Rich dynamics of a hepatitis C virus infection model with logistic proliferation and time delays

Abstract

In this paper, we study a dynamic model of hepatitis C virus (HCV) infection with density-dependent proliferation of uninfected and infected hepatocytes and two time delays, which is derived from a three-dimensional model by the quasi-steady-state approximation. The model can exhibit forward bifurcation or backward bifurcation, and an explicit control threshold parameter $R_c$ is obtained for the case of backward bifurcation. It is shown that if the proliferation rate of infected hepatocytes is greater than the proliferation rate of uninfected hepatocytes by a certain amount, it becomes more difficult for the virus to be removed. The model has rich dynamical properties: (i) In some parameter regions, bistability can occur; (ii) both time delays ${\tau }_1$ (virus-to-cell delay) and ${\tau }_2$ (cell-to-cell delay) can lead to Hopf bifurcations; (iii) same length of time delays ${\tau }_1$ and ${\tau }_2$ can lead to at most one stability switch, but different time delays can lead to multiple stability switches. Several sufficient conditions for the global stability of the disease-free equilibrium and the endemic equilibrium are obtained for both forward and backward bifurcation scenarios. In particular, several sharp results on global stability are obtained. Theoretical and numerical results portray the complexity of viral evolutionary dynamics in chronic HCV-infected patients.