Mutation and Control of the Human Immunodeficiency Virus · Mutation and Control of the Human Immunodeficiency Virus 2 Together with a high production rate, the continuing ap-pearance

Mutation and Control of the Human Immunodeficiency Virus

Robert F. Stengel

Princeton University and Institute for Advanced Study

Princeton, NJ e-mail: [email protected]

Abstract - We examine the dynamics of infection by the hu-man immunodeficiency virus (HIV), as well as therapies that minimize viral load, restore adaptive immunity, and use minimal dosage of anti-HIV drugs. Virtual therapies for wild-type infections are demonstrated; however, the HIV infection is never cured, requiring continued treatment to keep the condition in remission. With high viral turnover and mutation rates, drug-resistant strains of HIV evolve quickly. The ability of optimal therapy to contain drug-resistant strains is shown to depend upon the relative fitness of mutant strains. Index Terms – Human immunodeficiency virus, immune system dynamics, optimal control.

I. INTRODUCTION Tens of millions of people have been infected by the

human immunodeficiency virus (HIV) since it was first recognized in the early 1980s, and more than 20 million have died from ensuing disease [1-4]. The virus attacks CD4-presenting cells -- helper T cells, macrophages, den-dritic cells, eosinophils, microglia, and natural killer cells -- mainstays in the body’s immune response to pathogenic attack. Without treatment, immune cells and lymphatic tissue are destroyed, and the infection evolves into ac-quired immune deficiency syndrome (AIDS), an inevita-bly fatal condition for all but a few [5]. While AIDS it-self is rarely the cause of death, the body becomes vulner-able to a host of potentially lethal opportunistic infections and malignancies, including pneumonia, tuberculosis, dementia, bacterial sepsis, lymphoma, and Kaposi’s sar-coma. With multi-drug, highly active antiretroviral ther-apy (HAART), the rate of HIV infection can be dramati-cally slowed; fatalities brought about by the disease have plummeted since the inception of HAART [6].

presented at the 13th Yale Workshop on Adaptive and Learning Systems, Yale University, New Haven, CT, May 30 - June 1, 2005. Copyright 2005 by Robert F. Stengel. All rights reserved.

Nevertheless, HAART is not a cure because current drug regimens cannot eliminate diverse HIV strains in a broad population [6-13]. Furthermore, HAART can lead to serious and discomforting side effects. Cardiovascular disease, hepatitis, pancreas damage, glucose intolerance, kidney stones, rashes, depression, fat accumulation and redistribution, nausea, diarrhea, pain or numbness in the feet, mouth inflammation, blurred vision, headache, diz-ziness, anemia, weakness, insomnia, and adverse inter-action with non-HIV drugs are not uncommon [14].

There are a number of reasons why the human im-munodeficiency virus cannot be defeated with current therapy. The virus targets different immune-system cell types that are capable of replicating the virus; treatments that inhibit HIV replication in one cell type may not be as effective for another. Once infected by a single HIV par-ticle (or virion), an immune cell may produce over a thou-sand copies of the virion before the cell is destroyed; hence, unchecked infection is exponentially unstable. The virus is short-lived but prolific, with maximum daily turnover rates measured in the billions of virions. Be-cause all encounters between cells and molecules within the body’s circulatory systems are stochastic, both infec-tion and therapeutic effect are never certain. The virus sequesters itself in a variety of reservoirs throughout the body, some of which are physically resistant to therapy. Long-term remission requires continued stable control over the immune cells that are vulnerable to infection for the remainder of the patient’s life. There is no assurance that the last virion will not replicate unless the therapy induces stable decay to the HIV production process [15].

Perhaps most important, the infection is able to adapt to maintain its virulence in the presence of natural im-mune response and drug therapy. HIV is a retrovirus that transmits the genetic code contained in its dual strands of RNA by reverse transcription to cDNA [1]. This process is not monitored and controlled by the exonuclease proof reading that occurs in forward transcription from DNA to RNA. Thus, the DNA that the virus splices into the host cell’s genome is highly variable. In the cell’s subsequent “normal” transcription, many mutants of the virus are replicated, and some of these are likely to resist therapy.

Mutation and Control of the Human Immunodeficiency Virus 2

Together with a high production rate, the continuing ap-pearance of diverse mutants is assured. In the majority of patients, the mutant virus destroys the immune system and eludes treatment, progressing to AIDS and, finally, death.

In the remainder of the paper, we summarize HIV infection and immune response, present a mathematical model of HIV infection with drug-resistant mutation, and demonstrate the effects of mathematically optimal ther-apy. The virtual therapy that we derive is optimal in the sense that it minimizes the viral load and infected T cell count, trading off the beneficial effects of reduced con-centrations of the virus and infected cells against the ad-verse side effects of therapy. We see that normal compe-tition between wild-type and less-fit variant HIV strains plays an important role in providing such control; above a certain fitness level, the drug-resistant strain cannot be controlled. Opportunities to apply adaptive control con-cepts are limited because high-fitness mutation defeats the efficacy of known therapeutic agents.

II. INFECTION AND IMMUNE RESPONSE

The HIV infection of CD4-presenting cells begins

with transfer of the virus across the mucous membranes of sexual partners, transfusion of HIV-infected blood, transplantation of HIV-infected tissue, pre- or post-natal infection of a child by its mother, or injection using an in-fected needle. Macrophages and helper T cells (or Th cells) are generally among the first cells to be infected. The infection is then passed to other immune-system cells primarily within lymphoid tissue [16]. Th cells are de-pleted, and the functions of lymphoid organs are disrupted over the course of HIV infection [1, 5, 17, 18].

Once the HIV particle is fused to the target CD4 cell, the cell membrane is breached, and the contents of the virion enter the target [26]. Each viral RNA strand is accompanied by reverse transcriptase, an enzyme that facilitates the production of a corresponding cDNA strand, and integrase, an enzyme that aids the integration of the cDNA into the target cell’s genome. This inte-grated strand of DNA is called a provirus, and the in-fected cell is a proviral cell. Transcription of the modi-fied DNA genome to its RNA image cannot occur unless the CD4 cell has recognized a foreign antigen from the HIV, another pathogen, or immunizing vaccine directed at another pathogen. Upon activation, transcription of RNA derived from the original and proviral DNA is initiated [19], and the cell becomes productively infected, that is, it can produce new copies of HIV particles.

Transcription is followed by translation of the new RNA to original and HIV proteins. Through the action of a protease enzyme also carried in the infecting virion, proteins essential to creating new virions are produced. The HIV RNA and proteins move to the cell membrane, assemble into a viable unit, and “bud” from the target cell

as a new virion. This process occurs many times in a productively infected cell, producing exponential growth in the number of HIV particles. Ultimately, the cell un-dergoes programmed cell death (apoptosis), disintegrating and releasing additional HIV product into the compart-ment.

Although there is a multiplicity of T-cell types in-volved in HIV infection, activated Th cells, which have recognized a foreign antigen, are the primary producers of and targets for HIV particles [20]. Most Th cells are in-fected and reside in lymphoid tissue [21], where they can be observed only by biopsy. Just 1-2% of infected cells reside in the periphery, where observation by blood test is possible; hence, the balance between cell destruction and impaired production that leads to depletion over time is not well understood.

While Th cells produce most of the new virus, eosi-nophils [22], natural killer cells [64], microglia [23, 24], and macrophages that are within the central nervous sys-tem or have been co-infected by opportunistic pathogens [23, 25] can be productively infected, and lymphoid tissue can harbor HIV. Other immune system cell types may be infected but not killed by HIV, producing long-term reser-voirs for the virus [22, 26, 27]; however, they generally do not present antigens to the B and T cells of the adap-tive immune system [19] and, therefore, play a secondary role in the dynamics of HIV infection.

The immune system recognizes and mounts a strong response to HIV infection through innate, humoral, and adaptive pathways [16, 28]. However, the response is not uniformly protective, and chronic or hyper-activation of the immune system is associated with poor clinical out-come [29, 30]. During the first weeks following infec-tion, both CD4 and CD8 T cells proliferate at a high rate, there is a sharp drop in viral load, and antibodies for the virus’s antigens build to measurable levels in the blood. The innate response is not specific to the intruder’s anti-gen; it destroys virions and infected cells that are recog-nized as “non-self.” The humoral response is guided by antigen-specific antibodies, which bind the antigen and present a constant peptide region to invariant receptors on phagocytic cells.

The adaptive response is spearheaded by activated killer (CD8) T cells, which require the help of Th cells (the principal target of infection) to recognize foreign antigen [19]. Killer T cells initially destroy large numbers of HIV-infected cells [15, 19, 29, 31-33], but the deple-tion of helper T cells over the course of time degrades the ability of killer T cells to destroy infected cells. The cy-totoxic effects of antigen-specific CD8 T cells eliminate the HIV strains of early infection, but this action renders the trained cells obsolete and clears the way for mutant strains expressing different antigens to prevail. HIV in-fection has a profound effect not only on the functions of T cells but on the lymphoid tissue in which they are cre-ated and mature, the bone marrow and thymus. The pro-


duction of new T cells decreases, and the body is in-creasingly dependent on cell division in the periphery to maintain immune protection.

Thus, “the HIV-1-specific response becomes part of the problem as well as part of the solution” [34]. The human immunodeficiency virus is uniquely capable of immune system judo, leveraging the system’s signaling and effecting mechanisms to work against itself. Sig-naling and effector cells are depleted, organs atrophy, and immune function is lost at all levels, leaving the patient susceptible to the opportunistic infections and malignan-cies that characterize AIDS.

III. THERAPY, PERSISTENCE, AND MUTATION Because HIV has a high turnover rate, the main goal

of current HIV therapy is to break the replication cycle. Fundamental damage to the immune system occurs during the first weeks of infection, when the diversity of virions is small [34]. A “hit early, hit hard” approach to therapy is more likely to minimize damage to lymphoid tissue and to eradicate a higher percentage of virulent particles than delayed treatment would [35-38].

With ten to a hundred million productively infected copies circulating at any given time, Th cell infection is not only the disease’s predominant feature, it is at the core of the immune system’s dynamic response to HIV. Cur-rent HIV drugs enter the cells’ nucleus and cytoplasm via passive diffusion, and they operate at chokepoints in the replicative sequence. Reverse transcriptase inhibitors (RTIs) are intended to prevent viral RNA from producing viral cDNA, thereby preventing a target cell’s genome from being modified. The HIV’s reverse transcriptase cannot distinguish between viral nucleosides and nucleo-side-analogue RTIs (NRTIs), which contain a chain-ter-minating sequence that prevents the formation of com-plete HIV cDNA [39, 40]. Non-nucleoside RTIs (NNRTIs) use a different mechanism, blocking a binding pocket in the HIV reverse transcriptase to halt synthesis. Protease inhibitors (PIs) act farther downstream in the replication process, preventing protease from cleaving the long HIV peptide chain produced by translation into smaller proteins that are essential to fusing virions with new targets. A PI does not prevent the synthesis of viral components, but it does render the virions non-infectious. RTIs act primarily in newly infected cells, while PIs are also effective in chronically infected cells [41]. Dosage is a critical factor not only in treating the disease but in causing adverse side effects and creating selective pres-sure for mutation. PIs are generally less toxic than RTIs, and they are less likely to stimulate drug-resistant mu-tants.

The principal failing of these drugs is that they are vulnerable to mutation in the HIV genome. The genome contains about 104 base pairs, and the point mutation rate is about one per 103-104 base-pair replications, or one to

ten per virion replication [35]. Given the broad nature of mutation [42], only a small percentage of mutant quasis-pecies1 are viable [44]. However, the number of new virions produced each day is large, and it is just a matter of time before strains resistant to any monotherapy (i.e., single-drug therapy) appear. Without therapy, wild-type strains dominate most mutant strains; however, with ther-apy, wild-type quasispecies are repressed, giving less-fit (but potentially lethal) mutants the opportunity to survive [45].

While a single point mutation in the reverse-tran-scriptase genomic sequence is sufficient to cause resis-tance to an NNRTI [46], multiple mutations are required to escape NRTIs or PIs [43, 47]. Furthermore, resistance to one drug in a class (e.g., NRTI, NNRTI, or PI) can produce cross-resistance to other drugs in the same class [47]. Consequently, monotherapy is susceptible to early drug resistance. Because HAART employs several drugs of different types, many mutations are required to defeat the therapy. Adding new classes of drugs to the HAART “cocktail,” such as attachment, fusion, maturation, and integrase inhibitors or adjuvant agents (which augment natural immune functions), should further increase the number of mutations required to defeat treatment [48-51]. Tailoring therapeutic profiles through a better under-standing of HIV-immune system dynamics can improve overall efficacy and minimize side effects [52-54]. Pre-ventative vaccines are desperately sought, but there is lit-tle hope that they will be available soon [55, 56].

IV. MATHEMATICAL MODELS AND CONTROL OF

HIV-IMMUNE SYSTEM DYNAMICS As noted in [57], mathematical models of HIV-im-

mune system dynamics (or “host-pathogen interactions”) were proposed not later than 1986 [58], and the disease has become the subject of intense modeling efforts. A review of all of the models is beyond the present scope; however, we direct attention to a number of important papers. References [13, 32, 36, 59-88] investigate dy-namic models of host-pathogen interactions; these papers compare and correlate response characteristics with bio-logical and clinical observations. Several of the papers demonstrate computed effects of therapeutic alternatives, such as switching various drugs on or off during the course of infection. Systems and control theories are ap-plied to the specification of therapeutic protocols in [89-94], particularly through the use of open-loop optimiza-tion or stabilization via feedback. All of these models and design studies are based on ordinary differential equa-tions, typically of order four or less.

1 A quasispecies is a family of strains that are genetically “close” if not identical [43].


Wild-Type Model of Host-Pathogen Interaction

The present paper takes the fourth-order model of [89] as a starting point, with modifications to the ways in which control effects are expressed, with significant changes to the cost function that is minimized by optimal control, and with the addition of a drug-resistant com-partment. The four elements of the initial model's state represent concentrations of free, wild-type HIV particles (x1), uninfected Th cells (x2), proviral infected Th cells (x3), and productively infected Th cells (x4) in both the periph-ery and lymphoid organs. The model of host-pathogen dynamics (Fig. 1) is expressed by the equations,

!

˙ x 1 = "a1x1 " a2x1x2(1" u2) + a3a4 x4 (1" u1)

˙ x 2 =a5

1+ x1

" a2x1x2(1" u2) " a6x2 + a7 1"x2 + x3 + x4

a8

#

$ %

&

' ( x2(1+ u3)

˙ x 3 = a2x1x2(1" u2) " a9x3(1" u4 ) " a6x3

˙ x 4 = a9x3(1" u4 ) " a4 x4

(1-4) where the form, rationale, and parameter values for the equations are given in [89]. The degree to which these equations reflect current knowledge of HIV dynamics can be assessed by referring to the prior sections of this paper. Clearly, many effects are aggregated and simplified; the core notion is that infection of Th cells is the dominant dynamic characteristic. Explicit cytotoxic and phagocytic immune effects are not modeled but are incorporated in parameter definitions; a discussion of this assumption can be found in [69]. Equations 1-4 do not account for muta-tion, whose effects are considered in a later section.

Figure 1. Overview of the dynamic model for HIV-im-mune system interaction.

The control vector is different from that considered in

[89]; its elements are concentrations of protease inhibitor (u1), fusion inhibitor (u2), Th cell enhancer (u3), and re-verse transcription inhibitor (u4). The controls enter as factors that either attenuate or enhance naturally occurring

terms in the equations. If an inhibitor takes a value of one (100% efficacy), it eliminates the term’s effect. When u3 takes a value of one, the concentration-limited rate of healthy Th cell production is doubled. Here, we illustrate monotherapies that employ either PIs or RTIs.

A typical untreated wild-type response is shown in Fig. 2. The initial condition is taken at a point beyond sero-conversion where the immune system is still reduc-ing viral load. Uninfected Th cell concentration is climb-ing (a good response), but so are the concentrations of infected cells, indicating that the HIV replication process is progressing. Over a simulated period of a year-and-a-half, the concentrations of free virions and infected Th cells grow, while concentration of uninfected Th cells decays. Left unchecked, the patient generally progresses to AIDS when the total Th cell concentration falls below 200 cells per mm3 of blood.

Figure 2. Untreated response to wild-type HIV infec-

tion.

Optimal Therapy for Wild-Type Infection The optimal therapeutic protocol is derived by mini-

mizing a treatment cost function, J, that penalizes large values of HIV and infected Th cell concentrations at the end time and during the fixed time interval [to, tf], as well as excessive application of therapeutic agents during the interval. The scalar cost function takes the general form,

J =1

2xT( tf )S fx(t f ) + xT (t)Qx(t ) + uT (t)Ru(t)[ ]

to

t f

! dt" # $

% & '

= ( x(t f )[ ] + L x(t),u(t)[ ]to

t f

! dt

(5a)

!

J =1

2sf 11x1 f

2

+ sf 33x3 f2

+ sf 44x4 f

2( )

+1

2q11x1

2

+ q33x3

2

+ q44x4

2

+ rui2[ ]dt

to

t f

"

(5b)


where

!

xT = x

1x2x3x4[ ] , u = ui (i = 1 or 4), and the diago-

nal matrices S, Q, and R establish relative weights that penalize variations from zero of the final state and the state and control over the time interval of interest. The variables are squared to amplify the effects of large varia-tions and to de-emphasize contributions of small varia-tions. Each squared element is multiplied by a coefficient that establishes the relative importance of the factor in the treatment cost. The coefficients (sii, qii, and r) could re-flect financial cost, or they could represent physiological "cost," such as toxicity or discomfort. Thus, there is a mechanism for trading one variation against the others in defining the treatment protocol, balancing speed and effi-cacy of treatment against implicit side effects.

Methods for minimizing the cost function with re-spect to the control and subject to the dynamic constraint are well known; they are presented in [95] and applied to an immune response problem in [96]. A Hamiltonian is formed from the Lagrangian (the integrand of eq. 5) with the dynamic constraint adjoined by Lagrange multipliers, and the Euler-Lagrange equations (necessary conditions for optimality) are satisfied via steepest-descent numerical iteration. Further inequality constraints are applied to assure that all variables remain positive and do not exceed specified limits. The result is an optimal trajectory x*(t) in [to, tf] that is generated by the optimal control profile, u*(t), in the same time interval. An example based on protease inhibitor therapy is shown in Fig. 3. Here, the PI is assumed to have 100% efficacy. The maximum dosage is maintained for about two months, during which time the viral and infected Th cell concentrations are reduced dramatically. At the end of the period, the PI concentra-tion is decreased toward a maintenance level that reaches a non-zero steady state (not shown). If the PI efficacy is reduced to 90%, the duration of maximum dosage is dou-bled and the decay rate of infection is slowed, but the trends are the same (Fig. 4).

Figure 3. Effect of optimal protease inhibitor therapy

with 100% efficacy.

RTI monotherapy is about as effective in taming the wild-type infection as is PI therapy (Fig. 5). Because the agent enters at an earlier stage of the HIV replication se-quence, proviral Th cell concentration continues to grow for a few weeks after therapy has begun, then decays over the next several months. These results suggest that PI and RTI therapy would be quite effective in controlling HIV if there were no mutation.

Figure 4. Effect of optimal protease inhibitor therapy

with 90% efficacy.

Figure 5. Effect of optimal reverse transcriptase in-

hibitor therapy with 100% efficacy. Although not shown, two other therapeutic effects

have been calculated and are of interest. Because a fusion inhibitor (u2) impedes the replication process at an early stage, its optimal effect is similar to that of an RTI; hence, its dynamic history is similar to Fig. 5. Optimizing ther-apy with the cost function of eq. 5 and a notional Th cell enhancer (u3) has a counter-intuitive effect: it calls for killing healthy Th Cells (through negative enhancement)! The reason is that Th cell concentration does not appear in the cost function. Because healthy Th cells become HIV factories, the only way such a control can reduce viral and infected Th cell concentrations is to attack healthy Th cells. On further thought, this seemingly anomalous re-


sult suggests that temporary immune system suppression could play a role in HIV treatment, as it already does for recovery from organ transplants (as suggested by Paul Shapshak, University of Miami, in personal communica-tion). Penalizing deviation from a healthy Th cell popula-tion in the cost function gives rise to time-variations in both viral and healthy Th cell levels that require more study. Model of Host-Pathogen Interaction with Wild-Type and Mutant Infection

Regrettably, mutation does occur in HIV replication, and it leads to drug-resistant strains of the virus. Thus, by definition, the new strains do not respond to treatment, but it remains to be seen if their dynamic interactions with drug-sensitive strains can be used to good effect. To ex-amine this possibility, we examine a single mutant strain and its effects by appending three equations to our origi-nal fourth-order model:

!

˙ x 1 = "a1x1 " a2x1x2(1" u2) + a3a4 x4 (1" a10)(1" u1)

˙ x 2 =a5

1+ x1 + x5

" a2x1x2(1" u2) " a6x2 " a2a11x5x2

+a7 1"x2 + x3 + x4 + x6 + x7

a8

#

$ %

&

' ( x2(1+ u3)

˙ x 3 = a2x1x2(1" u2) " a9x3(1" u4 ) " a6x3

˙ x 4 = a9x3(1" u4 ) " a4 x4

˙ x 5 = a3a4a10x4 + a3a4 x7 " a1x5 " a2a11x5x2

˙ x 6 = a2a11x5x2 " a9x6 " a6x6

˙ x 7 = a9x6 " a4 x7

(6-12)

The new state elements represent concentrations of the mutant HIV strain (x5), proviral Th cells infected by the mutant strain (x6), and Th cells productively infected by the new strain (x7). The differential equations for these three variables (eq. 10-12) are modeled after eq. 1, 3, and 4, but they do not contain direct effects of therapy. They are, however, coupled to the previous equations by a mu-tation rate term (with multiplier a10), and they contain a decay term representing “fitness” that is moderated by a11. The wild-type strain can be assumed to have a fitness of one (after [97]), and the mutant strain has a fitness (a11) less than one (else it would be a wild-type strain through evolution). The equation governing the production of healthy Th cells (eq. 7) also is modified to account for the new populations of virions and infected cells. For illus-tration, we assume that the mutation rate is 10%, double the highest suggested value in the literature, and we allow the mutant-strain fitness to vary between 10% and 90%.

With a mutant fitness of 90% and no therapy, both HIV strains progress, and the healthy Th cell population

decays as before (Fig. 6). The rate of growth is about the same for both strains, although the mutant strain has lower net concentration. Lowering the mutant fitness to 10% produces a similar result (Fig. 7), but the mutant concentration is lower still. Throughout the simulated period, both mutant strains grow, but the wild-type strain remains responsible for the bulk of infection. In this model, the wild-type strain does not explicitly suppress the mutant strain through competition for resources, but it does remain the dominant factor.

Figure 6. Untreated response to wild-type and mu-

tant infection. Mutant fitness = 90%.

Figure 7. Untreated response to wild-type and mu-

tant infection. Mutant fitness = 10%. Optimal Therapy for Wild-Type Plus Mutant Infection

Therapy is optimized for the revised system. The cost function is reframed to penalize the growth of the mutant strain and its infection of Th cells:

!

J =1

2sf 55x5 f

2+ sf 66x6 f

2+ sf 77x7 f

2( )

+1

2q55x5

2+ q

66x6

2+ q

77x7

2+ rui

2[ ]dtto

t f

"

(13)

The wild-type strain and its effects are not considered in the cost, and the control cost is the same as in eq. 5. With a mutant fitness of 90%, PI effect on the wild-type HIV strain is quite similar to the unmutated case; the wild type


is virtually eliminated within the first few weeks through the action of maximum drug dosage. Nevertheless, the mutant strain grows; it contributes to reduction in the number of healthy Th cells, although at a lower rate than in the untreated case (Fig. 6). The optimal therapy re-mains at or near the full value for the simulated period.

Figure 8. Response of wild-type and mutant virus to op-timal protease inhibitor therapy. Mutant fitness = 90%.

Reducing the mutant fitness level to 70% has little

effect on the optimal PI therapy history (Fig. 9), but the healthy Th cell count is maintained at high levels for the period. The wild-type HIV is controlled as before, and the mutant strain grows at a much lower rate. A further 10% reduction in mutant fitness has a dramatic effect: the mutant concentration and its effects diminish over time. The optimal dosage is smaller toward the end of the pe-riod, suggesting that the situation is in control. Thus, there is a dividing point between success and failure in controlling the drug-resistant mutant strain at a fitness level between 60% and 70%.

Figure 9. Response of wild-type and mutant virus to

protease inhibitor. Mutant fitness = 70%.

With reduction of mutant fitness to 20%, there is sig-nificant reduction in optimal PI dosage, but the dosage is increasing toward the end of the period. This may sug-gest optimality of a structured reduction in therapy when

the mutant strain has low fitness. “Start-stop therapy” or “drug holidays” are largely discounted by clinical trials [14, 31], although there are somewhat mixed results. Dif-ferences in experimental findings could arise from dis-similar fitness levels in actual mutant strains.

There is, of course a major caveat: we have no con-trol over the fitness of mutant strains, and there is every reason to believe that mutations at every fitness level occur. Nevertheless, the present result shows us that monotherapy can have an effect on controlling at least some proportion of the drug-resistant mutant population. Multi-drug therapy targets different mutant strains, in-creasing overall efficacy. If fitness, drug resistance, and progression of the disease are related, as suggested in [44, 45, 97], optimizing multi-drug strategies may reveal syn-ergistic effects that are currently unknown, providing better control of the virus and reducing harmful side ef-fects of treatment.

Figure 10. Response of wild-type and mutant virus

to protease inhibitor. Mutant fitness = 60%.

Figure 11. Response of wild-type and mutant virus

to protease inhibitor. Mutant fitness = 20%.

V. REFLECTIONS AND EXTENSIONS

The suggestions for HIV therapy that are presented

here are optimal only to the extent that the mathematical models are accurate reflections of host-pathogen-pharma-


ceutical interactions. Complex issues of pharmacokinet-ics have not been addressed, under the assumption that prescribed levels of therapeutic agents could be main-tained in circulatory and lymphatic systems through un-specified processes. The circulatory and lymphatic sys-tems have been treated as a single control volume, even though we know that host-pathogen interactions and resi-dence times are quite different in the periphery and in various lymphoid tissues. Furthermore, no two individu-als respond to infection and treatment in quite the same way, in large part because the rest of the body’s systems couple into HIV response. The immune system’s active response is only inferred in the model, the dynamics of CD4 cells other than Th cells have been neglected, and details of viral reservoirs remain to be determined. There may never be answers to reasonable questions about the nature of important physiological traits and the potential direct and side effects of therapy.

Still, there is much room and cause for future work on optimal therapies for HIV. Some of the issues just raised have been treated in references mentioned earlier, so there is a rich foundation for further application of control-theoretic principles. References [98, 99] present an introduction to dealing with uncertainty and incom-pleteness in immune system models, measurements, and control action, and many other approaches to design and analysis remain to be investigated. Through the normal course of biomedical research, models will improve, al-lowing control theory to make significant contributions to clinical practice.

CONCLUSION

Numerical optimization of nonlinear models can sug-

gest improved therapies for treating HIV infection. The dynamics of infection are far more complex and varied than portrayed by the simple mathematical models used here, but the analytical results give structures and ration-ales for treatment that could prove foundational for future practice. In particular, the study shows that current drug classes can provide indirect control of low-fitness drug-resistant HIV strains through dynamic coupling to wild-type strains and immune-system dynamics. Extension of this approach to multi-drug HIV therapy is straightfor-ward and is the logical next step.

ACKNOWLEDGEMENT This research was initiated under a grant from the

Alfred P. Sloan Foundation. This paper was written while the author was on sabbatical leave from Princeton Univer-sity as a member of the Institute for Advanced Study, Princeton, NJ. Discussions with Alan Perelson, Los Ala-mos National Laboratory, Arnold Levine, director of the Institute’s Center for Systems Biology, and members of the Center have been invaluable. Paul Shapshak, Univer-

sity of Miami, has been the source of much information on the fundamentals of HIV infection. Undergraduate research by David Sachs, '02, and Hugh Strange, '04, provided preliminary analyses that were helpful in for-mulating the current study.

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