Skip to main content

Advertisement

Log in

The Impact of Hepatitis A Virus Infection on Hepatitis C Virus Infection: A Competitive Exclusion Hypothesis

  • Original Article
  • Published:
Bulletin of Mathematical Biology Aims and scope Submit manuscript

Abstract

We address the observation that, in some cases, patients infected with the hepatitis C virus (HCV) are cleared of HCV when super-infected with the hepatitis A virus (HAV). We hypothesise that this phenomenon can be explained by the competitive exclusion principle, including the action of the immune system, and show that the inclusion of the immune system explains both the elimination of one virus and the co-existence of both infections for a certain range of parameters. We discuss the potential clinical implications of our findings.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1

Similar content being viewed by others

References

  • Ajelli, M., Iannelli, M., Manfredi, P., & degli Atti, M. L. C. (2008). Basic mathematical models for the temporal dynamics of HAV in medium-endemicity Italian areas. Vaccine, 26, 1697–1707.

    Article  Google Scholar 

  • Amaku, M., Burattini, M. N., Coutinho, F. A. B., & Massad, E. (2010a). Modeling the dynamics of viral infection considering competition within individual hosts and at population levels: the effect of treatment. Bull. Math. Biol., 72(5), 1284–1314.

    Article  MathSciNet  Google Scholar 

  • Amaku, M., Burattini, M. N., Coutinho, F. A. B., & Massad, E. (2010b). Modeling the competition between viruses in a complex plant-pathogen system. Phytopathology, 100, 1042–1047.

    Article  Google Scholar 

  • Anderson, R. M., & May, R. M. (1991). Infectious diseases of humans: dynamics and control. Oxford: Oxford University Press.

    Google Scholar 

  • Andraud, M., Lejeune, O., Musoro, J. Z., Ogunjimi, B., Beutels, P., et al. (2012). Living on three time scales: the dynamics of plasma cell and antibody populations illustrated for hepatitis A virus. PLoS Comput. Biol., 8(3), e1002418. doi:10.1371/journal.pcbi.1002418.

    Article  MathSciNet  Google Scholar 

  • Bremermann, H. J., & Thieme, H. R. (1988). A competitive exclusion principle for pathogen virulence. J. Math. Biol., 27, 179–190.

    Article  MathSciNet  Google Scholar 

  • Burattini, M. N., Coutinho, F. A. B., & Massad, E. (2008). Viral evolution and the competitive exclusion principle. Biosci. Hypotheses, 1(3), 168–171.

    Article  Google Scholar 

  • Dahari, H., Ribeiro, R. M., & Perelson, A. S. (2007). Triphasic decline of hepatitis C virus RNA during antiviral therapy. Hepatology, 46(1), 16–21.

    Article  Google Scholar 

  • Dahari, H., Layden-Almer, J. E., Kallwitz, E., Ribeiro, R. M., Cotler, S. J., Layden, T. J., & Perelson, A. S. (2009). A mathematical model of hepatitis C virus dynamics in patients with high baseline viral loads or advanced liver disease. Gastroenterology, 136, 1402–1409.

    Article  Google Scholar 

  • Deterding, K., Tegtmeyer, T., Cornberg, M., Hadem, J., Potthoff, A., Böker, K. H. W., Tillmann, H. L., Manns, M. P., & Wedemeyer, H. (2006). Hepatitis A virus infection suppresses hepatitis C virus replication and may lead to clearance of HCV. J. Hepatol., 45, 770–778.

    Article  Google Scholar 

  • Edelstein-Keshet, L. (2005). Mathematical models in biology. Philadelphia: SIAM.

    Book  MATH  Google Scholar 

  • Gruener, N. H. Jung, M. C., Ulsenheimer, A., Gerlach, T. J., Diepolder, H. M., Schirren, C. A., Hoffman, R., Wächtler, M., Backmund, M., & Pape, G. R. (2002). Hepatitis C virus eradication associated with hepatitis B virus superinfection and development of a hepatitis B virus specific T cell response. J. Hepatol., 37, 866–869.

    Article  Google Scholar 

  • Jeger, M. J., van den Bosch, F., & Madden, L. V. (2011). Modelling virus and host-limitation in vectored plant disease epidemics. Virus Res., 159, 215–222.

    Article  Google Scholar 

  • Lecoq, H., Fabre, F., Joannon, B., Wipf-Scheibel, C., Chandeysson, C., Schoeny, A., & Desbiez, C. (2011). Search for factors involved in the rapid shift in watermelon mosaic virus (WMV) populations in South-eastern France. Virus Res., 159, 115–123.

    Article  Google Scholar 

  • Liaw, Y. F., Tsai, S. L., Chang, J. J., Sheen, I. S., Chien, R. N., Lin, D. G., & Chu, C. M. (1994). Displacement of hepatitis B virus by hepatitis C virus as the cause of continuing chronic hepatitis. Gastroenterology, 106, 1048–1053.

    Google Scholar 

  • Lopez, L. F., Coutinho, F. A. B., Burattini, M. N., & Massad, E. (2002). Threshold conditions for infection persistence in complex host-vectors interactions. C. R. Biol. Acad. Sci. Paris, 325, 1073–1084.

    Google Scholar 

  • Marcos, R., Monteiro, R. A. F., & Rocha, E. (2005). Design-based stereological estimation of hepatocyte number, by combining the smooth optical fractionator and immunocytochemistry with anti-carcinoembryonic antigen polyclonal antibodies. Liver Int., 26, 116–124.

    Article  Google Scholar 

  • May, R. M. (1974). Stability and complexity in model ecosystems (2nd ed.). Princeton: Princeton University Press.

    Google Scholar 

  • McGehee, R., & Armstrong, R. A. (1977). Some mathematical problems concerning the ecological principle of competitive exclusion. J. Differ. Equ., 23, 30–52.

    Article  MathSciNet  MATH  Google Scholar 

  • Nelson, D. R., Marousi, C. G., & Davis, G. L. (1997). The role of hepatitis C virus-specific cytotoxic T lymphocytes in chronic hepatitis C. J. Immunol., 158, 1473–1481.

    Google Scholar 

  • Nowak, M., & May, R. M. (2001). Virus dynamics. Mathematical principles of immunology and virology. Oxford: Oxford University Press.

    Google Scholar 

  • Power, A. (1996). Competition between viruses in a complex plant-pathogen system. Ecology, 77(4), 1004–1010.

    Article  MathSciNet  Google Scholar 

  • Rehermann, B., Chang, K. M., & McHutchinson, J. G. (1996). Quantitative analysis of the peripheral blood cytotoxic T lymphocyte response in patients with chronic hepatitis C virus infection. J. Clin. Invest., 98, 1432–1440.

    Article  Google Scholar 

  • Rong, L., & Perelson, A. S. (2010). Treatment of hepatitis C virus infection with interferon and small molecule direct antivirals: viral kinetics and modeling. Crit. Rev. Immunol., 30(2), 131–148.

    Article  Google Scholar 

  • Sagnelli, E., Coppola, N., Marrocco, C., Onofrio, M., Sagnelli, C., Coviello, G., Scolastico, C., & Filippini, P. (2006). Hepatitis C virus superinfection in hepatitis B virus chronic carriers: a reciprocal viral interaction and variable clinical course. J. Clin. Virol., 35, 317–320.

    Article  Google Scholar 

  • Thomas, D. L., Ray, S. C., & Lemon, S. M. (2005). Hepatitis C. In G. L. Mandell, J. E. Bennett & R. Dolin (Eds.), Principles and practice of infectious diseases (6th ed.). Philadelphia: Elsevier Churchill Livingstone.

    Google Scholar 

  • Vento, S., Garofano, T., Rensini, C., Cainelli, F., Casali, F., Ghironzi, G., Ferraro, T., & Concia, E. (1998). Fulminant hepatitis associated with hepatitis A virus superinfection in patients with chronic hepatitis C. N. Engl. J. Med., 338, 286–290.

    Article  Google Scholar 

Download references

Acknowledgements

This work was partially supported by LIM-01 HCFMUSP, CNPq and FAPESP.

M.A., F.A.B.C., E.C., and E.M. designed the work, performed the analysis, and wrote the paper.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Eduardo Massad.

Appendix

Appendix

In this Appendix, we deduce the expressions for the thresholds given by Eqs. (6) and (7).

1.1 A.1 Derivation of Threshold \(\beta_{1\hbox{-}\mathrm{threshold}}^{\mathit{II}}\)

At this threshold (see Fig. 1), we have an equilibrium in which \(H_{1}\approx 0, V_{1}\approx 0, H_{2}=H_{2}^{*}, V_{2}=V_{2}^{*}\) and \(T_{2}=T_{2}^{*}\). System (1) therefore reduces to

$$ \everymath{\displaystyle} \begin{array}{rll} \beta_{1}\bigl(N-H_{2}^{*}\bigr) V_{1}-\alpha_{1} H_{1}-\gamma_{1} H_{1} T_{1} &=& 0, \\[6pt] c_{1} H_{1}-\mu_{1} V_{1}-{\varGamma}_{1} T_{1} V_{1} &=& 0, \\[6pt] -\mu_{T} T_{1}+r_{1} T_{1}\biggl(1-\frac{T_{1}}{K_{T}}\biggr) &=& 0. \end{array} $$
(8)

From the second equation of system (8), we deduce that

$$ H_{1} = \frac{\mu_{1} + {\varGamma}_{1} T_{1}}{c_{1}} V_{1}. $$
(9)

Substituting Eq. (9) into the first equation of system (8), we have

$$ \beta_{1\hbox{-}\mathrm{threshold}}^{\mathit{II}} = \frac{(\mu_{1}+{\varGamma}_{1} T_{1})(\alpha_{1}+\gamma_{1}T_{1})}{c_{1}(N-H_{2}^{*})}, $$
(10)

where

$$ T_{1} = \frac{r_{1}-\mu_{T}}{r_{1}}K_{T}, $$
(11)

and \(H_{2}^{*}\) is derived from the third, fifth and seventh equations of system (1),

$$ \everymath{\displaystyle} \begin{array}{rll} \beta_{{2}}\bigl(N-H_{{2}}^{*}\bigr)V_{{2}}^{*}-\alpha_{{2}}H_{{2}}^{*} - \gamma_{{2}}H_{{2}}^{*}T_{{2}}^{*} &=& 0, \\[6pt] c_{2}H_{{2}}^{*} - \mu_{{2}}V_{{2}}^{*} - {\varGamma}_{{2}}T_{{2}}^{*}V_{{2}}^{*} &=& 0, \\[6pt] b_{{2}}H_{{2}}^{*} \biggl(1-\frac{T_{{2}}^{*}}{K_{T}}\biggr) - \mu_{T}T_{{2}}^{*} + r_{{2}}T_{{2}}^{*} \biggl(1 - \frac{T_{{2}}^{*}}{K_{T}}\biggr) &=& 0, \end{array} $$
(12)

resulting in a third-order equation for \(H_{2}^{*}\). It is therefore preferable to solve system (12) numerically.

1.2 A.2 Derivation of Threshold \(\beta_{1\hbox{-}\mathrm{threshold}}^{\mathit{III}}\)

The threshold \(\beta_{1\hbox{-}\mathrm{threshold}}^{\mathit{III}}\) can be deduced by noting that, at this point, the equilibrium values are \(H_{2}^{**}\approx 0, V_{2}^{**}\approx 0, H_{1}=H_{1}^{**}, V_{1}= V_{1}^{**}\) and \(T_{1}=T_{1}^{**}\). From the third and fifth equations of system (1),

$$ \begin{array}{rll} \beta_{2}\bigl(N-H_{1}^{**}\bigr)V_{2}^{**}-\alpha_{2}H_{2}^{**}-\gamma_{2}H_{2}^{**}T_{2}^{**} &=& 0, \\[6pt] c_{2}H_{2}^{**}-\mu_{2}V_{2}^{**}-{\varGamma}_{2}T_{2}^{**}V_{2}^{**} &=& 0, \end{array} $$
(13)

we obtain

$$ \beta_{2}\bigl(N-H_{1}^{**}\bigr) - \frac{(\alpha_{2}+\gamma_{2}T_{2}^{**}) (\mu_{2} + {\varGamma}_{2}T_{2}^{**})}{c_{2}} = 0, $$
(14)

where

$$ T_{2}^{**} = \frac{r_{2} - \mu_{T}}{r_{2}}K_{T}. $$
(15)

From system (13), we can calculate the value of \(H_{1}^{**}\) as a function of the parameters labelled “2”. Substituting \((N-H_{1}^{**})\) for \((N-H_{2}^{*})\) in Eqs. (8), we then have

$$ \beta_{1\hbox{-}\mathrm{threshold}}^{\mathit{III}} = \frac{(\alpha_{1}+\gamma_{1}T_{1}^{**})(\mu_{1}+{\varGamma}_{1}T_{1}^{**})}{c_{1} (N-H_{1}^{**})}. $$
(16)

Our only remaining task is to calculate \(T_{1}^{**}\) which can be obtained from the second and third equations of system (8),

$$ \begin{array}{rll} c_{1}H_{1}^{**}-\mu_{1}V_{1}^{**}-{\varGamma}_{1}T_{1}^{**}V_{1}^{**} &=& 0, \\[6pt] b_{1}V_{1}^{**}\bigl(K_{T}-T_{1}^{**}\bigr)-\mu_{1}K_{T}T_{1}^{**}+r_{1}\bigl(K_{T}-T_{1}^{**}\bigr) T_{1}^{**} &=& 0. \end{array} $$
(17)

Eliminating \(V_{1}^{**}\) from the system results in a third-order equation for \(T_{1}^{**}\) that can be solved numerically.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Amaku, M., Coutinho, F.A.B., Chaib, E. et al. The Impact of Hepatitis A Virus Infection on Hepatitis C Virus Infection: A Competitive Exclusion Hypothesis. Bull Math Biol 75, 82–93 (2013). https://doi.org/10.1007/s11538-012-9795-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11538-012-9795-0

Keywords

Navigation