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.
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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.
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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
From the second equation of system (8), we deduce that
Substituting Eq. (9) into the first equation of system (8), we have
where
and \(H_{2}^{*}\) is derived from the third, fifth and seventh equations of system (1),
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),
we obtain
where
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
Our only remaining task is to calculate \(T_{1}^{**}\) which can be obtained from the second and third equations of system (8),
Eliminating \(V_{1}^{**}\) from the system results in a third-order equation for \(T_{1}^{**}\) that can be solved numerically.
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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
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DOI: https://doi.org/10.1007/s11538-012-9795-0