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# Production of resistant HIV mutants during antiretroviral therapy

^{*}Wellcome Trust Centre for the Epidemiology of Infectious Diseases, Department of Zoology, University of Oxford, South Parks Road, OX1 3PS, Oxford, United Kingdom; and

^{†}Friedrich Miescher Institut, P.O. Box 2543, CH-4002 Basel, Switzerland

^{‡}To whom reprint requests should be addressed. E-mail: hc.imf@bes.

## Abstract

HIV drug therapy often fails because of the appearance of multidrug-resistant virus. There are two possible scenarios for the outgrowth of multidrug-resistant virus in response to therapy. Resistant virus may preexist at low frequencies in drug-naïve patients and is rapidly selected in the presence of drugs. Alternatively, resistant virus is absent at the start of therapy but is generated by residual viral replication during therapy. Currently available experimental methods are generally too insensitive to distinguish between these two scenarios. Here we use deterministic and stochastic models to investigate the origin of multidrug resistance. We quantify the probabilities that resistant mutants preexist, and that resistant mutants are generated during therapy. The models suggest that under a wide range of conditions, treatment failure is most likely caused by the preexistence of resistant mutants.

In recent years, management of HIV infection has greatly improved because of the development of new treatment protocols, involving the combination of highly potent drugs (1–5). However, combination therapy is not effective in all patients and may fail because of severe side effects, nonadherence to therapy protocol, lack of potency of drugs, or emergence of resistant virus (refs. 2 and 6; See www.hivatis.org/trtgdlns.html and refs. therein).

Generally, there are two main processes leading to resistance-related
treatment failure: preexisting resistant strains may be selected by the
drugs used, or resistant mutants are generated *de novo* by
residual virus replication during treatment. It is important to
distinguish between these processes, because they require different
actions to improve therapy. If treatment fails because of preexisting
resistant virus, than increasing the efficacy of the drugs (for example
by increasing the dosage) may not suffice to control virus replication.
Rather, several drugs with different resistance profiles need to be
combined, reducing the likelihood that strains resistant to combination
therapy are present in the first place. On the other hand, if
resistance arises *de novo* during treatment, then increasing
the dosage of the drug may lead to a more effective treatment. In this
case, the objective would be to minimize any residual replication of
the sensitive virus during therapy, because this would reduce the
probability of producing a resistant mutant.

Thus, determination of which of the two causes for treatment failure is
more likely may be helpful to find the best therapy regimen. Laboratory
testing for the presence of resistant strains in a patient has been
proposed (7–12). However, current methods either are not sensitive
enough to detect mutants at very low frequencies or are too laborious
to be used in clinical practice (7). In view of these difficulties,
this question has been addressed by using population dynamical models
(13–15). However, so far, these theoretical approaches have
underestimated the probability of treatment failure attributed to
*de novo* generation of resistance by the residual virus
replication during treatment. Here, we use both deterministic and
stochastic approaches to investigate the origin of drug-resistant
mutants and derive an upper limit to the probability of emergence of
resistant virus during therapy. On the basis of quantifiable parameters
such as the viral load and the viral mutation rate, we estimate the
likelihood of preexistence of resistant strains in comparison to the
likelihood of emergence of resistant virus during therapy. We emphasize
at the outset that we are not concerned with drug resistance in
patients who were infected by resistant carriers. Although a major
concern for the future, to date the spread of resistant strains seems
to account only for a minority of treatment failures caused by
resistance (16–18). Instead, we focus on the emergence of resistance
in drug-naïve patients who were infected with sensitive virus,
but who may develop resistant mutants at low frequency in a
mutation-selection equilibrium.

## Definition of the Model

We begin with the basic model of HIV dynamics (13, 14, 19–22):

For a detailed description of this model, see ref. 23. Here the
variables *x* and *y* denote the population densities
of susceptible and infected cells, respectively. The model has five
parameters: λ, the rate of immigration of susceptible cells from a
pool of precursor cells; δ, the death rate of susceptible cells;
*b*, the infectivity rate; *a*, the per capita death
rate of infected cells; and *r*, the inhibitory effect of drug
therapy on virus replication, which is between 0 and 1, with
*r* = 1 corresponding to no treatment.

This model has two equilibria corresponding to the uninfected and
infected steady states. The evolution of the system to one or the other
steady state is determined by the basic reproductive ratio,
*R*_{0} (24–26). In the present context,
*R*_{0} is defined as the average number of secondary
infected cells produced by the first infected cell introduced in a
wholly susceptible population. For the above model, the basic
reproductive rate before the start of treatment is given by
*R*_{b} = λ*b*/(δ*a*). If
*R*_{b} < 1, then on average one infected cell
produces less than one secondary infected cell, and hence the system
goes to the uninfected steady state given by *x*_{U}
= λ/δ and *y*_{U} = 0. Conversely, if
*R*_{b} > 1, the system goes to the infected
steady state given by *x*_{I} = *a*/*b* and
*y*_{I} = λ/*a* − δ/*b*.

In the absence of treatment (*r* = 1), the virus is
expected to have a basic reproductive ratio larger than 1, because
otherwise it is unable to cause or sustain an infection. During
treatment (*r* < 1), the new basic reproductive ratio,
*R*_{d} = *r*λ*b*/(δ*a*), can be smaller or
larger than 1, depending on whether the virus population is sensitive
or resistant to treatment. In the following, we define resistant
viruses as those viruses with a basic reproductive ratio during
treatment larger than one, *R*_{d} > 1. This
defines drug resistance not in terms of an *in vitro* assay,
but as a combined property of host, virus, and drug. Note, however,
that a basic reproductive ratio smaller than one does not imply that
there is no replication. The *de novo* production of a
resistant virus during therapy is possible so long as the basic
reproductive ratio is larger than zero.

## Analysis of the Model

We use the above model to distinguish between the two alternative
hypotheses for the cause of treatment failure caused by resistance:
(*i*) The *preexistence hypothesis*, according to
which treatment failure is caused by the existence of resistant mutants
in the patient's virus population before the start of therapy; and
(*ii*) the *emergence hypothesis*, according to which
resistant variants are absent before therapy, but treatment failure
occurs because resistant mutants are produced from the sensitive virus
population as it declines during therapy.

The likelihood of emergence of resistance during therapy depends
on the number of cells that become newly infected as the sensitive
virus population declines during therapy. This number is given by
*rb* ∫_{t=0}^{t=∞} *xydt*, where *t* =
0 is the start of therapy, and *r* < 1.
Unfortunately, this integral cannot be calculated in closed form,
because we do not have analytical solutions for *x*(*t*) and
*y*(*t*). However, we can approximate Eq. 1 by
neglecting the term *rbxy* in the first equation for
*x*(*t*). The simplified model is:

Because the simplified model has no nonvanishing steady
state for the infected cell population, we use the infected steady
state of Eq. 1 as the initial condition. Because Eq.
2 neglects the loss rate of susceptible cells because of
infection, (*t*) overestimates *x*(*t*), as
given by the full model (Eq. 1). As a consequence,
ỹ(*t*) also overestimates *y*(*t*), because the
per capita rate of production of new infected cells,
*rb*(*t*), overestimates *rbx*(*t*). For a
numerical comparison of the models, see Fig.
Fig.1.1. For the simplified model, we can
obtain analytical solutions for (*t*) and
ỹ(*t*) and thus can compute *rb*
∫_{0}^{∞}ỹ*dt*, which gives an upper
limit to *rb* ∫_{0}^{∞} *xydt*.

*A*, we compare the increase in the number of susceptible cells and in

*B*, the decrease in the infected cell population. Notice that in

**...**

In equilibrium, the probability of preexistence of resistant
virus depends on the total number of infected cells present at the
start of therapy, which is given by *y*_{I}. We thus
calculate the ratio, Θ, of the number of cells infected during
therapy (given by *rb* ∫_{0}^{∞}ỹ*dt*) and the number of infected cells present at the
start of therapy (given by *y*_{I}). A detailed
derivation in the Appendix yields:

where _{1}*F*_{1} is the generalized
hypergeometric function, γ = *a*/δ is the ratio of
the death rates of infected and uninfected cells, and
*R*_{b} and *R*_{d} are the basic
reproductive ratios of the sensitive virus before and during treatment,
respectively. Note also that the ratio, Θ, depends only on three
parameters, γ, *R*_{b}, and
*R*_{d}.

As the sensitive virus is able to maintain an infection in the
absence but not in the presence of treatment, we have
*R*_{b} > 1 and *R*_{d} <
1. Furthermore, it is reasonable to assume that the death rate of
infected cells exceeds that of uninfected cells, i.e. γ > 1.
Fig. Fig.22 shows the ratio Θ as a function
of *R*_{b} and *R*_{d} for a
conservative minimal value of γ = 10 (21, 27, 28). [Generally,
Θ(*R*_{b}, *R*_{d}, γ) is a decreasing
function of γ.] Interestingly, the ratio Θ is smaller than 1 for
most of the parameter region, implying that the number of infected
cells present at the start of therapy is larger than the number of
cells infected during therapy. Θ is larger than 1 only if
*R*_{d} is close to 1. However, this is also the
region where Θ greatly overestimates the corresponding ratio
(*rb*/*y*_{I} ∫_{0}^{∞} *xydt*) of the
full model defined by the system of Eq. 1.

*R*

_{d}and

*R*

_{b}, for γ = 10. For most of the parameter region, the ratio

**...**

Thus, typically the number of infected cells present at the start
of therapy exceeds the number of infected cells produced during
therapy, which suggests that the probability of preexistence of
resistance is larger than the probability of production of resistance
during therapy. Exceptions to this rule may occur in a narrow region,
where the basic reproductive ratio of the sensitive virus during
therapy, *R*_{d}, is very close to but below 1.
Nonetheless, numerical simulations of the full model, given by Eq.
1, suggest that this region is actually smaller than that
shown in Fig. Fig.22.

## Multistrain Model

Although the comparison of the number of infected cells present at
the start of therapy with the number of cells infected during therapy
provides an intuitive measure of the relative likelihood of the
preexistence and emergence hypothesis, this is only partially
satisfactory, because it ignores the full complexity of a heterogeneous
virus population before and during selection pressures imposed by drug
treatment. Therefore, we extend the above model and subdivide the
population of infected cells into *l* populations,
*y*_{i}, each infected with a different virus mutant
*i*. Again we calculate the ratio of expected production
during therapy and expected frequency before therapy, but this time we
calculate this ratio for each mutant, as a function of the number of
point mutation differences between the mutant and the predominant wild
type. The modified dynamical equations are:

Here the infectivity parameter *b* is multiplied by a
factor 1 − *s*_{i}, which accounts for the
selective disadvantage of mutant *i* in comparison to the wild
type. Hence, *s*_{0} = 0 for the wild type
(mutant 0), and 0 < *s*_{i} ≤ 1 for all
other mutants. The matrix μ_{ij} describes the
probability of mutation of strain *j* into strain *i*
during reverse transcription. Hence, cells infected by mutant
*i* are produced either by infection of a susceptible cell
with mutant *i* or by mutation of strain *j* into
strain *i* during infection.

We consider only the *n* nucleotide sites in conferring
resistance to a particular drug regimen. That is, we group all possible
strains in the viral population in classes according to their status at
the sites that confer resistance. At these *n* sites, each
strain either has the nucleotide necessary for resistance or not, which
we call 1 and 0, respectively. This represents a binary model for
*n* nucleotides. The resistant mutant is *n*-point
mutations away from the wild type, and for each class of
*k*-point mutants, there are (_{k}^{n})
strains. For instance, if we consider *n* = 3, then there
are three one-point mutants (001, 010, 100) and three two-point mutants
(011, 101, 110). The total number of strains, *l*, in Eq.
4 thus equals 2^{n}.

For simplicity, we assume that the point mutation rate per replication
cycle is the same at all sites, μ, such that
μ_{ij} = μ^{|i−j|} for
*i* ≠ *j*, and μ_{ii} = 1 −
∑_{i≠j} μ_{ij} = (1 −
μ)^{n} (where |*i* − *j*| is the
number of sites at which mutant *i* and *j* differ).
Assume further that all mutants have the same selective disadvantage in
comparison to the wild type (*s*_{i} = *s* 1
for 1 ≤ *i* ≤ *l*). In this case, a simple expression can
be obtained for the equilibrium frequency of a *k*-point
mutant (29):

where *y*^{*}_{0} stands for the
equilibrium frequency of the wild type.

For the resistant *n*-point mutant, we calculate the ratio
between the number of cells newly infected during therapy and the
number of cells infected with that strain present at the start of
therapy. To this end, we calculate the total production of the
*n*-point mutant during therapy, provided it did not exist
before therapy, but all 0 to *n* − 1 point mutants were
in the mutation-selection equilibrium given by expression **5**.
Again, at *t* = 0, we initiate therapy which reduces the
basic reproductive ratio of all strains present to 0 <
*R*_{d} < 1. The total production of
*n*-point mutants by a strain *k*(0 ≤ *k* <
*n*) during therapy is given by:

where *b*_{d} = *rb*(1 − *s*) is
the infectivity during therapy. Hence, the total production of
*n*-point mutants by all other strains
*k*(0 ≤ k ≤ n − 1) can be
approximated as:

This expression represents the sum of the contribution of all
preexisting strains according to Eq. 6, taking into account
that for each *k* there are (_{k}^{n})
strains, as explained above.

Evaluating the sum in Eq. 7 (see Appendix) and
dividing by expression **5** (with *k* = *n*), we
obtain for the ratio of the production of *n*-point mutants
during therapy and their frequency at the start of therapy:

where Θ(*R*_{b}, *R*_{d}, γ) is given
by Eq. 3. The approximation is valid for *s* <
0.2 and *n* > 3 (see Appendix).

Θ_{n} has a number of surprising
properties. First, it is independent of the mutation rate, μ. Thus,
counterintuitively, the relative likelihood of a mutant being produced
during therapy and being present at the start of therapy does not
depend on the mutation rate. Second, also contrary to expectation,
Θ_{n} increases with increasing selective
disadvantage of the mutants, *s* (Fig.
(Fig.3).3). This result arises because when
*s* increases, the probability of preexistence decreases
disproportionately in relation to the decrease in the likelihood of
emergence during therapy. Third, Θ_{n} depends
only very weakly on *n*. Hence, the relative likelihood of
emergence and preexistence is approximately the same regardless of the
number of point mutations by which the resistant differs from the wild
type (at least for *n* ≥ 3). Finally,
Θ_{n} is typically smaller than Θ, because for
small-to-moderate selective disadvantages (*s* < 0.6),
we have ∑_{i=1}^{n} *s*^{i}/*i*! ≈
(e^{s} − 1) < 1. This implies that mutants
are less likely to be produced (for the first time) during therapy than
to preexist when therapy is started. This conclusion holds so long as
the basic reproductive ratio of the sensitive virus during therapy is
not very close to one, and the selective disadvantages involved are
small.

## Stochastic Simulations

The results described so far were obtained by using
deterministic models to calculate the number of infected cells produced
during therapy and present at the start of therapy. In a deterministic
model, the frequency of a mutant will never go to zero, provided its
basic reproductive ratio is larger than one. However, this may not be
realistic. A particular strain might be created by mutation, but there
is some probability that subsequently it will be lost again because of
stochastic effects, even if its basic reproductive ratio is larger than
one, *R*_{0} > 1. This is an important
consideration, because the frequencies of mutants are often very small
and therefore subject to stochastic fluctuations. In this section, we
develop a stochastic approach to the problems addressed above. Instead
of the number of infected cells produced, we consider the actual
probabilities that a specific mutant exists in the population before
and during treatment. Although an analytic approximation for this
stochastic model is possible (30), here, in the interest of space, we
show only results of simulations.

In the stochastic framework, each event (production and death of susceptible and infected cells) is assigned a probability as opposed to a deterministic rate. The probability of each event occurring is related to the rates in the system of Eq. 1. Table Table11 shows the structure of the stochastic model with its events and associated probabilities.

For the following simulations, we assume a scenario where three
particular point mutations confer drug resistance. Hence we consider
8(=2^{3}) different mutants, corresponding to all possible
one- and two-point mutants at the three relevant sites, as well as the
wild type (000) and the strain (111) with all the required mutations
for resistance. For simplicity, we assume, as in the previous section,
that all intermediate one- and two-point mutants and the resistant
three-point mutant have the same selective disadvantage, *s*.
At the start of the simulation, all strains except the resistant strain
are present at their respective equilibrium frequencies, given by Eq.
5. The simulation is run for a period that allows the
generation of stochastic diversity around the equilibrium. Then
treatment is started. If the resistant mutant is present, the
simulation is stopped. Otherwise, the simulation is continued either
until the virus population is eradicated or until a resistant mutant
emerges.

We simulated all permutations of the following set of parameters:
selective disadvantage *s* = 0.002, 0.005, 0.009; ratio
of infected to uninfected cell death rates γ = 5, 10; basic
reproductive ratio before therapy *R*_{b} = 2,
10; and basic reproductive ratio during therapy
*R*_{d} = 0.9, 0.98. This makes a total of 24
sets of simulations. For each set of parameters, the simulation was run
100 times to calculate the average probability of emergence of a
resistant mutant and 500 times for the average probability of
resistance emerging during treatment. We calculated the probability of
emergence of the resistant strain as the proportion of runs the
resistant strain was present at the end of a sufficiently long time,
with or without treatment. We also calculated the fraction of the time
that a resistant mutant is present in an untreated patient. Because
these results are equivalent to the number of times a resistant strain
is present at the end of a sufficiently long run, these results are not
shown here.

In support of the conclusions drawn from the deterministic models, the
probability of the resistant strain emerging after treatment was in no
case higher than the probability of the resistant virus already being
present before treatment (data not shown). Even when
*R*_{d}(=0.98) and *R*_{b}(=2) are
close to 1, for which the analytical calculations are most inaccurate,
the relevant ratio is still smaller than 1.

To study the effect of the different parameters on the ratio of
probabilities, the results of further simulations are presented in
Table Table2.2. In the top of the table, we find
that for increasing *R*_{d}, the ratio increases, as
expected from the deterministic theory. This increase in the ratio is
because of an increase in the probability of the resistant strain
emerging after the start of therapy, because, as expected, the
corresponding probability before therapy stays roughly constant.
Although the ratio increases with *R*_{d}, even for
*R*_{d} = 0.99, the ratio of probabilities is
small.

In the middle of the table, the effect of γ is shown. Again, the
probability of the resistant mutant being present before therapy stays
roughly constant, suggesting that this probability is independent of
γ. On the other hand, for smaller γ, the probability of producing a
mutant during therapy is higher. Thus, the ratio of probabilities
increases for smaller γ, but remains below one even for an
unrealistic value of γ = 1 (implying that uninfected and
infected cells have the same life span). The bottom of the table shows
the effect of *s*. The selective disadvantage affects both
probabilities before and after treatment in the same way: the
probabilities decrease for larger values of *s*. This double
effect makes it difficult to discern a clear trend for the ratio of
probabilities. However, it seems that the ratio of probabilities
increases with *s*, because the probability of the resistant
strain being present before therapy decreases disproportionately in
relation to the decrease in the corresponding probability after
treatment. In any case, also in these simulations, the ratio of
probabilities is always smaller than one.

In summary, the stochastic simulations are in excellent agreement
with our analytical calculations. They strongly support the hypothesis
that the more likely cause of resistance-related treatment failure is
the presence of resistant strains before therapy. Moreover, even for
unrealistic sets of parameters, corresponding to cases where the
theoretical value of Θ_{n} is close to or above
one, the probability of producing a new resistant mutant is smaller
than the probability of this mutant preexisting in the population.

So far, we have assumed that after therapy, all sensitive strains have
the same basic reproductive ratio, *R*_{d}. If, on
the other hand, therapy reduces *R*_{d} of the wild
type more than it reduces *R*_{d} of the other
sensitive strains, such that *R*_{d(wt)} <
*R*_{d(otherstrains)}, the results obtained can be
quite different. In Table Table3,3, we show some
results for the limiting case of that inequality, i.e., the wild
type does not replicate at all after therapy,
*R*_{d(wt)} = 0, and
*R*_{d(otherstrains)} is close to one. In this
limiting case, the probability of producing a resistant mutant is
higher than that observed when the wild type also replicates.

When therapy is more efficient in relation to the wild type than in relation to the other preexisting sensitive strains, the probability of production of a resistant mutant during therapy may sometimes be higher than the probability of that mutant already existing before therapy.

## Discussion

The appearance of HIV strains resistant to a particular drug
regimen is the main problem during treatment of infected individuals
(refs. 2 and 7; www.hivatis.org/trtgdlns.html). In principle, there
are two ways in which resistance can emerge in response to therapy (13,
14): (*i*) resistant strains may already exist when therapy is
started; or (*ii*) all preexisting strains are sensitive, but
the drug regimen is not 100% effective, and the resistant mutant is
created *de novo* during treatment. Using various
deterministic and stochastic models, we have shown that almost
universally treatment fails because of the preexistence of resistant
strains in the drug naïve viral population. It is generally
less likely that resistant mutants are generated for the first time
during treatment. This suggests that efforts to reduce the risk of
treatment failure need to concentrate on the combination of drugs with
different resistance profiles in order to minimize the risk that
multidrug-resistant strains preexist in a drug-naïve viral
population. Increasing the efficacy of replication inhibition is only
of secondary concern.

It is important to emphasize at this point that this paper has not dealt with the evolution of resistance caused by poor adherence to the drug regimen. Clearly, if patients at certain periods take only a subset of the prescribed drugs, then resistance may evolve successively to each of the drugs used. Similarly, spatial heterogeneity in the distribution of the drugs used may facilitate the evolution of resistance as the virus may locally be controlled only by a single drug (31). Furthermore, this paper also does not consider the contribution of viral replication at sanctuary sites to the evolution of resistance. In the narrow sense of our definition of drug resistance, a virus capable of persisting at a sanctuary site is resistant, because its basic reproductive ratio is larger than one during treatment. However, the question is whether the virus replicating at these sanctuary sites is actually likely to produce a fully resistant virus that is capable of recolonizing the main sites of infection in the presence of treatment. Clearly, continuous replication of the virus at sanctuary sites increases the risk that a fully resistant mutant is produced during treatment. However, if such a fully resistant mutant is unlikely to be present at the start of therapy, then we expect that its production during treatment should take a long time, because the viral population replicating during treatment is typically orders of magnitude smaller than the viral population at the start of therapy.

Although our results show, as expected, that the likelihood of
creating a resistant mutant during therapy decreases with more
effective inhibition of replication (i.e., with smaller
*R*_{d}), this likelihood is generally smaller than
the likelihood that resistant viruses preexist. Furthermore, it is
interesting that the ratio of these probabilities increases for higher
selective disadvantages of the sensitive strains present at the start
of therapy. We must caution, however, that this is true only for the
relative risk. In absolute terms, the risk decreases with increasing
*s* (Table (Table2).2). Most surprisingly, the relative risk is
independent of both the mutation rate and the number of point mutations
necessary for resistance.

Although much is now known about HIV genotypic and phenotypic resistance (11), testing for the presence of resistant strains in the context of drug therapy is still not very effective. This is because of the lack of sensitive and reliable tests, which can rapidly track resistance mutations (7–10). However, results of this paper suggest a new reason why great care must be taken in the choice of therapies. If the drugs are specifically targeted to be more efficient against the wild type but are less effective against other sensitive strains present in the viral quasispecies, the risk of producing a mutant during therapy increases relative to that of preexistence of resistance. The reason is that intermediate mutants contribute disproportionately to the production of the resistant strain, even though they are present at very low frequencies. If drugs control the wild type selectively, the other sensitive strains present may have more target cells available for infection (32–35), as they decline under treatment. The increased opportunities to infect cells, in turn, increase the risk of mutation into the resistant virus.

In summary, under very general conditions, analytical and numerical analysis of the viral quasispecies' response to selection pressures imposed by drug therapy argues strongly that the resistant mutants that appear in patients who failed on therapy are most likely present already at the start of therapy. Even those patients who fail on triple combination therapy but were fully compliant most likely already harbored resistant virus when therapy was started. Thus the key to drug resistance lies in the diversity of the viral population at the start of therapy.

## Acknowledgments

Support by the Novartis Research Foundation is gratefully acknowledged (S.B.). R.M.R. is supported by the PRAXIS XXI program of Fundação para a Ciência e Técnologia.

## Appendix

Here we present in detail the analytical expressions for the production of infected cells during treatment in the basic and multistrain models.

#### Basic Model.

The solution of the first equation in system **2** is

where (0) = *x*_{I} = *a*/*b*
corresponds to the infected steady state of the full model at the start
of therapy. Substituting solution **9** into the second equation
of system **2** and solving for ỹ, we obtain

where ỹ(0) = *y*_{I} = λ/*a*
− δ/*b* is the steady-state frequency of infected cells at the
start of therapy for the full model. *R*_{b} =
λ*b*/(δ*a*) and *R*_{d} = *rR*_{b}
are the basic reproductive ratios of the virus before and during
treatment, and γ = *a*/δ is the ratio of the life
spans of infected and uninfected cells. Using these definitions, we can
transform the integral *rb* ∫_{0}^{∞}(*t*)ỹ(*t*)*dt* as follows:

Substituting *z* = *e*^{−δt}, ρ =
γ*R*_{d}(*R*_{b} − 1)/*R*_{b} and
= *a*(1 − *R*_{d}), and using Eq.
10, we obtain:

where Γ(/δ) is the gamma function, Γ(/δ, −ρ)
is the incomplete gamma function, and
_{1}*F*_{1}(1, 1 + /δ, −ρ) is
the generalized hypergeometric function. The integrals and functional
relationships used in the calculation can be found in ref. 37.
Expressed in terms of basic reproductive ratios, we obtain, by
back-substituting *z*, ρ, and ,

The ratio of the number of infected cells produced during therapy
(Eq. 13) and the number of infected cells before therapy
(*y*_{I}) is:

#### Multistrain Model.

The total production of cells infected with the resistant
*n*-point mutant during therapy is given by Eq. 7.
Using Eqs. 7 and 5 and substituting
*y*_{k}(0) =
*k*!(μ/*s*)^{k}*y*^{*}_{0}, we obtain:

where *y*_{0}(0) = *y*_{I}.
The ratio of production of *n*-point mutants during therapy
and their frequency of preexistence is obtained by dividing expression
**15** by expression **5**, with *k* replaced
by *n*:

For *s* < 0.2 and *n* > 3, we can
simplify further (with an error of less than about 10%):

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