Stochastic modeling of the dynamics of CD4+ T-cell infection by HIV and some Monte Carlo studies

Numerical approaches for the rapid analysis of prophylactic efficacy against HIV with arbitrary drug-dosing schemes

Pre-exposure prophylaxis (PrEP) is an important pillar to prevent HIV transmission. Because of experimental and clinical shortcomings, mathematical models that integrate pharmacological, viral- and host factors are frequently used to quantify clinical efficacy of PrEP. Stochastic simulations of these models provides sample statistics from which the clinical efficacy is approximated. However, many stochastic simulations are needed to reduce the associated sampling error. To remedy the shortcomings of stochastic simulation, we developed a numerical method that allows predicting the efficacy of arbitrary prophylactic regimen directly from a viral dynamics model, without sampling. We apply the method to various hypothetical dolutegravir (DTG) prophylaxis scenarios. The approach is verified against state-of-the-art stochastic simulation. While the method is more accurate than stochastic simulation, it is superior in terms of computational performance. For example, a continuous 6-month prophylactic profile is computed within a few seconds on a laptop computer. The method’s computational performance, therefore, substantially expands the horizon of feasible analysis in the context of PrEP, and possibly other applications.

Pre-exposure prophylaxis (PrEP) is an important tool to prevent HIV transmission. However, experimental identification of parameters that determine prophylactic efficacy is extremely difficult. Clues about these parameters could prove essential for the design of next-generation PrEP compounds. Integrative mathematical models can fill this void: Based on stochastic simulation, a sample statistic can be generated, from which the prophylactic efficacy is estimated. However, for this sample statistic to be accurate, many simulations need to be performed.

Here, we introduce a numerical method to directly compute the prophylactic efficacy from a viral dynamics model, without the need for sampling. Based on several examples with dolutegravir (DTG) -based short- and long-term PrEP, as well as post-exposure prophylaxis we demonstrate the correctness of the new method and its outstanding computational performance. Due to the method’s computational performance, a number of analyses, including formal sensitivity analysis, are becoming feasible with the proposed method.

The Utility of Efavirenz-based Prophylaxis Against HIV Infection. A Systems Pharmacological Analysis

Pre-exposure prophylaxis (PrEP) is considered one of the five “pillars” by UNAIDS to reduce HIV transmission. Moreover, it is a tool for female self-protection against HIV, making it highly relevant to sub-Saharan regions, where women have the highest infection burden. To date, Truvada is the only medication for PrEP. However, the cost of Truvada limits its uptake in resource-constrained countries. Similarly, several currently investigated, patent-protected compounds may be unaffordable in these regions. We set out to explore the potential of the patent-expired antiviral efavirenz (EFV) as a cost-efficient PrEP alternative. A population pharmacokinetic model utilizing data from the ENCORE1 study was developed. The model was refined for metabolic autoinduction. We then explored EFV cellular uptake mechanisms, finding that it is largely determined by plasma protein binding. Next, we predicted the prophylactic efficacy of various EFV dosing schemes after exposure to HIV using a stochastic simulation framework. We predicted that plasma concentrations of 11, 36, 1287 and 1486ng/mL prevent 90% sexual transmissions with wild type and Y181C, K103N and G190S mutants, respectively. Trough concentrations achieved after 600 mg once daily dosing (median: 2017 ng/mL, 95% CI:445–9830) and after reduced dose (400 mg) efavirenz (median: 1349ng/mL, 95% CI: 297–6553) provided complete protection against wild-type virus and the Y181C mutant, and median trough concentrations provided about 90% protection against the K103N and G190S mutants. As reduced dose EFV has a lower toxicity profile, we predicted the reduction in HIV infection when 400 mg EFV-PrEP was poorly adhered to, when it was taken “on demand” and as post-exposure prophylaxis (PEP). Once daily EFV-PrEP provided 99% protection against wild-type virus, if ≥50% of doses were taken. PrEP “on demand” provided complete protection against wild-type virus and prevented ≥81% infections in the mutants. PEP could prevent >98% infection with susceptible virus when initiated within 24 h after virus exposure and continued for at least 9 days. We predict that 400 mg oral EFV may provide superior protection against wild-type HIV. However, further studies are warranted to evaluate EFV as a cost-efficient alternative to Truvada. Predicted prophylactic concentrations may guide release kinetics of EFV long-acting formulations for clinical trial design.

Mechanistic framework predicts drug-class specific utility of antiretrovirals for HIV prophylaxis

Currently, there is no effective vaccine to halt HIV transmission. However, pre-exposure prophylaxis (PrEP) with the drug combination Truvada can substantially decrease HIV transmission in individuals at risk. Despite its benefits, Truvada-based PrEP is expensive and needs to be taken once-daily, which often leads to inadequate adherence and incomplete protection. These deficits may be overcome by next-generation PrEP regimen, including currently investigated long-acting formulations, or patent-expired drugs. However, poor translatability of animal- and ex vivo/in vitro experiments, and the necessity to conduct long-term (several years) human trials involving considerable sample sizes (N>1000 individuals) are major obstacles to rationalize drug-candidate selection. We developed a prophylaxis modelling tool that mechanistically considers the mode-of-action of all available drugs. We used the tool to screen antivirals for their prophylactic utility and identify lower bound effective concentrations that can guide dose selection in PrEP trials. While in vitro measurable drug potency usually guides PrEP trial design, we found that it may over-predict PrEP potency for all drug classes except reverse transcriptase inhibitors. While most drugs displayed graded concentration-prophylaxis profiles, protease inhibitors tended to switch between none- and complete protection. While several treatment-approved drugs could be ruled out as PrEP candidates based on lack-of-prophylactic efficacy, darunavir, efavirenz, nevirapine, etravirine and rilpivirine could more potently prevent infection than existing PrEP regimen (Truvada). Notably, some drugs from this candidate set are patent-expired and currently neglected for PrEP repurposing. A next step is to further trim this candidate set by ruling out compounds with ominous safety profiles, to assess different administration schemes in silico and to test the remaining candidates in human trials.

Pre-exposure prophylaxis (PrEP) is a novel, promising strategy to halt HIV transmission. PrEP with Truvada can substantially decrease the risk of infection. However, individuals often inadequately adhere to the once-daily regimen and the drug is expensive. These shortcomings may be overcome by next-generation PrEP compounds, including long-acting formulations. However, poor translatability of animal- and ex vivo/in vitro experiments, and difficulties in conducting long-term trials involving considerable sample sizes (N > 1000 individuals) make drug-candidate selection and optimization of administration schemes costly and often infeasible. We developed a simulation tool that mechanistically considers the mode-of-action of all antivirals. We used the tool to screen all available antivirals for their prophylactic utility and identified lower bound effective concentrations for designing PrEP dosing regimen in clinical trials. We found that in vitro measured drug potency may over-predict PrEP potency, for all antiviral classes except reverse transcriptase inhibitors. We could rule out a number of antivirals for PrEP repurposing and predicted that darunavir, efavirenz, nevirapine, etravirine and rilpivirine provide complete protection at clinically relevant concentrations. Further trimming of this candidate set by compound-safety and by assessing different implementation schemes is envisaged.

Early HIV infection predictions: role of viral replication errors

In order to prevent and/or control infections it is necessary to understand their early-time dynamics. However this is precisely the phase of HIV about which the least is known. To investigate the initial stages of HIV infection within a host we have developed a multi-type, continuous-time branching process model. This model is a stochastic extension of the standard viral dynamics model, under the assumption that the number of cell targets for viral infection is constant, biologically reasonable since, during the earliest stages of HIV infection, very few cells are infected relative to their total population size. We use our model to investigate three important clinical characteristics of early HIV infection following intravenous challenge: risk of infection, time to infection clearance (assuming failed infection), and time to infection detection. Our focus is on the impact of errors in viral replication that result in non-infectious virus production on these characteristics. Only a small fraction of circulating virus in any chronically infected individual is capable of infecting susceptible cells: estimates range from 1/10⁴ - 1/10³. Characterization and quantification of the processes by which virus becomes defective remains incomplete. We consider two mechanisms that result in defective virus: (1) Copying errors, i.e., lethal errors in reverse transcription, which introduce mutations into the HIV-1 proviral genome, some of which may cripple the viral genome produced, and (2) Packaging errors, i.e., errors during viral packaging, at the end of the viral replication cycle, which cause defective virus by packaging new virions without, for example, viral RNA or key proteins required for infectivity. We show that assumptions on mechanisms of defective virus production can significantly impact early HIV infection model predictions. For example, the risk of infection is orders of magnitude higher if all defective virus is associated with packaging errors, but infection is predicted to be detectable sooner following HIV exposure if all defective virus is associated with copying errors. Thus, in order to make reliable predictions of risk, clearance time, and detection time, better characterization of viral replication is required.

Hybrid stochastic framework predicts efficacy of prophylaxis against HIV: An example with different dolutegravir prophylaxis schemes

To achieve the 90-90-90 goals set by UNAIDS, the number of new HIV infections needs to decrease to approximately 500,000 by 2020. One of the 'five pillars' to achieve this goal is pre-exposure prophylaxis (PrEP). Truvada (emtricitabine-tenofovir) is currently the only medication approved for PrEP. Despite its advantages, Truvada is costly and requires individuals to adhere to the once-daily regimen. To improve PrEP, many next-generation regimen, including long-acting formulations, are currently investigated. However, pre-clinical testing may not guide candidate selection, since it often fails to translate into clinical efficacy. On the other hand, quantifying prophylactic efficacy in the clinic is ethically problematic and requires to conduct long (years) and large (N>1000 individuals) trials, precluding systematic evaluation of candidates and deployment strategies. To prioritize- and help design PrEP regimen, tools are urgently needed that integrate pharmacological-, viral- and host factors determining prophylactic efficacy. Integrating the aforementioned factors, we developed an efficient and exact stochastic simulation approach to predict prophylactic efficacy, as an example for dolutegravir (DTG). Combining the population pharmacokinetics of DTG with the stochastic framework, we predicted that plasma concentrations of 145.18 and 722.23nM prevent 50- and 90% sexual transmissions respectively. We then predicted the reduction in HIV infection when DTG was used in PrEP, PrEP 'on demand' and post-exposure prophylaxis (PEP) before/after virus exposure. Once daily PrEP with 50mg oral DTG prevented 99-100% infections, and 85% of infections when 50% of dosing events were missed. PrEP 'on demand' prevented 79-84% infections and PEP >80% when initiated within 6 hours after virus exposure and continued for as long as possible. While the simulation framework can easily be adapted to other PrEP candidates, our simulations indicated that oral 50mg DTG is non-inferior to Truvada. Moreover, the predicted 90% preventive concentrations can guide release kinetics of currently developed DTG nano-formulations.

Mathematical analysis and dynamic active subspaces for a long term model of HIV

a long-term model of HIV infection dynamics [8] was developed to describe the entire time course of the disease. It consists of a large system of ODEs with many parameters, and is expensive to simulate. In the current paper, this model is analyzed by determining all infection-free steady states and studying the local stability properties of the unique biologically-relevant equilibrium. Active subspace methods are then used to perform a global sensitivity analysis and study the dependence of an infected individual's T-cell count on the parameter space. Building on these results, a global-in-time approximation of the T-cell count is created by constructing dynamic active subspaces and reduced order models are generated, thereby allowing for inexpensive computation.

Accurate predictions of population-level changes in sequence and structural properties of HIV-1 Env using a volatility-controlled diffusion model

The envelope glycoproteins (Envs) of HIV-1 continuously evolve in the host by random mutations and recombination events. The resulting diversity of Env variants circulating in the population and their continuing diversification process limit the efficacy of AIDS vaccines. We examined the historic changes in Env sequence and structural features (measured by integrity of epitopes on the Env trimer) in a geographically defined population in the United States. As expected, many Env features were relatively conserved during the 1980s. From this state, some features diversified whereas others remained conserved across the years. We sought to identify “clues” to predict the observed historic diversification patterns. Comparison of viruses that cocirculate in patients at any given time revealed that each feature of Env (sequence or structural) exists at a defined level of variance. The in-host variance of each feature is highly conserved among individuals but can vary between different HIV-1 clades. We designate this property “volatility” and apply it to model evolution of features as a linear diffusion process that progresses with increasing genetic distance. Volatilities of different features are highly correlated with their divergence in longitudinally monitored patients. Volatilities of features also correlate highly with their population-level diversification. Using volatility indices measured from a small number of patient samples, we accurately predict the population diversity that developed for each feature over the course of 30 years. Amino acid variants that evolved at key antigenic sites are also predicted well. Therefore, small “fluctuations” in feature values measured in isolated patient samples accurately describe their potential for population-level diversification. These tools will likely contribute to the design of population-targeted AIDS vaccines by effectively capturing the diversity of currently circulating strains and addressing properties of variants expected to appear in the future.

HIV-1 is the causative agent of the global AIDS pandemic. The envelope glycoproteins (Envs) of HIV-1 constitute a primary target for antibody-based vaccines. However, the diversity of Envs in the population limits the potential efficacy of this approach. Accurate estimates of the range of variants that currently infect patients and those expected to appear in the future will likely contribute to the design of population-targeted immunogens. We found that different properties (features) of Env have different propensities for small “fluctuations” in their values among viruses that infect patients at any given time point. This propensity of each feature for in-host variance, which we designate “volatility”, is conserved among patients. We apply this parameter to model the evolution of features (in patients and population) as a diffusion process driven by their “diffusion coefficients” (volatilities). Using volatilities measured from a few patient samples from the 1980s, we accurately predict properties of viruses that evolved in the population over the course of 30 years. The diffusion-based model described here efficiently captures evolution of phenotypes in biological systems controlled by a dominant random component.

On the extinction probability in models of within-host infection: the role of latency and immunity

Not every exposure to virus establishes infection in the host; instead, the small amount of initial virus could become extinct due to stochastic events. Different diseases and routes of transmission have a different average number of exposures required to establish an infection. Furthermore, the host immune response and antiviral treatment affect not only the time course of the viral load provided infection occurs, but can prevent infection altogether by increasing the extinction probability. We show that the extinction probability when there is a time-dependent immune response depends on the chosen form of the model-specifically, on the presence or absence of a delay between infection of a cell and production of virus, and the distribution of latent and infectious periods of an infected cell. We hypothesise that experimentally measuring the extinction probability when the virus is introduced at different stages of the immune response, alongside the viral load which is usually measured, will improve parameter estimates and determine the most suitable mathematical form of the model.

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.

Modeling antiretroviral drug responses for HIV-1 infected patients using differential equation models

We review mathematical modeling and related statistical issues of HIV dynamics primarily in response to antiretroviral drug therapy in this article. We start from a basic model of virus infection and then review a number of more advanced models with consideration of pharmacokinetic factors, adherence and drug resistance. Specifically, we illustrate how mathematical models can be developed and parameterized to understand the effects of long-term treatment and different treatment strategies on disease progression. In addition, we discuss a variety of parameter estimation methods for differential equation models that are applicable to either within- or between-host viral dynamics.

Modeling the three stages in HIV infection

A typical HIV infection response consists of three stages: an initial acute infection, a long asymptomatic period and a final increase in viral load with simultaneous collapse in healthy CD4+T cell counts. The majority of existing mathematical models give a good representation of either the first two stages or the last stage of the infection. Using macrophages as a long-term active reservoir, a deterministic model is proposed to explain the three stages of the infection including the progression to AIDS. Simulation results illustrate how chronic infected macrophages can explain the progression to AIDS provoking viral explosion. Further simulation studies suggest that the proposed model retains its key properties even under moderately large parameter variations. This model provides important insights on how macrophages might play a crucial role in the long term behavior of HIV infection.

Excitability in the host-pathogen interactions of HIV infection and emergence of viral load blips

HIV viral blips are characterized by intermittent episodes of detectable low-level viraemia which return spontaneously to an undetectable level in patients with full suppression of viraemia (<50 copies/ml). The precise mechanisms responsible for viraemia blips and their clinical significance are not known. In this work, we analyze HIV blips using a mathematical model describing basic host-pathogen interactions, in particular regulatory processes involving CD4+, CD8+ T-cells and the virus. We show that under adequate conditions, this interaction system can be excitable and small perturbations of the system by external stimuli can generate robust viral load (VL) blips of regular or irregular frequency and peak amplitudes. Importantly, our analysis showed that direct perturbations of the viral load (by latent reservoirs or opportunistic diseases for example) more efficiently trigger VL blips on contrary to direct perturbations of the immune system, in particular the levels of uninfected CD4+ and cytotoxic CD8+ T-cells. This feature is shown to rely on specific stability properties in this interaction system. Our analysis moreover suggests that blips should be of low clinical significance since any other VL or immune system perturbations could trigger transient viraemia under adequate excitability conditions.

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.

Basic stochastic models for viral infection within a host

Stochastic differential equation (SDE) models are formulated for intra-host virus-cell dynamics during the early stages of viral infection, prior to activation of the immune system. The SDE models incorporate more realism into the mechanisms for viral entry and release than ordinary differential equation (ODE) models and show distinct differences from the ODE models. The variability in the SDE models depends on the concentration, with much greater variability for small concentrations than large concentrations. In addition, the SDE models show significant variability in the timing of the viral peak. The viral peak is earlier for viruses that are released from infected cells via bursting rather than via budding from the cell membrane.

Pharmacokinetics and pharmacodynamics of the reverse transcriptase inhibitor tenofovir and prophylactic efficacy against HIV-1 infection

Antiviral pre-exposure prophylaxis (PrEP) through daily drug administration can protect healthy individuals from HIV-1 infection. While PrEP was recently approved by the FDA, the potential long-term consequences of PrEP implementation remain entirely unclear. The aim of this study is to predict the efficacy of different prophylactic strategies with the pro-drug tenofovir-disoproxil-fumarate (TDF) and to assess the sensitivity towards timing- and mode of TDF administration (daily- vs. single dose), adherence and the number of transmitted viruses. We developed a pharmacokinetic model for TDF and its active anabolite tenofovir-diphosphate (TFV-DP) and validated it with data from 4 different trials, including 4 distinct dosing regimes. Pharmacokinetics were coupled to an HIV model and viral decay following TDF mono-therapy was predicted, consistent with available data. Subsequently, a stochastic approach was used to estimate the % infections prevented by (i) daily TDF-based PrEP, (ii) one week TDF started either shortly before, or -after viral exposure and (iii) a single dose oral TDF before viral challenge (sd-PrEP). Analytical solutions were derived to assess the relation between intracellular TFV-DP concentrations and prophylactic efficacy. The predicted efficacy of TDF was limited by a slow accumulation of active compound (TFV-DP) and variable TFV-DP half-life and decreased with increasing numbers of transmitted viruses. Once daily TDF-based PrEP yielded [Formula: see text]80% protection, if at least 40% of pills were taken. Sd-PrEP with 300 mg or 600 mg TDF could prevent [Formula: see text]50% infections, when given at least before virus exposure. The efficacy dropped to [Formula: see text]10%, when given 1 h before 24 h exposure. Efficacy could not be increased with increasing dosage or prolonged administration. Post-exposure prophylaxis poorly prevented infection. The use of drugs that accumulate more rapidly, or local application of tenofovir gel may overcome the need for drug administration long before virus exposure.

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.

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.

Within-host demographic fluctuations and correlations in early retroviral infection

In this paper we analyze the demographic fluctuations and correlations present in within-host populations of viruses and their target cells during the early stages of infection. In particular, we present an exact treatment of a discrete-population, stochastic, continuous-time master equation description of HIV or similar retroviral infection dynamics, employing Monte Carlo simulations. The results of calculations employing Gillespie's direct method clearly demonstrate the importance of considering the microscopic details of the interactions which constitute the macroscopic dynamics. We then employ the τ-leaping approach to study the statistical characteristics of infections involving realistic absolute numbers of within-host viral and cellular populations, before going on to investigate the effect that initial viral population size plays on these characteristics. Our main conclusion is that cross-correlations between infected cell and virion populations alter dramatically over the course of the infection. We suggest that these statistical correlations offer a novel and robust signature for the acute phase of retroviral infection.

A SEMI-STOCHASTIC MODEL FOR HIV POPULATION DYNAMICS

A semi-stochastic model is developed and investigated for human immunodeficiency virus type-1 (HIV-1) population dynamics. The model includes both stochastic parts (changes of CD4+ T cells) and deterministic parts (changes of free virions). Using the best currently available parameter values, we estimate the distributions of the time of occurrence and the magnitude of the early peak in virions. We investigated the effects of varying parameter values on mean solutions in order to assess the stochastic effects of between-patient variability. Numerical simulation shows that the lower the infection rate, the higher the death rate of the infected cells, more rapid clearance of virions and lower rate of virion emission by the infected cells result in lower speed of infection progression and magnitude but a greater variability in response. We also examine the probability that a small viral inoculum fails to establish an infection. Further, we theoretically quantify the expected variability around the infected equilibrium for each population by using diffusion approximation.

Stochastic models for virus and immune system dynamics

New stochastic models are developed for the dynamics of a viral infection and an immune response during the early stages of infection. The stochastic models are derived based on the dynamics of deterministic models. The simplest deterministic model is a well-known system of ordinary differential equations which consists of three populations: uninfected cells, actively infected cells, and virus particles. This basic model is extended to include some factors of the immune response related to Human Immunodeficiency Virus-1 (HIV-1) infection. For the deterministic models, the basic reproduction number, R0, is calculated and it is shown that if R0<1, the disease-free equilibrium is locally asymptotically stable and is globally asymptotically stable in some special cases. The new stochastic models are systems of stochastic differential equations (SDEs) and continuous-time Markov chain (CTMC) models that account for the variability in cellular reproduction and death, the infection process, the immune system activation, and viral reproduction. Two viral release strategies are considered: budding and bursting. The CTMC model is used to estimate the probability of virus extinction during the early stages of infection. Numerical simulations are carried out using parameter values applicable to HIV-1 dynamics. The stochastic models provide new insights, distinct from the basic deterministic models. For the case R0>1, the deterministic models predict the viral infection persists in the host. But for the stochastic models, there is a positive probability of viral extinction. It is shown that the probability of a successful invasion depends on the initial viral dose, whether the immune system is activated, and whether the release strategy is bursting or budding.

Using modeling to help understand vaginal microbicide functionality and create better products

A summary is presented of a range of mathematical models that relate to topical microbicidal molecules, applied vaginally to inhibit HIV transmission. These models contribute to the fundamental understanding of the functioning of those molecules, as introduced in different delivery systems. They also provide computational tools that can be employed in the practical design and evaluation of vaginal microbicide products. Mathematical modeling can be implemented, using stochastic principles, to understand the probability of infection by sexually transmitted HIV virions. This provides a frame of reference for the deterministic models of the various processes that underlie HIV transmission and its inhibition, including: the temporal and spatial history of HIV migration from semen to vaginal epithelial surfaces and thence to the underlying stroma; the time and spatial distribution of microbicidal drugs as delivered by various vehicles (e.g., gels, rings, films, and tablets)-this is central to understanding microbicide product pharmacokinetics; and the time and space history of the drug interactions with HIV directly and with host cells for HIV within the vaginal environment-this informs the understanding of microbicide pharmacodynamics. Models that characterize microbicide functionality and performance should and can interface with both in vitro and in vivo experimental studies. They can serve as a rapidly applied, inexpensive tool, to facilitate microbicide R&D, in advance of more costly and time consuming clinical trials.

Stochastic theory of early viral infection: continuous versus burst production of virions

Viral production from infected cells can occur continuously or in a burst that generally kills the cell. For HIV infection, both modes of production have been suggested. Standard viral dynamic models formulated as sets of ordinary differential equations can not distinguish between these two modes of viral production, as the predicted dynamics is identical as long as infected cells produce the same total number of virions over their lifespan. Here we show that in stochastic models of viral infection the two modes of viral production yield different early term dynamics. Further, we analytically determine the probability that infections initiated with any number of virions and infected cells reach extinction, the state when both the population of virions and infected cells vanish, and show this too has different solutions for continuous and burst production. We also compute the distributions of times to establish infection as well as the distribution of times to extinction starting from both a single virion as well as from a single infected cell for both modes of virion production.

The dynamics of HIV infection and treatment has been extensively studied using ordinary differential equation models. Recent work on HIV transmission has suggested that most sexually transmitted infections are started by a single virus or infected cell. This observation coupled with the fact that successful HIV transmission only occurs in 1 per 100 to 1 per 1000 coital acts suggests that early events in infection are stochastic. Here we develop a stochastic model of HIV infection and use it to characterize the dynamics of early infection when virus is released from cells either continuously or in a burst. We show that these mechanisms of viral production produce different early dynamics, with different probabilities of extinction and different distributions of time to establish infection. In deterministic models, these modes of viral production are indistinguishable.

Modeling sequence evolution in acute HIV-1 infection

We describe a mathematical model and Monte Carlo (MC) simulation of viral evolution during acute infection. We consider both synchronous and asynchronous processes of viral infection of new target cells. The model enables an assessment of the expected sequence diversity in new HIV-1 infections originating from a single transmitted viral strain, estimation of the most recent common ancestor (MRCA) of the transmitted viral lineage, and estimation of the time to coalesce back to the MRCA. We also calculate the probability of the MRCA being the transmitted virus or an evolved variant. Excluding insertions and deletions, we assume HIV-1 evolves by base substitution without selection pressure during the earliest phase of HIV-1 infection prior to the immune response. Unlike phylogenetic methods that follow a lineage backwards to coalescence, we compare the observed data to a model of the diversification of a viral population forward in time. To illustrate the application of these methods, we provide detailed comparisons of the model and simulations results to 306 envelope sequences obtained from eight newly infected subjects at a single time point. The data from 68 patients were in good agreement with model predictions, and hence compatible with a single-strain infection evolving under no selection pressure. The diversity of the samples from the other two patients was too great to be explained by the model, suggesting multiple HIV-1-strains were transmitted. The model can also be applied to longitudinal patient data to estimate within-host viral evolutionary parameters.

State estimation and optimal long period clinical treatment of HIV seropositive patients

Optimal control theory provides a very interesting quantitative method that can be used to assist the decision making process in several areas of application, such as engineering, biology, economics and sociology. The main idea is to determine the values of the manipulated variables, such as drug doses, so that some cost function is minimized, subject to physical constraints. In this work, the cost function reflects the number of CD4+T cells, viral particles and the drug doses. It is worth noticing that high drug doses are related to more intense side-effects, apart from the impact on the actual cost of the treatment. In a previous paper by the authors, the LQR - Linear Quadratic Regulator approach was proposed for the computation of long period maintenance doses for the drugs, which turns out to be of state feedback form. However, it is not practical to determine all the components of the state vector, due to the fact that infected and uninfected CD4+T cells are not microscopically distinguishable. In order to overcome this difficulty, this work proposes the use of Extended Kalman Filter to estimate the state, even though, because of the nonlinear nature of the involved state equations, the separation principle may not be valid. Extensive simulations were then carried out to investigate numerically if the control strategy consisting of the feedback of estimated states yielded satisfactory clinical results.

The probability of HIV infection in a new host and its reduction with microbicides

We use a simple mathematical model to estimate the probability and its time dependence that one or more HIV virions successfully infect target cells. For the transfer of a given number of virions to target cells we derive expressions for the probability P(inf), of infection. Thus, in the case of needlestick transfer we determine P(inf) and an approximate time window for post-exposure prophylaxis (PEP). For heterosexual transmission, where the transfer process is more complicated, a parameter gamma is employed which measures the strength of the infection process. For the smaller value of gamma, P(inf) is from 6 x 10(-5) to 0.93 or from 7.82 x 10(-6) to 0.29, where the lower figures are for the transfer of 100 virions and the upper figures are for the transfer of 4.4 million virions. We estimate the reductions in P(inf) which occur with a microbicide of a given efficacy. It is found that reductions may be approximately as stated when the number of virions transferred is less than about 10(5), but declines to zero for viral loads above that number. It is concluded that PEP should always be applied immediately after a needlestick incident. Further, manufacturers of microbicides should be encouraged to investigate and report their effectiveness at various transferred viral burdens.

Kin selection and evolution of infectious disease resistance

Discoveries of mutations conferring resistance to infectious diseases have led to increased interest in the evolutionary dynamics of disease resistance. Several recent papers have estimated the historical strength of selection for mutations conferring disease resistance. These studies are based on simple population genetic models that do not take account of factors such as spatial and family structure. Such factors may have a substantial impact on the strength of natural selection through inclusive fitness effects. That is, people have a strong tendency to live with relatives and therefore have a high probability of transmitting infectious diseases to them. Thus, an allele that protects an individual against disease infection also protects that individual's family members. Because some of these family members are likely to also be carrying the allele, selection for that allele is magnified by family structure. In this paper, I use mathematical modeling techniques to explore the impact of such kin selection on the strength of selection for infectious disease resistance alleles. I show that if the resistance allele has the same proportional effect on both within- and between-family transmission, then the impact of kin selection is relatively minor. Selection coefficients are increased by 5-35%, with a greater benefit for weaker alleles. The reason is that an individual with a strong resistance allele does not need much protection from infection by family members and thus does not benefit much from their alleles. The effect of kin selection can be dramatic, however, if the resistance allele has a larger effect on between-family transmission than within-family transmission (which can occur if between-family infection rates are much smaller than within-family rates), increasing selection coefficients by as much as two- to threefold. These results show conditions when it is important to consider family structure in estimates of the strength of selection for infectious disease resistance alleles.

Recombination favors the evolution of drug resistance in HIV-1 during antiretroviral therapy

We studied the relationship between recombination and the fixation time of multiple drug resistance mutations after HIV-1 drug therapy, under a set of different realistic scenarios. We have generalized a previous model by Bretscher et al. [Bretscher, Althaus, Muller, Bonhoeffer, 2004. Recombination in HIV and the evolution of drug resistance: for better or for worse? Bioessays 26(2), 180-188] in order to explore different implementations of phenotypic mixing and more realistic demographic and selective regimes. Using computer simulations we show that the effect of recombination on the evolution of drug resistance depends strongly on the intensity of selection, as well as on the viral population size. Under the high selection pressure expected during antiretroviral therapy, the strength of the Hill-Robertson effect increases and recombination favors the evolution of resistance under a wide range of population sizes, independently of the sign of the epistatic interaction. Our results suggest that recombination plays an important role in the evolution of drug resistance in HIV-1 under various realistic scenarios. These findings could be taken into account in order to develop optimal HIV-1 drug treatments.

Natural variation in HIV infection: Monte Carlo estimates that include CD8 effector cells

Viral load and CD4 T-cell counts in patients infected with the human immunodeficiency virus (HIV) are commonly used to guide clinical decisions regarding drug therapy or to assess therapeutic outcomes in clinical trials. However, random fluctuations in these markers of infection can obscure clinically significant change. We employ a Monte Carlo simulation to investigate contributing factors in the expected variability in CD4 T-cell count and viral load due solely to the stochastic nature of HIV infection. The simulation includes processes that contribute to the variability in HIV infection including CD4 and CD8 T-cell population dynamics as well as T-cell activation and proliferation. The simulation results may reconcile the wide range of variabilities in viral load observed in clinical studies, by quantifying correlations between viral load measurements taken days or weeks apart. The sensitivity of variability in T-cell count and viral load to changes in the lifetimes of CD4 and CD8 T-cells is investigated, as well as the effects of drug therapy.

Monte Carlo estimates of natural variation in HIV infection

We describe a Monte Carlo simulation of the within-host dynamics of human immunodeficiency virus 1 (HIV-1). The simulation proceeds at the level of individual T-cells and virions in a small volume of plasma, thus capturing the inherent stochasticity in viral replication, mutation and T-cell infection. When cell lifetimes are distributed exponentially in the Monte Carlo approach, our simulation results are in perfect agreement with the predictions of the corresponding systems of differential equations from the literature. The Monte Carlo model, however, uniquely allows us to estimate the natural variability in important parameters such as the T-cell count, viral load, and the basic reproductive ratio, in both the presence and absence of drug therapy. The simulation also yields the probability that an infection will not become established after exposure to a viral inoculum of a given size. Finally, we extend the Monte Carlo approach to include distributions of cell lifetimes that are less-dispersed than exponential.

Evaluation of HIV-1 kinetic models using quantitative discrimination analysis

MOTIVATION

Since the identification of human immunodeficiency virus (HIV) over twenty years ago, many mathematical models of HIV dynamics have been proposed. The purpose of this study was to evaluate intracellular and intercellular scale HIV models that best described the dynamics of viral and cell titers of a person, where parameters were determined using typically available patient data. In this case, 'best' was defined as the model most capable of describing experimental patient data and was determined by Bayesian-based model discrimination analysis and the ability to provide realistic results.

RESULTS

Twenty models of HIV-1 viral dynamics were initially evaluated to determine whether parameters could be obtained from readily available clinical data from established HIV-1 patients with stable disease. Based on this analysis, three models were chosen for further examination and comparison. Parameters were estimated using experimental data from a cohort of 338 people monitored for up to 2484 days. The models were evaluated using a Bayesian technique to determine which model was most probable. The model ultimately selected as most probable was overwhelmingly favored relative to the remaining two models, and it accounted for uninfected cells, infected cells and cytotoxic T lymphocyte dynamics. The authors developed a fourth model for comparison purposes by combining the features of the original three models. Parameters were estimated for the new model and the statistical analysis was repeated for all four models. The model that was initially favored was selected again upon model discrimination analysis.

CONTACT

srivasta@engr.uconn.edu.

A parameter sensitivity methodology in the context of HIV delay equation models

A sensitivity methodology for nonlinear delay systems arising in one class of cellular HIV infection models is presented. Theoretical foundations for a typical sensitivity investigation and illustrative computations are given.

On the behavior of solutions in viral dynamical models

We consider simple mathematical models for the early population dynamics of the human immunodefficiency type 1 virus (HIV-1). Although these systems of differential equations may be solved by numerical methods, few general theoretical results are available due to nonlinearities. We analyze a model whose components are plasma densities of uninfected CD4+ T-cells and infected cells (assumed in this model to be proportional to virion density). In addition to analyzing the nature of the equilibrium points, we show that there are no periodic or limit-cycle solutions. Depending on the values of the parameters, solutions either tend without oscillation to an equilibrium point with zero virion density or to an equilibrium point in which there are a nonzero number of virions. In the latter case the approach to equilibrium may be through damped oscillations or without oscillation.

When to initiate HIV therapy: a control theoretic approach

This paper shows an application of control theory to human immunodeficiency virus (HIV)/AIDS models. Minimum singular value decomposition is applied to HIV/AIDS models to measure the extent to which the different stages in the progression of HIV/AIDS disease are controllable and, consequently, when best to initiate therapy such that the general objectives of therapy are satisfied. Simulations will be used to demonstrate the effect of treatment at various stages. Comparisons will be made between mono-class and combination therapies and between when therapy is initiated at the acute infection, asymptomatic and the advanced stages.

Modeling HIV dynamics and antiviral response with consideration of time-varying drug exposures, adherence and phenotypic sensitivity

Highly active antiretroviral therapies consisting of reverse transcriptase inhibitor drugs and protease inhibitor drugs, which can rapidly suppress HIV below the limit of detection, are currently the most effective treatment for HIV infected patients. In spite of this, many patients fail to achieve viral suppression, probably due to existing or developing drug resistance, poor adherence, pharmacokinetic problems and other clinical factors. In this paper, we develop a viral dynamic model to evaluate how time-varying drug exposure and drug susceptibility affect antiviral response. Plasma concentrations, in turn, are modeled using a standard pharmacokinetic (PK) one-compartment open model with first order absorption and elimination as a function of fixed individual PK parameters and dose times. Imperfect adherence is considered as missed doses in PK models. We discuss the analytic properties of the viral dynamic model and study how time-varying treatment efficacies affect antiviral responses, specifically viral load and T cell counts. The relationship between actual failure time (the time at which the viral growth rate changes from negative to positive) and detectable failure time (the time at which viral load rebounds to above the limit of detection) is investigated. We find that an approximately linear relationship can be used to estimate the actual rebound failure time from the detectable rebound failure time. In addition, the effect of adherence on antiviral response is studied. In particular, we examine how different patterns of adherence affect antiviral response. Results suggest that longer sequences of missed doses increase the chance of treatment failure and accelerate the failure. Simulation experiments are presented to illustrate the relationship between antiviral response and pharmacokinetics, time-varying adherence and drug resistance. The proposed models and methods may be useful in AIDS clinical trial simulations.

Incorporation of variability into the modeling of viral delays in HIV infection dynamics

We consider classes of functional differential equation models which arise in attempts to describe temporal delays in HIV pathogenesis. In particular, we develop methods for incorporating arbitrary variability (i.e., general probability distributions) for these delays into systems that cannot readily be reduced to a finite number of coupled ordinary differential equations (as is done in the method of stages). We discuss modeling from first principles, introduce several classes of non-linear models (including discrete and distributed delays) and present a discussion of theoretical and computational approaches. We then use the resulting methodology to carry out simulations and perform parameter estimation calculations, fitting the models to a set of experimental data. Results obtained confirm the statistical significance of the presence of delays and the importance of including delays in validating mathematical models with experimental data. We also show that the models are quite sensitive to the mean of the distribution which describes the delay in viral production, whereas the variance of this distribution has relatively little impact.

Short and long period optimization of drug doses in the treatment of AIDS

Numerical optimization techniques are useful in solving problems of computing the best inputs for systems described by mathematical models and when the objectives can be stated in a quantitative form. This work concerns the problem of optimizing the drug doses in the treatment of AIDS in terms of achieving a balance between the therapeutic response and the side effects. A mathematical model describing the dynamics of HIV viruses and CD4 cells is used to compute the short term optimal drug doses in the treatments of patients with AIDS by a direct method of optimization using a cost function of Bolza type. The model parameters were fitted to actual published clinical data. In order to simplify the numerical procedures, the control law is expressed as a series and the sub-optimal control is obtained by truncating the higher terms. When the patient reaches a clinically satisfactory state, the LQR - Linear Quadratic Regulator technique is used to determine the long period maintenance doses for the drugs. The doses computed using the LQR technique tend to be smaller than equivalent constant-dose therapy in terms of increase in the counts of CD4+T cells and reduction of the density of free viruses.

Improving the fit of bivariate smoothing splines when estimating longitudinal immunological and virological markers in HIV patients with individual antiretroviral treatment strategies

CD4+ lymphocyte count and HIV RNA plasma viral load are longitudinally monitored in patients with HIV infection. Because data collection intervals may be unequally spaced and these markers experience high within-patient variability, they may be smoothed before use in subsequent models. Estimation strategies must be able to accommodate the drastic changes in viral load which may occur when an individual's treatment strategy is updated. Because these treatment changes are not regimented, these dynamics cannot be modelled using standard methods. We propose univariate and bivariate cubic smoothing splines to fit CD4+ count and viral load over time. The method is developed using state space equations, and the Kalman filter is used to calculate the log-likelihood. Non-linear optimization is used to obtain the maximum likelihood estimates. A modification of the Kalman filter allows non-informative or diffuse priors at the initial observation. Since treatment changes are expected to alter the shape of the curve, we further extend the Kalman filter to permit greater flexibility in the smoothing spline at these time points. The method produces smoothed estimates of the viral load and CD4+ count curves over time.

A stochastic modeling of early HIV-1 population dynamics

In a recent paper, Tuckwell and Le Corfec [J. Theor. Biol. 195 (1998) 450-463] applied the multi-dimensional diffusion process to model early human immunodeficiency virus type-1 (HIV-1) population dynamics. The purpose of this paper is to assess certain features and consequences of their model in the context of Tan and Wu's stochastic approach [Math. Biosci. 147 (1998) 173-205].

A competition model for viral inhibition of host cell proliferation

Some viruses encode proteins that promote cell proliferation while others, such as the human immunodeficiency virus (HIV), encode proteins that prevent cell division. It has been hypothesized that the selective advantage determining which strategy evolves depends on the ability of the virus to induce a cellular environment which maximizes both virus production and cell life span. In HIV, the protein that causes cell cycle arrest is Vpr. In this paper, we develop a mathematical model, based on difference equations, to study the competition between two genotypes of HIV - one that encodes this protein (Vpr+) and one that does not (Vpr-). In particular, we are interested in parameters that could be different between the in vitro condition, where the Vpr- genotype dominates, and the in vivo condition, where the Vpr+ genotype dominates. Our model indicates that the infected cell death-rate, the viral half-life, and the dynamics of the target cell population all effect the competition dynamics between the Vpr+ and Vpr- viral genotypes. Perturbing any of these parameters from the in vitro estimates while holding the others fixed has no affect on the competition outcome, i. e., the Vpr- genotype dominates. Perturbing the infected cell death-rate and the target cell source causes a switch in competitive outcome, although not necessarily at values, which represent the in vivo condition. Adding a perturbation in the viral half-life from in vitro to in vivo condition results in a switch of the competitive advantage from the Vpr- genotype to the Vpr+ genotype with parameters for all three mechanisms set to estimated in vivo values.

Population HIV-1 dynamics in vivo: applicable models and inferential tools for virological data from AIDS clinical trials

In this paper, we introduce a novel application of hierarchical nonlinear mixed-effect models to HIV dynamics. We show that a simple model with a sum of exponentials can give a good fit to the observed clinical data of HIV-1 dynamics (HIV-1 RNA copies) after initiation of potent antiviral treatments and can also be justified by a biological compartment model for the interaction between HIV and its host cells. This kind of model enjoys both biological interpretability and mathematical simplicity after reparameterization and simplification. A model simplification procedure is proposed and illustrated through examples. We interpret and justify various simplified models based on clinical data taken during different phases of viral dynamics during antiviral treatments. We suggest the hierarchical nonlinear mixed-effect model approach for parameter estimation and other statistical inferences. In the context of an AIDS clinical trial involving patients treated with a combination of potent antiviral agents, we show how the models may be used to draw biologically relevant interpretations from repeated HIV-1 RNA measurements and demonstrate the potential use of the models in clinical decision-making.

Some state space models of HIV pathogenesis under treatment by anti-viral drugs in HIV-infected individuals

In this paper we have extended the model of HIV pathogenesis under treatment by anti-viral drugs given by Perelson et al. [A.S. Perelson et al., Science 271 (1999) 1582] to a stochastic model. By using this stochastic model as the stochastic system model, we have developed a state space model for the HIV pathogenesis under treatment by anti-viral drugs. In this state space model, the observation model is a statistical model based on the observed numbers of RNA virus copies over different times. For this model we have developed procedures for estimating and predicting the numbers of infectious free HIV and non-infectious free HIV as well as the numbers of different types of T cells through extended Kalman filter method. As an illustration, we have applied the method of this paper to the data of patient Nos. 104, 105 and 107 given by Perelson et al. [A.S. Perelson et al., Science 271 (1999) 1582] under treatment by Ritonavir. For these individuals, it is shown that within two weeks since treatment, most of the free HIV are non-infectious, indicating the usefulness of the treatment. Furthermore, the Kalman filter method revealed a much stronger effect of the treatment within the first 10 to 20 h than that predicted by the deterministic model.