Complex patterns of viral load decay under antiretroviral therapy: influence of pharmacokinetics and intracellular delay

SAAM II: A general mathematical modeling rapid prototyping environment

Simulation Analysis and Modeling II (SAAM II) is a graphical modeling software used in life sciences for compartmental model analysis, particularly, but not exclusively, appreciated in pharmacokinetics (PK) and pharmacodynamics (PD), metabolism, and tracer modeling. Its intuitive "circles and arrows" visuals allow users to easily build, solve, and fit compartmental models without the need for coding. It is suitable for rapid prototyping of models for complex kinetic analysis or PK/PD problems, and in educating students and non-modelers. Although it is straightforward in design, SAAM II incorporates sophisticated algorithms programmed in C to address ordinary differential equations, deal with complex systems via forcing functions, conduct multivariable regression featuring the Bayesian maximum a posteriori, perform identifiability and sensitivity analyses, and offer reporting functionalities, all within a single package. After 26 years from the last SAAM II tutorial paper, we demonstrate here SAAM II's updated applicability to current life sciences challenges. We review its features and present four contemporary case studies, including examples in target-mediated PK/PD, CAR-T-cell therapy, viral dynamics, and transmission models in epidemiology. Through such examples, we demonstrate that SAAM II provides a suitable interface for rapid model selection and prototyping. By enabling the fast creation of detailed mathematical models, SAAM II addresses a unique requirement within the mathematical modeling community.

Spatial information allows inference of the prevalence of direct cell-to-cell viral infection

The role of direct cell-to-cell spread in viral infections-where virions spread between host and susceptible cells without needing to be secreted into the extracellular environment-has come to be understood as essential to the dynamics of medically significant viruses like hepatitis C and influenza. Recent work in both the experimental and mathematical modelling literature has attempted to quantify the prevalence of cell-to-cell infection compared to the conventional free virus route using a variety of methods and experimental data. However, estimates are subject to significant uncertainty and moreover rely on data collected by inhibiting one mode of infection by either chemical or physical factors, which may influence the other mode of infection to an extent which is difficult to quantify. In this work, we conduct a simulation-estimation study to probe the practical identifiability of the proportion of cell-to-cell infection, using two standard mathematical models and synthetic data that would likely be realistic to obtain in the laboratory. We show that this quantity cannot be estimated using non-spatial data alone, and that the collection of data which describes the spatial structure of the infection is necessary to infer the proportion of cell-to-cell infection. Our results provide guidance for the design of relevant experiments and mathematical tools for accurately inferring the prevalence of cell-to-cell infection in in vitro and in vivo contexts.

Incorporating Intracellular Processes in Virus Dynamics Models

In-host models have been essential for understanding the dynamics of virus infection inside an infected individual. When used together with biological data, they provide insight into viral life cycle, intracellular and cellular virus-host interactions, and the role, efficacy, and mode of action of therapeutics. In this review, we present the standard model of virus dynamics and highlight situations where added model complexity accounting for intracellular processes is needed. We present several examples from acute and chronic viral infections where such inclusion in explicit and implicit manner has led to improvement in parameter estimates, unification of conclusions, guidance for targeted therapeutics, and crossover among model systems. We also discuss trade-offs between model realism and predictive power and highlight the need of increased data collection at finer scale of resolution to better validate complex models.

Dynamical analysis of a general delayed HBV infection model with capsids and adaptive immune response in presence of exposed infected hepatocytes

The aim of this paper is to develop and investigate a novel mathematical model of the dynamical behaviors of chronic hepatitis B virus infection. The model includes exposed infected hepatocytes, intracellular HBV DNA-containing capsids, uses a general incidence function for viral infection covering a variety of special cases available in the literature, and describes the interaction of cytotoxic T lymphocytes that kill the infected hepatocytes and the magnitude of B-cells that send antibody immune defense to neutralize free virions. Further, one time delay is incorporated to account for actual capsids production. The other time delays are used to account for maturation of capsids and free viruses. We start with the analysis of the proposed model by establishing the local and global existence, uniqueness, non-negativity and boundedness of solutions. After defined the threshold parameters, we discuss the stability properties of all possible steady state constants by using the crafty Lyapunov functionals, the LaSalle's invariance principle and linearization methods. The impacts of the three time delays on the HBV infection transmission are discussed through local and global sensitivity analysis of the basic reproduction number and of the classes of infected states. Finally, an application is provided and numerical simulations are performed to illustrate and interpret the theoretical results obtained. It is suggested that, a good strategy to eradicate or to control HBV infection within a host should concentrate on any drugs that may prolong the values of the three delays.

A retrospective cohort study of Paxlovid efficacy depending on treatment time in hospitalized COVID-19 patients

Paxlovid, a SARS-CoV-2 antiviral, not only prevents severe illness but also curtails viral shedding, lowering transmission risks from treated patients. By fitting a mathematical model of within-host Omicron viral dynamics to electronic health records data from 208 hospitalized patients in Hong Kong, we estimate that Paxlovid can inhibit over 90% of viral replication. However, its effectiveness critically depends on the timing of treatment. If treatment is initiated three days after symptoms first appear, we estimate a 17% chance of a post-treatment viral rebound and a 12% (95% CI: 0-16%) reduction in overall infectiousness for non-rebound cases. Earlier treatment significantly elevates the risk of rebound without further reducing infectiousness, whereas starting beyond five days reduces its efficacy in curbing peak viral shedding. Among the 104 patients who received Paxlovid, 62% began treatment within an optimal three-to-five-day day window after symptoms appeared. Our findings indicate that broader global access to Paxlovid, coupled with appropriately timed treatment, can mitigate the severity and transmission of SARS-Cov-2.

Viral dissemination and immune activation modulate antiretroviral drug levels in lymph nodes of SIV-infected rhesus macaques

Introduction and methods

To understand the relationship between immunovirological factors and antiretroviral (ARV) drug levels in lymph nodes (LN) in HIV therapy, we analyzed drug levels in twenty-one SIV-infected rhesus macaques subcutaneously treated with daily tenofovir (TFV) and emtricitabine (FTC) for three months.

Results

The intracellular active drug-metabolite (IADM) levels (TFV-dp and FTC-tp) in lymph node mononuclear cells (LNMC) were significantly lower than in peripheral blood mononuclear cells (PBMC) (P≤0.005). Between Month 1 and Month 3, IADM levels increased in both LNMC (P≤0.001) and PBMC (P≤0.01), with a steeper increase in LNMC (P≤0.01). The viral dissemination in plasma, LN, and rectal tissue at ART initiation correlated negatively with IADM levels at Month 1. Physiologically-based pharmacokinetic model simulations suggest that, following subcutaneous ARV administration, ART-induced reduction of immune activation improves the formation of active drug-metabolites through modulation of kinase activity and/or through improved parent drug accessibility to LN cellular compartments.

Conclusion

These observations have broad implications for drugs that need to phosphorylate to exert their pharmacological activity, especially in the settings of the pre-/post-exposure prophylaxis and efficacy of antiviral therapies targeting pathogenic viruses such as HIV or SARS-CoV-2 replicating in highly inflammatory anatomic compartments.

An explanation for SARS-CoV-2 rebound after Paxlovid treatment

In a fraction of SARS-CoV-2 infected individuals treated with the oral antiviral Paxlovid, the virus rebounds following treatment. The mechanism driving rebound is not understood. Here, we show that viral dynamic models based on the hypothesis that Paxlovid treatment near the time of symptom onset halts the depletion of target cells, but may not fully eliminate the virus, which can lead to viral rebound. We also show that the occurrence of viral rebound is sensitive to model parameters, and the time treatment is initiated, which may explain why only a fraction of individuals develop viral rebound. Finally, the models are used to test the therapeutic effects of two alternative treatment schemes. These findings also provide a possible explanation for rebounds following other antiviral treatments for SARS-CoV-2.

Machine learning aided multiscale modelling of the HIV-1 infection in the presence of NRTI therapy

Human Immunodeficiency Virus (HIV) is one of the most common chronic infectious diseases in humans. Extending the expected lifetime of patients depends on the use of optimal antiretroviral therapies. Emergence of the drug-resistant strains can reduce the effectiveness of treatments and lead to Acquired Immunodeficiency Syndrome (AIDS), even with antiretroviral therapy. Investigating the genotype-phenotype relationship is a crucial process for optimizing the therapy protocols of the patients. Here, a mathematical modelling framework is proposed to address the impact of existing mutations, timing of initiation, and adherence levels of nucleotide reverse transcriptase inhibitors (NRTIs) on the evolutionary dynamics of the virus strains. For the first time, the existing Stanford HIV drug resistance data have been combined with a multi-strain within-host ordinary differential equation (ODE) model to track the dynamics of the most common NRTI-resistant strains. Overall, the D4T-3TC, D4T-AZT and TDF-D4T drug combinations have been shown to provide higher success rates in preventing treatment failure and further drug resistance. The results are in line with the genotype-phenotype data and pharmacokinetic parameters of the NRTI inhibitors. Moreover, we show that the undetectable mutant strains at the diagnosis have a significant effect on the success/failure rates of the NRTI treatments. Predictions on undetectable strains through our multi-strain within-host model yielded the possible role of viral evolution on the treatment outcomes. It has been recognized that the improvement of multi-scale models can contribute to the understanding of the evolutionary dynamics, and treatment options, and potentially increase the reliability of genotype-phenotype models.

Physiologically Based Pharmacokinetic Modeling of 3 HIV Drugs in Combination and the Role of Lymphatic System after Subcutaneous Dosing. Part 1: Model for the Free-Drug Mixture

Drug-combination nanoparticles (DcNP) allow the formulation of multiple HIV drugs in one injectable. In nonhuman primates (NHP), all drugs in DcNP have demonstrated long-acting pharmacokinetics (PK) in the blood and lymph nodes, rendering it suitable for a Targeted Long-acting Antiretroviral Therapy (TLC-ART). To support the translation of TLC-ART into the clinic, the objective is to present a physiologically based PK (PBPK) model tool to control mechanisms affecting the rather complex DcNP-drug PK. Two species contribute simultaneously to the drug PK: drugs that dissociate from DcNP (Part 1) and drugs retained in DcNP (Part 2, presented separately). Here, we describe the PBPK modeling of the nanoparticle-free drugs. The free-drug model was built on subcutaneous injections of suspended lopinavir, ritonavir, and tenofovir in NHP, and validated by external experiments. A novelty was the design of a lymphatic network as part of a whole-body PBPK system which included major lymphatic regions: the cervical, axillary, hilar, mesenteric, and inguinal nodes. This detailed/regionalized description of the lymphatic system and mononuclear cells represents an unprecedented level of prediction that renders the free-drug model extendible to other small-drug molecules targeting the lymphatic system at both the regional and cellular levels.

A diffusive virus model with a fixed intracellular delay and combined drug treatments

The method of administration of an effective drug treatment to eradicate viruses within a host is an important issue in studying viral dynamics. Overuse of a drug can lead to deleterious side effects, and overestimating the efficacy of a drug can result in failure to treat infection. In this study, we proposed a reaction-diffusion within-host virus model which incorporated information regarding spatial heterogeneity, drug treatment, and the intracellular delay to produce productively infected cells and viruses. We also calculated the basic reproduction number [Formula: see text] under the assumptions of spatial heterogeneity. We have shown that the infection-free periodic solution is globally asymptotically stable when [Formula: see text], whereas when [Formula: see text] there is an infected periodic solution and the infection is uniformly persistent. By conducting numerical simulations, we also revealed the effects of various parameters on the value of [Formula: see text]. First, we showed that the dependence of [Formula: see text] on the intracellular delay could be monotone or non-monotone, depending on the death rate of infected cells in the immature stage. Second, we found that the mobility of infected cells or virions could facilitate the virus clearance. Third, we found that the spatial fragmentation of the virus environment enhanced viral infection. Finally, we showed that the combination of spatial heterogeneity and different sets of diffusion rates resulted in complicated viral dynamic outcomes.

Systems and Clinical Pharmacology of COVID-19 Therapeutic Candidates: A Clinical and Translational Medicine Perspective

Over 50 million people have been infected with the SARS-CoV-2 virus, while around 1 million have died due to COVID-19 disease progression. COVID-19 presents flu-like symptoms that can escalate, in about 7-10 days from onset, into a cytokine storm causing respiratory failure and death. Although social distancing reduces transmissibility, COVID-19 vaccines and therapeutics are essential to regain socioeconomic normalcy. Even if effective and safe vaccines are found, pharmacological interventions are still needed to limit disease severity and mortality. Integrating current knowledge and drug candidates (approved drugs for repositioning among >35 candidates) undergoing clinical studies (>3000 registered in ClinicalTrials.gov), we employed Systems Pharmacology approaches to project how antivirals and immunoregulatory agents could be optimally evaluated for use. Antivirals are likely to be effective only at the early stage of infection, soon after exposure and before hospitalization, while immunomodulatory agents should be effective in the later-stage cytokine storm. As current antiviral candidates are administered in hospitals over 5-7 days, a long-acting combination that targets multiple SARS-CoV-2 lifecycle steps may provide a long-lasting, single-dose treatment in outpatient settings. Long-acting therapeutics may still be needed even when vaccines become available as vaccines are likely to be approved based on a 50% efficacy target.

Pre-existing resistance in the latent reservoir can compromise VRC01 therapy during chronic HIV-1 infection

Passive immunization with broadly neutralizing antibodies (bNAbs) of HIV-1 appears a promising strategy for eliciting long-term HIV-1 remission. When administered concomitantly with the cessation of antiretroviral therapy (ART) to patients with established viremic control, bNAb therapy is expected to prolong remission. Surprisingly, in clinical trials on chronic HIV-1 patients, the bNAb VRC01 failed to prolong remission substantially. Identifying the cause of this failure is important for improving VRC01-based therapies and unraveling potential vulnerabilities of other bNAbs. In the trials, viremia resurged rapidly in most patients despite suppressive VRC01 concentrations in circulation, suggesting that VRC01 resistance was the likely cause of failure. ART swiftly halts viral replication, precluding the development of resistance during ART. If resistance were to emerge post ART, virological breakthrough would have taken longer than without VRC01 therapy. We hypothesized therefore that VRC01-resistant strains must have been formed before ART initiation, survived ART in latently infected cells, and been activated during VRC01 therapy, causing treatment failure. Current assays preclude testing this hypothesis experimentally. We developed a mathematical model based on the hypothesis and challenged it with available clinical data. The model integrated within-host HIV-1 evolution, stochastic latency reactivation, and viral dynamics with multiple-dose VRC01 pharmacokinetics. The model predicted that single but not higher VRC01-resistant mutants would pre-exist in the latent reservoir. We constructed a virtual patient population that parsimoniously recapitulated inter-patient variations. Model predictions with this population quantitatively captured data of VRC01 failure from clinical trials, presenting strong evidence supporting the hypothesis. We attributed VRC01 failure to single-mutant VRC01-resistant proviruses in the latent reservoir triggering viral recrudescence, particularly when VRC01 was at trough levels. Pre-existing resistant proviruses in the latent reservoir may similarly compromise other bNAbs. Our study provides a framework for designing bNAb-based therapeutic protocols that would avert such failure and maximize HIV-1 remission.

Antiretroviral therapy (ART) can control but not eradicate HIV-1. Stopping ART leads to rapid viral resurgence and progressive disease. ART is therefore administered lifelong. Tremendous efforts are ongoing to devise strategies that will enable stopping ART and yet prevent viral resurgence. One such strategy involves the administration of broadly neutralizing antibodies (bNAbs) of HIV-1 at the time of stopping ART. This strategy is expected to delay if not prevent viral resurgence. Surprisingly, treatment with VRC01, a potent bNAb, resulted in hardly any improvement in viral remission. In this study, we elucidate the cause of this failure. We hypothesized that VRC01-resistant strains may pre-exist in latently infected cells, which are unaffected by ART. They can thus outlast ART and get reactivated, triggering VRC01 failure. We built a detailed mathematical model based on this hypothesis and showed that it quantitatively captured observations of VRC01 failure in clinical trials on chronic HIV-1 patients. Our study thus identifies a potential vulnerability of bNAbs, namely, bNAb-resistant strains pre-existing in latently infected cells. Our model offers a framework for predicting bNAb-based treatment protocols that would preclude failure due to pre-existing resistance and maximally prolong remission.

An Integrated Spatial Dynamics-Pharmacokinetic Model Explaining Poor Penetration of Anti-retroviral Drugs in Lymph Nodes

Although combined anti-retroviral therapy (cART) suppresses plasma HIV viremia below the limit of detection in a majority of HIV patients, evidence is emerging that the distribution of the anti-retroviral drugs is heterogeneous in tissue. Clinical studies measuring antiretroviral drug concentrations in lymph nodes (LNs) revealed lower concentrations compared to peripheral blood levels suggesting poor drug penetration properties. Our current study is an attempt to understand this poor anti-retroviral drug penetration inside lymph node lobules through integrating known pharmacokinetic and pharmacodynamic (PK/PD) parameters of the anti-retroviral drugs into a spatial model of reaction and transport dynamics within a solid lymph node lobule. Simulated drug penetration values were compared against experimental results whenever available or matched with data that is available for other drugs in a similar class. Our integrated spatial dynamics pharmacokinetic model reproduced the experimentally observed exclusion of antivirals from lymphoid sites. The strongest predictor of drug exclusion from the lymphoid lobule, independent of drug class, was lobule size; large lobules (high inflammation) exhibited high levels of drug exclusion. PK/PD characteristics associated with poor lymphoid penetration include high cellular uptake rates and low intracellular half-lives. To determine whether this exclusion might lead to ongoing replication, target CD4+ T cell, infected CD4+ T cell, free virus, and intracellular IC50 values of anti-retroviral drugs were incorporated into the model. Notably, for median estimates of PK/PD parameters and lobule diameters consistent with low to moderate inflammation, the model predicts no ongoing viral replication, despite substantial exclusion of the drugs from the lymphoid site. Monte-Carlo studies drawn from the prior distributions of the PK/PD parameters predicts increases in site-specific HIV replication in a small fraction of the patient population for lobule diameters greater than 0.2 mm; this fraction increases as the site diameter/ inflammation level increases. The model shows that cART consisting of two nRTIs and one PI is the most likely treatment combination to support formation of a sanctuary site, a finding that is consistent with clinical observations.

Application of Clinical Pharmacology Principles in Drug Development of Modified-Release Products: Leveraging Exposure-Response Information to Support Approval

The development of modified-release (MR) drug products aims to address a clinical need such as improving patient compliance. There are multiple pathways and development strategies for the registration and approval of MR products. The development strategy of an MR product is usually dependent on the availability and pharmacokinetic/pharmacodynamics (PK/PD) characteristics of the reference drug product, that is, an immediate-release (IR) product or a reference MR. Compared with a reference IR product, an MR product is likely to have a different PK profile over the least common dosing time due to unequal dosing intervals. In case of differences in PK profiles between the MR product and the reference product, confirmatory efficacy and safety studies may be needed to support registration. In some cases, however, a thorough clinical PK/PD characterization may provide sufficient basis to support the approval of the proposed MR product without the need for additional safety and efficacy studies. This article summarizes the US Food and Drug Administration experience and the regulatory considerations supporting the approval of MR products in the past 6 years and discusses cases in which clinical pharmacology and PK/PD information were leveraged to support approval without the need for additional clinical studies. Details of all these cases are available in the public domain. In 2 cases a well-characterized exposure-response relationship provided sufficient justification that differences in the shape of the PK profiles were not clinically relevant. In the remaining 3 cases a thorough characterization of the PK profile along with a risk-based approach provided bases for approval.

Bifurcation Analysis of a Delayed Infection Model with General Incidence Function

In this paper, an infection model with delay and general incidence function is formulated and analyzed. Theoretical results reveal that positive equilibrium may lose its stability, and Hopf bifurcation occurs when choosing delay as the bifurcation parameter. The direction of Hopf bifurcation and the stability of the periodic solutions are also discussed. Furthermore, to illustrate the numerous changes in the local stability and instability of the positive equilibrium, we conduct numerical simulations by using four different types of functional incidence, i.e., bilinear incidence, saturation incidence, Beddington–DeAngelis response, and Hattaf–Yousfi response. Rich dynamics of the model, such as Hopf bifurcations and chaotic solutions, are presented numerically.

Multidrug Therapy for HIV Infection: Dynamics of Immune System

A mathematical model of the dynamics of the immune system is considered to illustrate the effect of its response to HIV infection, i.e. on viral growth and on T-cell dynamics. The specific immune response is measured by the levels of cytotoxic lymphocytes in a human body. The existence and stability analyses are performed for infected steady state and uninfected steady state. In order to keep infection under control, roles of drug therapies are analyzed in the presence of efficient immune response. Numerical simulations are computed and exhibited to illustrate the support of the immune system to drug therapies, so as to ensure the decay of infection and to maintain the level of healthy cells.

Insight into treatment of HIV infection from viral dynamics models

The odds of living a long and healthy life with HIV infection have dramatically improved with the advent of combination antiretroviral therapy. Along with the early development and clinical trials of these drugs, and new field of research emerged called viral dynamics, which uses mathematical models to interpret and predict the time-course of viral levels during infection and how they are altered by treatment. In this review, we summarize the contributions that virus dynamics models have made to understanding the pathophysiology of infection and to designing effective therapies. This includes studies of the multiphasic decay of viral load when antiretroviral therapy is given, the evolution of drug resistance, the long-term persistence latently infected cells, and the rebound of viremia when drugs are stopped. We additionally discuss new work applying viral dynamics models to new classes of investigational treatment for HIV, including latency-reversing agents and immunotherapy.

Kinetic Modeling of Virus Growth in Cells

When a virus infects a host cell, it hijacks the biosynthetic capacity of the cell to produce virus progeny, a process that may take less than an hour or more than a week. The overall time required for a virus to reproduce depends collectively on the rates of multiple steps in the infection process, including initial binding of the virus particle to the surface of the cell, virus internalization and release of the viral genome within the cell, decoding of the genome to make viral proteins, replication of the genome, assembly of progeny virus particles, and release of these particles into the extracellular environment. For a large number of virus types, much has been learned about the molecular mechanisms and rates of the various steps. However, in only relatively few cases during the last 50 years has an attempt been made-using mathematical modeling-to account for how the different steps contribute to the overall timing and productivity of the infection cycle in a cell. Here we review the initial case studies, which include studies of the one-step growth behavior of viruses that infect bacteria (Qβ, T7, and M13), human immunodeficiency virus, influenza A virus, poliovirus, vesicular stomatitis virus, baculovirus, hepatitis B and C viruses, and herpes simplex virus. Further, we consider how such models enable one to explore how cellular resources are utilized and how antiviral strategies might be designed to resist escape. Finally, we highlight challenges and opportunities at the frontiers of cell-level modeling of virus infections.

An analysis of the delay-dependent HIV-1 protease inhibitor model

In this paper, we have studied about the sensitivity analysis of the human immunodeficiency virus (HIV) protease inhibitor (PI) model and estimated the length of the delay. We have fabricated an HIV PI model accompanied with humoral immunity. Stability analysis of the constructed model about its steady states has been deliberated. We have evaluated some numerical simulations for PI model with humoral immunity by using the existing patient data.

Bio-inspired computational heuristics to study models of HIV infection of CD4+ T-cell

In this work, biologically-inspired computing framework is developed for HIV infection of CD4[Formula: see text] T-cell model using feed-forward artificial neural networks (ANNs), genetic algorithms (GAs), sequential quadratic programming (SQP) and hybrid approach based on GA-SQP. The mathematical model for HIV infection of CD4[Formula: see text] T-cells is represented with the help of initial value problems (IVPs) based on the system of ordinary differential equations (ODEs). The ANN model for the system is constructed by exploiting its strength of universal approximation. An objective function is developed for the system through unsupervised error using ANNs in the mean square sense. Training with weights of ANNs is carried out with GAs for effective global search supported with SQP for efficient local search. The proposed scheme is evaluated on a number of scenarios for the HIV infection model by taking the different levels for infected cells, natural substitution rates of uninfected cells, and virus particles. Comparisons of the approximate solutions are made with results of Adams numerical solver to establish the correctness of the proposed scheme. Accuracy and convergence of the approach are validated through the results of statistical analysis based on the sufficient large number of independent runs.

Life cycle synchronization is a viral drug resistance mechanism

Viral infections are one of the major causes of death worldwide, with HIV infection alone resulting in over 1.2 million casualties per year. Antiviral drugs are now being administered for a variety of viral infections, including HIV, hepatitis B and C, and influenza. These therapies target a specific phase of the virus's life cycle, yet their ultimate success depends on a variety of factors, such as adherence to a prescribed regimen and the emergence of viral drug resistance. The epidemiology and evolution of drug resistance have been extensively characterized, and it is generally assumed that drug resistance arises from mutations that alter the virus's susceptibility to the direct action of the drug. In this paper, we consider the possibility that a virus population can evolve towards synchronizing its life cycle with the pattern of drug therapy. The periodicity of the drug treatment could then allow for a virus strain whose life cycle length is a multiple of the dosing interval to replicate only when the concentration of the drug is lowest. This process, referred to as "drug tolerance by synchronization", could allow the virus population to maximize its overall fitness without having to alter drug binding or complete its life cycle in the drug's presence. We use mathematical models and stochastic simulations to show that life cycle synchronization can indeed be a mechanism of viral drug tolerance. We show that this effect is more likely to occur when the variability in both viral life cycle and drug dose timing are low. More generally, we find that in the presence of periodic drug levels, time-averaged calculations of viral fitness do not accurately predict drug levels needed to eradicate infection, even if there is no synchronization. We derive an analytical expression for viral fitness that is sufficient to explain the drug-pattern-dependent survival of strains with any life cycle length. We discuss the implications of these findings for clinically relevant antiviral strategies.

Stability of HIV-1 infection with saturated virus-target and infected-target incidences and CTL immune response

This paper studies the dynamical behavior of an HIV-1 infection model with saturated virus-target and infected-target incidences with Cytotoxic T Lymphocyte (CTL) immune response. The model is incorporated by two types of intracellular distributed time delays. The model generalizes all the existing HIV-1 infection models with cell-to-cell transmission presented in the literature by considering saturated incidence rate and the effect of CTL immune response. The existence and global stability of all steady states of the model are determined by two parameters, the basic reproduction number ([Formula: see text]) and the CTL immune response activation number ([Formula: see text]). By using suitable Lyapunov functionals, we show that if [Formula: see text], then the infection-free steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text] [Formula: see text], then the CTL-inactivated infection steady state [Formula: see text] is globally asymptotically stable; if [Formula: see text], then the CTL-activated infection steady state [Formula: see text] is globally asymptotically stable. Using MATLAB we conduct some numerical simulations to confirm our results. The effect of the saturated incidence of the HIV-1 dynamics is shown.

Mathematical analysis of an HIV infection model including quiescent cells and periodic antiviral therapy

In this paper, we revisit the model by Guedj et al. [J. Guedj, R. Thibaut and D. Commenges, Maximum likelihood estimation in dynamical models of HIV, Biometrics 63 (2007) 198–206; J. Guedj, R. Thibaut and D. Commenges, Practical identifiability of HIV dynamics models, Bull. Math. Biol. 69 (2007) 2493–2513] which describes the effect of treatment with reverse transcriptase (RT) inhibitors and incorporates the class of quiescent cells. We prove that there is a threshold value [Formula: see text] of drug efficiency [Formula: see text] such that if [Formula: see text], the basic reproduction number [Formula: see text] and the infection is cleared and if [Formula: see text], the infectious equilibrium is globally asymptotically stable. When the drug efficiency function [Formula: see text] is periodic and of the bang–bang type we establish a threshold, in terms of spectral radius of some matrix, between the clearance and the persistence of the disease. As stated in related works [L. Rong, Z. Feng and A. Perelson, Emergence of HIV-1 drug resistance during antiretroviral treatment, Bull. Math. Biol. 69 (2007) 2027–2060; P. De Leenheer, Within-host virus models with periodic antiviral therapy, Bull. Math. Biol. 71 (2009) 189–210.], we prove that the increase of the drug efficiency or the active duration of drug must clear the infection more quickly. We illustrate our results by some numerical simulations.

Analysis of HIV models with two time delays

Time delays can affect the dynamics of HIV infection predicted by mathematical models. In this paper, we studied two mathematical models each with two time delays. In the first model with HIV latency, one delay is the time between viral entry into a cell and the establishment of HIV latency, and the other delay is the time between cell infection and viral production. We defined the basic reproductive number and showed the local and global stability of the steady states. Numerical simulations were performed to evaluate the influence of time delays on the dynamics. In the second model with HIV immune response, one delay is the time between cell infection and viral production, and the other delay is the time needed for the adaptive immune response to emerge to control viral replication. With two positive delays, we obtained the stability crossing curves for the model, which were shown to be a series of open-ended curves.

A model incorporating combined RTIs and PIs therapy during early HIV-1 infection

We develop a within host mathematical model of HIV-1 infection describing the effects of combined RTIs and PIs treatments on early HIV-1 infection when treatment is captured using periodic functions of pharmacokinetics type. We use an alternative of the basic reproduction number to analyze endemicity level of HIV-1 infection. Various treatment scenarios incorporating perfect and imperfect drug adherence in drug administration are explored. Our results show that pharmacokinetics treatment is a more realistic way of administering the treatment. Apart from confirming that PIs drugs are more effective than RTIs drugs and that combined RTIs and PIs therapy is more effective than monotherapy of RTIs or PIs, our results show that imperfect drug adherence leads to the increase of viral in the absence of mutation even though the drug is good.

Global stability of a delayed virus dynamics model with multi-staged infected progression and humoral immunity

In this paper, we propose a nonlinear virus dynamics model that describes the interactions of the virus, uninfected target cells, multiple stages of infected cells and B cells and includes multiple discrete delays. We assume that the incidence rate of infection and removal rate of infected cells are given by general nonlinear functions. The model can be seen as a generalization of several humoral immunity viral infection model presented in the literature. We derive two threshold parameters and establish a set of conditions on the general functions which are sufficient to establish the existence and global stability of the three equilibria of the model. We study the global asymptotic stability of the equilibria by using Lyapunov method. We perform some numerical simulations for the model with specific forms of the general functions and show that the numerical results are consistent with the theoretical results.

Models of Viral Population Dynamics

Models of viral population dynamics have contributed enormously to our understanding of the pathogenesis and transmission of several infectious diseases, the coevolutionary dynamics of viruses and their hosts, the mechanisms of action of drugs, and the effectiveness of interventions. In this chapter, we review major advances in the modeling of the population dynamics of the human immunodeficiency virus (HIV) and briefly discuss adaptations to other viruses.

Effects of combined drug therapy on HIV-1 infection dynamics

Infection of human immunodeficiency virus (HIV) is determined through the decay of healthy CD4+ T-cells in a well-mixed compartment, such as a bloodstream. A mathematical model is considered to illustrate the effects of combined drug therapy, i.e. reverse transcription plus protease inhibitor, on viral growth and T-cell population dynamics. This model is used to explain the existence and stability of infected and uninfected steady states in HIV growth. An analytical technique, called variational iteration method (VIM), is used to solve the mathematical model. This method is modified to obtain the rapidly convergent successive approximations of the exact solution. These approximations are obtained without any restrictions or the transformations that may change the physical behavior of the problem. Numerical simulations are computed and exhibited to illustrate the effects of proposed drug therapy on the growth or decay of infection.

Optimal Treatment Strategies in the Context of 'Treatment for Prevention' against HIV-1 in Resource-Poor Settings

An estimated 2.7 million new HIV-1 infections occurred in 2010. `Treatment-for-prevention’ may strongly prevent HIV-1 transmission. The basic idea is that immediate treatment initiation rapidly decreases virus burden, which reduces the number of transmittable viruses and thereby the probability of infection. However, HIV inevitably develops drug resistance, which leads to virus rebound and nullifies the effect of `treatment-for-prevention’ for the time it remains unrecognized. While timely conducted treatment changes may avert periods of viral rebound, necessary treatment options and diagnostics may be lacking in resource-constrained settings. Within this work, we provide a mathematical platform for comparing different treatment paradigms that can be applied to many medical phenomena. We use this platform to optimize two distinct approaches for the treatment of HIV-1: (i) a diagnostic-guided treatment strategy, based on infrequent and patient-specific diagnostic schedules and (ii) a pro-active strategy that allows treatment adaptation prior to diagnostic ascertainment. Both strategies are compared to current clinical protocols (standard of care and the HPTN052 protocol) in terms of patient health, economic means and reduction in HIV-1 onward transmission exemplarily for South Africa. All therapeutic strategies are assessed using a coarse-grained stochastic model of within-host HIV dynamics and pseudo-codes for solving the respective optimal control problems are provided. Our mathematical model suggests that both optimal strategies (i)-(ii) perform better than the current clinical protocols and no treatment in terms of economic means, life prolongation and reduction of HIV-transmission. The optimal diagnostic-guided strategy suggests rare diagnostics and performs similar to the optimal pro-active strategy. Our results suggest that ‘treatment-for-prevention’ may be further improved using either of the two analyzed treatment paradigms.

HIV-1 continues to spread globally. Antiviral treatment cannot cure patients, but it slows disease progression and may prevent HIV transmission by decreasing the amount of transmittable viruses in treated individuals. ‘Treatment-for-prevention’ argues for immediate treatment initiation and may reduce transmission by 96% (CI: 73–99%), according to the results of a large clinical study (HPTN052). In order to ensure long-lasting treatment success, early therapy initiation demands more sophisticated treatment strategies & exceeding funds. However, countries facing the highest HIV burden are among the poorest. Within this work, we provide a mathematical framework that allows assessing different treatment paradigms using optimal control theory together with stochastic modelling of within-host viral dynamics and drug resistance development. We use this framework to compute and evaluate two distinct optimal long-term treatment strategies for resource-constrained settings: (i) a diagnostic-guided and (ii) a pro-active treatment strategy. The cost of a strategy is evaluated from a national economic perspective, valuating a severe patient health status in terms of an economic loss. The optimal strategies are compared with current clinical treatment protocols and no treatment in terms of costs, life expectation and reduction of secondary cases. Our simulations indicate that the pro-active treatment strategy performs comparably to the diagnostic-guided treatment strategy. Both strategies perform better than current clinical protocols, suggesting directions for improvement.

Linking the population pharmacokinetics of tenofovir and its metabolites with its cellular uptake and metabolism

Empirical pharmacokinetic models are used to explain the pharmacokinetics of the antiviral drug tenofovir (TFV) and its metabolite TFV diphosphate (TFV-DP) in peripheral blood mononuclear cells. These empirical models lack the ability to explain differences between the disposition of TFV-DP in HIV-infected patients vs. healthy individuals. Such differences may lie in the mechanisms of TFV transport and phosphorylation. Therefore, we developed an exploratory model based on mechanistic mass transport principles and enzyme kinetics to examine the uptake and phosphorylation kinetics of TFV. TFV-DP median C_max from the model was 38.5 fmol/10⁶ cells, which is bracketed by two reported healthy volunteer studies (38 and 51 fmol/10⁶ cells). The model presented provides a foundation for exploration of TFV uptake and phosphorylation kinetics for various routes of TFV administration and can be updated as more is known on actual mechanisms of cellular transport of TFV.

Global properties of a cell mediated immunity in HIV infection model with two classes of target cells and distributed delays

In this paper, we study the global properties of a human immunodeficiency virus (HIV) infection model with cytotoxic T lymphocytes (CTL) immune response. The model is a six-dimensional that describes the interaction of the HIV with two classes of target cells, CD4+ T cells and macrophages. The infection rate is given by saturation functional response. Two types of distributed time delays are incorporated into the model to describe the time needed for infection of target cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is determined by two threshold numbers, the basic infection reproduction number R0 and the immune response activation number [Formula: see text]. We have proven that if R0 ≤ 1, then the uninfected steady state is globally asymptotically stable (GAS), if [Formula: see text], then the infected steady state without CTL immune response is GAS, and if [Formula: see text], then the infected steady state with CTL immune response is GAS.

A hepatitis C virus infection model with time-varying drug effectiveness: solution and analysis

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

Prodrug strategies for improved efficacy of nucleoside antiviral inhibitors

PURPOSE OF REVIEW

This review focuses on the chemical and pharmacological rationale behind the development of nucleoside antiviral prodrugs (NAPs).

RECENT FINDINGS

Highly efficacious NAPs have been developed that extend and improve the quality of lives of individuals infected with HIV and hepatitis B virus (HBV), herpes viruses, and adenovirus infection in immunocompromised individuals. A very high rate of hepatitis C virus (HCV) cure is now possible using NAPs combined with other direct acting antiviral agents (DAAs).

SUMMARY

Prodrug strategies can address the issues of poor oral bioavailability and delivery of active metabolites to the targeted cells. Additionally, NAPs demonstrate potential for improving deficiencies in oral absorption, metabolism, tissue distribution, cellular accumulation, phosphorylation, and overall potency, in addition to diminishing potential for in-vivo selection of resistant viruses. NAPs continue to be the backbone for the treatment of HIV and HBV, herpesviruses, and adenovirus infections because their active forms are potent, have long intracellular half-lives and are relatively safe with high barrier to resistance.

A DELAYED HIV-1 MODEL WITH MULTIPLE TARGET CELLS AND GENERAL NONLINEAR INCIDENCE RATE

We investigate a delayed HIV-1 infection model with general nonlinear incidence functions and two classes of target cells: CD4+ T-cells and macrophages. To account for the time lags between viruses' entry into the corresponding two types of target cells and the production of new virus particles, we incorporate four distributed intracellular delays into the model. We show that the basic reproduction number ℜ0 is the sum of the basic reproduction numbers of HIV-1 infection with CD4+ T-cells and that with macrophages; moreover, if ℜ0 is less than or equal to one, then the HIV-1 infection is cleared from the T-cell population and macrophages; whereas if ℜ0 is larger than one, then the viral concentration maintains at some constant level. It is shown, from both our analytic and numeric results, that ignoring the contributions of macrophages to HIV-1 infection and production will underestimate both the risk of HIV-1 infection and the viral load when persisting. This highlights the important effects of multiple target cells on HIV-1 infection.

Global stability of HIV infection of CD4+ T cells and macrophages with CTL immune response and distributed delays

We study the global stability of a human immunodeficiency virus (HIV) infection model with Cytotoxic T Lymphocytes (CTL) immune response. The model describes the interaction of the HIV with two classes of target cells, CD4⁺ T cells and macrophages. Two types of distributed time delays are incorporated into the model to describe the time needed for infection of target cell and virus replication. Using the method of Lyapunov functional, we have established that the global stability of the model is determined by two threshold numbers, the basic reproduction number R₀ and the immune response reproduction number R₀^∗. We have proven that, if R₀ ≤ 1, then the uninfected steady state is globally asymptotically stable (GAS), if R₀* ≤ 1 < R₀, then the infected steady state without CTL immune response is GAS, and, if R₀* > 1, then the infected steady state with CTL immune response is GAS.

Opinion: the pharmacometrics of infectious disease

The application of pharmacometric principles to the treatment of infectious diseases must address important biological issues across the diversity of pathogenic organisms. Recent applications of pharmacometric tools in this therapeutic area have had important translational impact not only in drug development but on real-world clinical practice. The fruitful fusion of preclinical and population methodologies promises increasingly personalized and mechanistic approaches.CPT: Pharmacometrics & Systems Pharmacology (2013) 2, e70; doi:10.1038/psp.2013.46; published online 28 August 2013.

HOW SELF-PROLIFERATION OF CD4+T CELLS AFFECT THE HIV DYNAMICS IN AN IN-HOST TARGET-CELL LIMITED HIV MODEL WITH SATURATION INFECTION RATE: A QUASI-STEADY-STATE APPROXIMATION ANALYSIS

In this study, we consider two target-cell limited models with saturation type infection rate and intracellular delay: one without self-proliferation and the other with self-proliferation of activated CD4+T cells. We discuss about the local and global behavior of both the systems in presence and absence of intracellular delay. It is shown that the endemic equilibrium of a target-cell limited model would be unstable in presence and absence of intracellular delay only when self-proliferation of activated CD4+T cell is considered. Otherwise, all positive solutions converge to the endemic equilibrium or disease-free equilibrium depending on whether the basic reproduction ratio is greater than or less than unity. Our study suggests that amplitude of oscillation is negatively correlated with the constant input rate of CD4+T cell when intracellular delay is absent or low. However, they are positively correlated if the delay is too high. Amplitude of oscillation, on the other hand, is always positively correlated with the proliferation rate of CD4+T cell for all delay. Our mathematical and simulation analysis also suggest that there are many potential contributors who are responsible for the variation of CD4+T cells and virus particles in the blood plasma of HIV patients.

Explaining the determinants of first phase HIV decay dynamics through the effects of stage-dependent drug action

A recent investigation of the effect of different antiretroviral drug classes on first phase dynamics of HIV RNA plasma virus levels has indicated that drugs acting at stages closer to viral production, such as the integrase inhibitor raltegravir, can produce a steeper first phase decay slope that may not be due to drug efficacy. Moreover it was found that for most drug classes the first phase transitions from a faster (phase IA) to a slightly slower decay region (phase IB) before the start of the usual second phase. Neither of these effects has been explained to date. We use a mathematical model that incorporates the different stages of the HIV viral life cycle in CD4+ T cells: viral entry, reverse transcription, integration, and viral production, to investigate the intracellular HIV mechanisms responsible for these complex plasma virus decay dynamics. We find differences in the phase IA slope across drug classes arise from a higher death rate of cells when they enter the productively infected stage post-integration, with a half-life of approximately 8 hours in this stage, whereas cells in earlier stages of the infection cycle have half-lives similar to uninfected cells. This implies any immune clearance is predominantly limited to the productive infection stage. We also show that the slowing of phase IA to phase IB at day 2 to 4 of monotherapy, depending on drug class, is a result of new rounds of infection. The level at which this slowing occurs is a better indicator of drug efficacy than the slope of the initial decay.

The infection of a cell by HIV proceeds through a series of stages and each stage can now be inhibited by an available antiretroviral drug class. It is known that different drug classes can result in different decay curves of plasma viral levels that are not well explained by current mathematical models of HIV dynamics. Here we develop a mathematical model that incorporates these stages of infection and show how it successfully reproduces plasma decay curves for the five classes of currently available antiretroviral drugs. Our modeling indicates that the efficacy of antiretroviral drugs is not solely described by the rate of decay of plasma viral levels as currently thought. Drugs such as the integrase inhibitor raltegravir will result in a faster initial decline of plasma viral levels compared to a drug that acts further from viral integration and production such as the CCR5 inhibitor maraviroc, even though they may have the same efficacy. Moreover, we find that infected cells only die at rates above the background level when they are in the productive phase, indicating that immune clearance is mostly absent from the early stages of HIV cellular infection. This is of particular concern given that most infected cells are in these early stages of infection.

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.

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.

A comparison of computational efficiencies of stochastic algorithms in terms of two infection models

In this paper, we investigate three particular algorithms: a stochastic simulation algorithm (SSA), and explicit and implicit tau-leaping algorithms. To compare these methods, we used them to analyze two infection models: a Vancomycin-resistant enterococcus (VRE) infection model at the population level, and a Human Immunodeficiency Virus (HIV) within host infection model. While the first has a low species count and few transitions, the second is more complex with a comparable number of species involved. The relative efficiency of each algorithm is determined based on computational time and degree of precision required. The numerical results suggest that all three algorithms have the similar computational efficiency for the simpler VRE model, and the SSA is the best choice due to its simplicity and accuracy. In addition, we have found that with the larger and more complex HIV model, implementation and modification of tau-Leaping methods are preferred.

Multiple endemic states in age-structured SIR epidemic models

SIR age-structured models are very often used as a basic model of epidemic spread. Yet, their behaviour, under generic assumptions on contact rates between different age classes, is not completely known, and, in the most detailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemic equilibrium only under a rather restrictive condition. Here, we show an example in the form of a 3x3 contact matrix in which multiple non-trivial steady states exist. This instance of non-uniqueness of positive equilibria differs from most existing ones for epidemic models, since it arises not from a backward transcritical bifurcation at the disease free equilibrium, but through two saddle-node bifurcations of the positive equilibrium. The dynamical behaviour of the model is analysed numerically around the range where multiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasi-periodic and chaotic attractors. It is also shown that, if the contact rates are in the form of a 2x2 WAIFW matrix, uniqueness of non-trivial steady states always holds, so that 3 is the minimum dimension of the contact matrix to allow for multiple endemic equilibria.