Abstract
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.
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References
Adiwijaya, B. S., Herrmann, E., Hare, B., Kieffer, T., Lin, C., et al. (2010). A multi-variant, viral dynamic model of genotype 1 HCV to assess the in vivo evolution of protease-inhibitor resistant variants. PLoS Comput. Biol., 6, e1000745.
Bacon, B. R., Gordon, S. C., Lawitz, E., Marcellin, P., Vierling, J. M., et al. (2011). Boceprevir for previously treated chronic HCV genotype 1 infection. N. Engl. J. Med., 364, 1207–1217.
Bartels, D. J., Zhou, Y., Zhang, E. Z., Marcial, M., Byrn, R. A., et al. (2008). Natural prevalence of hepatitis C virus variants with decreased sensitivity to NS3.4A protease inhibitors in treatment-naive subjects. J. Infect. Dis., 198, 800–807.
Bartenschlager, R., Frese, M., & Pietschmann, T. (2004). Novel insights into hepatitis C virus replication and persistence. Adv. Virus Res., 63, 71–180.
Chayama, K., Takahashi, S., Toyota, J., Karino, Y., Ikeda, K., et al. (2012). Dual therapy with the nonstructural protein 5A inhibitor, daclatasvir, and the nonstructural protein 3 protease inhibitor, asunaprevir, in hepatitis C virus genotype 1b-infected null responders. Hepatology, 55, 742–748.
Chu, H., Herrmann, E., Reesink, H., Forestier, N., Weegink, C., et al. (2005). Pharmacokinetics of VX-950, and its effect on hepatitis C viral dynamics. Hepatology, 42 Suppl 1, S694.
Cubero, M., Esteban, J. I., Otero, T., Sauleda, S., Bes, M., et al. (2008). Naturally occurring NS3-protease-inhibitor resistant mutant A156T in the liver of an untreated chronic hepatitis C patient. Virology, 370, 237–245.
Dahari, H., Ribeiro, R. M., & Perelson, A. S. (2007). Triphasic decline of hepatitis C virus RNA during antiviral therapy. Hepatology, 46, 16–21.
Davis, G. L., Esteban-Mur, R., Rustgi, V., Hoefs, J., Gordon, S. C., et al. (1998). Interferon alfa-2b alone or in combination with ribavirin for the treatment of relapse of chronic hepatitis C. International Hepatitis Interventional Therapy Group. N. Engl. J. Med., 339, 1493–1499.
De Leenheer, P., & Pilyugin, S. S. (2008). Multistrain virus dynamics with mutations: a global analysis. Math. Med. Biol., 25, 285–322.
Domingo, E. (1996). Biological significance of viral quasispecies. Viral Hepat. Rev., 2, 247–261.
Dore, G. J., Matthews, G. V., & Rockstroh, J. (2011). Future of hepatitis C therapy: development of direct-acting antivirals. Curr. Opin. HIV AIDS, 6, 508–513.
Duffy, S., Shackelton, L. A., & Holmes, E. C. (2008). Rates of evolutionary change in viruses: patterns and determinants. Nat. Rev. Genet., 9, 267–276.
Egger, D., Wolk, B., Gosert, R., Bianchi, L., Blum, H. E., et al. (2002). Expression of hepatitis C virus proteins induces distinct membrane alterations including a candidate viral replication complex. J. Virol., 76, 5974–5984.
Elahi, E., & Ronaghi, M. (2004). Pyrosequencing: A tool for DNA sequencing analysis. Methods Mol. Biol., 255, 211–219.
Evans, M. J., von Hahn, T., Tscherne, D. M., Syder, A. J., Panis, M., et al. (2007). Claudin-1 is a hepatitis C virus co-receptor required for a late step in entry. Nature, 446, 801–805.
Forestier, N., Reesink, H. W., Weegink, C. J., McNair, L., Kieffer, T. L., et al. (2007). Antiviral activity of telaprevir (VX-950) and peginterferon alfa-2a in patients with hepatitis C. Hepatology, 46, 640–648.
Foster, G. R. (2004). Past, present, and future hepatitis C treatments. Semin. Liver Dis., 24 Suppl 2, 97–104.
Foy, E., Li, K., Wang, C., Sumpter, J. R., Ikeda, M., et al. (2003). Regulation of interferon regulatory factor-3 by the hepatitis C virus serine protease. Science, 300, 1145–1148.
Fusco, D. N., & Chung, R. T. (2012). Novel therapies for hepatitis C: insights from the structure of the virus. Annu. Rev. Med., 63, 373–387.
Gillespie, J. (1998). Population genetics: a concise guide. Baltimore: Johns Hopkins University Press.
Handel, A., Regoes, R. R., & Antia, R. (2006). The role of compensatory mutations in the emergence of drug resistance. PLoS Comput. Biol., 2, e137.
Hezode, C., Forestier, N., Dusheiko, G., Ferenci, P., Pol, S., et al. (2009). Telaprevir and peginterferon with or without ribavirin for chronic HCV infection. N. Engl. J. Med., 360, 1839–1850.
Jazwinski, A. B. & Muir, A. J. (2011). Direct-acting antiviral medications for chronic hepatitis C virus infection. Gastroenterol. Hepatol. 7, 154–162.
Keeffe, E. B., Dieterich, D. T., Pawlotsky, J. M., & Benhamou, Y. (2008). Chronic hepatitis B: preventing, detecting, and managing viral resistance. Clín. Gastroenterol. Hepatol., 6, 268–274.
Khunvichai, A., Chu, H., Garg, V., McHutchison, J., Lawitz, E., et al. (2007). Predicting HCV treatment duration with an HCV protease inhibitor co-administered with PEG-IFN/RBV by modeling both wild-type virus and low level resistant variant dynamics. In Digestive disease week 2007. Washington, D.C., May 19–24, 2007.
Kieffer, T. L., Sarrazin, C., Miller, J. S., Welker, M. W., Forestier, N., et al. (2007). Telaprevir and pegylated interferon-alpha-2a inhibit wild-type and resistant genotype 1 hepatitis C virus replication in patients. Hepatology, 46, 631–639.
Lamarre, D., Anderson, P. C., Bailey, M., Beaulieu, P., Bolger, G., et al. (2003). An NS3 protease inhibitor with antiviral effects in humans infected with hepatitis C virus. Nature, 426, 186–189.
Lawitz, E., Rodriguez-Torres, M., Muir, A. J., Kieffer, T. L., McNair, L., et al. (2008). Antiviral effects and safety of telaprevir, peginterferon alfa-2a, and ribavirin for 28 days in hepatitis C patients. J. Hepatol., 49, 163–169.
Lin, K., Kwong, A. D., & Lin, C. (2004). Combination of a hepatitis C virus NS3-NS4A protease inhibitor and alpha interferon synergistically inhibits viral RNA replication and facilitates viral RNA clearance in replicon cells. Antimicrob. Agents Chemother., 48, 4784–4792.
Lin, C., Gates, C. A., Rao, B. G., Brennan, D. L., Fulghum, J. R., et al. (2005). In vitro studies of cross-resistance mutations against two hepatitis C virus serine protease inhibitors, VX-950 and BILN 2061. J. Biol. Chem., 280, 36784–36791.
Lin, K., Perni, R. B., Kwong, A. D., & Lin, C. (2006). VX-950, a novel hepatitis C virus (HCV) NS3-4A protease inhibitor, exhibits potent antiviral activities in HCV replicon cells. Antimicrob. Agents Chemother., 50, 1813–1822.
Lok, A. S., Gardiner, D. F., Lawitz, E., Martorell, C., Everson, G. T., et al. (2012). Preliminary study of two antiviral agents for hepatitis C genotype 1. N. Engl. J. Med., 366, 216–224.
Lu, L., Pilot-Matias, T. J., Stewart, K. D., Randolph, J. T., Pithawalla, R., et al. (2004). Mutations conferring resistance to a potent hepatitis C virus serine protease inhibitor in vitro. Antimicrob. Agents Chemother., 48, 2260–2266.
McCown, M. F., Rajyaguru, S., Le Pogam, S., Ali, S., Jiang, W. R., et al. (2008). The hepatitis C virus replicon presents a higher barrier to resistance to nucleoside analogs than to nonnucleoside polymerase or protease inhibitors. Antimicrob. Agents Chemother., 52, 1604–1612.
McHutchison, J. G., Gordon, S. C., Schiff, E. R., Shiffman, M. L., Lee, W. M., et al. (1998). Interferon alfa-2b alone or in combination with ribavirin as initial treatment for chronic hepatitis C. Hepatitis Interventional Therapy Group. N. Engl. J. Med., 339, 1485–1492.
McHutchison, J. G., Everson, G. T., Gordon, S. C., Jacobson, I. M., Sulkowski, M., et al. (2009). Telaprevir with peginterferon and ribavirin for chronic HCV genotype 1 infection. N. Engl. J. Med., 360, 1827–1838.
McPhee, F., Hernandez, D., Zhai, G., Friborg, J., Yu, F., et al. (2006). Pre-existence of substitutions conferring resistance to HCV NS3 protease inhibitors. In 1st international workshop on hepatitis C resistance and new compounds. Boston, MA, October 25–26, 2006.
Meylan, E., Curran, J., Hofmann, K., Moradpour, D., Binder, M., et al. (2005). Cardif is an adaptor protein in the RIG-I antiviral pathway and is targeted by hepatitis C virus. Nature, 437, 1167–1172.
Michalopoulos, G. K., & DeFrances, M. C. (1997). Liver regeneration. Science, 276, 60–66.
Neumann, A. U., Lam, N. P., Dahari, H., Gretch, D. R., Wiley, T. E., et al. (1998). Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy. Science, 282, 103–107.
Olsen, D. B., Davies, M. E., Handt, L., Koeplinger, K., Zhang, N. R., et al. (2011). Sustained viral response in a hepatitis C virus-infected chimpanzee via a combination of direct-acting antiviral agents. Antimicrob. Agents Chemother., 55, 937–939.
Pawlotsky, J. M. (2006). Hepatitis C virus population dynamics during infection. Curr. Top. Microbiol. Immunol., 299, 261–284.
Perni, R. B., Almquist, S. J., Byrn, R. A., Chandorkar, G., Chaturvedi, P. R., et al. (2006). Preclinical profile of VX-950, a potent, selective, and orally bioavailable inhibitor of hepatitis C virus NS3-4A serine protease. Antimicrob. Agents Chemother., 50, 899–909.
Pileri, P., Uematsu, Y., Campagnoli, S., Galli, G., Falugi, F., et al. (1998). Binding of hepatitis C virus to CD81. Science, 282, 938–941.
Ploss, A., Evans, M. J., Gaysinskaya, VA, Panis, M., You, H., et al. (2009). Human occludin is a hepatitis C virus entry factor required for infection of mouse cells. Nature, 457, 882–886.
Poordad, F., McCone, J. J., Bacon, B. R., Bruno, S., Manns, M. P., et al. (2011). Boceprevir for untreated chronic HCV genotype 1 infection. N. Engl. J. Med., 364, 1195–1206.
Powdrill, M. H., Tchesnokov, E. P., Kozak, R. A., Russell, R. S., Martin, R., et al. (2011). Contribution of a mutational bias in hepatitis C virus replication to the genetic barrier in the development of drug resistance. Proc. Natl. Acad. Sci. USA, 108, 20509–20513.
Powers, K. A., Dixit, N. M., Ribeiro, R. M., Golia, P., Talal, A. H., et al. (2003). Modeling viral and drug kinetics: Hepatitis C virus treatment with pegylated interferon alfa-2b. Semin. Liver Dis., 23 Suppl 1, 13–18.
Reesink, H. W., Zeuzem, S., Weegink, C. J., Forestier, N., van Vliet, A., et al. (2006). Rapid decline of viral RNA in hepatitis C patients treated with VX-950: a phase Ib, placebo-controlled, randomized study. Gastroenterology, 131, 997–1002.
Reluga, T. C., Dahari, H., & Perelson, A. S. (2009). Analysis of hepatitis C virus infection models with hepatocyte homeostasis. SIAM J. Appl. Math., 69, 999–1023.
Rong, L., Feng, Z., & Perelson, A. S. (2007). Emergence of HIV-1 drug resistance during antiretroviral treatment. Bull. Math. Biol., 69, 2027–2060.
Rong, L., Dahari, H., Ribeiro, R. M., & Perelson, A. S. (2010). Rapid emergence of protease inhibitor resistance in hepatitis C virus. Sci. Transl. Med., 2, 30ra32.
Sanjuan, R., Nebot, M. R., Chirico, N., Mansky, L. M., & Belshaw, R. (2010). Viral mutation rates. J. Virol., 84, 9733–9748.
Sarrazin, C., Kieffer, T. L., Bartels, D., Hanzelka, B., Muh, U., et al. (2007). Dynamic hepatitis C virus genotypic and phenotypic changes in patients treated with the protease inhibitor telaprevir. Gastroenterology, 132, 1767–1777.
Sarrazin, C., Rouzier, R., Wagner, F., Forestier, N., Larrey, D., et al. (2007). SCH 503034, a novel hepatitis C virus protease inhibitor, plus pegylated interferon alpha-2b for genotype 1 nonresponders. Gastroenterology, 132, 1270–1278.
Scarselli, E., Ansuini, H., Cerino, R., Roccasecca, R. M., Acali, S., et al. (2002). The human scavenger receptor class B type I is a novel candidate receptor for the hepatitis C virus. EMBO J., 21, 5017–5025.
Shen, L., Peterson, S., Sedaghat, A. R., McMahon, M. A., Callender, M., et al. (2008). Dose-response curve slope sets class-specific limits on inhibitory potential of anti-HIV drugs. Nat. Med., 14, 762–766.
Sherman, K. E., Flamm, S. L., Afdhal, N. H., Nelson, D. R., Sulkowski, M. S., et al. (2011). Response-guided telaprevir combination treatment for hepatitis C virus infection. N. Engl. J. Med., 365, 1014–1024.
Soriano, V., Perelson, A. S., & Zoulim, F. (2008). Why are there different dynamics in the selection of drug resistance in HIV and hepatitis B and C viruses? J. Antimicrob. Chemother., 62, 1–4.
Tong, X., Chase, R., Skelton, A., Chen, T., Wright-Minogue, J., et al. (2006). Identification and analysis of fitness of resistance mutations against the HCV protease inhibitor SCH 503034. Antivir. Res., 70, 28–38.
Vanwolleghem, T., Meuleman, P., Libbrecht, L., Roskams, T., De Vos, R., et al. (2007). Ultra-rapid cardiotoxicity of the hepatitis C virus protease inhibitor BILN 2061 in the urokinase-type plasminogen activator mouse. Gastroenterology, 133, 1144–1155.
Wang, L., & Li, M. Y. (2006). Mathematical analysis of the global dynamics of a model for HIV infection of CD4+T cells. Math. Biosci., 200, 44–57.
World Health Organization (2011) Hepatitis C. Fact sheet No. 164. Revised June 2011. http://www.who.int/mediacentre/factsheets/fs164/en/index.html.
Wyles, D. L., Kaihara, K. A., Vaida, F., & Schooley, R. T. (2007). Synergy of small molecular inhibitors of hepatitis C virus replication directed at multiple viral targets. J. Virol., 81, 3005–3008.
Wyles, D. L., Kaihara, K. A., & Schooley, R. T. (2008). Synergy of a hepatitis C virus (HCV) NS4A antagonist in combination with HCV protease and polymerase inhibitors. Antimicrob. Agents Chemother., 52, 1862–1864.
Yeni, P. G., Hammer, S. M., Carpenter, C. C., Cooper, D. A., Fischl, M. A., et al. (2002). Antiretroviral treatment for adult HIV infection in 2002: updated recommendations of the international AIDS society-USA panel. JAMA, 288, 222–235.
Yi, M., Tong, X., Skelton, A., Chase, R., Chen, T., et al. (2006). Mutations conferring resistance to SCH6, a novel hepatitis C virus NS3/4A protease inhibitor. Reduced RNA replication fitness and partial rescue by second-site mutations. J. Biol. Chem., 281, 8205–8215.
Zhou, Y., Bartels, D. J., Hanzelka, B. L., Muh, U., Wei, Y., et al. (2008). Phenotypic characterization of resistant Val36 variants of hepatitis C virus NS3-4A serine protease. Antimicrob. Agents Chemother., 52, 110–120.
Acknowledgements
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.
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Appendices
Appendix A: Several Inequalities Used in the Analysis of the Two-Strain Model
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1.
λ 3<λ 1<λ 2<λ 4<0.
From Δ 1=(c+δ)2−4ϵ s cδ it is clear that (c+δ)2>Δ 1>(c−δ)2. Thus, we have \(\lambda_{1}=-\frac{c+\delta+\sqrt{\varDelta _{1}}}{2}<0\) and \(\lambda_{2}=-\frac{c+\delta-\sqrt{\varDelta _{1}}}{2}<0\). Next, we show that Δ 2>Δ 1. Calculating the difference, we obtain \(\varDelta _{2}-\varDelta _{1}=4c\delta(1-\epsilon_{s}) [\frac{1}{(1-\mu)} \frac{\mathcal{R}_{r}'}{\mathcal{R}_{s}'}-1 ]\), where \(\mathcal{R}_{r}'=(1-\epsilon_{r})\mathcal{R}_{r}\) and \(\mathcal {R}_{s}'=(1-\epsilon_{s})\mathcal{R}_{s}\) are the reproductive ratios of the resistant and wild-type strains during therapy, respectively. Since we assume that drug-resistant virus is more fit than wild-type virus during therapy, we have that \(\mathcal{R}_{r}'>\mathcal{R}_{s}'\). Thus, Δ 2>Δ 1. Lastly, we show that (c+δ)2>Δ 2. It suffices to show that \(1-\frac{(1-\epsilon_{r})\mathcal{R}_{r}}{(1-\mu )\mathcal{R}_{s}}>0\), i.e., \(\frac{\mathcal{R}_{r}}{\mathcal{R}_{s}}<\frac{1-\mu }{1-\epsilon_{r}}\), which holds because wild-type virus is more fit than drug-resistant virus before treatment (\(\mathcal{R}_{r}<\mathcal{R}_{s}\)) and μ is very small (typically μ≪ϵ r ). Therefore, (c+δ)2>Δ 2>Δ 1>(c−δ)2. It follows that \(\lambda_{3}=-\frac{c+\delta+\sqrt{\varDelta _{2}}}{2}<0\) and \(\lambda_{4}=-\frac{c+\delta-\sqrt{\varDelta _{2}}}{2}<0\). Furthermore, from \(\sqrt{\varDelta _{2}}>\sqrt{\varDelta _{1}}\) we have that λ 3<λ 1<λ 2<λ 4<0.
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2.
C i >0, i=1,2,3,4.
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(i)
Notice that \(C_{1}=-\frac{c(1-2\epsilon_{s})+\delta-\sqrt{\varDelta _{1}}}{2\sqrt{\varDelta _{1}}}V_{s}(0)\) and Δ 1=(c+δ)2−4ϵ s cδ. C 1>0 is equivalent to \(-c(1-2\epsilon_{s})-\delta+\sqrt{\varDelta _{1}}>0\). Thus, for C 1>0, it suffices to prove that \(\sqrt{\varDelta _{1}}>c(1-2\epsilon_{s})+\delta\). 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−2cϵ s )2, which is equivalent to ϵ s <1. Hence, Δ 1>(c+δ−2cϵ s )2 and C 1>0.
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(ii)
Because \(\sqrt{\varDelta _{1}}>c-\delta\), we have \(c(1-2\epsilon_{s})+\delta+\sqrt{\varDelta _{1}}>c(1-2\epsilon_{s})+\delta +c-\delta=2c(1-\epsilon_{s})>0\). Thus, \(C_{2}=\frac{c(1-2\epsilon_{s})+\delta+\sqrt{\varDelta _{1}}}{2\sqrt{\varDelta _{1}}}V_{s}(0)>0\).
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(iii)
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 \(C_{3}=-\frac{c(1-2\theta)+\delta-\sqrt{\varDelta _{2}}}{2\sqrt{\varDelta _{2}}}\*V_{r}(0)\), \(C_{4}=\frac{c(1-2\theta)+\delta+\sqrt{\varDelta _{2}}}{2\sqrt{\varDelta _{2}}}V_{r}(0)\), and Δ 2=(c+δ)2−4θcδ, 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.
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3.
t r <t s and t r is an increasing function of ϵ r .
Since t s is the time at which two curves \(C_{1} e^{\lambda_{1}t}\) and \(C_{2} e^{\lambda_{2}t}\) intersect, we obtain that \(t_{s}=\frac{\ln{\frac{C_{1}}{C_{2}}}}{-\lambda_{1}+\lambda_{2}}= \frac{\ln{\frac{C_{1}}{C_{2}}}}{\sqrt{\varDelta _{1}}}\). Similarly, we have \(t_{r}=\frac{\ln{\frac{C_{3}}{C_{4}}}}{\sqrt{\varDelta _{2}}}\). Calculating the difference between \(\frac{C_{1}}{C_{2}}\) and \(\frac{C_{3}}{C_{4}}\), we obtain
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
which can be simplified to
Because drug-resistant virus is more fit than wild-type virus during treatment, we have \((1-\epsilon_{s})\mathcal{R}_{s}<(1-\epsilon_{r})\mathcal{R}_{r}\). Thus, \(\theta=1-\frac{1-\epsilon_{r}}{1-\mu}\frac{\mathcal {R}_{r}}{\mathcal{R}_{s}}<\epsilon_{s}\) since μ is very small. It follows from (15) that
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 (c≫δ). Therefore, \(\frac{C_{1}}{C_{2}}>\frac{C_{3}}{C_{4}}\). Also considering that \(\sqrt{\varDelta _{2}}>\sqrt{\varDelta _{1}}\), we have t s >t r .
Furthermore, we can prove that t r is an increasing function with respect to ϵ r , the efficacy of the protease inhibitor against the drug-resistant strain. As ϵ r decreases (corresponding to a more resistant viral strain), \(\theta=1-\frac{1-\epsilon_{r}}{1-\mu}\frac{\mathcal {R}_{r}}{\mathcal{R}_{s}}\) decreases and Δ 2=(c+δ)2−4θcδ increases. Rearranging \(\frac{C_{3}}{C_{4}}\), we have \(\frac{C_{3}}{C_{4}}=\frac{-c(1-2\theta)-\delta+\sqrt{\varDelta _{2}}}{ c(1-2\theta)+\delta+\sqrt{\varDelta _{2}}}=\frac{2\sqrt{\varDelta _{2}}}{c(1-2\theta)+\delta+\sqrt{\varDelta _{2}}}-1 =\frac{2}{1+f(\theta)}-1\), where \(f(\theta)=\frac{c(1-2\theta)+\delta}{\sqrt{\varDelta _{2}}} =\frac{c(1-2\theta)+\delta}{\sqrt{(c+\delta)^{2}-4\theta c\delta}}\). Taking the derivative of f(θ) with respect to θ, we obtain
The numerator of the above fraction can be simplified to 2c 2(2θδ−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 \(t_{r}=\ln{(C_{3}/C_{4})}/\sqrt{\varDelta _{2}}\) decreases. This shows that t r is an increasing function of ϵ r . Thus, for a mutant strain with high-level drug resistance (a small ϵ r ), t r is small.
Appendix B: The Mutant Frequency in the Model with Backward Mutation
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 V s equations, we obtain \(\mu p_{r} \beta_{r} \bar{T} \bar{V_{r}}=[c\delta-(1-\mu)p_{s}\beta_{s} \bar{T}]\bar{V_{s}}\), where \(\bar{T}\), \(\bar{V_{s}}\), and \(\bar{V_{r}}\) represent the steady states of uninfected target cells, wild-type, and drug-resistant virus, respectively. Similarly, from the I r and V r equations, we obtain \(\mu p_{s} \beta_{s} \bar{T} \bar{V_{s}}=[c\delta-(1-\mu)p_{r}\beta_{r} \bar{T}]\bar{V_{r}}\). Thus, the two strains coexist only when \((1-\mu)p_{s}\beta_{s}\bar{T}<c\delta\) and \((1-\mu)p_{r}\beta_{r}\bar{T}<c\delta\). From the above two equations, we obtain an equation that the steady state of uninfected hepatocytes, \(\bar{T}\), must satisfy \((1-2\mu)p_{s}\beta_{s} p_{r}\beta_{r} \bar{T}^{2}-(1-\mu)c\delta(p_{s}\beta_{s}+p_{r}\beta_{r})\bar{T}+(c\delta)^{2}=0\), which has two solutions:
Ignoring μ, we have two approximate solutions: \(\bar{T}_{1}\approx\frac{c\delta}{p_{r}\beta_{r}}\) (choosing “+” in (16)) and \(\bar{T}_{2}\approx\frac{c\delta}{p_{s}\beta_{s}}\) (choosing “−” in (16)). Because of the conditions for the existence of the coexistence steady state and the assumption that β r <β s and p r <p s , only \(\bar{T}_{2}\) is feasible. Thus, the mutant frequency before treatment is \(\varPhi=\frac{\bar{V}_{r}}{\bar{V}_{s}+\bar{V}_{r}}=\frac{1}{1+\frac{c\delta-(1-\mu )p_{r}\beta_{r}\bar{T}_{2}}{\mu p_{s}\beta_{s}\bar{T}_{2}}}\), where
Using \(\bar{T}_{2}\), Φ can be further simplified to
where \(r=\mathcal{R}_{r}/\mathcal{R}_{s}\). It is clear that Φ depends only on μ and r.
It follows from (17) that Φ can be approximated by
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 wo , where Φ w represents the mutant frequency with backward mutation (defined in (17)) and \(\varPhi_{wo}=\frac{\mu}{1-r}\) is the mutant frequency without backward mutation. For the proof, it suffices to show that \(\frac{(1-\mu)(1+r)+\sqrt{[(1-\mu)(1+r)]^{2}-4(1-2\mu)r}}{2}+\mu(1+r)>1\). This inequality is equivalent to r<1, which holds because resistant virus is less fit than wild-type virus in the absence of treatment (\(\mathcal{R}_{r}<\mathcal{R}_{s}\)). Therefore, Φ w <Φ wo . However, from the approximation of Φ w (Eq. (18)), we observe that the difference between Φ w and Φ wo is miniscule. This shows that backward mutation only plays a minor role in the pretreatment mutant frequency.
Appendix C: The Pretreatment Mutant Frequency in the Model with Hepatocyte Proliferation
From the V s and V r equations of model (5), we have \(\bar{V}_{s}=\frac{(1-\mu)p_{s} \bar{I}_{s}}{c}\) and \(\bar{V}_{r}=\frac{\mu p_{s}\bar{I}_{s}+p_{r} \bar{I}_{r}}{c}\). Substituting into the I s and I r equations, we obtain \(\frac{(1-\mu)\beta_{s}p_{s}\bar{T}}{c}+\rho_{s}(1-\frac{\bar{T}+\bar{I}_{s}+\bar{I}_{r}}{T_{\max}})=\delta\) and \(\frac{\beta_{r}(\mu p_{s}\bar{I}_{s}+p_{r} \bar{I}_{r})\bar{T}}{c\bar{I}_{r}}+\rho_{r}(1-\frac{\bar{T}+\bar{I}_{s}+\bar{I}_{r}}{T_{\max}})=\delta\). If ρ s =ρ r , from the above two equations we have \(\frac{(1-\mu)\beta_{s}p_{s}\bar{T}}{c}=\frac{\beta_{r}(\mu p_{s}\bar{I}_{s}+p_{r} \bar{I}_{r})\bar{T}}{c\bar{I}_{r}}\), which yields \(\bar{I}_{r}=\frac{\mu\beta_{r}p_{s}\bar{I}_{s}}{(1-\mu)\beta_{s} p_{s}-\beta_{r} p_{r}}\). Substituting into \(\bar{V}_{r}=\frac{\mu p_{s}\bar{I}_{s}+p_{r} \bar{I}_{r}}{c}\), we have \(\bar{V}_{r}=\frac{\mu p_{s}\bar{I}_{s}}{c}(1+\frac{r}{1-\mu-r})\), where r denotes the ratio \(\mathcal{R}_{r}/\mathcal{R}_{s}\).
Considering \(\bar{V}_{s}=(1-\mu)p_{s} \bar{I}_{s}/c\), we obtain the mutant frequency \(\varPhi=\frac{\bar{V}_{r}}{\bar{V}_{r}+\bar{V}_{s}}=\frac{\mu}{1-r}\), which is the same as the mutant frequency in the model without hepatocyte proliferation. It should be noted that although the mutant frequency is the same as that in the model without hepatocyte proliferation, the steady states of wild-type and resistant virus are not necessarily the same as the previous ones. They depend on ρ T , ρ s , ρ r , and other parameters.
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Rong, L., Ribeiro, R.M. & Perelson, A.S. Modeling Quasispecies and Drug Resistance in Hepatitis C Patients Treated with a Protease Inhibitor. Bull Math Biol 74, 1789–1817 (2012). https://doi.org/10.1007/s11538-012-9736-y
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DOI: https://doi.org/10.1007/s11538-012-9736-y