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329
Mathematical Approaches in the Study of Viral Kinetics and Drug
Resistance in HIV-1 Infection
V. Müller* and S. Bonhoeffer
Ecology & Evolution, ETH Zürich, ETH Zentrum NW, 8092 Zürich, Switzerland
Abstract: We review some crucial aspects of drug therapy and viral resistance that have been
investigated within the framework developed for the modelling of virus kinetics. First, we give a
general overview on the use of mathematical models in the field of HIV research. We seek to
identify the factors that determine the steady state virus load and show that stable reductions
during antiviral therapy are difficult to explain within the standard model of virus dynamics. We
discuss possible extensions that enable the models to account for the moderately reduced virus
loads during non-suppressive treatment and argue that the residual viremia under suppressive
treatment can probably be attributed to the survival of long-lived infected cells, rather than to
new rounds of replication. Next, we address the emergence of resistance during suppressive
therapy and demonstrate that the resistant virus is more likely to be present already at the start of treatment than to be
generated during therapy. The appearance of resistance after a prolonged period of initial suppression indicates that drug
efficacy is not continuously maintained over time. We investigate the potential risks and benefits of therapy interruptions.
Considering the effect of recombination, we argue that it probably decelerates, rather than accelerates the evolution of
multidrug-resistant virus. We also review state-of-the-art methods for the estimation of fitness, which is crucial to the
understanding of the emergence of resistance during therapy or the re-emergence of wild type upon the cessation of
therapy.
Key Words: HIV-1, drug resistance, mathematical modelling, viral fitness.
1. INTRODUCTION
Drug resistance poses a major challenge to the treatment
of HIV-1 infection. The response to the challenge is twofold: while new drugs are needed to combat viruses that are
resistant to existing drugs, it is also important to devise
optimal strategies for the application of the available tools, to
minimise the risk of the emergence of resistance in the first
place. Understanding the biological processes that underlie
drug resistance is clearly crucial for the latter aspect, and this
is where mathematical modelling can help.
Mathematical models have been instrumental in the
elucidation of the dynamical nature of HIV-1 infection [1-6].
The fitting of simple models to experimental data has
revealed that relatively stable virus loads reflect a dynamic
equilibrium between extremely fast production and
destruction of cells and virus particles, rather than true
latency. In this review, we first give an overview of the
general modelling framework used to describe the withinhost dynamics of the disease. We investigate what factors
might determine the steady-state virus load, which varies
several orders of magnitude between patients and has a
strong effect on disease outcome [7]. In particular, we
address how antiviral therapy and drug resistance influence
the virus load and demonstrate that the effect of new drug
*Address correspondence to this author at ELTE Department of Plant
Taxonomy and Ecology, Pázmány Péter Sétány 1/C, 1117 Budapest,
Hungary; Tel: (+41) 1 6327105; Fax: (+41) 16321271E-mail:
[email protected]
1568-0053/03 $41.00+.00
classes under development can be implemented in the
standard modelling framework without major modifications.
Next, we consider whether resistance is more likely to
emerge due to mutations occurring during drug treatment or
due to the low-level presence of mutants prior to therapy.
Recombination is also considered to be relevant to the
appearance of multidrug resistance—we present recent
results showing that the effect of recombination is strictly
dependent on the fitness relations of single and multidrugresistant mutants. In the next section, we review modelling
results on the risk of the emergence of drug resistance during
structured therapy interruptions (STIs). Under what conditions
do the benefits outweigh the risks? The replicative capacity
of resistant mutants is also highly relevant for the emergence
and maintenance of drug resistance. In the final section we
discuss recent methods for the estimation of viral fitness.
2. THE BASIC MODEL
In the basic model of viral dynamics we consider
uninfected target cells, T, productively infected cells, I, and
virus particles, V. Uninfected cells are produced at a constant
rate, σ, from a pool of precursor cells and die at rate δTT.
Virus reacts with uninfected cells to produce infected cells at
rate βTV, and infected cells die at rate δI. Virus is produced
from infected cells at rate pI and is cleared at rate cV.
Finally, we prefer a slight modification to the “basic
scheme” and introduce a factor f to describe the probability
that a newly infected cell progresses to virus production [8]
(and Bonhoeffer et al. in press). This gives rise to the
following system of ordinary differential equations:
© 2003 Bentham Science Publishers Ltd.
330 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4
dT
= σ δTT βTV,
dt
dI
= f βTV δI,
dt
dV
= pI cV
dt
(1-3)
The scheme of the model is shown in Fig. (1). Clearly,
such a simple model cannot cover the true complexity of
HIV-1 infection, but it can capture some important aspects of
the disease and has led to fundamental insights into HIV
replication kinetics in vivo [1, 2]. It can also serve as a
starting point for the inclusion of further biological details.
As a general rule, the complexity of a model should be kept
at the minimum that is necessary to answer the questions that
inspired the creation of the model. The equations are built
upon biological hypotheses of the infection, and these
diverge increasingly as more of the true biological
complexity is considered. For each detail, a choice has to be
made, and this will constrain the applicability of the results
obtained with the model. The basic model of HIV-1 infection
reflects a consensus of the field, and further details are
subject to controversies about alternative hypotheses. The
properties of the model have been discussed extensively
elsewhere [4, 5, 9, 10]. Here we provide an overview with an
emphasis on discussing possible extensions to the model.
A major advantage of mathematical modelling is that the
initial assumptions that limit the validity of the results are
clearly defined by the unambiguous mathematical formulation.
The basic model has several important simplifying
assumptions. It assumes that all processes occur in a wellmixed, spatially homogeneous system, and the rate of new
infections is proportional to the abundance of virus particles
and susceptible cells. This limitation can be partially
overcome by considering several spatial compartments [1113], but ordinary differential equations cannot describe
spatial heterogeneity within the compartments. Another
Müller and Bonhoeffer
assumption is the homogeneity of the cellular and viral
pools. The basic model considers a single virus-producing
cell population, productively infected CD4+ T lymphocytes,
which arise by the infection of a single target cell type,
activated CD4+ T cells. This simplification is justified in
untreated individuals, in whom this cell type is responsible
for the majority of virus production. Under therapy, the
population of productively infected cells dwindles fast, but
the decline of the virus level decelerates after the first few
weeks, indicating that other cell types are responsible for the
residual virus production. Models of long-term treatment are
therefore extended by the description of long-lived infected
cells and latently infected cells [14, 15]. Apart from
considering several cell types, some models have also
implemented heterogeneity within a single cell type by
employing a continuously distributed parameter (death rate)
instead of a fixed value [15, 16]. The virus is also described
as a single, homogeneous population in the basic model.
This approach is justified as long as only the total abundance
of virus (virus load) is considered and the processes can be
characterized by the average parameters of the swarm
(quasispecies) of diverse virus variants within the patient.
For instance, not all virus particles might be infectious, but
the infectivity parameter, β, in the model provides an
infection rate averaged over the total population of infectious
and non-infectious viruses. If there is need to keep track of
diversity or specific individual variants (e.g. wild type and
drug-resistant forms, see below), the virus load equation can
be duplicated to describe each variant separately. In practice,
the whole diversity of the viral quasispecies cannot be
followed with this method, but this is not needed for the
modelling of general virus dynamics. In some models of
disease progression, virus variants are distinguished on the
basis of their antigenic properties: each virus variable
describes the variants that can be targeted by a particular
CTL clone [17, 18]. A further general assumption of the
basic model is that infection occurs instantaneously, newly
infected cells start to produce virus immediately after
infection. This simplification does not affect the behaviour
of the model when the system is at steady state, but it might
Fig. (1). The scheme of the basic model of HIV-1 infection. Susceptible target cells arise from a pool of precursor cells at rate, σ, and die at
rate δTT. Virus reacts with uninfected cells to produce infected cells at rate βTV, but only a fraction f of newly infected cells progress to virus
production. Infected cells die at rate δI; virus is produced from infected cells at rate pI and is cleared at rate cV.
Mathematical Approaches in the Study of Viral Kinetics
Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4
become important when perturbations (e.g. drug therapy) are
considered. In the latter case, the assumption can be relaxed
by the use of delay differential equations [19, 20].
Whereas the assumptions listed above concern general
structural properties of the basic model and their possible
modifications, further biological details can also be
introduced into any specific process described by the model.
Each parameter can in fact reflect a complex biological
process, which would make it a function of other parameters
and variables. This approach has been explored in detail
elsewhere [8], where we have suggested the term “process
function” to capture this potential for hidden complexity.
Indeed, each parameter can be replaced by a function to
accommodate further biological details in the model. For
example, the process function σ by which target cells are
produced could be written as σ =aQ to represent activation
of quiescent cells, Q, with an activation rate a [21]. The loss
function of target cells, δT, could then be written explicitly as
a sum of death and reversion to resting state. Note that in
such a model increasing activation of CD4+ target cells can
alone result in their depletion, which is now a favoured
hypothesis of HIV pathogenesis [22-25].
An important extension is the implementation of an HIVspecific immune response that can react to varying levels of
infection. In the basic model the effect of the immune
response is hidden in the parameters, its strength is fixed. To
allow for an adaptable immune response, a new variable has
to be introduced to describe the level of the HIV-specific
effector cells [6, 26], e.g.:
dE
= αEI
dt
δEE
(4)
where α is the immune responsiveness, indicating the ability
of effector cells to respond by expansion to the presence of
infected cells, and δE is the death rate of these cells. The
process function approach can then help to implement the
action mechanism of the effector cells within the frame of
the basic model. If cytotoxic cells kill virus producing cells,
the death rate of infected cells can be expanded by the killing
term: δ = δ1 + kE, where δI is the intrinsic death rate and k is
the efficiency parameter of effector cells. However, if newly
infected cells can be killed before the onset of virus
production, then effector cells reduce the probability of progressing to virus-producing state: f = 1 +1kE . Further options
might involve non-cytotoxic mechanisms that influence the
infection rate, β, or the rate of virus production, p. The
immune responsiveness, α, and the effector efficiency, k, can
also be functions of the target cell or infected cell levels to
indicate CD4-help [27, 28] or saturated stimulation [6, 8,
26], respectively. In all, the basic model has immense
potential to accommodate further biological details. Note
that the expansion of parameters into process functions does
not affect the validity of the steady state results obtained by
the basic model. However, the new results obtained by the
expanded forms depend on the precise form of the
expansion. The basic model assumes that parameters are
constant in time. This assumption is justified if the period of
observation is sufficiently short, such that changes in the
parameters can be neglected, as is the case in the first few
331
days to weeks after initiation of therapy. To accommodate
the description of long-term dynamics in the context of the
basic model, the parameters need to be interpreted as process
functions that may change their values over time.
Finally, even the basic model can be simplified further if
this fits the purpose of the study. For instance, if there is no
need to keep track of the virus level explicitly, the equation
of the virus load (Eq 3) can be dropped and the rate of new
infections can be made a function of the infected cell load
directly, i.e. β*TI instead of βTV. This quasi-steady state
assumption is justified because the dynamics of the free
virus population are considerably faster than those of the
infected cells, and consequently the free virus can be
assumed to be always proportional to the infected cell
population (i.e. β*I = βV). We will see examples for both the
expansion and the simplification of the basic model at the
discussion of particular applications below.
3. STEADY STATE
Without treatment, virus load and the number of infected
and uninfected cells settle to a relatively stable level. This
situation can be well approximated by the steady state of the
mathematical models. Because differential equations
describe the rate of change of the variables, a steady state
corresponds to setting all equations to zero, i.e. to no change
in any of the variables. In the basic model, this procedure
yields two solutions. One equilibrium corresponds to the
uninfected state with no virus and T = σ / δT uninfected
target cells. The other reflects the infected steady state with
levels:
^
I =
σf
δ
p ^ ^
δT c , ^
δc
V = c I, T =
βp
βfp
(5)
If virus is introduced into an uninfected population, the
system will approach one of the two steady states depending
on the parameters. The criterion can be summarised in the
notion of the basic reproductive number, R0 [29, 30], which
is defined as the average number of infected cells that a
single infected cell produces when placed into an uninfected
cell population. In the basic model, it is obtained as
R0 = βσfp / δδ T c . If R0 < 1, the system approaches the
uninfected equilibrium and the infection dies out. If R0 > 1,
the system approaches the infected steady state, and infection
is established in the host. There are no unambiguously
documented cases in which the virus would have been
cleared completely, which indicates that R0 is almost always
above one. It is an open question whether highly exposed
uninfected individuals have an R0 below one, which would
indicate systemic resistance to the virus, or can prevent the
virus from accessing the main target cell population by a
form of local defence at the site of viral entry.
An interesting problem arises if one introduces an
explicit description of the HIV-specific immune response, as
given in Eq. (4) [6, 26]. The steady state level of infected and
uninfected cells and virus load assumes a new form:
^
δ
p
ασc
I = αE , V^ = c I,^ T^ =
αδT c + βδE p
(6)
332 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4
The steady state level of HIV-specific effectors will
depend on the precise form of the effector function. The
structurally different steady states in Eqs. (5-6) are difficult
to reconcile with each other [6, 8]. However, the transition
between the two forms can be established by allowing for
more biological details in the immune responsiveness,
following the method of process functions. We have shown
that in untreated patients the HIV-specific immune response
is likely to be largely insensitive to changes in the infected
cell level due to saturation in the presence of excess antigen
and competition between effector cells [8]. This implies that
the level of the HIV-specific immune response can be
regarded as constant, and needs not to be described by a
variable. The basic model thus provides a sufficient
description of an untreated, stable infection. Note, however,
that by influencing the process functions of the basic model,
the parameters of the immune response still have a role in
the setting of the steady state levels.
The debate about the factors that determine the steady
state is far from academic. The viral “set point”, which is
equivalent to the steady state virus load in the model, is a
strong prognostic marker of disease progression [7]. The set
point virus load can vary more than a thousand fold between
patients [31] and a higher virus load implies faster disease
progression in untreated infection. Remarkably, despite the
immense differences between individuals, the set point is
very stable within each patient and the virus load tends to
return to baseline after perturbations, e.g. when therapy is
discontinued [32-35]. This confirms that the set point is
indeed strictly determined by the biological parameters of
the system. It is of paramount importance to determine
which host and viral factors contribute to the setting of the
steady state and hence also the rate of disease progression.
Briefly, we have found that variation in the virus load
correlates strongly with variation in the level of infected cells
(Bonhoeffer et al., in press), as has been hypothesized before
[6, 26]. Variation in the level of infected cells, in turn,
probably reflects variation in the net production of
susceptible target cells (Bonhoeffer et al., in press). At closer
inspection, this variation can be dissected into variation in
multiple host and virus parameters [8]. We have used the
process function approach to expand those “parameters” that
had the largest variation as estimated in a well-studied
patient set. Those parameters or variables that show little
variation can be treated as constants. By this method we
have found that most of the variation in the virus load can be
broken down into variation in the total production of target
cells, the infection rate and the parameters that determine the
level of the HIV-specific immune response. Note that the
immune response is, as mentioned before, typically saturated
at a constant level, but this level varies between patients and
can thus affect the steady state virus load.
Finally, disease progression in this modelling framework
manifests itself as a slow shifting of the steady state towards
higher virus load and lower CD4 count, which implies a
change in the parameters that set the steady state.
4. DRUG THERAPY
Drug treatment perturbs the steady state. The effect of
drugs can be modelled by adjusting the parameters of the
Müller and Bonhoeffer
models. Reverse transcriptase inhibitors (RTIs) block new
infections, which can be modelled as a reduction in the
infection rate, such that the infection rate becomes β’ = (1ε)β, where ε is the efficacy of the treatment. Inhibitors of the
viral protease (PIs) render newly produced virus noninfectious, which can be modelled by introducing a new
equation for non-infectious virus. Note that the effect of the
two drug classes is very similar due to the short life span of
virus particles: infectious particles produced before the start
of protease treatment disappear quickly, which stops new
infections with only a negligible delay compared to the
direct inhibition by RTIs. To simplify the analysis, it is often
assumed that drugs act with perfect efficacy, i.e. the
infection rate is set to zero or all newly produced virions are
non-infectious. In this simplified case, virus load decays
exponentially, at a rate determined largely by the death rate
of virus producing cells, δ [1-3]. This has allowed the
estimation of the death rate and thus the average life span of
infected cells. However, intensified therapy regimens result
in faster initial decay, which implies that the efficacy of
common drug combinations cannot be perfect [36, 37] and
previous studies have overestimated the life span of virus
producing cells, which is now estimated to be one day at the
beginning of treatment. If imperfect efficacy is considered in
the models, it can be indeed shown that the initial rate of
virus decay becomes approximately the product of the true
death rate and the efficacy, and thus decreases with
decreasing efficacy [5].
After the first few weeks of therapy the decline of plasma
viremia begins to decelerate, indicating a change in the cell
population that is responsible for the bulk of virus
production [14]. Productively infected CD4+ T cells dwindle
quickly, and give way to other cell types with smaller initial
numbers but a longer life span, such as persistently and
latently infected cells. As mentioned before, the basic model
can be expanded to describe also these populations.
Remarkably, the deceleration of virus decline seems to be
gradual, indicating a range of life spans within the virus
producing cell population. We and others have proposed that
this phenomenon might reflect the heterogeneity of latently
infected cells with respect to the activation and death rates
[15, 38]. Latently infected cells are memory CD4+ T
lymphocytes that have a long life span, unless they are reactivated by antigenic stimulation. Some latently infected
cells might be specific for common antigens and might thus
have a high probability to receive stimulation and then die
quickly; other cells might be specific for rare antigens and
thus have a lower rate of re-activation and death. This
hypothesis has also received some experimental support
[38]. A consequence of decelerating virus decline is that,
after prolonged therapy, the average life span of virus
producing cells cannot be clearly defined, and hence
predictions for the time required for total eradication of the
virus have a high degree of uncertainty.
The estimation of the rate of virus clearance has also
proven to be notoriously difficult. The steep decline of
viremia after the initiation of antiviral therapy reflects mostly
the death rate of virus producing cells, and the estimation of
virus clearance is unreliable from these data [20]. Other
experimental settings based on plasma apheresis or infusion
Mathematical Approaches in the Study of Viral Kinetics
Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4
333
have yielded conflicting results [39-41]. Part of the
discrepancies can be explained by models that distinguish
between decay in the lymphoid tissue and in the blood
plasma compartment [13]. The decline of infectivity in the
blood plasma probably reflects the decay rate of virus in the
lymphoid tissues, whereas the estimate obtained by plasma
apheresis reflects the faster decay rate in the blood. However,
the even faster rate estimated in the infusion experiments
still remains unexplained (De Boer et al., in preparation).
Beyond estimating the turnover parameters of the
infection from the decline of viremia, mathematical
modelling has also provided interesting results concerning
the long-term outcome of therapy. In terms of the basic
model, there are two possible outcomes. The changes in viral
parameters due to therapy can either lower R0 below one,
and thus drive the virus to extinction, or R0 remains above
one despite the reduction and a new infected steady state is
attained. Remarkably, it is not always straightforward to
distinguish between the two scenarios. Accumulating
evidence indicates that HIV is likely to persist for life even
in the face of effective suppressive treatment [38, 42-46].
However, it is not clear whether this low-level persistence
results from ongoing low-level replication (corresponding to
a new infectious steady state) or from the survival of latently
or persistently infected cells. In the second case, the virus
can persist even though the new steady state would be free of
virus, because the uninfected steady state is not attained in
the lifetime of the patient.
In the age of HAART, the virus load in most patients
who do not harbour drug-resistant virus variants is
suppressed below the limit of detection of standard assays,
which makes the study of virus dynamics more difficult.
However, studies in earlier days have shown that suboptimal
therapy, e.g. a combination of lamivudine and zidovudine
can result in stable reductions in the virus load 10 to 100fold below the pre-treatment level [47]. Suppression is
achieved through a reduction in the infection rate of the
virus: the direct inhibition of the reverse transcriptase by the
drugs and the reduced replicative capacity of the emerging
lamivudine-resistant mutant are both likely to act on the
infection rate. Partial suppression can sometimes be
maintained also in patients who develop resistance mutations
under current combination therapy [48-50]. Moreover,
patients who acquire multidrug-resistant virus in the primary
infection have a lower untreated viral set point [51], which
also confirms that resistance has a price for the virus in terms
of lower replicative capacity. These observations suggest
that reductions in the infection rate can indeed result in a
stable infected steady state with a lower virus load.
Remarkably, however, the standard models of HIV-infection
cannot account for this plausible scenario. It has been shown
that reductions in the infection rate (or, more generally, in
the basic reproductive number) either result in the extinction
of the virus or they hardly affect the virus load at all [52, 53].
Fig. ( 2) depicts the dependence of the steady state virus
load on the efficacy of treatment, as obtained from the basic
model. The virus load is stable until the critical efficacy
(required for the eradication of the virus) is approached, and
the transition from the infected steady state to no infection
occurs in a very narrow parameter range. The intuitive
Fig. (2). Steady state virus load as a function of treatment efficacy
in the basic model.
reason for this phenomenon is that in response to the reduced
infectivity of the virus the level of target cells increases,
which compensates for the decreased infection rate. Thus,
treatment affects the target cell level, but not the virus load,
unless the virus gets close to extinction. It has been shown
that the difference between the critical efficacy and the
realized efficacy of a certain treatment cannot be greater than
the factor of reduction in the steady state virus load [53]. For
instance, a 100-fold reduction (factor of reduction: 0.01)
implies that the efficacy can be at most 0.01 less than the
critical efficacy. Such a close match is very unlikely given
that the efficacy can vary between zero and one. As
discussed above, the suppression achieved by current
combination therapy might be consistent with the case when
the virus would be driven to extinction after a time
exceeding the life span of an individual. However, the low
steady state observed with suboptimal or failing therapy, or
transmitted mutant virus, cannot be accounted for in the
frame of the basic model. The implementation of the HIVspecific immune response as in Eq. (4) cannot solve the
paradox, either. In this case, drug treatment reduces the level
of the anti-HIV immune response and does not affect the
virus load at all until the response is lost completely. As this
occurs very close to the critical efficacy that drives also the
virus to extinction, the virus load converges very steeply to
zero beyond this point, and the overall behaviour is very
similar to that observed in the basic model.
Several solutions have been proposed to overcome the
problem. Low but detectable steady state virus load can be
explained if virus induces the death of uninfected cells, or if
the proliferation of HIV-specific effector cells depends on
the level of infected cells, but not on the current effector
population size [52]. Both cases predict a linear dependence
of the virus load on treatment efficacy. This can explain the
low virus load observed under suboptimal therapy and with
multidrug-resistant viruses, but the extremely low virus load
under suppressive therapy would still require an efficacy
very close to the critical value. In the models, a steady state
virus load around the limit of detection (50 copies ml-1) can
only be attained if either the death rate of infected cells is
assumed to depend on the number of these cells, or there is a
compartment where the drugs cannot penetrate [53]. In the
former case, the death rate is modified as δ’=δIω. However,
the effect is very sensitive to the parameter ω, which expresses
the strength of the density dependence. The presence of a
drug sanctuary is therefore the most plausible hypothesis at
334 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4
present that accounts for a low steady state virus load
comparable to the virus load in well-suppressed patients. In
the models, this can be described by duplicating the equations
of the basic model (Eqs. (1-3.)) to describe virus load,
uninfected and infected target cells in the two compartments
separately. The connection between the two compartments is
established through an exchange of virus particles. Note that
to maintain the extremely low viremia observed in the main
compartment, the drugs have to act very efficiently. Otherwise,
the virus spilling out of the protected compartment could
initiate new rounds of infection in the main compartment and
thus result in detectable viremia.
We are thus left with two possibilities. If the low virus
load after prolonged suppressive therapy reflects a true
steady state, there must be a sanctuary site where the drugs
cannot take effect. Alternatively, residual virus loads below
the detection limit are maintained by long-lived cells, rather
than ongoing replication and thus do not reflect a true steady
state. In both cases, the moderately reduced virus load during
suboptimal or failing therapy, or after primary transmission
of multidrug-resistant virus, can be accounted for by virusinduced killing of uninfected cells or an HIV-specific
cytotoxic response.
What is the take-home message? In either case, the basic
model requires extensions to explain the changes in the virus
load during long-term antiviral therapy. Each extension
reflects a possible biological process, and our present
knowledge does not allow us to decide which scenario is
correct. However, the predictions of the models can be used
to test the underlying biological hypothesis. For example, if
unambiguous evidence for ongoing virus replication were
found in well-suppressed patients, the model would predict
the existence of drug sanctuaries, which should eventually be
found. Conversely, a prolonged failure to find drug
sanctuaries would strongly argue against the role of ongoing
virus replication in the maintenance of viremia. New rounds
of infection might still occur, but they cannot maintain a
stable level of viremia according to the model results. It has
indeed been observed that occasional viral ‘blips’ can slow
down the long-term virus decay in some patients [46, 54],
but are not needed for the persistence of residual viremia.
Since no drug sanctuaries have been identified as yet, we
would risk the prediction that viral persistence after longterm suppressive treatment is a consequence of the long-term
survival of infected cells, rather than a result of recurring
rounds of new infection. The background for the reduced
steady state virus load attained during suboptimal therapy is
more ambiguous. There is evidence for the virus-induced
killing of uninfected cells, for example by chronic
hyperactivation [55, 56]. However, there is also strong
evidence for the role of the virus-specific cytotoxic immune
responses in the control of viremia [57-60]. Moreover, in
previous work we have shown that the activation of the
immune response is likely to be saturated in untreated
chronic infection, which accounts for a similar behaviour as
in the implementation that has been shown to explain low
virus steady states [8]. Saturation due to competition
between effector cells corresponds to the independence of
immune activation of the current level of the effector
population as in [52]. Indeed, the model developed in [8] can
Müller and Bonhoeffer
also account for moderately reduced steady state virus loads.
The low steady state observed under suboptimal therapy and
with multidrug-resistant virus might thus reflect a
combination of the two possible factors.
Finally, we consider the effect of the new classes of
inhibitors that might become available for therapy in the
foreseeable future. Fusion inhibitors, chemokine analogues
and the inhibitors of the viral integrase all block new
infections, and thus act on the infection rate, β, in the
models. The implementation and the predicted effect of the
new treatments are thus identical to that of RTIs if only one
drug class is considered. However, a combination of drugs
from different classes can have tremendous impact, as the
effect of drugs acting on subsequent steps of the viral life
cycle is probably multiplicative. In this case, the infection
rate becomes β’ = (1-ε1)(1-ε2)(1-ε3)(1-ε4)β, where the
parameters ε1-4 denote the efficacy of the RTI, fusion
inhibitor, integrase inhibitor and coreceptor antagonist arm
of a drug combination. Drug classes not included in the
combination can simply be regarded as having zero efficacy.
The first clinical trials have indeed demonstrated that adding
a fusion inhibitor to standard combination therapy can
considerably improve the degree of virus control [61-63].
5. DRUG RESISTANCE
Antiviral drugs can suppress the replication of wild type
HIV, but mutations can render the virus resistant to the
drugs. Monotherapy in the early days of HIV treatment
invariably led to the fast emergence of resistant virus and the
rebound of viremia. Even current combination therapy can
fail due to the evolution of multidrug-resistant virus variants,
and a large fraction of patients have been “cycled” through
many drug regimens. It is of paramount importance to
identify the sources of resistance and devise optimal
strategies to prevent its development. Mathematical
modelling can be a useful tool also in this context.
The simplest implementation of drug resistance is to
duplicate the equation for infected cells to describe wild type
and resistant virus separately. As the emphasis is on the
spread of resistance and the explicit implementation of virus
levels is not relevant in this context, we can simplify the
model by making the rate of new infections a function of the
infected cell levels (see Basic Model). Thus we write:
dT
= σ δT T
dt
β1 TI1
β2 TI2 ,
dI1
= (1 µ )β1 TI1 + µβ2 TI2 δI1,
dt
dI2
= µβ 1TI1 + (1 µ)β2 TI2 δI2.
dt
(7-9)
I1 and I2 denote the populations of cells infected with
wild type and resistant virus, respectively. To be consistent
with the detailed studies in the field [64-69], we have
abandoned the factor of infected cells that progress to virus
production (in effect, we fix f = 1). Mutations occur at rate µ.
We assume that the two variants differ only in their infection
Mathematical Approaches in the Study of Viral Kinetics
rates, which are denoted by β1 and β2, respectively. The
basic reproductive number of the two variants can then be
written as R1 = β1σ / δδ T and R2 = β2σ / δδT . Under therapy,
‘resistance’ and ‘sensitivity’ can be defined on the basis of
the reproductive number. A virus variant is resistant if it can
maintain a steady state virus load in spite of therapy, which
corresponds to the criterion R2 > 1. Sensitive virus would
eventually be pushed to extinction, i.e. R1 < 1 during therapy.
The ‘wild type’, on the other hand, is defined as the
dominant variant in drug naïve untreated patients, which
implies that in the absence of therapy the sensitive wild type
virus has the highest basic reproductive number, i.e. R1 > R 2
> 1. (Note, that here we make also the assumption that the
basic reproductive number of the resistant virus is not lower
in the absence then in the presence of therapy). Importantly,
the basic reproductive number depends on both host and
viral factors; ‘resistance’ is therefore a property of the hostvirus system, not just that of the virus.
The outcome of treatment depends on its strength. Weak
drugs might be unable to push the basic reproductive ratio of
the wild type below one. In the narrow sense, even the wild
type is resistant in this case. Zidovudine monotherapy
reflects such a situation: it has been shown that the first
rebound of viremia after the start of therapy is actually
composed of wild type virus [70, 71]. As mentioned before,
the resurgence is fuelled by the increasing abundance of
target cells. Even in this case, however, mutant viruses can
have higher infection rates in the presence of the drug, i.e. R2
> R1 > 1. In this case the initial resurgence is followed by the
replacement of wild type virus with mutants of increasing
drug resistance, as has indeed been observed during longterm zidovudine therapy [72, 73]. The sequential emergence
of resistant variants has been described by mathematical
models [71, 74]. More efficient drugs can lower the basic
reproductive number of the wild type below one, while still
allowing the replication of the resistant mutants such that R2
> 1 > R1. In this case the wild type decays exponentially and
the resistant mutant is selected for. Stronger drugs result in
faster replacement of the wild type [64]. Interestingly, the
basic model predicts that the benefit of a larger initial
reduction in the virus load is cancelled out by the faster
resurgence. That is, the long-term benefits of therapy in
terms of increased CD4 count or decreased virus load are
independent of the efficacy of the drugs on the wild type
virus, as long as virus is not completely suppressed. Finally,
in the optimal scenario the drugs can suppress all virus
variants such that 1 > R2 > R1. In this case the virus is
suppressed and eradication is prevented only by replication
in drug sanctuaries (where R1 or R2 > 1) or the survival of
cells infected before the start of therapy. This might be the
situation with current combination therapy, contingent upon
perfect adherence.
An important question in the dynamics of drug resistance
is whether the resistant variants responsible for drug failure
are present at the start of therapy or arise during treatment
[64]. Resistant variants can appear by random mutations
even in the absence of therapy [75-77]. If their replicative
capacity is much lower than that of the wild type, they
cannot persist and the mutants present at any time point are
the result of recent mutations. The frequency of such mutants
Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4
335
is thus equivalent to the rate of the mutation that creates
them from the wild type. If the selective disadvantage of the
mutant is not profound, the mutant can persist at a higher
equilibrium frequency. If the infection rate of the mutant in
the absence of therapy is written as β 2 = (1 − s)β1 , where s is
the selective disadvantage of the mutant, then the pretreatment frequency can be approximated as µ/s [67]. This
gives the frequency of one-point mutations. However, to
escape current combination therapy the virus has to evolve
several mutations. If all mutation rates are equal and all
intermediate mutants have the same selective disadvantage,
the frequency of two-point mutants is 2(µ/s)2, while that of
three-point mutations is around 6(µ/s)3. With an estimated
mutation rate of µ = 3×10-5 [78] a mutant that has a selective
disadvantage of 0.01 can have a frequency around 3×10-3
prior to the start of therapy. Considering an estimated 107108 infected cells in a chronically infected patient [79], oneand two-point mutations might be relatively abundant and
even three-point mutations are likely to be present prior to
therapy. Such frequencies are not easy to detect in vivo, but
there is some experimental support for the pre-existence of
one-point mutants [76, 80] Higher selective disadvantage
will, of course, decrease the probability of pre-existence.
Importantly, the selective disadvantage of the intermediate
one- and two-point mutations also has a large impact on the
pre-existence of fully resistant mutants. A defective
intermediate creates a barrier to the generation of resistant
virus.
Note that in differential equation models the mutants are
always present at some low concentration. As an approximation, the mutant can be considered not present when its
population size is below one. Alternatively, stochastic
models can directly describe the probabilities of transitions,
such as the appearance of mutations. This approach yields
results that are compatible with the approximation by
deterministic differential equation systems [68].
The probability of de novo emergence of resistance
depends on the number of new cell infections during therapy.
If the efficacy of treatment is 100%, no cells are infected and
the probability of emergence is zero. The number of newly
infected cells in the case of imperfect efficacy cannot be
calculated precisely, but an analytical approximation based
on the modelling framework presented here has been
performed [69]. The results indicate that in the case of
suppressive therapy the number of cells infected during
treatment is smaller than the number present at the start of
treatment. This also holds for the number of mutants
produced. Interestingly, the relation between the probability
of pre-existence and emergence is independent of the
mutation rate, because the two probabilities increase
comparably with the mutation rate. This conclusion is also
insensitive to changes in other parameters. A further
important result is that drugs specifically targeted at the wild
type but less effective against intermediate sensitive strains
can actually increase the risk of the emergence of resistance.
The reason for this is that the suppression of the wild type
leads to an increase in the target cell level which benefits the
intermediate virus forms [66, 69]. It is thus not necessarily
the best strategy to treat with a regimen specifically
optimised against the wild type virus.
336 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4
Because under suppressive therapy pre-existence is more
likely than emergence, and the former depends on the
number of infected cells, the risk of drug resistance is the
lowest when the infected cell population is the smallest. This
supports the ‘hit early and hard’ strategy of treatment, to
attack the virus when the infected cell load is still low.
However, primary infection comprises a transient peak of the
virus load and the infected cell population, therefore also the
presence of resistance mutations is more probable in this
period [68]. Considering resistance, optimally timed therapy
should thus start after the decline of primary viremia.
We have concluded that the emergence of resistance
under suppressive therapy is less likely than its preexistence. Considering that combination therapy in drugnaïve patients is almost invariably successful for a
considerable time, one might ask the question: how can
multidrug resistance evolve at all? It is not present at the
start of therapy and it should be even less likely to appear
later. The key is that this result assumes continuous efficient
suppression of virus replication. If new rounds of replication
occur, even transiently in ‘blips’, the probability of
emergence increases. This problem will be discussed in the
next section.
Stochastic simulations indicate that multidrug resistance
evolves in a stepwise manner through intermediate forms
[68]. This brings up the possibility that recombination
between one-point mutants could accelerate the development
of fully resistant virus. Recombination is prevalent in
retroviruses [81, 82]. Virions produced by cells harbouring
multiple proviruses can contain RNA copies from two
different infecting strains. When such a “heterozygous”
virion infects the next cell, recombination during reverse
transcription can give rise to recombinant provirus. The high
rate of recombination per replication cycle (the reverse
transcriptase alternates on average about 3 times between the
two genomic RNA strands [81, 82]), combined with the high
prevalence of multiply infected cells [83] allows for a very
high rate of recombination at the cell population level. The
high prevalence of recombination in lentiviruses has also
gained support by the analysis of sequence sets from SIVinfected macaques [84]. However, we have shown that the
effect of recombination on the pre-existence and emergence
of resistance depends on the exact relations between the
infection rates of the various mutants (Bretscher et al., in
press). If the replication capacity of a one-point mutant is
closer to that of a fully resistant double mutant than to that of
the fully sensitive wild type, then recombination will indeed
accelerate the emergence of resistance during therapy.
However, if the viruses carrying one resistance mutation are
still strongly suppressed by the remaining drugs of the
combination therapy, then recombination will actually slow
down the emergence of resistance. Recombination can not
only create but also break up favourable gene combinations,
depending on the exact circumstances. As the latter scenario
is more likely, we arrive at the counterintuitive conclusion
that recombination might actually slow down the evolution
of drug resistance in this system. The pre-existence of double
mutant also depends on the relative replication capacity of
the intermediate mutant. However, in the absence of drugs it
is not clear whether the intermediate mutant is closer to the
Müller and Bonhoeffer
double mutant or to the wild type in replication capacity. The
direction of the effect on pre-existing frequencies can
therefore not be predicted yet.
A further interesting question is the disappearance or
maintenance of resistance after drug treatment is stopped. As
we have discussed before, in the absence of therapy the
original wild type virus is likely to have a higher replicative
capacity than the resistant mutant that emerges during
therapy. Accordingly, the resurgence of wild-type is often
observed when therapy is stopped [49, 50, 85-87]. Generally,
it is reasonable to expect that the disadvantage of drugresistant virus in the absence of therapy is smaller than its
advantage during treatment. Antiviral drugs suppress the
replication of wild type virus drastically, which resistant
viruses evade. The mutations responsible for resistance
might reduce the general replicative capacity of the mutant
virus, but this effect is probably much weaker than the effect
of drugs on wild type virus. The selection pressure driving
the outgrowth of wild type after the cessation of therapy is
thus probably smaller than that exerted by the drugs during
the primary emergence of resistance after the start of
treatment. Yet the number of resistance mutations often
begins to decrease as soon as six weeks after the interruption
of treatment [49, 50]. This suggests that wild type virus is
not reconstructed by backmutations, but is persistently
present also during therapy, probably in the “archive” of
latently infected cells [88]. Indeed, in patients who have been
infected by multidrug-resistant virus in the first place and
have never had wild type virus, the resistant virus can persist
for as long as five years in the absence of treatment [51]. In
this case, the genetic background of the primary resistance
mutations might also inhibit the appearance of wild type
virus (Fig. (3)). Multidrug-resistant virus is often the result
of long evolution under therapy, and the replicative capacity
impaired by primary resistance mutations can be partially
restored or sometimes even increased above the initial level
by secondary mutations [48, 89, 90]. However, the secondary
mutations that are optimised for the mutant reverse
transcriptase or protease might function less efficiently with
the wild type of these enzymes. In such a situation, the wild
type is not selected for even in the absence of therapy.
Reversion to wild type would require simultaneous
backmutation in the primary and secondary resistance genes,
which has a very low likelihood. The evolution of wild type
virus is thus inhibited by the low replicative capacity of the
intermediate virus variants that form the link between the
fully resistant virus and the pure wild type, both of which
contain well co-adapted genes. Recombination in this case
also acts to slow down the spread of wild type virus.
To understand the dynamics of replacement between
different virus types, one thus needs estimates on the relative
replicative capacity of the two virus types and also of the
intermediate forms between them. Replicative capacity is
described by the concept of fitness, which we review in
Section 7. Besides fitness, the speed of evolution also
depends on the mutation rate. Different types of point
mutations are generated at very different rates [78, 91] which
affects both the pre-existence and the emergence of resistant
variants. For instance, the M184I mutant is generated by a
high probability guanine→adenine mutation and is therefore
Mathematical Approaches in the Study of Viral Kinetics
Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4
337
Fig. (3). Fitness relations and the direction of evolution in the presence (A) and absence (B) of drug treatment. Rows contain wild type (WT)
or mutant (MUT) alleles of the primary resistance loci, and columns contain the alleles of compensatory loci that can restore replication
capacity in the presence of primary resistance mutations. The direction of evolution is indicated by arrows. During therapy (A), primary
mutations confer a selective advantage over the wild type, and compensatory mutations provide a further benefit in viruses with primary
mutations. Evolution in this case is therefore unidirectional from wild type to primary resistance and then to the double mutant genotype. In
the absence of therapy (B) we show fitness relations assuming that compensatory mutations co-adapt to primary resistance mutations and
thus perform better with the mutant than with the wild type allele of the primary resistance loci even in the absence of therapy. This creates a
barrier to the reversion of the MUT/MUT genotype to wild type, and thus the cessation of therapy cannot reverse the direction of evolution.
more likely to pre-exist than most other resistance mutations.
Moreover, the mutation rate can also be influenced by
interactions between the virus and the infected cell type [9295], by the direct effects of drug treatment [96-98] and by
resistance mutations in the reverse transcriptase gene [96].
Finally, note that the framework developed in this section
is independent of the action mechanism of the drugs. The
conclusions are thus also valid for the new classes of drugs
that are currently under development. Indeed, the emergence
of resistance has already been documented during
monotherapy (in phase one clinical trial) with the fusion
inhibitor enfuvirtide (T-20) [99]. Moreover, primary viruses
exhibit considerable variability in their susceptibility to T-20
before the start of treatment [100, 101]. The presence of less
susceptible variants can be regarded as an example for the
pre-existence of resistance.
An interesting property of entry inhibitors is that their
effect depends on the target cell tropism of the virus. X4 and
R5 viruses use the CXCR4 and the CCR5 coreceptor of
target cells, respectively. Each chemokine analogue can only
block one of the two coreceptors and will thus affect only
one virus type. Such drugs might therefore be able to
influence the evolution of the target cell tropism of the virus
population. This possibility has been explored in detail with
the help of mathematical modelling [102]. Compared to the
basic model, this study has duplicated the target cell equation
to distinguish between naïve and memory CD4+ T cells. The
former were assumed to carry CXCR4 only, while memory
cells carry both CXCR4 and CCR5. The study explored the
outcome of competition between virus variants with different
affinity towards the two coreceptors, and then investigated
the effect of coreceptor antagonists in this context. In the
model CXCR4 inhibitors select for the coexistence of R5
and X4 viruses, and cannot drive X4 virus to extinction,
because R5 viruses cannot utilise the resource of naïve target
cells. The effect of CCR5 inhibitors depends on whether X4
variants are present at the start of therapy. If they are not,
CC5 inhibitors do not facilitate the initial appearance of X4
variants. However, if X4 variants pre-exist, then treatment
with CCR5 inhibitors accelerates the switch, because X4
viruses can infect both memory and naïve cells. This advises
caution in the use of CCR5 inhibitors because X4 viruses
have been associated with accelerated disease progression
[103]. Similarly to drug-resistant and drug-sensitive virus
variants, X4 and R5 viruses can also persist in the latent
reservoir, and may thus never be truly lost [104]. This
implies that evolution driven by coreceptor antagonist
treatment can immediately reverse its direction when
treatment is stopped.
6. STRUCTURED THERAPY INTERRUPTIONS
The outgrowth of wild type virus in the absence of drugs,
as has been described in the previous section, is one of the
potential benefits that inspired the idea of controlled therapy
interruptions. Reversion to wild type restores sensitivity to
the previously employed drugs, which is especially
important in patients who have accumulated resistance
against most available drugs. Another hope was that
intermittent peaks of viremia would boost the immune
responses directed against HIV and facilitate the immune
control of the infection even in the absence of further
therapy. The method of structured therapy interruptions
(STIs), which involves a protocol of alternating periods on
and off therapy, was devised to avoid the inhibition of the
immune responses by the fast growing virus, and allow for a
gradual build-up of HIV-specific immunity. Besides the
potential benefits, however, this approach carries the danger
that renewed virus replication during treatment interruptions
might generate drug-resistant mutants. The risks and benefits
must therefore be assessed by careful experimental and
theoretical studies.
The modelling framework described in the preceding
sections can be used to investigate both sides of the coin.
Previously, we have argued that the probability of the
appearance of resistance mutations depends on the number
of cells that become newly infected. The relative risk of the
emergence of resistance thus depends on the extent of virus
replication occurring during treatment interruptions. Unless
338 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4
virus replicates to levels approaching the set point virus load
maintained before the start of antiviral therapy, the
probability of emergence is lower than that of pre-existence
before therapy [105]. However, the probability increases
with the height of the peaks and with the cumulative time off
therapy [105, 106]. Furthermore, the comparison of relative
risks used to predict the source of resistance during therapy
cannot be used as a guideline in this case. Even if mutations
are generated with a lower probability during STIs than
before the initial start of therapy, this gives a new chance for
the virus. Patients who do not harbour pre-existing mutants,
run a new risk during STIs, which should be evaluated on
absolute, rather than relative criteria.
Clearly, phenotypic resistance needs the fixation of the
resistance mutations after their initial appearance. However,
resistant mutants can only increase in frequency when the
levels of the drugs are low enough to allow their replication
but high enough to give them a selective advantage over the
wild type (reviewed in [107]). Since multidrug-resistance
develops most probably by the stepwise accumulation of
multiple mutations, the initial mutants are resistant to one
drug only, and are suppressed under combination therapy.
The time window for the selection of resistant variants is
thus probably short during each treatment cycle. However, a
modelling study has shown that partially resistant variants
can increase in frequency unless they have a large selective
disadvantage in the absence of therapy [106]. The model
used in the study was an expanded version of Eqs. (7-9.),
involving also latently infected cells and virus particles. The
growth of mutant virus is facilitated also by the abundance of
target cells at the beginning of each interruption period. In
contrast to chronic infection, there is no competition between
mutant and wild type viruses at this stage. This free
unlimited growth also implies that appearing resistance
mutations can be preserved even if they do not have a
selective advantage during STI. In such a case, the first
appearance of the mutant will establish its “pre-existence”,
which enables the mutant to grow immediately once the
conditions become favourable for it (cf. Drug Resistance).
In summary, the appearance of multidrug resistance is
improbable during STIs, but partial resistance might
develop, especially if the virus load is allowed to reach
higher levels and/or the protocol is continued for a long time.
Resistance mutations have indeed been detected in some
patients under STI [108] (Metzner et al., in press), which
might reflect the growth of pre-existing mutants in some
cases [109]. However, Metzner et al. have detected minor
populations of the M184V and L90M mutations in 14 and 3
out of 25 subjects who have been on their first combination
therapy before the STI and had no prior history of
suboptimal treatment. In this case, the mutations probably
arose de novo during the interruptions. Furthermore, even if
phenotypic resistance is not detected, hidden pre-existence of
mutations might be established due to the unlimited growth
of virus during short interruptions. Partially resistant viruses
might also enter the latent reservoir [106]. Importantly,
during repeated cycles of interruption, recurrent growth of
partially resistant viruses gives an opportunity for the
accumulation of further mutations and the development of
multidrug-resistance. This process is likely to occur in
Müller and Bonhoeffer
uncontrolled “interruptions” due to non-adherence. Irregular
pill intake can also lead to out of phase fluctuations in the
concentration of different drugs, which also facilitates the
consecutive acquisition of resistance mutations. As we have
concluded in the previous section, this is probably the most
important path to multidrug resistance.
The importance of pre-existence is also underlined by the
results of STI studies. Not only does the wild type outgrow
resistant mutants during interruptions, but also the mutants
return when continuous therapy is re-initiated [110, 111].
The growth of partially resistant variants during STIs might
also be attributed to pre-existence in some cases [109]. It
seems that once a variant had been present in the virus pool,
it will remain archived in the pool of latently infected cells,
and can re-grow under favourable selecting conditions. This
property abolishes the benefit of STIs in re-establishing drug
sensitivity. As soon as therapy is restarted, resistant mutants
that existed before the STIs can resurge—the virus
“remembers” all of its previous states.
The other objective of STI studies has been to enhance
the immune control of the infection, ideally to a point where
drug treatment can be stopped completely. Indeed, treatment
interruptions can induce an increase in the size and breadth
of the HIV-specific immune responses [34, 112-116].
Furthermore, the growth rate of the virus during the
interruptions decreases over the consecutive STI cycles [35,
113, 117]. Given that the target cells are abundant at the
beginning of the interruptions, it has been hypothesised that
the reduced growth rate might reflect the effect of a boosted
immune response. Unfortunately, in chronically infected
patients the increased immune responses and slower viral
growth rate are not associated with enhanced control of the
virus [35, 117, 118]. Virus levels increase in most patients
during the interruption periods [34, 112, 113]. After the end
of STIs some patients maintain reduced virus load for as
long as 12 months without therapy [112], others, however,
attain levels comparable to the pre-treatment baseline [114,
118]. A better immune control of the virus is achieved only
if treatment is initiated during primary infection and then
stopped [116, 119-121]. These phenomena can be
understood in terms of the steady states of the models.
In the basic model (Eqs. (1-3)), multiple parameters
determine the steady state virus load in the absence of
treatment. As we have shown, therapy disturbs this steady
state and either establishes a new equilibrium, or puts the
virus onto an asymptotic course towards extinction.
However, when therapy is stopped, all levels return to their
pre-treatment value, and the basic model returns to the
baseline virus load preceding treatment. Going through
cycles of interruptions cannot change this behaviour, unless
the parameters (process functions) are altered and this
change persists after the end of STIs. The basic model can
thus explain the effects of STIs observed in chronically
infected patients, but not in acutely infected patients. To
account for sustained effects one needs an extended version
of the model that can have two alternative infected steady
states with a possibility for early treatment to push the
system from one steady state to the other. We can devise
such a model extension by analysing the mathematical
possibilities to achieve such behaviour and/or by considering
Mathematical Approaches in the Study of Viral Kinetics
biologically plausible scenarios. In the former approach we
describe the rate of change of the infected cell population
with a simplified version of Eq. (2), again assuming that the
level of free virus follows that of infected cells at a quasi
steady state: dI / dt = f βTI − δI . In the absence of therapy the
system settles to a steady state, which implies that the rate of
change will be zero, the positive production term and the
negative death term are equal and balance each other
perfectly. During therapy, the level of susceptible target
cells, T, increases. If all other parameters remain unchanged,
this will increase the input term of infected cells and thus
result in a positive growth rate and an increased virus load
when the treatment blocking new infections is stopped [105].
To maintain the suppression of the infection after the
cessation of therapy, either the infection rate, β, or the
fraction of infected cells progressing to virus-production, f,
has to decrease, or the death rate of infected cells, δ, has to
increase during therapy. Having identified the mathematical
possibilities, we seek a biological explanation that could be
responsible for them. Potentially, all three effects can be
achieved if the level of HIV-specific effector cells increases.
By killing infected cells these cells can reduce the fraction
progressing to virus-production or increase the death rate of
virus-producing cells. Alternatively, non-cytotoxic effector
cells might reduce the infection rate.
However, in the standard models of HIV-specific effector
cells, e.g. in Eq. (4), the proliferation of the cells depends on
antigenic stimulation by HIV. Suppression of the viremia
results in reduced effector levels, which is consistent with
observations on chronically infected patients entering
therapy [122]. To account for the sustained control of
viremia in patients who were treated during acute infection,
one requires a mechanism either to increase the activation
rate of effector cells (immune responsiveness) or to decrease
their death rate [105]. The dependence of the effector
response on CD4-help has been proposed to account for the
former possibility [28, 123-125]. The basic model (Eqs. (14)) has been expanded by an equation for the precursors of
the effector cells, and the proliferation of precursors was
assumed to depend on both the infected cell load and the
level of CD4 target cells. In this model, an increase in
overall CD4 counts due to virus suppression also increases
the level of HIV-specific CD4 help, which might improve
either the proliferation or the effector function of HIVspecific CD8 T cells. There are two stable steady states in
the absence of therapy: one with a preserved HIV-specific
helper response, a strong CD8 effector response and low,
controlled virus load; the other with a severely reduced
helper population and high virus load in spite of the induced
CD8 response. Early antiviral therapy during acute infection
can preserve the CD4 population until the CD8 effector cells
proliferate to a level sufficient to control the virus [28, 123].
Interestingly, the analysis of a generalised model has
demonstrated that in such a system a single phase of welltimed drug therapy should be able to push the system from
the uncontrolled to the controlled steady state and thus
establish sustained immunity [126]. The failure of therapy to
do so in chronic HIV infection indicates that the HIV
specific immune response might be lost irrevocably at this
stage. We note, however, that too long therapy can restore
the system to the initial susceptible state with too few
Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4
339
effector cells to prevent the growth of the virus into the
uncontrolled steady state once therapy is stopped [105].
Similar results are obtained if we assume that the death
rate of effector cells increases with the virus load due to
some inhibitory effect exerted on the cells by HIV [105].
Writing the death rate as a process function, e.g. as δE’ = δE +
I / ( K + I ) to account for a saturating impairment at high
infected cell load, also allows for the existence of two stable
steady states. The behaviour of this model is similar to that
of the CD4 help version.
Finally, a further concern about treatment interruptions is
that they might refill the latent reservoir. It has been shown
that this process is not likely to be relevant if the kinetics of
latently infected cells is similar to that observed during
chronic infection [105]. Virus replication during interruptions
could refill the latent pool only if it reached a level
comparable to the pre-treatment baseline. However, the
turnover of latently infected cells might be much faster
during chronic infection, which is accompanied by immune
hyperactivation, than during therapy and even the STI cycles
[38]. In this case, low levels of virus replication during STIs
could indeed rapidly refill the reservoir.
In all, treatment interruptions cannot restore drug
sensitivity and they can only contribute to immune control if
the virus load has a strong and reversible effect on the HIVspecific effector cells. However, interruptions impose a small
but with time increasing risk for de novo emergence of drug
resistance, and might contribute to the re-seeding of the
latent reservoir.
7. ESTIMATION OF VIRAL FITNESS
Fitness in a broad sense is used as a synonym for
replicative capacity or growth rate in virology, especially in
the context of measuring and comparing the growth rate of
different virus variants. The stricter definition of the term
depends on the particular methods used for its estimation.
“Fitness” values from different studies are therefore only
comparable if they were obtained with the same method.
In the preceding sections we have shown that the
direction and speed of viral evolution during and after
therapy, and also the composition of the virus population
before therapy depends on the relative replicative capacity,
or fitness, of the competing virus variants under the given
circumstances. The probability that a resistant mutant is
present at the start of therapy depends on its fitness relative
to that of the drug sensitive wild type virus in the absence of
the drugs. The short-term outcome of treatment depends on
the fitness relations in the presence of the drugs. The
evolution of multidrug-resistant virus is influenced by the
fitness of the intermediate variants and a similar effect shapes
the reversion of complex mutants to wild type in the absence
of therapy. To understand and possibly predict viral evolution,
one thus needs to know the fitness relations of the relevant
virus variants. For detailed reviews on viral fitness see [127].
Fitness is sometimes quantified on the basis of direct
biochemical measurements of enzyme activity [89, 128] or
virus growth kinetics in pure cultures that contain only one
variant [50, 89, 129]. Experimental protocols have been
reviewed recently in [130]. However, the assays that operate
340 Current Drug Targets – Infectious Disorders 2003, Vol. 3, No. 4
on single virus variants cannot take the interactions between
different virus strains into account. Many studies have
therefore investigated the relative growth kinetics of two
virus variants growing in competition either in vivo [131134] or in vitro [128, 135-139]. The interpretation of such
experiments requires careful mathematical considerations
[140, 141].
A measure of fitness is typically derived by plotting the
ratio of the two competing variants on a logarithmic scale
against time and estimating the linear slope of this graph
(Fig. (4a)). The underlying mathematical model is written as
follows:
dW
= [r(t) δ]W(t),
dt
dM
= [(1 + s)r(t) δ]M(t).
dt
(10-11)
We use the notation employed in fitness estimations, but
note that these equations can be derived from the standard
modelling framework. W and M denote the cells infected
with wild type or mutant virus, respectively, and thus
correspond to I1 and I2 in Eqs. ( 8-9). Since we are concerned
with the relative abundance of the two variants, the level of
virus particles needs not be modelled explicitly. A further
simplification from the basic model is that the growth rate of
the infected cells is described by a single parameter, r, which
might be a function of time. Assuming a quasi steady state
for the virus, this generalised growth rate corresponds to r(t)
= βfpT(t)/c in the basic model and is thus a complex function
of viral parameters and target cell availability. It is assumed
that the different virus variants have the same death rate, δ,
and differ in the growth rate by a factor, s. This factor, the
so-called selection coefficient, denotes the relative fitness
difference. As a slightly confusing practice, in the modelling
of drug treatment s is typically used to describe the selective
Müller and Bonhoeffer
disadvantage of one variant compared to the other and is thus
interpreted as a reduction in the growth rate (or in the
infection rate, which is a linear component of the growth
rate). Note that the only difference is the sign of the factor: a
negative selection coefficient corresponds to a positive
selective disadvantage.
From Eqs. (10,11) the slope of the logarithmic plot of the
mutant to wild type ratio at time t can be derived as r(t)s.
Importantly, this measure reflects the absolute, rather than
the relative fitness difference between the two variants.
Under better conditions for growth, both variants can grow
faster (higher r) and the absolute fitness difference estimated
by the slope of the logarithmic plot will also be greater. This
implies that estimates obtained under different growth
conditions cannot be compared with each other. The selection
coefficient, s, which gives the relative fitness difference,
eliminates all factors that affect the replication of both
variants and thus enables a better comparison between data.
To calculate the selection coefficient, s, from the absolute
fitness difference, one needs an estimate for the replication
rate, r, under the given experimental conditions. Since this is
usually not available, many researchers have simply divided
the absolute fitness difference by the generation time of the
virus to obtain an estimate for the selection coefficient [131,
132, 135, 136, 138]. However, the reciprocal of the
generation time equals the replication rate only if the
populations are at steady state, which certainly does not hold
for the standard growth competition assays in vitro.
Moreover, the growth rate might change over the duration of
a single experiment, e.g. due to the progressing depletion of
susceptible target cells in the culture. Fluctuations in the
growth rate can also occur in vivo. In such a case, the
logarithmic plot is no longer linear, and a simple division by
the growth rate cannot solve the problem (Fig. (4b)).
Denoting the log mutant to wild type ratio by h =
ln(M/W) and the log wild type virus load by w = ln(W), one
Fig. (4). The estimation of fitness from competition experiments. The logarithm of the ratio of mutant to wild type virus is monitored over
time. In the conventional method (A), the fitness difference between the competing variants is estimated as the slope of the line fitted to the
data points by linear regression. However, the expected change in the log ratio of mutant to wild type is linear only if the basic growth rate of
the two variants is constant over the duration of the experiment (B). If the growth conditions that affect both variants change during the
experiment, simple linear regression cannot provide an appropriate fit to the data.
Mathematical Approaches in the Study of Viral Kinetics
can derive the following relation between the selection
coefficient and the values of h and w at the time-points t = 0
and t = T [141]: h(T) - h(0) = s (w(T) - w(0) + δ T). The
advantage of this equation is that it allows one to estimate
the selection coefficient without explicit knowledge of the
replication rate, and thus circumvents the problem of a timeand patient-dependent replication rate. A shortcoming of the
method is that it takes only two data points into account.
However, this problem can be circumvented by a non-linear
estimation method, which also provides confidence intervals
for the estimated selection coefficient [140]. The data needed
are the total virus load and the fraction of mutant virus at
each time point, and the death rate of virus-producing cells.
An implementation of the method is freely accessible at
http://www.eco.ethz.ch/fitness.html.
8. CONCLUDING REMARKS
We have attempted to give an overview of the most
important applications of mathematical modelling in drug
therapy and resistance, within the field of HIV-infection.
Beyond discussing particular problems our intention was
also to give an introduction to the methodology of
modelling. We have shown how the complexity of the
biological system must first be reduced to the bare bones,
which can then be fleshed out according to the varying
demands of particular problems. We have demonstrated the
power and the limitations of the basic, or “standard” model
of viral dynamics. The first extension was required to
account for stable low viral steady states.
We have concluded that the moderately reduced virus
loads during non-suppressive therapy and in patients infected
with multidrug-resistant virus can only be explained in the
models if we implement the virus induced killing of
uninfected cells or a cytotoxic response that is induced
proportional to the level of infected cells. This “requirement”
of the model implies that at least one of these two processes
are relevant in the setting of the steady state, i.e. of the
clinical status of patients. Similarly, the stable maintenance
of virus loads below detection level is only possible in the
models if we extend the basic scheme with a drug sanctuary
compartment. Since such a compartment has not been
identified yet, we are inclined to use the conclusion obtained
by modelling to argue against the existence, or at least the
relevance, of drug sanctuaries.
The second extension of the model was required to keep
track of drug resistant and sensitive virus variants separately
for the study of drug resistance. Quantitative modelling has
demonstrated that the resistant mutants that emerge during
suppressive therapy are more likely to arise by the selection
of pre-existing mutants than by de novo generation during
therapy. However, the reliable suppression of viremia in
drug naïve patients initiating therapy, which is typically
maintained for months if not years, indicates that there are
typically no pre-existing mutants that could grow in the face
of current combination treatments. The modelling results
then suggest that resistant mutants should be even less likely
to appear during treatment. In the light of these predictions,
the prevalence of eventual treatment failure due to emerging
multidrug resistance is puzzling. Combining the modelling
results with common sense we conclude that the reason for
Current Drug Targets – Infectious Disorder 2003, Vol. 3, No. 4
341
emerging resistance probably lies in temporal (and perhaps
spatial) troughs in drug concentrations, which relax
suppression and allow for new rounds of replication. It is of
vital importance to elucidate the causes of such drops in the
availability of drugs. A probable candidate in many cases is
non-adherence, which therefore comes at a very high price:
non-adherent patients can accumulate resistance to
consecutive drug regimens, eventually developing superresistant viruses. Non-adherence to the first regimen is
probably a strong predictor of non-adherence to later therapy
options. Furthermore, the transmission of resistant variants
has been rapidly increasing the pre-existence of resistance in
recently infected individuals [142, 143]. According to the
results presented in this review, these patients will never
fully revert to drug sensitivity and are therefore left with
reduced treatment options from the beginning. In all, every
effort should be made to ensure continuous viral suppression
during treatment through good adherence.
We have also shown that structured therapy interruptions
might only be beneficial under very strict conditions, namely
if the virus induces a strong but reversible impairment of the
HIV-specific immune responses in chronic infection. In this
case, the preservation or reconstitution of the immune
responses through well-timed antiviral therapy might shift
the system into a new steady state with a low virus load
controlled by a vigorous immune response. Such a steady
state might exist in long-term non-progressors and perhaps in
patients treated during primary infection, but in typical
chronically infected patients all attempts to establish an
immune controlled steady state have failed. The risks of STIs
on the other hand include a small but with time increasing
probability for the emergence of resistance and the reseeding
of the latent reservoir.
Finally, note that we have confined ourselves to the
discussion of the processes that occur within one infected
individual, i.e. to within-host dynamics. Mathematics can be
a useful tool also in the study of the spread of drug resistance
at the population level, but such epidemiological aspects lie
beyond the scope of the present review.
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