2013, 10(4): 1045-1065. doi: 10.3934/mbe.2013.10.1045

Model for hepatitis C virus transmissions

1. 

Merck Research Laboratories, UG1C-60, PO Box 1000, North Wales, PA 19454-1099, United States

Received  October 2012 Revised  March 2013 Published  June 2013

Hepatitis C virus (HCV) is a leading cause of chronic liver disease. This paper presents a deterministic model for HCV infection transmission and uses the model to assess the potential impact of antiviral therapy. The model is based on the susceptible-infective-removed-susceptible (SIRS) compartmental structure with chronic primary infection and possibility of reinfection. Important epidemiologic thresholds such as the basic and control reproduction numbers and a measure of treatment impact are derived. We find that if the control reproduction number is greater than unity, there is a locally unstable infection-free equilibrium and a unique, globally asymptotically stable endemic equilibrium. If the control reproduction number is less than unity, the infection-free equilibrium is globally asymptotically stable, and HCV will be eliminated. Numerical simulations suggest that, besides the parameters that determine the basic reproduction number, reinfection plays an important role in HCV transmissions and magnitude of the public health impact of antiviral therapy. Further, treatment regimens with better efficacy holds great promise for lowering the public health burden of HCV disease.
Citation: Elamin H. Elbasha. Model for hepatitis C virus transmissions. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1045-1065. doi: 10.3934/mbe.2013.10.1045
References:
[1]

M. J. Alter, H. S. Margolis, K. Krawczynski, F. N. Judson, A. Mares, W. J. Alexander, P. Y. Hu, J. K. Miller, M. A. Gerber and R. E. Sampliner, The natural history of community-acquired hepatitis C in the United States. The Sentinel Counties Chronic non-A, non-B Hepatitis Study Team,, N. Engl. J. Med., 327 (1992), 1899. Google Scholar

[2]

J. J. Amon, R. S. Garfein, L. Ahdieh-Grant, G. L. Armstrong, L. J. Ouellet, M. H. Latka, D. Vlahov, S. A. Strathdee, S. M. Hudson, P. Kerndt, D. Des Jarlais and I. T. Williams, Prevalence of hepatitis C virus infection among injection drug users in the United States, 1994-2004,, Clin. Infect. Dis., 46 (2008), 1852. Google Scholar

[3]

D. Amarapurkar, Natural history of hepatitis C virus infection,, J. Gastroenterol. Hepatol., 15 (2000). Google Scholar

[4]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1991). Google Scholar

[5]

G. L. Armstrong, A. Wasley, E. P. Simard, G. M. McQuillan, W. L. Kuhnert and M. J. Alter, The prevalence of hepatitis C virus infection in the United States, 1999 through 2002,, Ann. Intern. Med., 144 (2006), 705. Google Scholar

[6]

G. L. Armstrong, Injection drug users in the United States, 1979-2002: An aging population,, Arch. Intern. Med., 167 (2007), 166. Google Scholar

[7]

J. P. Bate, A. J. Colman, P. J. Frost, D. R. Shaw and H. A. J. Harley, High prevalence of late relapse and reinfection in prisoners treated for chronic hepatitis C,, J. Gastroenterol. Hepatol., 25 (2010), 1276. Google Scholar

[8]

W. G. Bennett, Y. Inoue, J. R. Beck, J. B. Wong, S. G. Pauker and G. L. Davis, Estimates of the cost-effectiveness of a single course of interferon-alpha 2b in patients with histologically mild chronic hepatitis C,, Ann. Intern. Med., 127 (1997), 55. Google Scholar

[9]

S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example,, Int. Stat. Rev., 62 (1994), 229. Google Scholar

[10]

S. Blower, K. Koelle and J. Mills, Health policy modeling: Epidemic control, HIV vaccines, and risky behavior,, in, (2002), 260. Google Scholar

[11]

A. A. Butt, A. C. Justice, M. Skanderson, M. O. Rigsby, C. B. Good and C. K. Kwoh, Rate and predictors of treatment prescription for hepatitis C,, Gut, 56 (2007), 385. Google Scholar

[12]

Centers for Disease Control and Prevention (CDC), Recommendations for prevention and control of hepatitis C virus (HCV) infection and HCV-related chronic disease,, MMWR Recomm Rep., 47 (1998), 1. Google Scholar

[13]

Centers for Disease Control and Prevention (CDC), "Hepatitis C Information for Health Professionals,", 2010. (accessed October 5, (2012). Google Scholar

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Centers for Disease Control and Prevention (CDC), Recommendations for the identification of chronic hepatitis C virus infection among persons born during 1945-1965,, Recommendations and Reports, 61 (2012), 1. Google Scholar

[15]

O. Dalgard, A. Egeland, K. Skaug, K. Vilimas and T. Steen, Health-related quality of life in active injecting drug users with and without chronic hepatitis C virus infection,, Hepatology, 39 (2004), 74. Google Scholar

[16]

A. M. Di Bisceglie, Natural history of hepatitis C: Its impact on clinical management,, Hepatology, 31 (2000), 1014. Google Scholar

[17]

P. Das, D. Mukherjee and A. K. Sarkar, Analysis of a disease transmission model of hepatitis C,, J. Biol. Sys., 13 (2005), 331. Google Scholar

[18]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar

[19]

O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The constructions of next-generation matrices fro compartmental epidemiologic models,, J. R. Soc. Interface, 7 (2010), 873. Google Scholar

[20]

E. Dorey, Competition intensifies around hepatitis C,, Nat. Biotechnology, 27 (2009), 305. Google Scholar

[21]

K. A. Dowd, R. C. Hershow, S. Yawetz, P. Larussa, C. Diaz, S. H. Landesman, M. E. Paul, J. S. Read, M. Lu, D. L. Thomas, D. M. Netski and S. C. Ray, Maternal neutralizing antibody and transmission of hepatitis C virus to infants,, J. Infect. Dis., 198 (2008), 1651. Google Scholar

[22]

E. H. Elbasha, Global stability of equilibria in a two-sex HPV vaccination model,, Bull. Math. Biol., 70 (2008), 894. doi: 10.1007/s11538-007-9283-0. Google Scholar

[23]

P. Ferenci and K. R. Reddy, Impact of HCV protease-inhibitor-based triple therapy for chronic HCV genotype 1 infection,, Antivir Ther., 16 (2011), 1187. Google Scholar

[24]

M. G. Ghany, D. B. Strader, D. L. Thomas and L. B. Seeff, Diagnosis, management, and treatment of hepatitis C: an update. American Association for the Study of Liver Diseases,, Hepatology, 49 (2009), 1335. Google Scholar

[25]

J. Grebely, B. Conway, J. D. Raffa, C. Lai, M. Krajden and M. W. Tyndall, Hepatitis C virus reinfection in injection drug users,, Hepatology, 44 (2006), 1139. Google Scholar

[26]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar

[27]

S. Kamal, Acute pepatitis C: A systematic review,, Am. J. Gastroenterol, 103 (2008), 1283. Google Scholar

[28]

H. Khalil, "Nonlinear Systems,", 3rd edn. Prentice Hall, (2002). Google Scholar

[29]

J. Kimber, L. Copeland, M. Hickman, J. Macleod, J. McKenzie, D. De Angelis and J. R. Robertson, Survival and cessation in injecting drug users: Prospective observational study of outcomes and effect of opiate substitution treatment,, BMJ, 341 (2010). Google Scholar

[30]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, part 1,, Proc. Roy. Soc. Lond. A., 115 (1927), 700. Google Scholar

[31]

S. H. Kleinman, N. Lelie and M. P. Busch, Infectivity of human immunodeficiency virus-1, hepatitis C virus, and hepatitis B virus and risk of transmission by transfusion,, Transfusion, 49 (2009), 2454. Google Scholar

[32]

M. Kretzschmar and L. Wiessing, Modelling the transmission of hepatitis C in injecting drug users,, in, (2004). Google Scholar

[33]

M. E. Major, K. Mihalik, M. Puig, B. Rehermann, M. Nascimbeni, C. M. Rice and S. M. Feinstone, Previously infected and recovered chimpanzees exhibit rapid responses that control hepatitis C virus replication upon rechallenge,, J. Virol., 76 (2002), 6586. Google Scholar

[34]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology,, J. Theor. Biol., 254 (2008), 178. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar

[35]

M. Martcheva and C. Castillo-Chavez, Disease with chronic stage in a population with varying size,, Math. Biosci., 182 (2003), 1. doi: 10.1016/S0025-5564(02)00184-0. Google Scholar

[36]

C. Matheï, S. Van Dooren, P. Lemey, P. Van Damme, F. Buntinx and A-M. Vandamme, The epidemic history of hepatitis C among injecting drug users in Flanders, Belgium,, J. Viral. Hepat., 15 (2008), 399. Google Scholar

[37]

S. H. Mehta, B. L. Genberg, J. Astemborski, R. Kavasery, G. D. Kirk, D. Vlahov, S. A. Strathdee and D. L. Thomas, Limited uptake of hepatitis C treatment among injection drug users,, J. Community Health, 33 (2008), 126. Google Scholar

[38]

A. McLean and S. Blower, Imperfect vaccines and herd immunity to HIV,, Proc. Roy. Soc. Lond. B., 253 (1993), 9. Google Scholar

[39]

S. H. Mehta, A. Cox, D. R. Hoover, X. H. Wang, Q. Mao, S. Ray, S. A. Strathdee, D. Vlahov and D. L. Thomas, Protection against persistence of hepatitis C,, Lancet., 359 (2002), 1478. Google Scholar

[40]

A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden and A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy,, Science, 282 (1998), 103. Google Scholar

[41]

H. Ohto, S. Terazewa and N. Sasaki, Transmission of hepatitis C virus from mothers to infants,, N. Engl. J. Med., 330 (1994), 744. Google Scholar

[42]

W. O. Osburn, B. E. Fisher, K. A. Dowd, G. Urban, L. Liu, S. C. Ray, T. L. Thomas and A. L. Cox, Spontaneous control of primary hepatitis C virus infection and immunity against persistent reinfection,, Gastroenterology, 138 (2010), 315. Google Scholar

[43]

O. G. Pybus, A. Cochrane, E. C. Holmes and P. Simmonds, The hepatitis C virus epidemic among injecting drug users,, Infec. Gene. Evol., 5 (2005), 131. Google Scholar

[44]

L. Rong and A. S. Perelson, Treatment of hepatitis C virus infection and small molecule direct antivirals: viral kinetics and modeling,, Crit. Rev. Immunol., 30 (2010), 131. Google Scholar

[45]

B. H. Singer and D. E. Kirschner, Influence of backward bifurcation on interpretation of $R_{0}$ in a model of epidemic Tuberculosis with reinfection,, Math Biosci. Eng., 1 (2004), 81. doi: 10.3934/mbe.2004.1.81. Google Scholar

[46]

A. Takaki, M. Wiese, G. Maertens, E. Depla, U. Seifert, A. Liebetrau, J. L. Miller, M. P. Manns and B. Rehermann, Cellular immune responses persist, humoral responses decrease two decades after recovery from a single source outbreak of hepatitis C,, Nat. Med., 6 (2000), 578. Google Scholar

[47]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[48]

M. L. Volk, R. Tocco, S. Saini and A. S. Lok,, Public health impact of antiviral therapy for hepatitis C in the United States,, Hepatology, 50 (2009), 1750. Google Scholar

[49]

J. J. Weusten, A. A. van Drimmelen and P. N. Lelie, Mathematical modeling of the risk of HBV, HCV, and HIV transmission by window phase donations not detected by NAT,, Transfusion, 42 (2002), 537. Google Scholar

[50]

World Health Organization (WHO), "Facts Sheet: Hepatitis C,", , (2012). Google Scholar

[51]

I. Zeiler, T. Langlands, J. M. Murray and A. Ritter, Optimal targeting of Hepatitis C virus treatment among injecting drug users to those not enrolled in methadone maintenance programs,, Drug Alcohol. Depend., 110 (2010), 228. Google Scholar

show all references

References:
[1]

M. J. Alter, H. S. Margolis, K. Krawczynski, F. N. Judson, A. Mares, W. J. Alexander, P. Y. Hu, J. K. Miller, M. A. Gerber and R. E. Sampliner, The natural history of community-acquired hepatitis C in the United States. The Sentinel Counties Chronic non-A, non-B Hepatitis Study Team,, N. Engl. J. Med., 327 (1992), 1899. Google Scholar

[2]

J. J. Amon, R. S. Garfein, L. Ahdieh-Grant, G. L. Armstrong, L. J. Ouellet, M. H. Latka, D. Vlahov, S. A. Strathdee, S. M. Hudson, P. Kerndt, D. Des Jarlais and I. T. Williams, Prevalence of hepatitis C virus infection among injection drug users in the United States, 1994-2004,, Clin. Infect. Dis., 46 (2008), 1852. Google Scholar

[3]

D. Amarapurkar, Natural history of hepatitis C virus infection,, J. Gastroenterol. Hepatol., 15 (2000). Google Scholar

[4]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans: Dynamics and Control,", Oxford University Press, (1991). Google Scholar

[5]

G. L. Armstrong, A. Wasley, E. P. Simard, G. M. McQuillan, W. L. Kuhnert and M. J. Alter, The prevalence of hepatitis C virus infection in the United States, 1999 through 2002,, Ann. Intern. Med., 144 (2006), 705. Google Scholar

[6]

G. L. Armstrong, Injection drug users in the United States, 1979-2002: An aging population,, Arch. Intern. Med., 167 (2007), 166. Google Scholar

[7]

J. P. Bate, A. J. Colman, P. J. Frost, D. R. Shaw and H. A. J. Harley, High prevalence of late relapse and reinfection in prisoners treated for chronic hepatitis C,, J. Gastroenterol. Hepatol., 25 (2010), 1276. Google Scholar

[8]

W. G. Bennett, Y. Inoue, J. R. Beck, J. B. Wong, S. G. Pauker and G. L. Davis, Estimates of the cost-effectiveness of a single course of interferon-alpha 2b in patients with histologically mild chronic hepatitis C,, Ann. Intern. Med., 127 (1997), 55. Google Scholar

[9]

S. M. Blower and H. Dowlatabadi, Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example,, Int. Stat. Rev., 62 (1994), 229. Google Scholar

[10]

S. Blower, K. Koelle and J. Mills, Health policy modeling: Epidemic control, HIV vaccines, and risky behavior,, in, (2002), 260. Google Scholar

[11]

A. A. Butt, A. C. Justice, M. Skanderson, M. O. Rigsby, C. B. Good and C. K. Kwoh, Rate and predictors of treatment prescription for hepatitis C,, Gut, 56 (2007), 385. Google Scholar

[12]

Centers for Disease Control and Prevention (CDC), Recommendations for prevention and control of hepatitis C virus (HCV) infection and HCV-related chronic disease,, MMWR Recomm Rep., 47 (1998), 1. Google Scholar

[13]

Centers for Disease Control and Prevention (CDC), "Hepatitis C Information for Health Professionals,", 2010. (accessed October 5, (2012). Google Scholar

[14]

Centers for Disease Control and Prevention (CDC), Recommendations for the identification of chronic hepatitis C virus infection among persons born during 1945-1965,, Recommendations and Reports, 61 (2012), 1. Google Scholar

[15]

O. Dalgard, A. Egeland, K. Skaug, K. Vilimas and T. Steen, Health-related quality of life in active injecting drug users with and without chronic hepatitis C virus infection,, Hepatology, 39 (2004), 74. Google Scholar

[16]

A. M. Di Bisceglie, Natural history of hepatitis C: Its impact on clinical management,, Hepatology, 31 (2000), 1014. Google Scholar

[17]

P. Das, D. Mukherjee and A. K. Sarkar, Analysis of a disease transmission model of hepatitis C,, J. Biol. Sys., 13 (2005), 331. Google Scholar

[18]

O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_{0}$ in models for infectious diseases in heterogeneous populations,, J. Math. Biol., 28 (1990), 365. doi: 10.1007/BF00178324. Google Scholar

[19]

O. Diekmann, J. A. P. Heesterbeek and M. G. Roberts, The constructions of next-generation matrices fro compartmental epidemiologic models,, J. R. Soc. Interface, 7 (2010), 873. Google Scholar

[20]

E. Dorey, Competition intensifies around hepatitis C,, Nat. Biotechnology, 27 (2009), 305. Google Scholar

[21]

K. A. Dowd, R. C. Hershow, S. Yawetz, P. Larussa, C. Diaz, S. H. Landesman, M. E. Paul, J. S. Read, M. Lu, D. L. Thomas, D. M. Netski and S. C. Ray, Maternal neutralizing antibody and transmission of hepatitis C virus to infants,, J. Infect. Dis., 198 (2008), 1651. Google Scholar

[22]

E. H. Elbasha, Global stability of equilibria in a two-sex HPV vaccination model,, Bull. Math. Biol., 70 (2008), 894. doi: 10.1007/s11538-007-9283-0. Google Scholar

[23]

P. Ferenci and K. R. Reddy, Impact of HCV protease-inhibitor-based triple therapy for chronic HCV genotype 1 infection,, Antivir Ther., 16 (2011), 1187. Google Scholar

[24]

M. G. Ghany, D. B. Strader, D. L. Thomas and L. B. Seeff, Diagnosis, management, and treatment of hepatitis C: an update. American Association for the Study of Liver Diseases,, Hepatology, 49 (2009), 1335. Google Scholar

[25]

J. Grebely, B. Conway, J. D. Raffa, C. Lai, M. Krajden and M. W. Tyndall, Hepatitis C virus reinfection in injection drug users,, Hepatology, 44 (2006), 1139. Google Scholar

[26]

H. W. Hethcote, The mathematics of infectious diseases,, SIAM Rev., 42 (2000), 599. doi: 10.1137/S0036144500371907. Google Scholar

[27]

S. Kamal, Acute pepatitis C: A systematic review,, Am. J. Gastroenterol, 103 (2008), 1283. Google Scholar

[28]

H. Khalil, "Nonlinear Systems,", 3rd edn. Prentice Hall, (2002). Google Scholar

[29]

J. Kimber, L. Copeland, M. Hickman, J. Macleod, J. McKenzie, D. De Angelis and J. R. Robertson, Survival and cessation in injecting drug users: Prospective observational study of outcomes and effect of opiate substitution treatment,, BMJ, 341 (2010). Google Scholar

[30]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics, part 1,, Proc. Roy. Soc. Lond. A., 115 (1927), 700. Google Scholar

[31]

S. H. Kleinman, N. Lelie and M. P. Busch, Infectivity of human immunodeficiency virus-1, hepatitis C virus, and hepatitis B virus and risk of transmission by transfusion,, Transfusion, 49 (2009), 2454. Google Scholar

[32]

M. Kretzschmar and L. Wiessing, Modelling the transmission of hepatitis C in injecting drug users,, in, (2004). Google Scholar

[33]

M. E. Major, K. Mihalik, M. Puig, B. Rehermann, M. Nascimbeni, C. M. Rice and S. M. Feinstone, Previously infected and recovered chimpanzees exhibit rapid responses that control hepatitis C virus replication upon rechallenge,, J. Virol., 76 (2002), 6586. Google Scholar

[34]

S. Marino, I. B. Hogue, C. J. Ray and D. E. Kirschner, A methodology for performing global uncertainty and sensitivity analysis in systems biology,, J. Theor. Biol., 254 (2008), 178. doi: 10.1016/j.jtbi.2008.04.011. Google Scholar

[35]

M. Martcheva and C. Castillo-Chavez, Disease with chronic stage in a population with varying size,, Math. Biosci., 182 (2003), 1. doi: 10.1016/S0025-5564(02)00184-0. Google Scholar

[36]

C. Matheï, S. Van Dooren, P. Lemey, P. Van Damme, F. Buntinx and A-M. Vandamme, The epidemic history of hepatitis C among injecting drug users in Flanders, Belgium,, J. Viral. Hepat., 15 (2008), 399. Google Scholar

[37]

S. H. Mehta, B. L. Genberg, J. Astemborski, R. Kavasery, G. D. Kirk, D. Vlahov, S. A. Strathdee and D. L. Thomas, Limited uptake of hepatitis C treatment among injection drug users,, J. Community Health, 33 (2008), 126. Google Scholar

[38]

A. McLean and S. Blower, Imperfect vaccines and herd immunity to HIV,, Proc. Roy. Soc. Lond. B., 253 (1993), 9. Google Scholar

[39]

S. H. Mehta, A. Cox, D. R. Hoover, X. H. Wang, Q. Mao, S. Ray, S. A. Strathdee, D. Vlahov and D. L. Thomas, Protection against persistence of hepatitis C,, Lancet., 359 (2002), 1478. Google Scholar

[40]

A. U. Neumann, N. P. Lam, H. Dahari, D. R. Gretch, T. E. Wiley, T. J. Layden and A. S. Perelson, Hepatitis C viral dynamics in vivo and the antiviral efficacy of interferon-alpha therapy,, Science, 282 (1998), 103. Google Scholar

[41]

H. Ohto, S. Terazewa and N. Sasaki, Transmission of hepatitis C virus from mothers to infants,, N. Engl. J. Med., 330 (1994), 744. Google Scholar

[42]

W. O. Osburn, B. E. Fisher, K. A. Dowd, G. Urban, L. Liu, S. C. Ray, T. L. Thomas and A. L. Cox, Spontaneous control of primary hepatitis C virus infection and immunity against persistent reinfection,, Gastroenterology, 138 (2010), 315. Google Scholar

[43]

O. G. Pybus, A. Cochrane, E. C. Holmes and P. Simmonds, The hepatitis C virus epidemic among injecting drug users,, Infec. Gene. Evol., 5 (2005), 131. Google Scholar

[44]

L. Rong and A. S. Perelson, Treatment of hepatitis C virus infection and small molecule direct antivirals: viral kinetics and modeling,, Crit. Rev. Immunol., 30 (2010), 131. Google Scholar

[45]

B. H. Singer and D. E. Kirschner, Influence of backward bifurcation on interpretation of $R_{0}$ in a model of epidemic Tuberculosis with reinfection,, Math Biosci. Eng., 1 (2004), 81. doi: 10.3934/mbe.2004.1.81. Google Scholar

[46]

A. Takaki, M. Wiese, G. Maertens, E. Depla, U. Seifert, A. Liebetrau, J. L. Miller, M. P. Manns and B. Rehermann, Cellular immune responses persist, humoral responses decrease two decades after recovery from a single source outbreak of hepatitis C,, Nat. Med., 6 (2000), 578. Google Scholar

[47]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[48]

M. L. Volk, R. Tocco, S. Saini and A. S. Lok,, Public health impact of antiviral therapy for hepatitis C in the United States,, Hepatology, 50 (2009), 1750. Google Scholar

[49]

J. J. Weusten, A. A. van Drimmelen and P. N. Lelie, Mathematical modeling of the risk of HBV, HCV, and HIV transmission by window phase donations not detected by NAT,, Transfusion, 42 (2002), 537. Google Scholar

[50]

World Health Organization (WHO), "Facts Sheet: Hepatitis C,", , (2012). Google Scholar

[51]

I. Zeiler, T. Langlands, J. M. Murray and A. Ritter, Optimal targeting of Hepatitis C virus treatment among injecting drug users to those not enrolled in methadone maintenance programs,, Drug Alcohol. Depend., 110 (2010), 228. Google Scholar

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