) T cells from HIV-positive (HIV+) subjects on suppressive regimens were found to upregulate cell-associated HIV-1 mRNA within 1 h of stimulation and produce extracellular virus as early as 6 h poststimulation. In spite of the rapid kinetics of virus production, we show that CD8(+) T cells from 2 out of 4 viremic controllers were capable of effectively eliminating reactivated autologous CD4(+) cells that upregulate cell-associated HIV-1 mRNA. The results have implications for devising strategies to prevent rebound viremia due to reactivation of rare latently infected cells that persist after potentially curative therapy.

IMPORTANCE

A prominent HIV-1 cure strategy termed "shock and kill" involves the induction of HIV-1 transcription in latently infected CD4(+) T cells with the goal of elimination of these cells by either the cytotoxic T lymphocyte response or other immune cell subsets. However, the cytotoxic T cell response may also be required after curative treatment if residual latently infected cells remain. The kinetics of HIV-1 reactivation indicate rapid upregulation of cell-associated HIV-1 mRNA and a 5-h window between transcription and virus release. Thus, HIV-specific CD8(+) T cell responses likely have a very short time frame to eliminate residual latently infected CD4(+) T cells that become reactivated after discontinuation of antiretroviral therapy following potentially curative treatment strategies.

An evolutionary role for HIV latency in enhancing viral transmission

HIV latency is the chief obstacle to eradicating HIV but is widely believed to be an evolutionary accident providing no lentiviral fitness advantage. However, findings of latency being "hardwired" into HIV's gene-regulatory circuitry appear inconsistent with latency being an evolutionary accident, given HIV's rapid mutation rate. Here, we propose that latency is an evolutionary "bet-hedging" strategy whose frequency has been optimized to maximize lentiviral transmission by reducing viral extinction during mucosal infections. The model quantitatively fits the available patient data, matches observations of high-frequency latency establishment in cell culture and primates, and generates two counterintuitive but testable predictions. The first prediction is that conventional CD8-depletion experiments in SIV-infected macaques increase latent cells more than viremia. The second prediction is that strains engineered to have higher replicative fitness—via reduced latency—will exhibit lower infectivity in animal-model mucosal inoculations. Therapeutically, the theory predicts treatment approaches that may substantially enhance "activate-and-kill" HIV-cure strategies.

Post-treatment control of HIV infection

Antiretroviral therapy (ART) for HIV is not a cure. However, recent studies suggest that ART, initiated early during primary infection, may induce post-treatment control (PTC) of HIV infection with HIV RNA maintained at <50 copies per mL. We investigate the hypothesis that ART initiated early during primary infection permits PTC by limiting the size of the latent reservoir, which, if small enough at treatment termination, may allow the adaptive immune response to prevent viral rebound (VR) and control infection. We use a mathematical model of within host HIV dynamics to capture interactions among target cells, productively infected cells, latently infected cells, virus, and cytotoxic T lymphocytes (CTLs). Analysis of our model reveals a range in CTL response strengths where a patient may show either VR or PTC, depending on the size of the latent reservoir at treatment termination. Below this range, patients will always rebound, whereas above this range, patients are predicted to behave like elite controllers. Using data on latent reservoir sizes in patients treated during primary infection, we also predict population-level VR times for noncontrollers consistent with observations.

Hybrid spreading mechanisms and T cell activation shape the dynamics of HIV-1 infection

HIV-1 can disseminate between susceptible cells by two mechanisms: cell-free infection following fluid-phase diffusion of virions and by highly-efficient direct cell-to-cell transmission at immune cell contacts. The contribution of this hybrid spreading mechanism, which is also a characteristic of some important computer worm outbreaks, to HIV-1 progression in vivo remains unknown. Here we present a new mathematical model that explicitly incorporates the ability of HIV-1 to use hybrid spreading mechanisms and evaluate the consequences for HIV-1 pathogenenesis. The model captures the major phases of the HIV-1 infection course of a cohort of treatment naive patients and also accurately predicts the results of the Short Pulse Anti-Retroviral Therapy at Seroconversion (SPARTAC) trial. Using this model we find that hybrid spreading is critical to seed and establish infection, and that cell-to-cell spread and increased CD4+ T cell activation are important for HIV-1 progression. Notably, the model predicts that cell-to-cell spread becomes increasingly effective as infection progresses and thus may present a considerable treatment barrier. Deriving predictions of various treatments' influence on HIV-1 progression highlights the importance of earlier intervention and suggests that treatments effectively targeting cell-to-cell HIV-1 spread can delay progression to AIDS. This study suggests that hybrid spreading is a fundamental feature of HIV infection, and provides the mathematical framework incorporating this feature with which to evaluate future therapeutic strategies.

Simple mathematical models do not accurately predict early SIV dynamics

Upon infection of a new host, human immunodeficiency virus (HIV) replicates in the mucosal tissues and is generally undetectable in circulation for 1–2 weeks post-infection. Several interventions against HIV including vaccines and antiretroviral prophylaxis target virus replication at this earliest stage of infection. Mathematical models have been used to understand how HIV spreads from mucosal tissues systemically and what impact vaccination and/or antiretroviral prophylaxis has on viral eradication. Because predictions of such models have been rarely compared to experimental data, it remains unclear which processes included in these models are critical for predicting early HIV dynamics. Here we modified the “standard” mathematical model of HIV infection to include two populations of infected cells: cells that are actively producing the virus and cells that are transitioning into virus production mode. We evaluated the effects of several poorly known parameters on infection outcomes in this model and compared model predictions to experimental data on infection of non-human primates with variable doses of simian immunodifficiency virus (SIV). First, we found that the mode of virus production by infected cells (budding vs. bursting) has a minimal impact on the early virus dynamics for a wide range of model parameters, as long as the parameters are constrained to provide the observed rate of SIV load increase in the blood of infected animals. Interestingly and in contrast with previous results, we found that the bursting mode of virus production generally results in a higher probability of viral extinction than the budding mode of virus production. Second, this mathematical model was not able to accurately describe the change in experimentally determined probability of host infection with increasing viral doses. Third and finally, the model was also unable to accurately explain the decline in the time to virus detection with increasing viral dose. These results suggest that, in order to appropriately model early HIV/SIV dynamics, additional factors must be considered in the model development. These may include variability in monkey susceptibility to infection, within-host competition between different viruses for target cells at the initial site of virus replication in the mucosa, innate immune response, and possibly the inclusion of several different tissue compartments. The sobering news is that while an increase in model complexity is needed to explain the available experimental data, testing and rejection of more complex models may require more quantitative data than is currently available.

Global stability of an infection-age structured HIV-1 model linking within-host and between-host dynamics

Although much evidence shows the inseparable interaction between the within-host progression of HIV-1 infection and the transmission of the disease at the population level, few models coupling the within-host and between-host dynamics have been developed. In this paper, we adopt the nested approach, viewing the transmission rate at each stage (primary, chronic, and AIDS stage) of HIV-1 infection as a saturated function of the viral load, to formulate an infection-age structured epidemic model. We explicitly link the individual and the host population scale, and derive the basic reproduction number R0 for the coupled system. To analyze the model and perform a detailed global dynamics analysis, two Lyapunov functionals are constructed to prove the global asymptotical stability of the disease-free and endemic equilibria. Theoretical results indicate that R0 provides a threshold value determining whether or not the disease dies out. Numerical simulations are presented to quantitatively investigate the influence of the within-host viral dynamics on between-host transmission dynamics. The results suggest that increasing the effectiveness of inhibitors can decrease the basic reproduction number, but can also increase the overall infected population because of a lower disease-induced mortality rate and a longer lifespan of HIV infected individuals.

The Effect of Interference on the CD8(+) T Cell Escape Rates in HIV

In early human immunodeficiency virus (HIV) infection, the virus population escapes from multiple CD8⁺ cell responses. The later an escape mutation emerges, the slower it outgrows its competition, i.e., the escape rate is lower. This pattern could indicate that the strength of the CD8⁺ cell responses is waning, or that later viral escape mutants carry a larger fitness cost. In this paper, we investigate whether the pattern of decreasing escape rates could also be caused by genetic interference among different escape strains. To this end, we developed a mathematical multi-epitope model of HIV dynamics, which incorporates stochastic effects, recombination, and mutation. We used cumulative linkage disequilibrium measures to quantify the amount of interference. We found that nearly synchronous, similarly strong immune responses in two-locus systems enhance the generation of genetic interference. This effect, combined with a scheme of densely spaced sampling times at the beginning of infection and sparse sampling times later, leads to decreasing successive escape rate estimates, even when there were no selection differences among alleles. These predictions are supported by empirical data from one HIV-infected patient. Thus, interference could explain why later escapes are slower. Considering escape mutations in isolation, neglecting their genetic linkage, conceals the underlying haplotype dynamics and can affect the estimation of the selective pressure exerted by CD8⁺ cells. In systems in which multiple escape mutations appear, the occurrence of interference dynamics should be assessed by measuring the linkage between different escape mutations.

Mathematical model of multivalent virus-antibody complex formation in humans following acute and chronic HIV infections

Antibodies that bind viral surface proteins can limit the spread of the infection through neutralizing and non-neutralizing functions. During both acute and chronic Human Immunodeficiency Virus infection, antibody-virion immune complexes are formed, but fail to ensure protection. In this study, we develop a mathematical model of multivalent antibody binding and use it to determine the dynamical interactions that lead to immune complexes formation and the role of complexes with increased numbers of bound antibodies in the pathogenesis of the disease. We compare our predictions with published temporal virus and immune complex data from acute infected patients. Finally, we derive quantitative and qualitative conditions needed for antibody-induced protection.

Antibody responses during hepatitis B viral infection

Hepatitis B is a DNA virus that infects liver cells and can cause both acute and chronic disease. It is believed that both viral and host factors are responsible for determining whether the infection is cleared or becomes chronic. Here we investigate the mechanism of protection by developing a mathematical model of the antibody response following hepatitis B virus (HBV) infection. We fitted the model to data from seven infected adults identified during acute infection and determined the ability of the virus to escape neutralization through overproduction of non-infectious subviral particles, which have HBs proteins on their surface, but do not contain nucleocapsid protein and viral nucleic acids. We showed that viral clearance can be achieved for high anti-HBV antibody levels, as in vaccinated individuals, when: (1) the rate of synthesis of hepatitis B subviral particles is slow; (2) the rate of synthesis of hepatitis B subviral particles is high but either anti-HBV antibody production is fast, the antibody affinity is high, or the levels of pre-existent HBV-specific antibody at the time of infection are high, as could be attained by vaccination. We further showed that viral clearance can be achieved for low equilibrium anti-HBV antibody levels, as in unvaccinated individuals, when a strong cellular immune response controls early infection.

A fractional-order model of HIV infection: Numerical solution and comparisons with data of patients

In this paper, a fractional-order model which describes the human immunodeficiency type-1 virus (HIV-1) infection is presented. Numerical solutions are obtained using a generalized Euler method (GEM) to handle the fractional derivatives. The fractional derivatives are described in the Caputo sense. We show that the model established in this paper possesses non-negative solutions. Comparisons between the results of the fractional-order model, the results of the integer model and the measured real data obtained from 10 patients during primary HIV-1 infection are presented. These comparisons show that the results of the fractional-order model give predictions to the plasma virus load of the patients better than those of the integer model.

MODELING AND OPTIMAL CONTROL FOR ANTIRETROVIRAL THERAPY

We consider a three-dimensional nonlinear control model based on the Wodarz HIV model. The model phase variables are populations of the uninfected and infected target cells and the concentration of an antiretroviral drug. The drug intake rate is assumed to be a bounded control function. An optimal control problem of minimizing the cumulative infection level (the infected cells population) on a given time interval is stated and solved, and the types of the optimal control for different model parameters are found by analytical methods. We thereby reduce the two-point boundary value problem (TPBVP) for the Pontryagin maximum principle to a problem of the finite-dimensional optimization. Numerical results are presented to illustrate the optimal solution.

Integrated population pharmacokinetic/viral dynamic modelling of lopinavir/ritonavir in HIV-1 treatment-naïve patients

BACKGROUND

Lopinavir (LPV)/ritonavir (RTV) co-formulation (LPV/RTV) is a widely used protease inhibitor (PI)-based regimen to treat HIV-infection. As with all PIs, the trough concentration (C trough) is a primary determinant of response, but the optimum exposure remains poorly defined. The primary objective was to develop an integrated LPV population pharmacokinetic model to investigate the influence of α-1-acid glycoprotein and link total and free LPV exposure to pharmacodynamic changes in HIV-1 RNA and assess viral dynamic and drug efficacy parameters.

METHODS

Data from 35 treatment-naïve HIV-infected patients initiating therapy with LPV/RTV 400/100 mg orally twice daily across two studies were used for model development and simulations using ADAPT. Total LPV (LPVt) and RTV concentrations were measured by high-performance liquid chromatography with ultraviolet (UV) detection. Free LPV (LPVf) concentrations were measured using equilibrium dialysis and mass spectrometry.

RESULTS

The LPVt typical value of clearance (CLLPVt/F) was 4.73 L/h and the distribution volume (VLPVt/F) was 55.7 L. The clearance (CLLPVf/F) and distribution volume (Vf/F) for LPVf were 596 L/h and 6,370 L, respectively. The virion clearance rate was 0.0350 h(-1). The simulated LPVLPVt C trough values at 90% (EC90) and 95% (EC95) of the maximum response were 316 and 726 ng/mL, respectively.

CONCLUSIONS

The pharmacokinetic-pharmacodynamic model provides a useful tool to quantitatively describe the relationship between LPV/RTV exposure and viral response. This comprehensive modelling and simulation approach could be used as a surrogate assessment of antiretroviral (ARV) activity where adequate early-phase dose-ranging studies are lacking in order to define target trough concentrations and possibly refine dosing recommendations.

The majority of CD4+ T-cell depletion during acute simian-human immunodeficiency virus SHIV89.6P infection occurs in uninfected cells

Untreated human immunodeficiency virus (HIV) infection is characterized by depletion of CD4(+) T cells, ultimately leading to the impairment of host immune defenses and death. HIV-infected CD4(+) T cells die from direct virus-induced apoptosis and CD8 T-cell-mediated elimination, but a broader and more profound depletion occurs in uninfected CD4(+) T cells via multiple indirect effects of infection. We fit mathematical models to data from experiments that tested an HIV eradication strategy in which five macaques with a proportion of CD4(+) T cells resistant to simian-human immunodeficiency virus (SHIV) entry were challenged with SHIV89.6P, a highly pathogenic dual-tropic chimeric SIV-HIV viral strain that results in rapid loss of both SHIV-susceptible and SHIV-resistant CD4(+) T cells. Our results suggest that uninfected (bystander) cell death accounts for the majority of CD4(+) T-lymphocyte loss, with at least 60% and 99% of CD4(+) T cell death occurring in uninfected cells during acute and established infection, respectively. Mechanisms to limit the profound indirect killing effects associated with HIV infection may be associated with immune preservation and improved long-term survival.

IMPORTANCE

HIV infection induces a massive depletion of CD4(+) T cells, leading to profound immunodeficiency, opportunistic infections, and eventually death. While HIV induces apoptosis (programmed cell death) by directly entering and replicating in CD4(+) T cells, uninfected CD4(+) T cells also undergo apoptosis due to ongoing toxic inflammation in the region of infection. In this paper, we use mathematical models in conjunction with data from simian-human immunodeficiency virus SHIV89.6P infection in macaques (a model of HIV infection in humans) to estimate the percentage of cell death that occurs in uninfected cells during the initial period of infection. We reveal that the vast majority of cell death occurs in these cells, which are not infected. The "bystander effects" that lead to enormous reductions in the number of uninfected CD4(+) T cells may be a target for future interventions that aim to limit the extent of damage caused by HIV.

Can non-lytic CD8+ T cells drive HIV-1 escape?

The CD8+ T cell effector mechanisms that mediate control of HIV-1 and SIV infections remain poorly understood. Recent work suggests that the mechanism may be primarily non-lytic. This is in apparent conflict with the observation that SIV and HIV-1 variants that escape CD8+ T cell surveillance are frequently selected. Whilst it is clear that a variant that has escaped a lytic response can have a fitness advantage compared to the wild-type, it is less obvious that this holds in the face of non-lytic control where both wild-type and variant infected cells would be affected by soluble factors. In particular, the high motility of T cells in lymphoid tissue would be expected to rapidly destroy local effects making selection of escape variants by non-lytic responses unlikely. The observation of frequent HIV-1 and SIV escape poses a number of questions. Most importantly, is the consistent observation of viral escape proof that HIV-1- and SIV-specific CD8+ T cells lyse infected cells or can this also be the result of non-lytic control? Additionally, the rate at which a variant strain escapes a lytic CD8+ T cell response is related to the strength of the response. Is the same relationship true for a non-lytic response? Finally, the potential anti-viral control mediated by non-lytic mechanisms compared to lytic mechanisms is unknown. These questions cannot be addressed with current experimental techniques nor with the standard mathematical models. Instead we have developed a 3D cellular automaton model of HIV-1 which captures spatial and temporal dynamics. The model reproduces in vivo HIV-1 dynamics at the cellular and population level. Using this model we demonstrate that non-lytic effector mechanisms can select for escape variants but that outgrowth of the variant is slower and less frequent than from a lytic response so that non-lytic responses can potentially offer more durable control.

The interplay between viruses and the immune system cannot always be studied with current experimental techniques or commonly used mathematical models. Consequently, many important questions remain unanswered. The questions we wished to address fall into this category. Recent evidence strongly suggests that CD8+ T cells control SIV, and potentially HIV-1, primarily by secreting anti-viral factors rather than by killing infected cells. However, this does not seem compatible with the common observation that HIV and SIV evolve to escape the immune response. Soluble anti-viral factors, like RANTES which protects uninfected cells from infection, would be expected to inhibit both wild-type and variant virus. Furthermore, the high speed and motility of T cells in lymphoid tissue will increase homogeneity and again decrease the likelihood that an escape variant can have a selective advantage in the face of non-lytic control. We wanted to understand whether viral escape is proof that HIV-1- and SIV-specific CD8+ T cells kill infected cells, determine the factors that facilitate viral escape, and investigate the comparative efficiency of lytic and non-lytic responses in controlling viral infections. Here we develop an elaborate but robust computational framework that captures T cell kinetics and spatial interactions in lymphoid tissue to addresses these important questions.

Modeling the within-host dynamics of HIV infection

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.

Contribution of follicular dendritic cells to persistent HIV viremia

HIV-1 infections cannot be completely eradicated by drug therapy, as the virus persists in reservoirs. Low-level plasma viremia has been detected in patients treated for over 7 years, but the cellular compartments that support this low-level viremia have not been identified. The decay of HIV-1 during treatment appears to occur in four phases, with the 3rd and 4th phases occurring when the virus is below the limit of detection of conventional assays. Here, we focus on the 3rd phase of decay, which has been estimated to have a half-life of 39 months. We show that follicular dendritic cells (FDC), which have been identified as an HIV reservoir, can be the main source of the low-level viremia detected during the 3rd phase of decay and contribute to viremia at even longer times. Our calculations show that the kinetics of leakage of virus from FDC is consistent with three types of available clinical data.

APOBEC3G-Augmented Stem Cell Therapy to Modulate HIV Replication: A Computational Study

The interplay between the innate immune system restriction factor APOBEC3G and the HIV protein Vif is a key host-retrovirus interaction. APOBEC3G can counteract HIV infection in at least two ways: by inducing lethal mutations on the viral cDNA; and by blocking steps in reverse transcription and viral integration into the host genome. HIV-Vif blocks these antiviral functions of APOBEC3G by impeding its encapsulation. Nonetheless, it has been shown that overexpression of APOBEC3G, or interfering with APOBEC3G-Vif binding, can efficiently block in vitro HIV replication. Some clinical studies have also suggested that high levels of APOBEC3G expression in HIV patients are correlated with increased CD4+ T cell count and low levels of viral load; however, other studies have reported contradictory results and challenged this observation. Stem cell therapy to replace a patient's immune cells with cells that are more HIV-resistant is a promising approach. Pre-implantation gene transfection of these stem cells can augment the HIV-resistance of progeny CD4+ T cells. As a protein, APOBEC3G has the advantage that it can be genetically encoded, while small molecules cannot. We have developed a mathematical model to quantitatively study the effects on in vivo HIV replication of therapeutic delivery of CD34+ stem cells transfected to overexpress APOBEC3G. Our model suggests that stem cell therapy resulting in a high fraction of APOBEC3G-overexpressing CD4+ T cells can effectively inhibit in vivo HIV replication. We extended our model to simulate the combination of APOBEC3G therapy with other biological activities, to estimate the likelihood of improved outcomes.

Computational inference methods for selective sweeps arising in acute HIV infection

During the first weeks of human immunodeficiency virus-1 (HIV-1) infection, cytotoxic T-lymphocytes (CTLs) select for multiple escape mutations in the infecting HIV population. In recent years, methods that use escape mutation data to estimate rates of HIV escape have been developed, thereby providing a quantitative framework for exploring HIV escape from CTL response. Current methods for escape-rate inference focus on a specific HIV mutant selected by a single CTL response. However, recent studies have shown that during the first weeks of infection, CTL responses occur at one to three epitopes and HIV escape occurs through complex mutation pathways. Consequently, HIV escape from CTL response forms a complex, selective sweep that is difficult to analyze. In this work, we develop a model of initial infection, based on the well-known standard model, that allows for a description of multi-epitope response and the complex mutation pathways of HIV escape. Under this model, we develop Bayesian and hypothesis-test inference methods that allow us to analyze and estimate HIV escape rates. The methods are applied to two HIV patient data sets, concretely demonstrating the utility of our approach.

A model of HIV drug resistance driven by heterogeneities in host immunity and adherence patterns

Background

Population transmission models of antiretroviral therapy (ART) and pre-exposure prophylaxis (PrEP) use simplistic assumptions – typically constant, homogeneous rates – to represent the short-term risk and long-term effects of drug resistance. In contrast, within-host models of drug resistance allow for more detailed dynamics of host immunity, latent reservoirs of virus, and drug PK/PD. Bridging these two levels of modeling detail requires an understanding of the “levers” – model parameters or combinations thereof – that change only one independent observable at a time. Using the example of accidental tenofovir-based pre-exposure prophyaxis (PrEP) use during HIV infection, we will explore methods of implementing host heterogeneities and their long-term effects on drug resistance.

Results

We combined and extended existing models of virus dynamics by incorporating pharmacokinetics, pharmacodynamics, and adherence behavior. We identified two “levers” associated with the host immune pressure against the virus, which can be used to independently modify the setpoint viral load and the shape of the acute phase viral load peak. We propose parameter relationships that can explain differences in acute and setpoint viral load among hosts, and demonstrate their influence on the rates of emergence and reversion of drug resistance. The importance of these dynamics is illustrated by modeling long-lived latent reservoirs of virus, through which past intervals of drug resistance can lead to failure of suppressive drug regimens. Finally, we analyze assumptions about temporal patterns of drug adherence and their impact on resistance dynamics, finding that with the same overall level of adherence, the dwell times in drug-adherent versus not-adherent states can alter the levels of drug-resistant virus incorporated into latent reservoirs.

Conclusions

We have shown how a diverse range of observable viral load trajectories can be produced from a basic model of virus dynamics using immunity-related “levers”. Immune pressure, in turn, influences the dynamics of drug resistance, with increased immune activity delaying drug resistance and driving more rapid return to dominance of drug-susceptible virus after drug cessation. Both immune pressure and patterns of drug adherence influence the long-term risk of drug resistance. In the case of accidental PrEP use during infection, rapid transitions between adherence states and/or weak immunity fortifies the “memory” of previous PrEP exposure, increasing the risk of future drug resistance. This model framework provides a means for analyzing individual-level risks of drug resistance and implementing heterogeneities among hosts, thereby achieving a crucial prerequisite for improving population-level models of drug resistance.

Linking immunological and epidemiological dynamics of HIV: the case of super-infection

In this paper, a two-strain model that links immunological and epidemiological dynamics across scales is formulated. On the within-host scale, the two strains eliminate each other with the strain with the larger immunological reproduction persisting. However, on the population scale superinfection is possible, with the strain with larger immunological reproduction number super-infecting the strain with the smaller immunological reproduction number. The two models are linked through the age-since-infection structure of the epidemiological variables. In addition, the between-host transmission and the disease-induced death rate depend on the within-host viral load. The immunological reproduction numbers, the epidemiological reproduction numbers and invasion reproduction numbers are computed. Besides the disease-free equilibrium, there are two population-level strain one and strain two isolated equilibria, as well as a population-level coexistence equilibrium when both invasion reproduction numbers are greater than one. The single-strain population-level equilibria are locally asymptotically stable suggesting that in the absence of superinfection oscillations do not occur, a result contrasting previous studies of HIV age-since-infection structured models. Simulations suggest that the epidemiological reproduction number and HIV population prevalence are monotone functions of the within-host parameters with reciprocal trends. In particular, HIV medications that decrease within-host viral load also increase overall population prevalence. The effect of the immunological parameters on the population reproduction number and prevalence is more pronounced when the initial viral load is lower.

A hunter virus that targets both infected cells and HIV free virions: implications for therapy

The design of ‘hunter’ viruses aimed at destroying human immunodeficiency virus (HIV) infected cells is an active area of research that has produced promising results in vitro. Hunters are designed to target exposed viral envelope proteins in the membranes of infected cells, but there is evidence that the hunter may also target envelope proteins of free HIV, inducing virus-virus fusion. In order to predict the effects of this fusion on therapy outcomes and determine whether fusion ability is advantageous for hunter virus design, we have constructed a model to account for the possibility of hunter-HIV fusion. The study was based on a target cell-limited model of HIV infection and it examined the hunter therapeutic effect on recovering the HIV main target cells, the activated CD4⁺ T lymphocytes. These cells assist in setting up an immune response to opportunistic infections. The study analyzed the hunter dual mechanisms to control infection and because of diverse estimates for viral production and clearance of HIV, simulations were examined at rates spanning an order of magnitude. Results indicate that without hunter-HIV fusion ability, hunters that kill HIV-infected cells lead to a substantial recovery of healthy cell population at both low and high HIV turnover rates. When hunter-HIV fusion is included, cell recovery was particularly enhanced at lower HIV turnover rates. This study shows that the fusion ability, in addition to hunter infection ability, could be a favorable attribute for improving the efficacy of hunter-viral therapy. These results provide support for the potential use of engineered viruses to control HIV and other viral infections.

Nested model reveals potential amplification of an HIV epidemic due to drug resistance

The use of antiretroviral therapy (ART) is the most efficient measure in controlling the HIV epidemic. However, emergence of drug-resistant strains can reduce the potential benefits of ART. The viral dynamics of drug-sensitive and drug-resistant strains at the individual level may play a crucial role in the emergence and spread of drug resistance in a population. We investigate the effect of the viral dynamics within an infected individual on the epidemiological dynamics of HIV using a nested model that links both dynamical levels. A time-dependent between-host transmission rate that receives feedback from a model of two-strain virus dynamics within a host is incorporated into an epidemiological model of HIV. We analyze the resulting dynamics of the model and identify model parameters such as time when ART is initiated, fraction of cases treated, and the probability that a patient develops drug resistance, as having the greatest impact on total infection and prevalence of drug resistance. Importantly, for small values of the risk of a patient developing drug resistance, increasing the fraction of cases treated can increase the cumulative number of infected individuals. Such a pattern is the result of the balance between not treating a patient and having future cases still sensitive to treatment, and treating the patient and increasing the chances for future (untreatable) drug-resistant infections. The current modeling framework incorporates important aspects of virus dynamics within a host into an epidemic model. This approach provides useful insights on the drug resistance dynamics of an epidemic of HIV, which may assist in identifying an optimal use of ART.

On the role of CD8 T cells in the control of persistent infections

The control of pathogen density during infections is typically assumed to be the result of a combination of resource limitation (loss of target cells that the pathogen can infect), innate immunity, and specific immunity. The contributions of these factors have been considered in acute infections, which are characterized by having a short duration. What controls the pathogen during persistent infections is less clear, and is complicated by two factors. First, specific immune responses become exhausted if they are subject to chronic stimulation. Exhaustion has been best characterized for CD8 T cell responses, and occurs through a combination of cell death and loss of functionality of surviving cells. Second, new nonexhausted T cells can immigrate from the thymus during the infection, and may play a role in the control of the infection. In this article, we formulate a partial-differential-equation model to describe the interaction between these processes, and use this model to explore how thymic influx and exhaustion might affect the ability of CD8 T cell responses to control persistent infections. We find that although thymic influx can play a critical role in the maintenance of a limited CD8 T cell response during persistent infections, this response is not sufficiently large to play a significant role in controlling the infection. In doing so, our results highlight the importance of resource limitation and innate immunity in the control of persistent infections.

Modelling the course of an HIV infection: insights from ecology and evolution

The Human Immunodeficiency Virus (HIV) is one of the most threatening viral agents. This virus infects approximately 33 million people, many of whom are unaware of their status because, except for flu-like symptoms right at the beginning of the infection during the acute phase, the disease progresses more or less symptom-free for 5 to 10 years. During this asymptomatic phase, the virus slowly destroys the immune system until the onset of AIDS when opportunistic infections like pneumonia or Kaposi’s sarcoma can overcome immune defenses. Mathematical models have played a decisive role in estimating important parameters (e.g., virion clearance rate or life-span of infected cells). However, most models only account for the acute and asymptomatic latency phase and cannot explain the progression to AIDS. Models that account for the whole course of the infection rely on different hypotheses to explain the progression to AIDS. The aim of this study is to review these models, present their technical approaches and discuss the robustness of their biological hypotheses. Among the few models capturing all three phases of an HIV infection, we can distinguish between those that mainly rely on population dynamics and those that involve virus evolution. Overall, the modeling quest to capture the dynamics of an HIV infection has improved our understanding of the progression to AIDS but, more generally, it has also led to the insight that population dynamics and evolutionary processes can be necessary to explain the course of an infection.

Time-course analysis of main markers of primary infection in cats with the feline immunodeficiency virus

Studies of the response of the immune system to feline immunodeficiency virus (FIV) during primary infection have shown that a subpopulation of CD8⁺ T-cells with an activated phenotype and reduced expression of the CD8β chain (denoted CD8β^low T cells) expands to reach up to 80% of the total CD8⁺ T cell count. The expansion of this subpopulation is considered to be a signature of FIV and an indicator of immune system alteration. We use a simple mathematical formalism to study the relationships over time between the dose of infection, the size of the CD8β^low population, and the circulating viral load in cats infected with FIV. Viremia profiles are described using a combination of two exponential laws, whereas the CD8β^low percentage (out of the total CD8⁺ population) is represented by a Gompertz law including an expansion phase and a saturation phase. Model parameters are estimated with a population approach using data from 102 experimentally infected cats. We examine the dose of infection as a potential covariate of parameters. We find that the rates of increase of viral load and of CD8β^low percentage are both correlated with the dose of infection. Cats that develop strong acute viremia also show the largest degree of CD8β^low expansion. The two simple models are robust tools for analysing the time course of CD8β^low percentage and circulating viral load in FIV-infected cats and may be useful for generating new insights on the disease and on the design of therapeutic strategies, potentially applicable to HIV infection.

Cell-free transmission of human adenovirus by passive mass transfer in cell culture simulated in a computer model

Viruses spread between cells, tissues, and organisms by cell-free and cell-cell transmissions. Both mechanisms enhance disease development, but it is difficult to distinguish between them. Here, we analyzed the transmission mode of human adenovirus (HAdV) in monolayers of epithelial cells by wet laboratory experimentation and a computer simulation. Using live-cell fluorescence microscopy and replication-competent HAdV2 expressing green fluorescent protein, we found that the spread of infection invariably occurred after cell lysis. It was affected by convection and blocked by neutralizing antibodies but was independent of second-round infections. If cells were overlaid with agarose, convection was blocked and round plaques developed around lytic infected cells. Infected cells that did not lyse did not give rise to plaques, highlighting the importance of cell-free transmission. Key parameters for cell-free virus transmission were the time from infection to lysis, the dose of free viruses determining infection probability, and the diffusion of single HAdV particles in aqueous medium. With these parameters, we developed an in silico model using multiscale hybrid dynamics, cellular automata, and particle strength exchange. This so-called white box model is based on experimentally determined parameters and reproduces viral infection spreading as a function of the local concentration of free viruses. These analyses imply that the extent of lytic infections can be determined by either direct plaque assays or can be predicted by calculations of virus diffusion constants and modeling.

DYNAMICAL MODEL OF IN-HOST HIV INFECTION: WITH DRUG THERAPY AND MULTI VIRAL STRAINS

In this paper, a mathematical model for the effect of drug therapy on the in-host dynamics of HIV is considered and analyzed. As the process of reverse transcription is highly error prone, it causes mutation of virus which results in the emergence of drug resistant virus. This is also accounted in the model and corresponding model with both drug resistant and drug sensitive viral strains is studied. We found that, if reproductive ratios for both the strains are less than one, the virus population goes to extinction. If the reproductive ratio of either strain is greater than one and the reproductive ratio of drug resistant virus is smaller than that of drug sensitive virus then both the virus strains persist and infection is not cleared. However if reproductive ratio of drug resistant virus is greater than that of drug sensitive virus then the drug resistant virus out-competes the drug sensitive virus and only drug resistant virus survives. Hence the ratio of two reproduction ratios works as invading capacity threshold value for drug resistant strain. We also noted that by increasing the effective efficacy of the drug, virus may be cleared. Numerical simulations are performed to support and elaborate the analytical findings.

GLOBAL DYNAMICS OF AN IN-HOST HIV-1 INFECTION MODEL WITH THE LONG-LIVED INFECTED CELLS AND FOUR INTRACELLULAR DELAYS

In this paper, we investigate global dynamics for an in-host HIV-1 infection model with the long-lived infected cells and four intracellular delays. Our model admits two possible equilibria, an uninfected equilibrium and infected equilibrium depending on the basic reproduction number. We derive that the global dynamics are completely determined by the values of the basic reproduction number: if the basic reproduction number is less than one, the uninfected equilibrium is globally asymptotically stable, and the virus is cleared; if the basic reproduction number is larger than one, then the infection persists, and the infected equilibrium is globally asymptotically stable.

The acute HIV infection: implications for intervention, prevention and development of an effective AIDS vaccine

Effective preventive measures against HIV must function near the time of virus transmission to prevent the establishment of a chronic infection. Low-dose SIV/SHIV infections by multiple routes lead to remarkably rapid systemic dissemination of virus and large numbers of infected cells during the initial weeks of the acute infection. Here we describe the narrow time-frame during which potent post-exposure interventions such as anti-retroviral therapy or the administration of high-titered neutralizing antibodies can block the establishment of the in vivo infection. This short window of opportunity is applicable to HIV infections and represents a formidable challenge for developing effective chemoprophylaxis and vaccine approaches.

Analysis of the virus dynamics model reveals that early treatment of HCV infection may lead to the sustained virological response

Considerable progress has been made towards understanding hepatitis C virus, its pathogenesis and the effect of the drug therapy on the viral load, yet around 50% of patients do not achieve the sustained virological response (SVR) by the standard treatment. Although several personalized factors such as patients’ age and weight may be important, by mathematical modeling we show that the time of the start of the therapy is a significant factor in determining the outcome. Toward this end, we first performed sensitivity analysis on the standard virus dynamics model. The analysis revealed four phases when the sensitivity of the infection to drug treatment differs. Further, we added a perturbation term in the model to simulate the drug treatment period and predict the outcome when the therapy is carried out during each of the four phases. The study shows that while the infection may be difficult to treat in the late phases, the therapy is likely to result in SVR if it is carried out in the first or second phase. Thus, development of newer and more sensitive screening methods is needed for the early detection of the infection. Moreover, the analysis predicts that the drug that blocks new infections is more effective than the drug that blocks the virus production.

Pairing experimentation and computational modeling to understand the role of tissue inducer cells in the development of lymphoid organs

The use of genetic tools, imaging technologies and ex vivo culture systems has provided significant insights into the role of tissue inducer cells and associated signaling pathways in the formation and function of lymphoid organs. Despite advances in experimental technologies, the molecular and cellular process orchestrating the formation of a complex three-dimensional tissue is difficult to dissect using current approaches. Therefore, a robust set of simulation tools have been developed to model the processes involved in lymphoid tissue development. Specifically, the role of different tissue inducer cell populations in the dynamic formation of Peyer’s patches has been examined. Utilizing approaches from systems engineering, an unbiased model of lymphoid tissue inducer cell function has been developed that permits the development of emerging behaviors that are statistically not different from that observed in vivo. These results provide the confidence to utilize statistical methods to explore how the simulator predicts cellular behavior and outcomes under different physiological conditions. Such methods, known as sensitivity analysis techniques, can provide insight into when a component part of the system (such as a particular cell type, adhesion molecule, or chemokine) begins to have an influence on observed behavior, and quantifies the effect a component part has on the end result: the formation of lymphoid tissue. Through use of such a principled approach in the design, calibration, and analysis of a computer simulation, a robust in silico tool can be developed which can both further the understanding of a biological system being explored, and act as a tool for the generation of hypotheses which can be tested utilizing experimental approaches.

HIV model parameter estimates from interruption trial data including drug efficacy and reservoir dynamics

Mathematical models based on ordinary differential equations (ODE) have had significant impact on understanding HIV disease dynamics and optimizing patient treatment. A model that characterizes the essential disease dynamics can be used for prediction only if the model parameters are identifiable from clinical data. Most previous parameter identification studies for HIV have used sparsely sampled data from the decay phase following the introduction of therapy. In this paper, model parameters are identified from frequently sampled viral-load data taken from ten patients enrolled in the previously published AutoVac HAART interruption study, providing between 69 and 114 viral load measurements from 3-5 phases of viral decay and rebound for each patient. This dataset is considerably larger than those used in previously published parameter estimation studies. Furthermore, the measurements come from two separate experimental conditions, which allows for the direct estimation of drug efficacy and reservoir contribution rates, two parameters that cannot be identified from decay-phase data alone. A Markov-Chain Monte-Carlo method is used to estimate the model parameter values, with initial estimates obtained using nonlinear least-squares methods. The posterior distributions of the parameter estimates are reported and compared for all patients.

Spontaneous clearance of viral infections by mesoscopic fluctuations

Spontaneous disease extinction can occur due to a rare stochastic fluctuation. We explore this process, both numerically and theoretically, in two minimal models of stochastic viral infection dynamics. We propose a method that reduces the complexity in models of viral infections so that the remaining dynamics can be studied by previously developed techniques for analyzing epidemiological models. Using this technique, we obtain an expression for the infection clearance time as a function of kinetic parameters. We apply our theoretical results to study stochastic infection clearance for specific stages of HIV and HCV dynamics. Our results show that the typical time for stochastic clearance of a viral infection increases exponentially with the size of the population, but infection still can be cleared spontaneously within a reasonable time interval in a certain population of cells. We also show that the clearance time is exponentially sensitive to the viral decay rate and viral infectivity but only linearly dependent on the lifetime of an infected cell. This suggests that if standard drug therapy fails to clear an infection then intensifying therapy by adding a drug that reduces the rate of cell infection rather than immune modulators that hasten infected cell death may be more useful in ultimately clearing remaining pockets of infection.

Immediate antiviral therapy appears to restrict resting CD4+ cell HIV-1 infection without accelerating the decay of latent infection

HIV type 1 (HIV-1) persists within resting CD4(+) T cells despite antiretroviral therapy (ART). To better understand the kinetics by which resting cell infection (RCI) is established, we developed a mathematical model that accurately predicts (r = 0.65, P = 2.5 × 10(-4)) the initial frequency of RCI measured about 1 year postinfection, based on the time of ART initiation and the dynamic changes in viremia and CD4(+) T cells. In the largest cohort of patients treated during acute seronegative HIV infection (AHI) in whom RCI has been stringently quantified, we found that early ART reduced the generation of latently infected cells. Although RCI declined after the first year of ART in most acutely infected patients, there was a striking absence of decline when initial RCI frequency was less than 0.5 per million. Notably, low-level viremia was observed more frequently as RCI increased. Together these observations suggest that (i) the degree of RCI is directly related to the availability of CD4(+) T cells susceptible to HIV, whether viremia is controlled by the immune response and/or ART; and (ii) that two pools of infected resting CD4(+) T cells exist, namely, less stable cells, observable in patients in whom viremia is not well controlled in early infection, and extremely stable cells that are established despite early ART. These findings reinforce and extend the concept that new approaches will be needed to eradicate HIV infection, and, in particular, highlight the need to target the extremely small but universal, long-lived latent reservoir.

HIV evolution and progression of the infection to AIDS

In this paper, we propose and discuss a possible mechanism, which, via continuous mutations and evolution, eventually enables HIV to break from immune control. In order to investigate this mechanism, we employ a simple mathematical model, which describes the relationship between evolving HIV and the specific CTL response and explicitly takes into consideration the role of CD4(+)T cells (helper T cells) in the activation of the CTL response. Based on the assumption that HIV evolves towards higher replication rates, we quantitatively analyze the dynamical properties of this model. The model exhibits the existence of two thresholds, defined as the immune activation threshold and the immunodeficiency threshold, which are critical for the activation and persistence of the specific cell-mediated immune response: the specific CTL response can be established and is able to effectively control an infection when the virus replication rate is between these two thresholds. If the replication rate is below the immune activation threshold, then the specific immune response cannot be reliably established due to the shortage of antigen-presenting cells. Besides, the specific immune response cannot be established when the virus replication rate is above the immunodeficiency threshold due to low levels of CD4(+)T cells. The latter case implies the collapse of the immune system and beginning of AIDS. The interval between these two thresholds roughly corresponds to the asymptomatic stage of HIV infection. The model shows that the duration of the asymptomatic stage and progression of the disease are very sensitive to variations in the model parameters. In particularly, the rate of production of the naive lymphocytes appears to be crucial.

Modeling adaptive regulatory T-cell dynamics during early HIV infection

Regulatory T-cells (Tregs) are a subset of CD4(+) T-cells that have been found to suppress the immune response. During HIV viral infection, Treg activity has been observed to have both beneficial and deleterious effects on patient recovery; however, the extent to which this is regulated is poorly understood. We hypothesize that this dichotomy in behavior is attributed to Treg dynamics changing over the course of infection through the proliferation of an 'adaptive' Treg population which targets HIV-specific immune responses. To investigate the role Tregs play in HIV infection, a delay differatial equation model was constructed to examine (1) the possible existence of two distinct Treg populations, normal (nTregs) and adaptive (aTregs), and (2) their respective effects in limiting viral load. Sensitivity analysis was performed to test parameter regimes that show the proportionality of viral load with adaptive regulatory populations and also gave insight into the importance of downregulation of CD4(+) cells by normal Tregs on viral loads. Through the inclusion of Treg populations in the model, a diverse array of viral dynamics was found. Specifically, oscillatory and steady state behaviors were both witnessed and it was seen that the model provided a more accurate depiction of the effector cell population as compared with previous models. Through further studies of adaptive and normal Tregs, improved treatments for HIV can be constructed for patients and the viral mechanisms of infection can be further elucidated.

PREDICTING THE PROBABILITY OF PERSISTENCE OF HIV INFECTION WITH THE STANDARD MODEL

It is not well understood why the transmission of HIV may have a small probability of occurrence despite frequent high risk exposures or ongoing contact between members of a discordant couple. We explore the possible contributions made by distributions of system parameters beginning with the standard three-component differential equation model for the growth of a HIV virion population in an infected host in the absence of drug therapy. The overall dynamical behavior of the model is determined by the set of values of six parameters, some of which describe host immune system properties and others which describe virus properties. There may be one or two critical points whose natures play a key role in determining the outcome of infection and in particular whether the HIV population will persist or become extinct. There are two cases which may arise. In the first case, there is only one critical point P1at biological values and this is an asymptotically stable node. The system ends up with zero virions and so the host becomes HIV-free. In the second case, there are two critical points P1and P2at biological values. Here P1is an unstable saddle point and P2is an asymptotically stable spiral point with a non-zero virion level. In this case the HIV population persists unless parameters change. We let the parameter values take random values from distributions based on empirical data, but suitably truncated, and determine the probabilities of occurrence of the various combinations of critical points. From these simulations the probability that an HIV infection will persist, across a population, is estimated. It is found that with conservatively estimated distributions of parameters, within the framework of the standard 3-component model, the chances that a within-host HIV population will become extinct is between 0.6% and 6.9%. With less conservative parameter estimates, the probability is estimated to be as high as 24%. The many complicating factors related to the transmission and possible spontaneous elimination of the virus and the need for experimental data to clarify whether transient infections may occur are discussed. More realistic yet complicated higher dimensional models are likely to yield smaller probabilities of extinction.