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- Abstract
- 1 Introduction
- 2 Rapid emergence of drug resistance
- 3 A multi-strain model and quasispecies dynamics
- 4 Conclusions and discussions
- References

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Bull Math Biol. Author manuscript; available in PMC 2013 August 1.

Published in final edited form as:

Published online 2012 May 26. doi: 10.1007/s11538-012-9736-y

PMCID: PMC3400267

NIHMSID: NIHMS386490

The publisher's final edited version of this article is available at Bull Math Biol

See other articles in PMC that cite the published article.

Telaprevir, a novel hepatitis C virus (HCV) NS3-4A serine protease inhibitor, has demonstrated substantial antiviral activity in patients infected with HCV. However, drug-resistant HCV variants were detected in vivo at relatively high frequency a few days after drug administration. Here we use a two-strain mathematical model to explain the rapid emergence of drug resistance in HCV patients treated with telaprevir monotherapy. We examine the effects of backward mutation and liver cell proliferation on the preexistence of the mutant virus and the competition between wild-type and drug-resistant virus during therapy. We also extend the two-strain model to a general model with multiple viral strains. Mutations during therapy only have a minor effect on the dynamics of various viral strains, although they are capable of generating low levels of HCV variants that would otherwise be completely suppressed because of fitness disadvantages. Liver cell proliferation may not affect the pretreatment frequency of mutant variants, but is able to influence the quasispecies dynamics during therapy. It is the relative fitness of each mutant strain compared with wild-type that determines which strain(s) will dominate the virus population. This study provides a theoretical framework for exploring the prevalence of preexisting mutant variants and the evolution of drug resistance during treatment with other HCV protease inhibitors or polymerase inhibitors.

Chronic infection with hepatitis C virus (HCV) has caused an epidemic with approximately 130 to 170 million people infected worldwide and 3 to 4 million individuals newly infected each year [1]. About 80% of newly infected patients develop chronic infection. Of those chronically infected, 60 – 70% develop chronic liver disease, 5 – 20% develop cirrhosis, and 1 – 5% die from cirrhosis or liver cancer [1]. A combination of pegylated interferon (PEG-IFN), administered once weekly, and daily oral ribavirin (RBV) has been used to treat HCV infection for 24 or 48 weeks [2]. Although the combination exerts synergistic antiviral effects [3, 4], it leads to sustained viral elimination in only some treated patients. The HCV genotype appears to be an important factor in predicting response. In patients infected with genotypes 2 and 3, about 80–90% of patients achieve a sustained viral response (SVR), defined as the absence of detectable serum HCV RNA 24 weeks after completion of treatment. In patients infected with genotype 1, the major genotype affecting North America, Europe and Japan, only about 40% of treated individuals achieve SVR [2]. Lack of a complete response, viral relapse following treatment, and premature termination of therapy due to adverse events that occur during dosing all contribute to this unsatisfactory response rate observed among HCV genotype 1 infected patients. Therefore, new antiviral drugs with higher efficacy, shorter treatment duration, and a more favorable side-effect profile as a monotherapy or in combination with other antivirals are highly desirable.

New treatment options are focused on the development of direct-acting antiviral agents (DAAs) that target different steps of the HCV life cycle [5–7]. An important target is the HCV-encoded NS3-4A serine protease. In clinical trials HCV protease inhibitors have been used to treat HCV genotype 1 infected patients. They have shown an impressive capacity to block the NS3-4A protease-dependent cleavage of the HCV polyprotein, which is an essential step in viral replication (HCV replication will be discussed in detail later). The first protease inhibitor, BILN 2061 (ciluprevir; Boehringer-Ingelheim), showed potent antiviral activity in patients infected with HCV genotype 1 [8], but clinical development was halted due to drug-induced cardiotoxicity [9]. Boceprevir (SCH 503034; Schering-Plough Pharmaceuticals), another oral HCV protease inhibitor, also demonstrated substantial antiviral effects when used in combination with PEG-IFN-alpha-2b in HCV genotype 1 infected patients, who were previously nonresponders to PEG-IFN-alpha-2b with/without RBV therapy [10]. Telaprevir (VX-950; Vertex Pharmaceuticals) is a reversible, selective, and specific peptidomimetic inhibitor of NS3-4A that is effective in inhibiting viral replication in HCV replicon cells [11]. It had a favorable pharmacokinetic profile with high exposure in the liver in several animal models [12], and in monotherapy induced a profound decline of plasma HCV RNA levels of the order of 3–4 logs in patients infected with HCV genotype 1 treated for 14 days [13]. Both boceprevir and telaprevir have recently been approved by the US Food and Drug Administration (FDA) to treat HCV infection when used in combination with PEG-IFN and RBV. The addition of either of them to therapy with PEG-IFN and RBV has significantly increased the rates of SVR [14–18].

Emergence of drug-resistant mutations is a problem challenging the development of direct-acting antiviral drugs. Like most RNA viruses, HCV evolves rapidly because of high-level viral replication through an error-prone RNA polymerase that lacks associated proofreading capacity. As a consequence, the viral population exists as a complex mixture of genetically distinct, but closely related, variants commonly referred to as a quasispecies, whose composition is subject to continuous change due to the competition between newly generated mutants and existing variants with different phenotypes and fitness [19]. During antiviral therapy pre-existing minor viral populations with reduced susceptibility to the administered drug or drugs will gain a growth advantage over wild-type and become the dominant genotype. The amino acid substitutions selected by protease inhibitors that confer drug resistance have been characterized *in vitro* in the HCV replicon system [20–22].

The initial selection and kinetics of telaprevir-resistant HCV variants have been further described in patients given the protease inhibitor alone [13, 23] or in combination with PEG-IFN-alpha-2a [24, 25]. Although 14 days of treatment resulted in substantial decreases in HCV RNA levels, there was evidence of viral breakthrough in some patients during the dosing period, which was believed to be associated with the selection of HCV variants with reduced sensitivity to telaprevir [13]. Using a highly sensitive sequencing assay, Sarrazin et al. [23] identified mutations that confer resistance to telaprevir in the NS3 protease catalytic domain and correlated them with virologic response. These mutations were further investigated in a subsequent study [25] that provides a detailed kinetic analysis of HCV variants in patients treated with telaprevir alone or in combination with PEG-IFN-alpha-2a for 14 days. The four HCV genotype 1a infected patients in the telaprevir monotherapy group all exhibited viral load rebound during the dosing period. Virus isolated from these patients at day 2 contained low levels (5%–20%) of single-mutant resistant variants, which increased in the population of virus isolated at days 6 and 10, and were replaced by more resistant double-mutant variants by day 13 and during the first follow-up week with PEG-IFN plus RBV [25]. Why drug-resistant viral variants emerged so rapidly following treatment with telaprevir is not fully understood.

In this paper, we study HCV quasispecies and drug resistance in patients treated with the protease inhibitor telaprevir. We begin with a simple two-strain model in which liver cells, e.g., hepatocytes, infected with wild-type virus are able to produce not only wild-type virus but also a small amount of drug-resistant variants. The two-strain model was studied numerically and was shown to fit the observed dynamics of both drug-sensitive and drug-resistant viruses in patients treated with telaprevir [26]. Here we study this model analytically. With reasonable simplifications, we obtain an analytical solution for the mutant frequency in patients given telaprevir alone, which is capable of explaining the rapid selection of pre-existing drug resistant variants after therapy initiation. We study the competition between wild-type and drug-resistant virus during treatment. We also examine the effects of backward mutation and hepatocyte proliferation on the pre-existing mutant frequency and the evolution of viral variants during therapy. Extending the two-strain model, we then develop a multi-strain model in which drug-resistant HCV variants that differ at more than one site are incorporated. We calculate the expected frequency of each viral strain in untreated patients. The results of the competition between multiple viral variants during therapy with telaprevir are also provided. Because telaprevir and boceprevir inhibit the same HCV protease, the analysis in this study with telaprevir can be applied to boceprevir or to other HCV protease inhibitors under development.

Before describing the model, we use a diagram of the HCV life cycle (Figure 1) as a framework for discussing our current knowledge of virus replication. The exact mechanism by which HCV enters hepatocytes, the primary targets of infection, is still largely unknown. It is presumably receptor-mediated and involves CD81 [27], the human scavenger receptor class B type 1 (SR-B1) [28], and other molecules such as claudin-1 [29] and occludin [30]. Following fusion of the viral and cellular membranes, nucleocapsid enters the cytoplasm of the host cell and releases a single-stranded, positive-sense RNA genome (uncoating). This genome serves, together with newly synthesized RNAs, multiple roles within the HCV life cycle: as a messenger RNA (mRNA) for translation to produce a large polyprotein, as a template for HCV RNA replication, and as a nascent genome that is packaged in progeny virus particles. The generated polyprotein is then cleaved by several enzymes including the NS3-4A serine protease to produce 10 viral proteins: the structural proteins (the core protein C, glycoproteins E1 and E2), a small integral membrane protein p7, and the nonstructural (NS) proteins NS2, NS3, NS4A, NS4B, NS5A and NS5B. This is followed by RNA replication that occurs in a specific cytoplasmic membrane alteration, termed the “membranous web”, whose formation is induced by the integral membrane protein NS4B [31]. The process of RNA synthesis is not fully characterized, but is likely to be semi-conservative and asymmetric [32]: the positive-strand genome RNA serves as a template for the synthesis of a negative-strand intermediate; the negative-strand RNA then serves as a template to produce multiple nascent genomes. Both of these steps are catalyzed by the NS5B RNA-dependent RNA polymerase (RdRp). In the meantime, structural proteins E1, E2 and C have matured. Together with progeny positive-strand genomes, they assemble and are ready for vesicle fusion at the host cell plasma membrane, after which new virions are released into the extracellular milieu by exocytosis.

The viral RdRp is an important enzyme that catalyzes the synthesis of both positive- and negative-strand RNAs. However, the HCV RdRp has a high error rate, with a misincorporation rate of 10^{−4}–10^{−5} per copied nucleotide [33]. Furthermore, since the RdRp is devoid of proofreading capacity and other postreplicative repair mechanisms, it cannot correct misincorporations that occur randomly during replication [19]. The high mutation rate, together with rapid HCV replication [34] and the large viral population size, results in progressive diversification of viral genotypes and subtypes in geographically or epidemiologically-linked populations, and in the quasispecies nature of the virus population in a given infected individual [19].

We adapt a mathematical model, which was used to study HIV-1 infection and drug resistance [35], to examine the quasispecies dynamics of HCV before and during treatment. Based on the error-prone nature of the HCV polymerase, hepatocytes infected with wild-type virus are expected to produce both wild-type virus and mutant variants. A simple model including two strains, wild-type and drug-resistant (assuming a single mutation confers a certain level of drug resistance), is described by the following equations:

$$\begin{array}{l}\frac{d}{dt}T(t)=s-dT-{\beta}_{s}{V}_{s}T-{\beta}_{r}{V}_{r}T,\\ \frac{d}{dt}{I}_{s}(t)={\beta}_{s}{V}_{s}T-\delta {I}_{s},\\ \frac{d}{dt}{I}_{r}(t)={\beta}_{r}{V}_{r}T-\delta {I}_{r},\\ \frac{d}{dt}{V}_{s}(t)=(1-\mu ){p}_{s}{I}_{s}-{cV}_{s},\\ \frac{d}{dt}{V}_{r}(t)=\mu {p}_{s}{I}_{s}+{p}_{r}{I}_{r}-{cV}_{r},\end{array}$$

(1)

where *T* is the number of target cells; *I _{s}* and

There are 3 possible steady states (, *Ī _{s}*,

The above existence conditions also provide threshold conditions for the stability of the steady states. Indeed, we can show that (i) when
< 1/(1 − *μ*) and
< 1, *E*_{0} is locally asymptotically stable; (ii) when
> 1 and *r* > 1 − *μ*, *E _{r}* is locally stable; and (iii) when
> 1/(1 −

We calculate the frequency of the pre-existing drug-resistant variants in the total virus population from the coexistence steady state *E _{c}*. The mutant frequency is given by Φ =

The HCV NS3-4A serine protease plays an important role in viral polyprotein processing, cleaving at the NS3-4A junction and all downstream sites. Telaprevir, a new protease inhibitor, has been developed to block this step in the viral life cycle [12] and has been shown to profoundly reduce the plasma viral load in infected individuals [13, 23]. This is not surprising since the products of polyprotein cleavage are needed to mediate viral RNA replication and virion assembly (Figure 1). Assuming *ε _{s}* and

$$\begin{array}{l}\frac{d}{dt}T(t)=s-dT-{\beta}_{s}{V}_{s}T-{\beta}_{r}{V}_{r}T,\\ \frac{d}{dt}{I}_{s}(t)={\beta}_{s}{V}_{s}T-\delta {I}_{s},\\ \frac{d}{dt}{I}_{r}(t)={\beta}_{r}{V}_{r}T-\delta {I}_{r},\\ \frac{d}{dt}{V}_{s}(t)=(1-\mu )(1-{\epsilon}_{s}){p}_{s}{I}_{s}-{cV}_{s},\\ \frac{d}{dt}{V}_{r}(t)=\mu (1-{\epsilon}_{s}){p}_{s}{I}_{s}+(1-{\epsilon}_{r}){p}_{r}{I}_{r}-{cV}_{r}.\end{array}$$

(2)

Assuming *T* remains at the pretreatment steady state level, *T*_{0} = *cδ*/[(1−*μ*)*p _{s}β_{s}*], over a short period of time after drug administration, and ignoring the term

The mutant frequency following treatment is then given by the following function of *t*,

$$\mathrm{\Phi}(t)=\frac{{V}_{r}(t)}{{V}_{s}(t)+{V}_{r}(t)}=\frac{{C}_{3}{e}^{{\lambda}_{3}t}+{C}_{4}{e}^{{\lambda}_{4}t}}{{C}_{1}{e}^{{\lambda}_{1}t}+{C}_{2}{e}^{{\lambda}_{2}t}+{C}_{3}{e}^{{\lambda}_{3}t}+{C}_{4}{e}^{{\lambda}_{4}t}},$$

(3)

which depends on *c*, *δ*, *μ*, *ε _{s}*,

To study the change of the mutant frequency Φ(*t*) after drug administration, we have to determine the drug efficacy of telaprevir for each strain. The effectiveness of a drug against wild-type virus can be approximated by a simple function [39]
${\epsilon}_{s}(t)=\frac{C{(t)}^{h}}{{{\text{IC}}_{50}}^{h}+C{(t)}^{h}}$, where *C*(*t*) is the drug concentration, IC_{50} is the concentration of drug needed to inhibit viral production by 50%, and *h* is the Hill coefficient. Based on the pharmacodynamics of telaprevir, the drug efficacy for the wild-type strain was calculated to be 0.9997 (median) [40], which is consistent with the 3–4 log first-phase drop of plasma HCV RNA levels when telaprevir was administered in monotherapy [13]. Similarly, for the mutant virus with n-fold resistance, i.e., an n-fold increase in IC_{50}, we have
${\epsilon}_{r}(t)=\frac{C{(t)}^{h}}{{(n\xb7{\text{IC}}_{50})}^{h}+C{(t)}^{h}}$. From the equations of *ε _{s}*(

Plotting Φ(*t*) with typical parameter values and constant drug efficacy shows that the mutant frequency increases substantially from the pre-existing low level (< 1%) to > 5% within ~ 2 days following treatment (Figure 2). This is in agreement with the results in [25]. The rapid increase in mutant frequency does not necessarily mean that the drug-resistant viral variant grows this rapidly following the telaprevir treatment. In fact, taking a close look at the eigenvalues of the system, we find *λ*_{3} < *λ*_{1} < *λ*_{2} < *λ*_{4} < 0 (Appendix A). This implies that both wild-type and drug resistant viruses experience a two-phase decline when we assume *T* = *T*_{0} over a short time interval following treatment (see below for a time-varying population of target cells). Furthermore, the drug-resistant strain decreases slightly more rapidly than the wild-type strain during the first phase because *λ*_{3} < *λ*_{1} < 0 and the difference between *λ*_{1} and *λ*_{3} is small (Appendix A) as *ε _{s}* is close to 1, whereas it decreases slightly slower than wild-type strain in the second-phase viral decline (

In the above, successful suppression of the pre-existing mutant virus regardless of its drug resistance level is due to the assumption that the number of susceptible target cells remains at a constant baseline level, *T*_{0} = *cδ*/[(1 − *μ*)*p _{s}β_{s}*], following telaprevir treatment. If we describe the dynamics of target cells as in model (2), then drug-resistant virus is able to emerge and dominate the virus population under certain conditions.

Considering model (2), we define the reproductive ratios under treatment,
${\mathcal{R}}_{s}^{\prime}=(1-{\epsilon}_{s}){\mathcal{R}}_{s}$ and
${\mathcal{R}}_{r}^{\prime}=(1-{\epsilon}_{r}){\mathcal{R}}_{r}$. Before treatment, both strains coexist, but resistant virus remains at a very low level (*R _{s}* >

In addition to forward mutation, a backward mutation may occur that restores the original viral sequence. We compare model (1) (no backward mutation) with the following model (before treatment, i.e., *ε _{s}* =

$$\begin{array}{l}\frac{d}{dt}T(t)=s-dT-{\beta}_{s}{V}_{s}T-{\beta}_{r}{V}_{r}T,\\ \frac{d}{dt}{I}_{s}(t)={\beta}_{s}{V}_{s}T-\delta {I}_{s},\\ \frac{d}{dt}{I}_{r}(t)={\beta}_{r}{V}_{r}T-\delta {I}_{r},\\ \frac{d}{dt}{V}_{s}(t)=(1-\mu )(1-{\epsilon}_{s}){p}_{s}{I}_{s}+\mu (1-{\epsilon}_{r}){p}_{r}{I}_{r}-{cV}_{s},\\ \frac{d}{dt}{V}_{r}(t)=\mu (1-{\epsilon}_{s}){p}_{s}{I}_{s}+(1-\mu )(1-{\epsilon}_{r}){p}_{r}{I}_{r}-{cV}_{r}.\end{array}$$

(4)

Before drug therapy (*ε _{s}* =

It is interesting to study the contribution of mutation to the dynamics of virus during therapy. Supposing that wild-type and mutant virus are both at their pretreatment baseline levels, we compare virus dynamics of the model given by Eq. (4) with the model in which both forward and backward mutations are ignored (*μ* = 0 in (4)). Figure 4 shows the dynamics of both wild-type and drug-resistant viruses during therapy. For a mutation that confers a low level of drug resistance (for example, the mutant V36M/A confers 3.5-fold resistance [23]), inclusion of mutation has a negligible effect on the dynamics of both viral strains (Figure 4, left column). Even if mutation confers high-level resistance (for example, the A156V/T mutant confers 466-fold resistance [23]), the contribution of mutation to the level of the drug-resistant viral variant is still minor. However, in this case, wild-type virus can be maintained by backward mutation at a low level rather than being completely suppressed (Figure 4, right column). These observations are not surprising because in the presence of effective therapy targeted against wild-type virus the mutation from wild-type to drug-resistant strain makes a negligible contribution to the mutant viral load since it occurs at rate *μ*(1 − *ε _{s}*). Therefore, mutations only play a minor role in the dynamics of drug resistant virus during treatment.

Without mutation (*μ* = 0), Eq. (4) represents a standard two-strain model in which the two strains of virus compete for the same resource (susceptible target cells). Thus, the competitive exclusion principle applies—when the drug-resistant strain has a higher fitness under treatment (
${\mathcal{R}}_{r}^{\prime}>{\mathcal{R}}_{s}^{\prime}$), it outcompetes the wild-type strain.

Hepatocyte proliferation is important in liver regeneration [42] and can compensate for loss of hepatocytes during HCV infection. It can generate new target cells and has been included in mathematical models [43, 44]. Models with proliferation can explain complex HCV RNA profiles, such as the triphasic viral decay observed during treatment of some patients [43]. Here we incorporate proliferation of both uninfected and infected hepatocytes into model (1) and study the effects on the pretreatment mutant frequency and the evolution of drug resistance during therapy. The model with hepatocyte proliferation is

$$\begin{array}{l}\frac{d}{dt}T(t)=s+{\rho}_{T}T(1-\frac{T+{I}_{s}+{I}_{r}}{{T}_{\mathit{max}}})-dT-{\beta}_{s}{V}_{s}T-{\beta}_{r}{V}_{r}T,\\ \frac{d}{dt}{I}_{s}(t)={\beta}_{s}{V}_{s}T+{\rho}_{s}{I}_{s}(1-\frac{T+{I}_{s}+{I}_{r}}{{T}_{\mathit{max}}})-\delta {I}_{s},\\ \frac{d}{dt}{I}_{r}(t)={\beta}_{r}{V}_{r}T+{\rho}_{r}{I}_{r}(1-\frac{T+{I}_{s}+{I}_{r}}{{T}_{\mathit{max}}})-\delta {I}_{r},\\ \frac{d}{dt}{V}_{s}(t)=(1-\mu ){p}_{s}{I}_{s}-{cV}_{s},\\ \frac{d}{dt}{V}_{r}(t)=\mu {p}_{s}{I}_{s}+{p}_{r}{I}_{r}-{cV}_{r},\end{array}$$

(5)

where uninfected hepatocytes (i.e. target cells), hepatocytes infected with wild-type virus, and hepatocytes infected with drug resistant virus can proliferate with maximum proliferation rates *ρ _{T}*,

We are interested in the pretreatment mutant frequency. In Appendix C, we show that the mutant frequency is the same as that of model (1) if *ρ _{s}* =

If we ignore mutations during treatment, then the model with hepatocyte proliferation becomes

$$\begin{array}{l}\frac{d}{dt}T(t)=s+{\rho}_{T}T(1-\frac{T+{I}_{s}+{I}_{r}}{{T}_{\mathit{max}}})-dT-{\beta}_{s}{V}_{s}T-{\beta}_{r}{V}_{r}T,\\ \frac{d}{dt}{I}_{s}(t)={\beta}_{s}{V}_{s}T+{\rho}_{s}{I}_{s}(1-\frac{T+{I}_{s}+{I}_{r}}{{T}_{\mathit{max}}})-\delta {I}_{s},\\ \frac{d}{dt}{I}_{r}(t)={\beta}_{r}{V}_{r}T+{\rho}_{r}{I}_{r}(1-\frac{T+{I}_{s}+{I}_{r}}{{T}_{\mathit{max}}})-\delta {I}_{r},\\ \frac{d}{dt}{V}_{s}(t)=(1-{\epsilon}_{s}){p}_{s}{I}_{s}-{cV}_{s},\\ \frac{d}{dt}{V}_{r}(t)=(1-{\epsilon}_{r}){p}_{r}{I}_{r}-{cV}_{r}.\end{array}$$

(6)

This is not a standard two-strain competition model because the two strains can coexist under certain conditions. Substituting * _{s}* = (1−

$$\left[\frac{(1-{\epsilon}_{s}){\beta}_{s}{p}_{s}\overline{T}}{c}+{\rho}_{s}(1-\frac{\overline{T}+{\overline{I}}_{s}+{\overline{I}}_{r}}{{T}_{\mathit{max}}})-\delta \right]{\overline{I}}_{s}=0$$

(7)

and

$$\left[\frac{(1-{\epsilon}_{r}){\beta}_{r}{p}_{r}\overline{T}}{c}+{\rho}_{r}(1-\frac{\overline{T}+{\overline{I}}_{s}+{\overline{I}}_{r}}{{T}_{\mathit{max}}})-\delta \right]{\overline{I}}_{r}=0.$$

(8)

If *ρ _{s}* =

Although the two strains can coexist under certain conditions, wild-type virus is in general successfully suppressed because the inhibitor telaprevir is very effective against wild-type virus. Whether drug-resistant virus will also be suppressed depends on its reproduction capacity, proliferation potential of cells infected with resistant virus, and the drug efficacy *ε _{r}*.

During treatment with telaprevir, mutations mainly occur at 4 positions in the HCV NS3 protease catalytic domain, i.e., at amino acids: 36, 54, 155, and 156 [23, 25]. Here we consider the mutations occurring at these 4 positions and develop a multi-strain viral dynamic model. We assume that there is no backward mutation and that the probability of a forward mutation occurring at each amino acid is identical, denoted by *μ*. A schematic diagram of the mutations between these viral variants is given in Figure 5. A different multi-variant viral dynamic model was used by Adiwijaya et al. [45] to quantify the antiviral response to telaprevir and the in vivo fitness of different variants in HCV patients.

The full model with 4 positions where mutation is possible, without considering backward mutations (*i.e.*, once a position mutates it can not mutate again), is given by

$$\begin{array}{l}\frac{d}{dt}T(t)=s-dT-\sum _{j}{\beta}_{j}{V}_{j}T,\\ \frac{d}{dt}{I}_{j}(t)={\beta}_{j}{V}_{j}T-\delta {I}_{j},\\ \frac{d}{dt}{V}_{0}(t)={(1-\mu )}^{4}{p}_{0}{I}_{0}-{cV}_{0},\\ \frac{d}{dt}{V}_{i}(t)=\mu {(1-\mu )}^{3}{p}_{0}{I}_{0}+{(1-\mu )}^{3}{p}_{i}{I}_{i}-{cV}_{i},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}i=1,2,3,4,\\ \frac{d}{dt}{V}_{ij}(t)={\mu}^{2}{(1-\mu )}^{2}{p}_{0}{I}_{0}+\mu {(1-\mu )}^{2}({p}_{i}{I}_{i}+{p}_{j}{I}_{j})+{(1-\mu )}^{2}{p}_{ij}{I}_{ij}-{cV}_{ij},\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}i,j=1,2,3,4\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}i<j,\\ \frac{d}{dt}{V}_{\mathit{ijk}}(t)={\mu}^{3}(1-\mu ){p}_{0}{I}_{0}+{\mu}^{2}(1-\mu )({p}_{i}{I}_{i}+{p}_{j}{I}_{j}+{p}_{k}{I}_{k})+\mu (1-\mu )({p}_{ij}{I}_{ij}+{p}_{ik}{I}_{ik}+{p}_{jk}{I}_{jk})+(1-\mu ){p}_{\mathit{ijk}}{I}_{\mathit{ijk}}-{cV}_{\mathit{ijk}},\phantom{\rule{0.38889em}{0ex}}i,j,k=1,2,3,4\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}i<j<k,\\ \frac{d}{dt}{V}_{1234}(t)={\mu}^{4}{p}_{0}{I}_{0}+{\mu}^{3}\sum _{i=1}^{4}{p}_{i}{I}_{i}+{\mu}^{2}\sum _{\begin{array}{c}i,j=1,2,3,4\\ i<j\end{array}}{p}_{ij}{I}_{ij}+\mu \sum _{\begin{array}{c}i,j,k=1,2,3,4\\ i<j<k\end{array}}{p}_{\mathit{ijk}}{I}_{\mathit{ijk}}+{p}_{1234}{I}_{1234}-{cV}_{1234}.\end{array}$$

(9)

In the first two equations, the strain index *j* is in the set Ω (*j* Ω), where Ω={0, 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 234, 1234}. Strain 0 represents wild-type virus; strains 1, 2, 3, 4 represent the viral strains with mutations occurring at positions 36, 54, 155, and 156, respectively. Strains *ij*, *i*, *j* = 1, 2, 3, 4 and *i* < *j*, are the strains with double mutations occurring at positions *i* and *j*. Strains *ijk*, *i*, *j*, *k* = 1, 2, 3, 4 and *i* < *j* < *k*, and strain 1234 can be defined similarly (Figure 5). In this model, there is conservation of viruses as they mutate within the self-contained system of 16 strains.

The basic reproductive ratio for each strain is
= *β _{i}p_{i}s*/(

A tedious but straightforward calculation yields the mutant frequency of the pre-existing viral variants before treatment. The viral load of each strain is

$$\begin{array}{l}{V}_{i}=\frac{\mu}{1-\mu -{r}_{i}}{V}_{0},\phantom{\rule{0.38889em}{0ex}}i=1,2,3,4,\\ {V}_{ij}=\frac{1}{{(1-\mu )}^{2}-{r}_{ij}}\left[{\mu}^{2}{V}_{0}+\mu ({r}_{i}{V}_{i}+{r}_{j}{V}_{j})\right],\phantom{\rule{0.38889em}{0ex}}i,j=1,2,3,4\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}i<j,\\ {V}_{\mathit{ijk}}=\frac{1}{{(1-\mu )}^{3}-{r}_{\mathit{ijk}}}\left[{\mu}^{3}{V}_{0}+{\mu}^{2}({r}_{i}{V}_{i}+{r}_{j}{V}_{j}+{r}_{k}{V}_{k})+\mu ({r}_{ij}{V}_{ij}+{r}_{ik}{V}_{ik}+{r}_{jk}{V}_{jk})\right],\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}i,j,k=1,2,3,4\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}i<j<k,\\ {V}_{1234}=\frac{1}{{(1-\mu )}^{4}-{r}_{1234}}\left[{\mu}^{4}{V}_{0}+{\mu}^{3}\sum _{i=1}^{4}{r}_{i}{V}_{i}+{\mu}^{2}\sum _{\begin{array}{c}i,j=1,2,3,4\\ i<j\end{array}}{r}_{ij}{V}_{ij}+\mu \sum _{\begin{array}{c}i,j,k=1,2,3,4\\ i<j<k\end{array}}{r}_{\mathit{ijk}}{V}_{\mathit{ijk}}\right],\end{array}$$

(10)

where *V*_{0} is the steady state of wild-type virus before treatment.

The above defines a recursive scheme, which allows us to orderly calculate the pretreatment steady states of the double-mutant variants (*V _{ij}*), 3-mutant variants (

$$\begin{array}{l}{V}_{ij}=\frac{{\mu}^{2}}{{(1-\mu )}^{2}-{r}_{ij}}\left[1+\frac{{r}_{i}}{1-\mu -{r}_{i}}+\frac{{r}_{j}}{1-\mu -{r}_{j}}\right]{V}_{0},\phantom{\rule{0.38889em}{0ex}}i,j=1,2,3,4\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}i<j,\\ {V}_{\mathit{ijk}}=\frac{{\mu}^{3}}{{(1-\mu )}^{3}-{r}_{\mathit{ijk}}}[1+\sum _{l=i,j,k}\frac{{r}_{l}}{1-\mu -{r}_{l}}+\sum _{\begin{array}{c}l,m=i,j,k\\ l<m\end{array}}\frac{{r}_{lm}}{{(1-\mu )}^{2}-{r}_{lm}}(1+\frac{{r}_{l}}{1-\mu -{r}_{l}}+\frac{{r}_{m}}{1-\mu -{r}_{m}})]{V}_{0},\phantom{\rule{0.38889em}{0ex}}i,j,k=1,2,3,4\phantom{\rule{0.16667em}{0ex}}\text{and}\phantom{\rule{0.16667em}{0ex}}i<j<k,\\ {V}_{1234}=\frac{{\mu}^{4}}{{(1-\mu )}^{4}-{r}_{1234}}[1+\sum _{i=1}^{4}\frac{{r}_{i}}{1-\mu -{r}_{i}}+\sum _{\begin{array}{c}i,j=1,2,3,4\\ i<j\end{array}}\frac{{r}_{ij}}{{(1-\mu )}^{2}-{r}_{ij}}(1+\frac{{r}_{i}}{1-\mu -{r}_{i}}+\frac{{r}_{j}}{1-\mu -{r}_{j}})\\ +\sum _{\begin{array}{c}i,j,k=1,2,3,4\\ i<j<k\end{array}}\frac{{r}_{\mathit{ijk}}}{{(1-\mu )}^{3}-{r}_{\mathit{ijk}}}(1+\sum _{l=i,j,k}\frac{{r}_{l}}{1-\mu -{r}_{l}}\\ +\sum _{\begin{array}{c}l,m=i,j,k\\ l<m\end{array}}\frac{{r}_{lm}}{{(1-\mu )}^{2}-{r}_{lm}}(1+\frac{{r}_{l}}{1-\mu -{r}_{1}}+\frac{{r}_{m}}{1-\mu -{r}_{m}}))]{V}_{0},\end{array}$$

(11)

It follows that the pretreatment steady state level of an *m*-mutant viral variant is of the order of *μ ^{m}V*

The frequency of the pre-existing viral strain *i* (*i* Ω) before treatment is then Φ* _{i}* =

The above formulation of the multi-strain model and the calculation of the pre-existing mutant frequency can be extended to include *n* possible mutations without difficulty. For example, suppose that mutations occur mainly at *n* positions and an *m*-mutant variant, *V _{m}*

$${V}_{m-\mathit{mutant}}=\frac{1}{{(1-\mu )}^{m}-{r}_{m-\mathit{mutant}}}\left[{\mu}^{m}{V}_{0}+{\mu}^{m-1}\sum _{i=1}^{m}{r}_{i}{V}_{i}+{\mu}^{m-2}\sum _{\begin{array}{c}i,j=1,\dots ,m\\ i<j\end{array}}{r}_{ij}{V}_{ij}+{\mu}^{m-3}\sum _{\begin{array}{c}i,j,k=1,\dots ,m\\ i<j<k\end{array}}{r}_{\mathit{ijk}}{V}_{\mathit{ijk}}+\dots \right].$$

In the above expression, the steady states of the other viral variants with fewer mutations can be obtained recursively as in the scheme of equation (10). In this way, we can calculate the frequency of all pre-existing variants in a general model with *n* possible mutations.

The multi-strain model (Eq. 9) includes all possible viral strains that bear mutations at the four positions. However, only a few mutant strains were detected during treatment with telaprevir [23–25]. Specifically, of all the strains with two or more mutations, only strain 13 (36/155) and strain 14 (36/156) were frequently observed. Considering only these observed strains, the strain index set Ω becomes Θ = {0, 1, 2, 3, 4, 13, 14}. Setting the appropriate *r _{i}* to 0 in Eqs. (10) and (11), the pretreatment steady states of the mutant strains are

$$\begin{array}{l}{V}_{i}=\frac{\mu}{1-\mu -{r}_{i}}{V}_{0},\phantom{\rule{0.38889em}{0ex}}i=1,2,3,4,\\ {V}_{13}=\frac{{\mu}^{2}}{{(1-\mu )}^{2}-{r}_{13}}(1+\frac{{r}_{1}}{1-\mu -{r}_{1}}+\frac{{r}_{3}}{1-\mu -{r}_{3}}){V}_{0},\\ {V}_{14}=\frac{{\mu}^{2}}{{(1-\mu )}^{2}-{r}_{14}}(1+\frac{{r}_{1}}{1-\mu -{r}_{1}}+\frac{{r}_{4}}{1-\mu -{r}_{4}}){V}_{0},\end{array}$$

(12)

where *V*_{0} is the pretreatment steady state of the wild-type virus and *r _{i}* =
/
,

Using the in vitro estimates of the relative fitness (assuming the fitness of wild-type virus is 1) of each mutant strain in [23], we obtain the mutant frequency of the pre-existing viral variants before therapy (Table 2). For the single-mutant variant, the frequency is determined by its relative fitness —the larger the relative fitness, the higher the frequency. For the strain with two mutations, the frequency also relies on the relative fitness of those single-mutant strains that can mutate to the double-mutant strain. Although various mutant variants may exist before drug treatment [36], they only account for a very small fraction of the entire virus population. New technologies such as pyrosequencing [46] may allow one to determine the frequency of rare mutants, but to our knowledge this has not been done for the patients in the study we have analyzed [25].

As shown in previous sections, mutation during treatment has a minor effect on the dynamics of viral variants. Mutation is capable of generating low levels of viral variants that would otherwise be completely suppressed due to their fitness disadvantage, but mutation itself cannot determine which strain(s) will dominate the virus population during therapy. Here we neglect all the mutations generated during therapy. Then the multi-strain model under treatment becomes

$$\begin{array}{l}\frac{d}{dt}T(t)=s-dT-\sum _{i}{\beta}_{i}{V}_{i}T,\\ \frac{d}{dt}{I}_{i}(t)={\beta}_{i}{V}_{i}T-\delta {I}_{i},\\ \frac{d}{dt}{V}_{i}(t)=(1-{\epsilon}_{i}){p}_{i}{I}_{i}-{cV}_{i},\end{array}$$

(13)

where *i* Θ = {0, 1, 2, 3, 4, 13, 14}.

The reproductive ratio for each strain under therapy is:
${\mathcal{R}}_{i}^{\prime}=[(1-{\epsilon}_{i}){\beta}_{i}{p}_{i}s]/(dc\delta )$, *i* Θ. The above model represents a multi-strain competition system. The competitive exclusion principle also applies here. Any two viral strains cannot coexist ultimately unless they have the same reproductive ratio. Furthermore, only the viral strain with the largest reproductive ratio will persist during therapy. All the other strains with lower reproductive ratios will die out. Mathematically, if
${\mathcal{R}}_{i}^{\prime}>max\phantom{\rule{0.16667em}{0ex}}(1,{\mathcal{R}}_{j}^{\prime})$, for any *j* Θ and *j* ≠ *i*, then the solution of the system will converge to the steady state *E _{i}*, in which only strain

Much of the recent HCV drug discovery effort has been focused on generating new therapies for HCV genotype 1 infection because of its prevalence and relatively poor response to therapy with PEG-IFN and RBV. Direct-acting antiviral agents for HCV have been suggested to be an attractive strategy whose objective is to achieve a greater response rate, with shorter treatment duration and better tolerability. However, the development of drug resistance has been a major limitation for such treatment options. The high HCV replication rate and the error-prone nature of viral RNA polymerases generate a large number of mutant viral variants, termed a quasispecies, from which variants resistant to specific drugs can be selected during treatment. Since these drug-resistant viral variants have a fitness advantage against wild-type virus in the presence of drug pressure, they are able to evolve quickly and dominate the virus population.

The HCV NS3-4A serine protease is not only involved in viral polyprotein processing but also contributes to HCV persistence by helping HCV escape the IFN antiviral response through its ability to block retinoic acid-inducible gene I and toll-like receptor-3 signaling [47, 48]. Therefore, the NS3-4A protease has become an ideal target for the development of new anti-HCV agents. Telaprevir, a new protease inhibitor, has demonstrated substantial antiviral activity in clinical studies [13, 23–25]. Administration of telaprevir even in monotherapy resulted in ~ 4-log reduction of the plasma viral load in HCV genotype 1 infected patients after 14 days [13]. However, drug-resistant viral variants were detected at high frequencies within a few days during the dosing period. The exact mechanisms underlying emergence of viral variants such a short time after initiation of therapy is not fully characterized.

This paper studies the prevalence of the pre-existing HCV variants and the evolution of drug resistance in patients treated with a direct-acting antiviral agent such as telaprevir. We began with a simple model including two viral strains: wild-type and drug-resistant. The host cell infected with wild-type virus can produce both wild-type virus and a small fraction of drug-resistant virus due to mutations. The two strains coexist before treatment, although drug-resistant virus only accounts for a very small proportion of the virus population. The pre-existing mutant frequency, defined as the ratio of the number of the mutant virus to the total virus before treatment, is Φ = *μ*/(1 − *r*), which is dependent only on *r*, the relative fitness between drug-resistant and wild-type virus, and *μ*, the mutation rate. Using a simplified two-strain model, we obtained an analytical solution of the mutant frequency following treatment with telaprevir. We showed that the rapid increase in the mutant frequency during therapy may not reflect the rapid replication of the pre-existing viral variants, but rather could be a consequence of the rapid and profound decline of wild-type virus, which uncovers the pre-existing mutant virus.

We studied the effects of mutation and hepatocyte proliferation on the pre-treatment mutant frequency and the evolution of drug resistance during therapy. Using the two-strain model, we showed that backward mutation has a negligible effect on the pretreatment mutant frequency. Because the protease inhibitor is highly effective against wild-type virus, both forward and backward mutations do not have a noticeable impact on the dynamics of drug-resistant virus. However, when drug-resistant virus dominates the virus population, backward mutation is able to maintain the wild-type virus at a very low level. Therefore, mutations during therapy do not contribute much to the dynamics of HCV variants. They cannot determine which strain will dominate the virus population. The dynamics of each viral strain are primarily determined by its relative fitness. Specifically, they are determined by the tradeoff between the reduced susceptibility to the protease inhibitor and resistance-associated fitness loss of the mutant virus. When hepatocyte proliferation is included in the models, the analysis becomes more complicated. In a specific case, we showed that the mutant frequency before treatment was not altered. During treatment, wild-type virus is usually suppressed by the effective agent. Whether drug-resistant virus will also be suppressed depends on the proliferation potential of cells infected with resistant virus, relative fitness of the mutant, as well as the drug efficacy. Even if the infection persists, the infected steady state can be unstable and periodic solutions may exist for an open set of parameter values [49].

We also developed a general multi-strain viral dynamic model that considers mutations among various viral strains. We derived the frequency of the pre-existing mutant variants before therapy. Without backward mutation, the frequency depends on the relative fitness of all strains that have fewer mutations. Even though including all the mutations can generate low levels of viral variants that would otherwise be significantly suppressed because of fitness disadvantage in the presence of drug pressure, the quasispecies dynamics are principally determined by the relative fitness of each strain. Mathematically, as De Leenheer and Pilyugin showed in [50], the steady state corresponding to the fittest strain in the model without mutations is globally asymptotically stable. With small perturbations due to mutations, global stability of this steady state is still preserved [50]. A few issues need also to be kept in mind when one discusses the dynamics of various viral variants during therapy. First, although only a few of all possible variants are frequently detected in clinical studies, failure to observe the other viral strains during treatment does not imply that they are not present. Indeed, even if certain viral strains are predicted to die out from the above analysis, in reality they may be present because they can be generated by mutation. However, such strains should remain at very low levels and may exist below the detection limit of assays. Second, when we say a viral strain will die out or survive, we are referring to its steady state level (a long-term behavior). During the short dosing period of telaprevir in clinical trials, even the viral variant with the lowest fitness might be observed.

The emergence of HCV variants at high frequencies is faster than that seen in HIV. This difference can be explained by several factors related to our calculations. (1) Preexistence of drug-resistant variants. The fidelity of reverse transcriptase for HIV may be higher than that of RdRp for HCV [51, 52]. Thus, the intrinsic HCV mutation rate may be higher than that of HIV [53]. The frequency of pre-existing HCV mutants (which is proportional to *μ ^{m}* for

The prevalence of the pre-existing HCV variants has been observed in experiments. McPhee et al. [55] examined the baseline prevalence of HCV variants resistant to protease inhibitors using a highly sensitive assay (limit of detection < 0.1% of the total population). In three of eight patients, they detected the A156T variant at a frequency of 0.36%-0.75%. Cubero et al. [56] reported a similar mutant frequency (0.78%) of the A156T mutant in a chronic hepatitis C patient never treated with NS3-protease inhibitors. This frequency is higher than what we obtained in Table 2. The discrepancy can be explained either by a larger in vivo fitness than that in Table 2 estimated from in vitro experiments or by compensatory mutations (not considered in our models, see [57] that studies how compensatory mutations affect the emergence of drug resistance), which allow partial fitness recovery of the mutant variants. It has been reported that three second-site mutations, P89L, Q86R, and G162R, were able to partially reverse A156T-associated defects in polyprotein processing and/or replicon fitness without significantly reducing resistance to the protease inhibitor SCH6 [58]. In the study of Cubero et al. [56], they also detected changes at positions 89 and 86 (P89Q and Q86P) along with the A156T mutant. The presence of these mutations might compensate for the A156T-associated fitness loss and result in a higher frequency in untreated patients. The contribution of compensatory mutations to the preexistence of HCV variants and the evolution of drug resistance during treatment requires more in vitro and in vivo studies.

Our calculations may have implications for developing treatment strategies for HCV infection. From expression (11), the steady state viral level of an *m*-mutant strain is of the order of *μ ^{m}*, which implies that a mutant variant will have a very low frequency if it carries more mutations conferring resistance to multiple drugs. In fact, in clinical trials, HCV viral variants with three or more drug-resistant mutations have seldom been identified so far. This raises the chance of success of a strategy that combines several specific HCV inhibitors targeting different steps of the HCV life cycle. The combination treatment strategy is, in theory, the same as for hepatitis B virus (HBV) [59] and HIV treatment [60]. This idea has been recently confirmed in

By calculating the fraction of all possible mutants produced per day, we estimated the number of mutations a combination of direct antivirals would need to overcome to be successful [26]. More clinical data on toxicity and drug-drug-interactions are needed in order to design combination therapies. In addition, *in vitro* data indicate that telaprevir and IFN act synergistically to inhibit HCV RNA replication and facilitate viral RNA clearance in replicon cells [67]. In clinical studies [25, 68], telaprevir was combined with PEG-IFN-alpha-2a and caused a continued antiviral response during the dosing period. More recent clinical trials [14, 15] showed that treatment with a telaprevir-based regimen significantly improved the SVR rate in patients with genotype 1 HCV. Even in patients with viral breakthrough following telaprevir alone, follow-up treatment with PEG-IFN-alpha-2a and RBV could inhibit growth of both wild-type and resistant variants [25]. These results suggest that HCV variants with reduced sensitivity to telaprevir may remain sensitive to IFN plus RBV. Based on this, both telaprevir and boceprevir were only approved by the FDA for use in combination with PEG-IFN and RBV. If we assume that IFN lowers the viral production rate by a factor (1 − *ε _{IF N}*), then the model including combination therapy of IFN and a protease inhibitor is the same as the model with the protease inhibitor monotherapy except that

Portions of this work were done under the auspices of the US Department of Energy under contract DE-AC52-06NA25396, and supported by NIH grants P30-EB011339, P20-RR018754, AI028433, OD011095, and NSF grant DMS-1122290. We also thank the reviewers for their comments that improved the manuscript.

From Δ_{1} = (*c* + *δ*)^{2} − 4*ε _{s}cδ*, it is clear that (

- Notice that ${C}_{1}=-\frac{c(1-2{\epsilon}_{s})+\delta -\sqrt{{\mathrm{\Delta}}_{1}}}{2\sqrt{{\mathrm{\Delta}}_{1}}}{V}_{s}(0)$ andΔ
_{1}= (*c*+*δ*)^{2}− 4*ε*_{s}cδ. C_{1}> 0 is equivalent to $-c(1-2{\epsilon}_{s})-\delta +\sqrt{{\mathrm{\Delta}}_{1}}>0$. Thus, for*C*_{1}> 0 it suffices to prove that $\sqrt{{\mathrm{\Delta}}_{1}}>c(1-2{\epsilon}_{s})+\delta $. If the right hand side is less than 0, then the inequality automatically holds. If the right hand side is greater than 0, then we only need to show that Δ_{1}> (*c*+*δ*−2*cε*)_{s}^{2}, which is equivalent to*ε*< 1. Hence, Δ_{s}_{1}> (*c*+*δ*− 2*cε*)_{s}^{2}and*C*_{1}> 0. - Because $\sqrt{{\mathrm{\Delta}}_{1}}>c-\delta $, we have $c(1-2{\epsilon}_{s})+\delta +\sqrt{{\mathrm{\Delta}}_{1}}>c(1-2{\epsilon}_{s})+\delta +c-\delta =2c(1-{\epsilon}_{s})>0$. Thus, ${C}_{2}=\frac{c(1-2{\epsilon}_{s})+\delta +\sqrt{{\mathrm{\Delta}}_{1}}}{2\sqrt{{\mathrm{\Delta}}_{1}}}{V}_{s}(0)>0$.
- For simplicity, we introduce a new parameter
*θ*, defined as $\theta =1-\frac{1-{\epsilon}_{r}}{1-\mu}\frac{{\mathcal{R}}_{r}}{{\mathcal{R}}_{s}}$. Thus,*θ*< 1. Then*C*_{3},*C*_{4}, and Δ_{2}can be simplified to ${C}_{3}=-\frac{c(1-2\theta )+\delta -\sqrt{{\mathrm{\Delta}}_{2}}}{2\sqrt{{\mathrm{\Delta}}_{2}}}{V}_{r}(0),{C}_{4}=\frac{c(1-2\theta )+\delta +\sqrt{{\mathrm{\Delta}}_{2}}}{2\sqrt{{\mathrm{\Delta}}_{2}}}{V}_{r}(0)$, and Δ_{2}= (*c*+*δ*)^{2}− 4*θcδ*, which have similar forms to*C*_{1},*C*_{2}, and Δ_{1}, respectively. Following the same arguments as in (i) and (ii), we can prove that*C*_{3}> 0 and*C*_{4}> 0.

Since *t _{s}* is the time at which two curves

$$\frac{{C}_{1}}{{C}_{2}}-\frac{{C}_{3}}{{C}_{4}}=\frac{-c(1-2{\epsilon}_{s})-\delta +\sqrt{{\mathrm{\Delta}}_{1}}}{c(1-2{\epsilon}_{s})+\delta +\sqrt{{\mathrm{\Delta}}_{1}}}-\frac{-c(1-2\theta )-\delta +\sqrt{{\mathrm{\Delta}}_{2}}}{c(1-2\theta )+\delta +\sqrt{{\mathrm{\Delta}}_{2}}},$$

(14)

where $\theta =1-\frac{1-{\epsilon}_{r}}{1-\mu}\frac{{\mathcal{R}}_{r}}{{\mathcal{R}}_{s}}$.

Using the common denominator to combine the two fractions in (14), we obtain the numerator

$$\left[-c(1-2{\epsilon}_{s})-\delta +\sqrt{{\mathrm{\Delta}}_{1}}\right]\phantom{\rule{0.16667em}{0ex}}\left[c(1-2\theta )+\delta +\sqrt{{\mathrm{\Delta}}_{2}}\right]-\left[c(1-2{\epsilon}_{s})+\delta +\sqrt{{\mathrm{\Delta}}_{1}}\right]\phantom{\rule{0.16667em}{0ex}}\left[-c(1-2\theta )-\delta +\sqrt{{\mathrm{\Delta}}_{2}}\right],$$

which can be simplified to

$$4c({\epsilon}_{s}\sqrt{{\mathrm{\Delta}}_{2}}-\theta \sqrt{{\mathrm{\Delta}}_{1}})-2(c+\delta )(\sqrt{{\mathrm{\Delta}}_{2}}-\sqrt{{\mathrm{\Delta}}_{1}}).$$

(15)

Because drug-resistant virus is more fit than wild-type virus during treatment, we have (1 − *ε _{s}*)
< (1 −

$$\begin{array}{l}4c({\epsilon}_{s}\sqrt{{\mathrm{\Delta}}_{2}}-\theta \sqrt{{\mathrm{\Delta}}_{1}})-2(c+\delta )(\sqrt{{\mathrm{\Delta}}_{2}}-\sqrt{{\mathrm{\Delta}}_{1}})\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}>\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}\phantom{\rule{0.38889em}{0ex}}4c({\epsilon}_{s}\sqrt{{\mathrm{\Delta}}_{2}}-{\epsilon}_{s}\sqrt{{\mathrm{\Delta}}_{1}})-2(c+\delta )(\sqrt{{\mathrm{\Delta}}_{2}}-\sqrt{{\mathrm{\Delta}}_{1}})\\ =2(\sqrt{{\mathrm{\Delta}}_{2}}-\sqrt{{\mathrm{\Delta}}_{1}})[2c{\epsilon}_{s}-(c+\delta )]>0.\end{array}$$

The last inequality holds because telaprevir is very effective in blocking production of wild-type virus (*ε _{s}* is close to 1) and virus has much faster dynamics than infected hepatocytes (

Furthermore, we can prove that *t _{r}* is an increasing function with respect to

$${f}^{\prime}(\theta )=\frac{-2c\left[{(c+\delta )}^{2}-4\theta c\delta \right]+2c\delta \left[c(1-2\theta )+\delta \right]}{{\left[{(c+\delta )}^{2}-4\theta c\delta \right]}^{{\scriptstyle \frac{3}{2}}}}.$$

The numerator of the above fraction can be simplified to 2*c*^{2}(2*θδ* − *c* − *δ*), which is less than 0 because *θδ* < *c* and *θδ*< *δ*. Thus, as *θ* decreases, *f*(*θ*) increases. Consequently, *C*_{3}/*C*_{4} decreases and
${t}_{r}=ln\phantom{\rule{0.16667em}{0ex}}({C}_{3}/{C}_{4})/\sqrt{{\mathrm{\Delta}}_{2}}$ decreases. This shows that *t _{r}* is an increasing function of

There are two possible steady states of Eq. (4) before treatment: the infection-free and infected (coexistence) steady states. We are interested in the latter one. From the *I _{s}* and

$${\overline{T}}_{1,2}=\frac{(1-\mu )({p}_{s}{\beta}_{s}+{p}_{r}{\beta}_{r})\pm \sqrt{{[(1-\mu )({p}_{s}{\beta}_{s}+{p}_{r}{\beta}_{r})]}^{2}-4(1-2\mu ){p}_{s}{\beta}_{s}{p}_{r}{\beta}_{r}}}{2(1-2\mu ){p}_{s}{\beta}_{s}{p}_{r}{\beta}_{r}}c\delta .$$

(16)

Ignoring *μ*, we have two approximate solutions:
${\overline{T}}_{1}\approx \frac{c\delta}{{p}_{r}{\beta}_{r}}$ (choosing “+” in (16)) and
${\overline{T}}_{2}\approx \frac{c\delta}{{p}_{s}{\beta}_{s}}$ (choosing “−” in (16)). Because of the conditions for the existence of the coexistence steady state and the assumption that *β _{r}* <

$${\overline{T}}_{2}=\frac{(1-\mu )({p}_{s}{\beta}_{s}+{p}_{r}{\beta}_{r})-\sqrt{{[(1-\mu )({p}_{s}{\beta}_{s}+{p}_{r}{\beta}_{r})]}^{2}-4(1-2\mu ){p}_{s}{\beta}_{s}{p}_{r}{\beta}_{r}}}{2(1-2\mu ){p}_{s}{\beta}_{s}{p}_{r}{\beta}_{r}}c\delta .$$

Using _{2}, Φ can be further simplified to

$$\mathrm{\Phi}=\frac{\mu}{\frac{(1-\mu )(1+r)+\sqrt{{[(1-\mu )(1+r)]}^{2}-4(1-2\mu )r}}{2}-r+\mu (1+r)},$$

(17)

where *r* =
/
. It is clear that Φ depends only on *μ* and *r*.

It follows from (17) that F can be approximated by

$$\mathrm{\Phi}=\frac{\mu}{1-r+\mu (1+r)},$$

(18)

which is less than
$\frac{\mu}{1-r}$, the mutant frequency in the model without considering backward mutation. In fact, it can be proved rigorously that Φ* _{w}* < Φ

From the *V _{s}* and

Considering * _{s}* = (1 −

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