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

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Abstract

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.

Introduction

Viruses are obligate intra-cellular parasites with a multitude of pathways for infecting and reproducing within their target hosts. The Human Immunodeficiency Virus (HIV) is a lentivirus that is the etiological agent for the slow, progressive, and fatal Acquired Immunodeficiency Syndrome (AIDS) to which there is currently no known cure. According to a Joint United Nations Programme on HIV/AIDS June 2000 report, there were ≈34.3 million individuals infected with HIV/AIDS worldwide at the end of 1999, including 24.5 million in sub-Saharan Africa [1]. Thus, HIV-related illness and death is and will continue to be an important clinical and public health issue as well as an international security, stability, and development issue. Clearly it is imperative that we attain a greater understanding of HIV/AIDS viral infection dynamics.

For HIV, the core of the virus is composed of single-stranded viral RNA and protein components. As depicted in Fig. 1, when an HIV virion comes into contact with an uninfected target cell, the viral envelope glycoproteins fuse to the cell’s lipid bilayer at a CD4 receptor site and the viral core is injected into the cell. Once inside, the protein components enable transcription and integration of the viral RNA into viral DNA and then incorporation into the cellular DNA (provirus). With its altered cellular DNA, the cell produces capsids and protein envelopes and transcribes multiple copies of viral RNA. The cell assembles a virion by then encasing the viral RNA in a capsid followed by a protein envelope. The new HIV virion pushes out through the cell membrane budding off in chains of virions (though sometimes single virions do float away into the plasma). Clearly the time from viral infection to viral production (sometimes called the eclipse phase [2]) is not instantaneous, and (as indicated in the figure) it is estimated that the first viral release occurs ≈24 h after the initial infection [3], [4]. As mentioned before, the development of mathematical models and the associated numerical techniques that incorporate these delays into models for HIV infection dynamics is a primary motivation for our efforts here.

Within the HIV modeling community, there is considerable debate upon the proper compartment definitions. It is not our goal to advocate one model over another, and as such, we will simply choose one with which we illustrate our methodology. The multi-compartment model that we use (and discuss in detail below) describes the moment a virion contacts the appropriate receptor site as the beginning of acute cellular infection. If the acutely infected cell survives through its first viral release, roughly 3 h later [3] the physiological characteristics of the cell change and it is subsequently classified as a chronically infected cell (see [5] for more discussion regarding the choice of these compartments). Note that in the chronic stage, it is possible for the cells to continue to divide and to produce virions, albeit at a much slower rate than acutely infected or non-infected cells.

Over the past seven years, the use of mathematical models as an aid in understanding features of HIV and other virus infection dynamics has been substantial. Several papers published in the mid-nineties provided strong evidence for the high rate of HIV-1 replication and clearance in infected individuals [6], [7], [8]. By the end of the decade, the general consensus was that, in vivo, on the order of 1010 virions are assembled and cleared every day [9], [10], [11]. In many of these papers, the viral clearance rate c was identified by modeling the disease pathogenesis with a system of deterministic differential equations, numerically calculating a solution, and then fitting the results with experimental data (using a non-linear least squares (NLS) approach), e.g., see [10], [7], [11]. The existence of such a high replication/clearance rate implies a high mutation rate, thus indicating that pharmacological mono-therapy will ultimately fail, since the virus can rapidly manifest a resistance to any one drug. More importantly, this knowledge directly contributed to the current practice of simultaneously administering multiple drugs to HIV positive individuals in an effort to counteract the high mutation rate of the virus.

Following its success in helping to identify this significant feature of the HIV pathogenesis, the use of mathematical modeling and parameter identification in the study of HIV experienced a dramatic increase. In particular, in the wake of the publication of [7], there were papers covering everything from additional and/or alternative compartment formulations [12], [13], [14], [15], [16], [17], [18], [19], [20], [21] to arguments for and against the use of delay differential equations in modeling the eclipse phase [2], [4], [22], [23], [24], [25], [26], [27] (including those that addressed the solution stability [26], [27], [28], [29]). As mentioned before, our goal here is to present and illustrate the use of tools which, hopefully, will allow one to develop new insights into HIV pathogenesis. Indeed, there is a precedence for this approach, as is evidenced by previous papers within the HIV modeling literature that make use of stochastic analysis and inference [30], [31], [32], [33], [34], [35], control theory [36], [37] and non-linear analysis [38], [39]. Note that the above survey is not intended to be comprehensive, as there already exist excellent and thorough reviews of the field presented in [40], [41], [42].

Within the context of delay equations, many of the aforementioned papers focused heavily on the inter-relationship between the parameters describing the drug efficacy η, the length of the eclipse phase τ, the infected T-cell death rate δ, and the virion clearance rate c [2], [4], [22], [24], [25], [26], [27], [29]. One approach to numerically simulating these systems with delays is sometimes referred to as the method of stages (MOS) and is described in [4], [25], [27], [43], [44], [45]. Indeed, several researchers have used this technique to simulate delay ODE models of HIV and argue that the inclusion of delays in the viral production of infected cells dramatically changes estimates of specific parameters. For example, Grossman et al., [22] argue that including a delay in the model for the death of infected cells leads to different conclusions regarding residual transmission of infection (during antiviral pharmaceutical therapy), while Lloyd [25] argues that an absence of delays in the model leads to (as we now know suspiciously) optimistic conclusions about treatment efficacy. Clearly an accurate model for the delay is important, and in Section 2, we carefully describe a way to properly develop a model for the delay that accounts for many of its biological aspects.

In all of the cited papers which include temporal delays in their models, the delay is either treated as a constant or represented as a distribution of delays. For the models with a constant delay, the resulting system of delay-differential equations may be readily simulated using the method of steps. For those models with a distributed delay, the gamma distribution (or Erlang distribution [44]) is used, which allows the resultant system of integro-differential equations to be reduced to a system of (non-delayed) ordinary differential equations [46]. This non-delayed system can then easily be simulated with standard numerical integration techniques (e.g., Runge–Kutta) in commonly used mathematical software. An alternative method (an implementation of which is presented in Section 4.2) that first converts a delay system into an abstract evolution equation (AEE) before numerical simulation is described in [47], [48], [49]. This approach allows for simulation of systems with general delay kernels describing the delay distributions and does not require that the model be reduced to a system of non-delayed ODEs.

In the course of developing our model, we employ a delay to mathematically represent the temporal lag between the initial viral infection and the first release of new virions. The study of delay equations has a long history in fields as disparate as economics [50] and ecology [51], with some early applications in engineering found in research concerning the stability of naval vessels [52]. Furthermore, there has also been extensive use of delay equations in modeling biological systems and indeed both May’s and Murray’s classic texts ([53] and [54] respectively) have a significant sections devoted to delay equations. For the interested reader, there are solid introductory texts [55], [56] (including those that focus heavily on applications to biological systems [57], [58]) and thorough (if somewhat theoretical) advanced texts [59], [60], [61].

In this paper, we concentrate on the mathematical modeling of viral dynamics, focusing in particular on the mathematical aspects and biological nature of the delays in primary infection. We also extend previous modeling work on HIV infection dynamics for in vitro laboratory experiments from the (continuous) delay differential equations developed in [62], which in turn were based on a discrete dynamical system from [5]. We use this model along with the in vitro data from [63] to illustrate the methods we discuss for both forward and inverse problems. Our primary contribution focuses on the use of this methodology in complex delay systems with data containing variability. While we do provide a fit-to-data with these data, it is the potential for use of such methodology in further HIV studies that should be of great interest to readers.

Section snippets

Models

We begin with a modification of the system of ordinary differential equations developed in [62] given byV̇(t)=−cV(t)+nAA(t)+nCC(t)−pV(t)T(t),Ȧ(t)=(rv−δA−γ−δX(t))A(t)+pV(t)T(t),Ċ(t)=(rv−δCδX(t))C(t)+γA(t),Ṫ(t)=(ru−δuδX(t)−pV(t))T(t)+Sfor 0⩽ttf with tf finite, where the parameters and the compartments are described in Table 1, Table 2, respectively, and t is the continuous independent time variable. In the first equation, the −pV(t)T(t) term is designed to account for the biological fact

Modeling of delays and variability

As mentioned in the Introduction, it is known that there exist temporal delays between viral infection and viral production and between productive acute infection and chronic infection.

A central focus of our modeling effort has been on attempting to obtain reasonable mathematical representations of these delays. The problem of how to mathematically represent these phenomena is decidedly non-trivial and includes issues such as how to account for intra-individual variability (e.g., inter-cellular

Numerical implementation

For those interested in the mathematical aspects of simulating an FDE system, this section contains an outline of the necessary mathematical and numerical analysis foundations. In Section 4.1, we describe the conversion of the FDE system to an AEE system as well as provide existence and uniqueness results for a solution to the FDE (proofs are given in [65]). Section 4.2 contains details of the numerical implementation along with the convergence results for our numerical method.

In order to

Numerical results

The number of data points in Fig. 2 is insufficient to carry out (with any degree of confidence) rigorous inverse problem investigations or to perform a legitimate statistical analysis with models such as those discussed above. However, our goal is to illustrate our methodology, and thus in this section, we perform the inverse problem calculations (fully aware of the shortcomings due to paucity of data points) to obtain an estimate of the delays and then compare these calculated values with the

Conclusions

We have mathematically modeled a biological system (using coupled FDE) that arises in the study of HIV infection dynamics and offered a derivation supporting a non-deterministic mathematical treatment of the biological delays. We converted these equations to an AEE to facilitate analysis and numerical approximation of the system. We used a χ2 statistical test to support our claim as to the significance of the presence of the delays in fitting experimental data. Additionally, our numerical

Acknowledgements

This research was supported in part by the Air Force Office of Scientific Research under grants AFOSR-F49620-01-1-0026 and AFOSR-F49620-98-1-0430, in part by the Joint DMS/NIGMS Initiative to Support Research in the Area of Mathematical Biology, under grant 1R01GM67299-01, in part by NIH grant R01AI42522, and in part through a GAANN Fellowship to D.M.B. under Department of Education grant P200A980801. We also wish to thank Dr Michael Emerman for the use of his experimental data as well as

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