Download Mutation and Control of the Human Immunodeficiency Virus

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 human 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 drugresistant 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, dendritic 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 acquired immune deficiency syndrome (AIDS), an inevitably fatal condition for all but a few [5]. While AIDS itself is rarely the cause of death, the body becomes vulnerable to a host of potentially lethal opportunistic infections
and malignancies, including pneumonia, tuberculosis,
dementia, bacterial sepsis, lymphoma, and Kaposi’s sarcoma. With multi-drug, highly active antiretroviral therapy (HAART), the rate of HIV infection can be dramatically 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, dizziness, anemia, weakness, insomnia, and adverse interaction with non-HIV drugs are not uncommon [14].
There are a number of reasons why the human immunodeficiency 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 particle (or virion), an immune cell may produce over a thousand 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. Because all encounters between cells and molecules within
the body’s circulatory systems are stochastic, both infection 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 immune 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
Together with a high production rate, the continuing appearance 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 therapy. 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 concentrations of the virus and infected cells against the adverse side effects of therapy. We see that normal competition 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 concepts 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 infected 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 depleted, 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 integrated strand of DNA is called a provirus, and the infected cell is a proviral cell. Transcription of the modified 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
2
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 undergoes programmed cell death (apoptosis), disintegrating
and releasing additional HIV product into the compartment.
Although there is a multiplicity of T-cell types involved 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 infected 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, eosinophils [22], natural killer cells [64], microglia [23, 24],
and macrophages that are within the central nervous system 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 reservoirs for the virus [22, 26, 27]; however, they generally
do not present antigens to the B and T cells of the adaptive 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 outcome [29, 30]. During the first weeks following infection, 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 antigen; it destroys virions and infected cells that are recognized 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 depletion of helper T cells over the course of time degrades the
ability of killer T cells to destroy infected cells. The cytotoxic 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 infection has a profound effect not only on the functions of
T cells but on the lymphoid tissue in which they are created and mature, the bone marrow and thymus. The pro-
Mutation and Control of the Human Immunodeficiency Virus
duction of new T cells decreases, and the body is increasingly 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. Signaling and effector cells are depleted, organs atrophy, and
immune function is lost at all levels, leaving the patient
susceptible to the opportunistic infections and malignancies 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. Current 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 nucleoside-analogue RTIs (NRTIs), which contain a chain-terminating sequence that prevents the formation of complete 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 pressure for mutation. PIs are generally less toxic than RTIs,
and they are less likely to stimulate drug-resistant mutants.
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
3
ten per virion replication [35]. Given the broad nature of
mutation [42], only a small percentage of mutant quasispecies1 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 therapy, 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-transcriptase genomic sequence is sufficient to cause resistance 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 understanding of HIV-immune system dynamics can improve
overall efficacy and minimize side effects [52-54]. Preventative vaccines are desperately sought, but there is little hope that they will be available soon [55, 56].
IV. M ATHEMATICAL MODELS AND CONTROL OF
HIV-IMMUNE SYSTEM DYNAMICS
As noted in [57], mathematical models of HIV-immune 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 dynamic models of host-pathogen interactions; these papers
compare and correlate response characteristics with biological 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 applied to the specification of therapeutic protocols in [8994], particularly through the use of open-loop optimization or stabilization via feedback. All of these models and
design studies are based on ordinary differential equations, typically of order four or less.
1
A quasispecies is a family of strains that are genetically
“close” if not identical [43].
Mutation and Control of the Human Immunodeficiency Virus
4
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 compartment. 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 periphery and lymphoid organs. The model of host-pathogen
dynamics (Fig. 1) is expressed by the equations,
x˙1 = "a1 x1 " a2 x1 x 2 (1" u2 ) + a3 a4 x 4 (1" u1 )
# x + x3 + x4 &
a
x˙ 2 = 5 " a2 x1 x 2 (1" u2 ) " a6 x 2 + a7 %1" 2
( x 2 (1+ u3 )
1+ x1
a8
$
'
˙x 3 = a2 x1 x 2 (1" u2 ) " a9 x 3 (1" u4 ) " a6 x 3
x˙ 4 = a9 x 3 (1" u4 ) " a4 x 4
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 reducing viral load. Uninfected Th cell concentration is climbing (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-ahalf, 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.
(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 mutation, whose effects are considered in a later section.
Figure 2. Untreated response to wild-type HIV infection.
Optimal Therapy for Wild-Type Infection
The optimal therapeutic protocol is derived by minimizing 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=
tf
%
1" T
#x (t f )S f x(t f ) + ! [x T (t)Qx(t ) + uT (t)Ru(t)]dt &
2$
to
'
tf
[
]
= ( x(t f ) + ! L[ x(t), u(t)]dt
Figure 1. Overview of the dynamic model for HIV-immune system interaction.
to
(5a)
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 reverse transcription inhibitor (u4). The controls enter as
factors that either attenuate or enhance naturally occurring
!
1
J = s f 11 x1 f 2 + s f 33 x 3 f 2 + s f 44 x 4 f 2
2
(
1
+
2
)
tf
" [q
2
2
2
]
x + q33 x 3 + q44 x 4 + rui dt
11 1
to
2
(5b)
Mutation and Control of the Human Immunodeficiency Virus
where x T = [ x1 x 2 x 3 x 4 ] , u = ui (i = 1 or 4), and the diagonal 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 variations and to de-emphasize contributions of small variations. 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 reflect 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 efficacy of treatment against implicit side effects.
Methods for minimizing the cost function with respect 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 concentration 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 doubled and the decay rate of infection is slowed, but the
trends are the same (Fig. 4).
5
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 sequence, 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 inhibitor therapy with 100% efficacy.
Figure 3. Effect of optimal protease inhibitor 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 therapy 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-
Mutation and Control of the Human Immunodeficiency Virus
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 communication). Penalizing deviation from a healthy Th cell population 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 examine this possibility, we examine a single mutant strain
and its effects by appending three equations to our original fourth-order model:
6
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.
x˙1 = "a1 x1 " a2 x1 x 2 (1" u2 ) + a3 a4 x 4 (1" a10 )(1" u1 )
a5
x˙ 2 =
" a2 x1 x 2 (1" u2 ) " a6 x 2 " a2 a11 x 5 x 2
1+ x1 + x 5
# x + x3 + x4 + x6 + x7 &
+a7 %1" 2
( x 2 (1+ u3 )
a8
$
'
x˙ 3 = a2 x1 x 2 (1" u2 ) " a9 x 3 (1" u4 ) " a6 x 3
x˙ 4 = a9 x 3 (1" u4 ) " a4 x 4
x˙ 5 = a3 a4 a10 x 4 + a3 a4 x 7 " a1 x 5 " a2 a11 x 5 x 2
x˙ 6 = a2 a11 x 5 x 2 " a9 x 6 " a6 x 6
x˙ 7 = a9 x 6 " a4 x 7
Figure 6. Untreated response to wild-type and mutant infection. Mutant fitness = 90%.
(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 mutation 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 illustration, 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
Figure 7. Untreated response to wild-type and mutant 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
+
2
1
s f 55 x 5 f 2 + s f 66 x 6 f 2 + s f 77 x 7 f 2
2
(
)
tf
" [q
2
2
2
(13)
]
x + q66 x 6 + q77 x 7 + rui dt
55 5
to
2
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
Mutation and Control of the Human Immunodeficiency Virus
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 remains at or near the full value for the simulated period.
7
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. Differences in experimental findings could arise from dissimilar fitness levels in actual mutant strains.
There is, of course a major caveat: we have no control 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, increasing 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 synergistic effects that are currently unknown, providing
better control of the virus and reducing harmful side effects of treatment.
Figure 8. Response of wild-type and mutant virus to optimal 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 period, 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 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%.
Figure 9. Response of wild-type and mutant virus to
protease inhibitor. Mutant fitness = 70%.
V. REFLECTIONS AND EXTENSIONS
With reduction of mutant fitness to 20%, there is significant reduction in optimal PI dosage, but the dosage is
increasing toward the end of the period. This may suggest optimality of a structured reduction in therapy when
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-
Mutation and Control of the Human Immunodeficiency Virus
ceutical interactions. Complex issues of pharmacokinetics have not been addressed, under the assumption that
prescribed levels of therapeutic agents could be maintained in circulatory and lymphatic systems through unspecified processes. The circulatory and lymphatic systems have been treated as a single control volume, even
though we know that host-pathogen interactions and residence times are quite different in the periphery and in
various lymphoid tissues. Furthermore, no two individuals 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 incompleteness 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, allowing control theory to make significant contributions to
clinical practice.
8
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 formulating the current study.
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
CONCLUSION
9.
Numerical optimization of nonlinear models can suggest 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 rationales 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 drugresistant HIV strains through dynamic coupling to wildtype strains and immune-system dynamics. Extension of
this approach to multi-drug HIV therapy is straightforward and is the logical next step.
10.
11.
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 University as a member of the Institute for Advanced Study,
Princeton, NJ. Discussions with Alan Perelson, Los Alamos National Laboratory, Arnold Levine, director of the
Institute’s Center for Systems Biology, and members of
the Center have been invaluable. Paul Shapshak, Univer-
12.
13.
Gallo, R. C., and Montagnier, L. (1988). “AIDS in 1988,”
Scientific American, 259, pp. 41-48.
Prusiner, S. B. (2002). “Discovering the Cause of AIDS,”
Science, 298, pp. 1726-1727.
Montagnier, L. (2002). “A History of HIV Discovery,”
Science, 298, pp. 1727-1728.
Gallo, R. C. (2002). “The Early Years of HIV/AIDS,” Science, 298, pp. 1728-1730.
Weiss, R. A. (1993). “How Does HIV Cause AIDS?” Science, 260, pp. 1273-1278.
Dornadula, G., Zhang, H., VanUitert, B., Stern, J., Livornese, Jr., L., Ingerman, M. J., Witek, J., Kedanis, R. J.,
Natkin, J., DeSimone, J., and Pomerantz, R. J. (1999). “Residual HIV-1 RNA in Blood Plasma of Patients Taking
Suppressive Highly Active Antiretroviral Therapy,” J.
American Medical Association, 282 (17), pp. 1627-1632.
Zhang, L., Ramratnam, B., Tenner-Racz, K., He, X., Vesanen, M., Lewin, S., Talal, A., Racz, P., Perelson, A. S.,
Korber, B. T., Markowitz, M., and Ho, D. D. (1999).
“Quantifying Residual HIV-1 Replication in Patients Receiving Combination Antiretroviral Therapy,” New England J. Medicine, 340 (21), pp. 1605-1613.
Havlir, D. V., Hellmann, N. S., Petropoulos, C. J., Whitcomb, J. M., Collier, A. C., Hirsch, M. S., Tebas, P., Sommadossi, J.-P., and Richman, D. D. (2000). “Drug Susceptibility in HIV Infection After Viral Rebound in Patients
Receiving Indinavir-Containing Regimens,” J. American
Medical Association, 283 (2), pp. 229-234.
Furtado, M. R., Callaway, D. S., Phair, J. P., Kunstman, K.
J., Stanton, J. L., Macken, C. A., Perelson, A. S., and
Wolinsky, S. M. (1999). “Persistence of HIV-1 Transcription in Peripheral-Blood Mononuclear Cells in Patients Receiving Potent Antiretroviral Therapy,” New England J.
Medicine, 340 (21), pp. 1614-1622.
Ramratnam, B., Mittler, J. E., Zhang, L., Boden, D., Hurley, A., Fang, F., Macken, C. A., Perelson, A. S., Markowitz, M., and Ho, D. D. (2000). “The Decay of the Latent
Reservoir of Replication-Competent HIV-1 is Inversely
Correlated with the Extent of Residual Viral Replication
during Prolonged Anti-Retroviral Therapy,” Nature Medicine, 6 (1), pp. 82-85.
Sharkey, M. E., Teo, I., Greenough, T., Sharova, N., Luzuriaga, K., Sullivan, J. L., Bucy, R. P., Kostrikis, L. G.,
Haase, A., Veryard, C., Davaro, R. E., Cheeseman, S. H.,
Daly, J. S., Bova, C., Ellison, R. T., III, Mady, B., Lai, K.
K., Moyle, G. Nelson, M., Gazzard, B., Shaunak, S., and
Stevenson, M. (2000). “Persistence of Episomal HIV-1 Infection Intermediates in Patients on Highly Active AntiRetroviral Therapy,” Nature Medicine, 6 (1), pp. 76-81.
Fraser, C., Ferguson, N. M., and Anderson, R. M. (2001).
“Quantification of Intrinsic Residual Viral Replication in
Treated HIV-Infected Patients,” Proc. National Academy of
Science, 98 (26), pp. 15167-15172.
Di Mascio, M., Ribeiro, R. M., Markowitz, M., Ho, D. D.,
and Perelson, A. S. (2004). “Modeling the Long-Term
Mutation and Control of the Human Immunodeficiency Virus
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
Control of Viremia in HIV-1 Infected Patients Treated with
Antiretroviral Therapy,” Mathematical Biosciences, 188,
pp. 47-62.
Bartlett, J. G., and Moore, R. D. (1998). “Improving HIV
Therapy,” Scientific American, 279, pp. 84-87, 89, 91-93.
Salk, J., Bretscher, P. A., Salk, P. L., Clerici, M., and
Shearer, G. M. (1993). “A Strategy for Prophylactic Vaccination Against HIV,” Science, 260, pp. 1270-1272.
Redfield, R. R., and Burke, D. S. (1988). “HIV Infection:
The Clinical Picture,” Scientific American, 259, pp. 90-98.
Gougeon, M.-L., and Montagnier, L. (1993). “Apoptosis in
AIDS,” Science, 260, pp. 1269-1270.
McCune, J. M. (2001). “The Dynamics of CD4+ T-Cell
Depletion in HIV Disease,” Nature, 410, pp. 974-979.
Janeway, C. A., Travers, P., Walport, M., and Shlomchik,
M. (2001). Immunobiology, Garland, London.
Douek, D. C., Brenchley, J. M., Betts, M. R., Ambrozak, D.
R., Hill, B. J., Okamoto, Y., Casazza, J. P., Kuruppu, J.,
Kunstman, K., Wollinsky, S., Grossman, Z., Dybul, M.,
Oxenius, A., Price, D. A., Connors, M. and Koup, R. A.
(2002). “HIV Preferentially Infects HIV-Specific CD4+ T
Cells,” Nature, 417, pp. 95-98.
Schacker, T., Little, S., Connick, E., Gebhard, K., Zhang,
Z.-Q., Krieger, J., Pryor, J., Havlir, D., Wong, J. K.,
Schooley, R. T., Richman, D., Corey, L., and Haase, A. T.
(2001). “Productive Infection of T Cells in Lymphoid Tissues during Primary and Early Human Immunodeficiency
Virus Infection,” J. Infectious Diseases, 183, pp. 555-562.
Klebanoff, S. J., and Coombs, R. W. (1997). “Virucidal
Effect of Stimulated Eosinophils on Human Immunodeficiency Virus Type 1,” AIDS Res. Hum. Retroviruses, 12
(1), pp. 25-29.
Shapshak, P., Fujimura, R. K., Srivastava, A., and Goodkin, K. (2000). “Dementia and the Neurovirulence of HIV1,” CNS Spectrums, 5 (4), pp. 31-42.
Garden, G. A. (2002). “Microglia in Human Immunodeficiency Virus-Associated Neurodegeneration,” Glia, 40 (2),
pp. 240-251.
Orenstein, J. A., Fox, C., and Wahl, S. M. (1997). “Macrophages as a Source of HIV During Opportunistic Infections,” Science, 276, pp. 1857-1861.
Ponath, P. D., Qin, S., Post, T. W., Wang, J., Wu, L.,
Gerard, N. P., Newman, W., Gerard, C., and Mackay, C. R.
(1996). “Molecular Cloning and Characterization of a Human Eotaxin Receptor Expressed Selectively on Eosinophils,” J. Experimental Medicine, 183, pp. 2437-2448.
de Paulis, A., De Palma, R., Di Gioia, L., Carfora, M., Prevete, N., Tosi, G., Accolla, R. S., and Marone, G. (2000).
“Tat Protein Is an HIV-1-Encoded β-Chemokine Homolog
That Promotes Migration and Up-Regulates CCR3 Expression on Human FceRI+ Cells,” J. Immunology, 165, pp.
7171-7179.
Lempicki, R. A., Kovacs, J. A., Baseler, M. W., Adelsberger, J. W., Dewar, R. L., Natarajan, V., Bosche, M. C.,
Metcalf, J. A., Stevens, R. A., Lambert, L. A., Alvord, W.
G., Polis, M. A., Davey, R. T., Dimitrov, D. S., and Lane,
H. C. (2000). “Impact of HIV-1 Infection and Highly Active Antriretroviral Therapy on the Kinetics of CD4+ and
CD8+ T Cell Turnover in HIV-Infected Patients,” Proc.
National Academy of Science, 97 (25), pp. 13778-13783.
Papagno, L., Spina, C. A., Marchant, A., Salio, M., Rufer,
N., Little, S., Dong, T., Chesney, G., Waters, A., Easterbrook, P., Dunbar, P. R., Shepherd, D., Cerundolo, V., Emery, V., Griffiths, P., Conlon, C., McMichael, A. J., Rich-
9
30.
31.
32.
33.
34.
35.
36.
37.
38.
39.
40.
41.
42.
43.
44.
45.
man, D. D., Rowland-Jones, S. L., and Appay, V. (2004).
“Immune Activation and CD8+ T-Cell Differentiation Towards Senescence in HIV-1 Infection,” PLoS Biology, 2
(2), pp. 173-185.
McMichael, A. J., and Rowland-Jones, S. L. (2001).
“Cellular Immune Responses to HIV,” Nature, 410, pp.
980-987.
McMichael, A. J., and Rowland-Jones, S. L. (2002). “Bad
News for Start-Stop Therapy?” Nature, 420, pp. 371-372.
Nowak, M. A., May, R. M., Phillips, R. E., Rowland-Jones,
S., Lalloo, D. G., McAdam, S., Klenerman, P., Köppe,
Sigmund, K., Bangham, C. R. M., and McMichael, A. J.
(1995). “Antigenic Oscillations and Shifting Immunodominance in HIV-1 Infections,” Nature, 375, pp. 606-611.
Grant, R. M., Hecht, F. M., Warmerdam, M., Liu, L.,
Liegler, T., Petropoulos, C. J., Hellman, N. S., Chesney,
M., Busch, M. P., and Kahn, J. O. (2002). “Time Trends in
Primary HIV-1 Drug Resistance Among Recently Infected
Persons,” J. American Medical Association, 288 (2), pp.
181-188.
Douek, D. C., Picker, L. J., and Koup, R. A. (2003). “T
Cell Dynamics in HIV-1 Infection,” Annual Review of
Immunology, 21, pp. 265-304.
Ho, D. D. (1995). “Time to Hit HIV, Hard and Early,” New
England J. Medicine, 333, pp. 450-451.
Bonhoeffer, S., May, R. M., Shaw, G. M., and Nowak, M.
A. (1997). “Virus Dynamics and Drug Therapy,” Proc.
National Academy of Science, 94, pp. 6971-6976.
Karlsson, A. C., Birk, M., Lindbäck, S., Gaines, H., Mittler,
J. E., and Sönnerberg, A. (2001). “Initiation of Therapy
during Primary HIV Type 1 Infection Results in a
Continuous Decay of Proviral DNA and a Highly
Restricted Viral Evolution,” AIDS Research and Human
Retroviruses, 17 (5), pp. 409-416.
Wang, C., Vlahov, D., Galai, N., Bareta, J., Strathdee, S.
A., Nelson, K. E., and Sterling, T. R. (2004). “Mortality in
HIV-Seropositive versus –Seronegative Persons in the Era
of Highly Active Antiretroviral Therapy: Implications for
When to Initiate Therapy,” J. Infectious Diseases, 190, pp.
1046-1054.
Richman, D. D. (2001). “HIV Chemotherapy,” Nature,
410, pp. 995-1001.
Hoffman, C., and Kamps, B. (2005). “HIV Drug Resistance
2005”
HIV
Medicine,
http://www.hivmedicine.com/textbook/rt.htm .
Batsis, J. A. (2000). “Clinical Pharmacology of Protease
Inhibitors in HIV Infection,” Trinity Student Medical
Journal, 1, pp. 60-65.
Watson, J. D., Baker, T. A., Bell, S. P., Gann, A., Levine,
M., and Losick, R. (2004). Molecular Biology of the Gene,
Pearson/Benjamin Cummings, San Francisco.
Eigen, M., and Bierbricher, C. K. (1988). “Sequence Space
and Quasispecies Distribution,” in DNA Genetics, Volume
3: Variability of RNA Genomes, E. Domingo, J. J. Holland,
and P. Ahlquist, ed., CRC Press, Boca Raton.
Quiñones-Mateu, M. E., and Arts, E. J. (2001). “HIV-1
Fitness: Implications for Drug Resistance, Disease Progression, and Global Epidemic Evolution,” HIV Sequence
Compendium, Los Alamos National Laboratory, Los Alamos, NM, pp. 134-170.
Quiñones-Mateu, M. E., Ball, S. C., Marozsan, A. J., Torre,
V. S., Albright, J. L., Vanham, G., van der Groen, G.,
Colebunders, R. L., and Arts, E. J. (2000). “A Dual Infection/Competition Assay Shows a Correlation Between Ex
Mutation and Control of the Human Immunodeficiency Virus
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
Vivo Human Immunodeficiency Virus Type 1 Fitness and
Disease Progression,” J. Virology, 74 (19), pp. 9222-9233.
Joly, V., and Yeni, P. (1999). “Non Nucleoside Reverse
Transcriptase Inhibitors,” AIDS Reviews, 1 (1), pp. 37-44.
Ciancio, B. C., Trotta, M. P., Lorenzini, P., Forbici, F.,
Visco-Comandini, U., Gori, C., Bonfigli, S., Bellohi, M. C.,
Sette, P., D’Arrigo, R., Tozzi, V., Zaccarelli, M., Boumis,
E., Narciso, P., Perno, C. F., and Antinoni, A. (2003). “The
Effect of Number of Mutations and of Drug-Class Sparing
on Virological Response to Salvage Genotype-Guided
Antiretroviral Therapy,” Antiviral Therapy, 8 (6), pp. 611616.
Kilby, J. M., and Eron, J. J. (2003). “Novel Therapies
Based on Mechanisms of HIV-1 Cell Entry,” New England
J. Medicine, 348 (22), pp. 2228-2238.
Witvrouw, M., Van Maele, B., Vercammen, J., Hantson,
A., Engelborghs, Y., De Clerq, E., and Debyser, Z.(2004).
“Novel Inhibitors of HIV-1 Integration,” Curr. Drug Metab., 5 (4), pp. 291-304.
Li, F., Goila-Gaur, R., Salzwedel, K., Kilgore, N. R., Reddick, M., Matallana, C., Castillo, A., Zoumplis, D., Martin,
D. E., Orenstein, J. M., Allaway, G. P., Freed, E. O., and
Wild, C. T. (2003). “PA-457: A Potent HIV Inhibitor that
Disrupts Core Condensation by Targeting a Late Step in
Gag Processing,” Proc. National Academy of Science, 100
(23), pp. 13555-13560.
Lum, J. L., Schnepple, D. J., Nie, Z., Sanchez-Dardon, J.,
Mbisa, G. L., Mihowich, J., Hawley, N., Narayan, S., Kim,
J. E., Lynch, D. H., and Badley, A. D. (2004). “Differential
Effects of Interleukin-7 and Interleukin-15 on NK Cell
Anti-Human Immunodeficiency Virus Activity,” J. Virology, 78 (11), pp. 6033-6042.
Casado, J. L., Moreno, A., Sabido, R., Marti-Belda, P.,
Antela, A., Drona, F., Perez-Elias, M. J., and Moreno, S.
(2003). “Individualizing Salvage Regimens: The Inhibitory
Quotient (Ctrough/IC50) as Predictor of Virological Response,” AIDS, 17(2), pp. 262-264.
Evans, W. E., McLeod, H. L. (2003). “Pharmacogenomics
– Drug Disposition, Drug Targets, and Side Effects,” New
England J. Medicine, 348, pp. 538-549.
Gallant, J. E., Gerondelis, P. Z., Wainberg, M. A., Shulman, N. S., Haubrich, R. H., St. Clair, M., Lanier, E. R.,
Hellman, N. S., and Richman, D. D. (2003). “Nucleoside
and Nucleotide Analogue Reverse Transcriptase Inhibitors:
A Clinical Review ofAntiretroviral Resistance,” Antiviral
Therapy, 8, pp. 489-506.
Burton, D. R., Desrosiers, R. C., Doms, R. W., Koff, W. C.,
Kwong, P. D., Moore, J. P., Nabel, G. J., Sodroski, J., Wilson, I. A., and Wyatt, R. T. (2004). “HIV Vaccine Design
and the Neutralizing Antibody Problem,” Nature Immunology, 5 (3), pp. 233-236.
Capdevila, G., “AIDS Vaccine Elusive – Even for Patents,”
http://www.aegis.com/news/ips/2005/IP050207.html, Inter
Press Service, Feb. 8. 2005.
Covert, D. J., and Kirschner, D. (2000) “Revisiting Early
Models of the Host-Pathogen Interactions in HIV Infection,” Comments on Theoretical Biology, 5 (6), pp. 383411.
Cooper, L. (1986). “Theory of an Immune System Retrovirus,” Proc. National Academy of Science, 83, pp. 91599163.
Perelson, A. S. (1989). Modeling the Interaction of the
Immune System with HIV. in Mathematical and Statistical
Approaches to AIDS Epidemiology (C. Castillo-Chavez,
ed.), Springer-Verlag, New York, pp. 350-370.
10
60. Kirschner, D. (1996). “Using Mathematics to Understand
HIV Dynamics,” Notices of the AMS, 43 (2), pp. 191-202.
61. de Jong, M. D., Veenstra, J., Stillianakis, N. I., Schuurman,
R., Lange, J. M. A., de Boer, R. J., and Boucher, C. A. B.
(1996). “Host-Parasite Dynamics and Outgrowth of Virus
Containing a Single K70R Amino Acid Change in Reverse
Transcriptase are Responsible for the Loss of Human Immunodeficiency Virus Type 1 RNA Load Suppression by
Zidovudine,” Proc. National Academy of Science, 93, pp.
5501-5506.
62. de Boer, R. J., and Boucher, C. A. B. (1996). “Anti-CD4
Therapy for AIDS Suggested by Mathematical Models,”
Proc.: Biological Sciences, The Royal Society, 263 (1372),
pp. 899-905.
63. Kirschner, D. E., and Webb, G. F. (1997). “A Mathematical
Model of Combined Drug Therapy of HIV Infection,” J.
Theoretical Medicine, 1, pp. 25-34.
64. Stillianakis, N. I., Boucher, C. A. B., de Jong, M. D., van
Leeuven, R., Schuurman, R., and de Boer, R. J. (1997).
“Clinical Data Sets of Human Immunodeficiency Virus
Type 1 Reverse Transcriptase-Resistant Mutants Explained
by a Mathematical Model,” J. Virology, 71 (1), pp. 161168.
65. Bonhoeffer, S., and Nowak, M. A. (1997). “Pre-Existence
and Emergence of Drug Resistance in HIV-1 Infection,”
Proc.: Biological Sciences, The Royal Society, 264 (1382),
pp. 631-637.
66. Nowak, M. A., Bonhoeffer, S., Shaw, G. M., and May, R.
M. (1997). “Anti-Viral Drug Treatment: Dynamics of Resistance in Free Virus and Infected Cell Population,” J.
Theoretical Biology, 184, pp. 203-217.
67. Kirschner, D. E., and Webb, G. F. (1998). “Immunotherapy
of HIV-1 Infection,” J. Biological Systems, 6 (1), pp. 7183.
68. Wein, L. M., D’Amato, R. M., and Perelson, A. S. (1998).
“Mathematical Analysis of Antiretroviral Therapy Aimed
at HIV-1 Eradication or Maintenance of Low Viral Loads,”
J. Theoretical Biology, 192, pp. 81-98.
69. de Boer R. J., and Perelson, A. S. (1998). “Target Cell
Limited and Immune Control Models of HIV Infection: A
Comparison,” J. Theoretical Biology, 190, pp. 201-214.
70. Perelson, A. S., and Nelson, P. W. (1999). “Mathematical
Analysis of HIV-1 Dynamics in Vivo,” SIAM Review, 41
(1), pp. 3-44.
71. Wodarz, D., Lloyd, A. L., Jansen, V. A. A., and Nowak, M.
A. (1999). “Dynamics of Macrophage and T Cell Infection
by HIV,” J. Theoretical Biology, 196, pp. 101-113.
72. Wodarz, D., and Nowak, M. A. (1999). “Specific Therapy
Regimes Could Lead to Long-Term Immunological Control
of HIV,” Proc. National Academy of Sciences, 96 (25), pp.
14464-14469.
73. Stafford, M. A., Cao., Y. C., Ho, D. D., Corey, L., and
Perelson, A. S. (2000). “Modeling Plasma Virus Concentration and CD4+ T Cell Kinetics during Primary HIV Infection,” J. Theoretical Biology, 203, pp. 285-301.
74. Kaufmann, G. R., Zaunders, J., Murray, J. Kelleher, A. D.,
Lewin, S. R., Solomon, A., Smith, D., and Cooper, D. A.
(2001). “Relative Significance of Different Pathways of
Immune Reconstitution in HIV Type 1 Infection as Estimated by Mathematical Modeling,” AIDS Research and
Human Retroviruses, 17 (2), pp. 147-159.
75. Wodarz, D., and Jansen, V. A. A. (2001). “The Role of T
Cell Help for Anti-viral CTL Responses,” J. Theoretical
Biology, 211, pp. 419-432.
Mutation and Control of the Human Immunodeficiency Virus
76. Callaway, D. S., and Perelson, A. S. (2002). “HIV-1 Infection and Low Steady State Viral Loads,” Bul. Mathematical
Biology, 64, pp. 29-64.
77. Bajaria, S. H., Webb, G., Cloyd, M., and Kirschner, D.
(2002). “Dynamics of Naïve and Memory CD4+ T Lymphocytes in HIV-1 Disease Progression,” J. Acquired Immune Deficiency Syndromes, 30 (1), pp. 41-58.
78. Nelson, P. W., and Perelson, A. S. (2002). “Mathematical
Analysis of Delay Differential Equation Models of HIV-1
Infection,” Mathematical Biosciences, 179, pp. 73-94.
79. Altes, H. K., Wodarz, D., and Jansen, V. A. A. (2002).
“The Dual Role of CD4 T Helper Cells in the Infection
Dynamics of HIV and Their Importance for Vaccination,”
J. Theoretical Biology, 214, pp. 633-646.
80. Culshaw, R. V., Ruan, S., and Webb, G. (2003). “A Mathematical Model of Cell-to-Cell Spread of HIV-1 that Includes a Time Delay,” J. Mathematical Biology, 46, pp.
425-444.
81. Bajaria, S. H., Webb, G., and Kirschner, D. E. (2004).
“Predicting Differential Responses to Structured Treatment
Interruptions During HAART,” Bul. Mathematical Biology,
66 (5), pp. 1093-1118,
82. Dixit, N., and Perelson, A. S. (2004). “Complex Patterns of
Viral Load Decay under Antiretroviral Therapy; Influence
of Pharmacokinetics and Intracellular Delay,” J. Theoretical Biology, 226, pp. 95-109.
83. Iwasa, Y., Michor, F., and Nowak, M. (2004). “Some Basic
Properties of Immune Selection,” J. Theoretical Biology,
229, pp. 179-188.
84. Kirschner, D. (1999). “Dynamics of Co-Infection with M.
Tuberculosis and HIV-1,” Theo. Population Biology, 55,
pp. 94-109.
85. Kirschner, D., Mehr, R., and Perelson, A. (1998). “Role of
the Thymus in Pediatric HIV-1 Infection,” J. Acq. Imm.
Def. Synd. And Hum. Retro., 18 (2), pp. 95-109.
86. Ye, P., and Kirschner, D. E. (2002). “Reevaluation of T
Cell Receptor Excision Circles as a Measure of Recent
Thymic Emigrants,” J. Immunology, 168 (10), pp. 49684979.
87. Ye,. P., Kourtis, A., and Kirschner, D. (2002). “The Effects
of Different HIV Type 1 Strains on Human Thymic Function,” AIDS Res. And Hum. Retr., 18(7), pp. 1239-1251.
88. Ye,. P., Kourtis, A., and Kirschner, D. (2003). “Reconstitution of Thymic Function in HIV-1 Patients Treated with
Highly Active Antiretroviral Therapy,” Clinical Immunology, 106, pp. 95-105.
89. Kirschner, D., Lenhart, S., and Serbin, S. (1997). “Optimal
Control of the Chemotherapy of HIV,” J. Mathematical Biology, 35, pp. 775-792.
90. Wein, L. M., Zenios, S. A., and Nowak, M. A. (1997).
“Dynamic Multidrug Therapies for HIV: A Control Theoretic Approach,” J. Theoretical Biology, 185, pp. 15-29.
91. Alvarez-Ramirez, J., Meraz, M., and Velasco-Hernández, J.
X. (2000). “A Feedback Control Strategy for Chemotherapy of HIV,” Intl. J. Bifurcations and Chaos, 10 (9), pp.
2207-2219.
92. Velasco-Hernández, J. X., Garcia, J. A., and Kirschner, D.
E. (2001). “Remarks on Modeling Host-Viral Dynamics
and Treatment,” in Vol.125: Mathematical Approaches for
Emerging and Reemerging Infectious Diseases Part 1: An
Introduction to Models, Methods, and Theory (C. CastillioChavez, S. Blower, P. van den Driessche, D. Kirschner, and
A.-A. Yakubu, ed.), pp. 338-411.
11
93. Brandt, M. E., and Chen, G. (2001). “Feedback Control of
a Biodynamical Model of HIV-1”, IEEE Trans. Biomedical
Engineering, 48 (7), pp. 754-759.
94. Culshaw, R. L., Ruan, S., and Spiteri, R. J. (2004). “Optimal HIV Treatment by Maximising Immune Response,” J.
Mathematical Biology, 48, pp. 542-564.
95. Stengel, R. F. (1994). Optimal Control and Estimation,
Dover Publications, New York.
96. Stengel, R. F., Ghigliazza, R., Kulkarni, N., and Laplace,
O. (2002). "Optimal Control of Innate Immune Response,"
Optimal Control Applications & Methods, 23, pp. 91-104.
97. Nowak, M. A., and May, R. M. (2000). Virus Dynamics,
Oxford University Press, Oxford.
98. Stengel, R. F., Ghigliazza, R., and Kulkarni, N. (2002).
"Optimal Enhancement of Immune Response," Bioinformatics, 18 (9), pp. 1227-1235.
99. Stengel, R. F., and Ghigliazza, R. (2004). "Stochastic Optimal Enhancement of Immune Response," Mathematical
Biosciences, 191, pp. 123-142.