Department of Applied Mathematics, University of Western Ontario, London, Ontario, Canada

Abstract

Background

Small, highly reactive molecules called reactive oxygen species (ROS) play a crucial role in cell signalling and infection control. However, high levels of ROS can cause significant damage to cell structure and function. Studies have shown that infection with the human immunodeficiency virus (HIV) results in increased ROS concentrations, which can in turn lead to faster progression of HIV infection, and cause CD4^{+ }T-cell apoptosis. To counteract these effects, clinical studies have explored the possibility of raising antioxidant levels, with mixed results.

Methods

In this paper, a mathematical model is used to explore this potential therapy, both analytically and numerically. For the numerical work, we use clinical data from both HIV-negative and HIV-positive injection drug users (IDUs) to estimate model parameters; these groups have lower baseline concentrations of antioxidants than non-IDU controls.

Results

Our model suggests that increases in CD4^{+ }T cell concentrations can result from moderate levels of daily antioxidant supplementation, while excessive supplementation has the potential to cause periods of immunosuppression.

Conclusion

We discuss implications for HIV therapy in IDUs and other populations which may have low baseline concentrations of antioxidants.

Background

Reactive oxygen species (ROS) are highly reactive byproducts of cellular respiration. As second messengers, they play an important role in cell signaling and in gene regulation (e.g., cytokine, growth factor, and hormone action and secretion; ion transport; transcription; neuromodulation; and apoptosis)

A variety of reactive oxygen species are produced throughout the body. One particular species of interest, superoxide (_{2}O_{2}), a mild oxidant, which further helps to destroy some pathogens. Intermediate concentrations of H_{2}O_{2 }(and certain other ROS) result in the activation of nuclear factor

Despite their positive role, reactive oxygen species can be harmful. At normal ROS concentrations, cell function and structure are protected from destructive interactions with ROS by various defence mechanisms. These include the use of both enzymatic and nonenzymatic antioxidants, substances that significantly delay or prevent the oxidation of a given substrate. Non-enzymatic antioxidants obtained directly from the diet (i.e., glutathione, vitamins A, C and E, and flavenoids) decrease oxygen concentrations, remove catalytic metal ions and eliminate radicals from the system

In the event that intracellular ROS levels increase moderately, cells respond by boosting antioxidant levels and by promoting proinflammatory gene expression

Individuals infected by the human immunodeficiency virus (HIV) exhibit heightened serum concentrations of ROS ^{+ }T cells may impair the immune system's response to HIV ^{+ }T cell concentration in the plasma, is further exacerbated by oxidative stress-induced apoptosis. Third, increased HIV transcription leading to faster disease progression results from an increased activation of NF-

The lowered antioxidant concentrations observed in HIV-positive individuals are associated with micronutrient deficiencies ^{+ }T cell counts, HIV-related diseases, and mortality

Despite many indications that antioxidant supplementation is beneficial in HIV-positive individuals

In short, studies have shown a range of potential implications of antioxidant supplementation. Some have found reasons for concern, others have shown negligible effects, and still others have been positive about the potential of antioxidant supplementation as a therapy or supplemental therapy for HIV-infected individuals. Despite this range of opinions, the 2007 review by Drain

Injection drug users form a particular group of interest due to the endemic nature of HIV infection in this population. According to the WHO, the global population of injection drug users (IDUs) consists of approximately 15.9 million people, of which 3 million are HIV-positive. The spread of the virus is particularly rampant in populations where injecting equipment is re-used and shared. Of the new HIV infections, one in ten are caused by the use of injection drugs. In Eastern Europe and Central Asia, drug use can be attributed to 80% of all HIV infections

A particular clinical study conducted by Jaruga

In the sections which follow, a mathematical model is developed to investigate the use of antioxidants as a treatment strategy for HIV. We use clinical data from Jaruga

Note

Despite the benefits that can be obtained from antioxidant supplementation, we maintain that the need for accessible and affordable antiretrovirals in developing countries is of utmost importance and must not be neglected.

Methods

As outlined in the Background, HIV-infected CD4^{+ }T cells can produce HIV virions via two ROS-independent pathways: either directly or through the activation of NF-

To model these processes, we propose a system of differential equations which consists of four populations: uninfected CD4^{+ }T cells (^{+ }T cells (

where

Schematic diagram of the model

**Schematic diagram of the model**. Reactive oxygen species, while not the sole means for transcription of HIV, directly increase the transcription rate. This results in an increased infection rate

Uninfected CD4^{+}

CD4^{+ }T cells are produced by the thymus at constant rate _{x}, are eliminated from the system at per-capita rate _{x }and become infected through mass-action kinetics at rate

Infected CD4^{+}

CD4^{+ }T cells become infected at rate _{y}.

Reactive oxygen species

ROS are naturally produced at constant rate _{r}. In the event of infection, ROS are also produced by infected cells at a rate proportional to the number of infected CD4^{+ }T cells, _{r}

Antioxidants

Antioxidants are introduced into the system via dietary intake at constant rate _{a}. Plasma antioxidant levels may be supplemented therapeutically at constant rate _{a}

Infectivity

To capture ROS-activated transcription in our model, we would like

While several other forms of ^{p}) points described in the Parameter Estimation section, and illustrated in Figure

The

**The β(r) curve**. The

Note

Many standard HIV models also incorporate an explicit virion population. While virions are not directly modelled in our system, the vital role that they play is not neglected: since they are in quasi-equilibrium with the infected cells, the concentration of virions in the system is roughly proportional to that of the infected cells

Results

Analytical results

Evaluating for the equilibria yields one biologically meaningful disease-free equilibrium:

where _{a}_{m }+ _{r}_{a}-_{r}_{r }>_{r}^{d}, or whenever the production rate of ROS exceeds their overall removal rate, an HIV-negative individual will exhibit a balanced ROS-antioxidant equilibrium.

Using the next-generation matrix method from

which makes intuitive sense since a single infected cell at the uninfected equilibrium will produce new infected cells at rate ^{d})(1 - ϵ)^{d}, for mean lifetime 1/_{y}. (We note that, in practice, ϵ is almost always zero in this situation.)

We next examine stability of the disease-free equilibrium using the following Jacobian:

This yields four eigenvalues

Therefore, the disease-free equilibrium is stable when _{0 }< 1 (from (8)).

In addition to the disease-free equilibrium, two biologically meaningful internal equilibria exist; we omit their analytical expressions here since their complicated form offers little insight. Instead, following parameter estimation, we complete a bifurcation analysis of all three biologically meaningful equilibria in the Numerical Results. We note that our model, and the analytical results described up to this point, could be generalized to other factors that are produced in proportion to infected T cells (

Parameter estimation

Developing reasonable (if uncertain) parameter estimates is one of the most difficult aspects of theoretical immunology, and yet can be an extremely worthwhile endeavour

The model described above includes a total of four populations and 15 parameters. Estimates of six of these parameters (_{x}, _{x}, _{y}, _{a}, _{a }and ϵ) were directly obtained from the literature and can be found in Table _{0 }and the seven clinically measured equilibrium levels from Table _{2}O_{2 }scavenging antioxidant

Parameter estimates from the literature

**Parameter**

**Value**

**Reference**

_{
x
}

60.76 cells ^{-1 }day^{-1}

estimated from _{x}/_{x }

_{
x
}

0.057 day^{-1}

_{
y
}

1 day^{-1}

5.99 × 10^{7 }day^{-1}

(half life = 1 ms)

5.47 × 10^{13 }molecules ^{-1 }day^{-1}

_{
a
}

2.74 × 10^{13 }molecules ^{-1 }day^{-1}

estimated from

_{
a
}

0.0347 day^{-1}

(half life = 8 - 40 days, choose 20 days)

_{0}

4.5

estimated from

ϵ

estimated from

Equilibrium populations from the literature

**Parameter**

**Value**

**Reference**

**Healthy control group**

1,066 cells/

51.5 ± 4.95

56.8 ± 4.5

**HIV(-) (IDU)**

1,066 cells/

12.1 ± 1.8

**HIV(+)P (IDU)**

^{p }+ ^{p}

360 cells/

^{
p
}

43 cells/

from equation (10)

^{
p
}

8.2 ± 1.8

**HIV(+)V (IDU)**

^{v }+ ^{v}

460 cells/

^{
v
}

37 cells/

from equation (10)

^{
v
}

49.0 ± 5.0

Parameter estimates

**Parameter**

**Value**

_{0}

0.000211 (cell/^{-1 }day^{-1}

_{max}

0.00621 (cell/^{-1 }day^{-1}

_{half}

3.57 × 10^{13 }molecules ^{-1}

_{
r
}

1.66 × 10^{7 }day^{-1}

_{
r
}

1.86 × 10^{21 }molecules ^{-1 }day^{-1}

1.49 × 10^{19 }molecules cell^{-1 }day^{-1}

1.27 × 10^{-6 }(molecule/^{-1 }day^{-1}

variable molecules ^{-1 }day^{-1}

5.04 × 10^{-14 }(molecule/^{-1 }day^{-1}

In this section and the work which follows, we also refer to four cases of the infected equilibrium, which differ only in their parameter values. Specifically, we denote (1) the uninfected, control diet case with a "hat" (i.e. ^{p}) and (4) the infected, vitamin supplementation case with a superscript ^{v}) (see Table

Literature estimates for _{a}

It has been recommended that dietary vitamin C intake for all individuals exceed 200 mg per day ^{13 }molecules ^{-1 }day^{-1 }_{a }= 2.74 × 10^{13 }molecules ^{-1 }day^{-1}. Both of these estimates have a high degree of uncertainty since the pharmacokinetics and bioavailability of ascorbic acid are complex

Finding ^{p}, ^{p}, ^{v }and ^{v}

The clinical data in Table ^{p }+ ^{p }and ^{v }+ ^{v}. To find each term independently, we combine equation (1) and equation (2), at equilibrium,

where _{x}, _{x }and _{y }are known. Thus, for the HIV(+)P case, we find ^{p }= 317 and ^{p }= 43. Likewise, for the HIV(+)V case, we find ^{v }= 423 and ^{v }= 37.

Estimating the function

The Jaruga ^{+ }T cell counts from 231 ± 87 cells/^{+ }T cells before and during therapy as a proxy to estimate effectiveness, and assuming that this effectiveness has reached equilibrium after twelve months, we set ϵ to be

We are ultimately interested in modelling the three IDU populations, HIV(-), HIV(+)P and HIV(+)V. Therefore, we take _{0 }to be defined at the HIV(-) case where ϵ = 0. Using (6), we find _{0 }at this equilibrium to be:

Given the parameter values in Table _{0 }and thus _{0 }= 0.000211. From the disease-free IDU equilibrium we therefore have two points with which to fit the _{y }= 1 and ^{p}) =

These three points on the _{max }= 0.00621 and _{half }= 3.57×10^{13}. This fixes the function _{r}. The procedure we use is to estimate a value of _{r}, then follow through the steps described for estimating _{r}, ^{v}, the concentration of ROS at the HIV(+)V equilibrium. We then iteratively adjust our initial estimate of _{r }such that ^{v}(1 - ϵ) = 1/^{v }lies along the dashed curve in Figure _{r }= 1.66 × 10^{7 }day^{-1.}

Estimating _{r},

Given that _{r }= 5.99 × 10^{7 }day^{-1 }(Table _{r}, we directly compute ^{-6. }In addition, since

We assume that _{r}, the rate at which ROS are naturally produced, is constant for all individuals. In contrast, _{a }represents the dietary influx of antioxidants, and we thus assume that _{a }is constant for the HIV(-) and HIV(+) groups, but may differ for the control group. Therefore, we use _{a }for the IDU groups. Thus, we must first find

The parameter

From equation (4) at the HIV(+)P equilibrium,

We find our final parameter,

Finally, from equation (3) at the HIV(+)V equilibrium:

Numerical results

Using the parameters in Tables ^{+ }T cell and antioxidant concentrations in Jaruga ^{+ }T cell concentration from 1066 cells/^{+ }T cell equilibrium of 460 cells/

The analytical results for the control, HIV(-) and HIV(+)P groups

**The analytical results for the control, HIV(-) and HIV(+)P groups**. We include the unstable equilibrium point of the HIV(+)V group at a total vitamin C supplementation level of approximately 116 mg/day. The respective levels of uninfected cells are denoted with circles, infected cells with squares, ROS with diamonds and antioxidants with triangles.

Before examining the benefits and limitations of vitamin supplementation, we test our analytical results using numerical integration (MATLAB ^{®}, The MathWorks Inc.) for HIV-negative IDUs who subsequently become infected with HIV. In the absence of vitamin supplementation, such an individual would display trends similar to those observed in Figure ^{+ }T cells is followed, upon infection, by a sharp decline in the number of uninfected CD4+ T cells which eventually equilibrates at a significantly lower concentration of 317 cells/

An initially uninfected IDU who subsequently becomes infected

**An initially uninfected IDU who subsequently becomes infected**. Here we observe a significant drop in uninfected CD4^{+ }T cell levels (solid line) characteristic of the infection. An equilibrium is eventually reached where ^{14 }molecules/^{12 }molecules/

Next, we examine the behaviour of our model when patients are given moderate daily vitamin supplementation. For this case, our model suggests that an HIV-positive IDU's T cell count can increase, with a concomitant reduction of ROS. However, the magnitude and nature of these changes are dependent upon the level of supplementation. Notice, for example, the outcomes of two different supplementation levels in Figure ^{+ }T cells (to 345 cells/^{v}, found in Jaruga ^{+ }count of 460 cells/

Uninfeted (solid line) and infected (dashed line) cell concentration for an initially infected IDU who begins vitamin supplementation on day 50

**Uninfected (solid line) and infected (dashed line) cell concentration for an initially infected IDU who begins vitamin supplementation on day 50**. In (a), a stable equilibrium results from a supplement of ^{13 }molecules/(^{13 }molecules/(

We further investigate this interesting behaviour through numerical bifurcation analysis, substituting our parameter values into the analytically-determined eigenvalues of the Jacobian. Using the vitamin supplementation level, _{c }= 2.63×10^{13 }molecules/_{c }becomes a stable limit cycle for _{c }(Figure _{c}; however, these are of little clinical relevance.

Bifurcation diagrams of our model of the uninfected T cells and ROS

**Bifurcation diagrams of our model of the uninfected T cells and ROS**. A solid line implies a stable equilibrium and a dashed line implies an unstable equilibrium. In (a), we indicate the maximum attainable stable concentration of uninfected cells, _{max }(dotted black line). The dotted horizontal line in (b) indicates the ROS level observed in non-IDU control individuals. We denote the ROS concentration for which the disease-free equilibrium becomes stable with the grey box.

These bifurcation diagrams also confirm what we found in the Analytical Results: the disease-free equilibrium is stable when _{0 }< 1. This occurs when ^{12 }molecules/(^{13 }molecules/

The behaviour of the limit cycle is further examined in the region where _{c }by integrating our system numerically for 600 days and measuring the time between the last two peaks. As shown in Figure _{c}, the period of the oscillations increases dramatically. Interestingly, as _{c}, the oscillation is moderate, with symmetrical peaks and troughs. Higher levels of ^{+ }T cell counts followed by sharp, short-lived periods in which the patient is in an immunocompromised state. Regardless of the shape of these oscillations, a therapeutic regimen which causes repeated periods of immunosuppression would not be clinically advisable. Thus, our model predicts the existence of a maximum vitamin supplementation level, _{c}, beyond which further supplementation might be detrimental.

The period of the limit cycle as a function of vitamin supplementation levels

**The period of the limit cycle as a function of vitamin supplementation levels**. As the vitamin supplementation level increases beyond _{c }= 2.63 × 10^{13 }molecules/^{13 }and ^{13}, respectively. In the insets, the uninfected and infected cell concentrations are respectively represented by the solid and dashed lines.

To better understand this threshold behaviour, we look at _{max}, which we define to be the maximum attainable stable equilibrium concentration of uninfected T cells; that is, the equilibrium value of _{c }(Figure _{max }= 369 cells/^{v }= 423 cells/_{max }is sensitive to assumptions regarding our parameter values.

Sensitivity analysis

We examine the sensitivity of our model to several parameters for which our assumed values have a high degree of uncertainty, or which may display significant interpatient variability. In particular, we look at how the maximum attainable uninfected CD4^{+ }T cell concentration, _{max}, changes as a result of varying parameters. In each case, to compute _{max}, we performed a numerical bifurcation analysis as illustrated in Figure

We test for sensitivity in two ways. First, we examine the sensitivity of _{max }to the parameter values from the literature which we initially assumed in the Parameter Estimation section and upon which further parameter estimates depend. In a second analysis, we look at the sensitivity of _{max }to interpatient parameter variation. In both sections, we examine the trends in _{max }as well as the corresponding concentrations of infected T cells, ROS and antioxidants when a parameter of interest is varied.

Sensitivity to initial parameter estimates

In this section, we vary five parameters which have a high degree of uncertainty in order to test the overall sensitivity of our results to these assumed parameter values. In cases where the values of other parameters depend on these initial estimates, we subsequently recompute all other dependent model parameters, using the method described in the Parameter Estimation section.

Dietary antioxidant intake of the controls

First, due to the natural variability surrounding the diet of control individuals and the uncertainty regarding the amount of antioxidants absorbed, we vary _{r}, _{max}, _{half }and ^{p }and ^{v }were altered. From Figure ^{v}, while _{max }increases only slightly: a 200% increase in _{max}.

Sensitivity analyses of (a) _{a}

**Sensitivity analyses of (a) **. The equilibrium concentration of uninfected cells is represented by the solid line, infected cells by the dashed line, ROS by the dashed-dotted line and antioxidants by the dotted line. We also include the level of antioxidant supplementation,

Dietary antioxidant intake of IDUs

For reasons similar to those posed above, we secondly analyse the sensitivity of _{a}, the amount of antioxidants absorbed from the diet of IDUs, and find that _{max }decreases modestly as _{a }increases (Figure _{a }changes, so do our estimates of parameters _{r}, _{max }and _{half }. Equilibria ^{p}and ^{v }were altered as well. This restricts the range we can examine; when _{a }< 0.048 g day^{-1}, the positivity of certain parameter values is lost. Importantly, close to the lowest possible value of _{a}, we are able to replicate the HIV(+)V Jaruga ^{v }+ ^{v }= 460.

Again, a very modest change is observed: a 220% parameter increase results in a 21% decrease in _{max}.

Drug effectiveness

Third, we vary drug effectiveness due to our uncertainty surrounding its estimate and its dependence upon the treatment regimen. When ϵ changes, so do our estimates of parameters _{r}, _{0}, _{max }and _{half }. Equilibria ^{v }were altered as well. In Figure _{max}, although again _{max }is moderately sensitive to this parameter: a 31% increase in ϵ causes a 14% decrease in _{max}. Note that, at higher values of ϵ than illustrated in Figure

Sensitivity analyses of (a) ϵ and (b) _{0}

**Sensitivity analyses of (a) ϵ and (b) R _{0}**. The concentration of uninfected cells is represented by the solid line, infected cells by the dashed line, ROS by the dashed-dotted line and antioxidants by the dotted line. We also include the level of antioxidant supplementation,

Basic reproductive ratio

Fourth, since there is uncertainty surrounding the value of _{0}, the results of a range of parameter values are analysed. Note that, when _{0 }changes, so do our estimates of parameters _{r}, _{0}, _{max }and _{half }. Equilibria ^{v }were altered as well. We observe in Figure _{0 }increases by 17%, _{max }increases by 24%; therefore, we find that _{max }is somewhat sensitive to changes in _{0}. Values of _{0 }lying below the range presented in Figure

ROS removal

Finally, since the removal rate of ROS is extremely rapid and is therefore difficult to compute, we analyse the system for varying removal rates, _{r},

Despite these cascading changes to subsequently computed parameters in response to changes in _{a }or ϵ, we find that _{max }is fairly insensitive. However, the value of _{max }is somewhat sensitive to our initial assumption of the in-host _{0 }for HIV, which is interesting given that the value of this parameter is not well known _{c }is very sensitive to our initial assumptions regarding these parameters. We are able to replicate clinical results under the assumption that the IDU group has a very low dietary intake of antioxidants, corresponding to 48 mg absorbed per day.

Sensitivity to interpatient variability

In this section, we quantify the sensitivity of our model to interpatient variation for several parameter values. Unlike in the previous section where dependent parameter values were recalculated in response to variation in an assumed parameter, here we only vary the parameter of interest and hold all other parameters constant, except _{c }as before.

Drug effectiveness

Our first parameter of interest is drug effectiveness, since ϵ varies from patient to patient due to differences in HIV progression and levels of adherence. As anticipated, our model is sensitive to the level of effectiveness, with _{max }(solid line) rising with increasing effectiveness (Figure _{max }and increases the chance of oversupplementation. Thus, our model predicts, interestingly, that antioxidant supplementation should be reduced in patients who exhibit strong adherence, although some level of supplementation would continue to be beneficial.

Sensitivity analyses of (a) ϵ and (b) _{0 }for interpatient variability

**Sensitivity analyses of (a) ϵ and (b) R _{0 }for interpatient variability**. The concentration of uninfected cells is represented by the solid line, infected cells by the dashed line, ROS by the dashed-dotted line and antioxidants by the dotted line. We also include the level of antioxidant supplementation,

Basic reproductive ratio

Second, we test the sensitivity of our results to _{0}, since this parameter could also display interpatient variability due to differences in immunocompetence, disease progression and other factors. As we observe in Figure _{max }is relatively insensitive to changes in _{0 }over an extremely wide range: an increase from 0 to 35 results in a mere 3% increase in _{max}.

Natural ROS production

In the formulation of our model, we made the assumption that the natural rate of ROS production, _{r}, was the same for all individuals. Therefore, we thirdly examine the effect of a varying interpatient _{r}. In Figure _{r}, ROS concentrations (dashed-dotted line) initially decrease and therefore _{max }(solid line) initially increases. This trend can be attributed to significant increases in antioxidant supplementation levels (thin, grey line); as _{r}increases, higher values of _{c }are possible without losing the stability of the equilibrium. However, since physiological constraints would presumably impose some limit on the degree of the vitamin supplementation possible, we set a maximum antioxidant supplementation level of 2.0 × 10^{14 }molecules ^{-1 }day^{-1}, which is approximately 586 mg/day, absorbed into the bloodstream. The quantitative value of this limit has been chosen arbitrarily to illustrate the qualitative effects of the physiological limit which presumably exists.

Sensitivity analysis of _{r }for interpatient variability

**Sensitivity analysis of λ _{r }for interpatient variability**. The concentration of uninfected cells is represented by the solid line, infected cells by the dashed line, ROS by the dashed-dotted line and antioxidants by the dotted line. We also include the level of antioxidant supplementation,

Thus, the increases in _{max }continue until _{r }= 5.65 × 10^{21 }molecules ^{-1 }day^{-1}. Further increasing _{r}, combined with a constant _{max }to decrease. We address this interesting qualitative prediction further in the Discussion.

Dietary antioxidant intake of IDUs

Lastly, we examine the effect of a varying dietary antioxidant intake and find that our results are insensitive to this variation, the only change being an alteration in the vitamin supplementation level required to achieve _{max }(data not shown).

Discussion

We have developed and analysed a simple model of the interactions between CD4^{+ }T cells, reactive oxygen species and antioxidants. Verifying the results of various clinical studies, our model predicts that moderate levels of antioxidant supplementation in HIV-positive IDUs can lead to an increase in uninfected CD4^{+ }T cell concentrations. However, our model also suggests that excessive supplementation could cause fluctuating T cell concentrations in these individuals. For example, consider the limit cycle in Figure ^{+ }T cells - leaving the individual vulnerable to opportunistic infections.

In an effort to understand this periodic behaviour, we take a closer look at the system dynamics when the level of antioxidant supplementation is above the critical level, _{c}, in Figure

A closer look at the dynamics of the stable limit cycle

**A closer look at the dynamics of the stable limit cycle**. The concentration of uninfected cells is represented by the solid black line, infected cells by the dashed line, ROS by the dashed-dotted line and antioxidants by the dotted line, each rescaled for comparison. The solid grey line denotes

Regardless of its cause, the appearance of a limit cycle in our model could explain why some clinical studies show no improvement in patients' average CD4+ T cell concentrations: it is plausible that high supplementation levels could cause fluctuating T cell counts which are then sensitive to the details of measurement timing, leading to the conclusion that antioxidant supplementation has no immunological benefit for HIV-positive patients.

Since antioxidant supplementation levels above a critical value, _{c}, have the potential to pose difficulties for patients, we turn our attention to the stable equilibria obtained when _{c}. We examined in particular the maximum concentration of uninfected CD4^{+ }T cells, _{max}, which could be obtained in principle as a stable equilibrium via antioxidant supplementation. We found _{max }to be relatively insensitive to moderate variation in five initial parameter estimates, particularly when subsequent parameter estimates were changed as a result of these alternative assumptions. This insensitivity is presumably because subsequent parameters act to compensate for alternative assumptions, since we set parameters to match the clinically-observed equilibria. These compensatory changes also explain why the results described in the analysis of the sensitivity to initial parameter estimates seem counter-intuitive; for example, as our initial assumption for the in-host _{0 }increases, _{max }also increases (Figure _{max }is unlikely to be achieved in practice, since the required level of precision in the supplementation level would be impossible.

Interestingly, our sensitivity analysis revealed that even when our initial parameter estimates were varied, the mean T cell count observed by Jaruga _{a}. In the region of instability, however, values equivalent to the clinical data were frequently observed. For example, in Figure ^{+ }T cells at six months, 479 cells/^{13 }molecules/

The oscillatory dynamics of the system when 84 mg of the daily vitamin supplement is absorbed

**The oscillatory dynamics of the system when 84 mg of the daily vitamin supplement is absorbed**. We see that six months after the start of supplementation, we reach CD4^{+ }T cell levels observed in the Jaruga

To further investigate the benefits of antioxidant supplementation, we hope that future work could see the model extended to include appropriate pharmacokinetics of antioxidants. In its present form, our model considers

It would also be interesting to explore the effects of enzymatic antioxidants: glutathione peroxidase and catalase, for example. Both of these enzymes are used in the elimination of hydrogen peroxide (H_{2}O_{2}), but are not consumed by these reactions. Their short half-lives (less than 10 minutes)

Conclusion

While antioxidant supplementation may not be a long term solution for HIV-positive IDUs, our model suggests that moderate doses of antioxidants may temporarily boost uninfected CD4^{+ }T cell concentrations. This might enable HIV-positive individuals to lengthen the interval before costly drugs with severe side effects become necessary. These results could have implications for infected individuals in HIV-endemic areas, since dietary antioxidant intake depends on the availability of adequate antioxidant-rich produce. Moreover, where access to antiretroviral therapy is limited or non-existent due to economic constraints, a significantly more affordable vitamin supplementation therapy could potentially provide some limited benefit. Of course we emphasize that this in no way reduces the need for accessible and affordable antiretrovirals in developing countries.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

RDvG and LMW developed the model. RDvG analyzed the model, analytically and numerically, and produced all figures. RDvG and LMW interpreted the results. RDvG drafted the manuscript.

Acknowledgements

The authors thank Pei Yu and two anonymous referees for their insightful comments. This work is supported by the Natural Sciences and Engineering Research Council of Canada and the Ontario Ministry of Training, Colleges and Universities.

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