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

A novel simulation-based analysis of a stochastic HIV model with the time delay using high order spectral collocation technique

The economic impact of Human Immunodeficiency Virus (HIV) goes beyond individual levels and it has a significant influence on communities and nations worldwide. Studying the transmission patterns in HIV dynamics is crucial for understanding the tracking behavior and informing policymakers about the possible control of this viral infection. Various approaches have been adopted to explore how the virus interacts with the immune system. Models involving differential equations with delays have become prevalent across various scientific and technical domains over the past few decades. In this study, we present a novel mathematical model comprising a system of delay differential equations to describe the dynamics of intramural HIV infection. The model characterizes three distinct cell sub-populations and the HIV virus. By incorporating time delay between the viral entry into target cells and the subsequent production of new virions, our model provides a comprehensive understanding of the infection process. Our study focuses on investigating the stability of two crucial equilibrium states the infection-free and endemic equilibriums. To analyze the infection-free equilibrium, we utilize the LaSalle invariance principle. Further, we prove that if reproduction is less than unity, the disease free equilibrium is locally and globally asymptotically stable. To ensure numerical accuracy and preservation of essential properties from the continuous mathematical model, we use a spectral scheme having a higher-order accuracy. This scheme effectively captures the underlying dynamics and enables efficient numerical simulations.

Iterative community-driven development of a SARS-CoV-2 tissue simulator

The 2019 novel coronavirus, SARS-CoV-2, is a pathogen of critical significance to international public health. Knowledge of the interplay between molecular-scale virus-receptor interactions, single-cell viral replication, intracellular-scale viral transport, and emergent tissue-scale viral propagation is limited. Moreover, little is known about immune system-virus-tissue interactions and how these can result in low-level (asymptomatic) infections in some cases and acute respiratory distress syndrome (ARDS) in others, particularly with respect to presentation in different age groups or pre-existing inflammatory risk factors. Given the nonlinear interactions within and among each of these processes, multiscale simulation models can shed light on the emergent dynamics that lead to divergent outcomes, identify actionable "choke points" for pharmacologic interventions, screen potential therapies, and identify potential biomarkers that differentiate patient outcomes. Given the complexity of the problem and the acute need for an actionable model to guide therapy discovery and optimization, we introduce and iteratively refine a prototype of a multiscale model of SARS-CoV-2 dynamics in lung tissue. The first prototype model was built and shared internationally as open source code and an online interactive model in under 12 hours, and community domain expertise is driving regular refinements. In a sustained community effort, this consortium is integrating data and expertise across virology, immunology, mathematical biology, quantitative systems physiology, cloud and high performance computing, and other domains to accelerate our response to this critical threat to international health. More broadly, this effort is creating a reusable, modular framework for studying viral replication and immune response in tissues, which can also potentially be adapted to related problems in immunology and immunotherapy.

Intra- and Inter-cellular Modeling of Dynamic Interaction between Zika Virus and Its Naturally Occurring Defective Viral Genomes

Here, we examine in silico the infection dynamics and interactions of two Zika virus (ZIKV) genomes: one is the full-length ZIKV genome (wild type [WT]), and the other is one of the naturally occurring defective viral genomes (DVGs), which can replicate in the presence of the WT genome, appears under high-MOI (multiplicity of infection) passaging conditions, and carries a deletion encompassing part of the structural and NS1 protein-coding region. Ordinary differential equations (ODEs) were used to simulate the infection of cells by virus particles and the intracellular replication of the WT and DVG genomes that produce these particles. For each virus passage in Vero and C6/36 cell cultures, the rates of the simulated processes were fitted to two types of observations: virus titer data and the assembled haplotypes of the replicate passage samples. We studied the consistency of the model with the experimental data across all passages of infection in each cell type separately as well as the sensitivity of the model's parameters. We also determined which simulated processes of virus evolution are the most important for the adaptation of the WT and DVG interplay in these two disparate cell culture environments. Our results demonstrate that in the majority of passages, the rates of DVG production are higher inC6/36 cells than in Vero cells, which might result in tolerance and therefore drive the persistence of the mosquito vector in the context of ZIKV infection. Additionally, the model simulations showed a slower accumulation of infected cells under higher activation of the DVG-associated processes, which indicates a potential role of DVGs in virus attenuation. IMPORTANCE One of the ideas for lessening Zika pathogenicity is the addition of its natural or engineered defective virus genomes (DVGs) (have no pathogenicity) to the infection pool: a DVG is redirecting the wild-type (WT)-associated virus development resources toward its own maturation. The mathematical model presented here, attuned to the data from interplays between WT Zika viruses and their natural DVGs in mammalian and mosquito cells, provides evidence that the loss of uninfected cells is attenuated by the DVG development processes. This model enabled us to estimate the rates of virus development processes in the WT/DVG interplay, determine the key processes, and show that the key processes are faster in mosquito cells than in mammalian ones. In general, the presented model and its detailed study suggest in what important virus development processes the therapeutically efficient DVG might compete with the WT; this may help in assembling engineered DVGs for ZIKV and other flaviviruses.

Modelling heterogeneity in viral-tumour dynamics: The effects of gene-attenuation on viral characteristics

The use of viruses as a cancer treatment is becoming increasingly more robust; however, there is still a long way to go before a completely successful treatment is formulated. One major challenge in the field is to select which virus, out of a burgeoning number of oncolytic viruses and engineered derivatives, can maximise both treatment spread and anticancer cytotoxicity. To assist in solving this problem, an in-depth understanding of the virus-tumour interaction is crucial. In this article, we present a novel integro-differential system with distributed delays embodying the dynamics of an oncolytic adenovirus with a fixed population of tumour cells in vitro, allowing for heterogeneity to exist in the virus and cell populations. The parameters of the model are optimised in a hierarchical manner, the purpose of which is not to obtain a perfect representation of the data. Instead, we place our parameter values in the correct region of the parameter space. Due to the sparse nature of the data it is not possible to obtain the parameter values with any certainty, but rather we demonstrate the suitability of the model. Using our model we quantify how modifications to the viral genome alter the viral characteristics, specifically how the attenuation of the E1B 19 and E1B 55 gene affect the system performance, and identify the dominant processes altered by the mutations. From our analysis, we conclude that the deletion of the E1B 55 gene significantly reduces the replication rate of the virus in comparison to the deletion of the E1B 19 gene. We also found that the deletion of both the E1B 19 and E1B 55 genes resulted in a long delay in the average replication start time of the virus. This leads us to propose the use of E1B 19 gene-attenuated adenovirus for cancer therapy, as opposed to E1B 55 gene-attenuated adenoviruses.

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.

Global dynamics of a delayed within-host viral infection model with both virus-to-cell and cell-to-cell transmissions

A within-host viral infection model with both virus-to-cell and cell-to-cell transmissions and three distributed delays is investigated, in which the first distributed delay describes the intracellular latency for the virus-to-cell infection, the second delay represents the intracellular latency for the cell-to-cell infection, and the third delay describes the time period that viruses penetrated into cells and infected cells release new virions. The global stability analysis of the model is carried out in terms of the basic reproduction number R0. If R0≤1, the infection-free (semi-trivial) equilibrium is the unique equilibrium and is globally stable; if R0>1, the chronic infection (positive) equilibrium exists and is globally stable under certain assumptions. Examples and numerical simulations for several special cases are presented, including various within-host dynamics models with discrete or distributed delays that have been well-studied in the literature. It is found that the global stability of the chronic infection equilibrium might change in some special cases when the assumptions do not hold. The results show that the model can be applied to describe the within-host dynamics of HBV, HIV, or HTLV-1 infection.

Mathematical analysis of a model for thymus infection with discrete and distributed delays

In this paper, the dynamics of mathematical model for infection of thymus gland by HIV-1 is analyzed by applying some perturbation through two different types of delays such as in terms of Hopf bifurcation analysis. Further, the conditions for the existence of Hopf bifurcation are derived by evaluating the characteristic equation. The direction of Hopf bifurcation and stability of bifurcating periodic solutions are determined by employing the center manifold theorem and normal form method. Finally, some of the numerical simulations are carried out to validate the derived theoretical results and main conclusions are included.

Theoretical foundations for traditional and generalized sensitivity functions for nonlinear delay differential equations

In this paper we present new results for differentiability of delay systems with respect to initial conditions and delays. After motivating our results with a wide range of delay examples arising in biology applications, we further note the need for sensitivity functions (both traditional and generalized sensitivity functions), especially in control and estimation problems. We summarize general existence and uniqueness results before turning to our main results on differentiation with respect to delays, etc. Finally we discuss use of our results in the context of estimation problems.

A division-dependent compartmental model for computing cell numbers in CFSE-based lymphocyte proliferation assays

Some key features of a mathematical description of an immune response are an estimate of the number of responding cells and the manner in which those cells divide, differentiate, and die. The intracellular dye CFSE is a powerful experimental tool for the analysis of a population of dividing cells, and numerous mathematical treatments have been aimed at using CFSE data to describe an immune response [30,31,32,37,38,42,48,49]. Recently, partial differential equation structured population models, with measured CFSE fluorescence intensity as the structure variable, have been shown to accurately fit histogram data obtained from CFSE flow cytometry experiments [18,19,52,54]. In this report, the population of cells is mathematically organized into compartments, with all cells in a single compartment having undergone the same number of divisions. A system of structured partial differential equations is derived which can be fit directly to CFSE histogram data. From such a model, cell counts (in terms of the number of divisions undergone) can be directly computed and thus key biological parameters such as population doubling time and precursor viability can be determined. Mathematical aspects of this compartmental model are discussed, and the model is fit to a data set. As in [18,19], we find temporal and division dependence in the rates of proliferation and death to be essential features of a structured population model for CFSE data. Variability in cellular autofluorescence is found to play a significant role in the data, as well. Finally, the compartmental model is compared to previous work, and statistical aspects of the experimental data are discussed.

A hybrid stochastic-deterministic computational model accurately describes spatial dynamics and virus diffusion in HIV-1 growth competition assay

We present a new hybrid stochastic-deterministic, spatially distributed computational model to simulate growth competition assays on a relatively immobile monolayer of peripheral blood mononuclear cells (PBMCs), commonly used for determining ex vivo fitness of human immunodeficiency virus type-1 (HIV-1). The novel features of our approach include incorporation of viral diffusion through a deterministic diffusion model while simulating cellular dynamics via a stochastic Markov chain model. The model accounts for multiple infections of target cells, CD4-downregulation, and the delay between the infection of a cell and the production of new virus particles. The minimum threshold level of infection induced by a virus inoculum is determined via a series of dilution experiments, and is used to determine the probability of infection of a susceptible cell as a function of local virus density. We illustrate how this model can be used for estimating the distribution of cells infected by either a single virus type or two competing viruses. Our model captures experimentally observed variation in the fitness difference between two virus strains, and suggests a way to minimize variation and dual infection in experiments.

DYNAMICAL MODEL OF VIVAX MALARIA INTERMITTENCE ATTACK IN VIVO

Dynamical behavior of a vivax malaria model is provided and regular recurrences of the symptoms of the tertian fever are described in the human body. We calculate the basic reproduction number R0 and explain the connection between the basic reproduction number and the parasite-threshold. If R0 < 1, then plasmodium vivax will be eliminated. If R0 > 1, then malarial parasites are survivable and there is a so called parasite-threshold. When the value of the parasites is larger than this parasite-threshold the symptoms of the tertian fever appear; otherwise, if the value of the parasites is less than this parasite-threshold the tertian fever cannot give signs of the symptoms suddenly in vivo. We illustrate that the gravity of infected baby is worse than that of infected grownup and also explain that the advancing of the vivax malaria can be arrested by eliminating malarial parasites in erythrocyte stage with clinical treatment by numerical simulations.

A model of HIV-1 infection with two time delays: mathematical analysis and comparison with patient data

Mathematical models have made considerable contributions to our understanding of HIV dynamics. Introducing time delays to HIV models usually brings challenges to both mathematical analysis of the models and comparison of model predictions with patient data. In this paper, we incorporate two delays, one the time needed for infected cells to produce virions after viral entry and the other the time needed for the adaptive immune response to emerge to control viral replication, into an HIV-1 model. We begin model analysis with proving the positivity and boundedness of the solutions, local stability of the infection-free and infected steady states, and uniform persistence of the system. By developing a few Lyapunov functionals, we obtain conditions ensuring global stability of the steady states. We also fit the model including two delays to viral load data from 10 patients during primary HIV-1 infection and estimate parameter values. Although the delay model provides better fits to patient data (achieving a smaller error between data and modeling prediction) than the one without delays, we could not determine which one is better from the statistical standpoint. This highlights the need of more data sets for model verification and selection when we incorporate time delays into mathematical models to study virus dynamics.

Exploring the effect of biological delays in kinetic models of influenza within a host or cell culture

BACKGROUND

For a typical influenza infection in vivo, viral titers over time are characterized by 1-2 days of exponential growth followed by an exponential decay. This simple dynamic can be reproduced by a broad range of mathematical models which makes model selection and the extraction of biologically-relevant infection parameters from experimental data difficult.

RESULTS

We analyze in vitro experimental data from the literature, specifically that of single-cycle viral yield experiments, to narrow the range of realistic models of infection. In particular, we demonstrate the viability of using a normal or lognormal distribution for the time a cell spends in a given infection state (e.g., the time spent by a newly infected cell in the latent state before it begins to produce virus), while exposing the shortcomings of ordinary differential equation models which implicitly utilize exponential distributions and delay-differential equation models with fixed-length delays.

CONCLUSIONS

By fitting published viral titer data from challenge experiments in human volunteers, we show that alternative models can lead to different estimates of the key infection parameters.

Design considerations in building in silico equivalents of common experimental influenza virus assays

Experimentation in vitro is a vital part of the process by which the clinical and epidemiological characteristics of a particular influenza virus strain are determined. We detail the considerations which must be made in designing appropriate theoretical/mathematical models of these experiments and show how modeling can increase the information output of such experiments. Starting from a traditional system of ordinary differential equations, common to infectious disease modeling, we broaden the approach by using an agent-based model, applicable to more general experimental geometries and assumptions about the biological properties of viruses, cell and their interaction. Within this framework, we explore the limits of the assumptions made by more traditional models and the conditions under which these assumptions begin to break down, requiring the use of more sophisticated models. We apply the agent-based model to experimental plaque growth of two influenza strains, one resistant to the antiviral oseltamivir, and extract the values of key infection parameters specific to each strain.

Estimation of cell proliferation dynamics using CFSE data

Advances in fluorescent labeling of cells as measured by flow cytometry have allowed for quantitative studies of proliferating populations of cells. The investigations (Luzyanina et al. in J. Math. Biol. 54:57-89, 2007; J. Math. Biol., 2009; Theor. Biol. Med. Model. 4:1-26, 2007) contain a mathematical model with fluorescence intensity as a structure variable to describe the evolution in time of proliferating cells labeled by carboxyfluorescein succinimidyl ester (CFSE). Here, this model and several extensions/modifications are discussed. Suggestions for improvements are presented and analyzed with respect to statistical significance for better agreement between model solutions and experimental data. These investigations suggest that the new decay/label loss and time dependent effective proliferation and death rates do indeed provide improved fits of the model to data. Statistical models for the observed variability/noise in the data are discussed with implications for uncertainty quantification. The resulting new cell dynamics model should prove useful in proliferation assay tracking and modeling, with numerous applications in the biomedical sciences.

The effects of distributed life cycles on the dynamics of viral infections

We explore the role of cellular life cycles for viruses and host cells in an infection process. For this purpose, we derive a generalized version of the basic model of virus dynamics (Nowak, M.A., Bangham, C.R.M., 1996. Population dynamics of immune responses to persistent viruses. Science 272, 74-79) from a mesoscopic description. In its final form the model can be written as a set of Volterra integrodifferential equations. We consider the role of distributed lifespans and a intracellular (eclipse) phase. These processes are implemented by means of probability distribution functions. The basic reproductive ratio R(0) of the infection is properly defined in terms of such distributions by using an analysis of the equilibrium states and their stability. It is concluded that the introduction of distributed delays can strongly modify both the value of R(0) and the predictions for the virus loads, so the effects on the infection dynamics are of major importance. We also show how the model presented here can be applied to some simple situations where direct comparison with experiments is possible. Specifically, phage-bacteria interactions are analyzed. The dynamics of the eclipse phase for phages is characterized analytically, which allows us to compare the performance of three different fittings proposed before for the one-step growth curve.

A dynamic model for induced reactivation of latent virus

We develop a deterministic mathematical model to describe reactivation of latent virus by chemical inducers. This model is applied to the reactivation of latent KSHV in BCBL-1 cell cultures with butyrate as the inducing agent. Parameters for the model are first estimated from known properties of the exponentially growing, uninduced cell cultures. Additional parameters that are necessary to describe induction are determined from fits to experimental data from the literature. Our initial model provides good agreement with two independent sets of experimental data, but also points to the need for a new class of experiments which are required for further understanding of the underlying mechanisms.

TIME DELAY SYSTEMS WITH DISTRIBUTION DEPENDENT DYNAMICS

General delay dynamical systems in which uncertainty is present in the form of probability measure dependent dynamics are considered. Several motivating examples arising in biology are discussed. A functional analytic framework for investigating well-posedness (existence, uniqueness and continuous dependence of solutions), inverse problems, sensitivity analysis and approximations of the measures for computational purposes is surveyed.

Model Selection and Mixed-Effects Modeling of HIV Infection Dynamics

We present an introduction to a model selection methodology and an application to mathematical models of in vivo HIV infection dynamics. We consider six previously published deterministic models and compare them with respect to their ability to represent HIV-infected patients undergoing reverse transcriptase mono-therapy. In the creation of the statistical model, a hierarchical mixed-effects modeling approach is employed to characterize the inter- and intra-individual variability in the patient population. We estimate the population parameters in a maximum likelihood function formulation, which is then used to calculate information theory based model selection criteria, providing a ranking of the abilities of the various models to represent patient data. The parameter fits generated by these models, furthermore, provide statistical support for the higher viral clearance rate c in Louie et al. [AIDS 17:1151-1156, 2003]. Among the candidate models, our results suggest which mathematical structures, e.g., linear versus nonlinear, best describe the data we are modeling and illustrate a framework for others to consider when modeling infectious diseases.

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.

Probabilistic methods for addressing uncertainty and variability in biological models: application to a toxicokinetic model

Population variability and uncertainty are important features of biological systems that must be considered when developing mathematical models for these systems. In this paper we present probability-based parameter estimation methods that account for such variability and uncertainty. Theoretical results that establish well-posedness and stability for these methods are discussed. A probabilistic parameter estimation technique is then applied to a toxicokinetic model for trichloroethylene using several types of simulated data. Comparison with results obtained using a standard, deterministic parameter estimation method suggests that the probabilistic methods are better able to capture population variability and uncertainty in model parameters.

Dynamics of human T-cell lymphotropic virus I (HTLV-I) infection of CD4+ T-cells

Stilianakis and Seydel (Bull. Math. Biol., 1999) proposed an ODE model that describes the T-cell dynamics of human T-cell lymphotropic virus I (HTLV-I) infection and the development of adult T-cell leukemia (ATL). Their model consists of four components: uninfected healthy CD4+ T-cells, latently infected CD4+ T-cells, actively infected CD4+ T-cells, and ATL cells. Mathematical analysis that completely determines the global dynamics of this model has been done by Wang et al. (Math. Biosci., 2002). In this note, we first modify the parameters of the model to distinguish between contact and infectivity rates. Then we introduce a discrete time delay to the model to describe the time between emission of contagious particles by active CD4+ T-cells and infection of pure cells. Using the results in Culshaw and Ruan (Math. Biosci., 2000) in the analysis of time delay with respect to cell-free viral spread of HIV, we study the effect of time delay on the stability of the endemically infected equilibrium. Numerical simulations are presented to illustrate the results.

Estimates of Intracellular Delay and Average Drug Efficacy from Viral Load Data of HIV-Infected Individuals under Antiretroviral Therapy

We analyse viral load data of five patients under ritonavir monotherapy using a model of HIV dynamics under antiretroviral therapy that includes both drug pharmacokinetics and the intracellular delay from the time of cell infection to viral production. Using this approach we separate pharamacokinetic from intracellular delays, and obtain new estimates of intracellular delay and the antiviral efficacy of ritonavir. We find the average intracellular delay to be 1 day, in agreement with experiments. The average viral generation time is now estimated at 2 days, resulting in ∼180 replication cycles per year. The model also reveals that ritonavir monotherapy is ∼65% as efficacious as a recently used potent four-drug therapy, suggesting that selection for drug resistance may be facilitated by the relatively low efficacy of individual drugs, contributing in part to the inherent limitations of current therapies in combating HIV-1 infection.