A comparison of a deterministic and stochastic model for Hepatitis C with an isolation stage

REFERENCES

• L.J.S. Allen, An Introduction to Stochastic Processes with Applications to Biology, 2nd ed., Chapman and Hall/CRC, Boca Raton, FL, 2003.
   Google Scholar
• L.J.S. Allen, An introduction to stochastic epidemic models, in Mathematical Epidemiology, F. Brauer, P. Van den Driessche, and J. Wu, eds., Springer-Verlag, Berlin, 2008, pp. 77–128.
   Google Scholar
• L.J.S. Allen and A.M. Burgin, Comparison of deterministic and stochastic SIS and SIR models in discrete time, Math. Biosci. 163 (2000), pp. 1–33. doi: 10.1016/S0025-5564(99)00047-4
   PubMed Web of Science ®Google Scholar
• L.J.S. Allen, D.A. Flores, R.K. Ratnayake, and J.R. Herbold, Discrete-time deterministic and stochastic models for the spread of rabies, Appl. Math. Comput. 132 (2002), pp. 271–292. doi: 10.1016/S0096-3003(01)00192-8
   Web of Science ®Google Scholar
• M.J. Alter, Hepatitis C: The clinical spectrum of the disease, NIH Consensus Development Conference on Management of Hepatitis C, Bethesda, MD, 1997.
   Google Scholar
• M.J. Alter, Prevention of spread of hepatitis C, Hepatology 36 (2002), pp. 93–98. doi: 10.1002/hep.1840360712
   PubMed Web of Science ®Google Scholar
• N. Bailey, A simple stochastic epidemic, Biometrika 37 (1950), pp. 193–202
   PubMed Web of Science ®Google Scholar
• U. Bandy, Hepatitis C virus (HCV): A silent epidemic, Med. Health R. I. 82 (1999), pp. 223–224.
   PubMedGoogle Scholar
• S. Busenberg and P. van den Driessche, Analysis of a disease transmission model in a population with varying size, J. Math. Biol. 28 (1990), pp. 257–270. doi: 10.1007/BF00178776
   PubMed Web of Science ®Google Scholar
• S. Busenberg and K. Hadeler, Demography and epidemics, Math. Biosci. 101 (1990), pp. 63–74. doi: 10.1016/0025-5564(90)90102-5
   PubMed Web of Science ®Google Scholar
• S.L. Chen and T.R. Morgan, The natural history of hepatitis C virus (HCV) infection, Int. J. Med. Sci. 3 (2006), pp. 47–52. doi: 10.7150/ijms.3.47
   PubMedGoogle Scholar
• R. Colina, C. Azambuja, R. Uriarte, C. Mogdasy, and J. Cristina, Evidence of increasing diversification of Hepatitis C viruses, J. Gen. Virol. 80 (1999), pp. 1377–1382.
   PubMed Web of Science ®Google Scholar
• A.M. Di Bisceglie, S.E. Order, J.L. Klein, J.G. Waggoner, M.H. Sjogren, G. Kuo, and J.H. Hoofnagle, The role of chronic viral hepatitis in hepatocellular carcinoma in the United States, Am. J. Gastroenterol. 86 (1991), pp. 335–338.
   PubMed Web of Science ®Google Scholar
• C.A. Donnelly, A.C. Ghani, G.M. Leung, A.J. Hedley, C. Fraser, S. Riley, and R.M. Anderson, Epidemiological determinants of spread of causal agent of severe acute respiratory syndrome in Hong Kong, Lancet 361 (2003), pp. 1761–1766. doi: 10.1016/S0140-6736(03)13410-1
   PubMed Web of Science ®Google Scholar
• I.K. Dontwi, N.K. Frempong, D.E. Bentil, I. Adetunde, and E. Owusu-Ansah, Mathematical modeling of Hepatitis C Virus transmission among injecting drug users and the impact of vaccination, Amer. J. Sci. Ind. Res. 1 (2010), pp. 41–46.
   Google Scholar
• P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci. 180 (2002), pp. 29–48. doi: 10.1016/S0025-5564(02)00108-6
   PubMed Web of Science ®Google Scholar
• N. Esposito and C. Rossi, A nested-epidemic model for the spread of hepatitis C among injecting drug users, Math. Biosci. 188 (2004), pp. 29–45. doi: 10.1016/j.mbs.2003.11.001
   PubMed Web of Science ®Google Scholar
• G. Fattovich, G. Giustina, F. Degos, F. Tremolada, G. Diodati, P. Almasio, and G. Realdi, Morbidity and mortality in compensated cirrhosis type C: A retrospective follow-up study of 384 patients, Gastroenterology 112 (1997), pp. 463–472. doi: 10.1053/gast.1997.v112.pm9024300
   PubMed Web of Science ®Google Scholar
• A.B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, and B.M. Sahai, Modelling strategies for controlling SARS outbreaks, Proc. R. Soc. Ser. B 271 (2003), pp. 2223–2232. doi: 10.1098/rspb.2004.2800
   Web of Science ®Google Scholar
• K. Hadeler and C. Castillo-Chavez, A core group model for disease transmission, Math. Biosci. 128 (1995), pp. 41–45. doi: 10.1016/0025-5564(94)00066-9
   PubMed Web of Science ®Google Scholar
• H.W. Hethcote, The mathematics of infectious diseases, SIAM Rev. 42 (2000), pp. 599–653. doi: 10.1137/S0036144500371907
   Web of Science ®Google Scholar
• H.W. Hethcote, M. Zhien, and L. Shengbing, Effects of quarantine in six endemic models for infectious diseases, Math. Biosci. 180 (2002), pp. 141–160. doi: 10.1016/S0025-5564(02)00111-6
   PubMed Web of Science ®Google Scholar
• Y. Hutin, M.E. Kitler, G.J. Dore, J.F. Perz, G.L. Armstrong, G. Dusheiko, and D. Lavanchy, Global burden of disease (GBD) for hepatitis C, J. Clin. Pharmacol. 44 (2004), pp. 20–29. doi: 10.1177/0091270003258669
   PubMed Web of Science ®Google Scholar
• J.A. Ilyas and J.M. Vierling, An overview of emerging therapies for the treatment of chronic hepatitis C, Clin. Liver Dis. 15 (2011), pp. 515–536. doi: 10.1016/j.cld.2011.05.002
   PubMed Web of Science ®Google Scholar
• A.J. Keeling and J.V. Ross, On methods for studying stochastic disease dynamics, J. R. Soc. Interface 5 (2008), pp. 171–181. doi: 10.1098/rsif.2007.1106
   PubMed Web of Science ®Google Scholar
• K. Kiyosawa, T. Sodeyama, E. Tanaka, Y. Gibo, K. Yoshizawa, Y. Nakano, and H.J. Alter, Interrelationship of blood transfusion, non-A, non-B hepatitis and hepatocellular carcinoma: Analysis by detection of antibody to hepatitis C virus, Hepatology 12 (1990), pp. 671–675. doi: 10.1002/hep.1840120409
   PubMed Web of Science ®Google Scholar
• M.A. Krasnoselskii, Positive Solutions of Operator Equations, Noordhoff Leyden, Groningen, 1964.
   Google Scholar
• M. Lipsitch, T. Cohen, B. Cooper, J.M. Robins, S. Ma, L. James, and M. Murray, Transmission dynamics and control of severe acute respiratory syndrome, Science 300 (2003), pp. 1966–1970. doi: 10.1126/science.1086616
   PubMed Web of Science ®Google Scholar
• J.O. Lloyd-Smith, A.P. Galvani, and W.M. Getz, Curtailing transmission of severe acute respiratory syndrome within a community and its hospital, Proc. R. Soc. Lond. Ser. B Biol. Sci. 170 (2003), pp. 1979–1989. doi: 10.1098/rspb.2003.2481
   Web of Science ®Google Scholar
• F. Luo and Z. Xiang, Global analysis of an endemic model with acute and chronic stages, Int. Math. Forum 7 (2012), pp. 75–81.
   Google Scholar
• A. Maheshwari and P.J. Thuluvath, Management of acute hepatitis C, Clin. Liver Dis. 14 (2010), pp. 169–176. doi: 10.1016/j.cld.2009.11.007
   PubMedGoogle Scholar
• M. Martcheva and C. Castillo-Chavez, Diseases with chronic stage in a population with varying size, Math. Biosci. 182 (2003), pp. 1–25. doi: 10.1016/S0025-5564(02)00184-0
   PubMed Web of Science ®Google Scholar
• J. Marx, R. Hockberger, and R. Walls, Rosen's Emergency Medicine – Concepts and Clinical Practice, 7th ed., Mosby/Elsevier, Philadelphia, PA, 2010.
   Google Scholar
• D. Mather and N. Crofts, A computer model of the spread of hepatitis C virus among injecting drug users, Eur. J. Epidemiol. 15 (1999), pp. 5–10. doi: 10.1023/A:1007548307196
   PubMed Web of Science ®Google Scholar
• R.G. McLeod, J.F. Brewster, A.B. Gumel, and D.A. Slonowsky, Sensitivity and uncertainty analyses for a SARS model with time-varying inputs and outputs, Math. Biosci. Eng. 3 (2006), pp. 527–544. doi: 10.3934/mbe.2006.3.527
   PubMedGoogle Scholar
• P.K. Nelson, B.M. Mathers, B. Cowie, H. Hagan, D. Des Jarlais, D. Horyniak, and L. Degenhardt, Global epidemiology of hepatitis B and hepatitis C in people who inject drugs: Results of systematic reviews, Lancet 378 (2011), pp. 571–583. doi: 10.1016/S0140-6736(11)61097-0
   PubMed Web of Science ®Google Scholar
• R. Purcell, Hepatitis C virus: An introduction, NIH Consensus Development Conference on Management of Hepatitis C, Bethesda, MD, 1997.
   Google Scholar
• S.C. Ray and D.L. Thomas, Chapter 154: Hepatitis C, in Mandell, Douglas, and Bennett's Principles and Practice of Infectious Diseases, 7th ed., G.L. Mandell, J.E. Bennett, and R. Dolin, eds., Churchill Livingstone, Philadelphia, PA, 2009.
   Google Scholar
• B, Reade, R.G. Bowers, M. Begon, and R. Gaskell, A model of disease and vaccination for infections with acute and chronic phases, J. Theor. Biol. 190 (1998), pp. 355–367. doi: 10.1006/jtbi.1997.0557
   PubMed Web of Science ®Google Scholar
• S. Riley, Large-scale spatial-transmission models of infectious disease, Science 316 (2007), pp. 1298–1301. doi: 10.1126/science.1134695
   PubMed Web of Science ®Google Scholar
• M.A. Safi and A.B. Gumel, Global asymptotic dynamics of a model for quarantine and isolation, Discret Contin. Dyn. Syst. 14 (2010), pp. 209–231. doi: 10.3934/dcdsb.2010.14.209
   Web of Science ®Google Scholar
• L.B. Seeff, Z. Buskell-Bales, E.C. Wright, S.J. Durako, H.J. Alter, F.L. Iber, and C.G. Hollingsworth, Long-term mortality after transfusion-associated non-A, non-B hepatitis, New Engl. J. Med. 327 (1992), pp. 1906–1911. doi: 10.1056/NEJM199212313272703
   PubMed Web of Science ®Google Scholar
• M.L. Shiffman, Chronic Hepatitis C Virus: Advances in Treatment, Promise for the Future, Springer, New York, 2011.
   Google Scholar
• S. Spencer, Stochastic epidemic models for emerging diseases, Ph.D. diss., University of Nottingham, 2008.
   Google Scholar
• J.H. Tien and D.J. Earn, Multiple transmission pathways and disease dynamics in a waterborne pathogen model, Bull. Math. Biol. 72 (2010), pp. 1506–1533. doi: 10.1007/s11538-010-9507-6
   PubMed Web of Science ®Google Scholar
• R.A. Tohme and S.D. Holmberg, Is sexual contact a major mode of hepatitis C virus transmission, Hepatology 52 (2010), pp. 1497–1505. doi: 10.1002/hep.23808
   PubMed Web of Science ®Google Scholar
• J. Torresi, D. Johnson, and H. Wedemeyer, Progress in the development of preventive and therapeutic vaccines for hepatitis C virus, J. Hepatol. 54 (2011), pp. 1273–1285. doi: 10.1016/j.jhep.2010.09.040
   PubMed Web of Science ®Google Scholar
• A.R. Tuite, J. Tien, M. Eisenberg, D.J. Earn, J. Ma, and D.N. Fisman, Cholera epidemic in Haiti, 2010: Using a transmission model to explain spatial spread of disease and identify optimal control interventions, Ann. Intern. Med. 154 (2011), pp. 593–601.
   Google Scholar
• G.H. Weiss, and M. Dishon, On the asymptotic behavior of the stochastic and deterministic models of an epidemic, Math. Biosci. 11 (1971), pp. 261–265. doi: 10.1016/0025-5564(71)90087-3
   Google Scholar
• S. Zhang and Y. Zhou, Dynamics and application of an epidemiological model for hepatitis C, Math. Comput. Model. 56 (2012), pp. 36–42. doi: 10.1016/j.mcm.2011.11.081
   Web of Science ®Google Scholar